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EDITED  BY J.  H.  MUIRHEAD,  LL.D. 


INTRODUCTION  TO   MATHEMATICAL 
PHILOSOPHY 


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London :    George  Allen  &**   Unwin,  Ltd. 


INTRODUCTION   TO 

MATHEMATICAL 

PHILOSOPHY 


BY 


BERTRAND  RUSSELL 


LONDON  :    GEORGE  ALLEN  &  UNWIN,  LTD. 
NEW    YORK:     THE     MACMILLAN     CO, 


First  published  May  1919 
Second  Edition  April  1920 


[All  rights  reserved} 


PREFACE 

THIS  book  is  intended  essentially  as  an  "  Introduction,"  and 
does  not  aim  at  giving  an  exhaustive  discussion  of  the  problems 
with  which  it  deals.  It  seemed  desirable  to  set  forth  certain 
results,  hitherto  only  available  to  those  who  have  mastered 
logical  symbolism,  in  a  form  offering  the  minimum  of  difficulty 
to  the  beginner.  The  utmost  endeavour  has  been  made  to 
avoid  dogmatism  on  such  questions  as  are  still  open  to  serious 
doubt,  and  this  endeavour  has  to  some  extent  dominated  the 
choice  of  topics  considered.  The  beginnings  of  mathematical 
logic  are  less  definitely  known  than  its  later  portions,  but  are  of 
at  least  equal  philosophical  interest.  Much  of  what  is  set  forth 
in  the  following  chapters  is  not  properly  to  be  called  "  philosophy," 
though  the  matters  concerned  were  included  in  philosophy  so 
long  as  no  satisfactory  science  of  them  existed.  The  nature  of 
infinity  and  continuity,  for  example,  belonged  in  former  days 
to  philosophy,  but  belongs  now  to  mathematics.  Mathematical 
philosophy,  in  the  strict  sense,  cannot,  perhaps,  be  held  to  include 
such  definite  scientific  results  as  have  been  obtained  in  this 
region ;  the  philosophy  of  mathematics  will  naturally  be  ex 
pected  to  deal  with  questions  on  the  frontier  of  knowledge,  as 
to  which  comparative  certainty  is  not  yet  attained.  But 
speculation  on  such  questions  is  hardly  likely  to  be  fruitful 
unless  the  more  scientific  parts  of  the  principles  of  mathematics 
are  known.  A  book  dealing  with  those  parts  may,  therefore, 
claim  to  be  an  introduction  to  mathematical  philosophy,  though 
it  can  hardly  claim,  except  where  it  steps  outside  its  province, 
to  be  actually  dealing  with  a  part  of  philosophy.  It  does  deal, 


vi  Introduction  to  Mathematical  Philosophy 

however,  with  a  body  of  knowledge  which,  to  those  who  accept 
it,  appears  to  invalidate  much  traditional  philosophy,  and  even 
a  good  deal  of  what  is  current  in  the  present  day.  In  this  way, 
as  well  as  by  its  bearing  on  still  unsolved  problems,  mathematical 
logic  is  relevant  to  philosophy.  For  this  reason,  as  well  as  on 
account  of  the  intrinsic  importance  of  the  subject,  some  purpose 
may  be  served  by  a  succinct  account  of  the  main  results  of 
mathematical  logic  in  a  form  requiring  neither  a  knowledge  of 
mathematics  nor  an  aptitude  for  mathematical  symbolism. 
Here,  however,  as  elsewhere,  the  method  is  more  important  than 
the  results,  from  the  point  of  view  of  further  research ;  and  the 
method  cannot  well  be  explained  within  the  framework  of  such 
a  book  as  the  following.  It  is  to  be  hoped  that  some  readers 
may  be  sufficiently  interested  to  advance  to  a  study  of  the 
method  by  which  mathematical  logic  can  be  made  helpful  in 
investigating  the  traditional  problems  of  philosophy.  But  that 
is  a  topic  with  which  the  following  pages  have  not  attempted 
to  deal. 

BERTRAND   RUSSELL. 


EDITOR'S   NOTE 

THOSE  who,  relying  on  the  distinction  between  Mathematical 
Philosophy  and  the  Philosophy  of  Mathematics,  think  that  this 
book  is  out  of  place  in  the  present  Library,  may  be  referred  to 
what  the  author  himself  says  on  this  head  in  the  Preface.  It  is 
not  necessary  to  agree  with  what  he  there  suggests  as  to  the 
readjustment  of  the  field  of  philosophy  by  the  transference  from 
it  to  mathematics  of  such  problems  as  those  of  class,  continuity, 
infinity,  in  order  to  perceive  the  bearing  of  the  definitions  and 
discussions  that  follow  on  the  work  of  "  traditional  philosophy." 
If  philosophers  cannot  consent  to  relegate  the  criticism  of  these 
categories  to  any  of  the  special  sciences,  it  is  essential,  at  any 
rate,  that  they  should  know  the  precise  meaning  that  the  science 
of  mathematics,  in  which  these  concepts  play  so  large  a  part, 
assigns  to  them.  If,  on  the  other  hand,  there  be  mathematicians 
to  whom  these  definitions  and  discussions  seem  to  be  an  elabora 
tion  and  complication  of  the  simple,  it  may  be  well  to  remind 
them  from  the  side  of  philosophy  that  here,  as  elsewhere,  apparent 
simplicity  may  conceal  a  complexity  which  it  is  the  business  of 
somebody,  whether  philosopher  or  mathematician,  or,  like  the 
author  of  this  volume,  both  in  one,  to  unravel. 


vii 


CONTENTS 


CHAP.  PAGE 

PREFACE               ........  V 

EDITOR'S  NOTE         .......  vii 

1.  THE    SERIES    OF    NATURAL    NUMBERS     ....  I 

2.  DEFINITION    OF   NUMBER     .  .  .  .  .  ,11 

3.  FINITUDE    AND    MATHEMATICAL    INDUCTION  2O 

4.  THE   DEFINITION    OF   ORDER           .....  29 

5.  KINDS   OF   RELATIONS            ......  42 

6.  SIMILARITY    OF    RELATIONS              .              .              .              .              •  S2 

7.  RATIONAL,    REAL,    AND    COMPLEX    NUMBERS  ...  63 

8.  INFINITE    CARDINAL    NUMBERS      .....  77 

9.  INFINITE    SERIES   AND    ORDINALS             ....  89 

10.  LIMITS   AND    CONTINUITY    ......  97 

11.  LIMITS   AND    CONTINUITY    OF   FUNCTIONS        .             .             .  107 

12.  SELECTIONS    AND   THE    MULTIPLICATIVE    AXIOM        .              •  IJ7 

13.  THE    AXIOM    OF    INFINITY    AND    LOGICAL   TYPES       .              .  131 

14.  INCOMPATIBILITY    AND   THE   THEORY    OF    DEDUCTION        .  144 

15.  PROPOSITIONAL    FUNCTIONS             .....  155 

16.  DESCRIPTIONS  ........  167 

17.  CLASSES  ....                                                      .  l8l 

18.  MATHEMATICS    AND    LOGIC  ......  194 

INDEX 207 


Viii 


Introduction  to 
Mathematical   Philosophy 

CHAPTER  I 

THE    SERIES    OF    NATURAL    NUMBERS 

MATHEMATICS  is  a  study  which,  when  we  start  from  its  most 
familiar  portions,  may  be  pursued  in  either  of  two  opposite 
directions.  The  more  familiar  direction  is  constructive,  towards 
gradually  increasing  complexity :  from  integers  to  fractions, 
real  numbers,  complex  numbers ;  from  addition  and  multi 
plication  to  differentiation  and  integration,  and  on  to  higher 
mathematics.  The  other  direction,  which  is  less  familiar, 
proceeds,  by  analysing,  to  greater  and  greater  abstractness 
and  logical  simplicity ;  instead  of  asking  what  can  be  defined 
and  deduced  from  what  is  assumed  to  begin  with,  we  ask  instead 
what  more  general  ideas  and  principles  can  be  found,  in  terms 
of  which  what  was  our  starting-point  can  be  defined  or  deduced. 
It  is  the  fact  of  pursuing  this  opposite  direction  that  characterises 
mathematical  philosophy  as  opposed  to  ordinary  mathematics. 
But  it  should  be  understood  that  the  distinction  is  one,  not  in 
the  subject  matter,  but  in  the  state  of  mind  of  the  investigator. 
Early  Greek  geometers,  passing  from  the  empirical  rules  of 
Egyptian  land-surveying  to  the  general  propositions  by  which 
those  rules  were  found  to  be  justifiable,  and  thence  to  Euclid's 
axioms  and  postulates,  were  engaged  in  mathematical  philos 
ophy,  according  to  the  above  definition ;  but  when  once  the 
axioms  and  postulates  had  been  reached,  their  deductive  employ 
ment,  as  we  find  it  in  Euclid,  belonged  to  mathematics  in  the 

I 


2  Introduction  to  Mathematical  Philosophy 

ordinary  sense.  The  distinction  between  mathematics  and 
mathematical  philosophy  is  one  which  depends  upon  the  interest 
inspiring  the  research,  and  upon  the  stage  which  the  research 
has  reached ;  not  upon  the  propositions  with  which  the  research 
is  concerned. 

