The human worth of rigorous thinking: essays and addresses

THE HUMAN WORTH OF RIGOROUS THINKING

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SCIENCE AND RELIGION: THE RATIONAL AND THE
SUPERRATIONAL The Yale University Press

THE NEW INFINITE AND THE OLD THEOLOGY

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THE HUMAN WORTH OF
RIGOROUS THINKING

ESSAYS AND ADDRESSES

BY

CASSIUS J. KEYSER, PH.D., LL.D.

ADRAIN PROFESSOR OF MATHEMATICS
COLUMBIA UNIVERSITY

fork
COLUMBIA UNIVERSITY PRESS

1916

All rights reserved

Copyright, 1916
BY COLUMBIA UNIVERSITY PRESS

Printed from type, May, 1916

Qf\

PREFACE

THE following fifteen essays and addresses have ap-
peared, in the course of the last fifteen years, as articles
in various scientific, literary, and philosophical journals.
For permission to reprint I have to thank the editors and
managers of The Columbia University Quarterly, The Co-
lumbia University Press, Science, The Educational Review,
The Bookman, The Monist, The Eibbert Journal, and The
Journal of Philosophy, Psychology and Scientific Methods.

The title of the volume indicates its subject. The fact
that one of the essays, the initial one, bears the same title
is hardly more than a mere coincidence, for all of the dis-
cussions deal with the subject in question and nearly all
of them deal with it directly, consciously, and in terms.

In passing from essay to essay the attentive reader will
notice a few repetitions of thought and possibly a few
in forms of expression. Such reiterations, which owe
their presence to the occasional character of the essays
and to the aims and circumstances that originally con-
trolled their composition, may, it is hoped, be regarded
by the charitable reader less as blemishes than as means
of emphasizing important considerations.

CASSIUS J. KEYSER.
April 14, 1916.

CONTENTS

CHAPTER PAGE

I. The Human Worth of Rigorous Thinking i

II. The Human Significance of Mathematics 26

III. The Humanization of the Teaching of Mathematics 61

IV. The Walls of the World; or Concerning the Figure and the

Dimensions of the Universe of Space 81

V. Mathematical Emancipations: Dimensionality and Hyperspace 101

VI. The Universe and Beyond: The Existence of the Hypercosmic 122

VII. The Axiom of Infinity: A New Presupposition of Thought . 139

VIII. The Permanent Basis of a Liberal Education 163

IX. Graduate Mathematical Instruction for Graduate Students not

Intending to Become Mathematicians 176

X. The Source and Functions of a University 201

XI. Research in American Universities 209

XII. Principia Mathematica 220

XIII. Concerning Multiple Interpretations of Postulate Systems and

the "Existence" of Hyperspace 233

XIV. Mathematical Productivity in the United States 257

XV. Mathematics 271

THE HUMAN WORTH OF RIGOROUS
THINKING1

But in the strong recess of Harmony
Established firm abides the rounded Sphere.

— EMPEDOCLES

NEXT to the peaceful pleasure of meeting genuine
curiosity, half-way, upon its own ground, comes the
joy of combat when an attack upon some valued right
or precious interest of the human spirit requires to be
repelled. Indeed, given a competent jury, hardly any
other undertaking could be more stimulating than to
defend mathematics from a charge of being unworthy
to occupy, in the hierarchy of arts and sciences, the
high place to which, from the earliest times, the judg-
ment of mankind has assigned it. But, unfortunately,
no such accusation has been brought, brought, that is,
by persons of such scientific qualifications as to give
their opinion in the premises weight enough to call
for serious consideration. Mathematics has been often
praised by the scientifically incompetent; it has not,
so far as I am aware, been dispraised, or its worth
challenged or denied, by the scientifically competent.
The age-long immunity of mathematics from authorita-

1 An address delivered before the Mathematical Colloquium of Columbia
University, October 13, 1913. Printed, with slight change, in Science,
December 5, 1913; also, with other slight changes, printed in The Columbia
University Quarterly, June, 1914, under the title "The Study of Mathe-
matics." The substance of the address was delivered before the mathe-
matics section of the California High School Teachers Association, August,
1915, at Berkeley, California.

2 THE HUMAN WORTH OF RIGOROUS THINKING

tive arraignment, and the high estimation in which
the science has been almost universally held in enlight-
ened times and places, unite to give it a position nearly,
if not quite, unique in the history of criticism. Perhaps
it were better not so. Mathematicians have a sense
of security to which, it may be, they are not entitled
in a critical age and a reeling world. Conceivably it
might have been to the advantage of mathematics and
not only of mathematics but of science in general, of
philosophy, too, and the general enlightenment, if in
course of the centuries mathematicians had been now
and then really compelled by adverse criticism of their
science to discover and to present not only to themselves
but acceptably to their fellow-men the deeper justifica-
tions, if such there be, of the world's approval and
applause of their work. However that may be, no
one is likely to dissent from the opinion of Mr. Bertrand
Russell that "in regard to every form of activity it is
necessary that the question should be asked from time
to time: what is its purpose and ideal? In what way
does it contribute to the beauty of human existence?"
An inquiry that is thus necessary for the general wel-
fare ought to be felt as a duty, unless, more fortunately,
it be felt as a pleasure.

Why study mathematics? What • are the rightful
claims of the science to human regard? What are the
grounds upon which a university may justify the annual
expenditure of thirty to fifty thousands of dollars to
provide for mathematical instruction and mathematical
research?

A slight transformation of the questions will help to
disclose their significance and may give a quicker sense
of their poignancy and edge. What is mathematics?
I hasten to say that I do not intend to detain the reader

TH:

THE HUMAN WORTH OF RIGOROUS THINKING 3

and thus perhaps to dampen his interest with a defini-
tion of mathematics, though it must be said that the
discovery of what mathematics is, is doubtless one of the
very great scientific achievements of the nineteenth
century. The question asks, not for a definition of the
science, but for a brief and helpful description of it —
for an obvious mark or aspect of it that will enable us
to know what it is that we are here writing or reading
about. Well, mathematics may be viewed either as an
enterprise or as a body of achievements. As an enter-
prise mathematics is characterized by its aim, and its
aim is to think rigorously whatever is rigorously think-
able or whatever may become rigorously thinkable in
course of the upward striving and refining evolution of
ideas. As a body of achievements mathematics con-
sists of all the results that have come, in the course
of the centuries, from the prosecution of that enter-
prise: the truth discovered by it; the doctrines created
by it; the influence of these, through their applications
and their beauty, upon the advancement of civiliza-
tion and the weal of man.

Our questions now stand: Why should a human being
desire to share in that spiritual enterprise which has
for its aim to think rigorously whatever is or may
become rigorously thinkable and to "frame a world
according to a rule of divine perfection"? Why should
men and women seek some knowledge of that variety
of perfection with which men and women have enriched
life and the world by rigorous thought? What are the
just claims to human regard of perfect thought and the
spirit of perfect thinking? Upon what grounds may a
university justify the annual expenditure of thirty to
fifty thousands of dollars to provide for the disciplining
of men and women in the art of thinking rigorously

4 THE HUMAN WORTH OF RIGOROUS THINKING

and for the promotion of research in the realm of exact
thought?

Such are the questions. They plainly sum themselves
in one: among the human agencies that ameliorate life,
what is the r61e of rigorous thinking? What is the r61e
of the spirit that always aspires to the attainment of
logical perfection?

Evidently that question is not one for adequate
handling in a brief magazine article by an ordinary
student of mathematics. Rather is it a subject for a
long series of lectures by a learned professor of the
history of civilization. Indeed so vast is the subject
that even an ordinary student of mathematics can
detect some of the more obvious tasks such a philosophic
historian would have to perform and a few of the dif-
ficulties he would doubtless encounter. It may be
worth while to mention some of them.

Certainly one of the tasks, and probably one of the
difficulties also, would be that of securing an audience
— an audience, I mean, capable of understanding the
lectures, for is not a genuine auditor a listener who
understands? To understand the lectures it would
seem to be necessary to know what that is which the
lectures are about — that is, it would be necessary to
know what is meant by rigorous thinking. To know
this, however, one must either have consciously done
some rigorous thinking or else, at the very least, have
examined some specimens of it pretty carefully, just
as, in order to know what good art is, it is, in general,
essential either to have produced good art or to have
attentively examined some specimens of it, or to have
done both of these things. Here, then, at the outset
our historian would meet a serious difficulty, unless
his audience chanced to be one of mathematicians,

THE HUMAN WORTH OF RIGOROUS THINKING 5

which is (unfortunately) not likely, inasmuch as the
great majority of mathematicians are so exclusively
interested in mathematical study or teaching or research
as to be but little concerned with the philosophical
question of the human worth of their science. It is,
therefore, easy to see how our lecturer would have to
begin.

Ladies and gentlemen, we have met, he would say,
we have met to open a course of lectures dealing with the
role of rigorous thinking in the history of civilization.
In order that the course may be profitable to you, in
order that it may be a course in ideas and not merely
or mainly a verbal course, it is essential that you should
know what rigorous thinking is and what it is not.
Even I, your speaker, he will own, might reasonably
be held to the obligation of knowing that.

It is reasonable, ladies and gentlemen, it is reasonable
to assume, he would say, that in the course of your
education you neglected mathematics, and it is there-
fore probable or indeed quite certain that, notwith-
standing your many accomplishments, you do not quite
know or rather, perhaps I should say, you are very far
from knowing what rigorous thinking is or what it is
not. Of course, as you know, it is, generally speaking,
much easier to tell what a thing is not than to tell what
it is, and I might, he would say, I might proceed by
way of a preliminary to indicate roughly what rigorous
thinking is not. Thus I might explain that rigorous
thinking, though much of it has been done in the world
and though it has produced a large literature, is never-
theless a relatively rare phenomenon. I might point
out that a vast majority of mankind, a vast majority of
educated men and women, have not been disciplined
to think rigorously even those things that are most

6 THE HUMAN WORTH OF RIGOROUS THINKING

available for such thinking. I might point out that,
on the other hand, most of the ideas with which men
and women have constantly to deal are as yet too nebu-
lous and vague, too little advanced in the course of
their evolution, to be available for concatenative think-
ing and rigorous discourse. I should have to say, he
would add, that, on these accounts, most of the think-
ing done in the world in a given day, whether done by
men in the street or by farmers or factory-hands or
administrators or historians or physicians or lawyers
or jurists or statesmen or philosophers or men of letters
or students of natural science or even mathematicians
(when not strictly employed in their own subject),
comes far short of the demands and standards of rig-
orous thinking.

I might go on to caution you, our speaker would say,
against the current fallacy, recently advanced by elo-
quent writers to the dignity of a philosophical tenet,
of regarding what is called successful action as the
touchstone of rigorous thinking. For you should know
that much of what passes in the world for successful
action proceeds from impulse or instinct and not from
thinking of any kind; you should know that no action
under the control of non-rigorous thinking can be
strictly successful except by the favor of chance or
through accidental compensation of errors; you should
know that most of what passes for successful action,
most of what the world applauds and even commem-
morates as successful action, so far from being really
successful, varies from partial failure to failure that,
if not total, would at all events be fatal in any universe
that had the economic decency to forbid, under pain
of death, the unlimited wasting of its resources. The
dominant animal of such a universe would be, in fact,

TB

THE HUMAN WORTH OF RIGOROUS THINKING 7

a superman. In our world the natural resources of
life are superabundant, and man is poor in reason
because he has been the prodigal son of a too opulent
mother. But, ladies and gentlemen, our speaker will
conclude, you will know better what rigorous thinking
is not when once you have learned what it is. This,
however, cannot well be learned in a course of lectures
in which that knowledge is presumed. I have, there-
fore, to adjourn this course until such time as you shall
have gained that knowledge. It cannot be gained by
reading about it or hearing about it. The easiest way,
for most persons the only way, to gain it is to examine
with exceeding patience and care some specimens, at
least one specimen, of the literature in which rigorous
thinking is embodied. Such a specimen, he could
add, is Dr. Thomas L. Heath's magnificent edition of
Euclid, where an excellent translation of the Elements
from the definitive text of Heiberg is set in the composite
light of critical commentary from Aristotle down to
the keenest logical microscopists and histologists of our
own day. If you think Euclid too ancient or too stale
even when seasoned with the wit of more than two
thousand years of the acutest criticism, you may find
a shorter and possibly a fresher way by examining
minutely such a work as Veronese's Grundzuge der
Geometric or Hilbert's famous Foundations of Geom-
etry or Peano's Sui Numeri Irrazionali. In works of
this kind and not elsewhere you will find in its nakedness,
purity, and spirit, what you have neglected and what
you need. You will note that in the beginning of such
a work there is found a system of assumptions or postu-
lates, discovered the Lord only and a few men of genius
know where or how, selected perhaps with reference to
simplicity and clearness, certainly selected and tested

8 THE HUMAN WORTH OF RIGOROUS THINKING

with respect to their compatibility and independence,
and, it may be, with respect also to categoricity. You
will not fail to observe with the utmost minuteness,
and from every possible angle, how it is that upon these
postulates as a basis there is built up by a kind of
divine masonry, little step by step, a stately struc-
ture of ideas, an imposing edifice of rigorous thought,
a towering architecture of doctrine that is at once
beautiful, austere, sublime, and eternal. Ladies and
gentlemen, our speaker will say, to accomplish that ex-
amination will require twelve months of pretty assiduous
application. The next lecture of this course will be
given one year from date.

On resuming the course what will our philosopher
and historian proceed to say? He will begin to say what,
if he says it concisely, will make up a very large vol-
ume. Room is lacking here, even if competence were
not, for so much as an adequate outline of the matter.
It is possible, however, to draw with confidence a few
of the larger lines that such a sketch would have to
contain.

What is it that our speaker will be obliged to deal
with first? I do not mean obliged logically nor obliged
by an orderly development of his subject. I mean
obliged by the expectation of his hearers. Every one
can answer that question. For presumably the audience
represents the spirit of the times, and this age is, at
least to a superficial observer, an age of engineering.
Now, what is engineering? Well, the Charter of the
Institution of Civil Engineers tells us that engineering
is the "art of directing the great sources of power in
Nature for the use and convenience of man." By
Nature here must be meant external or physical nature,
for, if internal nature were also meant, every good form

THE HUMAN WORTH OF RIGOROUS THINKING Q

of activity would be a species of engineering, and maybe
it is such, but that is a claim which even engineers
would hardly make and poets would certainly deny.
Use and convenience — these are the key-bearing words.
It is perfectly evident that our lecturer will have to deal
first of all with what the world would call the "utility"
of rigorous thinking, that is to say, with the applica-
tions of mathematics and especially with its applica-
tions to problems of engineering. If he really knows
profoundly what mathematics is, he will not wish to
begin with applications nor even to make applications
a major theme of his discourse, but he must, and he will
do so uncomplainingly as a concession to the external-
mindedness of his time and his audience. He will not only
desire to show his audience applications of mathematics
to engineering, but, being an historian of civilization, he
will especially desire to show them the development of
such applications from the earliest times, from the
building of pyramids and the mensuration of land in
ancient Egypt down to such splendid modern achieve-
ments as the designing and construction of an Eads
Bridge, an ocean Imperator or a Panama Canal. The
story will be long and difficult, but it will edify. The
audience will be amazed at the truth if they under-
stand. If they do not understand the truth fully, our
speaker must at all events contrive that they shall see
it in glimmers and gleams and, above all, that they shall
acquire a feeling for it. They must be led to some
acquaintance with the great engineering works of the
world, past and present; they must be given an intel-
ligible conception of the immeasurable contribution
such works have made to the comfort, convenience, and
power of man; and especially must they be convinced
of the fact that, not only would the greatest of such

10 THE HUMAN WORTH OF RIGOROUS THINKING

achievements have been, except for mathematics, utterly
impossible, but that such of the lesser ones as could
have been wrought without mathematical help could
not have been thus accomplished without wicked and
pathetic waste both of material resources and of human
toil. In respect to this latter point, the relation of
mathematics to practical economy in large affairs, our
speaker will no doubt invite his hearers to read and
reflect upon the ancient work of Frontinus on the Water
Supply of the City of Rome in order that thus they may
gain a vivid idea of the fact that the most practical
people of history, despising mathematics and the finer
intellectualizations of the Greeks, were unable to accom-
plish their own great engineering feats except through
appalling waste of materials and men. Our jecturer
will not be content, however, with showing the service
of mathematics in the prevention of waste; he will
show that it is indispensable to the productivity and
trade of the modern world. Before quitting this divi-
sion of his subject he will have demonstrated that, if
all the contributions which mathematics has made, and
which nothing else could make, to navigation, to the
building of railways, to the construction of ships, to the
subjugation of wind and wave, electricity and heat, and
many other forms and manifestations of energy, he
will have demonstrated, I say, and the audience will
finally understand, that, if all these contributions of
mathematics were suddenly withdrawn, the life and body
of industry and commerce would suddenly collapse as
by a paralytic stroke, the now splendid outer tokens
of material civilization would perish, and the face of
our planet would quickly assume the aspect of a ruined
and bankrupt world.

As our lecturer has been constrained by circumstances

THE HUMAN WORTH OF RIGOROUS THINKING II

to back into his subject, as he has, that is, been com-
pelled to treat first of the service that mathematics has
rendered engineering, he will probably next speak of
the applications of mathematics to the so-called natural
sciences — the more properly called experimental sciences
— of physics, chemistry, biology, economics, psychology,
and the like. Here his task, if it is to be, as it ought to
be, expository as well as narrative, will be exceedingly
hard. For how can he weave into his narrative an intel-
ligible exposition of Newton's Principia, Laplace's Me-
canique Celeste, Lagrange's Mecanique Analytique, Gauss's
Theoria Motus Corporum Coelestium, Fourier's Theorie
Analytique de la Chaleur, Maxwell's Electricity and
Magnetism, not to mention scores of other equally dif-
ficult and hardly less important works of a mathemat-
ical-physical character? Even if our speaker knew it
all, which no man can, he could not tell it all in-
telligibly to his hearers. These will have to be con-
tent with a rather general and superficial view, with a
somewhat vague intuition of the truth, with fragmentary
and analogical insights gained through settings forth of
great things by small; and they will have to help them-
selves and their speaker, too, by much pertinent read-
ing. No doubt the speaker will require his hearers,
in order that they may thus gain a tolerable perspective,
to read well not only the first two volumes of the
magnificent work of John Theodore Merz dealing with
the History of European Thought in the Nineteenth Cen-
tury, but also many selected portions of the kindred
literature there cited in richest profusion. The work
treats mainly of natural science, but it deals with it
philosophically, under the larger aspect, that is, of
science regarded as Thought. By the help of such
literature in the hands of his auditors, our lecturer will

12 THE HUMAN WORTH OF RIGOROUS THINKING

be able to give them a pretty vivid sense of the great
and increasing role of mathematics in suggesting, formu-
lating, and solving problems in all branches of natural
science. Whether it be with "the astronomical view
of nature" that he is dealing, or "the atomic view" or
"the mechanical view" or "the physical view" or
"the morphological view" or "the genetic view" or "the
vitalistic view" or "the psychophysical view" or "the
statistical view," in every case, in all these great at-
tempts of reason to create or to find a cosmos amid the
chaos of the external world, the presence of mathe-
matics and its manifold service, both as instrument and
as norm, illustrate and confirm the Kantian and Rie-
mannian conception-oL,natural science as „" the^-attempt j
to_understand nature by means of exact concepts."

In connection with this division of his subject, our
speaker will find it easy to enter more deeply into the
spirit and marrow of it. He will be able to make it
clear that there is a sense, a just and important sense,
in which all thinkers and especially students of natural
science, though their thinking is for the most part not
rigorous, are yet themselves contributors to mathematics.
I do not refer to the powerful stimulation of mathe-
matics by natural science in furnishing it with many of
its problems and in constantly seeking its aid. What
I mean is that all thinkers and especially students of
natural science are engaged, both consciously and un-
consciously, both intentionally and unintentionally,
in the mathematicization of concepts — that is to say,
in so transforming and refining concepts as to fit them
finally for the amenities of logic and the austerities of
rigorous thinking, We are dealing here, our speaker
will say, with a process transcending conscious design.
We are dealing with a process deep in the nature and

THE HUMAN WORTH OF RIGOROUS THINKING 13

being of the psychic world. Like a child, an idea, once
it is born, once it has come into the realm of spiritual
light, possibly long before such birth, enters upon a
career, a career, however, that, unlike the child's, seems
to be immortal. In most cases and probably in all,
an idea, on entering the world of consciousness, is vague,
nebulous, formless, not at once betraying either what
it is or what it is destined to become. Ideas, however,
are under an impulse and law of amelioration. The
path of their upward striving and evolution — often
a long and winding way — leads towards precision and
perfection of form. The goal is mathematics. Witness,
for example, our lecturer will say, the age-long travail
and aspiration of the great concept now known as mathe-
matical continuity, a concept whose inner structure is
even now known and understood only of mathematicians,
though the ancient Greeks helped in molding its form
and though it has long been, if somewhat blindly, yet
constantly employed in natural science, as when a
physicist, for example, or an astronomer uses such
numbers as e and TT in computation. Witness, again,
how that supreme concept of mathematics, the concept
of function, has struggled through thousands of years
to win at length its present precision of form out of
the nebulous sense, which all minds have, of the mere
dependence of things on other things. Witness, too, he
will say, the mathematical concept of infinity, which
prior to a half-century ago was still too vague for logical
discourse, though from remotest antiquity the great
idea has played a conspicuous role, mainly emotional,
in theology, philosophy, and science. Like examples
abound, showing that one of the most impressive and
significant phenomena in the life of the psychic world,
if we will but discern and contemplate it, is the process

14 THE HUMAN WORTH OF RIGOROUS THINKING

by which ideas advance, often slowly indeed but surely,
from their initial condition of formlessness and inde-
termination to the mathematical estate. The chemic-
ization of biology, the physicization of chemistry, the
mechanicization of physics, the mathematicization of
mechanics, the arithmeticization of mathematics, these
well-known tendencies and drifts in science do but illus-
trate on a large scale the ubiquitous process in question.

At length, ladies and gentlemen, our speaker will say,
in the light of the last consideration the deeper and
larger aspects of our subject are beginning to show
themselves and there is dawning upon us an impressive
vision. The nature, function, and life of the entire
conceptual world seem to come within the circle and
scope of our present enterprise. We are beginning to
see that to challenge the human worth of mathematics,
to challenge the worth of rigorous thinking, is to chal-
lenge the worth of all thinking, for now we see thatf:
mathematics is but the ideal to which all thinking, byj
an inevitable process and law of the human spirit,!
constantly aspires. We see that to challenge the worth!
of that ideal is to' arraign before the bar of values what
seerniT the deepest process and inmost law of the uni-
verse of thought. Indeed we see that in defending
mathematics we are really defending a cause yet more
momentous, the whole cause, namely, of the conceptual
procedure of science and the conceptual activity of the
human mind, for mathematics is nothing but such con-
ceptual procedure and activity come to its maturity,
purity, and perfection.

Now, ladies and gentlemen, our lecturer will say, I
cannot in this course deal explicitly and fully with this
larger issue. But, he will say, we are living in a day
when that issue has been raised; we happen to be living

I

THE HUMAN WORTH OF RIGOROUS THINKING 15

in a time when, under the brilliant and effective leader-
ship of such thinkers as Professor Bergson and the late
Professor James, the method of concepts, the method of
intellect, the method of science, is being powerfully
assailed; and, he will say, whilst I heartily welcome this
attack of criticism as causing scientific men to reflect
more deeply upon the method of science, as exhibiting
more clearly the inherent limitations of its method, and
as showing that life is so rich as to have many precious
interests and the world much truth beyond the reach
of that method, yet I cannot refrain, he will say, from
attempting to point out what seems to me a radical
error of the critics, a fundamental error of theirs, in
respect to what is the highest function of conception
and in respect to what is the real aim and ideal of the
life of intellect. For we shall thus be led to a deeper
view of our subject proper.

These critics find, as all of us find, that what we call
mind or our minds is, in some mysterious way, func-
tionally connected with certain living organisms known
as human bodies; they find that these living bodies
are constantly immersed in a universe of matter and
motion in which they are continually pushed and pulled,
heated and cooled, buffeted and jostled about — a
universe that, according to James, would, in the "ab-
sence of concepts," reveal itself as "a big blooming
buzzing confusion" — though it is hard to see how such
a revelation could happen to any one devoid of the
concept " confusion," but let that pass; our critics find
that our minds get into some initial sort of knowing
connection with that external blooming confusion through
what they call the sensibility of our bodies, yielding
all manner of sensations as of weights, pressures, pushes
and pulls, of intensities and extensities of brightness,

1 6 THE; HUMAN WORTH OF RIGOROUS THINKING

sound, time, colors, space, odors, tastes, and so on; they
find that we must, on pain of organic extinction, take
some account of these elements of the material world;
they find that, as a fact, we human beings constantly
deal with these elements through the instrumentality
of concepts; they find that the effectiveness of our
dealing with the material world is precisely due to our
dealing with it conceptually; they infer that, there-
fore, dealing with matter is exactly what concepts are
for, saying with Ostwald, for example, that the goal of
natural science, the goal of the conceptual method of
mind, "is the domination of nature by man"; not only,
our speaker will say, do our critics find that we deal
with the material world conceptually, and effectively
because conceptually, but they find also that life has
interests and the world values not accessible to the con-
ceptual method, and as this method is the method of
the intellect, they conclude, not only that the intellect
cannot grasp life, but that the aim and ideal of intellect
is the understanding and subjugation of matter, saying
with Professor Bergson "that our intellect is intended
to think matter," "that our concepts have been formed
on the model of solids," "that the essential function
of our intellect . . . is to be a light for our conduct,
to make ready for our action on things," that "the
intellect always behaves as if it were fascinated by the
contemplation of inert matter," that "intelligence . . .
aims at a practically useful end," th^t "the intellect is
never quite at its ease, . . . except when it is working
upon inert matter, more particularly upon solids," and
much more to the same effect.

Now, ladies and gentlemen, our speaker will ask,
what are we to think of this? What are we to think of
this evaluation of the science-making method of con-

TI

THE HUMAN WORTH OF RIGOROUS THINKING 17

cepts? What are we to think of the aim and ideal
here ascribed to the intellect and of the station assigned
it among the faculties of the human mind? In the first
place, he will say, it ought to be evident to the critics
themselves, and evident to them even in what they
esteem the poor light of intellect, that the above-
sketched movement of their minds is a logically unsound
movement. They do not indeed contend that, because
a living being in order to live must deal with the material
world, it must, therefore, do so by means of concepts.
The lower animals have taught them better. But
neither does it follow that, because certain bipeds in
dealing with the material world deal with it concep-
tually, the essential function of concepts is just to deal
with matter. Nor does such an inference respecting
the essential function of concepts follow from the fact
that the superior effectiveness of man's dealing with the
physical world is due to his dealing with it conceptually.
For it is obviously conceivable and supposable that
such conceptual dealing with matter is only an incident
or byplay or subordinate interest in the career of con-
cepts. It is conceivably possible that such employ-
ment with matter is only an avocation, more or less
serious indeed and more or less advantageous, yet an
avocation, and not the vocation, of intellect. Is it
not evidently possible to go even further? Is it not
logically possible to admit or to contend that, inasmuch
as the human intellect is functionally attached to a
living body which is itself plunged in a physical uni-
verse, it is absolutely necessary for the intellect to con-
cern itself with matter in order to preserve, not indeed
the animal life of man, but his intellectual life — is it
not allowable, he will say, to admit or to maintain that
and at the same time to deny that such concernment

1 8 THE HUMAN WORTH OF RIGOROUS THINKING

with matter is the intellect's chief or essential function
and that the subjugation of matter is its ideal and aim?

Of course, our lecturer will say, our critics might be
wrong in their logic and right in their opinion, just as
they might be wrong in their opinion and right in their
logic, for prjinion is often a matter, not of logic or proof,
bjil_££_^mperajnent, taste, and insight. But, he will
say, if the issue as to the^nief function of concepts and
the ideal of the intellect is to be decided in accordance
with temperament, taste, and insight, then there is room
for exercise of the preferential faculty, and alternatives
far superior to the choice of our critics are easy enough
to find. It may accord better with our insight and
taste to agree with Aristotle that "It is owing," not to
the necessity of maintaining animal life or the desire
of subjugating matter, but "it is owing to their wonder
that men both now begin and first began to philoso-
phize; they wondered originally at the obvious diffi-
culties, then advanced little by little and stated the
difficulties about the greater matters." The striking
contrast of this with the deliverances of Bergson is
not surprising, for Aristotle was a pupil of Plato and
the doctrine of Bergson is that of Plato completely
inverted. It may accord better with our insight and
taste to agree with trie great^TG. I." facobi. who, when
he had been reproached by Fourier for not devoting his
splendid genius to physical investigations instead of
pure mathematics, replied that a philosopher like his
critic "ought to know that the unique end of science
is," not public utility and application to natural phe-
nomena, but "is the honor of the human spirit." It
may accord better with our temperament and insight
to agree with the sentiment of Diotima: "I am per-
suaded that all men do all things, and the better they

I

THE HUMAN WORTH OF RIGOROUS THINKING IQ

are, the better they do them, in the hope," not of
subjugating matter, but "in the hope of the glorious
fame of immortal virtue."

But it is unnecessary, ladies and gentlemen, it is un-
necessary, our speaker will say, to bring the issue to
final trial in the court of temperaments and tastes.
We should gain there a too easy victory. The critics
are psychologists, some of them eminent psychologists.
Let the issue be tried in the court of psychology, for
it is there that of right it belongs. They know the
fundamental and relevant facts. What is the verdict
according to these? The critics know the experiments
that have led to and confirmed the psychophysical law
of Weber and Fechner and the doctrine of thresholds;
they know that, in accordance with that doctrine and
that law, an appropriate stimulus, no matter what the
department of sense, may be finite in amount and yet
too small, or finite and yet too large, to yield a sensa-
tion; they know that the difference between two stimuli
of a kind appropriate to a given sense department, no
matter what department, may be a finite difference and
yet too small for sensibility to detect, or to work a
change of sensation; they ought to know, though they
seem not to have recognized, much less to have weighed,
the fact that, owing to the presence of thresholds, the
greatest number of distinct sensations possible in any
department of sense is a finite number; they ought to
know that the number of different departments of sense
is also & finite number; they ought to know that, there-
fore, the total number of distinct or different sensations
of which a human being is capable is a finite number;
they ought to know, though they seem not to have
recognized the fact, that, on the other hand, the world
of concepts is of infinite multiplicity, that concepts, the

20 THE HUMAN WORTH OF RIGOROUS THINKING

fruit of intellect, as distinguished from sensations, the
fruit of sensibility, are infinite in number; they ought,
therefore, to see, our speaker will say, though none of
them has seen, that in attempting to derive intellect
out of sensibility, in attempting to show that (as James
says) " concepts flow out of percepts," they are con-
fronted with the problem of bridging the immeasurable
gulf between the finite and the infinite, of showing, that
is, how an infinite multiplicity can arise from one that
is finite. But even if they solved that apparently
insuperable problem, they could not yet be in position
to affirm that the function of intellect and its concepts
is, like that of sensibility, just the function of dealing
with matter, as the function of teeth is biting and
chewing. Far from it.

Let us have another look, the lecturer will say, at
the psychological facts of the case. Owing to the pres-
ence of thresholds in every department of sense it may
happen and indeed it does happen constantly in every
department, that three different amounts of stimulus of
a same kind give three sensations such that two of them
are each indistinguishable from the third and yet are dis-
tinguishable from one another. Now, for sensibility in
any department of sense, two magnitudes of stimulus
are unequal or equal according as the sensations given
by them are or are not distinguishable. Accordingly in
the world of sensible magnitudes, in the sensible universe,
in the world, that is, of felt weights and thrusts and
pulls and pressures, of felt brightnesses and warmths and
lengths and breadths and thicknesses and so on, in this
world, which is the world of matter, magnitudes are such
that two of them may each be equal to a third without being
equal to one another. That, our speaker will say, is a
most significant fact and it means that the sensible

THE HUMAN WORTH OF RIGOROUS THINKING 21

world, the world of matter, is irrational, infected with
contradiction, contravening the essential laws of thought.
No wonder, he will say, that old Heracleitus declared
the unaided senses "give a fraud and a lie."

Now, our speaker will ask, what has been and is the
behavior of intellect in the presence of such contra-
diction? Observe, he will say, that it is intellect, and
not sensibility, that detects the contradiction. Of the
irrationality in question sensibility remains insensible.
The data among which the contradiction subsists are
indeed rooted in the sensible world, they inhere in the
world of matter, but the contradiction itself is known
only to the logical faculty called intellect. Observe
also, he will say, and the observation is important, that
such contradictions do not compel the intellect to any
activity whatever intended to preserve the life of the
living organism to which the intellect is functionally
attached. That is a lesson we have from our physical
kin, the beasts. What, then, has the intellect done
because of or about the contradiction? Has it gone on
all these centuries, as our critics would have us believe,
trying to "think matter," as if it did not know that
matter, being irrational, is not thinkable? Far from it,
he will say, the intellect is no such ass.

What it has done, instead of endlessly and stupidly
besieging the illogical world of sensible magnitudes
with the machinery of logic, what it has done, our lec-
turer will say, is this: it has created for itself another
world. It has not rationalized the world of sensible
magnitudes. That, it knows, cannot be done. It has
discerned the ineradicable contradictions inherent in
them, and by means of its creative power of conception
it has made a new world, a world of conceptual magni-
tudes that, like the continue, of mathematics, are so

22 THE HUMAN WORTH OF RIGOROUS THINKING

constructed by the spiritual architect and so endowed
by it as to be free alike from the contradictions of the
sensible world and from all thresholds that could give
them birth. Indeed conception, to speak metaphorically
in terms borrowed from the realm of sense, is a kind of
infinite sensibility, transcending any finite distinction,
difference or threshold, however minute or fine. And
now, our speaker will say, it is such magnitudes, magni-
tudes created by intellect and not those discovered by
sense, though the two varieties are frequently not
discriminated by their names; it is such conceptual mag-
nitudes that constitute the subject-matter of science.
If the magnitudes of science, apart from their ration-
ality, often bear in conformation a kind of close resem-
blance to magnitudes of sense, what is the meaning of
the fact? It means, contrary to the view of Bergson
but in accord with that of Poincare, that the free crea-
ative artist, intellect, though it is not constrained, yet
has chosen to be guided, in so far as its task allows,
by facts of sense. Thus we have, for example, concep-
tual space and sensible space so much alike in conforma-
tion that, though one of them is rational and the other
is not, the undiscriminating hold them as the same.

And now, our lecturer will ask, for we are nearing the
goal, what, then, is the motive and aim of this creative
activity of the intellect? Evidently it is not to preserve
and promote the life of the human body, for animals
flourish without the aid of concepts, without " discourse
of reason," and despite the contradictions in the world
of sense. The aim is, he will say, to preserve and pro-
mote the life of the intellect itself. In a realm infected
with irrationality, with omnipresent contradictions of
the laws of thought, intellect cannot live, much less
flourish; in the world of sense, it has no proper subject-

TE

THE HUMAN WORTH OF RIGOROUS THINKING 23

matter, no home, no life. To live, to flourish, it must
be able to think, to think in accordance with the laws
of its being. It is stimulated and its activity is sus-
tained by two opposite forces: discord and concord.
By the one it is driven; by the other, drawn. Intel-
lect is a perpetual suitor. The object of the suit is,
not the conquest of matter, it is a thing of mind, it is
the music of the spirit, it is Harmonia, the beautiful
daughter of the Muses. The aim, the ideal, the beati-
tude of intellect is harmony. That is the meaning
of its endless talk about compatibilities, consistencies
and concords, and that is the meaning of its endless
battling and circumvention and transcendence of con-
tradiction. But what of the applications of science and
public service? These are by-products of the intellect's
aim and of the pursuit of its ideal. Many things it
regards as worthy, high, and holy — applications of
science, public service, the " wonder" of Aristotle,
Jacobi's " honor of the human spirit," Diotima's "glori-
ous fame of immortal virtue" — but that which, by
the law of its being, Intellect seeks above all and per-
petually pursues and loves, is Harmony. It is for a
home and a dwelling with her that intellect creates a
world; and its admonition is: Seek ye first the kingdom
of harmony, and all these things shall be added unto
you.

And the ideal and admonition, thus revealed in the
light of analysis, are justified of history. Inverting the
order of time, we have only to contemplate the great
periods in the intellectual life of Paris, Florence, and
Athens. If, among these mightiest contributors to the
spiritual wealth of man, Athens is supreme, she is also
supreme in her devotion to the intellect's ideal. It is
of Athens that Euripides sings:

24 THE HUMAN WORTH OF RIGOROUS THINKING

The sons of Erechtheus, the olden,

Whom high gods planted of yore
In an old land of heaven upholden,

A proud land untrodden of war:
They are hungered, and lo, their desire

With wisdom is fed as with meat:
In their skies is a shining of fire,

A joy in the fall of their feet:
And thither with manifold dowers,

From the North, from the hills, from the morn,
The Muses did gather their powers,

That a child of the Nine should be born;
And Harmony, sown as the flowers,

Grew gold in the acres of corn.

And thus, ladies and gentlemen, our lecturer will say,
what I wish you to ,seejiere is, that jsciejice. .. and espe-
cially mathematics, the ideal torm of science^re crea-
tions^oPthe intellect in^its^qu^st"n^ is

as"such creations thar they are to be judged and their
human worth appraised. Of the applications of mathe-
matics to engineering and its service in natural science,
I have spoken at length, he will say, in course of previous
lectures. Other great themes of our subject remain for
consideration. To appraise the worth of mathematics
as a discipline in the art of rigorous thinking and as a
means of giving facility and wing to the subtler imagina-
tion; to estimate and explain its value as a norm for
criticism and for the guidance of speculation and pioneer-
ing in fields not yet brought under the dominion of
logic; to estimate its esthetic worth as showing forth
in psychic light the law and order of the psychic world;
to evaluate its ethical significance in rebuking by its
certitude and eternality the facile scepticism that doubts
all knowledge, and especially in serving as a retreat for
the spirit when as at times the world of sense seems
madly bent on heaping strange misfortunes up and "to
and fro the chances of the years dance like an idiot in

TI

THE HUMAN WORTH OF RIGOROUS THINKING 25

the wind"; to give a sense of its religious value in "the
%5 contemplation of ideas under the form of eternity,"
in disclosing a cosmos of perfect beauty and everlasting
iorder and in presenting there, for meditation, endless
sequences traversing the rational world and seeming to
point to a mystical region above and beyond: these and
similar themes, our speaker will say, remain to be dealt
with in subsequent lectures of the course.

THE HUMAN SIGNIFICANCE OF
MATHEMATICS *

Homo sum; humani nil a me alienum puto.

— TERENCE

THE subject of this address is not of my choosing. It
came to me by assignment. I may, therefore, be allowed
to say that it is in my judgment ideally suited to the
occasion. This meeting is held here upon this beautiful
coast because of the presence of an international exposi-
tion, and we are thus invited to a befitting largeness and
liberality of spirit. An international exposition prop-
erly may and necessarily will admit many things of a
character too technical to be intelligible to any one but
the expert and the specialist. Such things, however,
are only incidental — contributory, indeed, yet inci-
dental — to pursuit of . the principal aim, which is, I
believe, or ought to be, the representation of human
things as human — an exhibition and interpretation
of industries, institutions, sciences and arts, not pri-
marily in their accidental or particular character as
illustrating individuals or classes or specific localities
or times, but primarily in their essential and universal
character as representative of man. A world-exposition
will, therefore, as far as practicable, avoid placing in
the forefront matters so abstruse as to be fit for the

1 An address delivered August 3, 1915, Berkeley, Calif., at a joint meet-
ing of the American Mathematical Society, the American Astronomical
Society, and Section A of the American Association for the Advancement
of Science. Printed in Science, November 12, 1915.

HUMAN SIGNIFICANCE OF MATHEMATICS 27

contemplation and understanding of none but special-
ists; it will, as a whole, and in all its principal parts,
address itself to the general intelligence; for it aims at
being, for the multitudes of men and women who avail
themselves of its exhibitions and lessons, an exposition
of humanity: an exposition, no doubt, of the activities
and aspirations and prowess of individual men and
women, but of men and women, not in their capacity,
as individuals, but as representatives of humankind. In-
dividual achievements are not the object, they are the
means, of the exposition. The object is humanity.

What is the human significance — what is the sig-
nificance for humanity — of "the mother of the sci-
ences"? And how may the matter be best set forth,
not for the special advantage of professional mathe-
maticians, for I shall take the liberty of having these
but little in mind, but for the advantage and under-
standing of educated men and women in general? I am
unable to imagine a more difficult undertaking, so tech-
nical, especially in its language, and so immense is the
subject. It is clear that the task is far beyond the
resources of an hour's discourse, and so it is necessary
to restrict and select. This being the case, what is it
best to choose? The material is superabundant. What
part of it or aspect of it is most available for the end in
view? "In abundant matter to speak a little with
elegance," says Pindar, "is a thing for the wise to listen
to." It is not, however, a question of elegance. It is
a question of emphasis, of clarity, of effectiveness. What
shall be our major theme?

Shall it be the history of the subject? Shall it be
the modern developments of mathematics, its present
status and its future outlook? Shall it be the utilities
of the science, its so-called applications, its service in

28 HUMAN SIGNIFICANCE OF MATHEMATICS

practical affairs, in engineering and in what it is cus-
tomary to call the sciences of nature? Shall it be the
logical foundations of mathematics, its basic principles,
its inner nature, its characteristic processes and struc-
ture, the differences and similitudes that come to light
in comparing it with other forms of scientific and philo-
sophic activity? Shall it be the bearings of the science
as distinguished from its applications — the bearings of
it as a spiritual enterprise upon the higher concerns of
man as man? It might be any one of these things.
They are all of them great and inspiring themes.

It is easy to understand that a historian would choose
the first. The history^ of mathematicsjisindeed im-
pressive. but isTit not too long and too technical? And
is it not already accessible in a~"feuge piitffisliecr litera-
ture of its own? I grant, the historian would say, that
its history is long, for in respect of antiquity mathematics
is a rival of art, surpassing nearly all branches of sci-
ence and by none of them surpassed. I grant that, for
laymen, the history is technical, frightfully technical,
requiring interpretation in the interest of general in-
telligence. I grant, too, that the history owns a large
literature, but this, the historian would say, is not
designed for the general reader, however intelligent, the
numerous minor works no less than the major ones,
including that culminating monumental work of Moritz
Cantor, being, all of them, addressed to specialists and
intelligible to them alone. And yet it would be pos-
sible to tell in one hour, not indeed the history of mathe-
matics, but a true story of it that would be intelligible
to all and would show its human significance to be
profound, manifold, and even romantic. Tt^.wQTilcl...t
pruialblg— ±*» Jiiiw 1iisl"wally that this science, which

now carries its hea^L^*-«ljghLin^J:h^

HUMAN SIGNIFICANCE OF MATHEMATICS 29

of pure abstractions, has always kept its fee_t_upon the
solid earth; it would be possible to show that it owns
indeed a lowly origin, in the familiar needs of common
life, in the homely necessities of counting herds and
measuring lands; it would be possible to show that,
notwithstanding its birth in the concrete things of sense
and raw reality, it yet so appealed to sheer intellect —
and we must not forget that creative intellect is the
hum^m^a£uh£jDa^^ — - it so appealed to this

distinctive and disinterested faculty of man that, long
before the science rose to the level of a fine art in
the great days of Euclid and Archimedes, Plato in the
wisdom of his maturer years judged it essential to the
education of freemen because, said he, there is in it a
necessary something against which even God can not
contend and without which neither gods nor demi-gods
can wisely govern mankind; it would be possible, our
historian could say, to show historically to educated
laymen that, even prior to the inventions of analytical
geometry and the infinitesimal calculus, mathematics
had played an indispensable role in the "Two New
Sciences" of physics and mechanics in which Galileo
laid the foundations of our modern knowledge of nature;
it would be possible to show not only that the analytical
geometry of Descartes and Fermat and the calculus
of Leibnitz and Newton have been and are essential
to our still advancing conquest of the sea, but that it
is owing to the power of these instruments that the
genius of such as Newton, Laplace and Lagrange has
been enabled to create for us a new earth and a new
heavens compared with which the Mosaic cosmogony or
the sublimest creation of the Greek imagination is but
"as a cabinet of brilliants, or rather a little jewelled
cup found in the ocean or the wilderness"; it would

30 HUMAN SIGNIFICANCE OF MATHEMATICS

be possible to show historically that, just because the
pursuit of mathematical truth has been for the most part
disinterested — led, that is, by wonder, as Aristotle says,
and sustained by the love of beauty with the joy of
discovery — it would be possible to show that, just
because of the disinterestedness of mathematical re-
search, this science has been so well prepared to meet
everywhere and always, as they have arisen, the mathe-
matical exigencies of natural science and engineering;
above all, it would be feasible to show historically that
to the same disinterestedness of motive operating through
the centuries we owe the upbuilding of a body of pure
doctrine so towering to-day and vast that no man, even
though he have the "Andean intellect" of a Poincare,
can embrace it all. This much, I believe, and perhaps
more, touching the human significance of mathematics,
a historian of the science might reasonably hope to
demonstrate in one hour.

More difficult, far more difficult, I think, would be
the task of a pure mathematician who aimed at an
equivalent result by expounding, or rather by delineat-
ing, for he could not in one hour so much as begin to
expound, the modern developments of the subject.
Could he contrive even to delineate them in a way to
reveal their relation to what is essentially humane? Do
but consider for a moment the nature of such an enter-
prise. Mathematics may bo Irgitimritrly pursued for
its own^sakeor for the sake of its application* nr
a viejvJjto^nderstajidiB^ ^ts^4ogical"^ouarla.tions
inlfitnal structure or in__the interest of_ magnanimity
or f or_the^~sake oTTts bearings^upon the supreme con-
ce~rjis of man akman or from^two or more of these
Qursupposed delineator is actuated

by the first of them: his interest in mathematics is an

HUMAN SIGNIFICANCE OF MATHEMATICS 31

jnterest in mathematics for the sake of mathematics;
for fatmjfre science is simply^rtaTge^ali^jrowing j>gj£^

of logical consistencies or compatibilities; he derives
his inspiration from the muse "01 intellectual harmony;
he is a pure mathematician. He knows that pure mathe-
matics is a house of many chambers; he knows that
its foundations lie far beneath the level of common
thought; and that the superstructure, quickly tran-
scending the power of imagination to follow it, ascends
higher and higher, ever keeping open to the sky; he
knows that the manifold chambers — each of them a
mansion in itself — are all of them connected in won-
drous ways, together constituting a fit laboratory and
dwelling for the spirit of men of genius. He has assumed
the task of presenting a vision of it that shall be worthy
of a world-exposition. Can he keep the obligation?
He wishes to show that the life and work of pure
mathematicians are human life and work: he desires
to show that these toilers and dwellers in the chambers
of pure thought are representative men. He would
exhibit the many-chambered house to the thronging
multitudes of his fellow men and women; he would
lead them into it; he would conduct them from chamber
to chamber by the curiously winding corridors, passing
now downward, now upward, by delicate passage-
ways and subtle stairs; he would_show them that the
wondrous castle is not vaT dead" or stati£_jffair_1^ea
structure of marble or steel, but a living architecture, ..a
living^ mansion of life, human as their own; he would
show them the "mathetic spirit at work, how it is ever

, fabrics of beauty, finer than

_

gossamer yet stronger than cables of steel; he would
show them how it is ever enlarging its habitation, deep-
ening its foundations, expanding more and more and

32 HUMAN SIGNIFICANCE OF MATHEMATICS

elevating the^sm>erstructure; and, what is even more
amazing, how it perpetually performs the curious miracle
of permanence combined with change, transforming,
that is, the older portions of the edifice without destroy-
ing it, for the structure is eternal: in a word, he would
show them a vision of the whole, and he would do it in
a way to make them perceive and feel that, in thus
beholding there a partial and progressive attainment of
the higher ideals of man, they were but gazing upon a
partial and progressive realization of their own appe-
titions and dreams.

That is what he would do. But how? Mengenlehre,
Zahlenlehre, algebras of many kinds, countless geometries
of countless infinite spaces, function theories, trans-
formations, invariants, groups and the rest — how can
these with all their structural finesse, with their heights
and depths and limitless ramifications, with their laby-
rinthine and interlocking modern developments — I will
not say how can they be presented in the measure and
scale of a great exposition — but how is it possible in
one hour to give laymen even a glimpse of the endless
array? Nothing could be more extravagant or more
absurd than such an undertaking. Compared with it,
the American traveler's hope of being able to see Rome
in a single forenoon was a most reasonable expectation.
But it is worth while trying to realize how stupendous
the absurdity is.

It is evident that our would-be delineator must com-
promise. He can not expound, he can not exhibit, he
can not even delineate the doctrines whose human
worth he would thus disclose to his fellow men and
women. The fault is neither his nor theirs. It must
be imputed to the nature of things. But he need not,
therefore, despair and he need not surrender. The

HUMAN SIGNIFICANCE OF MATHEMATICS 33

method he has proposed — the method of exposition —
that indeed he must abandon as hopeless, but not his
aim. He is addressing men and women who are no
doubt without his special knowledge and his special
discipline, as he in his turn is without theirs, but who
are yet essentially like himself. He would have them
as fellows and comrades persuaded of the dignity of his
Fach: he would have them feel that it is also theirs;
he would have them convinced that mathematics stands
for an immense body of human achievements, for a
diversified continent of pure doctrine, for a discovered
world of intellectual harmonies. He can not show it
to them as a painter displays a canvas or as an architect
presents a cathedral. He can not give them an imme-
diate vision of it, but he can give them intimations;
by appealing to their fantasie and, through analogy with
what they know, to their understanding, not only can
he convince them that his world exists, but he can give
them an intuitive apprehension of its living presence
and its meaning for humankind. This is possible be-
cause, like him, they, too, are idealists, dreamers and
poets — such essentially are all men and women. His
auditors or his readers have all had some experience of
ideas and of truth, they have all had inklings of more
beyond, they have all been visited and quickened by a
sense of the limitless possibilities of further knowledge
in every direction, they have all dreamed of the perfect
and have felt its lure. They are thus aware that the
small implies the large; having seen hills, they can
believe in mountains; they know that Euripides, Shake-
speare, Dante, Goethe, are but fulfillments of prophecies
heard in peasant tales and songs; they know that the
symphonies of Beethoven or the dramas of Wagner are
harbingered in the melodies and the sighs of those who

34 HUMAN SIGNIFICANCE OF MATHEMATICS

garner grain and in their hearts respond to the music of
the winds or the "solemn anthems of the sea"; they
sense the secret by which the astronomy of Newton and
Laplace is foretokened in the shepherd's watching of
the stars; and knowing thus this plain spiritual law
of progressiveness and implication, they are prepared
to grasp the truth that modern mathematics, though
they do not understand it, is, like the other great things,
but a sublime fulfillment, the realization of prophecies
involved in what they themselves, in common with
other educated folk, know of the rudiments of the sci-
ence. Indeed, they would marvel if upon reflection
it did not seem to be so. Our pure mathematician in
speaking to his fellow men and women of his science
will have no difficulty in persuading them that he is
speaking of a subject immense and eternal. As born
idealists they have intimations of their own — the
evidence of intuition, if you please — or a kind of insight
resembling that of the mystic — that in the world of
mind there must be something deeper and higher, stabler
and more significant, than the pitiful ideas in life's
routine and the familiar vocations of men. They are
thus prepared to believe, before they are told, that
behind the veil there exists a universum of exact thought,
an everlasting cosmos of ordered ideas, a stable world
of concatenated truth. In their study of the elements,
in school or~ cortege, they may have caught a shimmer
of it or, in rare moments of illumination, even a gleam.
Of the existence, the reality, the actuality, of our pure
mathematician's world they will have no doubt, and
they will have no doubt of its grandeur. They may
even, in a vague way, magnify it overmuch, feeling that
it is, in some wise, more than human, significant only
for the rarely gifted spirit that dwells, like a star,

HUMAN SIGNIFICANCE OF MATHEMATICS 3$

apart. The pure mathematician's difficulty lies in
showing, in his way, that such is not the case. For he
does not wish to adduce utilities and applications. He
is well aware of these. He knows that if he "would
tell them they are more in number than the sands."
Neither does he despise them as of little moment. On
the contrary, he values them as precious. But he wishes
to do his subject and his auditors the honor of speaking
from a higher level: he desires to vindicate the worth
of mathematics on the ground of its sheer ideality, on
the ground of its intellectual harmony, on the ground
of its beauty, "free from the gorgeous trappings" of
sense, pure, austere, supreme. To do this, which ought,
it seems, to be easy, experience has shown to be exceed-
ingly difficult. For the multitude of men and women,
even the educated multitude, are wont to cry,

Such knowledge is too wonderful for me,
It is too high, I can not attain unto it,

thus meaning to imply, What, then, or where is its
human significance? Their voice is heard in the chal-
lenge once put to me by the brilliant author of "East
London Visions." What, said he, can be the human
significance of "this majestic intellectual cosmos of
yours, towering up like a million-lustred iceberg into the
arctic night," seeing that, among mankind, none is
permitted to behold its more resplendent wonders save
the mathematician alone? What response will our pure
mathematician make to this challenge? Make, I mean,
if he be not a wholly nai've devotee of his science and
so have failed to reflect upon the deeper grounds of its
justification. He may say, for one thing, what Pro-
fessor Klein said on a similar occasion:

Apart from the fact that pure mathematics can not be supplanted by
anything else as a means for developing the purely logical faculties of the

36 HUMAN SIGNIFICANCE OF MATHEMATICS

mind, there must be considered here, as elsewhere, the necessity of the
presence of a few individuals in each country developed in far higher degree
than the rest, for the purpose of keeping up and gradually raising the general
standard. Even a slight raising of the general level can be accomplished
only when some few minds have progressed far ahead of the average.

That is doubtless a weighty consideration. But is
it all or the best that may be said? It is just and
important but it does not go far enough; it is not, I
fear, very convincing; it is wanting in pungence and
edge; it does not touch the central nerve of the chal-
lenge. Our pure mathematician must rally his sceptics
with sharper considerations. He may say to them:
You challenge the human significance of the higher
developments of pure mathematics because they are
inaccessible to all but a few, because their charm is
esoteric, because their deeper beauty is hid from nearly
all mankind. Does that consideration justify your
challenge? You are individuals, but you are also
members of a race. Have you as individuals no human
interest nor human pride in the highest achievements
of your race? Is nothing human, is nothing humane,
except mediocrity and the commonplace? Was Phidias
or Michel Angelo less human than the carver and
painter of a totem-pole? Was Euclid or Gauss or
Poincare less representative of man than the countless
millions for whom mathematics has meant only the
arithmetic of the market place or the rude geometry
of the carpenter? Does the quality of humanity in
human thoughts and deeds decrease as they ascend
towards the peaks of achievement, and increase in
proportion as they become vulgar, attaining an upper
limit in the beasts? Do you not know that precisely
the reverse is true? Do you not count aspiration hu-
mane? Do you not see that it is not the common
things that every one may reach, but excellences high-

HUMAN SIGNIFICANCE OF MATHEMATICS

37

dwelling among the rocks — do you not know that, in
respect of human worth, these things, which but few
can attain, are second only to the supreme ideals attain-
able by none?

How very different and how very much easier the
task of one who sought to vindicate the human sig-
nificance of mathematics on the ground of its applica-
tions! In resrjecJ^JLj£inipe*3^

ence between the pure and the
applied mathematician is profound. I
liken spjjjJjiiaJL^hings to things of sense — muchke
tKe "difference between one whcTgfeets *a new-born day
because of its glory and one who regards it as a time
for doing chores and values its light only as showing
the way. F^orthe formerT mathematics is justified by
its supreme beliutvT"t5£^ its pumifnlHj^se.

But are the two kinds oT value essentially incompatible?
They are certainly not. The difference is essentially
a difference of authority — a difference, that is, of
worth, of elevation, of excellence. The pure mathe-
matician and the applied mathematician sometimes may,
indeed they not infrequently do, dwell together har-
moniously in a single personality. If our spokesman be
such a one — and I will not suppose the shame of having
the utilities of the science represented on such an occa-
sion by one incapable of regarding it as anything but
a tool, for that would be disgraceful — if, then, our
spokesman be such a one as I have supposed, he might
properly begin as follows: In speaking to you of the
applications of mathematics I would not have you sup-
pose, ladies and gentlemen, that I am thus presenting
the highest claims of the science to your regard; for its
highest justification is the charm of its immanent
beauty; I do not mean, he will say, the beauty of ap-

38 HUMAN SIGNIFICANCE OF MATHEMATICS

pearances — the fleeting beauties of sense, though
these, too, are precious — even the outer garment, the
changeful robe, of reality is a lovely thing; I mean the
eternal beauty of the world of pure thought; I mean
intellectual beauty; in mathematics this nearly attains
perfection; and "intellectual beauty is self-sufficing ";
uses, on the other hand, are not; they wear an aspect
of apology; uses resemble excuses, they savor a little of
a plea in mitigation. Do you ask: Why, then, plead
them? Because, he will say, many good people have
a natural incapacity to appreciate anything else; be-
cause, also, many of the applications, especially the
higher ones, are themselves matters of exceeding beauty;
and especially because I wish to show, not only that
use and beauty are compatible forms of worth, but
that the more mathematics has been cultivated for the
sake of its inner charm, the fitter has it become for
external service.