We  may  state  the  same  distinction  in  another  way.  The 
most  obvious  and  easy  things  in  mathematics  are  not  those  that 
come  logically  at  the  beginning  ;  they  are  things  that,  from 
the  point  of  view  of  logical  deduction,  come  somewhere  in  the 
middle.  Just  as  the  easiest  bodies  to  see  are  those  that  are 
neither  very  near  nor  very  far,  neither  very  small  nor  very 
great,  so  the  easiest  conceptions  to  grasp  are  those  that  are 
neither  very  complex  nor  very  simple  (using  "  simple "  in  a 
logical  sense).  And  as  we  need  two  sorts  of  instruments,  the 
telescope  and  the  microscope,  for  the  enlargement  of  our  visual 
powers,  so  we  need  two  sorts  of  instruments  for  the  enlargement 
of  our  logical  powers,  one  to  take  us  forward  to  the  higher 
mathematics,  the  other  to  take  us  backward  to  the  logical 
foundations  of  the  things  that  we  are  inclined  to  take  for  granted 
in  mathematics.  We  shall  find  that  by  analysing  our  ordinary 
mathematical  notions  we  acquire  fresh  insight,  new  powers, 
and  the  means  of  reaching  whole  new  mathematical  subjects 
by  adopting  fresh  lines  of  advance  after  our  backward  journey. 
It  is  the  purpose  of  this  book  to  explain  mathematical  philos 
ophy  simply  and  untechnically,  without  enlarging  upon  those 
portions  which  are  so  doubtful  or  difficult  that  an  elementary 
treatment  is  scarcely  possible.  A  full  treatment  will  be  found 
in  Principia  Mathematica  ; *  the  treatment  in  the  present  volume 
is  intended  merely  as  an  introduction. 

To  the  average  educated  person  of  the  present  day,  the 
obvious  starting-point  of  mathematics  would  be  the  series  of 
whole  numbers, 

i,  2,  3,  4,  ...  etc. 

1  Cambridge  University  Press,  vol.  i.,  1910  ;  vol.  ii.,  1911  ;  vol.  iii.,  1913. 
By  Whitehead  and  Russell. 


The  Series  of  Natural  Numbers  3 

Probably  only  a  person  with  some  mathematical  knowledge 
would  think  of  beginning  with  o  instead  of  with  i,  but  we  will 
presume  this  degree  of  knowledge ;  we  will  take  as  our  starting- 
point  the  series  : 

o,  i,  2,  3,  .  .  .  n,  n+ 1,  .  .  . 

and  it  is  this  series  that  we  shall  mean  when  we  speak  of  the 
"  series  of  natural  numbers." 

It  is  only  at  a  high  stage  of  civilisation  that  we  could  take 
this  series  as  our  starting-point.  It  must  have  required  many 
ages  to  discover  that  a  brace  of  pheasants  and  a  couple  of  days 
were  both  instances  of  the  number  2  :  the  degree  of  abstraction 
involved  is  far  from  easy.  And  the  discovery  that  I  is  a  number 
must  have  been  difficult.  As  for  o,  it  is  a  very  recent  addition ; 
the  Greeks  and  Romans  had  no  such  digit.  If  we  had  been 
embarking  upon  mathematical  philosophy  in  earlier  days,  we 
should  have  had  to  start  with  something  less  abstract  than  the 
series  of  natural  numbers,  which  we  should  reach  as  a  stage  on 
our  backward  journey.  When  the  logical  foundations  of  mathe 
matics  have  grown  more  familiar,  we  shall  be  able  to  start  further 
back,  at  what  is  now  a  late  stage  in  our  analysis.  But  for  the 
moment  the  natural  numbers  seem  to  represent  what  is  easiest 
and  most  familiar  in  mathematics. 

But  though  familiar,  they  are  not  understood.  Very  few 
people  are  prepared  with  a  definition  of  what  is  meant  by 
"  number,"  or  "  o,"  or  "  I."  It  is  not  very  difficult  to  see  that, 
starting  from  o,  any  other  of  the  natural  numbers  can  be  reached 
by  repeated  additions  of  I,  but  we  shall  have  to  define  what 
we  mean  by  "  adding  I,"  and  what  we  mean  by  "  repeated." 
These  questions  are  by  no  means  easy.  It  was  believed  until 
recently  that  some,  at  least,  of  these  first  notions  of  arithmetic 
must  be  accepted  as  too  simple  and  primitive  to  be  defined. 
Since  all  terms  that  are  defined  are  defined  by  means  of  other 
terms,  it  is  clear  that  human  knowledge  must  always  be  content 
to  accept  some  terms  as  intelligible  without  definition,  in  order 


4  Introduction  to  Mathematical  Philosophy 

to  have  a  starting-point  for  its  definitions.  It  is  not  clear  that 
there  must  be  terms  which  are  incapable  of  definition  :  it  is 
possible  that,  however  far  back  we  go  in  defining,  we  always 
might  go  further  still.  On  the  other  hand,  it  is  also  possible 
that,  when  analysis  has  been  pushed  far  enough,  we  can  reach 
terms  that  really  are  simple,  and  therefore  logically  incapable 
of  the  sort  of  definition  that  consists  in  analysing.  This  is  a 
question  which  it  is  not  necessary  for  us  to  decide ;  for  our 
purposes  it  is  sufficient  to  observe  that,  since  human  powers 
are  finite,  the  definitions  known  to  us  must  always  begin  some 
where,  with  terms  undefined  for  the  moment,  though  perhaps 
not  permanently. 

All  traditional  pure  mathematics,  including  analytical  geom 
etry,  may  be  regarded  as  consisting  wholly  of  propositions 
about  the  natural  numbers.  That  is  to  say,  the  terms  which 
occur  can  be  defined  by  means  of  the  natural  numbers,  and 
the  propositions  can  be  deduced  from  the  properties  of  the 
natural  numbers — with  the  addition,  in  each  case,  of  the  ideas 
and  propositions  of  pure  logic. 

That  all  traditional  pure  mathematics  can  be  derived  from 
the  natural  numbers  is  a  fairly  recent  discovery,  though  it  had 
long  been  suspected.  Pythagoras,  who  believed  that  not  only 
mathematics,  but  everything  else  could  be  deduced  from 
numbers,  was  the  discoverer  of  the  most  serious  obstacle  in 
the  way  of  what  is  called  the  "  arithmetising  "  of  mathematics. 
It  was  Pythagoras  who  discovered  the  existence  of  incom- 
mensurables,  and,  in  particular,  the  incommensurability  of  the 
side  of  a  square  and  the  diagonal.  If  the  length  of  the  side  is 
I  inch,  the  number  of  inches  in  the  diagonal  is  the  square  root 
of  2,  which  appeared  not  to  be  a  number  at  all.  The  problem 
thus  raised  was  solved  only  in  our  own  day,  and  was  only  solved 
completely  by  the  help  of  the  reduction  of  arithmetic  to  logic, 
which  will  be  explained  in  following  chapters.  For  the  present, 
we  shall  take  for  granted  the  arithmetisation  of  mathematics, 
though  this  was  a  feat  of  the  very  greatest  importance. 


The  Series  of  Natural  Numbers  5 

Having  reduced  all  traditional  pure  mathematics  to  the 
theory  of  the  natural  numbers,  the  next  step  in  logical  analysis 
was  to  reduce  this  theory  itself  to  the  smallest  set  of  premisses 
and  undefined  terms  from  which  it  could  be  derived.  This  work 
was  accomplished  by  Peano.  He  showed  that  the  entire  theory 
of  the  natural  numbers  could  be  derived  from  three  primitive 
ideas  and  five  primitive  propositions  in  addition  to  those  of 
pure  logic.  These  three  ideas  and  five  propositions  thus  became, 
as  it  were,  hostages  for  the  whole  of  traditional  pure  mathe 
matics.  If  they  could  be  defined  and  proved  in  terms  of  others, 
so  could  all  pure  mathematics.  Their  logical  "  weight,"  if  one 
may  use  such  an  expression,  is  equal  to  that  of  the  whole  series 
of  sciences  that  have  been  deduced  from  the  theory  of  the  natural 
numbers  ;  the  truth  of  this  whole  series  is  assured  if  the  truth 
of  the  five  primitive  propositions  is  guaranteed,  provided,  of 
course,  that  there  is  nothing  erroneous  in  the  purely  logical 
apparatus  which  is  also  involved.  The  work  of  analysing  mathe 
matics  is  extraordinarily  facilitated  by  this  work  of  Peano's. 

The  three  primitive  ideas  in  Peano's  arithmetic  are  : 

o,  number,  successor. 

By  "  successor "  he  means  the  next  number  in  the  natural 
order.  That  is  to  say,  the  successor  of  o  is  I,  the  successor  of 
I  is  2,  and  so  on.  By  "  number  "  he  means,  in  this  connection, 
the  class  of  the  natural  numbers.1  He  is  not  assuming  that 
we  know  all  the  members  of  this  class,  but  only  that  we  know 
what  we  mean  when  we  say  that  this  or  that  is  a  number,  just 
as  we  know  what  we  mean  when  we  say  "  Jones  is  a  man," 
though  we  do  not  know  all  men  individually. 

The  five  primitive  propositions  which  Peano  assumes  are  : 

(1)  o  is  a  number. 

(2)  The  successor  of  any  number  is  a  number. 

(3)  No  two  numbers  have  the  same  successor. 

1  We  shall  use  "  number  "  in  this  sense  in  the  present  chapter.  After 
wards  the  word  will  be  used  in  a  more  general  sense. 


6  Introduction  to  Mathematical  Philosophy 

(4)  o  is  not  the  successor  of  any  number. 

(5)  Any  property  which  belongs  to  o,  and  also  to  the  successor 

of  every  number  which  has  the  property,  belongs  to  all 
numbers. 

The  last  of  these  is  the  principle  of  mathematical  induction. 
We  shall  have  much  to  say  concerning  mathematical  induction 
in  the  sequel ;  for  the  present,  we  are  concerned  with  it  only 
as  it  occurs  in  Peano's  analysis  of  arithmetic. 

Let  us  consider  briefly  the  kind  of  way  in  which  the  theory 
of  the  natural  numbers  results  from  these  three  ideas  and  five 
propositions.  To  begin  with,  we  define  I  as  "  the  successor  of  o," 
2  as  "  the  successor  of  I,"  and  so  on.  We  can  obviously  go 
on  as  long  as  we  like  with  these  definitions,  since,  in  virtue  of 
(2),  every  number  that  we  reach  will  have  a  successor,  and,  in 
virtue  of  (3),  this  cannot  be  any  of  the  numbers  already  defined, 
because,  if  it  were,  two  different  numbers  would  have  the  same 
successor ;  and  in  virtue  of  (4)  none  of  the  numbers  we  reach 
in  the  series  of  successors  can  be  o.  Thus  the  series  of  successors 
gives  us  an  endless  series  of  continually  new  numbers.  In  virtue 
of  (5)  all  numbers  come  in  this  series,  which  begins  with  o  and 
travels  on  through  successive  successors  :  for  (a)  o  belongs  to 
this  series,  and  (b)  if  a  number  n  belongs  to  it,  so  does  its  successor, 
whence,  by  mathematical  induction,  every  number  belongs  to 
the  series. 

Suppose  we  wish  to  define  the  sum  of  two  numbers.  Taking 
any  number  m,  we  define  m-\-o  as  m,  and  m-\-(n-{-i)  as  the 
successor  of  m-\-n.  In  virtue  of  (5)  this  gives  a  definition  of 
the  sum  of  m  and  n,  whatever  number  n  may  be.  Similarly 
we  can  define  the  product  of  any  two  numbers.  The  reader  can 
easily  convince  himself  that  any  ordinary  elementary  proposition 
of  arithmetic  can  be  proved  by  means  of  our  five  premisses, 
and  if  he  has  any  difficulty  he  can  find  the  proof  in  Peano. 

It  is  time  now  to  turn  to  the  considerations  which  make  it 
necessary  to  advance  beyond  the  standpoint  of  Peano,  who 


The  Series  of  Natural  Numbers  7 

represents  the  last  perfection  of  the  "  arithmetisation "  of 
mathematics,  to  that  of  Frege,  who  first  succeeded  in  "  logicising  " 
mathematics,  i.e.  in  reducing  to  logic  the  arithmetical  notions 
which  his  predecessors  had  shown  to  be  sufficient  for  mathematics. 
We  shall  not,  in  this  chapter,  actually  give  Frege's  definition  of 
number  and  of  particular  numbers,  but  we  shall  give  some  of  the 
reasons  why  Peano's  treatment  is  less  final  than  it  appears  to  be. 
In  the  first  place,  Peano's  three  primitive  ideas — namely,  "  o," 
"  number,"  and  "  successor  " — are  capable  of  an  infinite  number 
of  different  interpretations,  all  of  which  will  satisfy  the  five 
primitive  propositions.  We  will  give  some  examples. 

(1)  Let  "  o  "  be  taken  to  mean  loo,  and  let  "  number  "  be 
taken  to  mean  the  numbers  from  100  onward  in  the  series  of 
natural    numbers.     Then    all    our    primitive    propositions    are 
satisfied,  even  the  fourth,  for,  though  100  is  the  successor  of 
99,  99  is  not  a  "  number  "  in  the  sense  which  we  are  now  giving 
to  the  word  "  number."     It  is  obvious  that  any  number  may  be 
substituted  for  100  in  this  example. 

(2)  Let   "  o "   have  its  usual   meaning,   but  let   "  number " 
mean   what    we   usually    call    "  even    numbers,"    and   let    the 
"  successor  "  of  a  number  be  what  results  from  adding  two  to 
it.     Then  "  I  "  will  stand  for  the  number  two,  "  2  "  will  stand 
for  the  number  four,  and  so  on ;   the  series  of  "  numbers  "  now 
will  be 

o,  two,  four,  six,  eight  .  .  . 

All  Peano's  five  premisses  are  satisfied  still. 

(3)  Let  "  o "   mean  the  number  one,  let  "  number "   mean 
the  set 

!>  i>  1>  i  TV»  •  •  • 

and    let    "successor"    mean    "half."     Then    all    Peano's    five 
axioms  will  be  true  of  this  set. 

It  is  clear  that  such  examples  might  be  multiplied  indefinitely. 
In  fact,  given  any  series 


8  Introduction  to  Mathematical  Philosophy 

which  is  endless,  contains  no  repetitions,  has  a  beginning,  and 
has  no  terms  that  cannot  be  reached  from  the  beginning  in  a 
finite  number  of  steps,  we  have  a  set  of  terms  verifying  Peano's 
axioms.  This  is  easily  seen,  though  the  formal  proof  is  some 
what  long.  Let  "  o  "  mean  #0,  let  "  number  "  mean  the  whole 
set  of  terms,  and  let  the  "  successor  "  of  #n  mean  xn+l.  Then 

(1)  "  o  is  a  number,"  i.e.  x0  is  a  member  of  the  set. 

(2)  "  The  successor  of  any  number  is  a  number,"  i.e.  taking 
any  term  xn  in  the  set,  xn+l  is  also  in  the  set. 

(3)  "  No  two  numbers  have  the  same  successor,"  i.e.  if  xm 
and  xn  are  two  different  members  of  the  set,  xm+l  and  xn+l  are 
different  ;    this  results  from  the  fact  that  (by  hypothesis)  there 
are  no  repetitions  in  the  set. 

(4)  "  o  is  not  the  successor  of  any  number,"  i.e.  no  term  in 
the  set  comes  before  x0. 

(5)  This  becomes  :    Any  property  which  belongs  to  x09  and 
belongs  to  xn+l  provided  it  belongs  to  xn,  belongs  to  all  the  x's. 