Having thus at the outset put himself in proper light
and given his auditors a scholar's warning against what
would else, he fears, foster a disproportionment of
values, what will he go on to signalize among the utili-
ties of a science whose primary allegiance to logical
rectitude allies it to art, and which only incidentally
and secondarily shapes itself to the ends of instrumental
service? He knows that the applications of mathe-
matics, if one will but trace them out in their multi-
farious ramifications, are as many-sided as the industries
and as manifold as the sciences of men, penetrating
everywhere throughout the full round of life. What
will he select? He will not dwell long upon its homely
uses in the rude computations and mensurations of
counting-house and shop and factory and field, for this
indispensable yet humble manner of world-wide and

HUMAN SIGNIFICANCE OF MATHEMATICS 39

perpetual service is known of all men and women. He
will quickly pass to higher considerations — to naviga-
tion, to the designing of ships, to the surveying of
lands and seas, and the charting of the world, to the
construction of reservoirs and aqueducts, canals, tunnels
and railroads, to the modern miracles of the marine
cable, the telegraph, the telephone, to the multiform
achievements of every manner of modern engineering,
civil, mechanical, mining, electrical, by which, through
the advancing conquest of land and sea and air and
space and time, the conveniences and the prowess of
man have been multiplied a billionfold. It need not be
said that not all this has been done by mathematics
alone. Far from it. It is, of course, the joint achieve-
ment of many sciences and arts, but — and just this
is the point — the contributions of mathematics to the
great work, direct and indirect, have been indispensable.
And it will require no great skill in our speaker to show
to his audience, if it have a little imagination, that, as I
have said elsewhere, if all these mathematical contri-
butions were by some strange spiritual cataclysm to be
suddenly withdrawn, the life and body of industry and
commerce would suddenly collapse as by a paralytic
stroke, the now splendid outer tokens of material civiliza-
tion would quickly perish, and the face of our planet
would at once assume the aspect of a ruined and bank-
rupt world. For such is the amazing utility, such the
wealth of by-products, it you please, tHat comeTr<3m
a gripru^jjjprl arf^that^ywes it&^Hfe^.its continuity and
its ""power to ma.n's. Ipye of mtellectuaMiannpny and
as its

tion. Indeed it appears — contrary to popular belief —
trTaTin our world there is nothing else quite so practical
as the inspiration of a muse.

4O HUMAN SIGNIFICANCE OF MATHEMATICS

But this is not all nor nearly all to which our applied
mathematician will wish to invite attention. It is only
the beginning of it. Even if he does not allude to the
quiet service continuously and everywhere rendered by
mathematics in its role as a norm or standard or ideal
in every field of thought whether exact or inexact, he
will yet desire to instance forms and modes of applica-
tion compared with which those we have mentioned,
splendid and impressive as they are, are meager and
mean. For those we have mentioned are but the more
obvious applications — those, namely, that continually
announce themselves to our senses everywhere in the
affairs, both great and small, of the workaday world.
But the really great applications of mathematics — those
which, rightly understood, best of all demonstrate the
human significance of the science — are not thus obvious ;
they do not, like the others, proclaim themselves in the
form of visible facilities and visible expedients every-
where in the offices, the shops, and the highways of
commerce and industry; they are, on the contrary,
almost as abstract and esoteric as mathematics itself,
for they are the uses and applications of this science
in other sciences, especially in astronomy, in mechanics
and in physics, but also and increasingly in the newer
sciences of chemistry, geology, mineralogy, botany,
zoology, economics, statistics and even psychology, not
to mention the great science and art of architecture.
In the matter of exhibiting the endless and intricate
applications of mathematics to the natural sciences,
applications ranging from the plainest facts of crystal-
lography to the faint bearings of the kinetic theory of
gases upon the constitution of the Milky Way, our
speaker's task is quite as hopeless as we found the pure
mathematician's to be; and he, too, will have to com-

HUMAN SIGNIFICANCE OF MATHEMATICS 41

promise; he will have to request his auditors to ac-
quaint themselves at their leisure with the available
literature of the subject and especially to read atten-
tively the great work of John Theodore Merz dealing
with the " History of European Thought in the Nine-
teenth Century," where they will find, in a form fit for
the general reader, how central has been the r61e of
mathematics in all the principal attempts of natural
science to find a cosmos in the seeming chaos of the
natural world. Another many-sided work that in this
connection he may wish to commend as being in large
part intelligible to men and women of general education
and catholic mind is Enriques's "Problems of Science."

I turn now for a moment to the prospects of one who
might choose to devote the hour to an exposition or
an indication of modern developments in what it is
customary to call the foundations of mathematics — to
a characterization, that is, and estimate of that far-
reaching and still advancing critical movement which
has to do with the relations of the science, philosophi-
cally considered, to the sciences of logic and methodology.
What can he say on this great theme that will be in-
telligible and edifying to the multitudes of men and
women who, though mathematically inexpert, yet have
a genuine humane curiosity respecting even the pro-
founder and subtler life and achievements of science?
He can point out that mathematics, like all the other
sciences, like the arts too, for that matter, and like
philosophy, originates in the refining process of reflec-
tion upon the crude data of common sense ; he can point
out that this process has gradually yielded from out the
raw material and still continues to yield more and more
ideas of approximate perfection in the respects of pre-
cision and form; he can point out that such ideas, thus

42 HUMAN SIGNIFICANCE OF MATHEMATICS

disentangled and trimmed of their native vagueness and
indetermination, disclose their mutual relationships and
so become amenable to the concatenative processes of
logic; and he can point out that these polished ideas
with their mutual relationships become the bases or
the content of various branches of mathematics, which
thus tower above common sense and appear to grow
out of it and to stand upon it like trees or forests
upon the earth. He will point out, however, that this
appearance, like most other obvious appearances, is de-
ceiving; he will, that is, point out that these upward-
growing sciences or branches of science are found, in
the light of further reflection, to be downward-growing
as well, pushing their roots deeper and deeper into a
dark soil far beneath the ground of evident common
sense; indeed, he will show that common sense is thus,
in its relation to mathematics, but as a sense-litten mist
enveloping only the mid-portion of the stately structure,
which, like a towering mountain, at once ascends into
the limpid ether far above the shining cloud and rests
upon a base of subterranean rock far below; he will
point out that, accordingly, mathematicians, in respect
of temperamental interest, fall into two classes — the
class of those who cultivate the upward-growing of the
science, working thus in the upper regions of clearer
light, and the class of those who devote themselves to
exploring the deep-plunging roots of the science; and it
is, he will say, to the critical activity of the latter class
— the logicians and philosophers of mathematics — that
we owe the discovery of what we are wont to call the
foundations of mathematics — the great discovery, that
is, of an immense mathematical sub-structure, which
penetrates far beneath the stratum of common sense
and of which many of even the greatest mathematicians

HUMAN SIGNIFICANCE OF MATHEMATICS 43

of former times were not aware. But whilst such founda-
tional research is in the main a modern phenomenon,
it is by no means exclusively such; and to protect his
auditors against a false perspective in this regard and
the peril of an overweening pride in the achievements of
their own time, our speaker may recommend to them
the perusal of Thomas L. Heath's superb edition of
Euclid's " Elements" where, especially in the first vol-
ume, they will be much edified to find, in the rich
abundance of critical citation and commentary which
the translator has there brought together, that the re-
fined and elaborate logico-mathematical researches of our
own time have been only a deepening and widening of
the keen mathematical criticism of a few centuries im-
mediately preceding and following the great date of
Euclid. Indeed but for that general declension of Greek
spirit which Professor Gilbert Murray in his "Four
Stages of Greek Religion" has happily characterized as
"the failure of nerve," what we know as the modern
critical movement in mathematics might well have come
to its present culmination, so far at least as pure geom-
etry is concerned, fifteen hundred or more years ago.
It is a pity that the deeper and stabler things of science
and the profounder spirit of man can not be here
disclosed in a manner commensurate with the great
exposition, surrounding us, of the manifold practical
arts and industries of the world. It is a pity there is
no means by which our speaker might, in a manner
befitting the subject and the occasion, exhibit intelligibly
to his fellow men and women the ways and results of
the last hundred years of research into the groundwork
of mathematical science and therewith the highly im-
portant modern developments in logic and the theory
of knowledge. How astonished the beholders would

44 HUMAN SIGNIFICANCE OF MATHEMATICS

be, how delighted too, and proud to belong to a race
capable of such patience and toil, of such disinterested
devotion, of such intellectual finesse and depth of pene-
tration. I can think of no other spectacle quite so im-
pressive as the inner vision of all the manifold branches
of rigorous thought seen to constitute one immense
structure of autonomous doctrine reposing upon the
spiritual basis of a few select ideas and, superior to the
fading beauties of time and sense, shining there like a
celestial city, in "the white radiance of eternity." That
is the vision of mathematics that a student of its phi-
losophy would, were it possible, present to his fellow
men and women.

In view of the foregoing considerations it evidently is,
I think, in the nature of the case impossible to give an
adequate sense of the human worth of mathematics if
one choose to devote the hour to any one of the great
aspects of it with which we have been thus far con-
cerned. Neither the history of the subject nor its
present estate nor its applications nor its logical founda-
tions — no one of these themes lends itself well to the
purpose of such exposition, and still less do two or more
of them combined. Even if such were not the case I
should yet feel bound to pursue another course; for I
have been long persuaded that, in respect of its human
significance, mathematics invites to a point of view
which, unless I am mistaken, has not been taken and
held in former attempts at appreciation. I have al-
ready alluded to bearings of mathematics as distin-
guished from applications. It is with its bearings that
I wish to deal. I mean its bearings upon the higher
concerns of man as man — those interests, namely,
which have impelled him to seek, over and above the
needs of raiment and shelter and food, some inner

HUMAN SIGNIFICANCE OF MATHEMATICS 45

adjustment of life to the poignant limitations of life in
our world and which have thus drawn him to manifold
forms of wisdom, not only to mathematics and natural
science, but also to literature and philosophy, to religion
and art, and theories of righteousness. What is the
role of mathematics in this perpetual endeavor of the
human spirit everywhere to win reconciliation of its
dreams and aspirations with the baffling conditions and
tragic facts of life and the world? What is its relation
to the universal quest of man for some supreme and
abiding good that shall assuage or annul the discords
and tyrannies of time and limitation, withholding less
and less, as time goes by, the freedom and the peace
of an ideal harmony infinite and eternal?

In endeavoring to suggest, in the time remaining for
this address, a partial answer to that great question, in
attempting, that is, to indicate the relations of mathe-
matics to the supreme ideals of mankind, it will be
necessary to seek a perspective point of view and to
deal with large matters in a large way.

Of the countless variety of appetitions and aspirations
that have given direction and aim to the energies of
men and that, together with the constraining conditions
of life in our world, have shaped the course and deter-
mined the issues of human history, it is doubtless not
yet possible to attempt confident and thoroughgoing
classification according to the principle of relative dig-
nity or that of relative strength. If, however, we ask
whether, in the great throng of passional determinants
of human thought and life, there is one supreme passion,
one that in varying degrees of consciousness controls
the rest, unifying the spiritual enterprises of our race
in directing and converging them all upon a single
sovereign aim, the answer, I believe, can not be doubt-

46 HUMAN SIGNIFICANCE OF MATHEMATICS

ful: the activities and desires of mankind are indeed
subject to such imperial direction and control. And if
now we ask what the sovereign passion is, again the
answer can hardly admit of question or doubt. In order
to see even a priori what the answer must be, we have
only to imagine a race of beings endowed with our
human craving for stability, for freedom, and for per-
petuity of life and its fleeting goods, we have only to
fancy such a race flung, without equipment of knowledge
or strength, into the depths of a treacherous universe
of matter and force where they are tossed, buffeted and
torn by the tumultuous onward-rushing flood of the
cosmic stream, originating they know not whence and
flowing they know not why nor whither, we have, I say,
only to imagine this, sympathetically, which ought to
be easy for us as men, and then to ask ourselves what
would naturally be the controlling passion and dominant
enterprise of such a race — unless, indeed, we suppose
it to become strangely enamored of distress or to be
driven by despair to self -extinction. We humans re-
quire no Gotama nor Heracleitus to tell us that man's
lot is cast in a world where naught abides. The uni-
versal impermanence of things, the inevitableness of
decay, the mocking frustration of deepest yearnings
and fondest dreams, all this has been keenly realized
wherever men and women have had seeing eyes or been
even a little touched with the malady of meditation,
and everywhere in the literature of power is heard the
cry of the mournful truth. "The life of man," said
the Spirit of the Ocean, "passes by like a galloping
horse, changing at every turn, at every hour."

"Great treasure halls hath Zeus in heaven,
From whence to man strange dooms be given,
Past hope or fear."

HUMAN SIGNIFICANCE OF MATHEMATICS 47

Such is the universal note. Whether we glance at the
question in a measure a priori, as above, or look into
the cravings of our own hearts, or survey the history
of human emotion and thought, we shall find, I think,
in each and all these ways, that human life owns the
supremacy of one desire: it is the passion for emancipa-
tion, for release from life's limitations and the tyranny
of change: it is our human passion for some ageless
form of reality, some everlasting vantage-ground or rock
to stand upon, some haven of refuge from the all-
devouring transformations of the weltering sea. And
so it is that our human aims, aspirations, and toils
thus find their highest unity — their only intelligible
unity — in the spirit's quest of a stable world, in its
endless search for some mode or form of reality that
is at once infinite, changeless, eternal.

Does some one say: This may be granted, but what
is the point of it all? It is obviously true enough, but
what, pray, can be its bearing upon the matter in hand?
What light does it throw upon the human significance
of mathematics? The question is timely and just. The
answer, which will grow in fullness and clarity as we
proceed, may be at once begun.

How long our human ancestors, in remote ages, may
have groped, as some of their descendants even now
grope, among the things of sense, in the hope of finding
there the desiderated good, we do not know — past time
is long and the evolution of wisdom has been slow. We
do know that, long before the beginnings of recorded
history, superior men — advanced representatives of
their kind — must have learned that the deliverance
sought was not to be found among the objects of the
mobile world, and so the spirit's quest passed from
thence; passed from the realm of perception and sense

48 HUMAN SIGNIFICANCE OF MATHEMATICS

to the realm of concept and reason: thought ceased,
that is, to be merely the unconscious means of pursuit
and became itself the quarry — mind had discovered
mind; and there, in the realm of ideas, in the realm of
spirit proper, in the world of reason or thought, the
great search — far outrunning historic time — has been
endlessly carried on, with varying fortunes, indeed, but
without despair or breach of continuity, meanwhile
multiplying its resources and assuming gradually, as the
years and centuries have passed, the characters and
forms of what we know today as philosophy and science
and art. I have mentioned the passing of the quest from
the realm of sense to the realm of conception: a most
notable transition in the career of mind and especially
significant for the view I am aiming to sketch. For
thought, in thus becoming a conscious subject or object
of thought, then began its destined course in reason:
in ceasing to be merely an unconscious means of pursuit
and becoming itself the quarry, it definitely entered
upon the arduous way that leads to the goal of rigor.
And so it is evident that the way in question is not a
private way; it does not belong exclusively to mathe-
matics; it is public property; it is the highway of con-
ceptual research. For it is a mistake to imagine that
mathematics, in virtue of its reputed exactitude, is an
insulated science, dwelling apart in isolation from other
forms and modes of conceptual activity. It would be
such, were its rigor absolute; for between a perfection
and any approximation thereto, however close, there
always remains an infinitude of steps. But the rigor
of mathematics is not absolute — absolute rigor is an
ideal, to be, like other ideals, aspired unto, forever
approached, but never quite attained, for such attain-
ment would mean that every possibility of error or

HUMAN SIGNIFICANCE OF MATHEMATICS 49

indetermination, however slight, had been eliminated
from idea, from symbol, and from argumentation. We
know, however, that such elimination can never be
complete, unless indeed the human mind shall one day
lose its insatiable faculty for doubting. What, then,
is the distinction of mathematics on the score of exacti-
tude? Its distinction lies, not in the attainment of
rigor absolute, but partly in its exceptional devotion
thereto and especially in the advancement it has made
along the endless path that leads towards that perfec-
tion. But, as I have already said, it must not be
thought that mathematics is the sole traveler upon the
way. It is important to see clearly that it is far from
being thus a solitary enterprise. First, however, let
us adjust our imagery to a better correspondence with
the facts. I have spoken of the path. We know, how-
ever, that the paths are many, as many as the varieties
of conceptual subject-matter, all of them converging
towards the same high goal. We see them originate
here, there and yonder in the soil and haze of common
thought; we see how indistinct they are at first — how
ill-defined; we observe how they improve in that regard
as the ideas involved grow clearer and clearer, more and
more amenable to the use and governance of logic. At
length, when thought, in its progress along any one of
the many courses, has reached a high degree of refine-
ment, precision and certitude, then and thereafter, but
not before, we call it mathematical thought; it has
undergone a long process of refining evolution and
acquired at length the name of mathematics; it is
not, however, the creature of its name; what is called
mathematics has been long upon the way, owning at
previous stages other designations — common sense,
practical art perhaps, speculation, theology it may be,

5O HUMAN SIGNIFICANCE OF MATHEMATICS

philosophy, natural science, or it may be for many a
millennium no name at all. Is it, then, only a question
of names? In a sense, yes: the ideal of thought is
rigor; mathematics is the name that usage employs to
designate, not attainment of the ideal, for it can not
be attained, but its devoted pursuit and close approxi-
mation. But this is not the essence of the matter.
The essence is that all thought, thought in all its stages,
however rude, however refined, however named, owns
the unity of being human: spiritual activities are one.
Mathematics thus belongs to the great family of spiritual
enterprises of man. These enterprises, all the members
of the great family, however diverse in form, in modes
of life, in methods of toil, in their progress along the
way that leads towards logical rectitude, are alike chil-
dren of one great passion. In genesis, in spirit and
aspiration, in motive and aim, natural science, theology,
philosophy, jurisprudence, religion and art are one with
mathematics: they are all of them sprung from the
human spirit's craving for invariant reality in a world
of tragic change; they all of them aim at rescuing man
from "the blind hurry of the universe from vanity to
vanity": they seek cosmic stability — a world of abiding
worth, where the broken promises of hope shall be
healed and infinite aspiration shall cease to be mocked.

Such has been the universal and dominant aim and
such are the cardinal forms that time has given its
prosecution.

And now we must ask: What have been the fruits
of the endless toil? What has the high emprize won?
And what especially, have been the contributions of
mathematics to the total gain? To recount the story
of the spirit's quest for ageless forms of reality would
be to tell afresh, from a new point of view, the his-

HUMAN SIGNIFICANCE OF MATHEMATICS 5 1

tory of human thought, so many and so diverse are
the modes or aspects of being that men have found or
fancied to be eternal. Edifying indeed would be the
tale, but it is long, and the hour contracts. Even a
meager delineation is hardly possible here. Yet we
must not fail to glance at the endless array and to call,
at least in part, the roll of major things. But where
begin? Shall it be in theology? How memory responds
to the magic word. "The past rises before us like a
dream." As the long succession of the theological cen-
turies passes by, what a marvelous pageant do they
present of human ideals, contrivings and dreams, both
rational and superrational. Alpha and Omega, the be-
ginning and ending, which is, which was and which is
to come; I Am That I Am; Father of lights with which
is no variableness, neither shadow of turning; the
bonitas, unitas, infinitas, immutabilitas of Deity; the
undying principle of soul; the sublime hierarchy of
immortal angels, terrific and precious, discoursed of by
sages, commemorated by art, feared and loved by mil-
lions of men and women and children: these things may
suffice to remind us of the invariant forms of reality
found or invented by theology in her age-long toil and
passion to conquer the mutations of time by means of
things eternal.

But theology's record is only an immense chapter of
the vastly more inclusive annals of world-wide philo-
sophic speculation running through the ages. If we
turn to philosophy understood in the larger sense, if we
ask what answers she has made in the long course of
time to the question of what is eternal, so diverse and
manifold are the voices heard across the centuries, from
the East and from the West, that the combined response
must needs seem to an unaccustomed ear like an infinite

52 HUMAN SIGNIFICANCE OF MATHEMATICS

babel of tongues: the Confucian Way of Heaven; the
mystic Tao, so much resembling fate, of Lao Tzu and
Chuang Tzu; Buddhism's inexorable spiritual law of
cause and effect and its everlasting extinction of indi-
viduality in Nirvana — the final blowing out of con-
sciousness and character alike; Ahura Mazda, the holy
One, of Zarathustra; Fate, especially in the Greek
tragedies and Greek religion — the chain of causes in
nature, "the compulsion in the way things grow," a fine
thread running through the whole of existence and
binding even the gods; the cosmic matter, or TO
aTrtipov of Anaximander; the cosmic order, the rhythm
of events, the logos or reason or nous, of Heracleitus;
the finite, space-filling sphere, or One, of the deep
Parmenides; the four material and two psychic, six
eternal, elements, of Empedocles; the infinitude of ever-
lasting mind-moved simple substances of Anaxagoras;
the infinite multitude and endless variety of invariant
"seeds of things" of Leucippus, Democritus, Epicurus
and Lucretius, together with their doctrines of absolute
void and the conservations of mass and motion and
infinite room or space; Plato's eternal world of pure
ideas; the great Cosmic Year of a thousand thinkers,
rolling in vast endlessly repeated cycles on the beginning-
less, endless course of time from eternity to eternity;
the changeless thought-forms of Zeno, Gorgias and Aris-
totle; Leibnitz's indestructible, pre-established harmony;
Spinoza's infinite unalterable substance; the Absolute
of the Hegelian school; and so on and on far beyond
the limits of practicable enumeration. This somewhat
random partial list of things will serve to recall and to
represent the enormous motley crowd of answers that
the ages of philosophic speculation have made to the
supreme inquiry of the human spirit: what is there

HUMAN SIGNIFICANCE OF MATHEMATICS 53

that survives the mutations of time, abiding unchanged
despite the whirling flux of life and the world?

And now, in the interest of further representing salient
features in a large perspective view, let me next ask
what contribution to the solution of the great problem
has been made by jurisprudence. Jurisprudence is no
doubt at once a branch of philosophy and a branch of
science, but it has an interest, a direction and a char-
acter of its own. And for the sake of due emphasis it
will be well worth while to remind ourselves specifically
of the half-forgotten fact that, in its quest for justice
and order among men, jurisprudence long ago found an
answer to our oft-stated riddle of the world, an answer
which, though but a partial one, yet satisfied the greatest
thinkers for many centuries, and which, owing to the
inborn supernalizing proclivity of the human mind, still
exercises sway over the thought of the great majority of
mankind. I allude to the conception of jus naturale or
lex naturcz, the doctrine that in the order of Nature
there somehow exists a perfect, invariant, universally
and eternally valid system or prototype of law over and
above the imperfect laws and changeful polities of men
— a conception and doctrine long familiar in the juristic
thought of antiquity, dominating, for example, the An-
tigone of Sophocles, penetrating the Republic and the
Laws of Plato, proclaimed by Demosthenes in the Ora-
tion on the Crown, becoming, largely through the
Republic and the Laws of Cicero, the crowning con-
ception of the imperial jurisprudence of Rome, and still
holding sway, as I have said, except in the case of our
doubting Thomases of the law, who virtually deny
our world the existence of any perfection whatever
because they can not, so to speak, feel it with the
hand, as if they did not know that to suppose an

54

HUMAN SIGNIFICANCE OF MATHEMATICS

ideal to be thus realized would be a flat contradiction
in terms.

If we turn for a moment to art and enquire what has
been her relation to the poignant riddle, shall we not
thus be going too far afield? The answer is certainly
no. In aterniiatem pingo, said Zeuxis, the Greek painter.
"The purpose of art," says John La Farge, "is com-
memoration." In these two sayings, one of them ancient,
the other modern, we have, I think, the evident clue.
They do but tell us that art, like the other great enter-
prises of man, springs from our spirit's coveting of
worth that abides. Like, tljeoiogyjlike philosophy, like
jurisprudencjE^Jike natural _sciericerToo, as I'^imean^to
pomt__ojit-ftii thery-^anj^likeinathematics, art is born of

Her Cj^str-iike-JJigirs, has been^a^earcji for .invariants,
for goods tha1^aj£,£ve?totjfig! Snd what has she found?
THe answer is simple. "The idea of beauty in each
species of being," said Joshua Reynolds, "is perfect,
invariable, divine." We know that by a faculty of
imaginative, mystical, idealizing discernment there is
revealed to us, amid the fleeting beauties of Time, the
immobile presence of Eternal beauty, immutable arche-
type and source of the grace and loveliness beheld in the
shifting scenes of the flowing world of sense. Such, I
take it, is art's contribution to our human release from
the tyranny of change and the law of death.

And now what should be said of science? Not so
brief and far less simple would be the task of character-
izing or even enumerating the many things that in the
great drama of modern science have been assigned the
r61e of invariant forms of reality or eternal modes of
being. It would be necessary to mention first of all, as
most imposing of all, our modern form of the ancient

HUMAN SIGNIFICANCE OF MATHEMATICS 55

doctrine of fate. I mean the reigning conception of our
universe as an infinite machine — a powerful conception
that more and more fascinates scientific minds even
to the point of obsession and according to which it
should be possible, were knowledge sufficiently advanced,
to formulate, in a system of differential equations, the
whole of cosmic history from eternity to eternity in
minutest detail, not even excluding a skeptic's doubt
whether such formulation be theoretically possible nor
excluding the conviction, which some minds have, that
the doctrine, regarded as an ultimate creed, is an abomi-
nable libel against the character of a world where the
felt freedom of the human spirit is not an illusion. It
would be necessary to mention — as next perhaps in
order of impressiveness — another doctrine that is, curi-
ously enough, vividly reminiscent of old-time fate. I
allude this time to the doctrine of heredity, a tremendous
conception, in accordance with which — as Professor W.
B. Smith has said in his recent powerful address on
"Push or Pull"? — " the remotest past reaches out its
skeletal fingers and grapples both present and future in
its iron grip." And there is the conservation of energy
and that of mass — both of them, again, doctrines pre-
figured in the thought of ancient Greece — and numer-
ous other so-called natural laws, simple and complex,
familiar and unfamiliar, all posing as permanent forms
of reality — as natural invariants under the infinite
system of cosmic transformations — and thus together
constituting the enlarging contribution of natural science
towards the slow vindication of a world that has seemed
capricious, lawless and impermanent.

Such, then, is a conspectus, suggested rather than
portrayed, of the results which the great allies of mathe-
matics, operating through the ages, have achieved in

56 HUMAN SIGNIFICANCE OF MATHEMATICS

their passionate endeavor to transcend the tragic vicissi-
tudes and limitations of life in an "ever-growing and
perishing" universe and to win at length the freedom,
the dignity and the peace of a stable world where order
and harmony reign and spiritual goods endure. If we
are to arrive at a really just or worthy sense of the
human significance of mathematics, it is in relation with
those great results of her sister enterprises that the
achievements of this science must be appraised. Im-
mense indeed and high is the task of criticism as thus
conceived. How diverse and manifold the doctrines to
be evaluated, what depths to be plumbed, what heights
to be scaled, how various the relationships and digni-
ties to be assigned their rightful place in the hierarchy
of values. In the presence of such a task what can we
think or say in the remaining moments of the hour?
If we have succeeded in setting the problem in its
proper light and in indicating the sole eminence from
which the matter may be rightly viewed, we ought per-
haps to be content with that as the issue of the hour,
for it is worth while to sketch a worthy program of
criticism even if time fails us to perform fully the task
thus set. And yet I can not refrain from inviting you to
imagine, before we close, a few at least of the things that
one who essayed the great critique would submit to his
auditors for meditation. And what do you imagine the
guiding lines and major themes of his discourse would be?
I fancy he would say: The question before us, ladies
and gentlemen, is not a question of weighing utilities nor
of counting applications nor of measuring material gains;
it is a question of human ideals together with the vari-
ous means of pursuing them and the differing degrees
of their approximation; we are occupied with a ques-
tion of appreciation, with the problem of values. I am,

HUMAN SIGNIFICANCE OF MATHEMATICS 57

he would say, addressing you as representatives of man,
and in so doing, I am not regarding man as a mere prac-
tician, as a hewer of wood and drawer of water, as an
animal content to serve the instincts for shelter and
food and reproduction. I am contemplating him as a
spiritual being, as a thinker, poet, dreamer, as a lover
of knowledge and beauty and wisdom and the joy of
harmony and light, responding to the lure of an ideal
destiny, troubled by the mystery of a baffling world,
conscious subject of tragedy, yearning for stable reality,
for infinite freedom, for perpetuity and a thousand per-
fections of life. As representatives of such a being, you,
he would say, and I, even if we be not ourselves pro-
ducers of theology or philosophy or science or jurispru-
dence or art or mathematics, are nevertheless rightful
inheritors of all this manifold wisdom of man. The
question is: What is the inheritance worth? We are
the heirs and we are to be the judges of the great
responses that time has made to the spiritual needs of
humanity. What are the responses worth? What are
their values, joint and several, absolute and relative?
And what, especially, is the human worth of the re-
sponse of mathematics? It is, he would say, not only
our privilege, but, as educated individuals and especially
as representatives of our race, it is our duty, to ponder
the matter and reach, if we can, a right appraisement.
For the proper study of mankind is man, and it is
essential to remember that "La me de la science est la
critique" I have, he would say, tried to make it clear
that mathematics is not an isolated science. I have
tried to show that it is not an antagonist, nor a rival,
but is the comrade and ally of the other great forms
of spiritual activity, all aiming at the same high end.
I have reminded you of the principal answers made by

58 HUMAN SIGNIFICANCE OF MATHEMATICS

these to the spiritual needs of man, and I do not, he
would say, desire to underrate or belittle them. They
are a precious inheritance. Many of them have not,
indeed, stood the test of time; others will doubtless
endure for aye; all of them, for a longer or shorter
period, have softened the ways of life to millions of
men and women. Neither do I desire, he would say,
to exaggerate the contributions of mathematics to the
spiritual weal of humanity. What I desire is a fair
comparative estimate of its claims. "Truth is the be-
ginning of every good thing, both to gods and men."
I am asking you to compare, consider and judge for
yourselves. The task is arduous and long.

There are, our critic would say, certain paramount
considerations that every one in such an enterprise must
weigh, and a few of them may, in the moments that
remain, be passed in brief review. Consider, for ex-
ample, our human craving for a world of stable reality.
Where is it to be found? We know the answers of the-
ology, of philosophy, of natural science and the rest.
We know, too, the answer of literature and general
thought:

The cloud-capped towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve,
And, like the baseless fabric of this vision,
Leave not a rack behind.

And now what, he would ask, is the answer of mathe-
matics? The answer, he would have to say, is this:
Transcending the flux of the sensuous universe, there exists
a stable world of pure thought, a divinely ordered world of
ideas, accessible to man, free from the mad dance of time,
infinite and eternal.

Consider our human craving for freedom. Of free-
dom there are many kinds. Is it the freedom of limitless

HUMAN SIGNIFICANCE OF MATHEMATICS 59

room, where our passion for outward expression, for
externalization of thought, may attain its aim? It is
to mathematics, our critic would say, that man is in-
debted for that priceless boon; for it is the cunning of
this science that has at length contrived to release our
long imprisoned thought from the old confines of our
three-fold world of sense and opened to its wing the
interminable skies of hyperspace. But if it be a more
fundamental freedom that is meant, if it be freedom of
thought proper — freedom, that is, for the creative
activity of intellect — then again it is to mathematics
that our faculties must look for the definition and a right
estimate of their prerogatives and power. For, regard-
ing this matter, we may indeed acquire elsewhere a
suspicion or an inkling of the truth, but mathematics,
and nothing else, is qualified to give us knowledge of the
fact that our intellectual freedom is absolute save for a
single limitation — the law of non-contradiction, the
law of logical compatibility, the law of intellectual har-
mony— sole restriction imposed by "the nature of
things" or by logic or by the muses upon the creative
activity of the human spirit.

Consider next, the critic might say, our human craving
for a living sense of rapport and comradeship with a
divine Being infinite and eternal. Except through the
modern mathematical doctrine of infinity, there is, he
would have to say, no rational way by which we may
even approximate an understanding of the supernal
attributes with which our faculty of idealization has
clothed Deity — no way, except this, by which our
human reason may gaze understandingly upon the
downward-looking aspects of the overworld. But this
is not all. I need not, he would say, remind you of the
reverent saying attributed to Plato that "God is a

60 HUMAN SIGNIFICANCE OF MATHEMATICS

geometrician." Who is so unfortunate as not to know
something of the religious awe, the solace and the peace
that come from cloistral contemplation of the purity and
everlastingness of mathematical truth?

Mighty is the charm of those abstractions to a mind beset with images
and haunted by himself.

"More frequently," says Wordsworth, speaking of
geometry,

More frequently from the same source I drew

A pleasure quiet and profound, a sense

Of permanent and universal sway,

And paramount belief; there, recognized

A type, for finite natures, of the one

Supreme Existence, the surpassing life

Which to the boundaries of space and time,

Of melancholy space and doleful time,

Superior and incapable of change,

Not touched by welterings of passion — is,

And hath the name of God. Transcendent peace

And silence did wait upon those thoughts

That were a frequent comfort to my youth.

And so our spokesman, did time allow, might con-
tinue, inviting his auditors to consider the relations of
mathematics to yet other great ideals of humanity —
our human craving for rectitude of thought, for ideal
justice, for dominion over the energies and ways of the
material universe, for imperishable beauty, for the dig-
nity and peace of intellectual harmony. We know that
in all such cases the issue of the great critique would
be the same, and it is needless to pursue the matter
further. The light is clear enough. Mathematics is, in
many ways, the most precious response that the human
spirit has made to the call of the infinite and eternal.
It is man's best revelation of the "Deep Base of the
World."

THE HUMANIZATION OF THE TEACHING
OF MATHEMATICS1

WHEN the distinguished chairman of your mathe-
matical conference did me the honor to request me to
speak to you, he was generous enough, whether wisely
or unwisely, to leave the choice of a subject to my dis-
cretion, merely stipulating that, whatever the title might
be, the address itself should bear upon the professional
function of those men and women who are engaged
in teaching mathematics in secondary schools. Inex-
pertness, it has been said, is the curse of the world;
and one may, not unnaturally, feel some hesitance in
undertaking a task that might seem to resemble the
r61e of a physician when, as sometimes happens, he is
called upon to treat a patient whose health and medical
competence surpass his own. I trust I am not wanting
in that natural feeling. In the present instance two
considerations have enabled me to overcome it. One
of them is that, having had some experience in teaching
mathematics in secondary schools, I might, it seemed
to me, regard that experience, though it was gained
more than a score of years ago, as giving something
like a title to be heard in your counsels. The other
consideration is that, in regard to the teaching of mathe-
matics, whether in secondary schools or in colleges, I

1 Address given at the meeting of the Michigan School Masters' Club,
at Ann Arbor, March 28, 1912. Printed in Science, April 26, 1912; in
The Educational Review, September, 1912; and in the Michigan School
Masters' Magazine.

62

HUMANIZATION OF TEACHING MATHEMATICS

have acquired a certain conviction, a pretty firm con-
viction, which, were it properly presented, you would
doubtless be generous enough and perhaps ingenious
enough to regard as having some sort of likeness to a
message.

My conviction is, th^Jiope^ofimprovement in mathe-
matics teacning^wEether in/lsecondary schools *bY"""fai
colleges, lies mamlyin the possibility uf hunidiuziHg~irr
It fs worth while to remember that our pupils are hu-
man beings. What it means to be a human being we
all of us presumably know pretty well; indeed we know
it so well that we are unable to tell it to one another
adequately; and, just because we do so well know what
it means to be a human being, we are prone to forget
it as we forget, except when the wind is blowing, that
we are constantly immersed in the earth's atmosphere.
To humanize the teaching of mathematics means so to

present thesufrfeTl, so 10 iiiterptet its loreas

ot me]

putatory faculty or to the logical faculty but to all the
great powers ajid^nTeTes^^^tfae^lmman mind^ ThaT
nialEeniatical ideas and doctrinesptrhethcr they be more
elementary or more advanced, admit of such a manifold,
liberal and stimulating interpretation, and that there-
fore the teaching of mathematics, whether in secondary
schools or in colleges, may become, in the largest and
best sense, human, I have no doubt. That mathe-
matical ideas and doctrines do but seldom receive such
interpretation and that accordingly the teaching of
mathematics is but seldom, in the largest and best sense,
human, I believe to be equally certain. That the indi-
cated humanization of mathematical teaching, the bring-
ing of the matter and the spirit of mathematics to bear,
not merely upon certain fragmentary faculties of the

HUMANIZATION OF TEACHING MATHEMATICS 63

mind, but upon the whole mind, that this is a great
desideration is, I assume, beyond dispute.

How can such humanization be brought about? The
answer, I believe, is not far to seek. I do not mean that
the answer is easy to discover or easy to communicate.
I mean that the game is near at hand and that it is
not difficult to locate it, though it may not be easy to
capture it. The difficulty inheres, I believe, in our con-
ception of mathematics itself; not so much in our con-
ception of what mathematics, in a definitional sense, is,
for that sense of what mathematics is has become pretty
clear in our day, but in our sense or want of sense of
what mathematics, whatever it may be, humanly sig-
nifies. In ordex-to^hurnanize mathematical teaching it is
necessary^ and I believe it is sufficient, to come under
fhg^roj|trr>1 nf fl, right conception of the human sig-
limcance of ma. them a. tics. It is sufficient, I mean to say,
and Tt is necessary, greatly to enlarge, to enrich and to
vitalize our sense of what mathematics, regarded as
human enterprise, signifies.

What does mathematics, regarded as an enterprise of
the human spirit, signify? What is a just and worthy
sense of the human significance of mathematics?

To the extent in which any of us really succeeds in
answering that question worthily, his teaching will have
the human quality, in so far as his teaching is, in point
of external circumstance, free to be what it would. I
believe it is important to put the question, and it is
with the putting of it rather than with the proposing of
an answer to it that I am here at the outset mainly
concerned. For any one who is really to acquire pos-
session of an answer that is worthy must win the answer
for himself. I need not say to you that such an acqui-
sition as a worthy answer to this kind of question does

64 HUMANIZATION OF TEACHING MATHEMATICS

not belong to the category of things that may be lent
or borrowed, sold or bought, donated or acquired by
gift. No doubt the answers we may severally win will
differ as our temperaments differ. Yet the matter is
not solely a matter of temperament. It is much more
a matter first j>l fr"r>wlgdge and then oTthe evalu^tToTT
of tEeknowIedge and ofitssubjectT To^the winning"
of a worthy sense of the human significance of mathe-
matics two things are indispensable, knowledge and re-
flection: knowledge of mathematics and reflection upon
it. To the winning of such a sense it is essential to
have the kind of knowledge that none but serious
students of mathematics can gain. Equally essential
is another thing and this thing students of mathematics
in our day do not, or do but seldom, gain. I mean the
kind of insight and the liberality of view that are to be
acquired only by prolonged contemplation of the nature
of mathematics and by prolonged reflection upon its
relations of contrast and similitude to the other great
forms of spiritual activity.

The question, though it is a question about mathe-
matics, is not a mathematical question; it is a philo-
sophical question. And just because it is a philosophical
question, mathematicians, despite the fact that one
of the indispensable qualifications for considering it is
possessed by them alone, have in general ignored it.
They have, in general, ignored it, and their ignoring of
it may help to explain the curious paradox that whilst
the world, whose mathematical knowledge varies from
little to less, has always as if instinctively held mathe-
matical science in high esteem, it has at the same time
usually regarded mathematicians as eccentric and ab-
normal, as constituting a class apart, as being something
more or something less than human. It may explain,

.

HUMANIZATION OF TEACHING MATHEMATICS 65

too, I venture to believe it does partly explain, both
why it is that in the universities the number of students
attracted to advanced lectures in mathematics compared
with the numbers drawn to advanced courses in some
other great subjects not inherently more attractive, is
so small; and why it is that, among the multitudes who
pursue mathematics in the secondary schools, only a few
find in the subject anything like delight. For I do not
accept the traditional and still current explanation, that
the phenomenon is due to a well-nigh universal lack
of mathematical faculty. I maintain, on the contrary,
that a vast majority of mankind possess mathematical
faculty in a very considerable degree. That the average
pupil's interest in mathematics is but slight, is a matter
of common knowledge. .His lack of interest is, in my
opinion, due, not to a lack of the appropriate faculty
in him, but to th,e cirnim^aprp that VIP is a human
n^

interest, is not.pxeseatejd.jp him in its human guise.
~Tf you ask the world — represented, let us say, by
the man in the street or in the market place or the
field — to tell you its estimate of the human significance
of mathematics, the answer of the world will be, that
mathematics has given mankind a metrical and com-
putatory art essential to the effective conduct of daily
life, that mathematics admits of countless applications
in engineering and the naturalj^ienr.fiS, ar>rl finally_,±hat
mathematics is a most excellent instrumentality— for
giving mental discipline. Such will be the answer of the
world. The answer is intelligible, it is important, and
it is good so far as it goes; but it is far from going far
enough and it is not intelligent. That it is far from
going far enough will become evident as we proceed.
That the answer is not intelligent is evident at once,

66 HUMANIZATION OF TEACHING MATHEMATICS

for the first part of it seems to imply that the rudi-
mentary mathematics of the carpenter and the counting-
house is scientific, which it is not; the second part of
the answer is but an echo by the many of the voice
of the few; and, as to the final part, the world's con-
ception of intellectual discipline is neither profound nor
well informed but is itself in sorry need of discipline.

If, turning from the world to a normal mathematician,
you ask him to explain to you the human significance
of mathematics, he will repeat to you the answer of the
world, of course with far more appreciation than the
world has of what the answer means, and he will sup-
plement the world's response by an important addition.
He will add, that is, that mathematics is the exact
science, the science of exact thought or of rigorous
thinking. By this he will not mean what the world
would mean if the world employed, as sometimes it does
employ, the same form of words. He will mean some-
thing very different. Especially if he be, as I suppose
him to be, a normal mathematician of the modern
critical type, he will mean that mathematics is, in the
oft-cited language of Benjamin Peirce, "the science
that draws necessary conclusions;" he will mean that,
in the felicitous words of William Benjamin Smith,
" mathematics is the universal art apodictic;" he will
mean that mathematics is, in the nicely technical phrase
of Fieri, "a hypothetico-deductive system." If you ask
him whether mathematics is the science of rigorous
thinking about all the things that engage the thought of
mankind or only about a few of them, such as numbers,
figures, certain operations, and the like, the answer he
will give you depends. If he be a normal mathematician
of the elder school, he will say that mathematics is the
science of rigorous thinking about only a relatively few

HUMANIZATION OF TEACHING MATHEMATICS 67

things and that these are such as you have exemplified.
And if now, with a little Socratic persistence, you press
him to indicate the human significance of a science of
rigorous thinking about only a few of the countless
things that engage human thought, his answer will give
you but little beyond a repetition of the above-mentioned
answer of the world. But if he be a normal mathe-
matician of the modern critical type, he will say that
mathematics is the science of rigorous thinking about
all the things that engage human thought, about all of
them, he will mean, in the sense that thinking, as it
approaches perfection, tends to assume certain definite
forms, that these forms are the same whatever the
subject matter of the thinking may be, and that mathe-
matics is the science of these forms as forms. If you
respond, as you well may respond, that, in accordance
with this ontological conception of mathematics, this
science, instead of thinking about all, thinks about
none, of the concrete things of interest to human
thought, and that accordingly Mr. Bertrand Russell
was right in saying that " mathematics is the science
in which one never knows what one is talking about nor
whether what one says is true" — if you respond that,
from the point of view above assumed, that delicious
mot of Mr. Russell's must be solemnly held as true, and
then if, in accordance with your original purpose, you
once more press for an estimation of the human sig-
nificance of such a science, I fear that the reply, if your
interlocutor is a mathematician of the normal type, will
contain little that is new beyond the assertion that the
science in question is very interesting, where, by in-
teresting, he means, of course, interesting to mathe-
maticians. It is true that Professor Klein has said:
" Apart from the fact that pure mathematics can not be

68 HUMANIZATION OF TEACHING MATHEMATICS

supplanted by anything else as a means for developing
the purely logical faculties of the mind, there must be
considered here as elsewhere the necessity of the pres-
ence of a few individuals in each country developed in
a far higher degree than the rest, for the purpose of
keeping up and gradually raising the general standard.
Even a slight raising of the general level can be ac-
complished only when some few minds have progressed
far ahead of the average." Here indeed we have, in
these words of Professor Klein, a hint, if only a hint, of
something better. But Professor Klein is not a mathe-
matician of the normal type, he is hypernormal. If, in
order to indicate the human significance of mathematics
regarded as the science of the forms of thought as forms,
your normal mathematician were to say that these forms
constitute, of themselves, an infinite and everlasting
world whose beauty, though it is austere and cold, is
pure, and in which is the secret and citadel of whatever
order and harmony our concrete universe contains, it
would yet be your right and your duty to ask, as the
brilliant author of "East London Visions" once asked
me, namely, what is the human significance of "this
majestic intellectual cosmos of yours, towering up like
a million-lustered iceberg into the arctic night," seeing
that, among mankind, none is permitted to behold its
more resplendent wonders save the mathematician him-
self? But the normal mathematician will not say what
I have just now supposed him to say; he will not say it,
because he is, by hypothesis, a normal mathematician,
and because, being a normal mathematician, he is exclu-
sively engaged in exploring the iceberg. A farmer was
once asked why he raised so many hogs. "In order,"
he said, "to buy more land." Asked why he desired
more land, his answer was, "in order to raise more

HUMANIZATION OF TEACHING MATHEMATICS 69

corn." Being asked to say why he would raise more
corn, he replied that he wished to raise more hogs. If
you ask the normal mathematician why he explores the
iceberg so much, his answer will be, in effect at least,
"in order to explore it more." In this exquisite cir-
cularity of motive, the farmer and the normal mathe-
matician are well within their rights. They are within
their rights just as a musician would be within his
rights if he chanced to be so exclusively interested in
the work of composition as never to be concerned
with having his creations rendered before the public
and never to attempt a philosophic estimate of the
human worth of music. The distinction involved is
not the distinction between human and inhuman,
between social and anti-social; it is the distinction
between what is human or inhuman, social or anti-
social, and what is neither the one nor the other.
No one, I believe, may contest the normal mathema-
tician's right as a mathematical student or investigator
to be quite indifferent as to the social value or the
human worth of his activity. Such activity is to
be prized just as we prize any other natural agency
or force that, however undesignedly, yet contributes,
sooner or later, directly or indirectly, to the weal of
mankind. The fact is that, among motives in research,
scientific curiosity, which is neither moral nor immoral,
is far more common and far more potent than charity
or philanthropy or benevolence. But whgp JEJIJL
n'iiPFi from thp role of sti^fnt. nr invp

to the r61e of teacher,., that right of indifference ceases,
for he has passed toTan office whosfi fnnr.tjnTisji.rp. social
and whose _obUgatjonsjire Tiuman. It is not his privi-
lege to chill and depress with the encasing fogs of the
iceberg. It is his privilege and his duty, in so far as

7O HUMANIZATION OF TEACHING MATHEMATICS

he may, to disclose its " million-lustered " splendors in
all their power to quicken and illuminate, to charm and
edify, the whole mind.

The conception of mathematics as the science of the
forms of thought as forms, the conception of it as the
refinement, prolongation and elaboration of pure logic,
is, as you are doubtless aware, one of the great out-
comes, perhaps I should say it is the culminating philo-
sophical outcome, of a century's effort to ascertain what
mathematics, in its intimate structure, is. This concep-
tion of what mathematics is conies to its fullest expres-
sion and best defense, as you doubtless know, in such
works as Schroeder's " Algebra der Logik," White-
head's "Universal Algebra," Russell's " Principles of
Mathematics," Peano's "Formulario Matematico," and
especially in Whitehead and Russell's monumental
"Principia Mathematica." I cite this literature because
it tells us what, in a definitional sense, the science in
which the normal mathematician is exclusively engaged,
is. If we wish to be told what that science humanly
signifies, we must look elsewhere; we must look to a
mathematician like Plato, for example, or to a phi-
losopher like Poincare, but especially must we look to
our own faculty for discerning those fine connective
things — community of aim, interformal analogies, struc-
tural similitudes — that bind all the great forms of
human activity and aspiration — natural science, the-
ology, philosophy, jurisprudence, religion, art and mathe-
matics — into one grand enterprise of the human spirit.

In the autumn of 1906 there was published in Poet
Lore a short poem which, though it says nothing ex-
plicitly of mathematics, yet admits of an interpretation
throwing much light upon the human significance of the
science and indicating well, I think, the normal mathe-

HUMANIZATION OF TEACHING MATHEMATICS 71

matician's place in the world of spiritual interests. The
author of the poem is my excellent friend and teacher,
Professor William Benjamin Smith, mathematician, phi-
losopher, poet and theologian. I have not asked his
permission to interpret the poem as I shall invite you
to interpret it. What its original motive was I am not
informed — it may have been the exceeding beauty of
the ideas expressed in it or the harmonious mingling
of their light with the melody of their song. The title
of the poem is "The Merman and the Seraph." As
you listen to the reading of it, I shall ask youtto regard
the Merman as representing the normal mathematician
and the Seraph as representing, let us say, the life of
the emotions in their higher reaches and their finer
susceptibilities.

Deep the sunless seas amid,
Far from Man, from Angel hid,
Where the soundless tides are rolled
Over Ocean's treasure-hold,
With dragon eye and heart of stone,
The ancient Merman mused alone.

And aye his arrowed Thought he wings

Straight at the inmost core of things —

As mirrored in his Magic glass

The lightning-footed Ages pass, —

And knows nor joy nor Earth's distress,

But broods on Everlastingness.

"Thoughts that love not, thoughts that hate not,

Thoughts that Age and Change await not,

All unfeeling,

All revealing,

Scorning height's and depth's concealing,
These be mine — and these alone!" —
Saith the Merman's heart of stone.

Flashed a radiance far and nigh
As from the vertex of the sky, —
Lo! a Maiden beauty-bright

72 HUMANIZATION OF TEACHING MATHEMATICS

And mantled with mysterious might
Of every power, below, above,
That weaves resistless spell of Love.

Through the weltering waters cold
Shot the sheen of silken gold;
Quick the frozen Heart below
Kindled in the amber glow;
Trembling Heavenward Nekkan yearned
Rose to where the Glory burned.
"Deeper, bluer than the skies are,
Dreaming meres of morn thine eyes are

All that brightens

Smile or heightens

Charm is thine, all life enlightens,
Thou art all the soul's desire." —
Sang the Merman's Heart of Fire.
"Woe thee, Nekkan! Ne'er was given
Thee to walk the ways of Heaven;

Vain the vision,

Fate's derision,

Thee that raps to realms elysian,
Fathomless profounds are thine" —
Quired the answering voice divine.

Came an echo from the West,
Pierced the deep celestial breast;
Summoned, far the Seraph fled,
Trailing splendors overhead;
Broad beneath her flying feet
Laughed the silvered ocean-street.

On the Merman's mortal sight

Instant fell the pall of Night;

Sunk to the sea's profoundest floor

He dreams the vanished Vision o'er,

Hears anew the starry chime,

Ponders aye Eternal Time.

"Thoughts that hope not, thoughts that fear not,

Thoughts that Man and Demon veer not

Times unending

Comprehending,

Space and worlds of worlds transcending,
These are mine — but these alone!" —
Sighs the Merman's heart of stone.

HUMANIZATION OF TEACHING MATHEMATICS 73

I have said that the poem, if it receive the interpre-
tation that I have invited you to give it, throws much
light on the human significance of mathematics and
indicates well the place of the normal mathematician
in the world of spiritual interests. No doubt the
place of the merman and the place of the angel
are not the same: no doubt the world of whatsoever
in thought is passionless, infinite and everlasting, and
the world of whatsoever in feeling is high and beau-
teous and good are distinct worlds, and they are
sundered wide in the poem. But, though in the
poem they are held widely apart, in the poet they are
united. For the song is not the merman's song nor
are its words the words of the seraph. It is the voice
of the poet — a voice of man. The merman's world
and the world of the seraph are not the same, they are
very distinct; in conception they are sundered; they
may be sundered in life, but in life it need not be so.
The merman indeed is confined to the one world and
the seraph to the other, but man, a man unless he be
a merman, may inhabit them both. For the angel's
denial, the derision of fate, is not spoken of man, it is
spoken of the merman; and the merman's sigh is not
his own, it is a human sigh — so lonely seems the mer-
man in the depths of his abode.

No, the world of interests of the human spirit is not
the merman's world alone nor the seraph's alone. It is
not so simple. It is rather a cluster of worlds, of worlds
that differ among themselves as differ the lights by
which they are characterized. As differ the lights.
The human spirit is susceptible of a variety of lights
and it lives at once in a corresponding variety of worlds.
There is perception's light, commonly identified with
solar radiance or with the radiance of sound, for music,

74 HUMANIZATION OF TEACHING MATHEMATICS

too, is, to the spirit, a kind of illumination: percep-
tional light, in which we behold the colors, forms and
harmonies of external nature: a beautiful revelation —
a world in which any one might be willing to spend the
remainder of his days if he were but permitted to live
so long. And there is imagination's light, disclosing a
new world filled with wondrous things, things that may
or may not resemble the things revealed in perception's
light but are never identical with them: light that is
not superficial nor constrained to paths that are straight
but reveals the interiors of what it illuminates and
phases that look away. Again, there is the light of
thought, of reason, of logic, the light of analysis, far
dimmer than perception's light, dimmer, too, than that
of imagination, but far more penetrating and far more
ubiquitous than either of them, disclosing things that
curiously match the things that they disclose and count-
less things besides, namely, the world of ideas and the
relations that bind them: a cosmic world, in the center
whereof is the home of the merman. There remains to
be named a fourth kind of light. I mean the light of
emotion, the radiance and glory of things that, save by
gleams and intimations, are not revealed in perception
or in imagination or in thought: the light of the seraph's
world, the world of the good, the true and the beautiful,
of the spirit of art, of aspiration and of religion.

Such, in brief, is the cluster of worlds wherein dwell
the spiritual interests of the human beings to whom
it is our mission to teach mathematics. My thesis is
that it is our privilege to show, in the way of our teach-
ing it, that its human significance is not confined to
one of the worlds but, like a subtle and ubiquitous
ether, penetrates them all. Objectively viewed, con-
ceptually taken, these worlds, unlike the spheres of the

HUMANIZAT10N OF TEACHING MATHEMATICS 75

geometrician, do not intersect — a thing in one of
them is not in another; but the things in one of them
and the things in another may own a fine resemblance
serving for mutual recall and illustration, effecting
transfer of attention — transformation as the mathe-
maticians call it — from world to world; for whilst these
worlds of interest, objectively viewed, have naught in
common, yet subjectively they are united, united as
differing mansions of the house of the human spirit.
A relation, for example, between three independent
variables exists only in the grey light of thought, only
in the world of the merman; the habitation of the
geometric locus of the relation is the world of imagina-
tion; if a model of the locus be made or a drawing of
it, this will be a thing in the world of perception;
finally, the wondrous correlation of the three things, or
the spiritual qualities of them — the sensuous beauty
of the model or the drawing, the unfailing validity of
the given relation holding as it does throughout "the
cycle of the eternal year," the immobile presence of the
locus or image poised there in eternal calm like a figure
of justice — these may serve, in contemplating them, to
evoke the radiance of the seraph's world: and thus the
circuit and interplay, ranging through the world of
imagination and the world of thought from what is
sensuous to what is supernal, is complete. It would not
have seemed to Plato, as it may seem to us, a far cry
from the prayer of a poet to the theorem of Pythagoras,
for example, or to that of Archimedes respecting a sphere
and its circumscribing cylinder. Yet I venture to say,
that calm reflection upon the existence and nature of
such a theorem — cloistral contemplation, I mean, of the
fact that it is really true, of its serene beauty, of
its silent omnipresence throughout the infinite universe

76 HUMANIZATION OF TEACHING MATHEMATICS

of space, of the absolute exactitude and invariance of
its truth from everlasting to everlasting — such reflec-
tion will not fail to yield a sense of reverence and awe
akin to the feeling that, for example, pervades this
choral prayer by Sophocles:

"Oh! that my lot may lead me in the path of holy
innocence of word and deed, the path which august
laws ordain, laws that in the highest empyrean had
their birth, of which Heaven is the father alone, nor
did the race of mortal men beget them, nor shall oblivion
put them to sleep. The god is mighty in them and he
groweth not old."