This  follows  from  the  corresponding  property  for  numbers. 
A  series  of  the  form 


in  which  there  is  a  first  term,  a  successor  to  each  term  (so  that 
there  is  no  last  term),  no  repetitions,  and  every  term  can  be 
reached  from  the  start  in  a  finite  number  of  steps,  is  called  a 
progression.  Progressions  are  of  great  importance  in  the  princi 
ples  of  mathematics.  As  we  have  just  seen,  every  progression 
verifies  Peano's  five  axioms.  It  can  be  proved,  conversely, 
that  every  series  which  verifies  Peano's  five  axioms  is  a  pro 
gression.  Hence  these  five  axioms  may  be  used  to  define  the 
class  of  progressions  :  "  progressions  "  are  "  those  series  which 
verify  these  five  axioms."  Any  progression  may  be  taken  as 
the  basis  of  pure  mathematics  :  we  may  give  the  name  "  o  " 
to  its  first  term,  the  name  "  number  "  to  the  whole  set  of  its 
terms,  and  the  name  "  successor  "  to  the  next  in  the  progression. 
The  progression  need  not  be  composed  of  numbers  :  it  may  be 


The  Series  of  Natural  Numbers  9 

composed  of  points  in  space,  or  moments  of  time,  or  any  other 
terms  of  which  there  is  an  infinite  supply.  Each  different 
progression  will  give  rise  to  a  different  interpretation  of  all  the 
propositions  of  traditional  pure  mathematics ;  all  these  possible 
interpretations  will  be  equally  true. 

In  Peano's  system  there  is  nothing  to  enable  us  to  distinguish 
between  these  different  interpretations  of  his  primitive  ideas. 
It  is  assumed  that  we  know  what  is  meant  by  "  o,"  and  that 
we  shall  not  suppose  that  this  symbol  means  100  or  Cleopatra's 
Needle  or  any  of  the  other  things  that  it  might  mean. 

This  point,  that  "  o "  and  "  number "  and  "successor " 
cannot  be  defined  by  means  of  Peano's  five  axioms,  but  must 
be  independently  understood,  is  important.  We  want  our 
numbers  not  merely  to  verify  mathematical  formulae,  but  to 
apply  in  the  right  way  to  common  objects.  We  want  to  have 
ten  fingers  and  two  eyes  and  one  nose.  A  system  in  which  "  I  " 
meant  100,  and  "  2  "  meant  101,  and  so  on,  might  be  all  right 
for  pure  mathematics,  but  would  not  suit  daily  life.  We  want 
"  o  "  and  "  number  "  and  "  successor  "  to  have  meanings  which 
will  give  us  the  right  allowance  of  fingers  and  eyes  and  noses. 
We  have  already  some  knowledge  (though  not  sufficiently 
articulate  or  analytic)  of  what  we  mean  by  "  I  "  and  "  2  "  and 
so  on,  and  our  use  of  numbers  in  arithmetic  must  conform  to 
this  knowledge.  We  cannot  secure  that  this  shall  be  the  case 
by  Peano's  method  ;  all  that  we  can  do,  if  we  adopt  his  method, 
is  to  say  "  we  know  what  we  mean  by  *  o '  and  '  number '  and 
'  successor,'  though  we  cannot  explain  what  we  mean  in  terms 
of  other  simpler  concepts."  It  is  quite  legitimate  to  say  this 
when  we  must,  and  at  some  point  we  all  must ;  but  it  is  the 
object  of  mathematical  philosophy  to  put  off  saying  it  as  long 
as  possible.  By  the  logical  theory  of  arithmetic  we  are  able  to 
put  it  off  for  a  very  long  time. 

It  might  be  suggested  that,  instead  of  setting  up  "  o  "  and 
"  number  "  and  "  successor  "  as  terms  of  which  we  know  the 
meaning  although  we  cannot  define  them,  we  might  let  them 


io  Introduction  to  Mathematical  Philosophy 

stand  for  any  three  terms  that  verify  Peano's  five  axioms.  They 
will  then  no  longer  be  terms  which  have  a  meaning  that  is  definite 
though  undefined:  they  will  be  "variables,"  terms  concerning 
which  we  make  certain  hypotheses,  namely,  those  stated  in  the 
five  axioms,  but  which  are  otherwise  undetermined.  If  we  adopt 
this  plan,  our  theorems  will  not  be  proved  concerning  an  ascer 
tained  set  of  terms  called  "  the  natural  numbers,"  but  concerning 
all  sets  of  terms  having  certain  properties.  Such  a  procedure 
is  not  fallacious  ;  indeed  for  certain  purposes  it  represents  a 
valuable  generalisation.  But  from  two  points  of  view  it  fails 
to  give  an  adequate  basis  for  arithmetic.  In  the  first  place,  it 
does  not  enable  us  to  know  whether  there  are  any  sets  of  terms 
verifying  Peano's  axioms  ;  it  does  not  even  give  the  faintest 
suggestion  of  any  way  of  discovering  whether  there  are  such  sets. 
In  the  second  place,  as  already  observed,  we  want  our  numbers 
to  be  such  as  can  be  used  for  counting  common  objects,  and  this 
requires  that  our  numbers  should  have  a  definite  meaning,  not 
merely  that  they  should  have  certain  formal  properties.  This 
definite  meaning  is  defined  by  the  logical  theory  of  arithmetic. 


CHAPTER  II 

DEFINITION    OF    NUMBER 

THE  question  "  What  is  a  number  ?  "  is  one  which  has  been 
often  asked,  but  has  only  been  correctly  answered  in  our  own 
time.  The  answer  was  given  by  Frege  in  1884,  in  his  Grundlagen 
der  Arithmetik*  Although  this  book  is  quite  short,  not  difficult, 
and  of  the  very  highest  importance,  it  attracted  almost  no 
attention,  and  the  definition  of  number  which  it  contains  re 
mained  practically  unknown  until  it  was  rediscovered  by  the 
present  author  in  1901. 

In  seeking  a  definition  of  number,  the  first  thing  to  be  clear 
about  is  what  we  may  call  the  grammar  of  our  inquiry.  Many 
philosophers,  when  attempting  to  define  number,  are  really 
setting  to  work  to  define  plurality,  which  is  quite  a  different 
thing.  Number  is  what  is  characteristic  of  numbers,  as  man 
is  what  is  characteristic  of  men.  A  plurality  is  not  an  instance 
of  number,  but  of  some  particular  number.  A  trio  of  men, 
for  example,  is  an  instance  of  the  number  3,  and  the  number 
3  is  an  instance  of  number ;  but  the  trio  is  not  an  instance  of 
number.  This  point  may  seem  elementary  and  scarcely  worth 
mentioning ;  yet  it  has  proved  too  subtle  for  the  philosophers, 
with  few  exceptions. 

A  particular  number  is  not  identical  with  any  collection  of 
terms  having  that  number  :  the  number  3  is  not  identical  with 

1  The  same  answer  is  given  more  fully  and  with  more  development  in 
his  Grundgesetze  der  Arithmetik,  vol.  i.,  1893. 


12  Introduction  to  Mathematical  Philosophy 

the  trio  consisting  of  Brown,  Jones,  and  Robinson.  The  number 
3  is  something  which  all  trios  have  in  common,  and  which  dis 
tinguishes  them  from  other  collections.  A  number  is  something 
that  characterises  certain  collections,  namely,  those  that  have 
that  number. 

Instead  of  speaking  of  a  "  collection,"  we  shall  as  a  rule  speak 
of  a  "  class,"  or  sometimes  a  "  set."  Other  words  used  in 
mathematics  for  the  same  thing  are  "  aggregate  "  and  "  mani 
fold."  We  shall  have  much  to  say  later  on  about  classes.  For 
the  present,  we  will  say  as  little  as  possible.  But  there  are 
some  remarks  that  must  be  made  immediately. 

A  class  or  collection  may  be  defined  in  two  ways  that  at  first 
sight  seem  quite  distinct.  We  may  enumerate  its  members,  as 
when  we  say,  "  The  collection  I  mean  is  Brown,  Jones,  and 
Robinson."  Or  we  may  mention  a  defining  property,  as  when 
we  speak  of  "  mankind  "  or  "  the  inhabitants  of  London."  The 
definition  which  enumerates  is  called  a  definition  by  "  exten 
sion,"  and  the  one  which  mentions  a  defining  property  is  called 
a  definition  by  "  intension."  Of  these  two  kinds  of  definition, 
the  one  by  intension  is  logically  more  fundamental.  This  is 
shown  by  two  considerations  :  (i)  that  the  extensional  defini 
tion  can  always  be  reduced  to  an  intensional  one;  (2)  that  the 
intensional  one  often  cannot  even  theoretically  be  reduced  to 
the  extensional  one.  Each  of  these  points  needs  a  word  of 
explanation. 

(i)  Brown,  Jones,  and  Robinson  all  of  them  possess  a  certain 
property  which  is  possessed  by  nothing  else  in  the  whole  universe, 
namely,  the  property  of  being  either  Brown  or  Jones  or  Robinson. 
This  property  can  be  used  to  give  a  definition  by  intension  of 
the  class  consisting  of  Brown  and  Jones  and  Robinson.  Con 
sider  such  a  formula  as  "  x  is  Brown  or  x  is  Jones  or  x  is  Robinson." 
This  formula  will  be  true  for  just  three  x's,  namely,  Brown  and 
Jones  and  Robinson.  In  this  respect  it  resembles  a  cubic  equa 
tion  with  its  three  roots.  It  may  be  taken  as  assigning  a  property 
common  to  the  members  of  the  class  consisting  of  these  three 


Definition  of  Number  13 

men,  and  peculiar  to  them.  A  similar  treatment  can  obviously 
be  applied  to  any  other  class  given  in  extension. 

(2)  It  is  obvious  that  in  practice  we  can  often  know  a  great 
deal  about  a  class  without  being  able  to  enumerate  its  members. 
No  one  man  could  actually  enumerate  all  men,  or  even  all  the 
inhabitants  of  London,  yet  a  great  deal  is  known  about  each  of 
these  classes.  This  is  enough  to  show  that  definition  by  extension 
is  not  necessary  to  knowledge  about  a  class.  But  when  we  come 
to  consider  infinite  classes,  we  find  that  enumeration  is  not  even 
theoretically  possible  for  beings  who  only  live  for  a  finite  time. 
We  cannot  enumerate  all  the  natural  numbers  :  they  are  o,  I,  2, 
3,  and  so  on.  At  some  point  we  must  content  ourselves  with 
"  and  so  on."  We  cannot  enumerate  all  fractions  or  all  irrational 
numbers,  or  all  of  any  other  infinite  collection.  Thus  our  know 
ledge  in  regard  to  all  such  collections  can  only  be  derived  from  a 
definition  by  intension. 

These  remarks  are  relevant,  when  we  are  seeking  the  definition 
of  number,  in  three  different  ways.  In  the  first  place,  numbers 
themselves  form  an  infinite  collection,  and  cannot  therefore 
be  defined  by  enumeration.  In  the  second  place,  the  collections 
having  a  given  number  of  terms  themselves  presumably  form  an 
infinite  collection  :  it  is  to  be  presumed,  for  example,  that  there 
are  an  infinite  collection  of  trios  in  the  world,  for  if  this  were 
not  the  case  the  total  number  of  things  in  the  world  would  be 
finite,  which,  though  possible,  seems  unlikely.  In  the  third 
place,  we  wish  to  define  "  number  "  in  such  a  way  that  infinite 
numbers  may  be  possible ;  thus  we  must  be  able  to  speak  of 
the  number  of  terms  in  an  infinite  collection,  and  such  a  collection 
must  be  defined  by  intension,  i.e.  by  a  property  common  to  all 
its  members  and  peculiar  to  them. 

For  many  purposes,  a  class  and  a  defining  characteristic  of 
it  are  practically  interchangeable.  The  vital  difference  between 
the  two  consists  in  the  fact  that  there  is  only  one  class  having  a 
given  set  of  members,  whereas  there  are  always  many  different 
characteristics  by  which  a  given  class  may  be  defined.  Men 


14  Introduction  to  Mathematical  Philosophy 

may  be  defined  as  featherless  bipeds,  or  as  rational  animals, 
or  (more  correctly)  by  the  traits  by  which  Swift  delineates  the 
Yahoos.  It  is  this  fact  that  a  defining  characteristic  is  never 
unique  which  makes  classes  useful ;  otherwise  we  could  be 
content  with  the  properties  common  and  peculiar  to  their 
members.1  Any  one  of  these  properties  can  be  used  in  place 
of  the  class  whenever  uniqueness  is  not  important. 

Returning  now  to  the  definition  of  number,  it  is  clear  that 
number  is  a  way  of  bringing  together  certain  collections,  namely, 
those  that  have  a  given  number  of  terms.  We  can  suppose 
all  couples  in  one  bundle,  all  trios  in  another,  and  so  on.  In 
this  way  we  obtain  various  bundles  of  collections,  each  bundle 
consisting  of  all  the  collections  that  have  a  certain  number  of 
terms.  Each  bundle  is  a  class  whose  members  are  collections, 
i.e.  classes  ;  thus  each  is  a  class  of  classes.  The  bundle  con 
sisting  of  all  couples,  for  example,  is  a  class  of  classes  :  each 
couple  is  a  class  with  two  members,  and  the  whole  bundle  of 
couples  is  a  class  with  an  infinite  number  of  members,  each  of 
which  is  a  class  of  two  members. 

How  shall  we  decide  whether  two  collections  are  to  belong 
to  the  same  bundle  ?  The  answer  that  suggests  itself  is  :  "  Find 
out  how  many  members  each  has,  and  put  them  in  the  same 
bundle  if  they  have  the  same  number  of  members."  But  this 
presupposes  that  we  have  defined  numbers,  and  that  we  know 
how  to  discover  how  many  terms  a  collection  has.  We  are  so 
used  to  the  operation  of  counting  that  such  a  presupposition 
might  easily  pass  unnoticed.  In  fact,  however,  counting, 
though  familiar,  is  logically  a  very  complex  operation  ;  more 
over  it  is  only  available,  as  a  means  of  discovering  how  many 
terms  a  collection  has,  when  the  collection  is  finite.  Our  defini 
tion  of  number  must  not  assume  in  advance  that  all  numbers 
are  finite ;  and  we  cannot  in  any  case,  without  a  vicious  circle, 

1  As  will  be  explained  later,  classes  may  be  regarded  as  logical  fictions, 
manufactured  out  of  denning  characteristics.  But  for  the  present  it  will 
simplify  our  exposition  to  treat  classes  as  if  they  were  real. 


Definition  of  Number  1 5 

use  counting  to  define  numbers,  because  numbers  are  used  in 
counting.  We  need,  therefore,  some  other  method  of  deciding 
when  two  collections  have  the  same  number  of  terms. 

In  actual  fact,  it  is  simpler  logically  to  find  out  whether  two 
collections  have  the  same  number  of  terms  than  it  is  to  define 
what  that  number  is.  An  illustration  will  make  this  clear. 
If  there  were  no  polygamy  or  polyandry  anywhere  in  the  world, 
it  is  clear  that  the  number  of  husbands  living  at  any  moment 
would  be  exactly  the  same  as  the  number  of  wives.  We  do 
not  need  a  census  to  assure  us  of  this,  nor  do  we  need  to  know 
what  is  the  actual  number  of  husbands  and  of  wives.  We  know 
the  number  must  be  the  same  in  both  collections,  because  each 
husband  has  one  wife  and  each  wife  has  one  husband.  The 
relation  of  husband  and  wife  is  what  is  called  "  one-one." 

A  relation  is  said  to  be  "  one-one  "  when,  if  x  has  the  relation 
in  question  to  y,  no  other  term  x'  has  the  same  relation  to  y, 
and  x  does  not  have  the  same  relation  to  any  term  y'  other 
than  y.  When  only  the  first  of  these  two  conditions  is  fulfilled, 
the  relation  is  called  "  one-many  "  ;  when  only  the  second  is 
fulfilled,  it  is  called  "  many-one."  It  should  be  observed  that 
the  number  I  is  not  used  in  these  definitions. 

In  Christian  countries,  the  relation  of  husband  to  wife  is 
one-one ;  in  Mahometan  countries  it  is  one-many ;  in  Tibet 
it  is  many-one.  The  relation  of  father  to  son  is  one-many ; 
that  of  son  to  father  is  many-one,  but  that  of  eldest  son  to  father 
is  one-one.  If  n  is  any  number,  the  relation  of  n  to  «-|-i  is 
one-one  ;  so  is  the  relation  of  n  to  2n  or  to  3«.  When  we  are 
considering  only  positive  numbers,  the  relation  of  n  to  «2  is 
one-one ;  but  when  negative  numbers  are  admitted,  it  becomes 
two-one,  since  n  and  — n  have  the  same  square.  These  instances 
should  suffice  to  make  clear  the  notions  of  one-one,  one-many, 
and  many-one  relations,  which  play  a  great  part  in  the  princi 
ples  of  mathematics,  not  only  in  relation  to  the  definition  of 
numbers,  but  in  many  other  connections. 