But why should we think it strange that interests,
though they seem to cluster about opposite poles, are
yet united by a common mood? Of the great world of
human interests, mathematics is indeed but a part;
but is a central part, and, in a profound and precious
sense, it is "the eternal type of the wondrous whole."
For poetry and painting, sculpture and music — art
in all its forms — philosophy, theology, religion and
science, too, however passional their life and however
tinged or deeply stained by local or temporal circum-
stance, yet have this in common: they all of them aim
at values which transcend the accidents and limitations
of every time and place; and so it is that the passion-
lessness of the merman's thought, the infiniteness of
the kind of being he contemplates and the everlasting-
ness of his achievements enter as essential qualities
into the ideals that make the glory of the seraph's
world. I do not forget, in saying this, that, of all
theory, mathematical theory is the most abstract.
I do not forget that mathematics therefore lends
especial sharpness to the contrast in the Mephistophe-
lian warning:

HUMANIZATION OF TEACHING MATHEMATICS 77

Grey, my dear friend, is all theory,
Green the golden tree of life.

Yet I know that one who loves not the grey of a
naked woodland has much to learn of the esthetic re-
sources of our northern clime. A mathematical doctrine,
taken in its purity, is indeed grey. Yet such a doctrine,
a world-filling theory woven of grey relationships finer
than gossamer but stronger than cables of steel, leaves
upon an intersecting plane a tracery surpassing in fine-
ness and beauty the exquisite artistry of frost-work upon
a windowpane. Architecture, it has been said, is frozen
music. Be it so. Geometry is frozen architecture.

No, the belief that mathematics, because it is abstract,
because it is static and cold and grey, is detached from
life, is a mistaken belief. Mathematics, even in its
purest and most abstract estate, is not detached from
life. It is just the ideal handling of the problems of
life, as sculpture may idealize a human figure or as
poetry or painting may idealize a figure or a scene.
Mathematics is precisely the ideal handling of the prob-
lems of life, and the central ideas of the science, the
great concepts about which its stately doctrines have
been built up, are precisely the chief ideas with which
life must always deal and which, as it tumbles and
rolls about them through time and space, give it its
interests and problems, and its order and rationality.
That such is the case a few indications will suffice to
show. The mathematical concepts of constant and
variable are represented familiarly in life by the notions
of fixedness and change. The concept of equation or
that of an equational system, imposing restriction upon
variability, is matched in life by the concept of natural
and spiritual law, giving order to what were else chaotic

78 HUMANIZATION OF TEACHING MATHEMATICS

change and providing partial freedom in lieu of none
at all. What is known in mathematics under the name
of limit is everywhere present in life in the guise of
some ideal, some excellence high-dwelling among the
rocks, an "ever flying perfect" as Emerson calls it,
unto which we may approximate nearer and nearer,
but which we can never quite attain, save in aspiration.
The supreme concept of functionality finds its correlate
in life in the all-pervasive sense of interdependence
and mutual determination among the elements of the
world. What is known in mathematics as transforma-
tion — that is, lawful transfer of attention, serving to
match in orderly fashion the things of one system with
those of another — is conceived in life as a process of
transmutation by which, in the flux of the world, the
content of the present has come out of the past and in its
turn, in ceasing to be, gives birth to its successor, as
the boy is father to the man and as things, in general,
become what they are not. The mathematical concept
of invariance and that of infinitude, especially the im-
posing doctrines that explain their meanings and bear
their names — what are they but mathematicizations
of that which has ever been the chief of life's hopes
and dreams, of that which has ever been the object of
its deepest passion and of its dominant enterprise, I
mean the finding of worth that abides, the finding of
permanence in the midst of change, and the discovery
of the presence, in what has seemed to be a finite world,
of being that is infinite? It is needless further to mul-
tiply examples of a correlation that is so abounding and
complete as indeed to suggest a doubt which is the
juster, to view mathematics as the abstract idealiza-
tion of life, or to regard life as the concrete realization
of mathematics.

HUMANIZATION OF TEACHING MATHEMATICS 79

Finally, I wish to emphasize the fact that the great
concepts out of which the so-called higher mathematical
branches have grown — the concepts of variable and
constant, of function, class and relation, of transforma-
tion, invariance, and group, of finite and infinite, of
discreteness, limit, and continuity — I wish, in closing,
to emphasize the fact that these great ideas of the
higher mathematics, besides penetrating life, as we have
seen, in all its complexity and all its dimensions, are
omnipresent, from the very beginning, in the elements
of mathematics as well. The notion of group, for ex-
ample, finds easy and beautiful illustration, not only
among the simpler geometric notions amd configura-
tions, but even in the ensemble of the very integers
with which we count. The like is true of the distinc-
tion of finite and infinite, and of the ideas of transforma-
tion, of invariant, and nearly all the rest. Why should
the presentation of them have to await the uncertain
advent of graduate years of study? For life already
abounds, and the great ideas that give it its interests,
order and rationality, that is to say, the focal concepts
of the higher mathematics, are everywhere present in
the elements of the science as glistening bassets of
gold. It is our privilege, in teaching the elements, to
avail ourselves of the higher conceptions that are present
in them; it is our privilege to have and to give a lively
sense of their presence, their human significance, their
beauty and their light. I do not advocate the formal
presentation, in secondary schools, of the higher con-
ceptions, in the way of printed texts, for the printed
text is apt to be arid and the letter killeth. What I
wish to recommend is the presentation of them, as
opportunity may serve, in Greek fashion, by means of
dialectic, face to face, voice answering to voice, ani-

80 HUMANIZATION OF TEACHING MATHEMATICS

mated with the varying moods and motions and accents
of life — laughter, if you will, and the lightning of wit
to cheer and speed the slower currents of sober thought.
Of dialectic excellence, Plato at his best, as in the
"Phaedo" or the " Republic," gives us the ideal model
and eternal type. But Plato's ways are frequently
circuitous, wearisome and long. They are ill suited to
the manners of a direct and undeliberate age; and we
must find, each for himself, a shorter course. Somebody
imbued with the spirit of the matter, possessed of ample
knowledge and having, besides, the requisite skill and
verve ought to write a book showing, in so far as the
printed page can be made to show, how naturally and
swiftly and with what a delightful sense of emancipation
and power thought may pass by dialectic paths from the
traditional elements of mathematics both to its larger
concepts and to a vision of their bearings on the higher
interests of life. I need not say that such a handling
of ideas implies much more than a verbal knowledge of
their definitions. It implies familiarity with the doc-
trines that unfold the meanings of the ideas defined. It
is evident that, in respect of this matter, the scripture
must read: Knowing the doctrine is essential to living
the life.

THE WALLS OF THE WORLD: OR CONCERN-
ING THE FIGURE AND THE DIMENSIONS
OF THE UNIVERSE OF SPACE1

THERE is something a little incongruous in attempt-
ing to consider the subject of this address in a theater
or lecture hall whose roof and walls shut out from view
the wide expanses of the world and the azure deeps.
For how can we, amid the familiar finite scenes of a
closed and blinded room, command a fitting mood for
contemplating the infinite scenes without and beyond?
A subject that has sheer vastness for its central or
major theme demands for its appropriate contemplation
the still expanse of some vast and open solitude, such
as the peak of a lone and lofty mountain would afford,
where the gaze meets no wall save the far horizon and
no roof but the starry sky. Perhaps you will be good
enough for the time to transport yourselves, in imagina-
tion, into the stillness of such a solitude, so that in the
musing spirit of the place the questions to be propounded
for consideration here may arise naturally and give us
a due sense of their significance and impressiveness.
What are the dimensions and what is the figure of our
universe of space? How big is it and what is its shape?
What is the figure of it and what is its size?

1 An address delivered under the auspices of the local chapters of the
Society of Sigma Xi at the state universities of Minnesota, Nebraska and
Iowa, April 24, 28 and 30, 1913, respectively, and at a joint meeting of the
chapters of Sigma Xi and Phi Beta Kappa of Columbia University, May 8,
1913. Printed in Science, June 13, 1913.

82 THE WALLS OF THE WORLD

I do not mind owning that these questions have
haunted me a good deal from the days of my youth. It
happened in those days, though I was not aware of it
nor became aware of it till after many years, that there
were then coming into mathematics, just entering the
fringe, so to speak, or the vestibule of the science, certain
striking ideas which, as I venture to hope we may see,
were destined, if not indeed to enable us to answer the
questions with certainty, at all events to clarify them,
to enrich their meaning and to make it possible to
discuss them profitably. It has not been my fortune
to meet many persons who had seriously propounded
the questions to themselves or who seemed to be imme-
diately interested in them when propounded by others
— not many, even among astronomers, whose minds,
it may be assumed, are especially "accustomed to con-
templation of the vast." And so I have been forced
to the somewhat embarrassing conclusion that my own
long interest in the questions has been due to the fact
of my being of a specially practical turn of mind. Quite
seriously I venture to say that we are here engaged in
a practical enterprise. For even if the questions were
in the nature of the case unanswerable, which we do
not admit, who does not know how great the boons
that have come to men through pursuit of the unat-
tainable? And who does not know that, as Mr. Chester-
ton has said, if you wish really to know a man, the
most practical question to ask is, not about his occupa-
tion or his club membership or his party or church
affiliations, but what are his views of the all-embracing
world? What does he think of the universe. Do but
fancy for a moment that in somewise men should come
to know the exact shape or figure and especially the
exact size or dimensions of the all-immersing space of

THE WALLS OF THE WORLD 83

our universe. It requires but little imagination, not
much reflection, no extensive knowledge of cosmogonic
history and speculation, no very profound insight into
the ways of truth to men, it needs, I say, but little
philosophic sense to see that such knowledge would in
a thousand ways, direct and indirect, react powerfully
upon our whole intelligence, upon all our attitudes,
sentiments and views, transforming our theology, our
ethics, our art, our religion, our philosophy, our liter-
ature, our science, and therewith affecting profoundly
the whole sense and manner, the tone, color and mean-
ing, of all our institutions and the affairs of daily life.
Nothing is quite so practical, in the sense of being
effectual and influential, as the views men hold, con-
sciously or unconsciously, regarding the great locus
of their lives and their cosmic home.

In order to discuss the questions before us intelligibly
and profitably it is not necessary by way of clearing
the ground to enter far into metaphysical speculation
or into psychological analysis with a view to ascer-
taining what it is that we mean or ought to mean by
space. We are not obliged to dispute, much less decide,
whether space is subjective or objective or both or
indeed something that, as Plato in the "Timaeus"
acutely contends, is neither the one nor the other. We
may or may not agree with the contention of Kant that
space is, not an object, but the form, of outer sense;
we may or may not agree with the radically different
contention of Poincare that (geometric as distinguished
from sensible) space is nothing but what is known in
mathematics as a group, of which the concept "is im-
posed on us, not as form of our sense, but as form of
our understanding." It is, I say, not necessary for us,
in the interest of soundness and intelligibility, to try

84 THE WALLS OF THE WORLD

to compose such differences or to attempt a settlement
of these profound and important questions. As to the
distinction between sensible space and geometric space,
it would indeed be indispensable to draw it sharply
and to keep it always in mind, if we were undertaking
to ascertain what the subject (or the object) of geom-
etry is, or, what is tantamount, if we were seeking to
get clearly aware of what it is that geometry is about.
But in discussing the subject before us it is unnecessary
to be always guarding that distinction; for, whilst it
is the space of geometry, and not sensible space, that
we shall be talking about, yet it would be a hindrance
rather than a help if we did not allow, as we habitually
do allow, the two varieties of space — the imagery of
the one, the conceptual characters of the other — to
mingle freely in our thinking. There will be finesse
enough for the keenest arrows of our thought without
our going out of the way to find it. A procedure less
sophisticated will suffice. It will be sufficient to regard
space as being what, to the layman and to the student
of natural science, it has always seemed to be: a vast
region or room round about us, an immense exteriority,
locus of all suspended and floating objects of outer sense,
the whence, where and whither of motion, theater, in a
word, of the ageless drama of the physical universe.
In naturally so construing the term we do not commit
ourselves to the philosophy, so-called, of common sense;
we thus merely save our discourse from the encumbrance
of needless refinements; for it is obvious that, if space
be not indeed what we have said it seems to be, the
seeming is yet a fact, and our questions would remain
without essential change: what, then, we should ask, are
the dimensions and what is the figure of that seeming?
Though all the things contained within that triply

THE WALLS OF THE WORLD 85

extended spread or expanse which we call space are
subject to the law of ceaseless change, the expanse
itself, the container of all, appears to suffer no vari-
ation whatever, but to be, unlike time, a genuine
constant, the same yesterday, today and forever, sole
absolute invariant under the infinite host of trans-
formations that constitute the cosmic flux. Whether
it be so in fact, of course we do not know. We only
know that no good reason has ever been advanced for
holding the contrary as an hypothesis.

And yet there is a sense, which we ought I think to
notice, an interesting sense, in which space seems to be,
not a constant, but, like time, a variable. There is a
sense, deeper and juster perhaps than at first we suspect,
in which the space of our universe has in the course of
time alternately shrunken and grown. During the last
century, for example, it has, so it seems, greatly grown,
in response, it may be, to an increasing need of the
human mind. By grown I do not mean grown in the
usual sense, I do not mean the biological sense, I do not
mean the sense that was present to the mind of that
great man Leonardo da Vinci, when he wrote in effect
as follows: if you wish to know that the earth has been
growing, you have only to observe "how, among the
high mountains, the walls of ancient and ruined cities
are being covered over and concealed by the earth's
increase"; and, if you would learn how fast the earth
is growing, you have only to set a vase, filled with pure
earth, upon a roof; to note how green herbs will imme-
diately begin to shoot up; to note that these, when
mature, will cast their seeds; to allow the process to
continue through repetition; then, after the lapse of a
decade, to measure the soil's increase; and, finally,
to multiply, in order to have thus determined "how

86 THE WALLS OF THE WORLD

much the earth has grown in the course of a thousand
years." In this matter, Leonardo was doubtless wrong.
At all events current scientific views are against him.
The earth, we know, has grown, but the growth has
been by accretion, by addition from without, and not,
in biologic sense, by expansion from within (unless,
indeed, we adopt the beautiful hypothesis of the poet
and physicist, Theodor Fechner, for which so hard-
headed a scientific man as Bernhardt Riemann had so
much respect, the hypothesis, namely, that the plants,
the earth and the stars have souls). The myriad-
minded Florentine was, we of today think, in error,
his error being one of those brilliant mistakes that but
few men have been qualified to make. But in saying
that space has grown we do not mean that it has grown
in the biologic sense of Leonardo nor yet in the sense
of addition from without. We mean that it has grown
as a thing in mind may grow, as a thing in thought
may grow; we mean that it has grown in men's con-
ception of it. That space has, in this sense, been en-
larged prodigiously in the course of recent time is evident
to all. It has been often said that " the first grand dis-
covery of modern times is the immense extension of the
universe in space." It would be juster to say that the
first grand achievement of modern science has been
the immense extension of space itself, the prodigious
enlargement of it, in the imagination and especially in
the thought of men. If we will but take the trouble
to recall vividly the Mosaic cosmogony, in terms of
which most of us have but recently ceased to frame our
sublimest conceptions of the vast: if we remind our-
selves of Plato's "concentric crystal spheres, the ada-
mantine axis turning in the lap of necessity, the bands
that held the heaven together like a girth that clasps

THE WALLS OF THE WORLD 87

a ship, the shaft which led from earth to sky, and which
was paced by the soul in a thousand years"; if we
compare these conceptions with our own; if we think
of "the fields from which our stars fling us their light,"
fields that are really near and yet are so far that the
swiftest of messengers, capable of circling the earth
eight times in a second, requires for its journey hither
thousands of years; if we do but make some such com-
parisons, we shall begin to realize dimly that, compared
with modern space — the space of modern thought —
elder space — the space of elder thought — is indeed
"but as a cabinet of 'brilliants, or rather a little jewelled
cup found in the ocean or the wilderness."

Suppose that in fact space were thus, like time, not a
constant, but a variable; suppose it were a mental thing
growing with the growth of mind; an increasing function
of increasing thought; suppose it were a thing whose
enlargement is essential as a psychic condition or con-
comitant or effect of the progress of science; would not
our questions regarding its figure and its dimensions
then lose their meaning? The answer is, no; as rational
beings we should still be bound to ask: what are the
dimensions and what is the figure of space to date?
That is not all. If these questions were answered, we
could propound the further questions: whether the
space so characterized — the space of the present — is
adequate to the present needs of science, and whether
it is not destined to yet further expansion in response
to the future needs of thought.

Men do not feel, however, that such spatial enlarge-
ments as I have indicated are genuine enlargements of
space. In spite of whatever metaphysics or psychology
may seem obliged to say to the contrary, men feel that
what is new in such an enlargement is merely an in-

88 THE WALLS OF THE WORLD

crease of enlightenment regarding something old; they
feel that what is new is, not an added vastness, but a
discovery of a vastness that always was and always will
be. Let us trust this feeling and, regarding space as
constant from everlasting to everlasting, let us take the
questions in their natural intent and form: what are
the dimensions and what is the figure of our universe
of space?

If you propound these questions to a normal student
of natural science, say to a normal astronomer, his re-
sponse will be — what? If you appear to him to be
quite sincere and if, besides, he be in an amiable mood,
his response will, not improbably, be a significant shrug
of the shoulders, designed to intimate that his time is
too precious to be squandered in considering questions
that, if not meaningless, are at all events unanswerable.
I maintain, on the contrary, that this same student of
natural science, and indeed, all other normally educated
men and women, have, as a part of their intellectual
stock in trade, perfectly definite answers to both of the
questions. I do not mean that they are aware of pos-
sessing such wealth nor shall I undertake to say in
advance whether their answers be correct. What I
am asserting and what, with your assistance, I shall
endeavor to demonstrate, is that perfectly precise,
very intelligent and perfectly intelligible answers to
both of the questions are logically involved in what
every normally educated mind regards as the securest
of its intellectual possessions. In order to show that
such answers are to be found embedded in the content
of the normally educated mind and in order to lay them
bare, it will be necessary to have recourse to the process
of explication. Explication, however, is nothing strange
to an academic audience. It is true, indeed, that we

THE WALLS OF THE WORLD 89

no longer derive the verb, to educate, from educere, but
it is yet a fact, as every one knows, that a large part
of education is eduction — the leading forth into light
what is hidden in the familiar content of our minds.

What are those answers? I shall present them in the
familiar and brilliant words of one who in the span of a
short life achieved a seven-fold immortality: immortality
as a physicist, as a philosopher, as a mathematician, as
a theologian, as a writer of prose, as an inventor and as
a fanatic. From this brief but " immortal" characteriza-
tion I have no doubt that you detect the author at
once and at once recall the words: Space is an infinite
sphere whose center is everywhere and whose surface is
nowhere.

You will observe that, without change of meaning,
I have substituted " space" for "universe" and " sur-
face" for "circumference." This brilliant. mot of Blaise
Pascal, as every one knows, has long been valued
throughout the world as a splendid literary gem. I
am not aware that it has been at any time regarded
seriously as a scientific thesis. It may, however, be
so regarded. I propose to show, with your co-opera-
tion, that this exquisite saying of Pascal expresses with
mathematical precision the firm, albeit unconscious, con-
viction of the normally educated mind respecting the
size and the shape of the space of our universe. Be
good enough to note carefully at the outset the car-
dinal phrases: infinite sphere, center everywhere, surface
nowhere.

If you are told that there is an object completely
enclosed and that the object is equally distant from all
parts of the enclosing boundary or wall, you instantly
and rightly think of a sphere having that object as
center. Let me ask you to think of some point, any

90 THE WALLS OF THE WORLD

convenient point, P, together with all the straight lines
or rays — called a sheaf of lines or rays — that, begin-
ning at P, run out from it as far as ever the nature of
space allows. We ask: do all the rays of the sheaf
run out equally far? It seems perfectly evident that they
do, and with this we might be content. It will be worth
while, however, to examine the matter a little more
attentively. Denote by L any chosen line or ray of the
sheaf. Choose any convenient unit of length, say a
mile. We now ask: how many of our units, how many
miles can we, starting from P, lay off along L? Lay
off, I mean, not in fact, but in thought. In other
words: how many steps, each a mile long, can we,
in traversing L, take in thought? Hereafter let the
phrase "in thought" be understood. Can the question
be answered? It can. Can it be answered definitely?
Absolutely so. How? As follows. Before proceeding,
however, let me beg of you not to hesitate or shy if cer-
tain familiar ideas seem to get submitted to the logical
process — 'the mind-expanding process — of generalization.
There is to be no resort to any kind of legerdemain.
Let us be willing to transcend imagination, and, with-
out faltering, to follow thought, for thought, free as the
spirit of creation, owns no bar save that of inconsistence
or self-contradiction. Consider the sequence of cardinal
numbers,

(5) i, 2, 3, 4, 5, 6, 7, ....

The sequence is neither so dry nor so harmless as it
seems. It has a beginning; but it has no end, for, by
the law of its formation, after each term there is a next.
The difference between a sequence that stops somewhere
and one that has no end is awful. No one, unless spir-
itually unborn or dead, can contemplate that gulf
without emotions that take hold of the infinite and

THE WALLS OF THE WORLD 91

everlasting. Let us compare the sequence with the ray
L of our sheaf. Choose in (S) any number n, however
large. Can we go from P along L that number n
of miles? We are certain that we can. Suppose the
trip made, a mile post set up and on it painted the
number n to tell how far the post is from P. As n is
any number in (5), we may as well suppose, indeed we
have already implicitly supposed, mile posts, duly dis-
tributed and marked, to have been set up along L to
match each and every number in the sequence. Have
we thus set up all the mile posts that L allows? We
are certain that we have, for, if we go out from P along
L any possible but definite number of miles, we are
perfectly certain that that number is a number in the
sequence, and that accordingly the journey did but take
us to a post set up before. What is the upshot? It
is that L admits of precisely as many mile posts as
there are cardinal numbers, neither more nor less. How
long is L? The answer is: L is exactly as many miles
long as there are integers or terms in the sequence (S).
Can we say of any other line or ray L' of the sheaf
what we have said of L? We are certain that we can.
Indeed we have said it, for L was any line of the sheaf.
May we, then, say that any two lines, L and L', of the
sheaf are equal? We may and we must. For, just as
we have established a one-to-one correspondence between
the mile posts of L and the terms of (S), so we may
establish a one-to-one correspondence between the mile
posts of L and those of L' ', and what we mean by the
equality of two classes of things is precisely the possi-
bility of thus setting up a one-to-one correlation be-
tween them. Accordingly, all the lines or rays of our
sheaf are equal. We can not fail to note that thus
there is forming in our minds the conception of a sphere,

92 THE WALLS OF THE WORLD

centered at P, larger, however, than any sphere of slate
or wood or marble — a sphere, if it be a sphere, whose
radii are the rays of our sheaf. Is not the thing, how-
ever, too -vast to be a sphere? Obviously yes, if the
lines or rays of the sheaf have a length that is indefinite,
unassignable; obviously no, if their length be assignable
and definite. We have seen the length of a ray contains
exactly as many miles as there are integers or terms
in (5). The question, then, is: has the totality of these
terms a definite assignable number? The answer is,
yes. To show it, look sharply at the following fact,
a bit difficult to see only because it is so obvious, being
writ, so to speak, on the very surface of the eye. I
wish, in a word, to make clear what is meant by the
cardinal number of any given class of things. The
fingers of my right hand constitute a class of objects;
the fingers of my left hand, another class. We can set
up a one-to-one correspondence between the classes,
pairing the objects in the one with those in the other.
Any two classes admitting of such a correlation are
said to be equivalent. Now given any class K, there is
another class C composed of all the classes each of
which is equivalent to K. C is called the cardinal num-
ber of K, and the name of C, if it has received a name,
tells how many objects are in K. Thus, if K is the class
of the fingers of my right hand, the word five is the name
of the class of classes each equivalent to K. Now to
the application. The terms of (S) constitute a class K
(of terms). Has it a definite number? Yes. What is
it? It is the class of all classes each equivalent to K.
Has this number class received a name of its own? Yes,
and it has, like many other numbers, received a symbol,
namely, K, read- Aleph null. It is, then, this cardinal
number Aleph, not familiar, indeed, but perfectly definite

THE WALLS OF THE WORLD 93

as denoting a definite class, it is this that tells us how
many terms are in (S) and therewith tells us the length
of the rays of our sheaf. Herewith the concept that
was forming is now completely formed: space is a sphere
centered at P.

But is the sphere, as Pascal asserts, an infinite sphere?
We may easily see that it is. Again consider the se-
quence (S) and with it the similar sequence (,$")>

(S) i, 2, 3, 4, 5, 6,7,.. .,

CS") 2, 4, 6, 8, 10, 12, 14, ....

Observe that all the terms in (5") are in (S) and that
(5) contains terms that are not in (Sf). (£') is, then,
a proper part of (5). Next observe that we can pair
each term in (5) with the term below it in (S'). That
is to say: the whole, -(61), is equivalent to one of its
parts, (£')• A class that thus has a part to which it
is equivalent is said to be infinite, and the number of
things in such a class is called an infinite number. Aleph
is, then, an infinite number, and so we see that the rays
of our sheaf, the radii of our sphere, are infinite in
length: space is an infinite sphere centered at P.

Finally, what of the phrases, center everywhere, surface
nowhere? Can we give them a meaning consistent with
common usage and common sense? We can, as follows.
Let 0 be any chosen point somewhere in your neigh-
borhood. By saying that the center P is everywhere
we mean that P may be taken to be any point within
a sphere centered at O and having a finite radius, a
radius, that is, whose length in miles is expressed by any
integer in (S). And by saying that the surface of our
infinite sphere is nowhere we mean that no point of the
surface can be reached by traveling out from P any
finite number, however large, of miles, by traveling, that

94 THE WALLS OF THE WORLD

is, a number of miles expressed by any number, however
large, in (5).

Here we have touched our primary goal: we have
demonstrated that men and women whose education,
in respect of space, has been of normal type, believe
profoundly, albeit unawares, that the space of our uni-
verse is an infinite sphere of which the center is every-
where and the surface nowhere. Such is the beautiful
conception, the great conception — mathematically pre-
cise yet mystical withal and awful in its limitless reaches
— which is ever ready to form itself, in the normally
educated mind and there to stand a deep-rooted con-
scious conviction regarding the shape and the size
of the all-embracing world.

Is the conception valid? Does the conviction corre-
spond to fact? Is it true? It is not enough that it be
intelligible, which it is; it is not enough that it be noble
and sublime, which also it is. No doubt whatever is
noble and sublime is, in some sense, true. For we
mortals have to do with more than reason. Yet science,
science in the modern technical sense of the term, having
elected for its field the domain of the rational, allows
no superrational tests of truth to be sufficient or final.
We must, therefore, ask: are the dimensions and the
figure of our space, in fact, what, as we have seen,
Pascal asserts and the normally educated mind believes
them to be? Long before the days of Pascal, back
yonder in the last century before the beginning of the
Christian era, one of the acutest and boldest thinkers
of all time, immortal expounder of Epicurean thought,
answered the question, with the utmost confidence, in
the affirmative. I refer to Lucretius and his "De Rerum
Natura." In my view that poem is the greatest and
finest union of literary excellence and scientific spirit to

THE WALLS OF THE WORLD 95

be found in the annals of human thinking. I main-
tain that opinion of the work despite the fact that the
majority of its conclusions have been invalidated by
time, have perished by supersession; for we must not
forget that, in respect of knowledge, "the present is
no more exempt from the sneer of the future than the
past has been." I maintain that opinion of the work
despite the fact that the enterprise of Lucretius was
marvelously extravagant; for we must not forget that
the relative modesty of modern men of science is not
inborn, but is only an imperfectly acquired lesson.
Well, it is in that great work that Lucretius endeavors
to prove that our universe of space is infinite in the
sense that we have explained. His argument, which
runs to many words, may be briefly paraphrased as
follows. Conceive that, starting from any point of
space, you go out in any direction as far as you please,
and that then you hurl your javelin. Either it will
go on, in which case there is space ahead for it to move
in, or it will not go on, in which case there must be
space ahead to contain whatever prevents its going.
In either case, then, however far you may have gone,
there is yet space beyond. And so, he concludes, space
is infinite, and he triumphantly adds:

Therefore the nature of room and the space of the unfathomable void
are such as bright thunderbolts can not race through in their course though
gliding on through endless tract of time, no nor lessen one jot the journey
that remains to go by all their travel — so huge a room is spread out on
all sides for things without any bounds in all directions round.

Such is the argument, the great argument, of the
Roman poet. Great I call it, for it is great enough to
have fooled all philosophers and men of science for two
thousand years. Indeed only a decade ago I heard
the argument confidently employed by an American

96 THE WALLS OF THE WORLD

thinker of more than national reputation. But is the
argument really fallacious? It is. The conclusion may
indeed be quite correct — space may indeed be infinite,
as Lucretius asserts — but it does not follow from his
argument. To show the fallacy is no difficult feat.
Consider a sphere of finite radius. We may suppose
it to be very small or intermediate or very large — no
matter what its size so long as its radius is finite. By
sphere, in this part of the discussion, I shall mean sphere-
surface. Be good enough to note and bear that in mind.
Observe that this sphere — this surface — is a kind of
room. It is a kind of space, region or room where
certain things, as points, circle arcs and countless other
configurations can be and move. These things, con-
fined to this surface, which is their world, their universe
of space, if you please, enjoy a certain amount, an
immense amount, of freedom: the points of this world
can move in it hither, thither and yonder; they can
move very far, millions and millions of miles, even in
the same direction, if only the sphere be taken large
enough. I see no reason why we should not, for the
sake of vividness, fancy that spherical world inhabited
by two-dimensional intelligences conformed to their
locus and home just as we are conformed to our own
space of three dimensions. I see no reason why we
should not fancy those creatures, in the course of their
history, to have had their own Democritus and Epicurus,
to have had their own Roman republic or empire and
in it to have produced the brilliant analogues of our own
Virgil, Cicero and Lucretius. Do but note attentively
— for this is the point — that their Lucretius could
have said about their space precisely what our own
said about ours. Their Lucretius could have said to
his fellow-inhabitants of the sphere: " starting at any

THE WALLS OF THE WORLD 97

point, go as far as ever you please in any straight line"
— such line would of course (as we know) be a great
circle of the sphere — "and then hurl your javelin" —
the javelin would, as we know, be only an arc of a great
circle — "either it will go on, in which case, etc.; or
it will not, etc."; thus giving an argument identical
with that of our own Lucretius. But what could it
avail? We know what would happen to the javelin
when hurled as supposed in the surface: it would go
on for a while, there being nothing to prevent it. But
whether it went on or not, it could not be logically
inferred that the surface, the space in question, is infinite,
for we know that the surface is finite, just so many, a
finite number of, square miles. The fallacy, at length,
is bare. It consists — the fact has been recently often
pointed out — in the age-long failure to distinguish
adequately between unbegrenzt and unendlich — between
boundless and infinite as applied to space. What our
fancied Lucretius proved is, if anything, that the sphere
is boundless, but not that it is infinite. What our real
Lucretius proved is, if anything, that the space of our
universe is boundless, but not that it is infinite. That
a region or room may be boundless without being in-
finite is clearly shown by the sphere (surface). How
evident, once it is drawn, the distinction is. And yet
it was never drawn, in thinking about the dimensions
of space, until in 1854 it was drawn by Riemann in
his epoch-marking and epoch-making Habilitationschrift
on the foundations of geometry.

What, then, is the fact? Is space finite, as Riemann
held it may be? Or is it infinite, as Lucretius and Pascal
deliberately asserted, and as the normally educated
mind, however unconsciously, yet firmly believes? No
one knows. The question is one of the few great out-

98 THE WALLS OF THE WORLD

standing scientific questions that intelligent laymen may,
with a little expert assistance, contrive to grasp. Shall
we ever find the answer? Time is long, and neither
science nor philosophy feels constrained to haul down
the flag and confess an ignorabimus. Neither is it
necessary or wise for science and philosophy to camp
indefinitely before a problem that they are evidently
not yet equipped to solve. They may proceed to related
problems, always reserving the right to return with
better instruments and added light.

In the present instance, let us suppose, for the mo-
ment, that Lucretius, Pascal and the normally educated
mind are right: let us suppose that space is infinite, as
they assert and believe. In that case the bounds of the
universe are indeed remote, and yet we may ask: are
there not ways to pass in thought the walls of even so
vast a world? There are such ways. But where and
how? For are we not supposing that the walls to be
passed are distant by an amount that is infinite? And
how may a boundary that is infinitely removed be
reached and overpassed? The answer is that there are
many infinites of many orders; that infinites are sur-
passed by other infinites; that infinites, like the stars,
differ in glory. This is not rhetoric, it is naked fact.
One _pf the grand achievements of mathematics in the
nineteenth century is to have defined infinitude (as
above defined) and to have discovered that infinites
rise above infinites, in a genuine hierarchy without a
summit. In order to show how we can in thought pass
the Lucre tian and Pascal walls of our universe, I must
ask you to assume as a lemma a mathematical proposi-
tion which has indeed been rigorously established and is
familiar, but the proof of which we can not tarry here
to reproduce. Consider all the real numbers from zero

THE WALLS OF THE WORLD 99

to one inclusive, or, what is tantamount, consider all
the points in a unit segment of a continuous straight
line. The familiar proposition that I am asking you
to assume is that it is not possible to set up a one-to-
one correspondence between the points of that segment
and the positive integers (in the sequence above given),
but that, if you take away from the segment an infini-
tude (Aleph) of points matching all the positive integers,
there will remain in the segment more points, infinitely
more, than you have taken away. That means that
the infinitude of points in the segment infinitely surpasses
the infinitude of positive integers; surpasses, that is, the
infinitude of mile posts on the radius of our infinite
(Pascal) sphere. Now conceive a straight line containing
as many miles as there are points in the segment. You
see at once that in that conception you have overleaped
the infinitely distant walls of the Lucre tian universe.
Overleaped, did I say? Nay, you have passed beyond
those borders by a distance infinitely greater than the
length of any line contained within them. And thus
it appears that, not our imagination, indeed, but our
reason may gaze into spatial abysses beside which the
infinite space of Lucretius and Pascal is but a meager
thing, infinitesimally small. There remain yet other
ways by which we are able to escape the infinite con-
fines of this latter space. One of these ways is pro-
vided in the conception of hyperspaces enclosing our
own as this encloses a plane. But that is another story,
and the hour is spent.

The course we have here pursued has not, indeed,
enabled us to answer with final assurance the two ques-
tions with which we set out. I hope we have seen along
the way something of the possibilities involved. I hope
we have gained some insight into the meaning of the

100 THE WALLS OF THE WORLD

questions and have seen that it is possible to discuss
them profitably. And especially I hope that we have
seen afresh, what we have always to be learning again,
that it is not in the world of sense, however precious
it is and ineffably wonderful and beautiful, nor yet in
the still finer and ampler world of imagination, but
it is in the world of conception and thought that the
human intellect attains its appropriate freedom — a free-
dom without any limitation save the necessity of being
consistent. Consistency, however, is only a prosaic name
for a limitation which, in another and higher realm,
harmony imposes even upon the muses.

MATHEMATICAL EMANCIPATIONS:
DIMENSIONALITY AND HYPERSPACE l

AMONG the splendid generalizations effected by modern
mathematics, there is none more brilliant or more in-
spiring or more fruitful, and none more nearly commen-
surate with the limitless immensity of being itself, than
that which produced the great concept variously desig-
nated by such equivalent terms as hyperspace, multidi-
mensional space, w-space, w-fold or w-dimensional space,
and space of n dimensions.

In science as in life the greatest truths are the sim-
plest. Intelligibility is alike the first and the last de-
mand of the understanding. Naturally, therefore, those
scientific generalizations that have been accounted really
great, such as the Newtonian law of gravitation, or the
principle of the conservation of energy, or the all-
conquering concept of cosmic evolution, are, all of them,
distinguished by their simplicity and apprehensibility.
To that rule the notion of hyperspace presents no excep-
tion. For its fair understanding, for a live sensibility
to its manifold significance and quickening power, a
long and severe mathematical apprenticeship, however
helpful it would be, is not demanded in preparation,
but only the serious attention of a mature intelligence
reasonably inured by discipline to the exactions of ab-
stract thought and the austerities of the higher imag-

1 Printed in The Monist, January, 1906. For a deeper view of this subject
the reader may be referred to the i3th essay of this volume.

102 MATHEMATICAL EMANCIPATIONS

ination. And it is to the reader having this general
equipment, rather than to the professional mathe-
matician as such, that the present communication is
addressed.

To a clear understanding of what the mathematician
means by hyperspace, it is in the first place necessary
to conceive in its full generality the closely related notion
of dimensionality and to be able to state precisely what
is meant by saying that a given manifold has such and
such a dimensionality, or such and such a number of
dimensions, in a specified entity or element.

Discrimination, as the proverb rightly teaches, is the
beginning of mind. The first psychic product of that
initial psychic act is numerical: to discriminate is to
produce two, the simplest possible example of multi-
plicity. The discovery, or better the invention, better
still the production, best of all the creation, of multi-
plicity with its correlate of number, is, therefore, the
most primitive achievement or manifestation of mind.
Such creation is the immediate issue of intellection,
nay, it is intellection, identical with its deed, and, with-
out the possibility of the latter, the former itself were
quite impossible. Accordingly it is not matter for sur-
prise but is on the contrary a perfectly natural or even
inevitable phenomenon that explanations of ultimate
ideas and ultimate explanations in general should more
and more avail themselves of analytic as distinguished
from intuitional means and should tend more and more
to assume arithmetic form. Depend upon it, the uni-
verse will never really be understood unless it may
be sometime resolved into an ordered multiplicity and
made to own itself an everlasting drama of the calculus.

Let us, then, trust the arithmetic instinct as funda-
mental and, for instruments of thought that shall not

MATHEMATICAL EMANCIPATIONS 103

fail, repair at once to the domain of number. Every
one who may reasonably aspire to a competent knowl-
edge of the subject in hand is more or less familiar with
the system of real numbers, composed of the positive
and negative integers and fractions, such irrational
numbers as \/2 and TT and countless hosts of similar
numbers similarly definable. He may know that, for
reasons which need not be given here, the system of
real numbers is commonly described as the analytical
continuum of second order. He knows, too, at any rate
it is a fact which he will assume and readily appreciate,
that the distance between any two points of a right line
is exactly expressible by a number of the continuum;
that, conversely, given any number, two points may be
found whose distance apart is expressed by the numerical
value of that number; that, therefore, it is possible to
establish a unique and reciprocal, or one-to-one, corre-
spondence between the real numbers and the points
of a straight line, namely, by assuming some point of
the line as a fixed point of reference or origin of dis-
tances, by agreeing that a distance shall be positive or
negative according as it proceeds from the origin in
this sense or in the other and by agreeing that a point
and the number which by its magnitude reckoned in
terms of a chosen finite unit however great or small
serves to express the distance of the point from the
origin and by its sign indicates on which side of the
origin the point is situated, shall be a pair of corre-
spondents. Accordingly, if the point P glides along
the line, the corresponding number v will vary in such
a way that to each position of the geometric there
corresponds one value of the arithmetic element, and
conversely. P represents v; and v, P. No two P's
represent a same vj and no two v's, a same P. By

104 MATHEMATICAL EMANCIPATIONS

virtue of the correlation thus established with the analyt-
ical continuum, we may describe the line as a simple or
one-fold geometric continuum, namely, of points. The
like may in general be said, and for the same reason,
of any curve whatever, but we select the straight line
as being the simplest, for in matters fundamental we
should prefer clearness to riches of illustration, in the
faith that, if first we seek the former, the latter shall
in due course be added unto it. The straight line,
when it is regarded as the domain of geometric opera-
tion, as the region or room containing the configurations
or sets of elements with which we deal, is and is called
a space; and this space, viewed as the manifold or
assemblage of its points, is said to be 0we-dimensional
for the reason that, as we have seen, in order to deter-
mine the position of a point in it, in order, i.e., to pick
out or distinguish a point from all the other points of
the manifold, it is necessary and sufficient to know one
fact about the point, as, e.g., its distance from an as-
sumed point of reference. In other words, the line
is called a one-dimensional space of points because in
that space the point has one and but one degree of
freedom or, what is tantamount, because the position
of the point depends upon the value of a single v,
known as its coordinate.

Herewith is immediately suggested the generic con-
cept of dimensionality: if an assemblage of elements of
any given kind whatsoever, geometric or analytic or neither,
as points, lines, circles, triangles, numbers, notions, senti-
ments, hues, tones, be such that, in order to distinguish
every element of the assemblage from all the others, it is
necessary and sufficient to know exactly n independent facts
about the element, then the assemblage is said to be n-
dimensional in the elements of the given kind. It appears,

MATHEMATICAL EMANCIPATIONS 10$

therefore, that the notion of dimensionality is by no
means exclusively associated with that of space but on
the contrary may often be attached to the far more
generic concept of assemblage, aggregate or manifold.
For example, duration, the total aggregate of time-points,
or instants, is a simple or one-fold assemblage. On the
other hand, the assemblage of colors is three-dimensional
as is also that of musical notes, for in the former case,
as shown by Clerk Maxwell, Thomas Young and others,
every color is composable as a definite mixture of three
primary ones and so depends upon three independent
variables or coordinates expressing the amounts of the
fundamental components. And in the latter case a
similar scheme obtains, one note being distinguishable
from all others when and only when the three general
marks, pitch, length, and loudness, are each of them
specified. In passing it seems worth while to point out
the possibility of appropriating the name soul to signify
the manifold of all possible psychic experiences, in which
event the term would signify an assemblage of probably
infinite dimensionality, and the assemblage would be
continuous, too, if Oswald J be right in his contention
that every manifold of experience possesses the character
of continuity. That contention, however, if the much
abused term continuity be allowed to have its single
precise definitely seizable scientific meaning, is far less
easy to make good than that eminent chemist and
courageous philosopher seems to think.

Returning to the concept of space, an w-fold assem-
blage will be an w-dimensional space if the elements of
the assemblage are geometric entities of any given kind.
We have seen that the straight line is a owe-dimensional
space of points. But in studying the right line conceived

1 Cf. his Natur-Philosophie.

106 MATHEMATICAL EMANCIPATIONS

as a space, we are not compelled to employ the point
as element. Instead we may choose to assume as ele-
ment the point pair or triplet or quatrain, and so on. The
line would then be for our thought primarily a space,
not of points, but of point pairs or triplets and so on,
and it would accordingly be strictly a space of two
dimensions or of three, and so on; for, obviously, to
distinguish say a point pair from all other such pairs
we should have to know two independent facts about
the pair. The pair would have two degrees of freedom
in the line, its determination would depend upon two
independent variables as vi and %. These variables
might be the two independent ratios of the coefficients
and absolute term in a quadratic equation in one un-
known, as x, for to know the ratios is to know the equa-
tion and therewith its two roots, the two values of x.
These being laid off on the line give the point pair.
Conversely, a point pair gives two values of x, hence
definite quadratic equation and so values of vi and v%.
On its arithmetic side the shield presents a precisely
parallel doctrine. The simple analytical continuum
composed of the real numbers immediately loses its
simplicity and assumes the character of a 2- or 3- ...
or w-fold analytical continuum if, instead of thinking of
its individual numbers, we view it as an aggregate of
number pairs or triplets or, in general, as the totality
of ordered systems of n numbers each.

In the light of the preceding paragraph it is seen that
the dimensionality of a given space is not unique but
depends upon the choice of geometric entity for primary
or generating element. A space being given, its dimen-
sionality is not therewith determined but depends upon
the will of the investigator, who by a proper choice
of generating elements may endow the space with any

MATHEMATICAL EMANCIPATIONS 107

dimensionality he pleases. That fact is of cardinal
significance alike for science and for philosophy. I
reserve for a little while its further consideration in order
to present at once a kind of complementary fact of equal
interest and of scarcely less importance. It is that two
spaces which in every other respect are essentially un-
like, thoroughly disparate, may, by suitable choice of
generating elements, be made to assume equal dimen-
sionalities. Consider, for example, the totality of lines
contained in a same plane and containing a point in
common. Such a totality, called a pencil, of lines is a
simple geometric continuum, namely, of lines. It is,
then, and may be called, a one-dimensional space of
lines just as the line or range of points is a one-dimen-
sional space of points. The two spaces are equally
rich in their respective elements. And if, following
Desargues and his successors, we adjoin to the points
of the range a so-called "ideal" point or point at infinity,
thus rendering the range like the pencil, closed, it be-
comes obvious that two intelligences, adapted and
confined respectively to the two simple spaces in ques-
tion, would enjoy equal freedom; their analytical experi-
ences would be identical, and their geometries, though
absolutely disparate in kind, would be equally rich in
content. Just as the range-dweller would discover that
the dimensionality of his space is two in point pairs,
three in triplets, and so on, so the pencil-inhabitant would
find his space to be of dimensionality two in line pairs,
three in triplets, and so on without end. It was indi-
cated above that any line, straight or curved, is a one-
dimensional space of points. In that connection it
remains to say that, speaking generally, any curve,
literally and strictly conceived as the assemblage of its
(tangent) lines and so including the point or pencil

108 MATHEMATICAL EMANCIPATIONS

as a special case, is also a owe-dimensional space of lines.
It is, moreover, obvious that the foregoing considera-
tions respecting the range of points and the pencil of
lines are, mutatis mutandis, equally valid for any one
of an infinite variety of other analogous spaces, as, e. g.,
the axal pencil, a one-fold space of planes, consisting
of the totality of planes having a line in common.

If perchance some reader should feel an ungrateful
sense of impropriety in our use of the term space to
signify such common geometric aggregates as we have
been considering, I gladly own that his state of mind
is a perfectly natural one. But it is, besides and on
that account, a source of real encouragement. Dictional
sensibility is a hopeful sign, being conclusive evidence
of life, and, while there is life, there remains the pos-
sibility and therewith the hope of readjustment. In
the present case, I venture to assure the reader, on
grounds both of personal experience and of the experi-
ence of others, that whatever sense he may have of
injury received will speedily disappear in the further
course of his meditations. Only, let him not be im-
patient. Larger meanings must have time to grow;
the smaller ones, those that are most natural and most
provincial, being also the most persistent. In the
process of clarification, expansion and readjustment,
his fine old word, space, early come into his life and
gradually stained through and through with the re-
fracted partial lights and multi-colored prejudices of
his youth, is not to be robbed of its proper charms nor
to be shorn of its proper significance. More than it will
lose of mystery, it shall gain of meaning. Of this last
it has hitherto had for him but little that was of sci-
entific value, but little that was not vague and elusive
and ultimately unseizable. That was because the word

MATHEMATICAL EMANCIPATIONS IOQ

stood for something absolutely sui generis, i. e., for a
genus neither including species nor being itself included
in a class. But now, on the other hand, both of these
negatives are henceforth to be denied, and the hitherto
baffling term, perfect symbol of the unthinkable, always
promising and never presenting definable content,
immediately assumes the characteristic twofold aspect
of a genuine concept, being at once included as member
of a higher class, the more generic class of manifolds,
and including within itself an endless variety of indi-
viduals, an infinitude of species of space.

Of these species, the next in order of simplicity, to
those above considered, is the plane. To distinguish a
point of a plane from all its other points, it is necessary
and sufficient to know two independent facts about its
position, as, e. g., its distances from two assumed lines
of reference, most conveniently taken at right angles.
Viewed as the ensemble of its points, the plane is, there-
fore, a space of two dimensions. In that space, the
point enjoys a freedom exactly twice that of a point
in a range or of a line in a pencil, and exactly equal to
that of a pair of points or of lines in the last-mentioned
spaces. On the other hand, if the point pair be taken
as element of the plane, the latter becomes a space of
four dimensions.

What if the line be taken as generating element of the
plane? It is obvious that the plane is equally rich in
pencils and in ranges. It contains as many lines as
points, neither more nor less. Two points determine a
line; two lines, a point; if the lines be parallel, their
common point is a Desarguesian, a point at infinity.
We should therefore expect to find that in a plane the
position of a line depends upon two and but two inde-
pendent variables. And the expectation is realized, as

110 MATHEMATICAL EMANCIPATIONS

it is easy to see. For if the variables be taken to repre-
sent (say) distances measured from chosen points along
two lines of reference, it is immediately evident that a
given pair of values of the variables determines a line
uniquely and that, conversely, a given line uniquely
determines such a pair. The plane is, therefore, a two-
dimensional space of lines as well as of points. In line
pairs, as in point pairs, its dimensionality is four . We
may suppose the space in question to be inhabited by
two sorts of individuals, one of them capable of thinking
in terms of points but not of lines, the other in terms of
lines but not of points. Each would find his space
bi-dimensional. They would enjoy precisely the same
analytical experience. Between their geometries there
would subsist a fact-to-fact correspondence but not the
slightest resemblance. For example, the circle would
be for the former individual a certain assemblage of
points but devoid of tangent lines, and, for the latter,
a corresponding assemblage of (tangent) lines but devoid
of contact points.

Passing from the plane to a curved surface, to a
sphere, for example, a little reflection suffices to show
that the latter may be conceived in a thousand and one
ways, but most simply as the ensemble of its points or
of its (tangent) planes or of its (tangent) lines. These
various concepts are logically equivalent and in them-
selves are equally intelligible. And if to us they do
not seem to be also equally good, that is doubtless be-
cause we are but little traveled in the great domain of
Reason and therefore naturally prefer our familiar
customs and provincial points of view to others that are
strange. At all events, it is certain that on purely
rational grounds, none of the concepts in question is to
be preferred, while, from preference based on other

MATHEMATICAL EMANCIPATIONS III

grounds, it is the office alike of science and of philosophy
to provide the means of emancipation. Let us, then,
detach ourselves from the vulgar point of view and for
a moment contemplate the three concepts as coordi-
nate indeed but independent concepts of surface. And
for the sake of simplicity, we may think of a sphere.1
Suppose it placed upon a plane and imagine its highest
point, which we may call the pole, joined by straight
lines to all the points of the plane. Each line pierces
the sphere in a second point. It is plain that thus a
one-to-one correspondence is set up between the points
of the sphere and those of the plane, except that the
pole corresponds at once to all the Desarguesian points
of the plane — an exception, however, which is here of
no importance. The plane and the sphere are, then,
equally rich in points. Accordingly, the sphere con-
ceived as a plenum or locus or space of points is a space
of two dimensions. In that space the point has two
degrees of freedom. Its position depends upon two
independent variables, as latitude and longitude. But
we may conceive the surface quite otherwise: at each
of its points there is a (tangent) plane, and now, dis-
regarding points, we may think only of the assemblage
of those planes. These together constitute a sphere,
not, however, as a locus of points, but as an envelope
(as it is called) of planes. And what shall we say of
the surface as thus conceived? The answer obviously
is that it is a ^^-dimensional space of planes, admitting
of a geometry quite as rich and as definite as is the
theory of any other space of equal dimensionality. In
each of the planes there is a pencil of lines of which each
is tangent to the sphere. Thus we are led to a third

1 The term is here employed as in the higher geometry to denote, not a
solid, but a surface.

112 MATHEMATICAL EMANCIPATIONS

conception of our surface. We have merely to dis-
regard both points and planes and confine our atten-
tion to the assemblage of lines. The vision which thus
arises is that of a //zretf-dimensional space of lines.
In pencils, its dimensionality is two. In this space the
pencil has two and the line three degrees of freedom.

But let us return to the plane. We have seen that at
the geometrician's bidding it plays the r61e of a two-
fold space either in points or in lines. It is natural to
ask whether it may be conceived as a space of three
dimensions, like the sphere in its third conception. The
answer is affirmative: it may be so conceived, and that
in an infinity of ways. Of these the simplest is to as-
sume the circle as primary or generating element. Of
circles the plane contains a threefold infinity, an infinity
of infinities of infinities. It is a circle continuum of
third order. To distinguish any one of its circles from
all the rest, three independent data, two for position and
one for size, are necessary and sufficient. In the plane
the circle has three degrees of freedom, its determination
depends upon three independent variables. The plane
is, accordingly, a tri-dimensional space of circles. In
parabolas its dimensionality is four; in conies, five; and
so on without limit.

Before turning to space, ordinarily so-called, it seems
worth while to indicate another geometric continuum
which, although it presents no likeness whatever to
the plane, nevertheless matches it perfectly in every con-
ceptual aspect. The reference is to the sheaf, or bundle,
of lines, i. e., the totality of lines having a point in
common. The point is to be disregarded and the lines
viewed as non-decomposable entities, like points in a line
or plane regarded as an assemblage of points. Thus
conceived, the sheaf is literally a space, namely, of lines*

MATHEMATICAL EMANCIPATIONS 113

It is, in the vulgar sense of the term, just as big, occu-
pies precisely as much room, nay indeed the same room,
as the space in which we live. The sheaf as a space is
2w0-dimensional in lines, like the plane in points; two-
dimensional in pencils, like the plane in lines; four-
dimensional in line or pencil pairs, like the plane in
point or line pairs; //w^e-dimensional in ordinary cones,
like the plane in circles; and so on and on.

In the light of the foregoing considerations, any hith-
'erto uninitiated reader will probably suspect that ordi-
nary space is not, as it is commonly supposed and said
to be, an inherently and uniquely three-dimensional
affair. His suspicion is completely justified by fact.
The simple traditional affirmation of tri-dimensionality
is devoid of definite meaning. It is unconsciously elliptic,
requiring for its completion and precision the specifica-
tion of an appropriate geometric entity for generating
element. Merely to say that space is tri-dimensional
because a solid, e. g., a plank, has length, breadth and
thickness, is too crude for scientific purposes. More-
over, it betrays, quite unwittingly indeed as we shall
see, an exceedingly meager point of view. Not only
does it assume the point as element but it does so tacitly
because unconsciously, as if the point were not merely
an but the element of ordinary space. An element the
point may obviously be taken to be, and in that ele-
ment ordinary space is indeed tri-dimensional, for the
position of a point at once determines and is determined
by three independent data, as its distances from three
assumed mutually perpendicular planes of reference.
It must be admitted, too, that the point does, in a sense,
recommend itself as the element par excellence, at least
for practical purposes. For example, we prefer to do our
drawing with the point of a pencil to doing it with a

114 MATHEMATICAL EMANCIPATIONS

straight edge. But that is a matter of physical as dis-
tinguished from rational convenience. Preference for
the point has, then, a cause: in the order of evolution,
practical man precedes man rational and determines
for the latter his initial choices. Causes, however, are
extra-logical things, and the preference in question,
though it has indeed a cause, has no reason. Accord-
ingly, when in these modern times, the geometrician
became clearly conscious that he was in fact and had
been from time immemorial employing the point as
element and that it was this use that lent to space
its traditional triplicity of dimensions, he did not fail
to perceive almost immediately the logically equal
possibility of adopting at will for primary element any
one of an infinite variety of other geometric entities and
so the possibility of rationally endowing ordinary space
with any prescribed dimensionality whatever.

Thus, for example, the plane is no less available
for generating element than is the point. The plane
is logically and intuitionally just as simple, for, if from
force of habit, we are tempted -to analyze the plane
into an assemblage of points, the point is in its turn
equally conceivable as or analyzable into an assemblage
of planes, the sheaf of planes containing the point.
We may, then, regard our space as primarily a plenum
of planes. To determine, a plane requires three and
but three independent data, as, say, the distances to
it measured along three chosen lines from chosen points
upon them. It follows that ordinary space is three-
dimensional in planes as well as in points. But now
if (with Pliicker) we think of the line as element, we
shall find that our space has/0wr dimensions. That fact
may be seen in various ways, most easily perhaps as
follows. A line is determined by any two of its points.

MATHEMATICAL EMANCIPATIONS 115

Every line pierces every plane. By joining the points
of one plane to all the points of another, all the lines
of space are obtained. To determine a line it is, then,
enough to determine two of its points, one in the one
plane and one in the other. For each of these deter-
minations, two dafa, as before explained, are necessary
and sufficient. The position of the line is thus seen to
depend upon four independent variables, and the four-
dimensionality of our space in lines is obvious. Again,
we may (with Lie) view our space as an assemblage of
its spheres. To distinguish a sphere from all other
spheres, we need to know four and but four independent
facts about it, as, say, three that shall determine its
center and one its size. Hence our space is four-
dimensional also in spheres. In circles its dimensionality
is six; in surfaces of second order (those that are pierced
by a straight line in two points), nine; and so on ad
infinitum.