Two  classes  are  said  to  be  "  similar  "  when  there  is  a  one-one 


1 6  Introduction  to  Mathematical  Philosophy 

relation  which  correlates  the  terms  of  the  one  class  each  with 
one  term  of  the  other  class,  in  the  same  manner  in  which  the 
relation  of  marriage  correlates  husbands  with  wives.  A  few 
preliminary  definitions  will  help  us  to  state  this  definition  more 
precisely.  The  class  of  those  terms  that  have  a  given  relation 
to  something  or  other  is  called  the  domain  of  that  relation  : 
thus  fathers  are  the  domain  of  the  relation  of  father  to  child, 
husbands  are  the  domain  of  the  relation  of  husband  to  wife, 
wives  are  the  domain  of  the  relation  of  wife  to  husband,  and 
husbands  and  wives  together  are  the  domain  of  the  relation  of 
marriage.  The  relation  of  wife  to  husband  is  called  the  converse 
of  the  relation  of  husband  to  wife.  Similarly  less  is  the  converse 
of  greater,  later  is  the  converse  of  earlier,  and  so  on.  Generally, 
the  converse  of  a  given  relation  is  that  relation  which  holds 
between  y  and  x  whenever  the  given  relation  holds  between 
x  and  y.  The  converse  domain  of  a  relation  is  the  domain  of 
its  converse  :  thus  the  class  of  wives  is  the  converse  domain 
of  the  relation  of  husband  to  wife.  We  may  now  state  our 
definition  of  similarity  as  follows  : — 

One  class  is  said  to  be  "  similar  "  to  another  when  there  is  a 
one-one  relation  of  which  the  one  class  is  the  domain,  while  the 
other  is  the  converse  domain. 

It  is  easy  to  prove  (i)  that  every  class  is  similar  to  itself,  (2) 
that  if  a  class  a  is  similar  to  a  class  j3,  then  j3  is  similar  to  a,  (3) 
that  if  a  is  similar  to  j3  and  j8  to  y,  then  a  is  similar  to  y.  A 
relation  is  said  to  be  reflexive  when  it  possesses  the  first  of  these 
properties,  symmetrical  when  it  possesses  the  second,  and  transi 
tive  when  it  possesses  the  third.  It  is  obvious  that  a  relation 
which  is  symmetrical  and  transitive  must  be  reflexive  throughout 
its  domain.  Relations  which  possess  these  properties  are  an 
important  kind,  and  it  is  worth  while  to  note  that  similarity  is 
one  of  this  kind  of  relations. 

It  is  obvious  to  common  sense  that  two  finite  classes  have 
the  same  number  of  terms  if  they  are  similar,  but  not  otherwise. 
The  act  of  counting  consists  in  establishing  a  one-one  correlation 


Definition  of  Number  17 

between  the  set  of  objects  counted  and  the  natural  numbers 
(excluding  o)  that  are  used  up  in  the  process.  Accordingly 
common  sense  concludes  that  there  are  as  many  objects  in  the 
set  to  be  counted  as  there  are  numbers  up  to  the  last  number 
used  in  the  counting.  And  we  also  know  that,  so  long  as  we 
confine  ourselves  to  finite  numbers,  there  are  just  n  numbers 
from  I  up  to  n.  Hence  it  follows  that  the  last  number  used  in 
counting  a  collection  is  the  number  of  terms  in  the  collection, 
provided  the  collection  is  finite.  But  this  result,  besides  being 
only  applicable  to  finite  collections,  depends  upon  and  assumes 
the  fact  that  two  classes  which  are  similar  have  the  same  number 
of  terms  ;  for  what  we  do  when  we  count  (say)  10  objects  is  to 
show  that  the  set  of  these  objects  is  similar  to  the  set  of  numbers 
I  to  10.  The  notion  of  similarity  is  logically  presupposed  in 
the  operation  of  counting,  and  is  logically  simpler  though  less 
familiar.  In  counting,  it  is  necessary  to  take  the  objects  counted 
in  a  certain  order,  as  first,  second,  third,  etc.,  but  order  is  not 
of  the  essence  of  number  :  it  is  an  irrelevant  addition,  an  un 
necessary  complication  from  the  logical  point  of  view.  The 
notion  of  similarity  does  not  demand  an  order  :  for  example, 
we  saw  that  the  number  of  husbands  is  the  same  as  the  number 
of  wives,  without  having  to  establish  an  order  of  precedence 
among  them.  The  notion  of  similarity  also  does  not  require 
that  the  classes  which  are  similar  should  be  finite.  Take,  for 
example,  the  natural  numbers  (excluding  o)  on  the  one  hand, 
and  the  fractions  which  have  I  for  their  numerator  on  the  other 
hand  :  it  is  obvious  that  we  can  correlate  2  with  J,  3  with  J,  and 
so  on,  thus  proving  that  the  two  classes  are  similar. 

We  may  thus  use  the  notion  of  "  similarity  "  to  decide  when 
two  collections  are  to  belong  to  the  same  bundle,  in  the  sense 
in  which  we  were  asking  this  question  earlier  in  this  chapter. 
We  want  to  make  one  bundle  containing  the  class  that  has  no 
members  :  this  will  be  for  the  number  o.  Then  we  want  a  bundle 
of  all  the  classes  that  have  one  member  :  this  will  be  for  the 
number  I.  Then,  for  the  number  2,  we  want  a  bundle  consisting 

2 


1 8  Introduction  to  Mathematical  Philosophy 

of  all  couples ;  then  one  of  all  trios  ;  and  so  on.  Given  any  collec 
tion,  we  can  define  the  bundle  it  is  to  belong  to  as  being  the  class 
of  all  those  collections  that  are  "  similar  "  to  it.  It  is  very  easy 
to  see  that  if  (for  example)  a  collection  has  three  members,  the 
class  of  all  those  collections  that  are  similar  to  it  will  be  the 
class  of  trios.  And  whatever  number  of  terms  a  collection  may 
have,  those  collections  that  are  "  similar  "  to  it  will  have  the  same 
number  of  terms.  We  may  take  this  as  a  definition  of  "  having 
the  same  number  of  terms."  It  is  obvious  that  it  gives  results 
conformable  to  usage  so  long  as  we  confine  ourselves  to  finite 
collections. 

So  far  we  have  not  suggested  anything  in  the  slightest  degree 
paradoxical.  But  when  we  come  to  the  actual  definition  of 
numbers  we  cannot  avoid  what  must  at  first  sight  seem  a  paradox, 
though  this  impression  will  soon  wear  off.  We  naturally  think 
that  the  class  of  couples  (for  example)  is  something  different 
from  the  number  2.  But  there  is  no  doubt  about  the  class  of 
couples  :  it  is  indubitable  and  not  difficult  to  define,  whereas 
the  number  2,  in  any  other  sense,  is  a  metaphysical  entity  about 
which  we  can  never  feel  sure  that  it  exists  or  that  we  have  tracked 
it  down.  It  is  therefore  more  prudent  to  content  ourselves  with 
the  class  of  couples,  which  we  are  sure  of,  than  to  hunt  for  a 
problematical  number  2  which  must  always  remain  elusive. 
Accordingly  we  set  up  the  following  definition  : — 

The  number  of  a  class  is  the  class  of  all  those  classes  that  are 
similar  to  it. 

Thus  the  number  of  a  couple  will  be  the  class  of  all  couples. 
In  fact,  the  class  of  all  couples  will  be  the  number  2,  according 
to  our  definition.  At  the  expense  of  a  little  oddity,  this  definition 
secures  definiteness  and  indubitableness  ;  and  it  is  not  difficult 
to  prove  that  numbers  so  defined  have  all  the  properties  that  we 
expect  numbers  to  have. 

We  may  now  go  on  to  define  numbers  in  general  as  any  one  of 
the  bundles  into  which  similarity  collects  classes.  A  number 
will  be  a  set  of  classes  such  as  that  any  two  are  similar  to  each 


Definition  of  Number  1 9 

other,  and  none  outside  the  set  are  similar  to  any  inside  the  set. 
In  other  words,  a  number  (in  general)  is  any  collection  which  is 
the  number  of  one  of  its  members  ;  or,  more  simply  still : 

A  number  is  anything  which  is  the  number  of  some  class. 

Such  a  definition  has  a  verbal  appearance  of  being  circular, 
but  in  fact  it  is  not.  We  define  "  the  number  of  a  given  class  " 
without  using  the  notion  of  number  in  general ;  therefore  we  may 
define  number  in  general  in  terms  of  "  the  number  of  a  given 
class  "  without  committing  any  logical  error. 

Definitions  of  this  sort  are  in  fact  very  common.  The  class 
of  fathers,  for  example,  would  have  to  be  defined  by  first  defining 
what  it  is  to  be  the  father  of  somebody  ;  then  the  class  of  fathers 
will  be  all  those  who  are  somebody's  father.  Similarly  if  we  want 
to  define  square  numbers  (say),  we  must  first  define  what  we 
mean  by  saying  that  one  number  is  the  square  of  another,  and 
then  define  square  numbers  as  those  that  are  the  squares  of 
other  numbers.  This  kind  of  procedure  is  very  common,  and 
it  is  important  to  realise  that  it  is  legitimate  and  even  often 
necessary. 

We  have  now  given  a  definition  of  numbers  which  will  serve 
for  finite  collections.  It  remains  to  be  seen  how  it  will  serve 
for  infinite  collections.  But  first  we  must  decide  what  we  mean 
by  "  finite  "  and  "  infinite,"  which  cannot  be  done  within  the 
limits  of  the  present  chapter. 


CHAPTER  III 

FINITUDE    AND    MATHEMATICAL    INDUCTION 

THE  series  of  natural  numbers,  as  we  saw  in  Chapter  I.,  can  all 
be  defined  if  we  know  what  we  mean  by  the  three  terms  "  o," 
"  number,"  and  "  successor."  But  we  may  go  a  step  farther  : 
we  can  define  all  the  natural  numbers  if  we  know  what  we  mean 
by  "  o  "  and  "  successor."  It  will  help  us  to  understand  the 
difference  between  finite  and  infinite  to  see  how  this  can  be  done, 
and  why  the  method  by  which  it  is  done  cannot  be  extended 
beyond  the  finite.  We  will  not  yet  consider  how  "  o  "  and  "  suc 
cessor  "  are  to  be  defined  :  we  will  for  the  moment  assume  that 
we  know  what  these  terms  mean,  and  show  how  thence  all  other 
natural  numbers  can  be  obtained. 

It  is  easy  to  see  that  we  can  reach  any  assigned  number,  say 
30,000.  We  first  define  "  I  "  as  "  the  successor  of  o,"  then  we 
define  "  2  "  as  "  the  successor  of  I,"  and  so  on.  In  the  case  of 
an  assigned  number,  such  as  30,000,  the  proof  that  we  can  reach 
it  by  proceeding  step  by  step  in  this  fashion  may  be  made,  if  we 
have  the  patience,  by  actual  experiment :  we  can  go  on  until 
we  actually  arrive  at  30,000.  But  although  the  method  of 
experiment  is  available  for  each  particular  natural  number,  it 
is  not  available  for  proving  the  general  proposition  that  all  such 
numbers  can  be  reached  in  this  way,  i.e.  by  proceeding  from  o 
step  by  step  from  each  number  to  its  successor.  Is  there  any 
other  way  by  which  this  can  be  proved  ? 

Let  us  consider  the  question  the  other  way  round.  What  are 
the  numbers  that  can  be  reached,  given  the  terms  "  o  "  and 


Finitude  and  Mathematical  Induction  21 

"  successor  "  ?  Is  there  any  way  by  which  we  can  define  the 
whole  class  of  such  numbers  ?  We  reach  I,  as  the  successor  of  o ; 
2,  as  the  successor  of  I  ;  3,  as  the  successor  of  2  ;  and  so  on.  It 
is  this  "  and  so  on  "  that  we  wish  to  replace  by  something  less 
vague  and  indefinite.  We  might  be  tempted  to  say  that  "  and 
so  on  "  means  that  the  process  of  proceeding  to  the  successor 
may  be  repeated  any  finite  number  of  times ;  but  the  problem 
upon  which  we  are  engaged  is  the  problem  of  defining  "  finite 
number,"  and  therefore  we  must  not  use  this  notion  in  our  defini 
tion.  Our  definition  must  not  assume  that  we  know  what  a 
finite  number  is. 

The  key  to  our  problem  lies  in  mathematical  induction.  It  will 
be  remembered  that,  in  Chapter  I.,  this  was  the  fifth  of  the  five 
primitive  propositions  which  we  laid  down  about  the  natural 
numbers.  It  stated  that  any  property  which  belongs  to  o,  and 
to  the  successor  of  any  number  which  has  the  property,  belongs 
to  all  the  natural  numbers.  This  was  then  presented  as  a  principle, 
but  we  shall  now  adopt  it  as  a  definition.  It  is  not  difficult 
to  see  that  the  terms  obeying  it  are  the  same  as  the  numbers 
that  can  be  reached  from  o  by  successive  steps  from  next  to 
next,  but  as  the  point  is  important  we  will  set  forth  the  matter 
in  some  detail. 

We  shall  do  well  to  begin  with  some  definitions,  which  will  be 
useful  in  other  connections  also. 

A  property  is  said  to  be  "  hereditary  "  in  the  natural-number 
series  if,  whenever  it  belongs  to  a  number  «,  it  also  belongs  to 
n-j-i,  the  successor  of  n.  Similarly  a  class  is  said  to  be  "  heredi 
tary  "  if,  whenever  n  is  a  member  of  the  class,  so  is  n+i.  It  is 
easy  to  see,  though  we  are  not  yet  supposed  to  know,  that  to  say 
a  property  is  hereditary  is  equivalent  to  saying  that  it  belongs 
to  all  the  natural  numbers  not  less  than  some  one  of  them,  e.g. 
it  must  belong  to  all  that  are  not  less  than  100,  or  all  that  are 
less  than  1000,  or  it  may  be  that  it  belongs  to  all  that  are  not 
less  than  o,  i.e.  to  all  without  exception. 

A  property  is  said  to  be  "  inductive  "  when  it  is  a  hereditary 


22  Introduction  to  Mathematical  Philosophy 

property  which  belongs  to  o.  Similarly  a  class  is  "  inductive  " 
when  it  is  a  hereditary  class  of  which  o  is  a  member. 

Given  a  hereditary  class  of  which  o  is  a  member,  it  follows 
that  I  is  a  member  of  it,  because  a  hereditary  class  contains  the 
successors  of  its  members,  and  I  is  the  successor  of  o.  Similarly, 
given  a  hereditary  class  of  which  I  is  a  member,  it  follows  that 
2  is  a  member  of  it ;  and  so  on.  Thus  we  can  prove  by  a  step- 
by-step  procedure  that  any  assigned  natural  number,  say  30,000, 
is  a  member  of  every  inductive  class. 

We  will  define  the  "  posterity  "  of  a  given  natural  number 
with  respect  to  the  relation  "  immediate  predecessor  "  (which 
is  the  converse  of  "  successor  ")  as  all  those  terms  that  belong 
to  every  hereditary  class  to  which  the  given  number  belongs.  It 
is  again  easy  to  see  that  the  posterity  of  a  natural  number  con 
sists  of  itself  and  all  greater  natural  numbers  ;  but  this  also  we 
do  not  yet  officially  know. 

By  the  above  definitions,  the  posterity  of  o  will  consist  of  those 
terms  which  belong  to  every  inductive  class. 

It  is  now  not  difficult  to  make  it  obvious  that  the  posterity  of 
o  is  the  same  set  as  those  terms  that  can  be  reached  from  o  by 
successive  steps  from  next  to  next.  For,  in  the  first  place,  o 
belongs  to  both  these  sets  (in  the  sense  in  which  we  have  defined 
our  terms)  ;  in  the  second  place,  if  n  belongs  to  both  sets,  so  does 
n+i.  It  is  to  be  observed  that  we  are  dealing  here  with  the 
kind  of  matter  that  does  not  admit  of  precise  proof,  namely,  the 
comparison  of  a  relatively  vague  idea  with  a  relatively  precise 
one.  The  notion  of  "  those  terms  that  can  be  reached  from  o 
by  successive  steps  from  next  to  next  "  is  vague,  though  it  seems 
as  if  it  conveyed  a  definite  meaning ;  on  the  other  hand,  "  the 
posterity  of  o  "  is  precise  and  explicit  just  where  the  other  idea 
is  hazy.  It  may  be  taken  as  giving  what  we  meant  to  mean 
when  we  spoke  of  the  terms  that  can  be  reached  from  o  by 
successive  steps. 

We  now  lay  down  the  following  definition  : — 

The  "  natural  numbers  "  are  the  -posterity  of  o  with  respect  to  the 


Finitude  and  Mathematical  Induction  23 

relation  "  immediate  predecessor "  (which  is  the  converse  of 
"  successor  "  ). 