Doubtless the reader is prepared to say that, if the
foregoing account of hyperspace be correct, the notion
is after all a very simple one. Let him be assured, the
account is correct and his judgment is just: the notion
is simple. That property, as said in the beginning, is
indeed one of its merits. As presented the concept is
entirely free from mystery. To seize upon it, it is un-
necessary to pass the bounds, of the visible universe or
to transcend the limits of intuition. Its realization is
found even in the line, in the pencil, in the plane, in the
sheaf, here, there and yonder, everywhere, in fact. The
account, however, though quite correct, is not yet com-
plete. The term hyperspace has yet another meaning
and yet in strictness not another, as we shall see. It
will be noticed that among the foregoing examples of
hyperspace, none is presented of dimensionality exceed-

Il6 MATHEMATICAL EMANCIPATIONS

ing three in points. It is precisely this variety of hyper-
space that the term is commonly employed to signify,
particularly in popular enquiry and philosophical specu-
lation. And it is this variety, too, that just because it
baffles the ordinary visual imagination, proves to be, for
the non-mathematician at any rate, at once so tanta-
lizing, so mysterious and so fascinating.

It remains, then, to ask, what is meant by a hyper-
space of points? How is the notion formed and what
is its motivity and use? The path of enquiry is a fa-
miliar one and is free from logical difficulty. Granted
that a one-to-one correspondence can be established
between the real numbers and the points of a right line,
so that the geometric serve to represent the arithmetic
elements; granted that all (ordered) pairs of numbers are
similarly representable by the points of a plane, and all
(ordered) triplets by the points of ordinary space; the
suggestion then naturally presents itself that, whether
there really is or not, there ought to be a space whose
points would serve to represent, as in the preceding
cases, all ordered systems of values of n independent
variables; and especially to an analyst with a strong
geometric predilection, to one who is a born Vorstellender
for whom analytic abstractions naturally tend to take
on figure and assume the exterior forms of sense, that
suggestion comes with a force which he alone perhaps
can fully appreciate. And what does he do? Not find-
ing the desiderated hyperspace present to his vision or
intuition or visual imagination, he posits it, or if
you prefer, he creates it, in thought. The concept of
hyperspace of points is thus seen to be off-spring of
Arithmetic and Geometry. It is legitimate fruit of the
indissoluble union of the fundamental sciences.

Does such hyperspace exist? It does exist genuinely.

MATHEMATICAL EMANCIPATIONS 117

If not for intuition, it exists for conception; if not for
imagination, it exists for thought; if not for sense, it
exists for reason; if not for matter, it exists for mind.
These if's are ifs in fact. The question of imaginability
is really a question. We shall return to it presently.

The concept of hyperspace of points is generable in
various other ways. Of all ways the following is per-
haps the best because of its appeal at every stage to
intuition. Let there be two points and grant that these
determine a line, point-space of one dimension. Next
posit a point outside of this line and suppose it joined
by lines to all the points of the given line. The points
of the joining lines together constitute a plane, point-
space of two dimensions. Next posit a point outside of
this plane and suppose it joined by lines to all the
points of the plane. The points of all the joining lines
together constitute an ordinary space, point-space of
three dimensions. The clue being now familiar to our
hand, let us boldly pursue the opened course. Let us
overleap the limits of common imagination, transcend
ordinary intuition as being at best but a non-essential
auxiliary, and in thought posit an extra point that, for
thought at all events, shall be outside the space last
generated. Suppose that point joined by lines to all
the points of the given space. The points of the join-
ing lines together constitute a point-space of four
dimensions. The process here applied is perfectly clear
and obviously admits of endless repetition.

Moreover, the process is equally available for gener-
ating hyperspaces of other elements than points. For
example, let there be two intersecting lines and grant
that these determine a pencil, line-space of one dimen-
sion. Next posit a line (through the vertex) outside of
the given pencil and suppose it joined by pencils to all

1 1 8 MATHEMATICAL EMANCIPATIONS

the lines of the given pencil. The lines of the joining
pencils together constitute a sheaf, line-space of two
dimensions. Next posit a line (through the vertex)
outside of the sheaf and suppose it joined by pencils
to all the lines of the sheaf. The lines of the joining
pencils constitute a hypersheaf, line-space of three dimen-
sions. The next step plainly leads to a line-space of
four dimensions; and so on ad infinitum.

And now as to the question of imaginability. Is it
possible to intuit configurations in a hyperspace of
points? Let it be understood at the outset that that
is not in any sense a mathematical question, and mathe-
matics as such is quite indifferent to whatever answer it
may finally receive. Neither is the question primarily
a question of philosophy. It is first of all a psychological
question. Mathematicians, however, and philosophers
are also men and they may claim an equal interest per-
haps with others in the profounder questions concern-
ing the potentialities of our common humanity. The
question, as stated, undoubtedly admits of affirmative
answer. For the lower spaces, with which the imagina-
tion is familiar, exist in the higher, as the line in the
plane, and the plane in ordinary space. But that is not
what the question means. It means to ask whether
it is possible to imagine hyper-configurations of points,
i.e., point-configurations that are not wholly contained
in a point-space (like our own) of three dimensions.
It is impossible to answer with absolute confidence.
One reason is that the term imagination still awaits
precision of definition. Undoubtedly just as three-
dimensional figures may be represented in a plane, so
four-dimensional figures may be represented in space.
That, however, is hardly what is meant by imagining
them. On the other hand, a four-dimensional figure

MATHEMATICAL EMANCIPATIONS IIQ

may be rotated and translated in such a way that all
of its parts come one after another into the threefold do-
main of the ordinary intuition. Again, the structure of a
fourfold figure, every minutest detail of its anatomy, can
be traced out by analogy with its three-dimensional ana-
logue. Now in such processes, repetition yields skill,
and so they come ultimately to require only amounts
of energy and of time that are quite inappreciable. Such
skill once attained, the parts of a familiar fourfold con-
figuration may be made to pass before the eye of in-
tuition in such swift and effortless succession that the
configuration seems present as a whole in a single instant.
If the process and result are not, properly speaking,
fourfold imagination and fourfold image, it remains for
the psychologist to indicate what is lacking.

Certainly there is naught of absurdity in supposing
that under suitable stimulation the human mind may
in course of time even speedily develop a spatial in-
tuition of four or more dimensions. At present, as the
psychologists inform us and as every teacher of geometry
discovers independently, the spatial imagination, in
case of very many persons, comes distinctly short of
being strictly even tri-dimensional. On the contrary, it
is flat. It is not every one, even among scholars, that
with eyes closed can readily form a visual image of the
whole of a simple solid like a sphere, enveloping it com-
pletely with bent beholding rays of psychic light. In
such defect of imagination, however, there is nothing
to astonish. In the first place, man as a race is only a
child. He has been on the globe but a little while,
long indeed compared with the fleeting evanescents that
constitute the most of common life, but very short, the
merest fraction of a second, in the infinite stretch of
time. In the second place, circumstances have not, in

120 MATHEMATICAL EMANCIPATIONS

general, favored the development of his higher poten-
tialities. His chief occupation has been the destruction
and evasion of his enemies, contention for mere exist-
ence against hostile environment. Painful necessity,
then, has been the mother of his inventions. That, and
not the vitalizing joy of self-realization, has for the
most part determined the selection of the fashion of his
faculties. But it would be foolish to believe that these
have assumed their final form or attained the limits of
their potential development. The imperious rule of
necessity will relax. It will never pass quite away but
it will relax. It is relaxing. It has relaxed appreciably.
The intellect of man will be correspondingly quickened.
More and more will joy in its activity determine its
modes and forms. The multi-dimensional worlds that
man's reason has already created, his imagination may
yet be able to depict and illuminate.

It remains to ask, finally, what purpose the concept
of hyperspace subserves. Reply, partly explicit but
chiefly implicit, is not, I trust, entirely wanting in what
has been already said. Motivity, at all events, and
raison d'etre are not far to seek. On the one hand, the
great generalization has made it possible to enrich,
quicken and beautify analysis with the terse, sensuous,
artistic, stimulating language of geometry. On the
other hand, the hyperspaces are in themselves im-
measurably interesting and inexhaustibly rich fields of
research. Not only does the geometrician find light in
them for the illumination of many otherwise dark and
undiscovered properties of the ordinary spaces of in-
tuition, but he also discovers there wondrous struc-
tures quite unknown to ordinary space. These he
examines. He handles them with the delicate instru-
ments of his analysis. He beholds them with the eye

MATHEMATICAL EMANCIPATIONS 121

of the understanding and delights in the presence of
their supersensuous beauty.

Creation of hyperspaces is one of the ways by which
the rational spirit secures release from limitation. In
them it lives ever joyously, sustained by an unfailing
sense of infinite freedom.

THE UNIVERSE AND BEYOND: THE
EXISTENCE OF THE HYPERCOSMIC1

Ni la contradiction n'est marque de faussete, ni I'incontradiction n'est
marque de verite. — PASCAL

THE inductive proof of the doctrine of evolution
seems destined to be ultimately judged as the great
contribution of Natural Science to modern thought.
Among the presuppositions of that doctrine, among the
axioms, as one may call them, of science, are found the
following: —

(1) The assumption of the universal and eternal
reign of law: the assumption that the universe, the
theatre of evolution, the field of natural science, is and
eternally has been a genuine Cosmos, an incarnate
rational logos, an embodiment of reason, an organic
affair of order, a closed domain of invariant uniformities,
in which waywardness and chance have had nor part
nor lot: an infinitely intricate garment, ever changing,
yet always essentially the same, woven, warp and weft
alike, of mathetic relationships.

(2) The assumption, not merely that the universe is
cosmic through and through, but that it is the all con-
junctively — the all, that is, in the sense of naught
excluded; the assumption, in other words, that it is not
merely a but the cosmos, the sole system of law and
order and harmony, the complete and perfect embodi-
ment of the whole of truth.

1 Appeared in The Hibbert Journal, January, 1905.

Qtir>l

THE UNIVERSE AND BEYOND 123

Such, I take it, are among the principles, the articles
of faith, more or less consciously held by the great
majority of the men of science and their adherents.

As for myself, I am unable to hold these tenets either
as self-evident truths, or as established facts, or as prop-
ositions the proof of which may be confidently awaited.
Truth, for example, especially when contemplated in its
relations to curiosity — at once the psychic product and
psychic agency of evolution — less seems a completed
thing coeval with the world than a thing derived and
still becoming. Again, while the assumption of the
cosmic character of our universe is of the greatest value
as a working hypothesis, I am unable to find in the
method of natural science or in that of mathematics
any ground, even the slightest, for expecting conclusive
proof of its validity. In striking contrast, on the other
hand, with this negative thesis, there is found in the
realm of pure thought, in the domain of mathematics,
very convincing evidence, not to say indubitable proof
of the proposition, that no single cosmos, whether our
universe be such or not, can enclose every rationally
constructible system of truth, but that any universe is
a component of an extra-universal, that above every
nature is a super-natural, beyond every cosmos a
hypercosmic.

These are among the theses presented in the following
pages, not in a controversial spirit, let me add, nor
accompanied by the minuter arguments upon which
they ultimately rest.

We all must allow that truth is. To deny it denies
the denial. Such scepticism is cut away by the sweep-
ing blade of its own unsparing doubt. But what it is —
that is another matter. The assumption that truth is
an agreement or correspondence between concepts and

124 THE UNIVERSE AND BEYOND

things, between thought and object, is of very great
value in practical affairs; it very well serves, too, the
immediate purposes of natural science, especially in its
cruder stage, before it has learned by critical reflection
on its own processes and foundations to suspect its
limitations, and while, like the proverbial " chesty"
youth who disdains the meagre wisdom of his father, it is
apt to proclaim, innocently enough if somewhat boast-
fully, a lofty contempt for all philosophy and meta-
physics. Although the assumption has the undoubted
merit of being thus useful in high degree, it is, when
regarded as a definitive formulation of what we mean
by truth, hardly to be accepted. For, not only does it
imply — what may indeed be quite correct, but is far
from being demonstrated, and far from being uni-
versally allowed — namely, that " thing" is one and
"concept" another, that "object" and "thought" are
twain, but even if we grant such ultimate implied
duality, it remains to ask what that "agreement" is,
or "correspondence," that mediates the hemispheres
and gives the whole its truth. The assumption is
slightly too naive and unsophisticated, a little too redo-
lent of an untamed soil and primitive stage of cultiva-
tion. Much profounder is that insight of Hegel's, that
truth is the harmony which prevails among the objects
of thought. If, with that philosopher, we identify
object and thought, we have at once the pleasing utter-
ance that truth is the harmony of ideas. But here,
again, easy reflection quickly finds no lack of difficulties.
For what should we say an idea is? And is there really
nothing else, except, of course, their harmony? And
what is that? And is there no such thing as contra-
diction and discord? Is that, too, a kind of truth, a
kind of harmonious jangling, a melody of dissonance?

THE UNIVERSE AND BEYOND 125

The fact seems to be that truth is so subtle, diverse,
and manifold, so complex of structure and rich in as-
pect, as to defy all attempt at final definition. Nay,
more, the difficulty lies yet deeper, and is in fact irre-
soluble. Being a necessary condition thereto, truth can
not be an object of definition. To suppose it defined
involves a contradiction, for the definition, being some-
thing new, is something besides the truth defined, but
it must itself be true, and, if it be, in that has failed —
the enclosing definition is not itself enclosed, and
straightway asks a vaster line to take it in, and so ad
infinitum. To define truth would be to construct a
formula that should include the structure, to conceive
a water-compassed ocean, bounded in but shutting
nothing out, a self-immersing sea, without bottom or
surface or shore.

Happily, to be indefinable is not to be unknowable
and not to be unknown. And we are absolutely certain
that truth, whatever it may be, is somehow the com-
plement of curiosity, is the proper stuff, if I may so
express it, to answer questions with. Now a question,
once one comes to think of it, is a rather odd phenome-
non. Half the secret of philosophy, said Leibnitz, is
to treat the familiar as unfamiliar. So treated, curi-
osity itself is a most curious thing. How blind our
familiar assumptions make us! Among the animals,
man, at least, has long been wont to regard himself as
a being quite apart from and not as part of the cosmos
round about him. From this he has detached himself
in thought, he has estranged and objectified the world,
and lost the sense that he is of it. And this age-long
habit and point of view, which has fashioned his life
and controlled his thought, lending its characteristic
mark and colour to his whole philosophy and art and

126 THE UNIVERSE AND BEYOND

learning, is still maintained, partly because of its con-
venience no doubt, and partly by force of inertia and
sheer conservatism, in the very teeth of the strongest
probabilities of biologic science. Probably no other single
hypothesis has less to recommend it, and yet no other
so completely dominates the human mind. Suppose
we deny the assumption, as we seem indeed com-
pelled to do, in the name of science, and readjoin our-
selves in thought, as we have ever been joined in fact,
to this universe in which we live and have our being;
the other half of the secret of philosophy will be re-
vealed, or illustrated at all events, in the strangeness
of aspect presented by things before familiar. Note the
radical character of the transformation to be effected.
The world shall no longer be beheld as an alien thing,
beheld by eyes that are not its own. Conception of the
whole and by the whole shall embrace us as part,
really, literally, consciously, as the latest term, it may
be, of an advancing sequence of developments, as occu-
pying the highest rank perhaps in the ever-ascending
hierarchy of being, but, at all events, as emerged and
still emerging natura naturata from some propensive
source within. I grant that the change in point of view
is hard to make — old habits, like walls of rock, tend-
ing to confine the tides of consciousness within their
accustomed channels — but it can be made and, by
assiduous effort, in the course of time, maintained.
Suppose it done. By that reunion, the whole regains,
while the part retains, the consciousness the latter pur-
loined. I cannot pause to note even the most striking
consequences of such a change in point of view. Time
would fail me to follow far the opening lines of specu-
lation that issue thence and invite pursuit. But I
cannot refrain from pointing out how exceedingly curi-

THE UNIVERSE AND BEYOND 127

ous a thing curiosity itself becomes when beheld and
contemplated from the mentioned point of view. For
it is now the whole that meditates, the universe that
contemplates — a once mindless universe according to
its present understanding of the term, not then know-
ing that it was, unwittingly unwitting throughout a
beginningless eternal past what it had been or was or
was to be; lawless, too, perhaps, could the stream of
events be reascended, though blindly and slowly be-
coming lawful through habit-taking tendence: a self-
transforming insensate mass composed of parts without
likeness or distinction, continually undergoing change
without a purpose, devoid of passion, and neither ig-
norant nor having knowledge. At length a wondrous
crisis came, an event momentous — when or how is yet
unknown, perhaps through fortuitous concourse of part-
less, lawless, wayward elements. At all events, the un-
in tending tissues formed a nerve, the universe awoke
alive with wonder, mind was born with curiosity and
began to look about and make report of part to part
and thence to whole, the age of interrogation was at
hand, and what had been an eternal infinity of mindless
being began to question, and know itself, and have a
sense of ignorance. In the whole universe of events,
none is more wonderful than the birth of wonder, none
more curious than the nascence of curiosity itself,
nothing to compare with the dawning of consciousness
in the ancient dark and the gradual extension of psychic
life and illumination throughout a cosmos that before
had only been. An eternity of blindly acting, trans-
forming, unconscious existence, assuming at length,
through the birth of sense and intellect, without loss
or break of continuity, the abiding form of fleeting
time. Another eternity remains to follow, and one

128 THE UNIVERSE AND BEYOND

cannot but wonder whether there shall issue forth in
future from the marvel-weaving loom another event, or
form or mode of being, that shall be to the modern
universe that both is and knows, as the birth of soul and
curiosity to the ancient universe that was but did not
know. A speculation by no means idle, but let it pass.

I wish to point out next, briefly, that curiosity is not
only a principle that leads to knowing, but a principle
and process of growing. By it the universe comes not
merely to understand itself, but actually to get bigger
thereby. For if there be an invariant amount of matter,
there is also mind increasing; if there be objects that
total a constant sum, there are also ideas that multiply.
A new query and a new answer are new elements in the
world, by which the latter is added unto and enriched.
Curiosity is the aspect of the universe seeking to realise
itself, and the fruit of such activity is new reality, stimu-
lating to new research. Imagine a body with an inner
core of outward-striving impulses producing buds at
every radial terminus. Such is knowledge — • a kind of
proliferating sphere, expanding along divergent lines by
the outward-seeking of an inner life of wonder. Where-
fore, it appears again that truth, the complement of
curiosity, itself grows with the latter's growth, and,
being never a finished thing, but one that both is and
is becoming, is not to be compassed by definition nor
fully solved in knowledge.

In respect to truth, then, the upshot is: we are certain
that it is; not, however, as a closed or completed
scheme of relationships, but as a kind of reality charac-
terised by the phenomena of growth and of becoming;
it does not admit of ultimate definition; we know, how-
ever, in a super-verbal sense, through myriad mani-
festations of it to a faculty in us of feeling for it, what

THE UNIVERSE AND BEYOND I2Q

it is; we recognise it as the motive power, the elixir
vita, the sustaining spring of wonder; it discovers itself
as the wherewithal for the proper fulfilment of the
implicit predictions and intimations of curiosity; as the
thing presaged in a spiritual craving, confidently, per-
sistently proclaiming its needs by an infinitude of
questionings.

And now as to the remainder of my subject, the tale
is quite too long to be told in full. But room must
be found for a partial account, for important fragments
at all events.

What, then, shall we say mathematics is? A question
much discussed by philosophers and mathematicians in
the course of more than two thousand years, and espe-
cially with deepened interest and insight in our own
time. Many an answer has been given to it, but none
has approved itself as final. Naturally enough, con-
ception of the science has had to grow with the science
itself. For it must not be imagined that mathematics,
because it is so old, is dead. Old it is indeed, classic
already in Euclid's day, being surpassed in point of
antiquity by only one of the arts and by none of the
sciences; but it is also living and new, flourishing
to-day as never before, advancing in a thousand direc-
tions by leaps and bounds. It is not merely as a giant
tree throwing out and aloft myriad branching arms in
the upper regions of clearer light, and plunging deep
and deeper roots in the darker soil beneath. It is rather
an immense forest of such oaks, which, however, liter-
ally grow into each other, so that, by the junction and
intercresence of root with root and limb with limb, the
manifold wood becomes a single living organic whole.
A vast complex of interlacing theories — that the science
now is actually, but it is far more wondrous still poten-

130 THE UNIVERSE AND BEYOND

tially, its component theories continuing more and more
to grow and multiply beyond all imagination, and
beyond the power of any single genius, however gifted.
What is this thing so marvellously vital? What does
it undertake? What is its motive? How is it related
to other modes and interests of the human spirit?

One of the oldest and at the same time the most
familiar of the definitions conceived mathematics to be
the science of magnitude, where magnitude, including
multitude as a special kind, was whatever was capable
of increase and decrease and measurement. This last —
capability of measurement — was the essential thing.
That was a most natural definition of the science, for
magnitude is a singularly fundamental notion, not only
inviting but demanding consideration at every stage
and turn of life. The necessity of finding out how many
and how much was the mother of counting and measure-
ment, and mathematics, first from necessity and then
from joy, so busied itself with these things that they
came to seem its whole employment. But now the
notion of measurement as the repeated application of
a constant unit has been so refined and generalised, on
the one hand through the creation of imaginary and
irrational numbers, and on the other by use of a scale,
as in non-Euclidian geometry, where the unit suffers a
lawful change from step to step of its application, that
to retain the old words and call mathematics the science
of measurement seems quite inept as no longer telling
what the spirit of mathesis is really bent upon. More-
over, the most striking measurements, as of the volume
of a planet, the swiftness of thought, the valency of
atoms, the velocity of light, the distance of star from
star, are not achieved by direct repeated application
of a unit. They are all accomplished by indirection.

THE UNIVERSE AND BEYOND 131

And it was perception of this fact which led to the
famous definition by the philosopher and mathematician,
Auguste Comte, that mathematics is the science of
indirect measurement. Doubtless we have here a finer
insight and a larger view, but the thought is yet too
narrow, nor is it deep enough. For it is obvious that
there is much mathematical activity which is not at all
concerned with measurement, either direct or indirect.
In projective geometry, for example, it was observed
that metric considerations were by no means chief. As
a simplest illustration, the fact that two points deter-
mine a line, or the fact that a plane cuts a sphere in a
circle, is not a metric fact, being concerned with neither
size nor magnitude. Here it was position rather than
size that seemed to some to be the central idea, and
so it was proposed to call mathematics the science of
magnitude, or measurement, and position.

Even as thus expanded, the definition yet excludes
many a mathematical realm of vast, nay, infinite extent.
Consider, for example, that immense class of things
familiarly known as operations. These are limitless,
alike in number and in kind. Now it so happens that
there are systems of operations such that any two opera-
tions of a given system which follow one another pro-
duce the same effect as some other single operation of
the system. For an illustration, think of all possible
straight motions in space. The operation of going
from A to B followed by the operation of going from
B to C is equivalent to the single operation of going
from A to C. Thus, the system of such straight opera-
tions is a closed system. Combination of any two of
them yields another operation, not without, but within
the system. Now the theory of such closed systems —
called groups of operations — is a mathematical theory,

132 THE UNIVERSE AND BEYOND

already of colossal proportions, and still growing with
astonishing rapidity. But, and this is the point, an
abstract operation, though a very real thing, is neither
a position nor a magnitude.

This way of trying to come at an adequate conception
of mathematics, viz., by naming its different domains,
or varieties of content, is not likely to prove successful.
For it demands an exhaustive enumeration not only of
the fields now occupied by the science, but also of the
realms destined to be conquered by it in the future, and
such an achievement would require a prevision that none
perhaps could claim.

Fortunately there are other paths of approach that
seem more promising. Everyone has observed that
mathematics, whatever it may be, possesses a certain
mark, namely, a degree of certainty not found else-
where. So it is, proverbially, the exact science par
excellence. Exact, you say, but in what sense? To this
an excellent answer is contained in a definition given
by an American mathematician, Professor Benjamin
Peirce: Mathematics is the science which draws necessary
conclusions, a formulation something more than finely
paraphrased by one 1 of my own teachers thus : Mathe-
matics is the universal art apodictic. These statements,
though neither of them may be entirely satisfactory, are
both of them telling approximations. Observe that they
place the emphasis on the quality of being correct. Noth-
ing is said about the conclusions being true. That is
another matter, to which I will return presently. But
why are the conclusions of mathematics correct? Is it
that the mathematician has an essentially different
reasoning faculty from other folks? By no means.
What, then, is the secret? Reflect that conclusion im-

1 Professor W. B. Smith.

THE UNIVERSE AND BEYOND 133

plies premises, and premises imply terms, and terms
stand for ideas or concepts, and that these, namely,
concepts, are the ultimate material with which the
spiritual architect, which we call the Reason, designs
and builds. Here, then, we may expect to find light.
The apodictic quality of mathematical thought, the
certainty and correctness of its conclusions, are due,
not to a special mode of ratiocination, but to the char-
acter of the concepts with which it deals. What is that
distinctive characteristic? I answer: precision, sharp-
ness, completeness, of definition. But how comes your
mathematician by such completeness? There is no mys-
terious trick involved; some ideas admit of such pre-
cision, others do not; and the mathematician is one who
deals with those that do. Law, says Blackstone, is a
rule of action prescribed by the supreme power of a
state commanding what is right and prohibiting what
is wrong. But what are a state and supreme power and
right and wrong? If all such terms admitted of com-
plete determination, then the science of law would be
a branch of pure, and its practice a branch of applied,
mathematics. But does not the lawyer sometimes arrive
at correct conclusions? Undoubtedly he does some-
times, and, what may seem yet more astonishing, so
does your historian and even your sociologist, and that
without the help of accident. When this happens, how-
ever, when these students arrive, I do not say at truth,
for that may be by lucky accident or happy chance or a
kind of intuition, but when they arrive at conclusions
that are correct, then that is because they have been
for the moment in all literalness acting the part of
mathematician. I do not say that for the aggrandise-
ment of mathematics. Rather is it for credit to all
thinkers that none can show you any considerable gar-

134 THE UNIVERSE AND BEYOND

ment of thought in which you may not find here and
there, rarely enough sometimes, a golden fibre woven in
some, it may be, exceptional moment, of precise con-
ception and rigorous reasoning. To think right — that
is no characteristic striving of a class of men. It is a
common aspiration. Only, the stuff of thought is mostly
intractable, formless, like some milky way waiting to
be analysed into distinct star-forms of definite ideas.
All thought aspires towards the character and condition
of mathematics.

The reality of this aspiration and the distinction it
implies admit of many illustrations, of which here a
single one must suffice. There is no more common or
more important notion than that of function, the term
being applied to either of two variable things such that
to any value or state of either there correspond one or
more values or states of the other. Of such function
pairs, examples abound on every hand, as the radius
and the area of a circle, the space traversed and the
rate of going, progress of knowledge and enthusiasm of
study, elasticity of medium and velocity of sound or
other undulation, the amount of hydrogen chloride
formed and the time occupied, the prosperity of a
given community and the intelligence of its patriotism.
Indeed, it may very well be that there is nothing which
is not in some sense a function of every other. Be that
as it may, one thing is very certain, namely, a very
great part and probably all of our thinking is concerned
with functional relationships, deals, that is, with pairs
of systems of corresponding values or states or changes.
Behold, for example, how the parallelistic psychology
searches for correlations between psychical and physical
phenomena. Witness, too, the sociologist trying to de-
termine the correspondence between the peacefulness

THE UNIVERSE AND BEYOND 135

and the homogeneity of a population, or, again, be-
tween manifestations of piety or the spread of populism
and the condition of the crops. It is then here, in the
wondrous domain of correspondence, the answering of
value to value, of change to change, of condition to
condition, of state to state, that the knowing activity
finds its field.

What is it precisely that we seek in a correlation?
The answer is: when one or more facts are given, to pass,
with absolute certainty, to the correlative fact or facts. To
do this obviously requires formulae or equations which
precisely define the manner of correlation, or the law
of interdependence. Where do such formulae come from?
I answer that, strictly speaking, they are never found,
they are always assumed. Now, nothing is easier than
to write down a perfectly definite formula that does not
tell, for example, how cheerfulness depends on climate,
or how pressure affects the volume of a gas. Nay, a
given formula may be perfectly intelligible in itself, it
may state, that is, a perfectly intelligible law of cor-
respondence, which, nevertheless, may have no validity
at all in the physical universe and none elsewhere than
in the formula itself. What, then, guides in the choice
of formulae? That depends upon your kind of curiosity,
and curiosity is not a matter of choice.

Just here we are in a position where we have only to
look steadily a little in order to see the sharp distinc-
tion between mathematics and natural science. These
are discriminated according to the kind of curiosity
whence they spring. The mathematician is curious about
definite abstract correspondences, about perfectly-defined
functional relationships in themselves. These are more
numerous than the sands of the seashore, they are as
multitudinous as the points of space. It is this as-

136 THE UNIVERSE AND BEYOND

semblage of pure, precisely-defined relationships which
constitute the mathematician's universe, an indefinitely
infinite universe, worlds of worlds of wonders, incon-
ceivably richer than the outer world of sense. This
latter is indeed immense and marvellous, with its rolling
seas and stellar fields and undulating ether, but, com-
pared with the hyperspaces explored by the genius of the
geometrician, the whole vast extent of the sensuous uni-
verse is a merest point of light in a blazing sky.

Now this mere speck of a physical universe, in which
the chemist and the physicist, the biologist and the so-
ciologist, and the rest of nature devotees, find their
great fields, may be, as it seems to be, an organic thing,
connected into an ordered whole by a tissue of definable
functional relationships, and it may not. The nature
devotee assumes that it is and tries to find the relation-
ships. The mathematician does not make that assump-
tion and does not seek for relationships in the outer
world. Is the assumption correct? As man, the mathe-
matician does not know, although he greatly cares. As
mathematician, man neither knows nor cares. The
mathematician does know, however, that, if the assump-
tion be correct, every definite relationship that is valid
in nature, every type of order and mode of correlation
obtaining there, is, in itself, a thing for his thought, an
essential element in his domain of study. He knows, too,
that, if the assumption be not correct, his domain re-
mains the same absolutely. The two realms, of mathe-
matics, of nature science, are fundamentally distinct
and disparate forever. To think the thinkable — that is
the mathematician's aim. To assume that nature is
thinkable, an incarnate rational logos, and to seek the
thought supposed incarnate there — these are at once
the principle and the hope of the nature student. Sci-

THE UNIVERSE AND BEYOND 137

ence, said Riemann,1 is the attempt to comprehend nature
by means of concepts. Suppose the nature student is
right, suppose the physical universe really is an enfleshed
logos of reason, does that imply that all the thinkable
is thus incorporated? It does not. A single ordered
universe, one that through and through is self-com-
patible, cannot be the whole of reason materialised and
objectified. There is many a rational logos, and the
mathematician has high delight in the contemplation
of ^consistent systems of consistent relationships. There
are, for example, a Euclidean geometry and more than
one species of non-Euclidean. As theories of a given
space, these are not compatible. If our universe be,
as Plato thought, and nature science takes for granted,
a space-conditioned, geometrised affair, one of these
geometries may be, none of them may be, not all of
them can be, valid in it. But in the vaster world of
thought, all of them are valid, there they co-exist, and
interlace among themselves and others, as differing com-
ponent strains of a higher, strictly supernatural, hyper-
cosmic, harmony.

It is, then, in the inner world of pure thought, where
all entia dwell, where is every type of order and manner
of correlation and variety of relationship, it is in this
infinite ensemble of eternal verities whence, if there be
one cosmos or many of them, each derives its character
and mode of being, — it is there that the spirit of
mathesis has its home and its life.

Is it a restricted home, a narrow life, static and cold
and grey with logic, without artistic interest, devoid of
emotion and mood and sentiment? That world, it is true,
is not a world of solar light, not clad in the colours that

1 Cf. Riemann: "Fragmente Philosophischen Inhalts," in Gesammeltc
Werke. These fragments, which are published in English by the Open
Court Pub. Co., Chicago, are exceedingly suggestive.

138 THE UNIVERSE AND BEYOND

liven and glorify the things of sense, but it is an illu-
minated world, and over it all and everywhere through-
out are hues and tints transcending sense, painted there
by radiant pencils of psychic light, the light in which it
lies. It is a silent world, and, nevertheless, in respect
to the highest principle of art — the interpenetration of
content and form, the perfect fusion of mode and mean-
ing — it even surpasses music. In a sense, it is a static
world, but so, too, are the worlds of the sculptor and
the architect. The figures, however, which reason con-
structs and the mathematical vision beholds, transcend
the temple and the statue, alike in simplicity and in
intricacy, in delicacy and in grace, in symmetry and in
poise. Not only are this home and this life thus rich
in aesthetic interests, really controlled and sustained by
motives of a sublimed and supersensuous art, but the
religious aspiration, too, finds there, especially in the
beautiful doctrine .of invariants, the most perfect sym-
bols of what it seeks — the changeless in the midst of
change, abiding things in a world of flux, configurations
that remain the same despite the swirl and stress of
countless hosts of curious transformations. The domain
of mathematics is the sole domain of certainty. There
and there alone prevail the standards by which every
hypothesis respecting the external universe and all ob-
servation and all experiment must be finally judged. It
is the realm to which all speculation and all thought must
repair for chastening and sanatation — the court of
last resort, I say it reverently, for all intellection what-
soever, whether of demon or man or deity. It is there
that mind as mind attains its highest estate, and the
condition of knowledge there is the ultimate object, the
tantalising goal of the aspiration, the Anders-Streben,
of all other knowledge of every kind.

THE AXIOM OF INFINITY: A NEW
PRESUPPOSITION OF THOUGHT.1

IT so happened that when the first number of The
Hibbert Journal appeared, containing an article by Pro-
fessor Royce on the Concept of the Infinite, I had been
myself for some tiineuneditatin& on thej^gical_bearings^
and philosophi^l^iniport of that concept, and was
actually then engaged in marking out the course which
it seemed to me a first discussion of the matter might
best follow. The order and scope of his treatment were
so like those I had myself decided upon that I should
naturally have felt a pardonable pride in the coinci-
dence, had not this feeling been at the same time quite
lost in a stronger one, namely, that of the evident
superiority of his manner to any which I could have
hoped to attain. Indeed, so patient is his exposition
of elements, so rich is it in suggestiveness, so intimately
and instructively, according to his wont, has he con-
nected the most abstruse and recondite of doctrines
with the most obvious and seemingly trivial of things,
and so luminous and stimulating is it all, that one must
admire the ingenuity it betrays, and cannot but wonder
whether after all there really are in science or philosophy
any notions too remote and obscure to be rendered
intelligible even to common sense, if only a sufficiently
cunning pen be engaged in the service.

1 Appeared in The Hibbert Journal, April, 1904.

140 THE AXIOM OF INFINITY

While his paper is thus replete with inspiring intima-
tions of the "glorious depths" and near-lying interests
of the doctrine treated, and is, in point of clearness
and vivid portrayal of its central thought, a model be-
yond the art of most, it is not, I believe, equally happy
when judged on the severer ground of its critico-logical
estimates. Even on this ground, I do not hesitate, after
close examination, to adjudge it the merit of general
soundness. That, however, it is thoroughly sound, com-
pletely mailed against every possible assault of criti-
cism, is a proposition I am by no means prepared to
maintain. Quite the contrary, in fact. Nor can the
defects be counted as trivial. One of them especially,
which it has in common with other both earlier and
later discussions of the subject, notably that by Dede-
kind himself and, more recently, that by Mr. Bertrand
Russell in his imposing treatise on The Principles of
Mathematics, is of the most radical nature, concerning
as it does no less a question than, I do not say merely
that of the validity, but that of the possibility, of
existence-proofs of the infinite.

And here I may as well state at once, lest there should
be some misapprehension in respect to purpose, that
the present writing is not primarily designed to be a
review of Professor Royce's or of other recent discus-
sions of the infinite. Reviewed to some extent they
will be, but only incidentally, and mainly because they
have declared themselves, erroneously as I think, upon
that most fundamental of questions, namely, whether it
is possible, by aid of the modern concept, to demonstrate
the existence, of the infinite. Argument would seem
superfluous to show the immeasurable import of this
problem, whether it be viewed solely in its immediate
logical bearings, or also mediately, through the latter,

THE AXIOM OF INFINITY 141

in its bearings upon philosophy, upon theology, and,
only more remotely, upon religion itself. It is chief
among the aims of this essay, to open that problem
anew, to appeal from the prevailing doctrine concerning
it, in the hope of securing, if possible, a readjudication
of the matter which shall be final.

This subject of the infinite, how it baffles approach!
How immediate and how remote it seems, how it abides
and yet eludes the grasp, how familiar it appears,
mingling with the elemental simplicities of the heart,
continuously weaving itself into the intimate texture
of common life, and yet how austere and immense and
majestic, outreaching the sublimest flights of the im-
agination, transcending the stellar depths, immeasurable
by the beginningless, endless chain of the ages! Com-
prehend the infinite! No wonder we hear that none
but the infinite itself is adequate to that. Du gleichst
dem Geist, den du begreifst. Be it so. Perhaps, then,
we are infinite. If not,

"'Wie' fass' ich dich, unendliche Natur?"

Or is it finally a mere illusion? And is there after all no
infinite reality to be seized upon? Again, if not, what
signifies the finite? Is that to be for ever without
definition, except as reciprocal of that which fails to
be? Is the All really enclosed in some vast ellipsoid,
without a beyond, incircumscriptible, devoid alike of
tangent plane and outer point? Are we eternally con-
demned to seek therein for the meaning and end of
processes that refuse to terminate? And is, then, this
region, too, but a locus of deceptions, "of false alluring
jugglery "? Is analysis but the victim of hallucination
when it thinks to detect the existence of realms that
underlie and overarch and compass about the domain of

142 THE AXIOM OF INFINITY

the countable and measurable? And does the spirit,
in its deeper musings, in its pensive moods, only seem to
feel the tremulous touch of transfinite waves, of vitalising
undulations from beyond the farthest shore of the sea
of sense?

One fact at once is clear, namely, that, whatever ulti-
mate justification the hypothesis may find, thought has
never escaped the necessity of supposing the universe
of things to be intrinsically somehow cleft asunder into
the two Grand Divisions, or figured, if you will, under
the two fundamental complementary all-inclusive Forms,
which, from motives 'more or less distinctly felt and also
just, as we shall see, though not quite justified, have
been, from time immemorial, designated as the Finite
and the Infinite. And these great terms or their verbal
equivalents — • for concepts in any strict sense they have
not been — though always vague and shifting, for ever
promising but never quite delivering the key to their
identities into the hand of Definition, have, neverthe-
less, in every principal scene, together played the gravest
role in the still unfolding drama of speculation. Or, to
change the figure, they have been as Foci, one of them
seemingly near, the other apparently remote, neither
of them quite itself determinate, but the two con-
jointly serving always to determine the ever-varying
eccentricity of the orbit of thought; and doubtless the
vaster lines that serve to bind the differing epochs of
speculation into a single continuous system can best be
traced by reference to these august terms as co-ordinate
poles of interest.

As a simple historical fact, then, philosophy has indeed,
with but negligible exception, throughout assumed the
existence of both the finite and the infinite. That is one
thing. Another fact of distinct and equal weight, no

THE AXIOM OF INFINITY 143

matter whether or how we may account for it, is that
man, in accord with the deeper meaning of the Pro-
tagorean maxim, h^"aJw^^s_idFliimseIf to have within,
or to be somehow, the-potential measurejot^inHaTls:
Is ft Insignificant that this faith^^Tor that is what it
seems to be — as if an indestructible character of the
race, as if an invariant defining property of the germ
plasm itself whence man springs and derives his con-
tinuity, should have survived every vicissitude of human
fortune? that it should have been indeed, if not the
substance, at least the promise, of things hoped for, the
evidence, too, of things not seen, marking and sustain-
ing metaphysical research from the earliest times? And,
what is more, the spirit of such research, curiosity I
mean, fit companion and counterpart of that abiding
faith, unlike "experience and observation," has known
no bounds, but, on the contrary, finding within itself
no fatal principle of limitation, it has ever disdained the
scale of finite things as competent to take its measure,
and boldly asserted claim to the entire realm of being.

These questions, however, have been something more
than fascinating. Perhaps their rise, but not their mani-
fold development, much less their profound significance
for life and thought, is to be adequately explained on
the hypothesis of insatiate curiosity alone. It must be
granted that their presence, especially in the arena of
dialectic, has been often due simply to their intrinsic
magical charm for "summit-intellects." And doubtless
the play-instinct, deep-dwelling in the constitution of
the mind, has often made them serve the higher faculties
merely as intricate puzzles, to beguile the time withal.
But, in general, the questions have worn a sterner
aspect. Philosophy has been not merely allured, it has
been constrained, to their consideration; constrained not

144 THE AXIOM OF INFINITY

only because of their inherence in problems of the con-
science, especially in that most radical problem of find-
ing the simplest system of postulates that shall be at
once both necessary and sufficient to explain the moral
feeling; but constrained still more powerfully by the
insistent demands that issue from the religious con-
sciousness. But this is yet not all. For man cannot
live by these august interests alone. And it is pro-
foundly significant, both as witnessing to the final inter-
blending, the fundamental unity, of all the concerns of
the human spirit, and as revealing the ultimate depth
and dignity of all its interests, that questions about the
infinite quite similar to those that claim so illustrious
parentage in Ethics and Philosophy, admit elsewhere of
humbler derivation, and readily own to the lowliest of
origins. Man, indeed, merely to live, has had to meas-
ure and to count, and this homely necessity, fruitful
mother of mystery and doubt, independently set the
problems of the indefinitely small and the indefinitely
great; and so it was that needs quite as immediate and
austere as those of Morals and Religion — I mean the
exigencies of Science, and especially of Mathematics —
demanded on their own ground, in the very beginnings
of exact knowledge, that the understanding transcend
every possible sequence of observations, pass the utter-
most limit of "experience," which, refine and enlarge
it as you may, remains but finite, and literally lay hold
on infinity itself.

To this ancient irrevocable demand, thus urged upon
the reason from every cardinal point of human interest,
genius has responded as to a challenge from the gods,
and I submit that the response, the endeavour of the
reason actually to subjugate extra-finite being and com-
pel surrender of its secrets by the organon of though t>

THE AXIOM OF INFINITY 145

constitutes the most sublime and strenuous and inspiring
enterprise of the human intellect in every age.

What of it? Long centuries of gigantic striving, age
on age of philosophic toil, immeasurable devotion of
time and energy and genius to a single end, the intel-
lectual conquest of transfinite being — • what has it all
availed? What triumphs have been won? I speak,
narrowly, of the conquest, and demand to know, not
whether it has been accomplished — for that were a
foolish query — but whether, strictly speaking, it has
been begun. Let not the import of the question be mis-
taken. No answer is sought in terms of such moral or
"spiritual" gains as may be incident even to efforts that
miss their aim. Everyone knows that seeking has com-
pensations of its own, which indeed are ofttimes better
than any which finding itself can give. And it seems
sometimes as if the higher life were chiefly sustained by
unsought gains incident to the unselfish pursuit of the
unattainable. The circle has not been squared, nor the
quintic equation solved,1 nor perpetual motion invented;
neither indeed can be; yet it would show but meagre
understanding of the ways of truth to men, did one
suppose all the labour devoted to such problems to have
been without reward. So, conceivably, it might be
with this problem of the infinite. It may be granted
that, even supposing no solution to be attainable, the
ceaseless search for one, the unwearied high endeavor
of the reason through the ages, presents a spectacle
ennobling to behold, and of which mankind, it may be,
could ill afford to be deprived. It may be granted that
incidentally many insights have been won which, though
not solutions, have nevertheless permanently enriched
the literature of the world and are destined to improve

1 That is, by means of radicals.

146 THE AXIOM OF INFINITY

its life. It may be granted that in every time some
doctrine of infinity, some philosophy of it, has been at
least effective, has helped, that is, for better or worse,
to fashion the forms of human institutions and to de-
termine the course of history. Concerning none of
these things is there here any question. As to what the
question precisely is, there need not be the slightest mis-
apprehension. The fact is that for thousands of years
philosophy has recognised the presence of a certain
definite Problem, namely, that of extending the dominion
of logic, the reign of exact thought, out beyond the utmost
reach of finite things into and over the realm, of infinite
being, and this problem, by far the greatest and most
impressive of her strictly intellectual concernments,
philosophy has, for thousands of years, arduously striven
to solve. And now I ask — not, has it been worth while?
for that is conceded, but — has she advanced the solu-
tion in any measure, and, if so, in what respect, and to
what extent?

We are here upon the grounds of the rational logos.
The whole force and charge of the question is directed
to matter of concept and inference. Fortunately, the
answer is to be as unmistakable as the question. It
must be recognised, of course, that the " problem," as
stated, is exceedingly, almost frightfully, generic, com-
prising a host of interdependent problems. One of these,
however, is pre-eminent: without its solution none other
can be solved; with its solution, any other may be
eventually. That problem is the problem of concep-
tion, of definition in the unmitigated rigour of its
severest meaning; it is the problem of discovering a
certain principle, of finding, without the slightest possi-
bility of doubt or indetermination, the intrinsic line
of cleavage that parts the universe of being into its

THE AXIOM OF INFINITY 147

two grandest divisions, and so of telling finally and
once for all precisely what, for thought, the infinite is
and what, for thought, the finite is.

And now, thanks to the subtle genius of the modern
Teutonic mind, this ancient problem, having baffled the
thought of all the centuries, has been at last com-
pletely solved, and therein our original question finds
its answer: J^hejopnguest has been begun. Bernhard
Riemann, profoundinathematiaan ancT^^Tmportant
fact, of which, strangely enough, too many philosophers
seem invincibly unaware — • profound metaphysician too,
having pointed out, in his famous Habilitationschrift,1
the epoch:m£kin£_c]j^Jiii^^

ness anHJnfinij^iHgj^Tna.Tijfn1^s siTrnMqjMr>Jl>fltnf space,
the greater glory was reserved for three contemporary

compatriots of his — Bernhard Bolzanb^ jjichard Deder

kind,3 and 'Georg^ Cantor,4 the first an acute and learned
philosopher and theologian, Vith deep mathematical
insight, the other two brilliant mathematicians, with a
strong bent for metaphysics — to win independently
and about the same time the long-coveted insight into
the intrinsic nature of infinity. And thus it is a dis-
tinction of our own time that within the memory of
living men the defining mark of the infinite first failed
to elude the grasp, and that that august term, after
the most marvellous career of any in the history of
speculation, has been finally made to assume the prosaic
form of an exact and completely determined concept,
and so at length to become available for the purposes of
rigorously logical discourse.

1 "Ueber die Hypothesen, welche die Geometric zu Grunde liegen,"
Ges. Werke. Also in English by W. K. Clifford.

2 "Paradoxien des Unendlichen."

8 " Was sind und was sollen die Zahlen."

4 Memoirs in Ada Mathematica, vol. ii., and elsewhere.

148 THE AXIOM OF INFINITY

Pray, then, what is this concept? Of various equiva-
lent forms of statement, I choose the following: An
assemblage (ensemble, collection, group, manifold) of ele-
ments (things, no matter what) is infinite or finite accord-
ing as it has or has not a PART to which the whole is just
EQUIVALENT in the sense that between the elements com-
posing that part and those composing the whole there 'subsists
a unique and reciprocal (one-to-one) correspondence.

If we may trust to intuition in questions about reality,
assemblages,1 infinite as defined, actually abound on
every hand. I need not pause to indicate examples.
Those pointed out in Professor Royce's mentioned paper
may suffice; they will, at all events, furnish the reader
with the "clew, which, once familiar to his hand, will
lengthen as he goes, and never break." The concept
itself I regard as a great achievement, one of the very
greatest in the history of thought. Not only does it
mark the successful eventuation of a long and toilsome
search; it furnishes criticism with a new standard of
judgment, it at once creates, and gives the means of
meeting, the necessity for a re-examination and a juster
evaluation of historic doctrines of infinity; and it is
greater still, I believe, as a destined instrument of ex-
ploration in that realm which it has opened to the under-
standing and whose boundary it defines.

Is that judgment not extravagant? For the concept
seems so simple, is so apparently independent of difficult
presuppositions, that one cannot but wonder why it was
not formed long ago. Had the concept in question been
early formed, the history and present status of philoso-

1 The very simplest possible example of such a manifold is that of the
count-numbers. The whole collection can be paired in one-to-one fashion
with, for example, half the collection, thus: i, 2; 2, 4; 3, 6; . . . ; the
totality of even and odd being just equivalent to the even.

THE AXIOM OF INFINITY 149

phy and theology, and of science too, had doubtless
been different. But it was not then conceived. Now
that we have it, is it too unbewildering to be impress-
ive? Shall we esteem it lightly just because we can
comprehend it, because it does not mystify? Simple
it is indeed, almost as simple as the Newtonian law of
gravitation, nearly as easy to understand as the geo-
metric interpretation of imaginary quantities, hardly
more difficult to grasp than the notion of the conserva-
tion of energy, the Mendelian principle of inheritance,
or than a score of other central concepts of science.
But shallow indeed and foolish is that criticism which
values ideas according to their complexity, and con-
founds the simple with the trivial.

As an immense city or a vast complex of mountain
masses, seen too near, is obscured as a whole by the
prominence of its parts, so the larger truth about any
great subject is disclosed only as one beholds it at a
certain remove which permits the assembling of principal
features in a single view, and a proportionate mingling
of reflected light from its grander aspects. Accordingly
it has seemed desirable, in the foregoing preliminary
survey, to hold somewhat aloof, to conduct the move-
ment, in the main, along the path of perspective centres,
in order to allow the vision at every point the amplest
range. It is now proposed to draw a little closer to the
subject and to examine some of its phases more minutely.
In respect to the modern concept of infinity, we desire
to know more fully what it really signifies, we wish to
be informed how it orients itself among cardinal prin-
ciples and established modes of thought. But recently
born to consciousness, it has already been advanced to
conspicuous and commanding station among funda-
mental notions, and we are concerned to know what, if

150 THE AXIOM OF INFINITY

any, transformations of existing doctrine, what read-
justments of attitude towards the universe without us
or within, what changes in our thought on ultimate
problems of knowledge and reality, it seems to demand
and may be destined to effect. In a word, and speak-
ing broadly, we wish to know not merely in a narrow
sense what the new idea is, but, in the larger meaning
of the term, what it "can."

I shall first speak briefly of the so-called "positive"
character of the definition, an alleged essential quality
of it, a seeming property which criticism is wont to
signalise as a radical or intrinsic virtue of the concept
itself. Quite independently of the mathematicians
Dedekind and Cantor, who, we have seen, were the
independent originators of the new formulation, the
then old philosopher, Bolzano, bringing to the subject
another order of training and of motive, arrived at
notions of the finite and infinite, which on critical
examination are found to be essentially the same as
theirs, though greatly differing in point alike of view
and of form. Bolzano's procedure is virtually as fol-
lows:— Suppose given a class C of elements, or things,
of any kind whatsoever, as the sands of the seashore,
or the stars of the firmament, or the points of space,
or the instants in a stretch of time, or the numbers
with which we count, or the total manifold of truths
known to an omniscient God. Out of any such class C,
suppose a series formed by taking for first term one of
the elements of C, for second term two of them, and
so on. Any term so obtainable is itself obviously a
class or group of things, and is defined to be finite. The
indicated process of series formation, if sufficiently pro-
longed, will either exhaust C or it will not. If it will,
C is itself demonstrably finite; if it will not, C is, on

THE AXIOM OF INFINITY 151

that account, defined to be infinite. Now, say Professor
Royce and others, a definition like the latter, being
dependent on such a notion as that of inexhaustibility
or endlessness or boundlessness, is negative; a certain
innate craving of the understanding remains unsatisfied,
we are told, because the definition presents the notion,
not in a positive way by telling us what the infinite
actually is, but merely in a negative fashion by telling
us what it is not. Undoubtedly the claim is plausible,
but is it more? Bolzano affirmed and exemplified a
certain proposition, in itself of the utmost importance,
and throwing half the needed light upon the question
in hand. That proposition is: Any class or assemblage
(of elements), if infinite according to his own definition
of the term, enjoys the property of being equivalent, in the
sense above explained, to some proper part of itself. Though
he did not himself demonstrate the proposition, it
readily admits of demonstration, and, since his time,
has in fact been repeatedly and rigorously proved.
Not only that, but the converse proposition, giving the
other half of the needed light, has been established too:
Every assemblage that HAS a part "equivalent" to the
whole, is infinite in the Bolzano sense of the term.

It so appears, in the conjoint light of those two
theorems, that the property seized upon and pointed
out by the ingenious theologian is in all strictness a
characteristic, though derivative, mark of the infinite
as he conceived and defined it. It is sufficiently obvious,
therefore, that this derivative property might logically
be regarded as primitive, made to serve, that is, as a
ground of definition. Precisely this fact it is which was
independently perceived by Dedekind and Cantor, with
the result that, as they have presented the matter, a
collection, or manifold, is infinite if it has a certain

152 THE1 AXIOM OF INFINITY

property, and finite if it has it not. And now, the critics
tell us, it is the infinite which is positive and the finite
which is negative.

The distinction appears to me to be entirely devoid
of essential merit. It seems rather to be only another
interesting example of that verbal legerdemain for which
a certain familiar sort of philosophising has long been
famous. For what indeed is positive and what negative?
Are we to understand that these terms have absolute
as distinguished from relative meaning? The distinc-
tion, I take it, is without external validity, is entirely
subjective, a matter quite at will, being dependent
solely on an arbitrary ordering of our thought. That
which is first put in thought is positive: the opposite,
being subsequently put, is negative; but the sens of the
time-vector joining the two may be reversed at the
thinker's will. It is sometimes contended that that
which generally happens in the world, and so constitutes
the rule, is intrinsically positive. As a matter of fact
a moving body "in general" continuously changes its
distance from every object. Such change of distance
from every other object would accordingly be a positive
something. Then it would follow that the classic defi-
nition of a sphere-surface as the locus of a moving
point which does not change its distance from a certain
specified point, is really negative. Obviously it avails
nothing essential to disguise the negativity by some
such seemingly positive phrase as "constant" distance.
The trick is an easy one. If, again, it be allowed that,
a process being once started, its continuation is positive,
its termination negative, then it would result that in-
exhaustibility is positive and exhaustibility negative,
whence we should have to own that it is Bolzano's
definition which is positive and that by Dedekind and

THE AXIOM OF INFINITY 153

Cantor negative. It hardly admits of doubt that the
matter is purely one of an arbitrarily chosen point of
view. The distinction is here of no importance. What
is important is that, no matter which of the definitions
be adopted as such, the other then states a derivable prop-
erty of the thing defined. In either case the concept of
the infinite remains the same, it is merely its garb that is
changed. I am very far from intending, however, to assert
herewith that, because the definitions are logically equiva-
lent, they must needs be or indeed are so practically, that
is, as instruments of investigation. That is another
matter, which, I regret to say, our somewhat pretentious
critiques of scientific method furnish no better means of
settling than the wasteful way of trial. Everyone will
recall from his school-days Euclid's definition of a plane
as being a surface such that a line joining any two
points of the surface lies wholly in the surface. Logi-
cally that is equivalent to saying: A plane is such an
assemblage of points that, any three independent points
of the assemblage being given, one and only one third
point of the assemblage can be found which is equi-
distant from the given three. But, despite their logical
equivalence, who would contend that, for elementary
purposes, the latter notion is " practically " as good as
the Greek? And so in respect to the infinite, I am free
to admit, or rather I affirm, that, on the score of
usability, the Dedekind-Cantor definition is greatly
superior to its Bolzanoan equivalent. Professor Royce
has indeed ingeniously shown how readily it lends itself
to philosophic and even to theologic uses.

I turn now to the current assertion by Professor
Royce and Mr. Russell, that the modern concept of the
infinite, of which I have given above in italics an exact
statement, to which the reader is referred, in fact denies

154 THE AXIOM OF INFINITY

a certain ancient axiom of common sense, namely, the
axiom of whole and part. I am not about to submit
a brief in behalf of the traditional conception of axioms
as self-evident truths. That conception, as is well
known, has been once for all abandoned by philosophy
and science alike, while to mathematicians in particular
no phenomenon is more familiar than that of the co-
existence of self-coherent bodies of doctrine constructed
on distinct and self-consistent but incompatible systems
of postulates. The co-ordination of such incompatible
theories is quite legitimate and presents no cause for
regret or alarm. The forced recession of the axioms
from the high ground of absolute authority, so far from
indicating chaos of intellection or ultimate dissolution
of knowledge, signifies a corresponding deepening of
foundation; it means an ascension of mind, the procla-
mation of its creative power, the assertion of its own
supremacy. And henceforth the denial of specific
axioms, or the deliberate substitution of one set for
another, is to be rightly regarded as an inalienable
prerogative of a liberated spirit. The question before
us, then, is one merely of fact, namely, whether a certain
axiom is indeed denied or contradicted by the modern
concept of the infinite.

It is in the first place to be observed that the statement
itself of that concept avoids the expression, " equality
of whole and part," but instead of it deliberately employs
the term "equivalence." The word actually used by
Dedekind himself is dhnlichkeit (similarity). But, says
Professor Royce, "equivalence" is just what the axiom
really means by equality. It is precisely this statement
which I venture to draw in question. If we know that
each soldier of a company marching along the street
has one and but one gun on his shoulder, then, we are

THE AXIOM OF INFINITY 155

told, even if we do not know how many soldiers or guns
there are, we do know that there are "as many" soldiers
as guns. What the definition in question, taken severely,
itself affirms in this case, is that the assemblage of guns
is "equivalent or similar" to that of the soldiers. Let
us now suppose that in place of soldiers we write, for
example, "all positive integers," and in place of guns,
"all even positive integers " — the integers are plainly
susceptible of unique and reciprocal association with
the even integers, — then the definition again asserts, as
before, "equivalence" of these assemblages. Note that
thus far nothing has been said about number as an
expression of how many. If there be a number that tells
how many things there are in one assemblage, that same
number doubtless tells how many there are in any
"equivalent" assemblage, and just because the number,
if there be one, is the same for both, the two are said
to be equal by axiom. In this view, equality of groups
means more than mere "equivalence"; it means, besides,
sameness of their numbers, and so applies only in case
there be numbers. But common sense, whose axiom is
here in court, has neither found, nor affirmed the ex-
istence of, a number telling, for example, how many
integers there are. On the other hand, in case of
assemblages for which common sense has known a
number, the axiom of whole and part is admittedly
valid without exception. It thus appears that the
axiom supposed, regarded, however unconsciously but
nevertheless in intention, as applicable only in case
there be a number telling how many, is, in all strictness,
not denied by the concept in question. Numbers
designed to tell how many elements there are in an
assemblage having a part "equivalent" to the whole
are of recent invention, and it may be remarked in

156 THE AXIOM OF INFINITY

passing that this invention bears immediate favourable
witness to the fruitfulness of the new idea. Such trans-
finite numbers once created, then undoubtedly, and
not before, the question naturally presents itself whether
" equivalence " shall be translated "equality," or, what
is tantamount, whether the latter term shall be gen-
eralised into the former; "generalised," I say, for,
though it is true that, as soon as the transfinite numbers
are created, there is, in case of an infinite collection
and some of its parts, a conjunction of "equivalence"
and "sameness of number," yet equality does not of
itself deductively attach, for the transfinite numbers
are in genetic principle,1 i.e., radically, different from the
number notion which the concept of equality has hith-
erto connoted. The question as to the mentioned trans-
lation or generalisation is, therefore, a question, and it
is to be decided, not under spur or stress of logic, but
solely from motives of economy acting on grounds of
pure expedience. If the decision be, as seems likely
because of its expedience and economy favourable to
such translation or generalisation, then indeed the old
axiom, as above construed, still remains uncontradicted,
is yet valid within the domain of its asserted validity.
It is merely that a new number-domain has been ad-
joined which the old verity never contemplated, and
in which, therefore, though it does not apply, it never
essentially pretended to; but on account of which ad-
junction, nevertheless, for the sake of good neighbour-
ship, it is constrained, not indeed to retract its ancient
claims, but merely to assert them more cautiously and
diplomatically, in preciser terms. Even then, in case
of quarrel, it is the generaliser who should explain, and
not a defender of the generalised.

1 Cf. Couturat, Ulnfini mathematique. Appendix.

THE AXIOM OF INFINITY 157

And now to my final thesis I venture to invite the
reader's special attention, and beg to be held with utmost
strictness accountable for my words. The question is,
whether it is possible, by means of the new concept, to
demonstrate the existence of the infinite; whether, in
other words, it can be proved that there are infinite
systems. That such demonstration is possible is affirmed
by Bolzano, by Dedekind, by Professor Royce, by Mr.
Russell, and in fact by a large and swelling chorus of
authoritative utterance, scarcely relieved by a dis-
senting voice. After no little pondering of the matter,
I have been forced, and that, too, I must own, against
my hope and will, to the opposite conviction. Candour,
then, compels me to assert, as I have elsewhere l briefly
done, not only that the arguments which have been
actually adduced are all of them vitiated by circularity,
but that, in the very nature of conception and inference,
by virtue of the most certain standards of logic itself,
every potential argument, every possible attempt to
prove the proposition, is foredoomed to failure, destined
before its birth to take the fatal figure of the wheel.

The alleged demonstrations are essentially the same,
being all of them but variants under a single type. It
is needless, therefore, in support of my first contention,
to present separate examination of them all. Analysis
of one or two specimens will suffice. I will begin with
one from Bolzano's offering, both because it marks the
beginning of the new era of thought about the subject
and because subsequent writers have nearly all of them
either cited or quoted it, and that, as far as I am aware,
always with approval. Bolzano 2 undertakes to demon-

1 "The Axiom of Infinity and Mathematical Induction," Bulletin of the
American Mathematical Society, vol. ix., May, 1903.

2 " Paradoxien," sect. 14.

158 THE AXIOM OF INFINITY

strate, among similar statements, the proposition that
die Menge der Sdtze und Wahrheiten an sich is infinite
(unendlicti) , this latter term being understood, of course,
in accordance with his own definition above given.
The attempt, as anyone may find who is willing to
examine it minutely, informally postulates as follows:
the proposition, There are such truths (as those con-
templated in the proposition), is such a truth, T; T is
true, is another such truth, T; so on; and, the indicated
process is inexhaustible. Now, these assumptions, which
are essential to the argument, and which any careful
reader cannot fail to find implicit in it, are, possibly, all of
them, correct, but the last is so evident a petitio principii
as to make one look again and again lest his own thought
should have played him a trick.

In case of Dedekind's demonstration, which has been
heralded far and wide, the fallacy is less glaring. The
argument is far subtler, more complicate, and the ver-
steckter Zirkel lies deeper in the folds. But it is un-
doubtedly there, and its presence may be disclosed by
careful explication. Let the symbol / stand for thought,
any thought, and denote by /' the thought that / is a
thought. For convenience, /' may be called the image
of t. On examination, Dedekind's proof is found to
postulate as certainties: (i) If there be a t, there is a t',
image of t; (2) if there be two distinct t's, the corre-
sponding t"s are distinct; (3) there is a t; (4) there is a
t which is not a t' ; (5) every / is other than its t' ' . These
being granted, it is easy to see, by supposing each / to be
paired with its t' , as object with image, that the assem-
blage 6 of all the t's and the assemblage 6' of all t"s are
"equivalent." But by (4) there is a / not in 6', which
latter is, therefore, a part of 6. Hence 6 is infinite, by
definition of the term.

THE AXIOM OF INFINITY 159

Let this matter be scrutinised a little. Assuming
only the mentioned postulates and, of course, the pos-
sibility of reflection, it is obvious that by pairing the /
of (4) with its image /', then the latter with its image,
and so on, a sequence S of t's is started which, because
of (i) and (5), is incapable of termination. This 5,
too, by Dedekind's proof, is an infinite assemblage.
Accordingly, postulate (i), without which, be it ob-
served, the proof is impossible, postulates, in advance
of the argument, certainty which, if the argument's
conclusion be true, transcends the finite before the infer-
ence that an infinite exists either is or can be drawn.
The reader may recall how the Russian mathematician
Lobatschewsky said, "In the absence of proof of the
Euclidian postulate of parallels, I will assume that it is
not true"; and how thereupon there arose a new science
of space. Suppose that, in like manner, we say here,
"In the absence of proof that an act once found to be
mentally performable is endlessly so performable, we
will assume that such is not the case," then, whatever
else might result — and of that we shall presently speak
— one thing is at once absolutely certain: Dedekind's
"argument" would be quite impossible. The fact is
that a more beautiful circle than his is hardly to be
found in the pages of fallacious speculation, or admits
of construction by the subtlest instruments of self-
deceiving dialectic, though it must be frankly allowed
that Mr. Russell's l more recent movement about the
same centre is equally round and exquisite.

And this disclosure of the fatal circle in the attempted

demonstration serves at once to introduce and exemplify

the truth of my second contention, which is that all

logical discourse, of necessity, ex vi termini, presupposes

1 Principles of Mathematics^ chap, xliii.

l6o THE AXIOM OF INFINITY

certainty that transcends the finite, where by logical
discourse I mean such as consists of completely deter-
mined concepts welded into a concatenated system by
the ancient hammer of deductive logic. The fact of this
presupposition, of course, cannot be proved, but, and
that is good enough, it can be exhibited and beheld. To
attempt to "prove" it would be to stultify oneself by
assuming the possibility of a deductive argument A
to prove that the conclusion of A cannot be drawn
unless it is assumed in advance. The fact, then, if it
be a fact, and of that there need not be the slightest
doubt, is to be added to that small group of fundamental
simplicities which can at best be seen, if the eye be fit.

Consider, for example, this simplest of syllogistic
forms: Every element e of the class c is an element e
of the class c' ; every e of cf is an element e of the class
c" ; .'. every e of c is an e of c" . I appeal now to the
reader's own subjective experience to witness to the fol-
lowing facts: (i) Our apodictic feeling is the sole justi-
fication of the inference as such; (2) that felt justifica-
tion is absolute, neither seeking nor admitting of appeal;
(3) that sole and absolute justification, namely, the
apodictic feeling, is in no slightest degree contingent
upon the answer to any question whether the multitude
of elements e or e or e is or is not, may or may not be
found to be, " equivalent" to some part of itself. The
feeling of validity here undoubtedly transcends the finite,
undoubtedly holds naught in reserve against any possi-
bility of the inference failing as an act should the system
of elements turn out to be infinite.

At some risk of excessive clearness and accentuation,
for the matter is immeasurably important, I venture
to ask the reader to witness how the transcendence or
transfiniteness of certainty shows itself in yet another

THE AXIOM OF INFINITY l6l

way, not merely in formal deductive inference, but also
in conception. When any concept, as that of Parabola,
for example, is formed or defined, it is found that the
concept contains implicitly a host of properties not
given explicitly in the definition. Properly speaking, the
thing defined is a certain organic assemblage of proper-
ties, of which the totality is implied in a properly se-
lected few of them. Now the act which it is decisive
here to note is that by conception we mean, among
other things, that whenever the definition may present
itself, even though it may be endlessly, a certain in-
variant assemblage of properties implicitly accompanies
the presentation. Without such transfinite certainty
of such invariant uncontingent implication, conception
would be devoid of its meaning.

The upshot, then, is this: that conception and logical
inference alike presuppose absolute certainty that an
act which the mind finds itself capable of performing is
intrinsically performable endlessly, or, what is the same
thing, that the assemblage of possible repetitions of a
once mentally performable act is equivalent to some
proper part of the assemblage. This certainty I name
the Axiom of Infinity, and this axiom being, as seen,
a necessary presupposition of both conception and
deductive inference, every attempt to " demonstrate "
the existence of the infinite is a predestined begging of
the issue.

What follows? Do we, then, know by axiom that the
infinite is? That depends upon your metaphysic. If
you are a radical a-priorist, yes; if not, no. If the latter,
and I am now speaking as an a-priorist, then you are
agnostic in the deepest sense, being capable, in utmost
rigour of the terms, of neither conceiving nor inferring.
But if we do not know the axiom to be true, and so

162 THE AXIOM OF INFINITY

cannot deductively prove the existence of the infinite,
what, then, is the probability of such existence? The
highest yet attained. Why? Because the inductive test
of the axiom, regarded now as a hypothesis, is trying to
conceive and trying to infer, and this experiment, which
has been world-wide for aeons, has seemed to succeed
in countless cases, and to fail in none not explainable on
grounds consistent with the retention of the hypothesis.
Finally, to make briefest application to a single con-
crete case. Do the stars constitute an infinite mul-
titude? No one knows. If the number be finite, that
fact may some time be ascertained by actual enumera-
tion, and, if and only if there be infinite ensembles of
possible repetitions of mental processes, it may also be
known by proof. But if the multitude of stars be in-
finite, that can never be known except by proof; this
last is possible only if the axiom of infinity be true, and
even if this be true, the actual proof may never be
achieved.

THE PERMANENT BASIS OF A LIBERAL
EDUCATION l

Is it possible to find a principle or a set of principles
qualified to serve as a permanent basis for a theory of
liberal education? If so, what is the principle or set
of principles? These are old questions. We are living
in a time when they must be considered anew.

If our world were a static affair, if our environment,
physical, spiritual and institutional, were stable, then
we should none of us have difficulty in agreeing that a
liberal education would be one that gave the student
adjustment and orientation in the world through dis-
ciplining his faculties in their relation to its cardinal
static facts. Such a world could be counted upon.
No one doubts that in such a world it would be possible
to find a permanent basis for a theory of liberal edu-
cation — a principle or a set of principles that would
be adequate and sound, not merely to-day, but to-day,
yesterday and to-morrow.

But we are reminded by certain rather numerous
educational philosophers that our world is not a static
affair. We are told that it is a scene of perpetual
change, of endless and universal transformation — phys-
ical flux, institutional flux, social flux, spiritual flux: all
is flux. These philosophers tell us of the rapid and
continued advancement and multiplication of knowledge.
They do not cease to remind us that knowledge goes

1 Printed in The Columbia University Quarterly, June, 1916.

1 64 PERMANENT BASIS OF LIBERAL EDUCATION

on building itself out, not only in all the old directions,
but also in an endlessly increasing variety of new direc-
tions. They remind us that the ever-augmenting volume
of knowledge is continually breaking up into new divi-
sions or kinds, and that each of these quickly asserts,
and sooner or later demonstrates, its parity with any
other division in respect of utility and dignity and dis-
ciplinary value. They remind us that a striking con-
comitant phenomenon, which is partly the effect of the
multiplication and differentiation of knowledge, partly
a cause of it and partly owing to other agencies and
influences, is the fact that new occupations constantly
spring into being on every hand and that the needs,
the desires and the habits of men, and therewith the
drifts and forms of social and institutional life, suffer
perpetual mutation. Nothing, they tell us, is perma-
nent except change itself. All things, material, mental,
moral, social, institutional, are tossed in an infinite and
endless welter of transformations — evolution, involu-
tion, revolution, all going on at once and forever.

It is evident, we are assured, that in such a world
the search for abiding principles is vain, whether we
seek a permanent basis for a liberal education or a per-
manent basis for anything else. The doctrine is that in
our world permanent bases do not exist. Permanence,
stability, invariance, immutability, there is none. It
exists only in rationalistic dreams. It exists only in
the insubstantial musings of the tender-minded. It
exists only in the cravings of such as have not the prag-
matistic courage or constitution to deal with reality
as it is in the welter and the raw. We are told that
there is in matters educational no such thing as eternal
wisdom. Wisdom is at best a transitory thing, depend-
ing on time and place, and constantly changing with

PERMANENT BASIS OF LIBERAL EDUCATION 165

them. A prescription that is wise to-day will be foolish
to-morrow. What was a, liberal education is not such
now. What is a liberal education to-day will not be
liberal in the future. Greek has gone, theology is gone,
religion is gone, Latin is almost gone, mathematics,
we are told, is going, and so on and on. Each branch
of knowledge will have its day, and then will cease to
be essential. Liberal curricula, it is contended, must
change with the times.

This doctrine, logically conceived and carried out,
means that as the years and generations follow endlessly,
time and change will beget an endless succession of
so-called liberal curricula. It means that, if, in this un-
ending sequence, we observe a finite number of success-
ive curricula, these will indeed be found to resemble each
other, overlapping, interpenetrating, and thus seeming
to be held together in a kind of unity by a more or
less vague and elusive bond; but that this must be
appearance only. For if the observed succession be
prolonged, as it is bound to be, the seeming principle
of unity must become dimmer and dimmer; the terms
or curricula of the endless succession of them can have,
in fact, nothing in common, no lien, no unity whatever,
save that pale variety which serves merely to constitute
the succession of curricula an infinite series of terms.
It is not unlikely that the educational philosophers in
question may not be aware that this is what their
doctrine means. Nevertheless, that is what it does
mean.

Is the doctrine sound? To me it seems not to be so.
The question is a question of fact. The denial of per-
manent principle and the assertion of its concomitant
theory of education seek to justify themselves by point-
ing to the fiuctuance of the world. I do not deny the

1 66 PERMANENT BASIS OF LIBERAL EDUCATION

fluctuance of the world. One must be blind to do that.
Here, there and yonder, in the world of matter, in the
world of mind, in thought, in religion, in morals, in
conventions, in institutions, everywhere are evident
the drif tings and shif tings of events: everywhere
course the hasting streams of change. I admit the
storm and stress, the tumult and hurly-burly of it all.
I do not deny that impermanence is a permanent and
mighty fact in our world. What I do deny is that
impermanence is universal. Its sweep is not clean.
Far from it. If it is, man has indeed been a colossal
fool, for the quest of Constance, the search for invari-
ance, for things that abide, for forms of reality that
are eternal, has been in all times and places the dom-
inant concern of man, uniting his philosophy, his religion,
his science, his art and his jurisprudence into one mani-
fold enterprise of mankind. Not permanence alone,
nor impermanence alone, but the two together, one
of them drawing and the other driving, it is these two
working together that have shaped the course of human
history and moulded the form of its content. I admit
that impermanence is more evident and obtrusive
than permanence, but I contend that a philosophy which
finds in the world nothing but change is a shallow phi-
losophy and false. The instinct that perpetually drives
man to seek the fixed, the stable, the everlasting, has
its root deep in nature. It is a cosmic thing. Must we
say that this instinct, this most imperious of human
cravings, has no function except that of qualifying man
to be eternally mocked? It cannot be admitted. The
sweep of mutation is indeed deep and wide, but it is
not universal. It would be possible, in a contest before
a committee of competent judges, to show that tem-
poralities are, in respect of number, more than matched

PERMANENT BASIS OF LIBERAL EDUCATION 167

by eternalities, and that, in respect of relative impor-
tance, changes are as dancing wavelets on an infinite
and everlasting sea.

In our environment there exist certain great invariant
massive facts that now are and always will be necessary
and sufficient to constitute the basis of a curriculum or
a theory of liberal education. These facts are obvious
and on that account they require to be pointed out,
just because, in the matter of escaping attention, what
is very obvious is a rival of what is obscure.

What are these facts? One of them is the fact that
every human being has behind him an immense human
past, the past of mankind. Of course, I do not mean
that what we call the human past is itself a fixed or
permanent thing. It is not. It is a variable, constantly
changing by virtue of perpetual additions to it as the
years and centuries empty the volume of their events
into that limitless sea. What is permanent is the fact
— it was so yesterday and it will be so to-morrow —
that behind each one of us there is a human past so
immense as to be practically infinite. That fact, I say,
is permanent. It can be counted on. It is as nearly
eternal as the race of man. Out of that past we have
come. Into it we are constantly passing. Meanwhile,
it is of the utmost importance to our lives. It contains
the roots of all we are, and of all we have of wisdom,
of science, of philosophy, of art, of jurisprudence, of
customs and institutions. It contains the record or
ruins of all the experiments that man has made during
a quarter or a half million years in the art of living in
this world. This great stable fact of an immense human
past behind every human being that now is or is to be,
obviously makes it necessary for any theory of liberal
education to provide for discipline in human history

1 68 PERMANENT BASIS OF LIBERAL EDUCATION

and in the literature of antiquity. How much? A
reasonable amount — enough, that is, to orient the stu-
dent in relation to the past, to give him a fair sense of
the continuity of the life of mankind, a decent appre-
ciation of ancient works of genius, and sense and knowl-
edge enough to guide his energies and to control his
enthusiasms in the light of human experience. As the
centuries go by, ancient literature and human history
will increase more and more. What is a reasonable
prescription will, therefore, become less and less in its
relation to the increasing whole, but it will never vanish.
It will never cease to be indispensable.

In this connection, the following question is certain
to be asked. From the point of view of this inquiry,
which aims at indicating an enduring basis for a theory
of liberal education, does it follow that Greek or Latin
or any other language that may be destined to become
" classic" and "dead" at some remote future time, —
does it follow that these or any of them must enter
as essential into the curriculum of a liberal education?
It does not. It would indeed be a grave misfortune if
there should ever come a time when there were no longer
a goodly number of scholars devoted to the great lan-
guages of antiquity. Some of the thought, of the sci-
ence, of the wisdom, of the beauty originally expressed
in these tongues, is, we have said, essential; but it is
precisely the chief function of those who master the
ancient languages to make their precious content avail-
able, through translations and critical commentaries,
for the great body of their fellow men to whom the lan-
guages themselves must remain unknown. It is not
denied that the scholars in question will know and
appreciate such content as no others can, but neither
will these scholars continue forever to deny the possi-

PERMANENT BASIS OF LIBERAL EDUCATION 169

bility of rendering most of the content reasonably well
in the living languages of their fellow men. The con-
trary cannot be much longer maintained. Indeed the
layman already knows that Euclid, Plato, Aristotle,
Aeschylus, Sophocles, Euripides, Demosthenes, Virgil,
Cicero, Lucretius, and many others, have already
learned, or are rapidly learning, to speak, beautifully
and powerfully, all the culture languages of the modern
world.

Another of the massive facts that transcends the flux
of the world, and that, therefore, must contribute basic-'
ally to any permanent theory of liberal education,
is the fact that every human being is encompassed by
a physical or material universe. Again I do not mean,
of course I do not mean, that the universe remains
always the same. What is permanent is the fact that
human beings always have been, now are, and always
will be, surrounded on every hand by an infinite objec-
tive world of matter and force. In that world we are
literally immersed. Our bodies are parts of it; they
are composed of its elements and will be resolved into
them again. If our minds, too, be not part of it, they
must at all events, on pain of our physical incompe-
tence or extinction, gain and maintain continuous and
intelligent relations with it. The great fact in question,
like the fact of the human past, can be counted on.
It survives all vicissitudes. The immersing universe
may be a chaos or a cosmos, or partly chaotic and partly
cosmic, preserving its character in that regard or tending
along an asymptotic path to chaos complete or to
cosmic perfection. But if it is chaotic, we humans
sufficiently match it in that regard to be able to treat
it more and more successfully as if it were an infinite
locus of order and law. And we know that to do this

170 PERMANENT BASIS OF LIBERAL EDUCATION

is immensely advantageous. In a strict sense, it is
absolutely indispensable. Merely to live, it is necessary
to treat nature as having some order.

These considerations show that any theory which
aims to orient and discipline the faculties of men and
women in their relation to the great permanent facts
of the world must make basal provision for discipline
in what we call natural, or physical, science. Again,
how much? Again the answer is, a reasonable amount.
But how much, pray, is that? Enough to give the
student a fair acquaintance with the heroes of natural
science, a fair understanding of what scientific men mean
by natural order or law, a decent insight into scientific
method, the role of hypothesis, and the processes of
experimentation and verification. But there are so
many branches of natural science and their number is
increasing. A liberal curriculum cannot require them
all. Which shall be chosen? It does not matter much.
These branches differ a good deal in content and in a
less degree in method, but they have enough in common
to make a claim of superiority for any one of them
mainly a partisan claim. The spirit of science, its
methods, some of its chief results, these are the essen-
tials. To give these, physics is competent, so is chem-
istry, so is botany, so is zoology, and so on. The choice
is a temporal detail, but the principle requiring the
choice is everlasting. A hundred or a thousand years
hence, there will be other details to choose from — sci-
entific branches not yet named, nor even dreamed of.
But — and this is the point — a theory of liberal edu-
cation will not cease to demand some discipline in
natural science so long as human beings are immersed
in an infinite world of matter and force.

Nor will such a theory fail to take account funda-

PERMANENT BASIS OF LIBERAL EDUCATION 171

mentally of a third great fact that persists despite the
flux of things and the law of death. I refer to the
fact that every human being's fortune depends vitally
upon what may be called the world of ideas. It is
evident that of the total environment of man, the
human Gedankenwelt is a stupendous and mighty com-
ponent. Like the other great components already
named, or namable, the world of ideas is, in respect of
its existence, a permanent datum amid the weltering
sea of change. Not only may it be counted on, but
it must be reckoned with. Some thinking everyone must
do. The formation and combination of ideas is not
merely indispensable to welfare, it is more fundamental
than that: it is essential to human life. The world of
ideas contains countless possibilities that are not actual-
ized or realized or validated or incarnated, as we say,
in the order of the material world, nor in any existing
social or institutional order. It is plain that discipline
in the ways and forms of abstract thinking, of dealing
with ideas as ideas, is essential to a liberal education,
not merely because the world of ideas is itself a thing
of supreme and eternal worth, but because those who
are incapable of constructing ideal orders may not hope
to have the imagination requisite for ascertaining or
for appreciating the frame and order actualized in ex-
ternal nature. From all of this it is clear that any
enduring theory of liberal education must provide for
the discipline of logic and mathematics, for it is in
these and these alone that rigorous or cogent thinking
finds its standard and its realization. It is true that
most of the thinking that the exigencies of life compel
us to do is not cogent thinking. We are obliged con-
stantly to deal with ideas that are too nebulous to
admit of rigorously logical handling. But to argue

172 PERMANENT BASIS OF LIBERAL EDUCATION

that consequently discipline in rigorous thinking is not
essential, is stupid. It is to ignore the value of stand-
ards and ideals. It is, in other words, to be spiritually
blind. I am making no partisan plea for my own sub-
ject. Mathematics happens to be the name that time
has given to rigorous or cogent thinking, and so it
happens that mathematics is the name of the one art
or science that is qualified to give men and women a
perfect standard of thinking and to bring them into the
thrilling presence of indestructible bodies of thought.
Call the science by any other name — • anathematics
or logostetics. The thing itself and its functions would
be the same.

Another cardinal fact among the permanent consid-
erations that a theory of liberal education must rest
upon is the fact that human beings are social beings.
It is only in dreams and romances that a human being
lives apart in isolation. Men, said Aristotle, are made
for co-operation. Every man and every woman is a
born member of a thousand teams. Not one is pure
individual. Each one is many. None can extricate
himself from the generic web of man. This fact sur-
vives the flux. It is as nearly everlasting as the human
race. It is a rock to build upon. And so it was true
yesterday, is true to-day, and will be true to-morrow,
that an education whose function it is to discipline the
faculties of man in their relation to the great abiding
facts of life and the world, must provide for discipline
in the fundamentals of political science. Moreover,
as it is essential to the health and to the effectiveness
of the individual, and also essential to the welfare of
society that men and women be able to express them-
selves acceptably and effectively, a liberal education
will provide for discipline in the greatest of all the arts

PERMANENT BASIS OF LIBERAL EDUCATION 173

— the art of rhetoric. No term has been more abused,
especially by amorphous men of science. Yet the late
Henri Poincare was made a member of the French
Academy, not because he was a great mathematician,
astronomer, physicist and philosopher, but because of
his masterful control of the resources of the French
language as an instrument of human expression.

I have spoken of the invariant fact of the human past.
Its complement is the fact of the human future. That,
too, is a great abiding fact. It is, in practice, to be
treated as eternal, for, if the race of man be doomed
to extinction, then, in that far off event, human edu-
cation itself will cease. Does it follow that a theory of
liberal education must provide for instruction in proph-
ecy? It does follow. But is it not foolish to speak
of instruction in prophecy? For is not prophecy a
thing of the past? Is it not a dead or a dying office
of priests? It is not foolish, it is not a thing of the
past, it is not a dead or dying office of priests. Proph-
ecy is a thing of the present, destined to increase with
the advancement of knowledge. Every department of
study is a department of prophecy. It is the function
of science to foretell. Prophecy is not the opposite of
history, it is history's main function. As W. K. Clif-
ford long ago pointed out, every proposition in physics
or astronomy or chemistry or zoology or mathematics,
or other branch of science, is a rule of conduct facing
the future — a rule saying that, if such-and-such be
true, then such-and-such must be true; if such-and-such
a situation be present, then such-and-such things will
happen; if we do thus-and-thus, then certain statable
consequences may be expected. Foretelling, indeed,
is not the exclusive office of knowledge, for musing,
meditation, pensiveness, pure contemplation, have their

174 PERMANENT BASIS OF LIBERAL EDUCATION

legitimate place; but man is mainly and primarily an
active animal; and in relation to action, the business
of knowledge is prophecy, forecasting what to do and
what to expect.

Finally it remains to mention another fundamental
matter that must contribute in a paramount measure
to any just theory of a liberal education. It is not a
matter strictly co-ordinate with the other matters
mentioned, but it touches them all, penetrates them
all and transfigures them all. I refer to the discipline
of beauty. Beauty is the most vitalizing thing in the
world. It is beauty that makes life worth living and
makes it possible. If, by some fiendish cataclysm,
all the beauty of art and all the beauty of nature were
to be suddenly blotted out, the human race would
quickly perish through depression caused by the ubiq-
uitous presence of ugliness. Does it follow that a
liberal curriculum must provide for the instruction of
every student in all the arts? No. Like the natural
sciences, the arts are enough alike to make any one of
them a representative of them all. Besides, all sub-
jects of study are penetrated with beauty, and any
one of them may be so administered as to enlarge
and refine the sense of what is beautiful in life and
the world.

Such I take to be major considerations among the
great permanent massive facts that together suffice
and are essential to constitute an enduring basis for a
theory of liberal education. Ought discipline to be
prescribed in all the indicated fields? The answer
would seem to be that a liberally educated man or
woman is one who has been instructed in them all. It
follows that there be seekers who are by nature not
qualified to find. But in the case of these, as in the

PERMANENT BASIS OF LIBERAL EDUCATION 175

case of their more gifted fellows, it must be remem-
bered that not the least service a program of liberal
study should render, is that of disclosing to men and
women and to their fellows their respective powers
and limitations.

GRADUATE MATHEMATICAL INSTRUCTION FOR
GRADUATE STUDENTS NOT INTENDING TO
BECOME MATHEMATICIANS1

IN his "Annual Report" under date of November
last, the President of Columbia University speaks in
vigorous terms of what he believes to be the increasing
failure of present-day advanced instruction to fulfil one
of the chief purposes for which institutions of higher
learning are established and maintained.

In the course of an interesting section devoted to
college and university teaching, President Butler says:

A matter that is closely related to poor teaching is found in the grow-
ing tendency of colleges and universities to vocationalize all their instruc-
tion. A given department will plan all its courses of instruction solely
from the point of view of the student who is going to specialize in that
field. It is increasingly difficult for those who have the very proper desire
to gain some real knowledge of a given topic without intending to become
specialists in it. A university department is not well organized and is not
doing its duty until it establishes and maintains at least one strong sub-
stantial university course designed primarily for students of maturity and
power, which course will be an end in itself and will present to those who
take it a general view of the subject-matter of a designated field of knowl-
edge, its methods, its literature and its results. It should be possible for
an advanced student specializing in some other field to gain a general
knowledge of physical problems and processes without becoming a physicist;
or a general knowledge of chemical problems and processes without becoming
a chemist; or a general knowledge of zoological problems and processes
without becoming a zoologist; or a general knowledge of mathematical
problems and processes without becoming a mathematician.

1 An address delivered before Section A of the American Association
for the Advancement of Science, December 30, 1914. Printed in Science,
March 26, 1915.

GRADUATE MATHEMATICAL INSTRUCTION 177

This is a large matter, involving all the cardinal
divisions of knowledge. I have neither time nor com-
petence to deal with it fully or explicitly in all its bear-
ings. As. indicated by the title of this address it is my
intention to confine myself, not indeed exclusively but
in the main, to consideration of the question in its
relation to advanced instruction in mathematics. The
obvious advantages of this restriction will not, I believe,
be counterbalanced by equal disadvantages. For, much
as the principal subjects of university instruction differ
among themselves, it is yet true that as instruments
of education they have a common character and for
their efficacy as such depend fundamentally upon the
same educational principles. A discussion, therefore,
of an important and representative part of the general
question will naturally derive no little of whatever
interest and value it may have from its implicit bearing
upon the whole. It is not indeed my intention to
depend solely upon such implicit bearings nor upon the
representative character of mathematics tojnfetate^ my
opinion respecting the question in its relation to other
subjects. On the contrary, I am going to assume that
specialists in other fields will allow me, as a lay neigh-
bor fairly inclined to minding his own affairs, the priv-
ilege of some quite explicit preliminary remarks upon
the larger question.

I suspect that my interest in the matter is in a meas-
ure temperamental; and my conviction in the premises,
though it is not, I believe, an unreasoned one, may be
somewhat colored by inborn predilection. At all events
I own that a good many years of devotion to one field
of knowledge has not destroyed in me a certain fondness
for avocational studies, for books that deal with large
subjects in large ways, and for men who, uniting the

178 GRADUATE MATHEMATICAL INSTRUCTION

generalist with the specialist in a single gigantic per-
sonality, can show you perspectives, contours and reliefs,
a great subject or a great doctrine in its principal
aspects, in its continental bearings, without first com-
pelling you to survey it pebble by pebble and inch by
inch. I can- not remembej the time when it did not
seem to me to be the very first obligation of universities
to dtiferisK-iftstruction of .the kind that is given Imclr
received in the avocational as distinguished from" the
vocational spirit — the kind of instruction that has for
its^aim7~riot action but understanding, not utilities but
ideas,, not efficiency but enlightenment, not prosperity
but magnanimity. For without intelligence and mag-
nanimity — without light and soul — no form of being
can be noble and every species of conduct is but a kind
of blundering in the night. I could hardly say more
explicitly that I agree heartily and entirely with the
main contention of President Butler's pronouncement.
Indeed I should go a step further than he has gone. He
has said that a university department is not well organ-
ized and is not doing its duty until it establishes and
maintains the kind of instruction I have tried to char-
acterize. To that statement I venture to add explicitly
— what is of course implicit in it — that a university is
not well organized and is not doing its duty until it
makes provision whereby the various departments are
enabled to foster the kind of instruction we are talking
about. That in all major subjects of university instruc-
tion there ought to be given courses designed for stu-
dents of "maturity and power" who, whilst specializing
in one subject or one field, desire to generalize in others,
appears to me to be from every point of view so rea-
sonable and just a proposition that it would not occur
to me to regard it as questionable or debatable were it

GRADUATE MATHEMATICAL INSTRUCTION 179

not for the fact that it actually is questioned and debated
by teachers of eminence and authority.

What is there in the contention about which men may
differ? Dr. Butler has said that there is a "growing
tendency of college and university departments to vo-
cationalize all their instruction." Is the statement
erroneous? It may, I think, be questioned whether the
tendency is growing. I hope it is not. Of course
spaQJalization is not a new thing in the world. It
is faT^ogg^TKajThistory. Le^it be granted that it
is here to stay, for ft^is indispensable to the advance-
ment of knowledge and to the conduct of human affairs.
Every "Tone EnDws — that. There_JjS7 iJowever, some
evidence that speclairzation~is becoming, indeed that it
has become, wiser, less exclusive, more temperate.
The symptoms of what not long ago promised to become
a kind of specialism mania appear to be somewhat less
pronounced. Recognition of the fact that specializa-
tion is in constant peril of becoming so minute and
narrow as to defeat its own ends is now a commonplace
among specialists themselves, many of whom have
learned the lesson through sad experience, others from
observation. Specialists are discoverers. One of our
recent discoveries is the discovery of a very old truth:
we have discovered that no work can be really great
which does not contain some element or touch of the
universal, and that is not exactly a new insight. Leo-
nardo da Vinci says:

We may frankly admit that certain people deceive themselves who
apply the title "a good master" to a painter who can only do the head or
the figure well. Surely it is no great achievement if by studying one thing
only during his whole lifetime he attain to some degree of excellence therein!

The conviction seems to be gaining ground that in
the republic of learning the ideal citizen is neither

l8o GRADUATE MATHEMATICAL INSTRUCTION

the ignorant specialist, however profound he may be,
nor the shallow generalist, however wide the range of
his interest and enlightenment. It is not important,
however, in this connection to ascertain whether the
vocationalizing tendency is at present increasing or de-
creasing or stationary. What is important is to recog-
nize the fact that the tendency, be it waxing or waning,
actually exists, and that it operates in such strength
as practically to exclude all provision for the student
who, if I may so express it, would qualify himself to
gaze into the heavens intelligently without having to
pursue courses designed for none but such as would
emulate a Newton or a Laplace. If any one doubts that
such is the actual state of the case, the remedy is very
simple: let him choose at random a dozen or a score of
the principal universities and examine their bulletins
of instruction in the major fields of knowledge.
h Another element — an extremely important element — •
/ of President Butler's contention is present in the form
of a double assumption : it is assumed that in any uni-
versity community there are serious and capable students
whose primary aim is indeed the winning of mastery
, in a chosen field of knowledge but who at the same time
desire to gain some understanding of other fields —
some intelligence of their enterprises, their genius, their
methods and their achievements; it is further assumed
that this non-vocational or avocational propensity is
legitimate and laudable. Are the assumptions correct?
The latter one involves a question of values and will
1 be dealt .with presently. In respect of the former
we have to do with what mathematicians call an exist-
ence theorem: Do the students described exist? They
do. Can ±he fact be demonstrated — deductively
proved? It can not. How, then, may we know it to be

GRADUATE MATHEMATICAL INSTRUCTION l8l

true? The answer is: partly by observation, partly
by experience, partly by inference and partly by being
candid with ourselves. Who is there among us that is
unwilling to admit that he himself now is or at least
once was a student of the kind? Where is the univer-
sity professor to whom such students have not revealed
themselves as such in conversation? Who is it that has
not learned of their existence through the testimony of
others? No doubt some of us not only have known
students of the kind, but have tried in a measure to
serve them. We may as well be frank. I have myself
for some years offered in my subject a course designed
in large part for students having no vocational interest
in mathematics. I may be permitted to say, for what
the testimony may be worth, that the response has been
good. The attendance has been composed about equally
of students who were not looking forward to a career
in mathematics and of students who were. And this
leads me to say, in passing, that, if the latter students
were asked to explain what value such instruction could
have for them, they would probably answer that it
served to give them some knowledge about a great sub-
ject which they could hardly hope to acquire from
courses designed solely to give knowledge of the subject.
Every one knows that it often is of great advantage to
treat a subject as an object. One of the chief values
of w-dimensional geometry is that it enables us to con-
template ordinary space from the outside, as even those
who have but little imagination can contemplate a
plane because it does not immerse them. Returning
from this digression, permit me to ask: if, without
trying to discover the type of student in question, we
yet become aware, quite casually, that the type actually
exists, is it not legitimate to infer that it is much more

182 GRADUATE MATHEMATICAL INSTRUCTION

numerously represented than is commonly supposed?
And if such students occasionally make their presence
known even when we do not offer them the kind of
instruction to render their wants articulate, is it not
reasonable to infer that the provision of such instruction
would have the effect of revealing them in much greater
numbers?

Indeed it does not seem unreasonable to suppose
that a " strong substantial course" of the kind in ques-
tion, in whatever great subject it were given, would be
attended not only by considerable numbers of regular
students but in a measure also by officers of instruction
in other subjects and even perhaps by other qualified
residents of an academic community. Only the other
day one of my mathematical colleagues said to me
that he would rejoice in an opportunity to attend such
a course in physics. The dean of a great school of law
not long ago expressed the wish that some one might
write a book on mathematics in such a way as would
enable students like himself to learn something of the
innerness of this science, something of its spirit, its
range, its ways, achievements and aspiration. I have
known an eminent professor of economics to join a
beginner's class in analytical geometry. Very recently
one of the major prophets of philosophy declared it to
be his intention to suspend for a season his own special
activity in order to devote himself to acquiring some
knowledge of modern mathematics. Similar instances
abound and might be cited by any one not only at
great length, but in connection with every cardinal
division of knowledge. Their significance is plain.
They are but additional tokens of the fact that the
race of catholic-minded men has not been extinguished
by the reigning specialism of the time, but that among

GRADUATE MATHEMATICAL INSTRUCTION 183

students and scholars there are still to be found those
whose curiosity and intellectual interests surpass all
professional limits and crave instruction more generic
in kind, more liberal, if you please, and ampler in its
scope, than our vocationalized programs afford.

As to the question of values, I maintain that the desire
of such men is entirely legitimate, that it is wholesome
and praiseworthy, that it deserves to be stimulated,
and that universities ought to meet it, if they can.
Indeed, all this seems to me so obvious that I find it a
little difficult to treat it seriously as a question. If the
matter must be debated, let it be debated on worthy
ground. To say, as proponents sometimes say, that,
inasmuch as all knowledge turns out sooner or later to
be useful, students preparing for a given vocation by
specializing in a given field may profitably seek some
general acquaintance with other fields because such
general knowledge will indirectly increase their voca-
tional equipment, is to offer a consideration which,
though in itself it is just enough, yet degrades the dis-
cussion from its appropriate level, which is that of an
ideal humanity, down to the level of mere efficiency and
practicianism. No doubt one engaged in minutely
studying the topography of a given locality because he
intends to reside in it might be plausibly advised to
study also the general geography of the globe on the
ground that his special topographical knowledge would
be thus enhanced, and that, moreover, he might some
time desire to travel. But if we ventured to counsel
him so, he might reply: What you say is true. But
why do you ply me with such low considerations? Why
do you regard me as something crawling on its belly?
Don't you know that I ought to acquire a general knowl-
edge of geography, not primarily because it may be

1 84 GRADUATE MATHEMATICAL INSTRUCTION

useful to me as a resident here or as a possible traveler,
but because such knowledge is essential to me in my
character as a man? The rebuke, if we were fortunately
capable of feeling it, would be well deserved. A man
building a bridge is greater than the engineer; a man
planting seed is greater than the farmer; a man teach-
ing calculus is greater than the mathematician; a man
presiding at a faculty meeting is greater than the dean
or the president. We may as well remember that man
is superior to any of his occupations. His supreme
vocation is not law nor medicine nor theology nor com-
merce nor war nor journalism nor chemistry nor physics
nor mathematics nor literature nor any specific science
or art or activity; it is intelligence, and it is this supreme
vocation of man as man that gives to universities
their supreme obligation. It is unworthy of a university
to conceive of man as if he were created to be the servant
of utilities, trades, professions and careers: these things
are for him: not ends but means. It is said that intel-
ligence is good because it prospers us in our trades,
industries and professions; it ought to be said that these
things are good because and in so far as they prosper
intelligence. Even if we do not conceive the office of
intelligence to be that of contributing to being in its
highest form, which consists in understanding, even if
we conceive its function less nobly as that of enabling
us to adjust ourselves to our environment, the same con-
clusion holds. For what is our environment? Is it
wholly or mainly a matter of sensible circumstance —
sea and land and sky, heat and cold, day and night,
seasons, food, raiment, and the like? Far from it. It
is rather a matter of spiritual circumstances — • ideas,
sentiments, doctrines, sciences, institutions, and arts.
It is in respect of this ever-changing and ever-devel-

GRADUATE MATHEMATICAL INSTRUCTION 185

oping world of spiritual things, it is in respect of this
invisible and intangible environment of life, that uni-
versities, whilst aiming to give mastery in this part or
that, are at the same time under equal obligation to
give to such as can receive it some general orientation
in the whole.

And now as to the question of feasibility. Can the
thing be done? So far as mathematics is concerned I
am confident that it can, and I have a strong lay sus-
picion that it can be done in all other subjects.

It is my main purpose to show, with some regard to
concreteness and detail, that the thing is feasible in
mathematics. Before doing so, however, I desire to
view the matter a little further in its general aspect
and in particular to deal with some of the considera-
tions that tend to deter many scientific specialists from
entering upon the enterprise.

One of the considerations, and one, too, that is often
but little understood, and so leads to wrong impu-
tations of motive, though it is in a sense distinctly
creditable to those who are influenced by it, is the con-
sideration that relates to intricacy and technicality of
subject-matter and doctrine. Every specialist knows
that the principal developments in his branch of science
are too intricate, too technical and too remote from the
threshold of the matter to be accessible to laymen,
whatever their abilities and attainments in foreign
fields. Not only does he know that there is thus but
relatively little of his science which laymen can under-
stand but he knows also that the portions which they
can not understand are in general precisely those of
greatest interest and beauty. And knowing this, he
feels, sometimes very strongly, that were he to endeavor
by means of a lecture course to give laymen a general

1 86 GRADUATE MATHEMATICAL INSTRUCTION

acquaintance with his subject, he could not fail to incur
the guilt of giving them, not merely an inadequate
impression, but an essentially false impression, of the
nature, significance and dignity of a great field of knowl-
edge. His hesitance, therefore, is not due, as it is some-
times thought to be, to indifference or to selfishness.
Rather is it due to a sense of loyalty to truth, to a
sense of veracity, to an unwillingness to mislead or de-
ceive. Of course strange things do sometimes happen,
and it is barely conceivable that once in a long time
nature may, in a sportive mood, produce a kind of
specialist whose subject affects him much as the pos-
session of an apple or a piece of candy affects the boy
who goes round the corner in order to have it all him-
self. But if the type exist, not many men could claim
the odd distinction of belonging to it. Specialists are
as generous and humane as other men. Their subjects
affect them as that same boy is affected when, if he
chance to come suddenly upon some strange kind of
flower or bird, he at once summons his sister or brother
or father or mother or other friend to share in his
surprise and joy. There is this difference, however —
the specialist must, unfortunately, suffer his joy in
solitude unless and until he finds a comrade in kind.
I admit that the deterrent consideration in question is
thoroughly intelligible. I contend that the motive it
involves presents an attractive aspect. But I can not
think it of sufficient weight to be decisive. It involves,
I believe, an erroneous estimate of values, a fallacious
view of the ways of truth to men. A few years ago,
when making a railway journey through one of the most
imposing parts of the Rocky Mountains, I was tempted
like many another passenger to procure some photo-
graphs of the scenery in order to convey to far-away

GRADUATE MATHEMATICAL INSTRUCTION 187

friends some notion of the wonders of it. So far,
however, did the actual scenery surpass the pictures of
it, excellent as these were, that I decided not to buy
them, feeling it were better to convey no impression
at all than to give one so inferior to my own. No
doubt the decision might be defended on the ground of
its motive. Did it not originate in a certain laudable
sense of obligation to truth? Nevertheless, as I am now
convinced, the decision was silly. For in accordance
with the same principle it is plain that I ought to have
wished to have my own impressions erased, seeing that
they must have been quite inferior to those of a widely
experienced mountaineer as those which the pictures
could have given were inferior to mine. Who is so
foolish as, to argue that no one should learn anything
about, say London, unless he means to master all its
plans, its architecture and its history in their every
phase, feature and detail? Who would contend that
because we are permitted to know only so little of
what is happening in the European war, we ought to
remain in total ignorance of it? Who would say that
no_one may with propriety seek to learn "somethmg
about ancjent^Romeiinle^a frfr^js bent^on becoming a
Gibbon/^a^M^msenr It is undoubtedly true that
an endeavor to present a body of doctrine or a science
to such as can not receive it fully must result in giving
a false impression of the truth. But the notion that
such an endeavor is therefore wrong is a notion which,
if consistently and thoroughly carried out, would put
the human mind entirely out of commission. All im-
pressions, all views, all theories, all doctrines, all sciences
are false in the sense of being partial, imperfect, incom-
plete. "II n'y a plus des problemes resolus et d'autres
qui ne le sont pas, il y a seulement des problemes

1 88 GRADUATE MATHEMATICAL INSTRUCTION

plus ou moins resolus," said Henri Poincare. Every
one must see that, but for the helpfulness of views
which because incomplete are also in a measure false,
even the practical conduct of life, not to say the advance-
ment of science, would be impossible. There is no
other choice: either we must subsist upon fragments or
perish.

Again, many a specialist shrinks from trying to pre-
sent his subject to laymen because he looks upon such
activity as a species of what is called popularization of
science, alibr~fae~je1ieveT.that such popularization, even
in its best sense^ clojejy_j^§e^ible£^vulgarization in its
wofsfT^He fancies that there is a~sEarp line bounding
ofFTmowledge that is mere knowledge from knowledge
that is scientific. In his view science is for specialists
and for specialists only. He declines, on something
like moral and esthetic grounds, to engage in what he
calls playing to the gallery. It might, of course, be
said that there is more than one way of playing to the
gallery. It could be said that one way consists in
acting the r61e of one who imagines that his intellectual
interests are so austere and elevated and his thought
so profound that a just sense of the awful dignity of
his vocation imposes upon him, when in presence of
the vulgar multitude, the solemn law of silence. It
would be ungenerous, however, if not unfair, to insist
upon the justice of such a possible retort. Rather let
it be granted, for it is true, that much so-called popu-
larization of science is vicious, relieving the ignorant of
their modesty without relieving them of their ignorance,
equipping them with the vocabulary of knowledge
without its content and so fostering not only a vain and
empty conceit, but a certain facility of speech that is
seemly, impressive and valuable only when, as is too

GRADUATE MATHEMATICAL INSTRUCTION 189

seldom the case, it is accompanied by solid attainments.
To say this, however, is not to lay an indictment against
that kind of scientific popularization which was so
happily illustrated by the very greatest men of antiquity,
which was not disdained even by Galileo in the begin-
nings of modern science nor by Leonardo da Vinci, and
which in our own time has engaged the interest and
skill of such men as Clifford and Helmholtz, Haeckel
and Huxley, Mach, Ostwald, Enriques and Henri Poin-
care. It is not to arraign that variety of popularization
which any one may behold in the constant movement of
ideas, once reserved exclusively for graduate students,
down into undergraduate curricula and which has, for
example, made the doctrine of limits, analytical geom-
etry, projective geometry, and the notions of the deriva-
tive and the integral available for presentation to college
freshmen or even to high-school pupils. It is not to
condemn that kind of popularization which is so nat-
ural a process that it actually goes on in a thousand
ways all about us without our deliberate cooperation,
without our intention or our consent, and has enriched
the common sense and common knowledge of our time
with countless precious elements from among the sci-
entific and philosophic discoveries made by other
generations of men.

Finally it remains to mention the important type of
specialist in whom strongly predominates the predilection
for research as distinguished from exposition. He knows,
as every one knows, that through what is called practical
applications of science many a scientific discovery is
made to serve innumerable human beings who do not
understand it and innumerable others who never can.
He may or may not believe in avocational instruction;
he may or may not regard intelligence as an ultimate

1 90 GRADUATE MATHEMATICAL INSTRUCTION

good and an end in itself; he may or may not think
that the arts and agencies for the dissemination of
knowledge, as distinguished from the discovery and
practical applications of truth, are important; he may
or may not know that the art and the gifts of the great
expositor are as important and as rare as those of the
great investigator and less often owe their success to the
favor of accident or chance. He may not even have
seriously considered these things. He does know his
own predilection; and so strong is his inclination towards
research that for him to engage in exposition, especially
in popular exposition, in avocational instruction for
laymen, would be to sin against the authority of his
vocation. This man, if he have intellectual powers
fairly corresponding to the seeming authority and ur-
gence of his inner call, belongs to a class whose rights
are peculiarly sacred and whose freedom must be guarded
in the interest of all mankind. It is not contended
that every representative of a given subject is under
obligation to expound it for the avocational interest
and enlightenment of laymen. The contention is that
such exposition is so important a service that any uni-
versity department should contain at least one man who
is at once willing and qualified to render it.

I come now to the keeping of my promise. It is to
be shown that the service is practicable in the subject
of mathematics and how it is so. Let us get clearly
in mind the kind of persons for whom the instruction
is to be primarily designed. They are to be students
of "maturity and power"; they do not intend to become
teachers, much less producers, of mathematics; they
are probably specializing in other fields; they do not
aim at becoming mathematicians; their interest in
mathematics is not vocational, it is avocational; it is

GRADUATE MATHEMATICAL INSTRUCTION IQI

the interest of those whose curiosity transcends the
limits of any specific profession or any specific form or
field of activity; each of them knows that, whatever
his own field may be, it is penetrated, overarched, com-
passed about by an infinitely vaster world of human
interests and human achievements; they feel its im-
mense presence, the poignant challenge of it all; as
specialists they will win mastery over a little part, but
they have heard the call to intelligence and are seeking
orientation in the whole; this they know is a thing of
mind; they are aware that the essential environment
of a scholar's life is a spiritual environment — the in-
visible and intangible world of ideas, doctrines, institu-
tions, sciences and arts; they know or they suspect
that one of the great components of that world is mathe-
matics; and so, not as candidates for a profession or a
degree, but in their higher capacity as men and women,
theydesire to learn something of this science viewed as
a^ human enterSnsH^as-fl. hody of human achievements ;
and they are willing to pay the price; they are not seek-
ing entertainment, they are prepared to work — to
listen, to read and to think.

And now we must ask: What measure of mathe-
matical training is to be required of them as a prepara-
tion? In view of what has just been said it is evident
that such training is not to be the whole of their equip-
ment nor even the principal part of it, but it is an
indispensable part. And the question is: How much
mathematical knowledge and mathematical discipline
is to be demanded? I have no desire to minimize my
present task. I, therefore, propose that only so much
mathematical preparation shall be demanded as can
be gained in a year of collegiate study. Most of them
will, of course, have had more; but I propose as a hy-

I Q2 GRADUATE MATHEMATICAL INSTRUCTION

pothesis that the amount named be regarded as an
adequate minimum. But it does not include the differ-
ential and integral calculus. And is it not preposterous
to talk of offering graduate instruction in mathematics
to students who have not had a first course in the
calculus? I am far from thinking so. A little reflec-
tion will suffice to show that in the case of such stu-
dents as I have described it is very far from preposterous.
In my opinion the absurdity would rather lie in demand-
ing the calculus of them. No one is so foolish as to
contend that a first course in the calculus is a sufficient
preparation for undertaking the pursuit of graduate
mathematical study. But to suppose it necessary is
just as foolish as to suppose it sufficient. There was
a time when it was necessary, and the belief that it is
necessary now owes its persistence and currency to the
inertia then acquired. Formerly it was necessary,
because formerly all advanced courses, at least all
initial courses of the kind, were either prolongations of
the calculus, like differential equations, for example,
or else courses in which the calculus played an essential
instrumental role as in rational mechanics, or the usual
introductions to function theory or to higher geometry
or algebra. But, as every mathematician knows, that
time has passed. It is true that courses for which a
preliminary training in the calculus is essential still
constitute and will continue to constitute the major
part of the graduate offer of any department of mathe-
matics. And quite apart from that consideration, it
seems wise, in the case of intending graduate students
who purpose to specialize in mathematics, to enforce
the usual calculus requirement as affording some slight
protection against immaturity and the lack of serious-
ness. But every mathematician knows that it is now

GRADUATE MATHEMATICAL INSTRUCTION 193

practicable to provide a large and diversified body of
genuinely graduate mathematical instruction for which
the calculus is strictly not prerequisite.

Fortunately it is just the material that is thus avail-
able which is in itself best suited for the avocational
instruction we are contemplating. As the calculus is
not to be presupposed it goes without saying that this
subject must find a place in the scheme. For evidently
an advanced mathematical course devised and con-
ducted in the interest of general intelligence can not
be silent respecting "the most powerful weapon of
thought yet devised by the wit of man." Technique
is not sought and can not be given. The subject is
not to be presented as to undergraduates. For the most
part these gain facility with but little comprehension. It
is to be presented to mature and capable students who
seek, not facility, but understanding. Their desire is to
acquire a general conception of the nature of the cal-
culus and of its place in science and the history of
thought — such a conception as will at least enable them
as educated men to mention the subject without a
feeling of sham or to hear it mentioned without a feeling
of shame. A few well-considered lectures should suffice.
At all events it would not require many to show the
historical background of the calculus, to explain the
nascence and nature of the scientific exigencies that
gave it birth, to make clear the concepts of derivative
and integral as the two central notions of its two great
branches, and to present a few simple applications of
these notions to intelligible problems of typical signifi-
cance. Even the idea of a differential equation could
be quickly reached, the nature of a solution explained,
and simple examples given of physical and geometric
interpretations. As to the range and power of the

194 GRADUATE MATHEMATICAL INSTRUCTION

calculus, a sense and insight can be given, in some
measure of course by a reference to its literature, but
much more effectively by a few problems carefully
selected from various fields of science and skillfully
explained with a view to showing wherein the methods
of the calculus are demanded and how they serve. Is
not all this elementary and undergraduate? In point
of nomenclature, yes. It is not necessary, however, to
let words deceive us. We teach whole numbers to
young children, but even Weierstrass was not aware of
the logico-mathematical deeps that underlie cardinal
arithmetic.

The calculus, however, is hardly the topic with which
the course would naturally begin. A principal aim of
the course should be to show what mathematics, in its
inner nature, is — to lay bare its distinctive character^
Its ^distincHve j:harjLct£r, its^atructural nature, is that
of ajiiiyp^n^tico-deju^tiyej^^ystem. Probably, there-
fore, it would be well to begin with an exposition of the
nature and function of postulate systems and of the
great role such systems have always played in the sci-
ence, especially in the illustrious period of Greek mathe-
matics and even more consciously and elaborately in
our own time. It is plain that such an exposition can
be made to yield fundamental insight into many matters
of interest and importance not only in mathematics,
but in logic, in psychology, in philosophy, and in the
methodology of natural science and general thought.
The material is almost superabundant, so numerous
are the postulate systems that have been devised as
foundations for many different branches of geometry,
algebra, analysis, Mengenlehre and logic. A general
survey of these, were it desirable to pass them all in
review, would not be sufficient. It will be necessary

GRADUATE MATHEMATICAL INSTRUCTION 1 95

to select a few systems of typical importance for minute
examination with reference to such capital points as
convenience, simplicity, adequacy, independence, com-
patibility and categoricalness. The necessity and pres-
ence of undefined terms in any and all systems will
afford a suitable opportunity to deal with the highly
important, much neglected and little understood subject
of definition, its nature, varieties and function, in light
of the recent literature, especially the suggestive han-
dling of the matter by Enriques in his " Problems of
Science." A given system once thus examined, the
easy; deduction of a few theorems will suffice to show
the possibility and the process of erecting upon it a
perfectly determinate and often imposing superstructure.
And so will arise clearly the just conception of a mathe-
matical doctrine as a body of thought composed of a
few undefined together with many defined ideas and a
few primitive or postulated propositions with many
demonstrated ones, all concatenated and welded into a
form independent of will and temporal vicissitudes.
Revelation of the charm of the science will have been
begun. A new revelation will result when next the
possibility is shown of so interchanging undefined with
defined ideas and postulates with demonstrated proposi-
tions that, despite such interchange of basal with super-
structural elements, the doctrine as an autonomous
whole will remain absolutely unchanged. But this is
not all nor nearly all. It is only the beginning of what
may be made a veritable apocalypse. Of great interest
to any intellectual man or woman, of very great interest
to students of logic, psychology, or philosophy, should
be the light which it will be possible in this connection
to throw upon the economic r61e of logic and upon the
constitution of mind or the world of thought. I refer

196 GRADUATE MATHEMATICAL INSTRUCTION

especially to the recently discovered fact that in inter-
preting a system of postulates we are not restricted to
a single possibility, but that, on the contrary, such a
system admits in general of a literally endless variety
of interpretations; which means, for such is the make-
up of our Gedankenwelt, that an infinitude of doctrines,
widely different in respect of their psychological char-
acter and interest, have nevertheless a common form,
being isomorphic, as we say, logically one, though
spiritually many, reposing on a single base. And how
foolish the instructor would be not to avail himself of
the opportunity of showing, too, in the same connec-
tion, how various mathematical doctrines that differ
not only psychologically, but logically also, are yet
such that, by virtue of a partial agreement in their
bases, they intersect one another, owning part of their
content jointly, whilst being, in respect of the rest,
mutually exclusive and incompatible. If, for example,
it be some Euclidean system that he has been expound-
ing, he will be able readily to show upon how seemingly
slight changes of base there arise now this or that
variety of non-Euclidean geometry, now a projective or
an inversion geometry or some species or form of higher
dimensionality. I need not say that analogous phe-
nomena will in like manner present themselves in other
mathematical fields. And it is of course obvious that
as various doctrines are thus made to pass along in
deliberate panorama it will be feasible to point out some
of their salient and distinctive features, to indicate
their historic settings, and to cite the more accessible
portions of their respective literatures. Naturally in
this connection and in the atmosphere of such a course
the question will arise as to why it is that, or wherein,
the hypothetico-deductive method fails of universal

GRADUATE MATHEMATICAL INSTRUCTION 197

applicability. So there will be opportunity to teach the
great lesson that this method is not rudimentary, but
is an ideal, the ideal of intellect and science; to teach
that mathematics is but the name of its occasional
realization; and that, though the ideal is, relatively
speaking, but seldom attained, yet its lure is universal,
manifesting itself in the most widely differing domains,
in the physical and mechanical assumptions of Newton,
in the ethical postulates of Spinoza, in our federal con-
stitution, even in the ten commandments, in every field
where men have sought a body of principles to serve
them as a basis of doctrine, conduct or achievement.
And if it shall thus appear that mathematics is very
high-placed as being, in respect of its method and its
form, the ideal and the lure of thought in general, the
fault must be imputed, not to the instructor, but to the
nature of things.

In all this study of the postulational method the
impression will be gained that the science of mathe-
matics consists of a large and increasing number of
more or less independent, somewhat closely related
and often interpenetrating branches, constituting, not a
jungle, but rather an immense, diversified, beautifully
ordered forest; and that impression is just. At the
same time another impression will be gained, namely,
that the various branches rest, each of them, upon a
foundation of its own. This impression will have to be
corrected. It will have to be shown that the branch-
foundations are not really fundamental in the science
but are literally and genuinely component parts of the
superstructure. It will have to be shown that mathe-
matics as a whole, as a single unitary body of doctrine,
rests upon a basis of primitive ideas and primitive
propositions that lie far below the so-called branch-

198 GRADUATE MATHEMATICAL INSTRUCTION

foundations and, in supporting the whole, support
these as parts. The course will, therefore, turn to the
task of acquainting its students with those strictly
fundamental researches which we associate with such
names as C. S. Peirce, Schroeder, Peano, Frege, Russell,
Whitehead and others, and which have resulted in
building underneath the traditional science a logico-
mathematical sub-structure that is, philosophically,
the most important of modern mathematical develop-
ments.

It must not be supposed, however, that the instruc-
tion must needs be, nor that it should preferably be,
confined to questions of postulate and foundation,
and I will devote the remainder of the time at my
disposal to indicating briefly how, as it seems to me,
a large or even a major part of the course may
concern itself with matters more traditional and more
concrete.

Any one can see that there is an abundance of avail-
able material. There is, for example, the history and
significance of the great concept of function, a concept
which mathematics has but slowly extracted and grad-
ually refined from out the common content and experi-
ence of all minds and which on that account can be
not only defined precisely and intelligibly to such lay-
men as are here concerned, but can also be clarified in
many of its forms by means of manifold examples drawn
from elementary mathematics, from the elements of
other sciences, and from the most familiar phenomena
of the work-a-day world.

Another-aYailable topic is the nature and role of the
sovereign notion of limit. This, too, as every mathe-
matician knows, -admits of countless illustration and
application within the radius of mathematical knowl-

GRADUATE MATHEMATICAL INSTRUCTION 199

edge here presupposed. In this connection the structure
and importance of what Sylvester called "the Grand
Continuum," which so many scientific and other folk
talk about unintelligently, will offer itself for explanation.
And if the class fortunately contain students of phil-
osophic mind, they will be edified and a little aston-
ished perhaps when they are led to see that the method
and the concept of limits are but mathematicized forms
of a process and notion familiar in all domains of
spiritual activity and known as idealization. Not
improbably some of the students will be sufficiently
enterprising to trace the mentioned similitude in
some of its manifestations in natural science, in psy-
chology, in philosophy, in jurisprudence, in literature
and in art.

I have not mentioned the modern doctrine variously
known as Mengenlehre, Mannigfaltigkeitslehre, the theory
of point-sets, assemblages, manifolds,, ^or aggregates :
a live and growing doctrine in which expert and layman
are about equally interested and which, like a subtle and
illuminating ether, is more and more pervading mathe-
matics in all its branches. For the_ avocational in-
struction of lay students of ^"maturity and power" how
rich a body of material is here, with all its fascinating
distinctions of discrete and ' continuous, finite and in-
finite, v denumerable - and non-denumerable, orderless,
ordered, and well-orderedT^nor^wTth its teeming host
of near-lying propositions, so Interesting, so illuminating,
often so amazing.

Finally, but far from exhausting the list, it remains
to mention the great subjects of invariants and groups.
Both of them admit of definition perfectly intelligible to
disciplined laymen; both admit of endless elementary
illustration, of having their mutual relations simply

200 GRADUATE MATHEMATICAL INSTRUCTION

exemplified, of Joeing shown in historic perspective,
and of being strikingly connected, especially the notion
of invariance, with the dominant enterprise of man:
his ceaseless quest for the changeless amid the turmoil
and transformation of the cosmic flux.

THE SOURCE AND FUNCTIONS OF A
UNIVERSITY1

IN returning hither from near and far to join in cele-
brating the seventy-fifth anniversary of the founding of
their academic birthplace and home, the alumni, the
sons and daughters of this institution, have not come to
congratulate an eld-worn mother upon the continuance
of her years beyond the Psalmist's allotment of three
score and ten nor to comfort her in the sorrows of age.
Their assembling is due to other sentiments and owns
another mood. They have come as beneficiaries in
order to pay, for themselves and for the many absent
ones whom they have the honor to represent, a tribute
of gratitude, loyalty and love to a noble benefactress
who, notwithstanding her wisdom and fame, yet is
literally in the early morning of her life. For it is not
written, nor ordained in the scheme of things, that, in
respect of years, the life of a university shall be as a
tale that is told or a watch in the night. It is indeed a
living demonstration of the greatness of man, bearing
witness to his superiority even over death, that men and
women, though they themselves must die, yet may,
whilst they live, create ideals and institutions that
survive. A college or a university may indeed have
been as a benignant mother to a thousand academic

1 An address delivered June 3, 1914, at the celebration of the seventy-
fifth anniversary of the founding of the University of Missouri. Printed
in The Columbia University Quarterly, March, 1915.

202 SOURCE AND FUNCTIONS OF A UNIVERSITY

generations and yet be younger than her youngest
child. Unlike man the individual, a university is, like
man the race, immortal. The age of three score and
fifteen in the life of an immortal institution is a mere
beginning. In emphasizing this consideration it is not
my intention to suggest or imply that the services ren-
dered by the University of Missouri have necessarily
been, because of her youth, meagre or ineffectual or
immature. On the contrary I maintain that her serv-
ices to the people of this state have been beyond com-
putation and that already her spiritual achievements
constitute the chief glory of a great commonwealth.
Is it the alumni only who owe her grateful allegiance?
Is the beneficence of an institution of learning exclu-
sively or even mainly confined to the relatively few who
dwell for a season in her immediate presence, who touch
the hem of her garment, come into personal contact
with her scholars and teachers and receive her degrees?
Far from it. Far from being the sole or the principal
beneficiaries of a university, the alumni are simply
among the more potent instrumentalities for extending
her ministrations to ever wider and wider circles. The
sun, we say, is far off yonder in the heavens. But
strictly speaking the sun really is wherever he shines.
Where is the University of Missouri? At Columbia, we
say, and the speech is convenient. But it is juster to
say that, owing to the pervasiveness of her light and
inspiration, the University of Missouri in a measure
now is, and in larger and larger measure will come to
be, in every home and school, in every factory and field,
in every mine and shop, in every council chamber, in
every office of charity, or medicine, or law, in all the
places near or remote where within the borders of this
beautiful state children play and men and women

SOURCE AND FUNCTIONS OF A UNIVERSITY 203

think and love, suffer and hope, aspire and toil. Nay,
by the researches and publications of her scholars and
by the migrations of those she has inspired and dis-
ciplined, the University of Missouri to-day lives and
moves abroad, mingling her presence with that of kin-
dred agencies, not only in every state of the union but
in many other quarters of the civilized world.

It is not my purpose to review the history of her
aspirations and struggles nor to relate the thrilling
story of her triumphs. I conceive that the central
motive of our assembling here is not so much to praise
the University for what she has already accomplished as
to renew our devotion to her high emprize, to congratu-
late her upon her solid attainments, to rejoice in her
divine discontent and spirit of progressiveness, to deepen
and enlarge our conception of her mission and destiny,
and especially to remind ourselves of the principles,
the faith and, above all, the ideals to which she owes
her birth, her continuity, her responsibilities, and her
power.

What is a university? How shall we conceive that
marvelous thing which, though having a local habitation
and a name and seeming to dwell in houses made by
human hands, yet contrives to be omnipresent; per-
vading the abodes of men everywhere throughout a
state, a nation or a world, like a divine ether; subtly,
gently, unceasingly, increasingly ministering to their
hearts and minds healing counsels and the mysterious
grace of light and understanding? What is it? Is it
something, an agency or an influence, that can be
denned? We know that it is not. We know that the
really great things of the world, the things that live
and grow and shine, the things that give to life its
interests and its worth, one and all elude formulation.

204 SOURCE AND FUNCTIONS OF A UNIVERSITY

Yet it is just these things, beauty and love, poetry and
thought, religion and truth and mind, it is precisely
these great indefinables of life that we may learn,
through experience and discipline, to know best of all.
And so it is with what we mean or ought to mean by
a university. What a university is no one can define,
but all may in a measure come to know. By pon-
dering its principles, by contemplating its ideals, by
examining its aims, activities and fruits, above all by
sharing in its spirit and aspiration, we may at length
win a conception of it that will fill our minds with light
and our hearts with devotion.

Where such a conception reigns a university will
flourish. But there is no conception more difficult for
a people to acquire. It is not a spontaneous growth,
springing up like a weed, but requires careful planting
and cultivation. Such is the husbandry to which a
university must perpetually devote itself as the essen-
tial precondition to the prosperous exercise and advance-
ment of all its other functions, and the husbandry is
not easy. Especially in our American communities
where universities must appeal for support to the in-
telligence of a democratic people, there is no service
more important or more difficult to render than that
which consists in teaching us to know what a university
really is and what it signifies alike for developing the
material resources of the world and for the spiritualizing
of man. And thus there devolves upon a university,
especially in the beginning of its career, the necessity
of performing a kind of miracle: without adequate
support, either material or moral, it must yet find
strength to teach us to give it both. The lesson is
one that takes long and long to teach because it is one
that takes long and long to learn.

SOURCE AND FUNCTIONS OF A UNIVERSITY 205

It is a great mistake to imagine that a university is
an essentially modern thing. In spirit, in idea and
essence, it is modern only in the sense in which forces
and ideals that are eternal are always modern, as they
are always ancient. We should not forget that even
the name University — so suggestive of the infinite
world which it is the aim of these institutions to sub-
jugate to the understanding and uses of man — even
the name, in its modern scholastic sense, has had a
history of more than a thousand years. But we know
that the institution itself, the thing that bears the name,
owns an antiquity far more remote. A few years ago,
standing upon the Acropolis of Athens, gazing pensively
about upon the hallowed scene where culminated the
genius of the ancient world, a friend, pointing towards
the spot near by where for fifty years Plato taught in
the grove of Academe, said to me, yonder, yonder
is the holy ground where was made the first attempt
to organize higher education in the western world. The
remark, which was just enough, was indeed impressive. It
is easy, however, to misunderstand its significance and
to exaggerate its importance. So many of the most
precious elements of our civilization trace their lineage
back to the creative activity of ancient Greece that we
are naturally tempted to imagine we may find there
also the source and origin of those aims, activities and
ideals which constitute what we today call a univer-
sity. Such imagining, however, is vain. The originals,
the first organizations, we may possibly find there at a
definite time and place, but not the origin, not the
source, not the nascence of the birth-giving and life-
sustaining power. For this must account, not only for the
universities of our time, but for the great school of Plato
as well. What, then, and where is the secret spring?

206 SOURCE AND FUNCTIONS OF A UNIVERSITY

Shall we seek it in a sense of need? Necessity is
indeed a keen spur to invention and is the mother of
many things. But necessity is not the mother of uni-
versities. The beasts flourish and propagate their kind
without the help of institutions of learning, and without
such help a similar existence is possible to men. Uni-
versities are not essential to life nor to animal pros-
perity. They are not creatures, they are creators,
of need. We do indeed nowadays hear much of the
services they render, and it is right that we should, for
they minister constantly and everywhere to countless
forms of need. But the needs they supply are in the
main needs that they have first produced, multiplied
desires and aspirations, new propensions of mind awak-
ened to new life, lifted by education to higher levels
and ampler possibilities of being. No, the origin, the
source we are seeking, the principle of explanation, is
no human contrivance nor institution nor sense of need.
It is that sovereign urgency, at once so strange and so
familiar, that drives us to seek it; it is the lure of
wisdom and understanding, of beauty and light; a
certain divine energy in the world, at once a cosmic
force and a human faculty, constituting man divine
in constituting him a seeker of truth and a lover of
harmony and illumination.

Has it an epoch and a name? It has both. In
accordance with the modern doctrine of evolution the
greatest events upon our planet occurred long before
the beginnings of recorded history. For according to
that doctrine there must have come a time, long, long
ago, when in what was a world of matter there began
to be mind, in what was a world of motion there began
to be emotion, and the blind dominion of force was
invaded by personality. Among all those marvels of

SOURCE AND FUNCTIONS OF A UNIVERSITY 207

prehistoric history, the supreme event was that one but
for which this world had been a world devoid of mystery
and devoid of truth — I mean the advent of Wonder.
With the advent of wonder came the sense of mystery,
the lure of truth, the sheen of ideality, the dream of the
perfect and, with these, the potence and promise of
research and creativeness with all their endless progeny
of knowledge and wisdom and science and art and
philosophy and religion. These things, children of the
spirit, offspring of wonder, these things are the interests
which it is the divine prerogative of universities to serve,
and the universities ultimately derive their own exist-
ence, their sustenance and their power from the same
mother that gives their charges birth. A_ genuine_uni-
versity is thus the offspring and the appointed agent of
the spirit of inquiry; it is the offspring, expression and"
servant of that imperious curiosity which in a. measure
impels ~~all men and women, but with an urgency like
destiny~literally drives^men and women of genius, to

^

seek tJXlEQw anoTTo teacfi to their jellows whatsoever
is worthy in all that has been discovered or thought,
spoken or done in the world, and at the same time
seeks to extend the empire of understanding endlessly
in all directions throughout the infinite, domain of the
yet uncharted and unknown. That high commission
is at once a university's charter of freedom and the
definition of her functions and her obligations. These
are, on the one hand, to teach — to teach with
no restrictions save those prescribed by decency and
candor — and, on the other hand, to foster and
prosecute research — research in any and all subjects
or fields to which the leading or the stress of
curiosity may draw or impel. In so far as the
great commonwealth of Missouri makes ample pro-

208 SOURCE AND FUNCTIONS OF A UNIVERSITY

vision for the exercise of these functions and for the
discharge of these obligations, to that extent she
may be said to cooperate with the divine energy
of the world in the maintenance of a genuine uni-
versity.

RESEARCH IN AMERICAN UNIVERSITIES1

THE present writer has been asked to deal briefly
with the question of research in American universities.
The subject is an immense one, and the following dis-
cussion makes no pretense of being exhaustive. It
aims merely to present the problem again, to emphasize
again its importance, and to point out once more some
of its harder conditions and some of the principles and
distinctions involved in any serious attempt at its
solution.

The problem may not be easy to appreciate, but it
is at all events easy to state. It is the problem of
securing in our universities suitable provision for the
work of research or investigation and productivity. For
a generation the great majority of the ablest men in
our universities have regarded that problem as the
most urgent and important educational problem con-
fronting these institutions and the American people.
Meanwhile, something has been done towards a solu-
tion. But none of the universities has secured ade-
quate provision, and the majority of them but little or
none at all. In the abstract, the problem is simple and
the solution is easy: given a body of able and enthu-
siastic men, provide them with proper facilities, afford
them opportunity to devote their powers continuously
to the prosecution of research, and the thing is done.
But in the concrete it is exceedingly difficult, being

1 Printed in The Bookman, May, 1906.

210 RESEARCH IN AMERICAN UNIVERSITIES

frightfully complicated with our whole institutional
history and life, in particular with our educational
traditions and tendencies, with the prevailing plan of
university organisation, and especially with the char-
acteristic temper, ideals and ambitions of the American
people.

Somebody besides our foreign friends and critics
ought to tell the truth about American education and
American universities. Our people have never ceased
to believe in education. Our belief has not always been
intelligent. We have been prone to ascribe to educa-
tion efficacies and potencies that do not belong to any
human agency or institution. But our faith in it,
though not always critical or enlightened, has been deep,
implicit and abiding; and we have diligently pursued
it, generally as a means no doubt, but sometimes as
an end, and occasionally as a thing in itself more pre-
cious than power and gold. In all this we have been,
quite unconsciously and contrary to all appearances,
very humble. We have been content to educate our-
selves with knowledge discovered by others and to
nourish ourselves with doctrines and truths produced
only by the spiritual activity of other lands. We may
have been vain but we have not been proud. Besides
a marvelous practical sense we have had, in degree
quite unsurpassed, two of the elements of genius, -
intellectual energy and intellectual audacity; and by
means of these we have created a material civilisation
so obtrusive, so elaborate and so efficient as to amaze
the world. But now at length there begin to appear
the indicia of change, of change for the better. A new
day has dawned. The sun is not yet risen high, but
it is rising. We have begun to suspect that genuine
civilisation is essentially an affair of the spirit, that it

RESEARCH IN AMERICAN UNIVERSITIES 211

can not be borrowed nor imported nor improvised nor
appropriated from without, but that it is a growth from
within, an efflorescence of mind and soul, and that its
highest tokens are not soldiers but savants, not the
purchasers and admirers of art but artists, not mere
retailers of knowledge nor teachers of the familiar and
the known, but discoverers of the unknown, not mere
inventors but men of science. And so we have begun
to feel our way towards the establishment of true
universities, that is to say of institutional centres for
the activity of the human spirit, and of organs, the
most potent yet invented by human society, for giving
effect to the noblest instinct of man, "the civilisation-
producing instinct of truth for truth's sake."

Just here we encounter a great danger. For a gen-
eration our progress in the matter has been so swift
that both the universities themselves and the edu-
cated public opinion upon which in our democratic
society their support and advancement ultimately de-
pend, are in danger of greatly overestimating it, and
that would be a misfortune. Absolutely the progress
has indeed been great, but relatively and judged by the
very highest standards, it has not. It is not first nor
mainly a question of achievements, of things done. It
is a quesion of ideals, of standards and aspirations. A
clear concept of a great university unconsciously serving
the highest interests of man by absolute devotion to
Truth for its own sake and without extraneous motive,
end or aim, does not yet exist in the mind of the Amer-
ican public and is not yet incarnate in any of its institu-
tions. Our universities are young, strong and robust.
They are full of potence and promise. But they have
not yet impressed their own imperfect ideals upon the
people; they have not yet given forth the light neces-

212 RESEARCH IN AMERICAN UNIVERSITIES

sary for their own proper beholding and appreciation.
Their perfections and their imperfections alike, remain
obscure. The old colleges about which as about nuclei
some of our universities have been formed have done
much to leaven and temper the American mind and to
subdue it to the influences of beauty and truth. Cor-
responding services have not yet been rendered by our
universities as such. No one can doubt that they are
destined to assume in future the permanent leadership,
and to exercise a controlling formative influence, in all
that goes to deepen thought and to exalt and refine
standards, character, and taste. At present, however,
they are themselves in the formative and impressionable
stage, resembling improvisations in some respects; and
to understand them, to see clearly both what they are
and what they are not, it is necessary to regard them
as being at the present time less the producers than
the products of our civilisation.

So regarded, they are seen to embody and to reflect
alike the merits and the defects of their progenitor.
Like the latter they are unsurpassed in boldness, in
energy and in enthusiasm, and their genius has been
mainly directed to material and outer ends. Their
first and chief concern has been with the physical and
exterior, with buildings and grounds and instruments
and laboratories, and while their material equipment is
still far from adequate, it has already evoked astonished
and admiring commentary from visiting scholars of
European seats of learning. Like the civilisation whence
they have sprung, our universities are intensely modern
and up-to-date, and they are intensely democratic in
everything but management; they set great store by
organisation, exalt the function of administration, and
tend to be regarded, to regard themselves, and in fact

RESEARCH IN AMERICAN UNIVERSITIES 213

to be, as vast and complicate machines or industrial
plants naturally demanding the control of centralised
authority. They have but little sentiment; they are
almost devoid of sacred and hallowing traditions, of
great and illustrious recollections; there is in and about
them nothing or but little of "the shadow and the hush
of a haunted past." They have no antiquity. In them
the utilitarian spirit, having learned the lingo of service,
contrives to receive an ample share of honour, and the
Genius of Industry that has transformed our land into
an abode of wealth and for generations assigned an
attainable upper limit to a people's aspiration, shapes
educational policy, holds and wields the balance of
power. The classic distinctions of good, better and best
in subjects and motives of study receive but slight re-
gard. The traditional hierarchy of educational values
and the ascending scale of spiritual worths have fallen
into disrepute. All things have been leveled up or
leveled down to a common level; so that the workshop
and the laboratory, schools of engineering, of agri-
culture and of the classics, the library, the model dairy
and departments of architecture and music, exist side
by side. In at least one institution, so it is reported,
the professor of poetry rubs shoulders with the pro-
fessor of poultry. No wonder that a distinguished
critic has said that some of our biggest universities
appear as hardly more than episodes in the wondrous
maelstrom of our industrial life.

Thus it appears that the American university, child
of a predominantly material and industrial civilisa-
tion half-blindly aspiring to higher things, strikingly
resembles its parent. Begotten in the hope that it
would be as a saviour and rescue us from our national
idols and respectable sins, it straightway became their

214 RESEARCH IN AMERICAN UNIVERSITIES

most enlightened servant and lent them the sanction
and the support of its honoured name. It is by no
means contended that this fact is the whole truth.
Our universities are not entirely devoted to the service
of industry; they are not wholly committed to teaching
youth the known from utilitarian motives and for imme-
diate and practical ends; they are not exclusively
concerned with the applications of science; out of gen-
eral devotion to the Useful, something is saved for the
True; science is not always regarded as a commodity;
the judgment of the great Jacobi is sometimes recog-
nised as just: "The unique end of science is the honour
of the human spirit." And it is a pleasure to be able
to proclaim the fact that in a few of our universities
something like a home has been provided for the spirit
of research and that by its activity there, American
genius has had a share in extending the empire of light,
in enlarging the domain of the known, in astronomy, in
physics, in mathematics, in the science of mind, in biol-
ogy, in criticism, in economics, in letters, in almost all
of the great fields where the instinct of truth for the
sake of truth contends against the dark. In this clear
evidence of our growing freedom and exaltation, let us
rejoice; but let us be candid also. Let us admit that
we have only begun the higher service of the soul; let
us confess in becoming humility that, in comparison
with our wealth, our numbers, our energies and our
talents, in comparison, too, with the intellectual achieve-
ments of some other peoples and other lands, the service
we have rendered to Science and Art and Truth is
meagre.

Why such emptiness, such poverty, such meagreness
in the fruits of the highest activity? The immediate
cause is easy to find. It is not incompetence nor lack

RESEARCH IN AMERICAN UNIVERSITIES 21$

of genius in our university faculties. These are not
inferior to the best in the world. It is not mainly due,
as is often said, to inadequacy of material compensa-
tion, though one of the greatest of living physicists,
Professor J. J. Thompson, has told us truly that Amer-
ican men of science receive less remuneration than their
colleagues in any other part of the world. The cause
in question is simple: lack of opportunity. The diffi-
culty is near at hand. It inheres in the composition
and organisation of our universities. Most of these are
built about and upon, and largely consist of, immense
undergraduate schools thronged by young men mainly
bent upon practical aims and neither qualified nor
intending to qualify for the work of investigation. The
interests of these schools are naturally the paramount
concern. The great and growing burdens of adminis-
tration tend to distribute themselves among the pro-
fessors. These have, besides, to give the most and the
best of their energies to elementary teaching, to teach-
ing, that is, which does not pertain to a university
proper but to gymnasia and lycees — a worthy, impor-
tant, necessary kind of work, but a kind that drains
off the energy in non-productive channels and tends to
form and harden the mind of those engaged in it about
a small group of simpler ideas. What is left, what can
be left, of spirit, of energy, of opportunity, for the
arduous work of research? One man attempting the
enterprise of three: administration, elementary teach-
ing, discovery and creative work. Who can suitably
characterise the absurdity? Who can compute the
wickedness of the waste in the impossible attempt to
effect daily the demanded transition from mood to mood?
A mind, by prolonged effort, at length immersed in
the depths of a profound and difficult investigation —

2l6 RESEARCH IN AMERICAN UNIVERSITIES

how poignant the pain of interruption, the rending of
continuity, the rude disturbance of poise and concentra-
tion. How easy to fail of due respect for, because it
is so easy not to understand, the creative mood, obliv-
ious to the outer world, the brooding " maternity of
mind," more delicate than fabric of gossamer, of infinite
subtlety, of infinite sensitiveness, a woven psychic struc-
ture finer than ether threads; and how easy to forget
that a sudden alien call may disturb and jar and even
destroy the structure.

Little excuse, then, have we to wonder at the recent
words of Professor Bjerknes, of the chair of mechanics
and mathematical physics in the University of Stock-
holm, and non-resident lecturer in mathematical physics
in Columbia University, who, in his farewell address
to his American colleagues, assembled to do him honour,
spoke substantially as follows:

"I have been much impressed with the material equipment of your uni-
versities, with your splendid buildings, with the fine instruments you have
placed in them, and with the enthusiasm of the men I have found at work
there. But I hope you will pardon me, gentlemen, for saying, as I must say,
that, when I found you attempting serious investigation with the remnants
of energy left after your excessive teaching and administrative work, I
could not help thinking you did not appreciate the fact that the finest
instruments in those buildings are your brains. I heard one of you counsel
his colleagues to care for the astronomical instruments lest these become
strained and cease to give true results. Allow me to substitute brain for
telescope, and to exhort you to care for your brains. I have been aston-
ished to find that some of you, in addition to much executive work, teach
from ten to fifteen and even more hours per week. I myself teach two hours
per week, and I can assure you that, if I had been required to do so much
of it as you do, you never would have invited me to lecture here in a diffi-
cult branch of science. That, gentlemen, is the most important message I
can leave with you."

Such, then, is the situation. No need that we should
behold it in picture drawn by foreign hand. We need
no copy. The original lies before us in all its proper-

RESEARCH IN AMERICAN UNIVERSITIES 217

tions. The challenge addresses itself at once to our pride
and to our practical sense. Of all peoples, we, it would
seem, should feel the challenge most keenly, for the
problem is a problem in freedom. It demands the
emancipation of American genius; it demands pro-
vision of free and ample opportunity for the highest
activity of our highest talent.

Hope of solution lies in division of labour. Our uni-
versities and the people they represent must reduce
their exactions. For three men's work, three must be
provided. There must be men to administer and men
to teach and men to investigate. Three varieties of
service, entirely compatible in kind, entirely incompat-
ible as co-ordinate vocations combined in one. Any
one of them may be as an avocation to another of the
three, but only so of choice and not by compulsion.
No invidious comparisons are implied. The distinctions
are not of greater and less; they are matters of economy
in the domain of mind. The great administrator is not
a clerk nor an amanuensis; he is a man of constructive
genius, a creator. The great teacher is not a pedagogue;
he is a source of inspiration and of aspiration, produc-
ing children of the spirit by "the urge and ardor" of
a deep and rich and enlightened personality; he was
in the mind of Goethe when he said of Winckelmann
that "from him you learned nothing, but you became
something." And the great investigator is not a mere
collector and recorder of facts; he is a discoverer, a dis-
closer, of the harmonies and the invariance hid beneath
the surface of seeming disorder and of ceaseless change.
The three great powers are compatible, and are usually
found united in a single gigantic personality, just as the
ordinary administrator and ordinary teacher and ordi-
nary investigator compose one unit of mediocrity.

2l8 RESEARCH IN AMERICAN UNIVERSITIES

It is perfectly evident that the total service demanded
of the universities will not diminish. On the contrary,
it will continue as now to increase in response to grow-
ing need. The case, then, is clear: the number of
servants must be increased, the number of those who
are to do the work must be greatly multiplied. And
thus the problem becomes a financial one. But a uni-
versity is not a money-making institution. Its function
is to convert the physical into the spiritual, to transform
the things of matter into the things of mind. It has,
however, a physical body, without which it may not
dwell among men; and, for the support of it, it depends
and must depend, whether through legislative appro-
priation or the benefaction of individuals, ultimately
upon the people. These now possess the means in
ample measure, and the promptings of generosity are
in the hearts of many wealthy and sagacious men.

And so the problem revolves upon itself and once
more turns full upon us its theoretic aspect. Its solu-
tion awaits public appreciation of its significance and
its terms. It is above all else a question of enlighten-
ment. Just here, if I am not mistaken, is the measure-
less opportunity of the university president. Beyond
all others, he is spokesman and representative before
the people of their highest spiritual interests. Their
ideals and aspirations will scarcely surpass his own.
The problem must be conceived boldly in truth and
presented in its larger aspects. It must be seen and be
felt to be the supreme problem of our civilisation. As
a people we have yet to learn the lesson deeply that
research, the competent application in any field what-
ever of human interest of any effective method whatever
for the discovery of truth and enlarging the bounds of
knowledge, is the highest form of human activity. We

RESEARCH IN AMERICAN UNIVERSITIES 2IQ

have yet to learn that a nation, a state, a university
without investigators, is a community without men of
profoundest conviction. For this can not be gained by
conning books; it can not be inherited; it is not merely
a pious hope or a pleasing superstition. It is not an
obsession.

As Helmholz has said, a teacher "who desires to
give his hearers a perfect conviction of the truth of his
principles must, first of all, know from his own experi-
ence how conviction is acquired and how not. He
must have known how to acquire conviction where no
predecessor had been before him — that is, he must
have worked at the confines of knowledge and have con-
quered new regions." We have yet to learn that the
value of a university professor can not be estimated by
counting the hours he stands before his classes. We
have yet to learn to prefer standards of quality to units
of quantity. We have yet to learn that the spirit of
pure research, the highest productive genius, has no
direct concern whatever with the useful; that, while
it does without intention create an atmosphere in which
utilities most greatly flourish, it is itself concerned solely
with the true; we have yet to learn that "the action
of faculty is imperious and always excludes the reflec-
tion why it acts." When these and kindred lessons
shall have been taken to heart, our emancipation, now
well begun, will advance towards completion; the Amer-
ican university will come to its own; and our present
civilisation will speedily pass to the rank of the highest
and best.

PRINCIPIA MATHEMATICA l

MATHEMATICIANS, many philosophers, logicians and
physicists, and a large number of other people are aware
of the fact that mathematical activity, like the activity
in numerous other fields of study and research, has been
in large part for a century distinctively and increasingly
critical. Every one has heard of a critical movement
in mathematics and of certain mathematicians distin-
guished for their insistence upon precision and logical
cogency. Under the influence of the critical spirit of
the time mathematicians, having inherited the tradi-
tional belief that the human mind can know some propo-
sitions to be true, convinced that mathematics may
not contain any false propositions, and nevertheless
rinding that numerous so-called mathematical proposi-
tions were certainly not true, began to re-examine the
existing body of what was called mathematics with a
view to purging it of the false and of thus putting an
end to what, rightly viewed, was a kind of scientific
scandal. Their aim was truth, not the whole truth,
but nothing but truth. And the aim was consistent
with the traditional faith which they inherited. They
believed that there were such things as self-evident prop-
ositions, known as axioms. They believed that the
traditional logic, come down from Aristotle, was an
absolutely perfect machinery for ascertaining what was
involved in the axioms. At this stage, therefore, they

1 An account of Messrs. Whitehead and Russell's great work bearing
this title. Printed in Science, vol. XXV.

PRINCIPIA MATHEMATICA 221

believed that, in order that a given branch of mathe-
matics should contain truth and nothing but truth, it
was sufficient to find the appropriate axioms and then,
by the engine of deductive logic, to explicate their mean-
ing or content. To be sure, one might have trouble
to "find" the axioms and in the matter of explication
one might be an imperfect engineer; but by trying hard
enough all difficulties could be surmounted for the
axioms existed and the engine was perfect. But mathe-
maticians were destined not to remain long in this
comfortable position. The critical demon is a restless
and relentless demon; and, having brought them thus
far, it soon drove them far beyond. It was discovered
that an axiom of a given set could be replaced by its
contradictory and that the consequences of the new set
stood all the experiential tests of truth just as well as
did the consequences of the old set, that is, perfectly.
Thus belief in the self-evidence of axioms received a
fatal blow. For why regard a proposition self-evident
when its contradictory would work just as well? But
if we do not know that our axioms are true, what about
their consequences? Logic gives us these, but as to
their being true or false, it is indifferent and silent.

Thus mathematics has acquired a certain modesty.
The critical mathematician has abandoned the search
for truth. He no longer flatters himself that his proposi-
tions are or can be known to him or to any other human
being to be true; and he contents himself with aiming
at the correct, or the consistent. The distinction is
not annulled nor even blurred by the reflection that
consistency contains immanently a kind of truth. He
is not absolutely certain, but he believes profoundly
that it is possible to find various sets of a few proposi-
tions each such that the propositions of each set are

222 PRINCIPIA MATHEMATICA

compatible, that the propositions of such a set imply
other propositions, and that the latter can be deduced
from the former with certainty. That is to say, he
believes that there are systems of coherent or consist-
ent propositions, and he regards it his business to dis-
cover such systems. Any such system is a branch of
mathematics. Any branch contains two sets of ideas
(as subject matter, a third set of ideas are used but
are not part of the subject matter) and two sets of
propositions (as subject matter, a third set being used
without being part of the subject): that is, any branch
contains a set of ideas that are adopted without defini-
tion and a set that are defined in terms of the others;
and a set of propositions adopted without proof and
called assumptions or principles or postulates or axioms
(but not as true or as self-evident) and a set deduced
from the former. A system of postulates for a given
branch of mathematics — a variety of systems may be
found for a same branch — is often called the founda-
tion of that branch. And that is what the layman
should think when, as occasionally happens, he meets
an allusion to the foundation of the theory of the real
variable, or to the foundation of Euclidean geometry
or of projective geometry or of Mengenlehre or of some
other branch of mathematics. The founding, in the sense
indicated, of various distinct branches of mathematics
is one of the great outcomes of a century of critical
activity in the science. It has engaged and still en-
gages the best efforts of men of genius and men of
talent. Such activity is commonly described as funda-
mental. It is very important, but fundamental in a
strict sense it is not. For one no sooner examines the
foundations that have been found for various mathemat-
ical branches and thereby as well as otherwise gains

PRINCIPIA MATHEMATICA 223

a deep conviction that these branches are constituents
of something different from any one of them and dif-
ferent from the mere sum or collection of all of them
than the question supervenes whether it may not be
possible to discover a foundation for mathematics itself
such that the above-indicated branch foundations would
be seen to be, not fundamental to the science itself,
but a genuine part of the superstructure. That ques-
tion and the attempt to answer it are fundamental
strictly. The question was forced upon mathematicians
not only by developments within the traditional field
of mathematics, but also independently from develop-
ments in a field long regarded as alien to mathematics,
namely, the field of symbolic logic. The emancipation
of logic from the yoke of Aristotle very much resembles
the emancipation of geometry from the bondage of
Euclid; and, by its subsequent growth and diversifica-
tion, logic, less abundantly perhaps but not less cer-
tainly than geometry, has illustrated the blessings of
freedom. When modern logic began to learn from such
a man as Leibnitz (who with the most magnificent
expectations devoted much of his life to researches in
the subject) the immense advantage of the systematic
use of symbols, it soon appeared that logic could state
many of its propositions in symbolic form, that it could
prove some of these, and that the demonstration could
be conducted and expressed in the language of symbols.
Evidently such a logic looked like mathematics and
acted like it. Why not call it mathematics? Evidently
it differed from mathematics in neither spirit nor form.
If it differed at all, it was in respect of content. But
where was the decree that the content of mathematics
should be restricted to this or that, as number or space?
No one could find it. If traditional mathematics could

224 PRINCIPIA MATHEMATICA

state and prove propositions about number and space,
about relations of numbers and of space configurations,
about classes of numbers and of geometric entities,
modern logic began to prove propositions about proposi-
tions, relations and classes, regardless of whether such
propositions, relations and classes have to do with
number and space or with no matter what other spe-
cific kind of subject. At the same time what was
admittedly mathematics was by virtue of its own inner
developments transcending its traditional limitations
to number and space. The situation was unmistakable:
traditional mathematics began to look like a genuine
part of logic and no longer like a separate something to
which another thing called logic applied. And so modern
logicians by their own researches were forced to ask a
question, which under a thin disguise is essentially the
same as that propounded by the bolder ones among the
critical mathematicians, namely, is it not possible to
discover for logic a foundation that will at the same
time serve as a foundation for mathematics as a whole
and thus render unnecessary (and strictly impossible)
separate foundations for separate mathematical branches?
It is to answer that great question that Messrs.
Whitehead and Russell have written "Principia Mathe-
matica" — a work consisting of four large volumes,
the first and second being in hand, the third soon to
appear — and the answer is affirmative. The thesis
is: it is possible to discover a small number of ideas
(to be called primitive ideas) such that all the other
ideas in logic (including mathematics) shall be defin-
able in terms of them, and a small number of propo-
sitions (to be called primitive propositions) such that
all other propositions in logic (including mathematics)
can be demonstrated by means of them. Of course,

PR1NC1PIA MATHEMATICA 22$

not all ideas can be defined — some must be assumed
as a working stock — and those called primitive are
so called merely because they are taken without defini-
tion; similarly for propositions, not all can be proved,
and those called primitive are so called because they are
assumed. It is not contended by the authors (as it was
by Leibnitz) that there exist ideas and propositions that
are absolutely primitive in a metaphysical sense or in
the nature of things; nor do they contend that but one
sufficient set of primitives (in their sense of the term)
can be discovered. In view of the immeasurable wealth
of ideas and propositions that enter logic and mathe-
matics, the authors' thesis is very imposing; and their
work borrows some of its impressiveness from the mag-
nificence of the undertaking. It is important to observe
that the thesis is not a thesis of logic or of mathematics,
but is a thesis about logic and mathematics. It can
not be proved syllogistically; the only available method
is that by which one proves that one can jump through
a hoop, namely, by actually jumping through it. If
the thesis be true, the only way to establish it as such
is to produce the required primitives and then to show
their adequacy by actually erecting upon them as a
basis the superstructure of logic (and mathematics) to
such a point of development that any competent judge of
such architecture will say: "Enough! I am convinced.
You have proved your thesis by actually performing
the deed that the thesis asserts to be possible."

And such is the method the authors have employed.
The labor involved — or shall we call it austere and
exalted play? — was immense. They had predecessors,
including themselves. Among their earlier works Rus-
sell's " Principles of Mathematics" and Whitehead's
" Universal Algebra" are known to many. The related

226 PRINCIPIA MATHEMATICA

works of their predecessors and contemporaries, modern
critical mathematicians and modern logicians, Weier-
strass, Cantor, Boole, Peano, Schroder, Peirce and many
others, including their own former selves, had to be
digested, assimilated and transcended. All this was
done, in the course of more than a score of years; and
the work before us is a noble monument to the authors'
persistence, energy, acumen and idealism. A people
capable of such a work is neither crawling on its belly
nor completely saturated with commercialism nor wholly
philistine. There are preliminary explanations in ordi-
nary language and summaries and other explanations
are given in ordinary language here and there through-
out the book, but the work proper is all in symbolic
form. Theoretically the use of symbols is not necessary.
] A sufficiently powerful god could have dispensed with
them, but unless he were a divine spendthrift, he
would not have done so, except perhaps for the reason
that whatever is feasible should be done at least once
in order to complete the possible history of the world.
But whilst the employment of symbols is theoretically
dispensable, it is, for man, practically indispensable.
Many of the results in the work before us could not
have been found without the help of symbols, and even
if they could have been thus found, their expression in
ordinary speech, besides being often unintelligible, owing
to complexity and involution, would have required at
least fifteen large volumes instead of four. Fortunately
the symbology is both interesting and fairly easy to
master. The difficulty inheres in the subject itself.

The initial chapter, devoted to preliminary explana-
tions that any one capable of nice thinking may read
with pleasure and profit, is followed by a chapter of
30 pages dealing with "the theory of logical types."

PRINCIPIA MATHEMATICA 227

Mr. Russell has dealt with the same matter in volume
30 of the American Journal of Mathematics (1908).
One may or may not judge the theory to be sound or
adequate or necessary and yet not fail to find in the
chapter setting it forth both an excellent example of
analytic and constructive thinking and a worthy model
of exposition. The theory, which, however, is recom-
mended by other considerations, originated in a desire
to exclude from logic automatically by means of its
principles what are called illegitimate totalities and
therewith a subtle variety of contradiction and vicious
circle fallacy that, owing their presence to the non-
exclusion of such totalities, have always infected logic
and justified skepticism as to the ultimate soundness
of all discourse, however seemingly rigorous. (Such
theoretic skepticism may persist anyhow, on other
grounds.) Perhaps the most obvious example of an
illegitimate totality is the so-called class of all classes.
Its illegitimacy may be shown as follows. If A is a
class (say that of men) and E is a member of it, we
say, E is an A. Now let W be the class of all classes
such that no one of them is a member of itself. Then,
whatever class x may be, to say that x is a W is equiva-
lent to saying that x is not an x, and hence to say that W
is a W is equivalent to saying that W is not a W! Such
illegitimate totalities (and the fallacies they breed) are
in general exceedingly sly, insinuating themselves under
an endless variety of most specious disguises, and that,
not only in the theory of classes but also in connection
with every species of logical subject-matter, as proposi-
tions, relations and prepositional functions. As the
prepositional function — any expression containing a
real (as distinguished from an apparent) variable and
yielding either non-sense or else a proposition whenever

228 PRINCIPIA MATHEMATICA

the variable is replaced by a constant term — is the
basis of our authors' work, their theory of logical types
is fundamentally a theory of types of propositional
functions. It can not be set forth here nor in fewer
pages than the authors have devoted to it. Suffice it to
say that the theory presents propositional functions as
constituting a summitless hierarchy of types such that
the functions of a given type make up a legitimate
totality; and that, in the light of the theory, truth and
falsehood present themselves each in the form of a
systematic ambiguity, the quality of being true (or
false) admitting of distinctions in respect of order, level
above level, without a summit. When Epimenides,
the Cretan, says that all statements of Cretans are
false, and you reply that then his statement is false,
the significance of " false" here presents two orders or
levels; and logic must by its machinery automatically
prevent the possibility of confusing them.

Next follows a chapter of 20 pages, which all phi-
losophers, logicians and grammarians ought to study,
a chapter treating of Incomplete Symbols wherein by
ingenious analysis it is shown that the ubiquitous expres-
sions of the form "the so and so" (the "the" being
singular, as "the author of Waverley," "the sine of a,"
"the Athenian who drank hemlock," etc.) do not of
themselves denote anything, though they have con-
textual significance essential to discourse, essential in
particular to the significance of identity, which, in the
world of discourse, takes the form of "a is the so and
so" and not the form of the triviality, a is a.

After the introduction of 88 pages, we reach the work
proper (so far as it is contained in the Volume I.),
namely, Part I.: Mathematical Logic. Here enuncia-
tion of primitives is followed by series after series of

PRINCIPIA MATHEMATICA 2 29

theorems and demonstrations, marching through 578
pages, all matter being clad in symbolic garb, except
that the continuity is interrupted here and there by
summaries and explanations in ordinary language.
Logic it is called and logic it is, the logic of propositions
and functions and classes and relations, by far the
greatest (not merely the biggest) logic that our planet
has produced, so much that is new in matter and in
manner; but it is also mathematics, a prolegomenon to
the science, yet itself mathematics in the most genuine
sense, differing from other parts of the science only in
the respects that it surpasses these in fundamentally,
generality and precision, and lacks traditionally. Few
will read it, but all will feel its effect, for behind it is
the urgence and push of a magnificent past: two thousand
five hundred years of record and yet longer tradition of
human endeavor to think aright.

Owing to the vast number, the great variety and the
mechanical delicacy of the symbols employed, errors
of type are not entirely avoidable and Volume II. opens
with a rather long list of "errata to Volume I." The
second volume is composed of three grand divisions:
Part III., which deals with cardinal arithmetic; Part IV.,
which is devoted to what is called relation-arithmetic;
and Part V., which treats of series. The theory of types,
which is presented in Volume I., is very important in the
arithmetic of cardinals, especially in the matter of
existence-theorems, and for the convenience of the
reader Part III. is prefaced with explanations of how
this theory applies to the matter in hand. In the initial
section of this part we find the definition and logical
properties of cardinal numbers, the definition of car-
dinal number being the one that is due to Frege, namely,
the cardinal number of a class C is the class of all classes

230 PRINCIPIA MATHEMAT1CA

similar to C, where by " similar" is meant that two
classes are similar when and only when the elements
of either can be associated in a one-to-one way with
the elements of the other. This section consists of
seven chapters dealing respectively with elementary
properties of cardinals; o and i and 2; cardinals of
assigned types; homogeneous cardinals; ascending
cardinals; descending cardinals; and cardinals of rela-
tional types. Then follows a section treating of addi-
tion, multiplication and exponentiation, where the
logical muse handles such themes as the arithmetical
sum of two classes and of two cardinals; double simi-
larity; the arithmetical sum of a class of classes; the
arithmetical product of two classes and of two cardinals;
next, of a class of classes; multiplicative classes and
arithmetical classes; exponentiation; greater and less.
Thus no less than 186 large symbolically compacted
pages deal with properties common to finite and infinite
classes and to the corresponding numbers. Nevertheless
finites and infinites do differ in many important re-
spects, and as many as 116 pages are required to present
such differences under such captions as arithmetical
substitution and uniform formal numbers; subtraction;
inductive cardinals; intervals; progressions; Aleph
null, Ko; reflexive classes and cardinals; the axiom of
infinity; and typically indefinite inductive cardinals.

As indicating the fundamental character of the "Prin-
cipia" it is noteworthy that the arithmetic of relations
is not begun earlier than page 301 of the second huge
volume. In this division the subject of thought is
relations including relations between relations. If RI
and R2 are two relations and if F\ and F2 are their
respective fields (composed of the things between which
the relations subsist), it may happen that FI and F*

PRINCIPIA MATHEMATICA 23!

can be so correlated that, if any two terms of FI have
the relation RI, their correlates in F2 have the relation
R2, and vice versa. If such is the case, RI and R2 are
said to be like or to be ordinally similar. Likeness of
relations is analogous to similarity of classes, and, as
cardinal number of classes is denned by means of class
similarity, so relation-number of relations is denned by
means of relation likeness. And 209 pages are devoted
to the fundamentals of relation arithmetic, the chief
headings of the treatment being ordinal similarity and
relation-numbers; internal transformation of a rela-
tion; ordinal similarity; definition and elementary
properties of relation-numbers; the relation-numbers,
on 2r and i5; relation-numbers of assigned types; homo-
geneous relation-numbers; addition of relations and the
product of two relations; the sum of two relations;
addition of a term to a relation; the sum of the rela-
tions of a field; relations of mutually exclusive rela-
tions; double likeness; relations of relations of couples;
the product of two relations; the multiplication and
exponentiation of relations; and so on.

The last 259 pages of the volume deal with series. A
large initial section is concerned with such properties
as are common to all series whatsoever. From this
exceedingly high and tenuous atmosphere, the reader is
conducted to the level of sections, segments, stretches and
derivatives of series. The volume closes with 58 pages
devoted to convergence, and the limits of functions.

To judge the "Principia," as some are wont to do,
as an attempt to furnish methods for developing exist-
ing branches of mathematics, is manifestly unfair; for
it is no such attempt. It is an attempt to show that
the entire body of mathematical doctrine is deducible
from a small number of assumed ideas and propositions.

232 PRINCIPIA MATHEMATICA

As such it is a most important contribution to the theory
of the unity of mathematics and of the compendence
of knowledge in general. As a work of constructive
criticism it has never been surpassed. To every one and
especially to philosophers and men of natural science,
it is an amazing revelation of how the familiar terms
with which they deal plunge their roots far into the dark-
ness beneath the surface of common sense. It is a
noble monument to the critical spirit of science and to
the idealism of our time.

CONCERNING MULTIPLE INTERPRETATIONS
OF POSTULATE SYSTEMS AND THE
"EXISTENCE" OF HYPERSPACE *

WHAT do we mean when we speak of w-dimensional
space and w-dimensional geometry, where n is greater
than 3? The question refers to talk about space and
geometry that are w-dimensional in points, for ordinary
space, as is well known, is 4-dimensional in lines, 4-di-
mensional in spheres, 5-dimensional in flat line-pencils, 6-
dimensional in circles, etc., and there is naturally no
mystery involved in speaking of these latter varieties
of multi-dimensional manifolds and their geometries,
no matter how high the dimensionality may be. No
mystery for the reason that in these geometries every-
thing lies within the domain of intuition in the same
sense in which everything in ordinary (point) geometry
lies in that domain. In other words, these w-dimen-
sional geometries are nothing but theories or geometries
of ordinary space, that arise when we take for element,
not the point, but some other entity, as the line or the
sphere, . . . whose determination in ordinary space
requires more than 3 independent data. Of these
varieties of w-dimensional geometry, the inventor was
Julius Pliicker (d. 1868), but Plucker declined to con-
cern himself with spaces and geometries of more than
four dimensions in points.

1 Printed in The Journal of Philosophy, Psychology and Scientific Method,
May 8, 1913.

234 INTERPRETATIONS OF POSTULATE SYSTEMS

Since Pliicker's time, however, such hyper-theories
of points have invaded not only almost every branch
of pure mathematics, but also — strangely enough -
certain branches of physical science, as, for example,
the kinetic theory of gases. As to the manner of this
latter invasion a hint may be instructive. Given N gas
molecules enclosed, say, in a sphere. These molecules
are, it is supposed, flying about hither and thither, all
of them in motion. Each of them depends on six co-
ordinates, x, y, z, u, v, w, where x, y, z, are the usual
positional coordinates of the molecule regarded as a
point in ordinary space, and u, v, w are the components
of the molecule's velocity along the three coordinate
axes. Knowing the six things about a given molecule,
we know where it is and the direction and rate of its
going. The N molecules making up the gas depend on
6N coordinates. At any instant these have definite
values. These values together define the " state" of
the gas at that instant. Now these 6N values are said
to determine a point in space of 6N dimensions. Thus
is set up a one-one correspondence between such points
and the varying gas states. As the state of the gas
qhanges, the corresponding point generates a locus in
the space of 6N dimensions. In this way the behavior
or history of the gas gets geometrically represented by
loci in the hyperspace in question.

Is such geometric w-dimensional phraseology merely
a geometric way of speaking about non-spatial things?
Even if there exists a space, Sn, one may employ the
language appropriate to the geometry of the space
without having the slightest reference to it, and, indeed,
without knowing or even enquiring whether it exists.
This use of geometric speech in discourse about non-
spatial things is not only possible, but in fact very com-

INTERPRETATIONS OF POSTULATE SYSTEMS 235

mon. An easily accessible example of it may be found
in Bocher1 where, in speaking of a set of values of n in-
dependent variables as a point in space of n dimensions
the reader is told that the author's use of geometric
language for the expression of algebraic facts is due to
certain advantages of that language compared with the
language of algebra or of analysis; he is told that the
geometric terms will be employed "in a wholly conven-
tional algebraic sense" and that "we do not propose
even to raise the question whether in any geometric
sense there is such a thing as space of more than three
dimensions."

It is held by many, including perhaps the majority
of mathematicians, that there are no hyperspaces of
points and that n- dimensional geometries are, rightly
speaking, not geometries at all, but that the facts
dealt with in such so-called geometries are nothing but
algebraic or analytic or numeric facts expressed in
geometric language. If this opinion be correct, then
the extensive and growing application of geometric
language to analytical theories of higher dimensionality
indicates a high superiority of geometric over analytic
speech, and it becomes a problem for psychology to
ascertain whether the mentioned superiority is ade-
quate to explain the phenomenon in question and, if
it be adequate, to show wherein the superiority resides.

No doubt geometric language has a kind of esthetic
value that is lacking in the speech of analysis, for the
former, being transfused with the rich reminiscences
of sensibility, constantly awakens a delightful sense,
as thinking proceeds, of the colors, forms, and motions
of the sensuous world. This is an emotional value. No
doubt, too, geometric language has, in its distinctive

1 "Introduction to Higher Algebra," page 9.

236 INTERPRETATIONS OF POSTULATE SYSTEMS

conciseness, an economic superiority, as when, for ex-
ample, one speaks of the points of the 4-dimensional
sphere, x2- + y* + z2 + w* = r2, instead of speaking of the
various systems of values of the variables x, y, z, w that
satisfy the equation x2 + . . . = r2. Additional advan-
tages of geometric over analytic speech are brought to
light in the following remarks by Poincare in his ad-
dress, "L'Avenir des Mathematiques " (1908):

"Un grand avantage de la geometric, c'est precisement
que les sens y peuvent venir au secours de 1'intelligence,
et aident a deviner la route a suivre, et bien des esprits
preferent ramener les problemes d'analyse a la forme
geometrique. Malheureusement nos sens ne peuvent
nous mener bien loin, et ils nous faussent compagnie
des que nous voulons nous envoler en dehors des trois
dimensions classiques. Est-ce a dire que, sortis de ce
domaine restreint ou ils semblent vouloir nous enfermer,
nous ne devons plus compter que sur 1'analyse pure et
que toute geometric a plus de trois dimensions est vaine
et sans objet? Dans la generation qui nous a precedes,
les plus grands maitres auraient repondu 'oui'; nous
sommes anjourd'hui tellement familiarises avec cette
notion que nous pouvons en parler, meme dans un cours
d'universite, sans provoquer trop d'etonnement.

"Mais a quoi peut-elle servir? II est aise de le voir:
elle nous donne d'abord un langage tres commode, qui
exprime en termes tres concis ce que le langage analytique
ordinaire dirait en phrases prolixes. De plus, ce langage
nous fait nommer du meme nom ce qui se ressemble et
affirme des analogies qu'il ne nous permet plus d'oublier.
H nous permet done cenore de nous diriger dans cet
espace qui est trop grand pour nous et que nous ne
pouvons voir, en nous rappelant sans cesse Pespace
visible qui n'en est qu'une image imparfaite sans doute,

INTERPRETATIONS OF POSTULATE SYSTEMS 237

mais que en est encore une image. Ici encore, comme
dans tous les exemples precedents, c'est Panalogie avec
ce qui est simple qui nous permet de comprendre ce qui
est complexe."

The question of determining the comparative advan-
tages and disadvantages of the languages of geometry
and analysis is a very difficult one. It is evidently in
the main a psychological problem. It appears that no
serious and systematic attempt has ever been made to
solve it. Here, it seems, is an inviting opportunity for
a properly qualified psychologist, it being understood
that proper qualification would include a familiar knowl-
edge of the languages in question. The interest and
manifold utility of such a study are obvious. In the
course of such an investigation it would probably be
found that the superiority of geometric over analytic
speech is alone sufficient to account for the extensive
and rapidly increasing literature of what is called n-
dimensional geometry and that, in order to account for
the rise of such literature, it is therefore not necessary
to suppose the existence of w-dimensional spaces, Sn,
the facts dealt with in the literature being, it could be
supposed, nothing but analytic facts expressed in geo-
metric language.

If such a result were found, would it follow that Sn
does not exist and that consequently w-dimensional geom-
etry must be nothing but analysis in geometric garb?
The answer is, no; for we may and we often do assign
an adequate cause of a phenomenon or event without
assigning the actual cause; and so the possibility would
remain that ^-dimensional geometry has an appropriate
object or subject, namely, a space Sn, which, though
without sensuous existence, yet has every kind of exist-
ence that may warrantably be attributed to ordinary

238 INTERPRETATIONS OF POSTULATE SYSTEMS

geometric space, S3. For this last, though it is imitated
by (or imitates) sensible space, as an ideal model or
pattern is imitated by (or imitates) an imperfect copy,
it is not identical with it. Sz is not tactile space, nor
visual space, nor that of muscular sensation, nor the
space of any other sense, nor of all the senses — it is
a conceptual space; and whether there are or are not
spaces £4, £5, etc., which have every sort of existence
rightly attributable to ordinary geometric space, S3, and
which differ from the latter only in the accident of
dimensionality and in the further accident that S3 ap-
pears in the role of an ideal prototype for an actual
sensible space, whilst 6*4, £5, etc., do not present such
an appearance, — that is the question which remains
for consideration.

A friend called at my study, and, finding me at work,
asked, "What are you doing?" My reply was: "I am
trying to tell how a world which probably does not exist
would look if it did." I had been at work on a chapter
of what is called 4-dimensional geometry. The incident
occurred ten years ago. The reply to my friend no
longer represents my conviction. Subsequent reflection
has convinced me that a space, Sn, of four or more
dimensions has every kind of existence that may be
rightly ascribed to the space, 53, of ordinary geometry.

The following paragraphs present — merely in out-
line, for space is lacking for a minute presentation — the
considerations that have led me to the conclusion above
stated.

Let sensible space be denoted by sS3. We know that
sSs is discontinuous (in the mathematical sense of the
term) and that it is irrational. By saying that it is
irrational I mean what common experience as well as
the results of experimental psychology prove: that

INTERPRETATIONS OF POSTULATE SYSTEMS 239

three sensible extensions of a same type, let us for
definiteness say three sensible lengths, 11} k, /3, may be
such that

(i) h = k,h= /3, l\ * k-

Because sS3 is thus irrational, because it is radically
infected with such contradictions as (i), this space is
not, and can not be, the subject or object of geometry,
for geometry is rational; it does not admit three such
extensions as those in (i). Not only do such contra-
dictions as (i) render sSs impossible as a subject or
object of geometry, but, when encountered, they pro-
duce intolerable intellectual pain — nay, if they could
not in somewise be transcended or overcome, they would
produce intellectual death, for, unless the law of non-
contradiction be preserved, concatentative thinking, the
life of intellect, must cease. In case of intellect we may
say that its struggle for existence is a struggle against
contradictions. But mere existence is not the character-
istic aim or aspiration of intellect. Its aim, its aspira-
tion, its joy, is compatibility. Indeed, intellect seems
to be controlled by two forces, a vis a tergo and a vis
a fronte: it is driven by discord and drawn by concord.
Intellect is a perpetual suitor, the object of the suit
being harmony, the beautiful daughter of the muses.
Its perpetual enemy is the immortal demon of discord,
ever being overcome, but never vanquished.

The victory of intellect over the characteristic con-
tradictions inherent in sSs is won through what we call
conception. That is to say that either we find or else we
create another kind of space which, in order to distin-
guish it nominally and symbolically from sS3, we may
call conceptual space, and denote by cS3. Unlike sS^
cSs is mathematically continuous and it is rational. Like
sSz, cSs is extended, it has room, but the room and

240 INTERPRETATIONS OF POSTULATE SYSTEMS

the extensions are not sensible, they are conceptual; and
these extensions are such that, if /i, k, 13 be three amounts
of a given type of extension, as length, say, and if /i
= k and k = k, then /i = /3. The space cSs, whether we
regard it as found by the intellect or as created by it,
is the subject or object of geometry. The current
vulgar confusion of sS3 and cS3 is doubtless due to the
fact that the former imitates the latter, or the latter
the former, as a sensible thing imitates its ideal, or as
an ideal (of a sensible thing) may be said to imitate
that thing; for it is precisely such alternative or mutual
imitation that enables us in a measure to control the
sensible world through its conceptual counterpart; and
so the exigencies of practical affairs and the fact that
reciprocally imitating things each reminds us of the
other cooperate to cause the sensible and the ideal, the
perceptual and the conceptual, to mingle constantly
and to become confused in that part of our mental life
that belongs to the sensible and the conceptual worlds
of three dimensions. Nevertheless, it is a fact to be
borne in mind that cSz is a subject or object of geom-
etry and that sS3 is not.

Now, in order to construct the geometry in question,
we start with a suitable system of postulates or axioms
expressing certain relations among what are called the
elements of cSs. These postulates, together with such
propositions as are deducible from them, constitute the
geometry of cSz. I shall call it pure geometry, for a
reason to be given later, and shall denote it by pG3.
For definiteness let us refer to the famous and familiar
postulates of Hilbert. Any other system would do as
well. In the Hilbert system, the elements are called
points, lines, and planes. It is customary and just to
point out that the terms point, line, and plane are not

INTERPRETATIONS OF POSTULATE SYSTEMS 241

defined, and in critical commentary it is customary to
add:

(A) That, consequently, these terms may be taken to
be the names of any things whatsoever with the single
restriction that the things must satisfy the relations
stated by the postulates;

(B) That, when some admissible or possible interpre-
tation / has been given to the element-names, the postu-
lates P together with their deducible consequences C
constitute a definite theory or doctrine D;

(C) That replacing 7 by a different interpretation /'
produces no change whatever in D;

(D) That this invariant D is Euclidean geometry
of three dimensions; and

(£) That, if we are to speak of D as a theory or geom-
etry of a space, this space is nothing but the ensemble
of any kind of things that may serve for an interpreta-
tion of P.

That the view expressed in that so-called " critical
commentary" does not agree with common sense or
with traditional usage is obvious. That it will not bear
critical reflection can, I believe, be made evident. Let
us examine it a little. In order to avoid the prejudicial
associations of the terms point, line, and plane, we may
replace them by the terms "roint," "rine," and "rane,"
so that the first postulate, or axiom, as Hilbert calls it,
will read: Two distinct roints always completely deter-
mine a rine. Or, better still, we may replace them by
the symbols e\, e^ £3, so that the reading will be: Two
distinct e\s always completely determine an £2; and sim-
ilarly for the remaining postulates.

We will suppose the phrasing of (A), (B), (C), (D),
(E), slightly changed to agree with the indicated new
phrasing of the postulates.

242 INTERPRETATIONS OF POSTULATE SYSTEMS

It seems very probable that there are no termless
relations, i. e., relations that do not relate. It seems
very probable that a relation to be a relation must be
something actually connecting or subsisting between
at least two things or terms. A postulate expressing
a relation having terms is at all events ostensibly a
statement about the terms, and so it would seem that,
if the relation be supposed to be termless, the statement
ceases to be a statement about something and, in so
ceasing, ceases to be a statement that is true or else
is false. In discourse, it is true, there is frequent seem-
ing evidence that relations are often thought of as
termless, as when, for example, we speak of "a relation
and its terms"; but then we speak also of a neckless
fiddle without intending to imply by such locution
that there can be a fiddle without a neck. As, however,
we do not wish the -validity of the following criticism to
depend on the denial of the possibility of termless rela-
tions, the discussion will be conducted in turn under
each of the alternative hypotheses: (hi) There are term-
less relations; (hz) There are no termless relations. We
will begin with

HYPOTHESIS fa

To 04) we make no objection.

Let us now suppose given to P some definite inter-
pretation /. Let us grant that we now have a definite
doctrine D, consisting of P and C. Either the things
which in / the e's denote have or they have not con-
tent, character, or meaning, m, in excess of the fact
that they satisfy P.

(i) Suppose they have not an excessive meaning m.
Denote the interpretation by I\ and the doctrine by
D\. This DI is a queer doctrine. We may ask: what

INTERPRETATIONS OF POSTULATE SYSTEMS 243

does Di relate or refer to? That is, what is it a doctrine
of or about? The question seems to admit of no intel-
ligent or intelligible answer. For if the doctrine is
about something, it is, it seems natural to say, a doc-
trine about the /i-things (denoted by the e's); but,
by (i), these /i-things can not be characterized or in-
dicated otherwise than by the fact of their satisfying
P; and so it appears that such attempted natural an-
swer is reducible and equivalent to saying (a) that the
doctrine D\ is about the things which it is about. In
order not to be thus defeated, one might try to give an
informing answer by saying that D\ is a doctrine, not
about the /i-things, i. e., not about terms of relations,
but about the relations themselves. Such an answer is
suspicious on account of its unnaturalness, and it is
unnatural because the propositions of DI wear the ap-
pearance of talking explicitly, not about relations, but
about terms of relations. Moreover, the answer is not
an informing one unless the relations that the doctrine
Di is alleged to be about can be characterized otherwise
than by the fact of their being satisfied by the /i-things,
for, if they can not be otherwise characterized, evidently
by (i) the answer reduces to a form essentially like
that of (a). May not one escape by saying that the
relations which DI is alleged to be a doctrine about are
just the relations expressed by the propositions in £>i?
Does this attempted characterization make the answer
in question an informing one? If DI is a doctrine about
the relations expressed by its propositions, then DI says
or teaches something about these relations, for every
doctrine, if it be about something, must teach or say
something about that which it is about. In the case
supposed, what does D\ teach about the relations?
Nothing except that they are satisfied by the /i-things.

244 INTERPRETATIONS OF POSTULATE SYSTEMS

In other words, what D\ teaches about the relations
expressed by its propositions is, by (i), that these are
satisfied by things that satisfy them — a not very
nutritious lesson. It is possible to make a yet further
attempt so to indicate the relations as to render the
answer, that the doctrine D\ is about relations, an in-
forming one. It is known that P may receive an inter-
pretation I' different from /i in that the I'- things do
not satisfy (i), but have an excessive content, character,
or meaning m. May we not give the required indica-
tion of the relations that DI is said to be a doctrine
about by saying that they are relations satisfied by the
/'-things, the presence of the m involved making the
indication genuine or effective? It seems so at first.
But if again we ask what D\ teaches about the relations
thus indicated, we are led into the same difficulty as
above. Moreover, when we ask what D\ is a doctrine
about we expect an answer in terms in somewise men-
tioned or intimated in the D\ discourse, whilst in the
case in hand the required indication has depended on
m, a thing expressly excluded from the D\ discourse
by (i).

So, I repeat, D\ is a queer doctrine.

It must be added that if there be an interpretation /i,
it is unique of its kind; for if I\ were an interpreta-
tion satisfying (i), the //-things would have no excess-
ive meaning m; hence they would be simply the /i-
things, and I\ and // would be merely two symbols
for a same interpretation.

Accordingly, if there were an interpretation /i, but no
other, i. e., no interpretation / in which the /-things
did not satisfy (i), then (C) would be pointless; by (D),
DI would be Euclidean geometry of three dimensions;
and, by (£), Euclidean space, if we wished to speak of

INTERPRETATIONS OF POSTULATE SYSTEMS 245

Di as a geometry of a space, would be the ensemble of
the /i- things; but, if we wished to characterize the I\-
things, the elements of Euclidean space, we could only
say that they are the things satisfying certain relations,
and, if we wished to indicate what relations, we could
only say, the relations satisfied by those things: a very
handsome circle.

In the following it will be seen that we are in fact not
imprisoned within that circle.

(2) Suppose the /-things of the above-assumed inter-
pretation / do not satisfy (i), but have an excessive
meaning m. (It is known that such an / is possible,
an example being found by taking for an e\ any ordered
triad of real numbers (x, y, z); for an e2 the ensemble
of triads satisfying any two distinct equations,

+ Biy + dz + Di = o, A2x + B2y + C2z + D2 =o,

in neither of which the coefficients are all of them zero;
and for an e* the ensemble of triads satisfying any one
such equation; the presence of m being evident in count-
less facts such as the fact, for example, that an e\ is
composed of numbers studied by school-boys or useful
in trade without regard to their ordered triadic rela-
tionship.) Denote the assumed definite interpretation
/ by 72 to remind us that it satisfies (2), and denote
the corresponding doctrine by A- It is immediately
evident that there is an interpretation I\ and hence a
doctrine DI, for to obtain A it is sufficient to abstract
from the m of the 72-things and to take the abstracts
(which plainly satisfy (i)) for Ii-things.

Are Di and A but two different symbols for one and
the same doctrine, as asserted by (C)? Evidently not.
For, in respect of A, we can give an informing answer
to the question, what is A a doctrine about? Owing

246 INTERPRETATIONS OF POSTULATE SYSTEMS

to the presence of the m in the /2-things, the answer
will be an informing one whether it be the natural answer
that A is a doctrine about the /2-things, or one of the
less natural answers, that A is about the relations having
the /2-things for terms, that A is about the relations
expressed by its propositions; whilst, as we have seen,
owing to the absence of m in the /i-things no such
answers were, in respect of A, informing answers.

Can not (C) be saved by refusing to admit that there
is an interpretation I\, and so refusing to admit that
there is a A? If there is no I\ and hence no A, then
(C) is pointless unless there is an 72' and so a A' in
which the /2'-things have an m' different from the m
of the /2-things. But if there is an /2' thus different
from /2, then obviously Ar and A are, contrary to (C),
different doctrines, for they are respectively doctrines
about the /2-things and the //-things, and these thing-
systems are different by virtue of the difference of m
and m '. Now, it is known that there are two such differ-
ing interpretations /2 and /2'. For we may suppose /2
to be the possible interpretation indicated in the above
parenthesis. And for /2' we may take for e\ any ordered
triad of real numbers, except a specified triad (a, b, c),
and including ( <» } oo } oo ) ; f or e2 the ensemble of triads,
except (a, b, c), that satisfy any pair of equations,

z + z2) + 2Bi(x -a)+ 2Ci(y - b)

+ 2AO - c) - A,(a2 + b*+ c2) = o,
A2(x2 + y2 + 22) + 2£2(* - a) + 2C2(;y - b)

+ 2A(z - c) - A2(a2+ b2+ c2) = o;

and for es the ensemble satisfying any one such equation.
Just as when we compared A and A, so here the con-
clusion is, that (C) is not valid.

INTERPRETATIONS OF POSTULATE SYSTEMS 247

As a matter of fact mathematicians know that there
are possible infinitely many different interpretations of
P. It follows from the foregoing that there are corre-
spondingly many different doctrines. For the sake of
completeness we may include DI among these, although,
for the purpose of answering a hypothetical objec-
tion, we momentarily supposed D\ to be disputable or
inadmissible.

Which one of the Z)'s is (or should be called) Euclidean
geometry of three dimensions? I say which "one"?
For, as no two are identical, it would be willful courting
of ambiguity to allow that two or more of them should
be so denominated. Which one, then? Evidently not
one of the numerical ones, such, for example, as the
two above specified. For who has ever really believed
that a point, for example, is a triad of numbers? We
know that the Greeks did arrive at geometry; we know
that they did not arrive at it through numbers; and
we know that, in their thought, points were not number
triads, nor were planes and lines, for them, certain
ensembles of such triads. The confusion, if anybody
ever was really thus confused, is due to the modern
discovery that number triads and certain ensembles of
them happen to satisfy the same relations as the Greeks
found to be satisfied by what they called points, lines,
and planes. There is really no excuse for the confusion,
for, if Smith is taller than Brown, and yonder oak is
taller than yonder beech, it obviously does not follow
that Smith is the oak and Brown the beech.

Evidently Euclidean geometry of three dimensions
is that particular D for which the /-things are points,
lines, and planes. Here it is certain to be asked:
What, then, are points, lines, and planes? And the
asker will mean to imply that, in order to maintain the

248 INTERPRETATIONS OF POSTULATE SYSTEMS

proposition, it is necessary to define these terms. The
proper reply is that is it not necessary to define them.
All that can be reasonably required is that they be
indicated, pointed out, sufficiently described for purposes
of recognition, for what we desire is to be able to say
or to recognize what Euclidean geometry is about. To
the question one might, not foolishly, reply that the
terms in question denote things that you and I, if we
have been disciplined in geometry, converse under-
standingly about when we converse about geometry,
though neither of us is able to say with absolute pre-
cision what the terms mean. For who does not know
that it is possible to write an intelligent and intelligible
discourse about cats, for example, without being able
to tell (for who can tell?) precisely what a cat is? And
if it be asked what the discourse is about, who does
not know that it is an informing answer to say that
it is about cats? It is informing because the term cat
has an excessive meaning, a meaning beyond that of
satisfying the propositions (or relations) of the discourse.
Just here it is well worth while to point out an im-
portant lesson in the procedure of Euclid. Against
Euclid it is often held as a reproach that he attempted
to define the element-names, point, line, and plane,
since no definitions of them could render any logical
service, that is, in the strictly deductive part of the
discourse. But to render no logical service is not to
render no service. And the lesson is that the definitions
in question, which it were perhaps better to call de-
scriptions, do render an extralogical service. They
render such service not only in guiding the imagination
in the matter of invention, but also in serving to indi-
cate, with a goodly degree of success, the excessive
meaning m of the elements denoted by the terms in

INTERPRETATIONS OF POSTULATE SYSTEMS 249

question and in thus serving to make known what it is
that the deductive part of the discourse is about. One
should not forget that no discourse, no doctrine, not
even so-called pure logic itself, is exclusively deductive,
for in any doctrine there is reference, implicit or ex-
plicit, to something extradeductive or extralogical,
reference, that is, to something which the doctrine is
about.

Are the three Euclidean "definitions," thus viewed
as descriptions, sufficient or adequate to the service
that they are here viewed as rendering? If by suffi-
cient or adequate be meant exhaustive, the answer is,
of course, no. For we may confidently say that no
possible description, that is, no description involving
only a finite number of words, can exhaust the meaning
of a system of terms except, possibly, in the special
case where these have no meaning beyond what they
must have in order merely to satisfy a finite number of
postulates. But exhaustive is not what is meant by
adequate. To employ a previous illustration, it is not
necessary to give or to attempt an exhaustive descrip-
tion of "cat" in order to tell adequately what it is
that a discourse ostensibly about cats is ostensibly about.
It is a question of intent. A description is nearly, if
not quite, adequate if it enables us to avoid thinking
that terms are intended to denote what they are not
intended to denote. And, whilst we may not admit
that the three Euclidean "descriptions" are the best that
can be invented for the purpose, yet we must allow
that they have long served the end in question pretty
effectively and that they are qualified to continue such
service. They have been and they are good enough,
for example, to save us from thinking that the things
which in geometry have been denoted by the terms,

250 INTERPRETATIONS OF POSTULATE SYSTEMS

points, lines, and planes, are identical with number
triads, etc. The open secret of their thus saving us is
no doubt in their causing us to think of points, lines,
and planes in terms of, or in essential connection with,
what we know as extension, whilst numbers and number
ensembles are not things naturally so conceived. For
evidently the notions of "length" and " breadth" in-
volved in the Euclidean "descriptions" are not metric
in meaning; they do not signify definite or numeric
quantities or amounts of something (as when we say
the length of this or that thing is so and so much);
but plainly they are generic notions connoting extension.
It is safe to say that a mind devoid of the concept or
the sense of extension could not know what things the
"descriptions" aim at describing. It is true that
Euclid's "description" of a point as "that which has
no part" implies a denial of extension, but the denial
is one of extension, and, in its contextual atmosphere,
it is felt to be essential to an adequate indication of
what is meant by point. On the other hand, if one
were (and how unnatural it would be!) to describe an
ordered triad of numbers as "that which has no part,"
it would be immediately necessary to explain away the
seeming falsity of the description by saying that the
triad is not the ordered multiplicity (of three numbers)
as a multiplicity, but is merely the uniphase of the
multiplicity, and that it is this uniphase which has no
part. If, next, we were to say that thus extension is
denied to the uniphase, the statement, though true,
would be felt to be inessential to an adequate indica-
tion of what is meant by a triad of numbers. Such
felt difference is alone sufficient to make any one pause
who is disposed to adopt the current creed that a point
is nothing but an ordered triad of numbers. It is not

INTERPRETATIONS OF POSTULATE SYSTEMS 251

contended that a point is composed of extension; the
contention is that point and extension are so connected
that a mind devoid of the latter notion would be devoid
of the former, just as a mind devoid of the notion of
variable or variation would be devoid of the notion
of constant, though a constant is not a thing consisting
of variation; just as the notion of limit would not be
intelligible except for the notion of something that may
have a limit, though the limit is not composed of it;
and just as an instant, which is not composed of time,
would not be intelligible except for the notion of time.
In a discussion of such matters it is foolish and futile
to talk about " proofs." The question, as said, is one
of intent; it is a question of self -veracity, of getting
aware of and owning what it is that we mean by the
terms and symbols of our discourse. If, despite the
Euclidean " descriptions " and despite any and all others
that may supplement or supplace them, one fails to see
that extension is essentially involved in the meaning that
the terms points, lines, and planes, are intended to have,
the failure will be because "as the eyes of bats are
to the blaze of day, so is the reason in our soul to the
things which are by nature most evident of all." Noth-
ing is more evident than that there is something that
is called extension. We have but to open our eyes to
get aware that we are beholding an expanse, something
extended. We see things as extended: things as ex-
tended are revealed to the tactile sense; a region or
room involving extension is a datum of the muscular
sensations connected with our bodily movements;
and so on. So much is certain. But it is said and
rightly said that these are sensible things; that the
extension they are revealed as having is sensible exten-
sion; that these sensibles are infected with contra-

252 INTERPRETATIONS OF POSTULATE SYSTEMS

diction, above noted, revealed in common experience,
and confirmed by the psychophysical law of Weber and
Fechner; that geometry is free from contradiction;
that, therefore, geometry is not a doctrine about these
sensibles; that among these sensibles are not the things
which in geometry are denoted by the terms point,
line, and plane; and that, if these terms imply or con-
note extension, as asserted, this extension is not sensible
extension. Granted. The " connoted extension" is not
sensible, it is conceptual. How know, however, that
there is conceptual extension? The answer is, by
arriving at it. (We need not here debate whether such
"arriving" is best called creating or is best called find-
ing.) But how does the mind arrive at it? By doing
certain things to the sensibles, the raw material of mental
architecture. What things? An exhaustive answer is
unnecessary — perhaps impossible. The things are of
two sorts: the mind gives to the sensibles; it takes
away from them. Consider for example a sensible
line. From it the conceptualizing intellect takes away
(abstracts from, disregards) certain things that the
sensible in question has or may have, as color, weight,
temperature, etc., including part of the extension, thus,
I mean, narrowing and thinning away all breadth and
thickness. What of the extension called length? Have
the narrowing and thinning taken it away? It was not
so intended, the opposite was intended. Yet no sensible
length (extension) remains. Does the narrowing and
thinning involve shortening? We are absolutely certain
that it does not. What, then, is it that has happened?
Evidently that, by the indicated taking away, the mind
has arrived at insensible length, one kind of insensible
extension, that is, at conceptual length, one kind of
conceptual extension. A stretch, we are sure, remains,

INTERPRETATIONS OF POSTULATE SYSTEMS 253

but it is not a sensible stretch. The extension thus
arrived at is yet not the extension connoted by or in-
volved in the things that geometry is about, for in the
taking-away process of arriving at it there is nothing
to disinfect it of the contradictions inherent in the
sensible with which we started. It remains, then, to
follow the indicated process of taking away by a process
of giving, that is to say, it remains to endow the con-
ceptual extension (arrived at) with continuity so as to
render it free from the mentioned contradictions. This
done, the kind of extension meant in ordinary geom-
etry or ordinary geometric space is arrived at. Such
is, in kind, the conceptual extension that, it is here
held, is essential to what the geometric terms, point,
line, plane, are intended to mean. Without further
talk we may say that such extension is essential in the
conceptual space that, we may say, ordinary Euclidean
geometry is about in being about the elements of the
space.

If we denote this conceptual space by cSz to distin-
guish it from (non-geometrizable) sensible space $£3,
then the geometry of cSs, if constructed by means of
postulates P making no indispensable use of algebraic
analysis, may be called pure geometry, pGs. If, as in
the Cartesian method, we use ordered number triads,
etc., as we may use them, not to be points, etc., but to
represent points, etc., then we get analytical geometry,
aGz, of cSz. On the other hand, if, as we may, we inter-
pret the P by allowing the /-things to be number triads,
etc., as above indicated, the resulting doctrine is, not
geometry, but a pure algebra or analysis, pAs. If we
use points, etc., not to be, but to represent, number
triads, etc., and so employ geometric language in con-
structing pAz, we get by this kind of an ti- Cartesian

254 INTERPRETATIONS OF POSTULATE SYSTEMS

procedure, not a geometry, but geometrical analysis,

gAz.

HYPOTHESIS hi

It is unnecessary to say anything and is not worth
while to say much under this hypothesis. For if the
e's in P do not denote something, then as the relations
(if there be any) are termless, the doctrine D (if there
be one) is not about anything, unless about the relations,
but about these it says nothing, for, if it says aught
about them, what it says is that they are satisfied by
certain terms whose presence in the discourse is excluded
by hi. We may profitably say, however, that, in the
case supposed where the e's do not denote something
but are merely uninterpreted variables ready, so to
speak, to denote something — in this case we may say
that, though there is no doctrine D, there is a doctrinal
function, A (e\, e%, £3). Then we should add that the
doctrines that do arise from actualized possible interpre-
tations of the e's are so many values of A. This func-
tion A, if we give some warning mark as A' to its
symbol, may be further conveniently employed in talk-
ing about an ambiguous one of the doctrines in question,
i. e., about "any value," an ambiguous value, of the
function. As above argued, these values, these doctrines
are identical in form, they are isomorphic, all of them
having the form of A, but no two of them are the same
in respect of content, reference, or meaning. In this
conclusion, analysis, happily, agrees with traditional
usage, intuition, and common sense.

CONCLUDING CONSIDERATIONS

We are, I believe, now prepared to answer definitively
the long- vexed question: What, if any, sort of existence
have point spaces of four or more dimensions?

INTERPRETATIONS OF POSTULATE SYSTEMS 255

As we have seen, the conceptual space cS$ of ordinary
geometry is an affair involving extension; it is a triply
extended conceptual spread or expanse: three independ-
ent linear extensions in it may be chosen; these suffice
to determine all the others. So much is as certain as
anything can be. It is equally certain that we can, for
we do without meeting contradiction, by means of postu-
lates or otherwise, conceive (not perceive or imagine) a
quadruply extended spread or expanse, one, that is, in
which it is possible to choose four independent linear
extensions, and then by reference to these to determine
all the rest There is not the slightest difference in kind
among the four independents and not the slightest differ-
ence between any three of these and the three of cSs.
The spread or expanse thus set up is a cS±; like cSz, it
is purely conceptual; the extension it involves is, in kind,
identical with that of cSz', it contains spreads of the
type of cSz as elements just exactly as a cSs contains
planes or spreads of type cSz as elements; it differs
not at all from cSz except in being one degree higher
in respect of dimensionality. In a word, cS± (and, of
course, cSa, and so on) has the same kind of existence as
cSs. It is true that cS$ is "imitated" by our sensible
space sSz, whilst there is no sSi thus imitating cS*.
But this writing is not intended for one who is capable
of thinking that the mentioned sensible imitation or
imitability of cSs confers upon the latter a new or
peculiar kind of existence.

But one thing remains to be said, and it is impor-
tant. If one denies that cSs has the conceptually exten-
sional existence, above alleged, then, of course, the
denial extends also to cS4, and the two spaces are, in
respect of existence, still on a level. If the denier then
asserts, and such is the alternative, that cSs is only the

256 INTERPRETATIONS OF POSTULATE SYSTEMS

ensemble of number triads, etc., as above explained,
then, if he be right, cS* is only, but equally, the ensemble
of ordered quatrains, etc., of numbers. Here, again,
cSz and cS* have precisely the same kind of existence.
The conclusion is that hyperspaces have every kind of
existence that may be warrantably attributed to the space
of ordinary geometry.

MATHEMATICAL PRODUCTIVITY IN THE
UNITED STATES1

BOTH on its own account and in its relation to the
general question of research, this subject is naturally
interesting to the specialists immediately concerned;
and it seems a happy augury that not long ago several
western college and university presidents, in convention,
considered the problem how to secure that officers of
instruction shall become, in addition, investigators and
producers. A complete solution will be found when,
and only when, the nature and importance of the prob-
lem shall be appreciated, not only by scientific special-
ists and university presidents, but by educators and the
educated public in general, and this condition will be
satisfied in proportion as the interdependence of all
grades and varieties of educational and scientific activity
shall come to be generally understood, and especially
in' proportion as we learn to value the things of mind,
not merely for their utility, but for their spiritual worth,
and to seek, as a community, in addition to comfort
and happiness, the glory of the sublimer forms of knowl-
edge and intellectual achievement.

Except when the contrary may be indicated or clearly
implied, the discussion will confine itself to pure mathe-
matics as distinguished from applied mathematics, such
as mechanics and mathematical physics.

And first as to the significance of terms. According
to the usage that has long prevailed among foreign

1 Printed in the Educational Review, November, 1902.

258 MATHEMATICAL PRODUCTIVITY

mathematicians, and which, during the last quarter of
a century, has come to prevail also in this country, the
term mathematical productivity is restricted to dis-
covery, successful research, extension in some sense of
the boundaries of mathematical knowledge; and such
productive activity includes and ranges thru the estab-
lishment of important new theorems, the critical ground-
ing of classical doctrines, the discovery or invention of
new methods of attack, and, in its highest form, the
opening and exploration of new domains.

Not only does the term productivity now signify here
what it signifies abroad, but the prevailing standards in
the United States agree with those of Europe. It is
not meant that the best work in this country is yet
equal to the very best of the European, nor that the
averages coincide, but that the Americans judge home
and foreign products by the same canons of value, and
that these are as rigorous as the French, German, or
British rules of criticism.

Time was when productivity meant, in the United
States, the writing and publishing of college text-books in
algebra, geometry, trigonometry, analytical geometry
and the calculus, not to mention arithmetic. That time
has gone by. At present the term neither signifies such
work nor, except in rare instances, includes it. With
reverence for the olden time when the college professor,
especially in comparison with the average of his suc-
cessors of the present time, was apt to be a man of
general attainments and diversified learning, it may be
said that, judged by modern standards of specialized
scholarship, the special attainments of American mathe-
maticians previous to a generation ago, except in the case
of a few illustrious men, were exceedingly meager —
a fact which, as it could hardly have been suspected

MATHEMATICAL PRODUCTIVITY 259

owing to their isolation by the mathematicians them-
selves, was even less known to their colleagues in other
branches of learning or to the educated public in general.
The writer of a college text-book in mathematics was
naturally regarded as a great mathematician, despite
the circumstance that, in general, the book contained
the sum of the author's knowledge of the subject
treated, much more than the average teacher's knowl-
edge, and quite as much as the most capable youth was
expected to master under the most favoring conditions.
In general, neither author nor teacher nor pupil had
knowledge of the fact that their most advanced instruc-
tion dealt only with the rudiments and often even with
these in an obsolete or obsolescent manner; in general,
there was no suspicion that, on the other side of the
Atlantic, mathematics was a vast and growing science,
much less that it was developing so rapidly and in so
manifold a manner that the greatest mathematical
genius found it necessary to specialize, even in his own
domain. As a natural consequence American mathe-
matical instruction depended almost exclusively on the
use of text-books. What was thus at first a necessity
became a tradition, and, accordingly, in striking con-
trast with French and German practice in schools of
corresponding grade, American college and undergraduate
university instruction in mathematics, with some excep-
tions, of which Harvard is the most notable, continues
still to make the text-book the basis of instruction, even
where it is not regarded as a sine qua non of the classroom.
One result of this practice and tradition is that the text-
book, which early assumed in the public estimation what
now seems to be an exaggerated importance, continues
still to be often regarded as an indispensable instrument
for the systematic impartation of knowledge.

260 MATHEMATICAL PRODUCTIVITY

The text-book method in undergraduate mathematical
instruction undoubtedly has some peculiar merits and
is recommended by considerations of weight. I am not
about to advocate its abandonment. That question,
moreover, is in a sense alien to the subject, here under
discussion. But I may say in passing that the notion,
so firmly lodged in many of our colleges and still more
firmly established in the mind of the general educated
public, that the text-book is indispensable, is an erro-
neous one. That, as already said, has been amply proved
both here and abroad, at Harvard, in some American
normal schools, and in the schools of Germany and
France, by the best, if not the only, method available
for settling such questions, namely, by trial. And I
could wish it were better known, particularly to teachers
in secondary schools, that some of the ablest mathe-
maticians and teachers of mathematics deprecate, not
the use of the text-book, for that use has been suffi-
ciently justified by the practice of most eminent and
effective teachers, but our traditional dependence upon
it, believing that this dependence often hampers the
competent teacher's freedom and so prevents a full
manifestation of the proper life of the subject. For
the subject has indeed a deep and serene and even a
joyous life, and, contrary to popular feeling, it is capable
of being so interpreted and administered as to have,
not merely for the few, but for the many, for the
majority indeed of those who find their way to college,
not only the highest disciplinary value, which is gen-
erally conceded, but a wonderful quickening power and
inspiration as well. And it may very well be that the
very great, tho not generally suspected, human signifi-
cance and cultural value of mathematics, the fact that
not merely in its elements it is highly useful and appli-

MATHEMATICAL PRODUCTIVITY 261

cable, but that throughout the entire immensity and
wondrous complex of its development it is informed
with beauty, being sustained indeed by artistic interest,

— it may indeed be that all this will in some larger
measure come to be felt and understood when teaching
shall depend less on the text-book, at best a relatively
dead thing, tending to bear the spirit of instruction
down, and shall instead be more by living men, speak-
ing immediately to living men, out of masterful knowl-
edge of their science and with a clear perception of its
spiritual significance and worth.

To return from this digression, it is fully recognized
by all that, as undergraduate mathematical instruction
is now carried on in our country, the text-book writer
is a pretty valuable citizen, nor is there any disposition
to detract from the dignity of his activity. Indeed,
though some of the older books compare favorably in
important respects with the best of the new, it may be
said that, in general, to write a highly acceptable mathe-
matical book for college use requires to-day an order
of attainment far superior to that which was sufficient
even a score or two of years ago. The training and
scholarship which such work presupposes are, in re-
spect to amount and more especially in respect to qual-
ity, not only relatively great, but very considerable
absolutely. The author of the kind of book in question

— and happily there is no lack of competition in this
field of writing — may be certain of intelligent, if not
always generous, appreciation; he may be able thereby
to lengthen his purse, his book stands some chance of
being briefly noticed in reputable journals, and he may
even gain local fame, but, however excellent the quality
of his workmanship, it will seldom secure him a place
in the ranks of the investigator or producer. The ser-

262 MATHEMATICAL PRODUCTIVITY

vice of the text-book writer has not been degraded. It
receives a more discriminating appreciation than ever
before. It is merely that this kind of work has received
a more critical appraisement. Not a few mathematicians
decline to undertake the work of text-book writing, for
the reason that they do not wish to be classed as text-
book authors. If one who has published several original
papers yields to the temptation to write a book for
college use, the chances are his reputation will suffer
loss rather than gain. Possibly such ought not to be
the case, but nevertheless it is the case.

In regard to mathematical productivity proper, it is
probably true that during the last twenty-five years,
especially during the latter half of this period, there has
been greater improvement in research work and output
in this country than elsewhere in the world. Such
sweeping statements are of course hazardous, and I
make this one subject to correction. At all events,
the gain in question has been great and is full of prom-
ise. Just about twenty-five years ago the American
Journal oj Mathematics was founded at the Johns Hopkins
University, where it is still published as a quarterly.
Previous to that time two or three attempts had been
made to publish journals of mathematics in this country,
but they met with little success, and are now scarcely
remembered. The American Journal of Mathematics, in
the beginning, sought contributions from abroad, and
reference to the early volumes will show that these are
to a considerable extent occupied by foreign products.
A second journal, the Annals of Mathematics, was founded
in 1884, and published at the University of Virginia.
This journal, a quarterly, still flourishes, being now
published at Cambridge, Mass., under the auspices of
Harvard University. In 1888 was founded the New

MATHEMATICAL PRODUCTIVITY 263

York Mathematical Club, which soon became the New
York Mathematical Society and began the publication
of a monthly Bulletin. In 1894 this society became the
American Mathematical Society which now has a
membership of nearly four hundred, including, with few
exceptions, every American mathematician of standing,
besides some members from Canada, England, and the
Continent. This society has a rapidly growing library,
and publishes two journals, the Bulletin, already men-
tioned, and the Transactions, a quarterly journal, re-
cently founded, and devoted to the publication of the
more important results of research. The four journals
named are, all of them, of good standing and exchange
with some of the best British and Continental journals.
Not by any means all the members of the society are
producing mathematicians, but a large precentage of
them are sufficiently interested to attend one or more
meetings of the society each year. These meetings are
bi-monthly meetings, held in New York, and a summer
meeting at a place chosen from year to year. To meet
growing demands, a Chicago section has been organized,
which holds regular meetings in that city, and a second
section on the Pacific Coast, whose business will be
conducted perhaps at San Francisco. These sections
report to the society proper, which has its offices in
New York City.

At these meetings there are presented annually several
scores of papers, a percentage of which deal with applied
mathematics. Of course not all of these papers are
important, but some of them possess very considerable,
a few of them distinctly great, value, and a large major-
ity of them fall properly within the category of original
investigation as defined. In addition to such more
regular contributions, a considerable number of mathe-

264 MATHEMATICAL PRODUCTIVITY

matical papers are annually presented before other
American scientific organizations, as, for example, before
Section A of the American Association for the Advance-
ment of Science. The majority of all these articles are
found to be available for publication, and the result is
that, altho foreign contributions are no longer invited
as formerly, and few of them received, the four journals
above mentioned are, nevertheless, taxed beyond their
capacity; and, for want of room, papers are sometimes
rejected by the American journals which, if produced
abroad, would probably be published there, where the
facilities for publication are ampler. In fact, the number
of American memoirs published abroad exceeds perhaps
the number of foreign contributions published here.

It is greatly to be regretted that our facilities for
publication, tho recently so greatly enhanced, are still
distinctly inadequate. For mathematicians are also
men, and, as such, one of their most powerful incentives
to research is the prospect of the recognition that comes
from having the results of their labors properly placed
before the scientific world.

While the picture thus drawn of American mathe-
matical activity is a pleasing one and is full of encour-
agement and hope, still we must not disguise from
ourselves the fact that, in view of the vast extent and
resources of our country and of the large number of
professional mathematicians connected with our numer-
ous colleges and universities, the amount and the average
quality of the American mathematical output are not
only distinctly inferior to that of the more scientific
countries of Europe, for which, not without some justice
and plausibility, we are wont to plead our youth in
defense and explanation, but this average and amount
are by no means a measure of our native ability nor in

MATHEMATICAL PRODUCTIVITY 265

keeping with our achievements in some other scarcely
worthier, if less ethereal, domains.

The reasons for this state of case are not far to seek,
and come readily to light on a minuter study of the
necessary and sufficient conditions for the vigorous
prosecution of mathematical research.

We may recall the philosopher's insight that "there
is but one poet and that is Deity." The poet is indeed
born, we all agree; and it is equally, if not so obviously,
true that the great mathematician or financier or admin-
istrator is born. But the mathematician is not born
trained or born with knowledge of the state of the
science, and hence it goes without saying that to native
ability, which we presuppose throughout as absolutely
essential and which is not so rare as is often thought,
training must be superadded, years of austere training
under, or still better, in co-operation with, competent
masters in a suitable atmosphere. Formerly, it was in
general necessary to seek such training abroad; that
is no longer the case, now that our better universities
are manned with scholars of the best American and
European training. Indeed the mathematical doctorate
of a few of our own, institutions now represents quite
as much as, if not more than, the average German
doctorate, though less, we must still confess, than the
French, which probably has the highest significance of
any in the world. Several of the most highly productive
mathematicians in the country have not received for-
eign training, while a still larger number of non-pro-
ducers studied abroad for years — a fact showing that
such training is neither a necessary nor a sufficient
condition. It is not intended to depreciate the absolute
value of foreign training, but only its relative value —
its value as compared with that of the best which our

266 MATHEMATICAL PRODUCTIVITY

own country now affords. It is still desirable, when
not too inconvenient, to spend a year in the atmosphere
of foreign universities, and many avail themselves of
the opportunity, largely for the sake of the prestige
which, owing partly to a tradition, it still affords in
many American communities. It is, of course, a mere
truism to say that training, though necessary, is not suf-
ficient. Unless there be the "gift of originality, training
can at best result in receptive and critical scholarship,
but not in productive power.

There are in our country a goodly number of men
having the requisite ability and training, who, never-
theless, produce but little or nothing at all — a fact
to be accounted for by the absence in their case of other
essential conditions.

In some cases library facilities are lacking. Mathe-
matical science is a growth. The new rises out of the
old, whence the necessity that the investigator have at
hand the major part at least of the literature of his
subject from the earliest times. Even more exacting,
if possible, is the necessity of having ready access to
the leading journals of England, Germany, France, and
Italy, besides those of America. One takes special
pleasure in mentioning Italy, because she has been
recently making rapid advances and in two important
directions, the geometry of hyperspace and mathemat-
ical logic, the ontology of pure thought, she comes well-
nigh leading the van. Of journals there are at least
a dozen which are absolutely indispensable to the re-
search worker and as many more that are highly desir-
able. In addition, the producing mathematician will
not infrequently have occasion to refer to memoirs
which, because of their length or for other reasons, have
not appeared in the journals, and are to be found only

MATHEMATICAL PRODUCTIVITY 267

in the proceedings of the leading general scientific and
philosophical societies of Europe. The lack of such
facilities, which in some cases a few hundred and in
others a few thousand dollars would suffice to make good,
is in itself sufficient to explain the non-productivity
of not a few American mathematicians.

Again, there are cases where able men have not the
necessary leisure to engage successfully in investiga-
tion. Our universities are for the most part so organ-
ized that the energies of scientific men are largely
expended in undergraduate teaching and in adminis-
trative work. We, as a people, have yet to learn that
the value of a professor to a community can be rightly
estimated, not by counting the number of hours he
actually stands before his classes, but rather, if we must
count at all, by reckoning the number of hours devoted
to the preparation of his lectures, and more particularly
by the fruit of quiet study and research. In Germany
the ordinary professor lectures from four to six hours a
week, to which if we add in some cases two hours
Seminar ubungen, we have a total of six to eight hours
of presence in the lecture room. In France the pro-
fessor is expected to give one course of lectures. These
take place twice a week and last from one to one and a
half hours. To this duty should be added that of
holding a large number of examinations — a rather
wearisome service from which the German escapes.
When we contrast this with the ten to fifteen and often
even twenty or more hours of actual teaching demanded
of the American professor, to say nothing of faculty
meetings, committee meetings, and the multitudinous
examinations, and when we do not fail to reflect that
ten hours are much more than twice five in their tax
upon energy, it is little wonder that in productivity our

268 MATHEMATICAL PRODUCTIVITY

most brilliant men are often so greatly outclassed by
their foreign competitors. Moreover, "our universities
are at present, for the first two years, gymnasia and
lycees, and our professors are accordingly obliged to
devote themselves largely to what is properly secondary
instruction" — a kind of work which, however worthy,
important, and necessary, has the effect, not merely of
drawing off the energy in non-productive channels, but
also eventually of forming and hardening the mind
about a relatively small group of simpler ideas.

Again, scientific activity is not infrequently rendered
impossible by the amount of administrative work which
professors of notable administrative ability are called
upon to perform. Indeed "the problem presses for
solution, how to retain the many peculiar excellences
of our college and university life and at the same time
to create for certain men of talent and training a suit-
able environment for the highest scientific activity."

Once more, it is very desirable, indeed it is really
necessary, for men working in a branch of science to
attend the meetings of scientific bodies, in order to
meet their fellow-men, to take counsel of them, to cre-
ate and share in a wholesome esprit de corps, to catch
the inspiration and enthusiasm, and to gain the sus-
taining impulses which can come only from personal
contact and co-operation. But our country is so vast,
the distances so long, and traveling so expensive, that
many mathematicians, owing to smallness of income,
find themselves hopelessly condemned to a life of iso-
lation, of which the result is a loss first of interest and
then of power. It is in vain that one counsels such
men to wake up and be strong and active, for their
state of inactivity is less a defect of will than an effect
of circumstances.

MATHEMATICAL PRODUCTIVITY 269

There is a second phase of this question of remunera-
tion which is, happily, beginning to attract attention
and to receive consideration in university circles. I
refer to the proposition that a wise economy will pro-
vide, for university service, remuneration, not such as
would attract men whose first ambition is to acquire
the ease that wealth is supposed to afford, but such as
will not, by its inadequacy to the reasonable demands
of modern social life, deter men of ability and predilec-
tion for scientific pursuits from entering upon them.
I know personally of six young men, not all of them
mathematicians, who have sufficiently demonstrated
that they possess such ability and predilection, five of
whom have recently relinquished the pursuit of science
and the fifth of whom told me only yesterday that
he seriously contemplates doing so, all of them, for
the reason that, as they allege, the university career
furnishes either not at all, or too tardily, a financial
competence and consequent relief from practical condem-
nation to celibacy. It matters little whether they be
mistaken to a degree or not, so long as the contrary
conviction determines choice. There is, indeed, more
than a bare suspicion that for reasons akin to those
actuating in the cases cited, the university career,
particularly in case of the more abstract sciences, such
as pure mathematics, whose doctrines have little or no
market value, fails to attract a due proportion of the
best intellects of the country. For it should be under-
stood that successful investigation in such sciences
demands men of intellectual resource, of power, of per-
sistence, in a word, men of strenupsity of life and char-
acter. Such men are indeed the intellectual peers of
the great financier, or soldier, or statesman, or admin-
istrator, and they are aware of it; so that if too many

270 MATHEMATICAL PRODUCTIVITY

such men are not to be drawn away from scientific fields
by the prospect of achieving elsewhere not only fame
but fortune also, it stands to reason that the university
career must promise at least a competence and the
peace of mind it brings. That such is the case, and that
the future will condemn the present for a too tardy
recognition of the fact, is a matter which can hardly
admit of doubt.

That the conditions above indicated are those which
determine the matter of mathematical productivity is a
proposition which not only commends itself a priori
to the reason, but is justified also a posteriori by experi-
ence, for statistics show that those institutions, both
foreign and domestic, where such productivity has
flourished best are also those where the conditions named
are most fully satisfied, and that where one, at least,
of the conditions is not fulfilled, there investigation
proceeds but feebly or is wholly wanting.

It remains to mention another condition which in a
sense includes all others, and whose fulfillment will come
gradually as at once the cause and the effect of the
fulfillment of all others. I mean, of course, a public
sentiment which will demand, because it has learned
to appreciate, knowledge, not merely because of its
applications and utility, but for its beauty, as one appre-
ciates the moon and the stars without regard to their
aid in navigation — a public sentiment that shall seek
every provision and regard as sacred every instrumental-
ity for the advancement and ministration of knowledge,
not only as a means to happiness, but as a glory, for
its own sake, as a self-justifying realization of the dis-
tinctive ambition of man, to understand the universe
in which he lives and the wondrous possibilities of the
Reason unto which it constantly makes appeal.

MATHEMATICS l

IN the ea^^^ait^QLlheJ^Lj^ntury a philosophic

French mathematician, addressing himself to the ques-
tion of the perfectibility of scientific doctrines, expressed
the opinion that one may not imagine the last word has
been said of a given theory so long as it can not by
a brief explanation be made clear to the man of the^-
jstreet^ Doubtless that conception of doctrinal per-
fectibility, taken literally, can never be realized. For
doubtless, just as there exist now, so in the future
there will abound, even in greater and greater variety
and on a vaster and vaster scale, deep-laid and high-
towering scientific doctrines that, in respect to their
infinitude of detail and in their remoter parts and more
recondite structure, shall not be intelligible to any but
such as concentrate their life upon them. And so the
noble dream of Gergonne can never literally come true.
Nevertheless, as an ideal, as a goal of aspiration, it
is of the highest value, and, though in no case can it
be quite attained, it yet admits in many, as I believe,
of a surprisingly high degree of approximation. I do

1 An address delivered in 1907 at Columbia University, the University
of Virginia, Washington and Lee University, the University of North Caro-
lina, Tulane University, the University of Arkansas, the University of
Nebraska, the University of Missouri, the University of Chicago, North-
western University, the University of Illinois, Vanderbilt University, the
University of Minnesota, the University of Michigan, the University of
Cincinnati, the Ohio State University, Vassar College, the University of
Vermont, Purdue University, and the University Club of New York City.
Printed by the Columbia University Press, 1907.

272 MATHEMATICS

not mind frankly owning that I do not share in the
feeling of those, if there be any such, who regard their
special subjects as so intricate, mysterious and high,
that in all their sublimer parts they are absolutely in-
accessible to the profane man of merely general culture
even when he is led by the hand of an expert and con-
descending guide. For scientific theories are, each and
all of them, and they will continue to be, built upon and
about notions which, however sublimated, are never-
theless derived from common sense. These etherealized
central concepts, together with their manifold bearings
on the higher interests of life and general thought, can
be measurably assimilated to the language of the com-
mon level from which they arose. And, in passing, I
should like to express the hope that here at Columbia
or other competent center there may one day be estab-
lished a magazine that shall have for its aim to mediate,
by the help, if it may be found, of such pens as those
of Huxley and Clifford, between the focal concepts and
the larger aspects of the technical doctrines of the
specialist, on the one hand, and the teeming curiosity,
the great listening, waiting, eager, hungering con-
sciousness of the educated thinking public on the other.
Such a service, however, is not to be lightly undertaken.
An hour, at all events, is hardly time enough in which
to conduct an excursion even of scientific folk through
the mazes of more than twenty hundred years of mathe-
matical thought or even to express intelligibly, if one
were competent, the significance of the whole in a
critical estimate.

Indeed, such is the character of mathematics in its
profounder depths and in its higher and remoter zones
that it is well nigh impossible to convey to one who has
not devoted years to its exploration a just impression

MATHEMATICS 273

of the scope and magnitude of the existing body of the.
science. An imagination formed by other disciplines
and accustomed to the interests of another field may
scarcely receive suddenly an apocalyptic vision of that
infinite interior world. But how amazing and how edi-
fying were such a revelation, if only it could be made.
To tell the story of mathematics from Pythagoras and
Plato to Hilbert and Lie and Poincare; to recount and
appraise the achievements of such as Euclid and Archi-
medes, Apollonius and Diophantus; to display and
estimate the creations of Descartes and Leibnitz and
Newton; to dispose in genetic order, to analyze, to
synthesize and evaluate, the discoveries of the Ber-
noullis and Euler, of Desargues and Pascal and Monge
and Poncelet, of Steiner and Mobius and Pliicker and
Staudt, of Lobatschewsky and Bolyai, of W. R. Ham-
ilton and Grassmann, of Laplace, Lagrange and Gauss,
of Boole and Cayley and Hermite and Gordan, of Bol-
zano and Cauchy, of Riemann and Weierstrass, of
Georg Cantor and Boltzmann and Klein, of the Peirces
and Schroder and Peano, of Helmholtz and Maxwell
and Gibbs; to explore, and then to map for perspective
beholding and contemplation, the continent of doctrine
built up by these immortals, to say nothing of the count-
less refinements, extensions and elaborations meanwhile
wrought by the genius and industry of a thousand
other agents of the mathetic spirit; — to do that would
indeed be to render an exceeck'ng service to the higher
intelligence of the world, but a service that would re-
quire the conjoint labors of a council of scholars for
the space of many years. Even the three immense
volumes of Moritz Cantor's Geschichte der Mathematik,
though they do not aspire to the higher forms of elab-
orate exposition and though they are far from exhaust-

274 MATHEMATICS

ing the material of the period traversed by them, yet
conduct the narrative down only to I758.1 That date,
however, but marks the time when mathematics, then
schooled for over a hundred eventful years in the un-
folding wonders of Analytic Geometry and the Calculus
and rejoicing in the possession of these the two most
powerful among the instruments of human thought, had
but fairly entered upon her modern career. And so
fruitful have been the intervening years, so swift the
march along the myriad tracks of modern analysis and
geometry, so abounding and bold and fertile withal
has been the creative genius of the time, that to record
even briefly the discoveries and the creations since the
closing date of Cantor's work would require an addition
to his great volumes of a score of volumes more.

Indeed the modern developments of mathematics
constitute not only one of the most impressive, but one
of the most characteristic, phenomena of our age. It
is a phenomenon, however, of which the boasted intelli-
gence of a "universalized" daily press seems strangely
unaware; and there is no other great human interest,
whether of science or of art, regarding which the mind
of the educated public is permitted to hold so many
fallacious opinions and inferior estimates. The golden
age of mathematics — that was not the age^oTEucM,
itj^ours^-42l^§_Js-fehe' age in which no less than six
international congresses of mathematics have been held
jn the course of nine years.2 It is in our day that more
than a dozen mathematical societies contain a growing
membership of over two thousand men representing
the centers of scientific light throughout the great cul-

1 The work is now being carried forward by younger men.

2 International congresses of mathematicians are held at intervals of
four years. Since the date of this address two have been held, one at Rome
in 1908 and one in Cambridge, England, in 1912.

MATHEMATICS 275

ture nations of the world. It is in our time that over
five hundred scientific journals are each devoted in
part, while more than two score others are devoted
exclusively, to the publication of mathematics. It is
in our time that the Jahrbuch iiber die Fortschritte der
Mathematik, though admitting only condensed abstracts
with titles, and not reporting on all the journals, has,
nevertheless, grown to nearly forty huge volumes in as
many years. It is in our time that as many as two
thousand books and memoirs drop from the mathemat-
ical press of the world in a single year, the estimated
number mounting up to fifty thousand in the last
generation. Finally, to adduce yet another evidence of
similar kind, it requires no less than the seven ponder-
ous tomes of the forthcoming Encyklopadie der Mathe-
matischen Wissenschaften to contain, not expositions,
not demonstrations, but merely compact reports and
bibliographic notices sketching developments that have
taken place since the beginning of the nineteenth cen-
tury. The, Elements of Euclid is as small a part of
mathematics as the Iliad is of literature; or as the
sculpture of Phidias is of the world's total art. Indeed
if Euclid or even Descartes were to return to the abode
of living men and repair to a university to resume pur-
suit of his favorite study, it is evident that, making
due allowance for his genius and his fame, and pre-
supposing familiarity with the modern scientific lan-
guages, he would yet be required to devote at least a
year to preparation before being qualified even to begin
a single strictly graduate course.

It is not, however, by such comparisons nor by sta-
tistical methods nor by any external sign whatever,
but only by continued dwelling within the subtle radi-
ance of the discipline itself, that one at length may

276 MATHEMATICS

catch the spirit and learn to estimate the abounding
life of modern mathesis: oldest of the sciences, yet
flourishing to-day as never before, not merely as a giant
tree throwing out and aloft myriad branching arms in
the upper regions of clearer light and plunging deeper
and deeper root in the darker soil beneath, but rather
as an immense mighty forest of such oaks, which, how-
ever, literally grow into each other so that by the junc-
tion and intercrescence of limb with limb and root with
root and trunk with trunk the manifold wood becomes
a single living organic growing whole.

What is this thing so marvelously vital? What does
it undertake? What is its motive? What its signifi-
cance? How is it related to other modes and forms and
interests of the human spirit?

What is mathematics? I inquire, not about the word,
but about the thing. Many have been the answers of
former years, but none has approved itself as final. All
of them, by nature belonging to the "literature of
knowledge," have fallen under its law and "perished by
supersession." Naturally conception of the science
has had to grow with the growth of the science itself.

A traditional conception, still current everywhere
except in critical circles, has held mathematics to be the
science of quantity or magnitude, where magnitude
including multitude (with its correlate of number) as a
special kind, signified whatever was "capable of increase
and decrease and measurement." Measurability was
the essential thing. That definition of the science was
a very natural one, for magnitude did appear to be a
singularly fundamental notion, not only inviting but
demanding consideration at every stage and turn of
life. The necessity of finding out how many and how
much was the mother of counting and measurement,

MATHEMATICS 277

and mathematics, first from necessity and then from
pure curiosity and joy, so occupied itself with these
things that they came to seem its whole employment.

Nevertheless, numerous great events of a hundred
years have been absolutely decisive against that view.
For one thing, the notion of continuum — the " Grand
Continuum" as Sylvester called it — that great central
supporting pillar of modern Analysis, has been con-
structed by Weierstrass, Dedekind, Georg Cantor and
others, without any reference whatever to quantity, so
that number and magnitude are not only independent,
they are essentially disparate. When we attempt to
correlate the two, the ordinary concept of measurement
as the repeated application of a constant finite unit,
undergoes such refinement and generalization through
the notion of Limit or its equivalent that counting no
longer avails and measurement retains scarcely a vestige
of its original meaning. And when we add the further
consideration that non-Euclidean geometry employs a
scale in which the unit of angle and distance, though it
is a constant unit, nevertheless appears from the Euclid-
ean point of view to suffer lawful change from step to
step of its application, it is seen that to retain the old
words and call mathematics the science of quantity or
magnitude, and measurement, is quite inept as no
longer telling either what the science has actually
become or what its spirit is bent upon.

Moreover, the most striking measurements, as of the
volume of a planet, the growth of cells, the valency of
atoms, rates of chemical change, the swiftness of thought,
the penetrative power of radium emanations, are none
of them done by direct repeated application of a unit or
by any direct method whatever. They are all of them
accomplished by one form or another of indirection. It

278 MATHEMATICS

was perception of this fact that led the famous phi-
losopher and respectable mathematician, Auguste Comte,
to define mathematics as "the science of indirect meas-
urement." Here doubtless we are in presence of a
finer insight and a larger view, but the thought is not
yet either wide enough or deep enough. For it is
obvious that there is an immense deal of admittedly
mathematical activity that is not in the least concerned
with measurement whether direct or indirect. Con-
sider, for example, that splendid creation of the nine-
teenth century known as Protective Geometry: a
boundless domain of countless fields where reals and
imaginaries, finites and infinites, enter on equal terms,
where the spirit delights in the artistic balance and
symmetric interplay of a kind of conceptual and log-
ical counterpoint, — an enchanted realm where thought
is double and flows throughout in parallel streams.
Here there is no essential concern with number or
quantity or magnitude, and metric considerations are
entirely absent or completely subordinate. The fact,
to take a simplest example, that two points determine
a line uniquely, or that the intersection of a sphere and
a plane is a circle, or that any configuration whatever —
the reference is here to ordinary space — presents two
reciprocal aspects according as it is viewed as an en-
semble of points or as a manifold of planes, is not a
metric fact at all: it is not a fact about size or quantity
or magnitude of any kind. In this domain it was
position rather than size that seemed to some the central
matter, and so it was proposed to call mathematics
the science of measurement and position.

Even as thus expanded, the conception yet excludes
many a mathematical realm of vast extent. Consider
that immense class of things known as Operations.

MATHEMATICS 279

These are limitless alike in number and in kind. Now
it so happens that there are many systems of operations
such that any two operations of a given system, if
thought as following one another, together thus produce
the same effect as some other single operation of the
system. Such systems are infinitely numerous and
present themselves on every hand. For a simple illus-
tration, think of the totality of possible straight motions
in space. The operation of going from point A to
point B, followed by the operation of going from B
to point C, is equivalent to the single operation of going
straight from A to C. Thus the system of such opera-
tions is a closed system: combination, i. e., of any two
of the operations yields a third one, not without, but
within, the system. The great notion of Group, thus
simply exemplified, though it had barely emerged into
consciousness a hundred years ago, has meanwhile
become a concept of fundamental importance and pro-
digious fertility, not only affording the basis of an
imposing doctrine — the Theory of Groups — but there-
with serving also as a bond of union, a kind of connec-
tive tissue, or rather as an immense cerebro-spinal
system, uniting together a large number of widely dis-
similar doctrines as organs of a single body. But —
and this is the point to be noted here — the abstract
operations of a group, though they are very real things,
are neither magnitudes nor positions.

This way of trying to come to an adequate conception
of mathematics, namely, by attempting to characterize
in succession its distinct domains, or its varieties of
content, or its modes of activity, in the hope of finding
a common definitive mark, is not likely to prove suc-
cessful. For it demands an exhaustive enumeration,
not only of the fields now occupied by the science, but

280 MATHEMATICS

also of those destined to be conquered by it in the future,
and such an achievement would require a prevision that
none may claim.

Fortunately there are other paths of approach that
seem more promising. Everyone has observed that
mathematics, whatever it may be, possesses a certain
mark, namely, a degree of certainty not found elsewhere.
So it is, proverbially, the exact science par excellence.
Exact, no doubt, but in what sense? An excellent
answer is found in a definition given about one genera-
tion ago by a distinguished Amerkan mathematician,
Professor Benjamin Peirce: "Mathematics is the science
which draws necessary conclusions" — a formulation of
like 'significance with the following fine mot by Professor
William Benjamin Smith: " Mathematics is the , uni-
versal art apodictic." These statements, though neither
of them is adequate, are both of them telling approxima-
tions, at once foreshadowing and neatly summarizing
for popular use, the epoch-making thesis established by
the creators of modern logic, namely, that mathematics
is included in, and, in a profound sense, may be said
to be identical with, Symbolic Logic. Observe that the
emphasisr falls on the quality of being "necessary,"
i. e., correct logically, or valid formally.

But why are mathematical conclusions correct? Is
it that the mathematician has a reasoning faculty essen-
tially different in kind from that of other men? By
no means. What, then, is the secret? Reflect that con-
clusion implies premises, that premises involve terms,
that terms stand for ideas or concepts or notions, and
that these latter are the ultimate material with which
the spiritual architect, called the Reason, designs and
builds. Here, then, one may expect to find light. The
apodictic quality of mathematical thought, the correct-

MATHEMATICS 281

ness of its conclusions as conclusions, are due, not to
any special mode of ratiocination, but to the character
of the concepts with which it deals. What is that dis-
tinctive characteristic? The answer is: precision and
completeness of determination. But how comes the
mathematician by such completeness? There is no
mysterious trick involved: some concepts admit of
such precision and completeness, others do not; the
mathematician is one who deals with those that do.

The matter, however, is not quite so simple as it
sounds, and I bespeak your attention to a word of
caution and of further explanation. The ancient maxim,
ex nihilo nihil fit, may well be doubted where it seems
most obviously valid, namely, in the realm of matter,
for it may be that matter has evolved from something
else; but the maxim cannot be ultimately denied where
its application is least obvious, namely, in the realm of
mind, for without principia in the strictest sense, doc-
trine is, in the strictest sense, impossible. And when
the mathematician speaks of complete determination of
concepts and of rigor of demonstration, he does not
mean that the undefined and the undemonstrated have
been or can be entirely eliminated from the foundations
of his science. He knows that such elimination is im-
possible; he knows, too, that it is unnecessary, for some
undefinable ideas are perfectly clear and some undemon-
strable propositions are perfectly precise and certain.
It is in terms of such concepts that a definable notion,
if it is to be mathematically available, must admit of
complete determination, and in terms of such proposi-
tions that mathematical discourse secures its rigor. It
is, then, of such indefmables among ideas and such in-
demonstrables among propositions — paradoxical as the
statement may appear — that the foundations of mathe-

282 MATHEMATICS

matics in its ideal conception are composed; and what-
ever doctrine is logically constructive on such a basis
is mathematics either actually or potentially. I am not
asserting that the substructure herewith characterized
has been brought to completion. It is on the concep-
tion of it that the accent is here designed to fall, for
it is the conception as such that at once affords to
fundamental investigation a goal and a guide and fur-
nishes the means of giving the science an adequate
definition.

On the other hand, actually to realize the conception
requires that the foundation to be established shall both
include every element that is essential and exclude every
one that is not. For a foundation that subsequently
demands or allows superfoetation of hypotheses is in-
complete; and one that contains the non-essential is
imperfect. Of the two problems thus presented, it is
the latter, the problem of exclusion, of reducing prin-
ciples to a minimum, of applying Occam's Razor to the
pruning away of non-essentials, — it is that problem
that taxes most severely both the analytic and the con-
structive powers of criticism. And it is to the solution
of that problem that the same critical spirit of our time,
which in other fields is reconstructing theology, burning
out the dross from philosophy, and working relentless
transformations of thought on every hand, has directed
a chief movement of modern mathematics.

Apart from its technical importance, which can
scarcely be overestimated, the power, depth and com-
prehensiveness of the modern critical movement in
mathematics, make it one of the most significant sci-
entific phenomena of the last century. Double in
respect to origin, the movement itself has been com-
posite. One component began at the very center of

MATHEMATICS 283

mathematical activity, while the other took its rise in
what was then erroneously regarded as an alien do-
main, the great domain of symbolic logic.

A word as to the former component. For more than
a hundred years after the inventions of Analytical
Geometry and the Calculus, mathematicians may be
said to have fairly rioted in applications of these instru-
ments to physical, mechanical and geometric problems,
without concerning themselves about the nicer questions
of fundamental principles, cogency, and precision. In
the latter part of the eighteenth century the efforts of
Euler, Lacroix and others to systematize results served
to reveal in a startling way the necessity of improving
foundations. Constructive work was not indeed arrested
by that disclosure. On the contrary new doctrines
continued to rise and old ones to expand and flourish.
But a new spirit had begun to manifest itself. The
science became increasingly critical as its towering edi-
fices more and more challenged attention to their foun-
dations. Manifest already in the work of Gauss and
Lagrange, the new tendence, under the powerful impulse
and leadership of Cauchy, rapidly develops into a
momentous movement. The Calculus, while its instru-
mental efficacy is meanwhile marvelously improved, is
itself advanced from the level of a tool to the rank and
dignity of a science. The doctrines of the real and of
the complex variable are grounded with infinite patience
and care, so that, owing chiefly to the critical construc-
tive genius of Weierstrass and his school, that stateliest
of all the pure creations of the human intellect — the
Modern Theory of Functions with its manifold branches
— rests to-day on a basis not less certain and not less
enduring than the very integers with which we count.
The movement still sweeps on, not only extending to

284 MATHEMATICS

all the cardinal divisions of Analysis but, through the
agencies of such as Lobatschewsky and Bolyai, Grass-
mann and Riemann, Cayley and Klein, Hilbert and Lie,
recasting the foundations of Geometry also. And there
can scarcely be a doubt that the great domains of
Mechanics and Mathematical Physics are by their
need destined to a like invasion.

In the light of all this criticism, mathematics came to
appear as a great ensemble of theories, compendent
no doubt, interpenetrating each other in a wondrous
way, yet all of them distinct, each built up by logical
processes on its own appropriate basis of pure hy-
potheses, or assumptions, or postulates. As all the
theories were thus seen to rest equally on hypothetical
foundations, all were seen to be equally legitimate; and
doctrines like those of Quaternions, non-Euclidean
geometry and Hyperspace, for a time suspected because
based on postulates not all of them traditional, speedily
overcame their heretical reputations and were admitted
to the circle of the lawful and orthodox.

It is one thing, however, to deal with the principal
divisions of mathematics severally, underpinning each
with a foundation of its own. That, broadly speaking,
has been the plan and the effect of the critical move-
ment as thus far sketched. But it is a very different
and a profounder thing to underlay all the divisions at
once with a single foundation, with a foundation that
shall serve as a support, not merely for all the divisions
but for something else, distinct from each and from the
sum of all, namely, for the whole, the science itself,
which they constitute. It is nothing less than that
achievement which, unconsciously at first, consciously
at last, has been the aim and goal of the other compo-
nent of the critical movement, that component which,

MATHEMATICS 285

as already said, found its origin and its initial interest
in the field of symbolic logic. The advantage of em-
ploying symbols in the investigation and exposition
of the formal laws of thought is not a recent discovery.
As everyone knows, symbols were thus employed to a
small extent by the Stagirite himself. The advantage,
however, was not pursued; because for two thousand
years the eyes of logicians were blinded by the blazing
genius of the " master of those that know." With the
single exception of the reign of Euclid, the annals of
science afford no match for the tyranny that has been
exercised by the logic of Aristotle. Even the important
logical researches of Leibnitz and Lambert and their
daring use of symbolical methods were powerless to
break the spell. It was not till 1854 when George
Boole, having invented an algebra to trace and illumi-
nate the subtle ways of reason, published his symbolical
"Investigation of the Laws of Thought," that the revo-
lution in logic really began. For, although for a time
neglected by logicians and mathematicians alike, it
was Boole's work that inspired and inaugurated the
scientific movement now known and honored throughout
the world under the name of Symbolic Logic.

It is true, the revolution has advanced in silence.
The discoveries and creations of Boole's successors, of
C. S. Peirce, of Schrb'eder, of Peano and of their dis-
ciples and peers, have not been proclaimed by the daily
press. Commerce and politics, gossip and sport, acci-
dent and crime, the shallow and transitory affairs of
the exoteric world, — these have filled the columns and
left no room to publish abroad the deep and abiding
things achieved in the silence of cloistral thought. The
demonstration by symbolical means of the fact that the
three laws of Identity, Excluded Middle and Non-

286 MATHEMATICS

contradiction are absolutely independent, none of them
being derivable from the other two; the discovery that
the syllogism is not deducible from those laws but has
to be postulated as an independent principle; the dis-
covery of the astounding and significant fact that false
propositions imply all propositions and that true ones,
though not implying, are implied by, all; the discovery
that most reasoning is not syllogistic, but is asyllogistic,
in form, and that, therefore, contrary to the teaching
of tradition, the class-logic of Aristotle is not adequate
to all the concerns of rigorous thought; the discovery
that Relations, no less than Classes, demand a logic of
their own, and that a similar claim is valid in the case
of Propositions: no intelligence of these events nor of
the immense multitude of others which they but mea-
gerly serve to hint and to exemplify, has been cabled
round the world and spread broadcast by the flying
bulletins of news. Even the scientific public, for the
most part accustomed to viewing the mind as only the
instrument and not as a subject of study, has been
slow to recognize the achievements of modern research
in the minute anatomy of thought. Indeed it has
been not uncommon for students of natural science to
sneer at logic as a stale and profitless pursuit, as the
barren mistress of scholastic minds. These men have
not been aware of what certainly is a most profound,
if indeed it be not the most significant, scientific move-
ment of our time. In America, in England, in Germany,
in France, and especially in Italy — supreme histolo-
gist of the human understanding — the deeps of mind
and logical reality have been explored in our genera-
tion as never before in the history of the world. Owing
to the power of the symbolic method, not only the founda-
tions of the Aristotelian logic — the Calculus of Classes —

MATHEMATICS 287

have been recast, but side by side with that everlasting
monument of Greek genius, there rise today other struc-
tures, fit companions of the ancient edifice, namely, the
Logic of Relations and the Logic of Propositions.

And what are the entities that have been found to
constitute the base of that triune organon? The answer
is surprising: a score or so of primitive, indemonstrable,
propositions together with less than a dozen undefinable
notions, called logical constants. But what is more
surprising — for here we touch the goal and are enabled
to enunciate what has been justly called "one of the
greatest discoveries of our age" — is the fact that the
basis of logic is the basis of mathematics also. Thus
the two great components of the critical movement,
though distinct in origin and following separate paths,
are found to converge at last in the thesis: Symbolic
Logic is Mathematics, Mathematics is Symbolic Logic,
the twain are one.

Is it really so? Does the identity exist in fact? Is
it true that so simple a unifying foundation for what has
hitherto been supposed two distinct and even mutually
alien interests has been actually ascertained? The
basal masonry is indeed not yet completed but the work
has advanced so far that the thesis stated is beyond
dispute or reasonable doubt. Primitive propositions
appear to allow some freedom of choice, questions still
exist regarding relative fundamentality, and statements
of principles have not yet crystallized into settled and
final form; but regarding the nature of the data to be
assumed, the smallness of their number and their ade-
quacy, agreement is substantial. In England, Russell
and Whitehead l are successfully engaged now in forging

1 This work has been projected in four immense volumes bearing the
title, Principia Mathematica, of which three volumes have appeared.

288 MATHEMATICS

" chains of deduction" binding the cardinal matters
of Analysis and Geometry to the premises of General
Logic, while in Italy the Formulaire de Mathematiques
of Peano and his school has been for some years grow-
ing into a veritable encyclopedia of mathematics wrought
by the means and clad in the garb of symbolic logic.

But is it not incredible that the concept of number
with all its distinctions of cardinal and ordinal, frac-
tional and whole, rational and irrational, algebraic and
transcendental, real and complex, finite and infinite,
and the concept of geometric space, in all its varieties
of form and dimensionality, is it not incredible that
mathematical ideas, surpassing in multitude the sands
of the sea, should be precisely definable, each and all
of them, in terms of a few logical constants, in terms,
i. e., of such indefinable notions as such that, implica-
tion, denoting, relation, class, prepositional function, and
two or three others? And is it not incredible that by
means of so few as a score of premises (composed of
ten principles of deduction and ten other indemonstrable
propositions of a general logical nature), the entire
body of mathematical doctrine can be strictly and for-
mally deduced?

It is wonderful, indeed, but not incredible. Not in-
credible in a world where the mustard seed becometh a
tree, not incredible in a world where all the tints and
hues of sea and land and sky are derived from three
primary colors, where the harmonies and the melodies
of music proceed from notes that are all of them but
so many specifications of four generic marks, and where
three concepts — energy, mass, motion, or mass, time,
space — apparently suffice for grasping together in
organic unity the mechanical phenomena of a universe.

But the thesis granted, does it not but serve to justify

MATHEMATICS 289

the cardinal contentions of the depreciators of mathe-
matics? Does it not follow from it that the science
is only a logical grind, suited only to narrow and strait-
ened intellects content to tramp in treadmill fashion
the weary rounds of deduction? Does it not follow that
Schopenhauer was right in regarding mathematics as
the lowest form of mental activity, and that he and our
own genial and enlightened countryman, Oliver Wendell
Holmes, were right in likening mathematical thought
to the operations of a calculating machine? Does it
not follow that Huxley's characterization of mathematics
as "that study which knows nothing of observation,
nothing of induction, nothing of experiment, nothing
of causation," is surprisingly confirmed by fact? Does
it not follow that Sir William Hamilton's famous and
terrific diatribe against the science finds ample warrant
in truth? Does it not follow, as the Scotch philosopher
maintains, that mathematics regarded as a discipline,
as a builder of mind, is inferior? That devotion to it
is fatal to the development of the sensibilities and the
imagination? That continued pursuit of the study
leaves the mind narrow and dry, meagre and lean,
disqualifying it both for practical affairs and for those
large and liberal studies where moral questions inter-
vene and judgment depends, not on nice calculation by
rule, but on a wide survey and a balancing of
probabilities?

The answer is, No. Those things not only do not
follow but they are not true. Every count in the in-
dictment, whether explicit or only implied, is false.
Not only that, but the opposite in each case is true.
On that point there can be no doubt; authority, reason
and fact, history and theory, are here in perfect accord.
Let me say once for all that I am conscious of no desire

2 QO MATHEMATICS

to exaggerate the virtues of mathematics. I am willing
to admit that mathematicians do constitute an important
part of the salt of the earth. But the science is no
catholicon for mental disease. There is in it no power
for transforming mediocrity into genius. It cannot
enrich where nature has impoverished. It makes no
pretense of creating faculty where none exists, of open-
ing springs in desert minds. "Du bist am Ende — was
du bist." The great mathematician, like the great
poet or great naturalist or great administrator, is born.
My contention shall be that where the mathetic endow-
ment is found, there will usually be found associated
with it, as essential implications in it, other endowments
in generous measure, and that the appeal of the science
is to the whole mind, direct no doubt to the central
powers of thought, but indirectly through sympathy
of all, rousing, enlarging, developing, emancipating all,
so that the faculties of will, of intellect and feeling,
learn to respond, each in its appropriate order and
degree, like the parts of an orchestra to the "urge and
ardor" of its leader and lord.

As for Hamilton and Schopenhauer, those detractors
need not detain us long. Indeed but for their fame and
the great influence their opinions have exercised over
"the ignorant mass of educated men," they ought not
in this connection to be noticed at all. Of the subject
on which they presumed to pronounce authoritative
judgment of condemnation, they were both of them
ignorant, the former well nigh proudly so, the latter
unawares, but both of them, in view of their pretensions,
disgracefully ignorant. Lack of knowledge, however, is
but a venial sin, and English-speaking mathematicians
have been disposed to hope that Hamilton might be
saved in accordance with the good old catholic doc-

MATHEMATICS 2QI

trine of invincible ignorance. But even that hope, as
we shall see, must be relinquished. In 1853 William
Whewell, then fellow and tutor of Trinity College,
Cambridge, published an appreciative pamphlet entitled
"Thoughts on the Study of Mathematics as a Part of
a Liberal Education." The author was a brilliant
scholar. " Science was his forte," but "omniscience his
foible," and his reputation for universal knowledge was
looming large. That reputation, however, Hamilton re-
garded as his own prerogative. None might dispute the
claim, much less share the glory of having it acknowl-
edged on his own behalf. Whewell must be crushed.
In the following year Sir William replies in the Edin-
burg Review, and such a show of learning! The reader
is apparently confronted with the assembled opinions
of the learned world, and — what is more amazing —
they all agree. Literati of every kind, of all nations
and every tongue, orators, philosophers, educators,
scientific men, ancient and modern, known and unknown,
all are made to support Hamilton's claim, and even the
most celebrated mathematicians seem eager to declare
that the study of mathematics is unworthy of genius
and injures the mind. Whewell was overwhelmed,
reduced to silence. His promised rejoinder failed to
appear. The Scotchman's victory was complete, his
fame enhanced, and his alleged judgment regarding a
great human interest of which he was ignorant has
reigned over the minds of thousands of men who have
been either willing or constrained to depend on borrowed
estimates. But even all this may be condoned. Jeal-
ousy, vanity, parade of learning, may be pardoned even
in a philosopher. Hamilton's deadly sin was none of
these, it was sinning against the light. In October,
1877, A. T. Bledsoe, then editor of the Southern Review

2 9 2 MATHEMATICS

— unfortunately too little known — published an article
in that journal in which he proved beyond a reasonable
doubt — I have been at the pains to verify the proof -
that Hamilton by studied selections and omissions de-
liberately and maliciously misrepresented the great
authors from whom he quoted — d'Alembert, Blaise
Pascal, Descartes and others — distorting their express
and unmistakable meaning even to the extent of com-
plete inversion. This same verdict regarding Hamilton's
vandalism, in so far as it relates to the works of
Descartes, was independently reached by Professor
Pringsheim and in 1904 announced by him in his
Festrede before the Munich Academy of Sciences. As
for Schopenhauer, I regret to say that a similar charge
and finding stand against him also. For not only did
he endorse without examination and re-utter Hamilton's
tirade in the strongest terms, thus reinforcing it and
giving it currency on the continent, but, as Pringsheim
has shown, the German philosopher, by careful excision
from the writings of Lichtenberg, converts that
distinguished physicist's just strictures on the then flour-
ishing but wayward Combinatorial School of mathe-
matics into a severe condemnation of mathematicians
in general and of the science itself, which, nevertheless,
in the opening but omitted line of the very passage
from which Schopenhauer quotes, is characterized by
Lichtenberg as "eine gar herrliche Wissenschajt." Re-
garding the question of the intrinsic merit of the esti-
mate of mathematics which these two most famous and
influential enemies of the science have made so largely
current in the world that it fairly fills the atmosphere
and people take it in unconsciously as by a kind of cere-
bral suction, I shall speak in another connection. What
I desire to emphasize here is the fact that neither the

MATHEMATICS 293

vast, splendid, superficial learning of the pompous
author of "The Philosophy of the Conditioned" nor
the pungence and pith, brilliance and intrepidity of
the author of "Die Welt als Wille" can avail to con-
stitute either of them an authority in a subject in which
neither was informed and in which both stand convicted
falsifiers of the judgments and opinions of other men.

As to Huxley and Holmes, the case is different. Both
of them were generous, genial and honest, and to their
opinions on any subject we gladly pay respect qualified
only as the former's judgment regarding mathematics
was qualified by Sylvester himself:

"Verstandige Leute kannst du irren sehn
In Sachen namlich, die sie nicht verstehn."

In relation to Huxley's statement that mathematical
study knows nothing of observation, induction, exper-
iment, and causation, it ought to be borne in mind
that there are two kinds of observation: outer and
inner, objective and subjective, material and immaterial,
sensuous and sense- transcending; observation, that is,
of physical things by the bodily senses, and observation,
by the inner eye, by the subtle touch of the intellect,
of the entities that dwell in the domain of logic and
constitute the objects of pure thought. For, phrase it
as you will, there is a world that is peopled with ideas,
ensembles, propositions, relations, and implications,
in endless variety and multiplicity, in structure ranging
from the very simple to the endlessly intricate and
complicate. That world is not the product but the object,
not the creature but the quarry of thought, the entities
composing it — propositions, for example, — being no
more identical with thinking them than wine is identical
with the drinking of it. Mind or no mind, that world

2Q4 MATHEMATICS

exists as an extra-personal affair, — Pragmatism to the
contrary notwithstanding. It appears to me to be a
radical error of pragmatism to blink the fact that the
most fundamental of spiritual things, namely, curiosity,
never poses as a maker of truth but is found always and
only in the attitude of seeking it. Indeed truth might
be defined to be the presupposition or the complement
of curiosity — as that without which curiosity would
cease to be what it is. The constitution of that extra-
personal world, its intimate ontological make-up, is
logic in its essential character and substance as an inde-
pendent and extra-personal form of being, while the
study of that constitution is logic pragmatically, in its
character, i. e., as an enterprise of mind. Now — and
this is the point I wish to stress — just as the astron-
omer, the physicist, the geologist, or other student of
objective science looks abroad in the world of sense,
so, not metaphorically speaking but literally, the mind
of the mathematician goes forth into the universe of
logic in quest of the things that are there; exploring
the heights and depths for facts — ideas, classes, rela-
tionships, implications, and the rest; observing the
minute and elusive with the powerful microscope of his
Infinitesimal Analysis; observing the elusive and vast
with the limitless telescope of his Calculus of the In-
finite; making guesses regarding the order and internal
harmony of the data observed and collocated; testing
the hypotheses, not merely by the complete induction
peculiar to mathematics, but, like his colleagues of the
outer world, resorting also to experimental tests and
incomplete induction; frequently finding it necessary,
in view of unforeseen disclosures, to abandon a once
hopeful hypothesis or to transform it by retrenchment
or by enlargement : — thus, in his own domain, matching,

MATHEMATICS 295

point for point, the processes, methods and experience
familiar to the devotee of natural science.

Is it replied that it was not observation of the objects
of pure thought but the other kind, namely, sensuous
observation, that Huxley had in mind, then I rejoin
that, nevertheless, observation by the inner eye of the
things of thought is observation, not less genuine, not
less difficult, not less rich in its objects and disciplinary
value, than is sensuous observation of the things of
sense. But this is not all, nor nearly all. Indeed for
direct beholding, for immediate discerning, of the things
of mathematics there is none other light but one, namely,
psychic illumination, but mediately and indirectly they
are often revealed or at all events hinted by their sensu-
ous counterparts, by indications within the radiance of
day, and it is a great mistake to suppose that the
mathetic spirit elects as its agents those who, having
eyes, yet see not the things that disclose themselves in
solar light. To facilitate eyeless observation of his
sense-transcending world, the mathematician invokes
the aid of physical diagrams and physical symbols in
endless variety and combination; the logos is thus
drawn into a kind of diagrammatic and symbolical in-
carnation, gets itself externalized, made flesh, so to
speak; and it is by attentive physical observation of
this embodiment, by scrutinizing the physical frame and
make-up of his diagrams, equations and formulae, by
experimental substitutions in, and transformations of,
them, by noting what emerges as essential and what as
accidental, the things that vanish and those that do not,
the things that vary and the things that abide un-
changed, as the transformations proceed and trains
of algebraic evolution unfold themselves to view, — it
is thus, by the laboratory method, by trial and by

296 MATHEMATICS

watching, that often the mathematician gains his best
insight into the constitution of the invisible world thus
depicted by visible symbols. Indeed the importance to
the mathematician of such sensuous observation cannot
be overrated. It is not merely that the craving to see
has led to the construction of the manifold models,
ingenious and noble, of Schilling and others, illustrating
important parts of Higher Geometry, Analysis Situs,
Function Theory and other doctrines, but the annals
of the science are illustrious with achievements made
possible by facts first noted by the physical eye. To
take a simple example from ancient days, it was by
observation of the fact that the squares of certain
numbers are each the sum of two other squares, the
detection and collection of these numbers by the method
of trial, observation of the fact that apparently all and
only the numbers of such triplets are measures of the
sides of right triangles, — it was thus, by observation
and experiment, by the method of incomplete induction,
common to the experimental sciences, that the Pyth-
agorean theorem, now familiar throughout the world,
was discovered. It was by Leibnitz's observation of
the definitely lawful manner in which the coefficients of
a system of equations enter their solution that the
suggestion came of a notion on the basis of which
there has grown up in our time an imposing theory,
an algebra built up on algebra — the colossal doctrine
of Determinants. It was the observation, the detection
by the eye of Lagrange and Boole and Eisenstein, of the
fact that linear transformation of certain algebraic
expressions leaves certain functions of their coefficients
absolutely undisturbed in form, unaltered in frame of
constitution, that gave rise to the concept, and there-
with to the morphological doctrine, of Invariants, a

MATHEMATICS 297

theory filling the heavens like a light-bearing ether,
penetrating all the branches of geometry and analysis,
revealing everywhere abiding configurations in the midst
of change, everywhere disclosing the eternal reign of the
law of Form. It was in order to render evident to
sensuous observation and to keep constantly before the
physical eye the pervasive symmetry of mathematical
thought that Hesse in the employment of homogeneous
coordinates set the example, since then generally fol-
lowed, of replacing a variety of different letters by
repetitions of a single one distinguished by indices or
subscripts, — a practice yet further justified on grounds
both of physical and of intellectual economy. It was
by sensuous observation that Clerk Maxwell, in the
beginning of his wondrous career, detected a lack of
symmetry in the then recognized equations of electro-
dynamics and by that observed fact together with a
discriminating sense of the scientific significance of
esthetic intimations, that he was led to remove the
seeming blemish by the addition of a term, antedating
experimental justification of his daring deed by twenty
years: an example of prescience not surpassed by that
of Adams and Leverrier who, while engaged in the study
of planetary disturbance, each of them about the same
time and independently of the other, felt the then un-
known Neptune "trembling on the delicate thread of
their analysis" and correctly informed the astronomer
where to point his telescope in order to behold the
planet. One might go on to cite the theorem of Sturm
in Equation Theory, the "Diophantine theorems of
Fermat" in the Theory of Numbers, the Jacobian "doc-
trine of double periodicity" in Function Theory, Le-
gendre's law of reciprocity, Sylvester's reduction of
Euler's problem of the Virgins to the form of a question

298 MATHEMATICS

in Simple Partitions, and so on and on, thus continuing
indefinitely the story of the great role of observation,
experiment and incomplete induction, in mathematical
discovery. Indeed it is no wonder that even Gauss,
"facile princeps matematicorum," even though he dwelt
aloft in the privacy of a genius above the needs and
ways of other minds, yet pronounced mathematics "a
science of the eye."

Indeed the time is at hand when at least the academic
mind should discharge its traditional fallacies regarding
the nature of mathematics and thus in a measure pro-
mote the emancipation of criticism from inherited
delusions respecting the kind of activity in which the
life of the science consists. Mathematics is no more
the art of reckoning and computation than architecture
is the art of making bricks or hewing wood, no
more than painting is the art of mixing colors on a
palette, no more than the science of geology is the art
of breaking rocks, or the science of anatomy the art of
butchering.

Did not Babbage or somebody invent an adding
machine? And does it not follow, say Holmes and
Schopenhauer, that mathematical thought is a merely
mechanical process? Strange how such trash is occa-
sionally found in the critical offering of thoughtful men
and thus acquires circulation as golden coin of wisdom.
It would not be sillier to argue that, because Stanley
Jevons constructed a machine for producing certain forms
of logical inference, therefore all thought, even that of
a philosopher like Schopenhauer or that of a poet like
Holmes, is merely a thing of pulleys and levers and
screws, or that the pianola serves to prove that a sym-
phony by Beethoven or a drama by Wagner is reducible
to a trick of mechanics.

MATHEMATICS 299

But far more pernicious, because more deeply im-
bedded and persistent, is the fallacy that the mathe-
matician's mind is but a syllogistic mill and that his
life resolves itself into a weary repetition of A is B, B
is C, therefore A is C; and Q.E.D. That fallacy is the
Carthago delenda of regnant methodology. Reasoning,
indeed, in the sense of compounding propositions into
formal arguments, is of great importance at every stage
and turn, as in the deduction of consequences, in the
testing of hypotheses, in the detection of error, in pur-
ging out the dross from crude material, in chastening
the deliverances of intuition, and especially in the final
stages of a growing doctrine, in welding together and
concatenating the various parts into a compact and co-
herent whole. But, indispensable in all such ways as
syllogistic undoubtedly is, it is of minor importance and
minor difficulty compared with the supreme matters
of Invention and Construction. Begriffbildung, the
resolution of the nebula of consciousness into star-forms
of definite ideas; discriminating sensibility to the log-
ical significances, affinities and bearings of these; sus-
ceptibility to the delicate intimations of the subtle or the
remote; sensitiveness to dim and fading tremors sent
below by breezes striking the higher sails; the ability
to grasp together and to hold in steady view at once a
multitude of ideas, to transcend the individuals and,
compounding their forces, to seize the resultant mean-
ing of them all; the ability to summon not only concepts
but doctrines, marshalling them and bringing them to
bear upon a single point, like great armies converging
to a critical center on a battle field. These and such
as these are the powers that mathematical activity in
its higher roles demands. The power of ratiocination,
as already said, is of exceeding great importance but

300 MATHEMATICS

it is neither the base nor the crown of the faculties
essential to " Mathematicised Man." When the greatest
of American logicians, speaking of the powers that con-
stitute the born geometrician, had named Conception,
Imagination, and Generalization, he paused. There-
upon from one in the audience there came the challenge,
"What of Reason?" The instant response, not less
just than brilliant, was "Ratiocination — that is but
the smooth pavement on which the chariot rolls."
When the late Sophus Lie, great comparative anatomist
of geometric theories, creator of the doctrines of Contact
Transformations, and Infinite Continuous Groups, and
revolutionizer of the Theory of Differential Equations,
was asked to name the characteristic endowment of the
mathematician, his answer was the following quaternion:
Phantasie, Energie, Selbstvertrauen, Selbstkritik. Not a
word, you observe, about ratiocination. Phantasie, not
merely the fine frenzied fancy that gives to airy nothings
a local habitation and a name, but the creative imagina-
tion that conceives ordered realms and lawful worlds
in which our own universe is as but a point of light
in a shining sky; Energie, not merely endurance and
doggedness, not persistence merely, but mental vis viva,
the kinetic, plunging, penetrating power of intellect;
Selbstvertrauen and Selbstkritik, self-confidence aware of
its ground, deepened by achievement and reinforced
until in men like Richard Dedekind, Bernhard Bolzano
and especially Georg Cantor it attains to a spiritual bold-
ness that even dares leap from the island shore of the
Finite over into the all-surrounding boundless ocean of
Infinitude itself, and thence brings back the gladdening
news that the shoreless vast of Transfinite Being differs
in its logical structure from that of our island home only
in owning the reign of more generic law.

MATHEMATICS 301

Indeed it is not surprising, in view of the polydynamic
constitution of the genuinely mathematical mind, that
many of the major heroes of the science, men like
Desargues and Pascal, Descartes and Leibnitz, Newton,
Gauss, and Bolzano, Helmholtz and Clifford, Riemann
and Salmon and Pliicker and Poincare, have attained
to high distinction in other fields not only of science
but of philosophy and letters too. And when we reflect
that the very greatest mathematical achievements have
been due, not alone to the peering, microscopic, histo-
logic vision of men like Weierstrass, illuminating the
hidden recesses, the minute and intimate structure of
logical reality, but to the larger vision also of men like
Klein who survey the kingdoms of geometry and analysis
for the endless variety of things that flourish there, as
the eye of Darwin ranged over the flora and fauna of
the world, or as a commercial monarch contemplates
its industry, or as a statesman beholds an empire; when
we reflect not only that the Calculus of Probability is a
creation of mathematics but that the master mathe-
matician is constantly required to exercise judgment —
judgment, that is, in matters not admitting of cer-
tainty — balancing probabilities not yet reduced nor
even reducible perhaps to calculation; when we reflect
that he is called upon to exercise a function analogous
to that of the comparative anatomist like Cuvier, com-
paring theories and doctrines of every degree of similar-
ity and dissimilarity of structure; when, finally, we
reflect that he seldom deals with a single idea at a time,
but is for the most part engaged in wielding organized
hosts of them, as a general wields at once the divisions
of an army or as a great civil administrator directs from
his central office diverse and scattered but related groups
of interests and operations; then, I say, the current

302 MATHEMATICS

opinion that devotion to mathematics unfits the devotee
for practical affairs should be known for false on a
priori grounds. And one should be thus prepared to
find that as a fact Gaspard Monge, creator of descrip-
tive geometry, author of the classic "Applications de
Panalyse a la geometric "; Lazare Carnot, author of the
celebrated works, " Geometric de position," and " Re-
flexions sur la Metaphysique du Calcul infinitesimal";
Fourier, immortal creator of the "Theorie analytique
de la chaleur"; Arago, rightful inheritor of Monge 's
chair of geometry; and Poncelet, creator of pure pro-
jective geometry; one should not be surprised, I say,
to find that these and other mathematicians in a land
sagacious enough to invoke their aid, rendered, alike
in peace and in war, eminent public service.

To speak at length, if that were necessary, of Huxley's
deliverance that the study of mathematics "knows
nothing of causation," the "law of my song and the
hastening hour forbid." Suffice it to say in passing
that when the mathematician seeks the consequences
of given suppositions, saying 'when these precede,
those will follow,' and when, having plied a circle, a
sphere or other form chosen from among infinitudes of
configurations, with some transformation among infinite
hosts at his disposal, he speaks of, its 'effect,' then, I
submit, he is employing the language of causation
with as nice propriety as it admits of in a world where,
as everyone knows, except such as still enjoy the bless-
ings of a juvenile philosophy, the best we can say is
that the ceaseless shuttles fly back and forth, and
streams of events without original source flow on with-
out ultimate termination. Indeed it is a certain and
signal lesson of science in all its forms everywhere that
the language of cause and effect, except in the sense of

MATHEMATICS

303

facts being lawfully implied in other facts, has no
indispensable use.

I have not spoken of " Applied Mathematics," and
that for the best of reasons: there is, strictly speaking,
no such thing. The term indeed exists, and, in a con-
servative practical world that cares but little for "The
nice sharp quillets of the law," it will doubtless persist
as a convenient designation for something that never
existed and never can. It is of the very essence of the
practician type of mind not to know aught as it is in
itself nor aught as self-justified but to mistake the
secondary and accidental for the primary and essential,
to blink and elude the presence of immediate worth,
and being thus blind to instant and immanent ends,
to revel in means and uses and applications, requiring
all things to excuse their being by extraneous and
emanant effects, — vindicating the stately elm by its
promise of lumber, or the lily by its message of purity,
or the flood of Niagara by its available energy, or even
knowledge itself by the worldly advantage and the power
which it gives. I am told that even the deep and ex-
quisite terminology of art has been to some extent
invaded by such barbarous and shallow phrases as
'applied music,' 'applied architecture,' 'applied sculp-
ture,' 'applied painting,' as if Beauty, virgin mother of
art, could, without dissolution of her essential char-
acter, consciously become the willing drudge and para-
mour of Use. And I suppose we are fated yet to hear
of applied glory, applied holiness, applied poetry —
i. e., poetry that is consciously pedagogic or that aims
at a moral and thereby sinks or rises to the level of a
sermon — of applied joy, applied ontology, yea, of
applied inapplicability itself.

It is in implications and not in applications that

304 MATHEMATICS

mathematics has its lair. Applied mathematics is mathe-
matics simply or is not mathematics at all. To think
aright is no characteristic striving of a class of men; it
is a common aspiration; and Mechanics, Mathematical
Physics, Mathematical Astronomy, and the other chief
Anwendungsgebiete of mathematics, as Geodesy, Geo-
physics, and Engineering in its various branches, are all
of them but so many witnesses of the truth of Riemann's
saying that "Natural science is the attempt to com-
prehenoT nature by means of exact concepts." A gas
molecule regarded as a minute sphere or other geometric
form, however complicate; stars and planets conceived as
ellipsoids or as points, and their orbits as loci; time and
space, mass and motion and impenetrability; velocity,
acceleration and energy; the concepts of norm and
average; — what are these but mathematical notions?
And the wondrous garment woven of them in the loom
of logic — • what is that but mathematics? Indeed
every branch of so-called applied mathematics is a
mixed doctrine, being thoroughly analyzable into two
disparate parts: one of these consists of determinate
concepts formally combined in accordance with the
canons of logic, i. e., it is mathematics and not natural
science viewed as matter of observation and experiment;
the other is such matter and is natural science in that
conception of it and not mathematics. No fibre of
either component is a filament of the other. It is a
fundamental error to regard the term Mathematicisa-
tion of thought as the importation of a tool into a
foreign workshop. It does not signify the transition of
mathematics conceived as a thing accomplished over into
some outlying domain like physics, for example. Its
significance is different radically, far deeper and far
wider. It means the growth of mathematics itself, its

MATHEMATICS 305

extension and development from within; it signifies
the continuous revelation, the endlessly progressive
coming into view, of the static universe of logic; or,
to put it dynamically, it means the evolution of intel-
lect, the upward striving and aspiration of thought
everywhere, to the level of cogency, precision and exacti-
tude. This self-propagation of the rational logos, the
springing up of mathetic rigor even in void and formless
places, in the very retreats of chaos, is to my mind the
most impressive and significant phenomenon in the
history of science, and never so strikingly manifest as
in the last half hundred years. Seventy-two years ago,
even Comte, the stout advocate of mathematics as
constituting "the veritable point of departure for all
rational scientific education, general or special," ex-
pressed the opinion that we should never "be in posi-
tion by any means whatever to study the chemical
composition of the stars." In less than twenty-five
years thereafter that negative prophecy was falsified
by the chemical genius of Bunsen fortified by the mathe-
matics of Kirchoff. Not only has mathematics grown,
in the domain of Physics, into the vast proportions of
Rational Dynamics, but the derivative and integral of
the Calculus, and Differential Equations, are more and
more finding subsistence in Chemistry also, and by the
work of Nernst and others even the foundations of the
latter science are being laid in mathematico-physical
considerations. Merely to sketch most briefly the
mathematical literature that has grown up in the field
of Political Economy requires twenty-five pages of the
above mentioned Encyklopadie of mathematics. Similar
sketches for Statistics and Life Insurance require no
less than thirty and sixty-five pages respectively. Even
in the banking and elusive matter of Psychology, the

306 MATHEMATICS

work of Herbart, Fechner, Weber, Wundt and others
confirms the hope that the soil of that great field will
some day support a vigorous growth of mathematics.
It seems indeed as if the entire surface of the world of
human consciousness were predestined to be covered
over, in varying degrees of luxuriance, by the flora of
mathetic science.

But while mathematics may spring up and flourish
in any and all experimental and observational fields, it
is by no means to be expected that 'experiment and
observation' will ever thus be superseded. Such domains
are rather destined to be occupied at the same time by
two tenants, mathematical science and science that is
not mathematical. But while the former will serve as
an ideal standard for the latter, mathematics has
neither the power nor the disposition to disseize experi-
ment and observation of any holdings that are theirs
by the rights of conquest and use. Between mathe-
matics on the one hand and non-mathematical science
on the other, there can never occur collision or quarrel,
for the reason that the two interests are ultimately
discriminated by the kind of curiosity whence they
spring. The mathematician is curious about definite
naked relationships, about logically possible modes of
order, about varieties of implication, about completely
determined or determinable functional relationships,
considered solely in and of themselves, considered, that
is, without the slightest concern about any question
whether or no they have any external or sensuous
validity or other sort of validity than that of being
logically thinkable. It is the aggregate of things think-
able logically that constitutes the mathematician's
universe, and it is inconceivably richer in mathetic
content than can be any outer world of sense such as the

MATHEMATICS 307

physical universe according to which we chance to have
our physical being.

This mere speck of a physical universe in which the
chemist, the physicist, the astronomer, the biologist,
the sociologist, and the rest of nature students, find
their great fields and their deep and teeming interests,
may be a realm of invariant uniformities, or laws; it
may be a mechanically organic aggregate, connected
into an ordered whole by a tissue of completely defin-
able functional relationships; and it may not. It may
be that the universe eternally has been and is a genuine
cosmos; it may be that the external sea of things im-
mersing us, although it is ever changing infinitely,
changes only lawfully, in accordance with a system
of immutable rules of order that constitute an invariant
at once underived and indestructible and securing
everlasting harmony through and through; and it may
not be such. The student of nature assumes, he rightly
assumes, that it is; and, moved and sustained by char-
acteristic appropriate curiosity, he endeavors to find
in the outer world what are the elements and what the
relationships assumed by him to be valid there. The
mathematician as such does not make that assumption
and does not seek for elements and relationships in
the outer world.

Is the assumption correct? Undoubtedly it is admis-
sible, and as a working hypothesis it is undoubtedly
exceedingly useful or even indispensable to the student
of external nature; but is it true? The mathematician
as man does not know although he cares. Man as
mathematician neither knows nor cares. The mathe-
matician does know, however, that, if the assumption
be correct, every relationship that is valid in nature
is, in abstractu, an element in his domain, a subject for

308 MATHEMATICS

his study. He knows, too, at least he strongly suspects,
that, if the assumption be not correct, his domain
remains the same absolutely, and the title of mathe-
matics to human regard " would remain unimpeached
and unimpaired" were the universe without a plan or,
having a plan, if it "were unrolled like a map at our
feet, and the mind of man qualified to take in the
whole scheme of creation at a glance."

The two realms, of mathematics, of natural science,
like the two curiosities and the two attitudes, the mathe-
matician's and the nature student's, are fundamentally
distinct and disparate. To think logically the logically
thinkable — that is the mathematician's aim. To as-
sume that nature is thus thinkable, an embodied rational
logos, and to discover the thought supposed incarnate
there — these are at once the principle and the hope of
the student of nature.

Suppose the latter student is right and that the outer
universe really is an embodied logos of reason, does it
follow that all the logically thinkable is incorporated
in it? It seems not. Indeed there appears to be many
a rational logos. A cosmos, a harmoniously ordered
universe, one that through and through is self-com-
patible, can hardly be the whole of reason materialized
and objectified. At all events the mathematician has
delight in the conceptual construction and in the con-
templation of divers systems that are inconsistent
with one another though each is thoroughly self -coherent.
He constructs in thought a summitless hierarchy of
hyperspaces, an endless series of ordered worlds, worlds
that are possible and logically actual. And he is con-
tent not to know if any of them be otherwise actual
or actualized. There is, for example, a Euclidean
geometry and there are infinitely many kinds of non-

MATHEMATICS 309

Euclidean. These doctrines, regarded as true descrip-
tions of some one actual space, are incompatible. In
our universe, to be specific, if it be as Plato thought
and natural science takes for granted, a geometrized
or geometrizable affair, then one of these geometries
may be, none of them may be, not all of them can be,
objectively valid. But in the infinitely vaster world of
pure thought, in the world of mathesis, all of them are
valid; there they co-exist, there they interlace and blend
among themselves and others as differing strains of
a hypercosmic harmony.

It is from some such elevation, not the misty lowland
of the sensuously and materially Actual, but from
a mount of speculation lawfully rising into the azure
of the logically Possible, that one may glimpse the
dawn heralded by the avowal of Leibnitz: "Ma meta-
physique est toute mathematique" Time fails me to deal
fittingly with the great theme herewith suggested, but
I cannot quite forbear to express briefly my conviction
that, apart from its service to kindred interests of
thought as a standard of clarity, rigor and certitude,
mathematics is and will be found to be an inexhaust-
ible quarry of material — of ontologic types, of ideas
and problems, of distinctions, discriminants and hints,
evidences, analogies and intimations — all for the ex-
ploitation and use of Philosophy, Psychology, and
Theology. The allusion is not to such celebrated alli-
ances of philosophy and mathesis as flourished in the
school of Pythagoras and in the gigantic personalities
of Plato, Descartes, Spinoza, and Leibnitz, nor to the
more technical mathematico-philosophical researches
and speculations of our own time by such as C. S.
Peirce, Russell, Whitehead, Peano, G. Cantor, Couturat
and Poincare, glorious as were those alliances and

310 MATHEMATICS

important as these researches are. The reference is
rather to the unappreciated fact that the measureless
accumulated wealth of the realm of exact thought is
at once a marvelous mine of subject matter and a rich
and ready arsenal for those great human concerns of
reflective and militant thought that is none the less
important because it is not exact.

For the vindication of that claim, a hint or two must
here suffice. The modern mathematical concepts of
number, time, space, order, infinitude, finitude, group,
manifold, functionality, and innumerable hosts of others,
the varied processes of mathematics, and the principles
and modes of its growth and evolution, all of these or
nearly all still challenge and still await those kinds of
analysis that are proper to the philosopher and the
psychologist. The psychology of Euclidean, non-Eu-
clidean, and hyperspaces, the question of the intuita-
bility of the latter, the secret of their having become
not only indispensable in various branches of mathe-
matics but instrumentally useful in other fields also,
as in the kinetic theory of gases; the question, for
example, why it is that while thought maintains a
straightforward course through four-dimensional space,
imagination travels through it on a zigzag path, of
two logically identical configurations, being partially
or completely blind to the one, yet perfectly beholding
the other; the evaluation and adjustment of the con-
tradictory claims of Poincare and his school on the one
hand and of Mach and his disciples on the other, the
former contending that Modern Analysis is a "free
creation of the human spirit" guided indeed but not
constrained by experience of the external world, being
merely kept by this from aimless wandering in wayward
paths; while the latter maintain that mathematical

MATHEMATICS 31 1

concepts, however tenuous or remote or recondite, have
been literally evolved continuously in accordance with
the needs of the animal organism and with environ-
mental conditions out of the veriest elements (feelings)
of physical life, and accordingly that the purest offspring
of mathematical thought may trace a legitimate lineage
back and down to the lowliest rudiments of physical
and physiological experience : — these problems and
such as these are, I take it, problems for the student
of mind as mind and for the student of psycho-physics.
Regarding the relations of mathesis to the former
"queen of all the sciences," I have on this occasion
but little to say. I do not believe that the declined
estate of Theology is destined to be permanent. The
present is but an interregnum in her reign and her
fallen days will have an end. She has been deposed
mainly because she has not seen fit to avail herself
promptly and fully of the dispensations of advancing
knowledge. The aims, however, of the ancient mistress
are as high as ever, and when she shall have made good
her present lack of modern education and learned to
extend a generous and eager hospitality to modern
light, she will reascend, and will occupy with dignity as
of yore an exalted place in the ascending scale of human
interests and the esteem of enlightened men. And
mathematics, by the character of her inmost being, is
especially qualified, I believe, to assist in the restoration.
It was but little more than a generation ago that the
mathematician, philosopher and theologian, Bernhard
Bolzano, dispelled the clouds that throughout all the
foregone centuries had enveloped the notion of Infini-
tude in darkness, completely sheared the great term of
its vagueness without shearing it of its strength, and
thus rendered it forever available for the purposes of

312 MATHEMATICS

logical discourse. Whereas, too, in former times the
Infinite betrayed its presence not indeed to the faculties
of Logic but only to the spiritual Imagination and Sensi-
bility, mathematics has shown, even during the life of
the elder men here present, — and the achievement
marks an epoch in the history of man, — that the
structure of Transfinite Being is open to exploration
by the organon of Thought. Again, it is in the mathe-
matical doctrine of Invariance, the realm wherein are
sought and found configurations and types of being
that, amid the swirl and stress of countless hosts of
transformations, remain immutable, and the spirit dwells
in contemplation of the serene and eternal reign of the
subtile law of Form, it is there that Theology may find,
if she will, the clearest conceptions, the noblest symbols,
the most inspiring intimations, the most illuminating
illustrations, and the surest guarantees of the object of
her teaching and her quest, an Eternal Being, unchan-
ging in the midst of the universal flux.

It is not, however, by any considerations or estimates
of utility in any form however high it be or essential
to the worldly weal of man; it is not by evaluating
mastery of the processes of measurement and compu-
tation, though these are continuously vital everywhere
to the conduct of practical life; nor is it by strengthen-
ing the arms of natural science and speeding her con-
quests in a thousand ways and a hundred fields; nor yet
by extending the empire of the human intellect over the
realms of number and space and establishing the do-
minion of thought throughout the universe of logic; it is
not even by affording argument and fact and light to
theology and so contributing to the advancement of
her supreme concerns; — it is not by any of these
considerations nor by all of them that Mathematics,

MATHEMATICS 313

were she called upon to do so, would rightly seek to
vindicate her highest claims to human regard. It
requires indeed but little penetration to see that no
science, no art, no doctrine, no human activity whatever,
however humble or high, can ultimately succeed in
justifying itself in. terms of measurable fruits and ema-
nant effects, for these remain always to be themselves
appraised, and the process of such attempted vindica-
tion is plainly fated to issue only in regression without
an end. Such Baconian apologetic, when offered as
final, quite mistakes the finest mood of the scientific
spirit and is beneath the level of academic faith. Sci-
ence does not seek emancipation in order to become a
drudge, she consents to serve indeed but her service
aims at freedom as an end.

Man has been so long a slave of circumstance and
need, he has been so long constrained to seek license
for his summit faculties, in lower courts without appeal,
that a sudden transitory moment of release sets him
trembling with distrust and fear, an occasional imperfect
vision of the instant dignity of his spiritual enterprises
is at once obscured by doubt, and he straightway
descends into the market places of the world to excuse
or to justify his illumination, pleading some mere utility
against the ignoring or the condemnation of an insight
or an inspiration whose worth is nevertheless immediate
and no more needs and no more admits of utilitarian
justification than the breaking of morning light on
mountain peaks or the bounding of lambs in a meadow.

The solemn cant of Science in our day and her
sombre visage are but the lingering tone and shade of
the prisonhouse, and they will pass away. Science is
destined to appear as the child and the parent of freedom
blessing the earth without design. Not in the ground

314 MATHEMATICS

of need, not in bent and painful toil, but in the deep-
centred play-instinct of the world, in the joyous mood
of the eternal Being, which is always young, Science
has her origin and root; and her spirit, which is the
spirit of genius in moments of elevation, is but a sub-
limated form of play, the austere _and lofty^ analogue
of the kitten playing with the entangled skein or of the
eaglet sporting with the mountain winds.

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