We  have  thus  arrived  at  a  definition  of  one  of  Peano's  three 
primitive  ideas  in  terms  of  the  other  two.  As  a  result  of  this 
definition,  two  of  his  primitive  propositions — namely,  the  one 
asserting  that  o  is  a  number  and  the  one  asserting  mathematical 
induction — become  unnecessary,  since  they  result  from  the  defini 
tion.  The  one  asserting  that  the  successor  of  a  natural  number 
is  a  natural  number  is  only  needed  in  the  weakened  form  "  every 
natural  number  has  a  successor." 

We  can,  of  course,  easily  define  "  o  "  and  "  successor  "  by  means 
of  the  definition  of  number  in  general  which  we  arrived  at  in 
Chapter  II.  The  number  o  is  the  number  of  terms  in  a  class 
which  has  no  members,  i.e.  in  the  class  which  is  called  the  "  null- 
class."  By  the  general  definition  of  number,  the  number  of  terms 
in  the  null-class  is  the  set  of  all  classes  similar  to  the  null-class, 
i.e.  (as  is  easily  proved)  the  set  consisting  of  the  null-class  all 
alone,  i.e.  the  class  whose  only  member  is  the  null-class.  (This 
is  not  identical  with  the  null-class  :  it  has  one  member,  namely  ? 
the  null-class,  whereas  the  null-class  itself  has  no  members.  A 
class  which  has  one  member  is  never  identical  with  that  one 
member,  as  we  shall  explain  when  we  come  to  the  theory  of 
classes.)  Thus  we  have  the  following  purely  logical  definition : — 

o  is  the  class  whose  only  member  is  the  null-class. 

It  remains  to  define  "  successor."  Given  any  number  n,  let 
a  be  a  class  which  has  n  members,  and  let  x  be  a  term  which 
is  not  a  member  of  a.  Then  the  class  consisting  of  a  with  x 
added  on  will  have  n-\-i  members.  Thus  we  have  the  following 
definition  : — 

The  successor  of  the  number  of  terms  in  the  class  a  is  the  number 
of  terms  in  the  class  consisting  of  a  together  with  x,  where  x  is  any 
term  not  belonging  to  the  class. 

Certain  niceties  are  required  to  make  this  definition  perfect, 
but  they  need  not  concern  us.1  It  will  be  remembered  that  we 
1  See  Principia  Mathematical,  vol.  ii.  *  no, 


24  Introduction  to  Mathematical  Philosophy 

have  already  given  (in  Chapter  II.)  a  logical  definition  of  the 
number  of  terms  in  a  class,  namely,  we  defined  it  as  the  set  of  all 
classes  that  are  similar  to  the  given  class. 

We  have  thus  reduced  Peano's  three  primitive  ideas  to  ideas 
of  logic  :  we  have  given  definitions  of  them  which  make  them 
definite,  no  longer  capable  of  an  infinity  of  different  meanings, 
as  they  were  when  they  were  only  determinate  to  the  extent  of 
obeying  Peano's  five  axioms.  We  have  removed  them  from  the 
fundamental  apparatus  of  terms  that  must  be  merely  appre 
hended,  and  have  thus  increased  the  deductive  articulation  of 
mathematics. 

As  regards  the  five  primitive  propositions,  we  have  already 
succeeded  in  making  two  of  them  demonstrable  by  our  definition 
of  "  natural  number."  How  stands  it  with  the  remaining  three  ? 
It  is  very  easy  to  prove  that  o  is  not  the  successor  of  any  number, 
and  that  the  successor  of  any  number  is  a  number.  But  there 
is  a  difficulty  about  the  remaining  primitive  proposition,  namely, 
"  no  two  numbers  have  the  same  successor."  The  difficulty 
does  not  arise  unless  the  total  number  of  individuals  in  the 
universe  is  finite ;  for  given  two  numbers  m  and  n,  neither  of 
which  is  the  total  number  of  individuals  in  the  universe,  it  is 
easy  to  prove  that  we  cannot  have  m-\-i=n-{-i  unless  we  have 
m—n.  But  let  us  suppose  that  the  total  number  of  individuals 
in  the  universe  were  (say)  10 ;  then  there  would  be  no  class  of 
II  individuals,  and  the  number  1 1  would  be  the  null-class.  So 
would  the  number  12.  Thus  we  should  have  11  =  12  ;  therefore 
the  successor  of  10  would  be  the  same  as  the  successor  of  n, 
although  10  would  not  be  the  same  as  n.  Thus  we  should  have 
two  different  numbers  with  the  same  successor.  This  failure  of 
the  third  axiom  cannot  arise,  however,  if  the  number  of  indi 
viduals  in  the  world  is  not  finite.  We  shall  return  to  this  topic 
at  a  later  stage.1 

Assuming  that  the  number  of  individuals  in  the  universe  is 
not  finite,  we  have  now  succeeded  not  only  in  defining  Peano's 
*  See  Chapter  XIH, 


Finitude  and  Mathematical  Induction  25 

three  primitive  ideas,  but  in  seeing  how  to  prove  his  five  primitive 
propositions,  by  means  of  primitive  ideas  and  propositions  belong 
ing  to  logic.  It  follows  that  all  pure  mathematics,  in  so  far 
as  it  is  deducible  from  the  theory  of  the  natural  numbers,  is  only 
a  prolongation  of  logic.  The  extension  of  this  result  to  those 
modern  branches  of  mathematics  which  are  not  deducible  from 
the  theory  of  the  natural  numbers  offers  no  difficulty  of  principle, 
as  we  have  shown  elsewhere.1 

The  process  of  mathematical  induction,  by  means  of  which 
we  defined  the  natural  numbers,  is  capable  of  generalisation. 
We  defined  the  natural  numbers  as  the  "  posterity  "  of  o  with 
respect  to  the  relation  of  a  number  to  its  immediate  successor. 
If  we  call  this  relation  N,  any  number  m  will  have  this  relation 
to  w+i.  A  property  is  "hereditary  with  respect  to  N,"  or 
simply  "  N-hereditary,"  if,  whenever  the  property  belongs  to  a 
number  m,  it  also  belongs  to  m-fi,  i.e.  to  the  number  to  which 
m  has  the  relation  N.  And  a  number  n  will  be  said  to  belong  to 
the  "  posterity  "  of  m  with  respect  to  the  relation  N  if  n  has 
every  N-hereditary  property  belonging  to  m.  These  definitions 
can  all  be  applied  to  any  other  relation  just  as  well  as  to  N.  Thus 
if  R  is  any  relation  whatever,  we  can  lay  down  the  following 
definitions  : 2  — 

A  property  is  called  "  R-hereditary  "  when,  if  it  belongs  to 
a  term  x,  and  x  has  the  relation  R  to  y,  then  it  belongs  to  y. 

A  class  is  R-hereditary  when  its  defining  property  is  R- 
hereditary. 

A  term  x  is  said  to  be  an  "  R-ancestor  "  of  the  term  y  if  y  has 
every  R-hereditary  property  that  x  has,  provided  x  is  a  term 
which  has  the  relation  R  to  something  or  to  which  something 
has  the  relation  R.  (This  is  only  to  exclude  trivial  cases.) 

1  For  geometry,  in  so  far  as  it  is  not  purely  analytical,  see  Principles  of 
Mathematics,  part  vi.  ;    for  rational  dynamics,  ibid.,  part  vii. 

2  These  definitions,  and  the  generalised  theory  of  induction,  are  due  to 
Frege,  and  were  published  so  long  ago  as  1879  in  his  Begriffsschrift.     In 
spite  of  the  great  value  of  this  work,  I  was,  I  believe,  the  first  person  who 
ever  read  it— more  than  twenty  years  after  its  publication. 


26  Introduction  to  Mathematical  Philosophy 

The  "  R-posterity  "  of  x  is  all  the  terms  of  which  x  is  an  R- 
ancestor. 

We  have  framed  the  above  definitions  so  that  if  a  term  is  the 
ancestor  of  anything  it  is  its  own  ancestor  and  belongs  to  its  own 
posterity.  This  is  merely  for  convenience. 

It  will  be  observed  that  if  we  take  for  R  the  relation  "  parent," 
"  ancestor "  and  "  posterity "  will  have  the  usual  meanings, 
except  that  a  person  will  be  included  among  his  own  ancestors 
and  posterity.  It  is,  of  course,  obvious  at  once  that  "  ancestor  " 
must  be  capable  of  definition  in  terms  of  "  parent,"  but  until 
Frege  developed  his  generalised  theory  of  induction,  no  one  could 
have  defined  "  ancestor  "  precisely  in  terms  of  "  parent."  A 
brief  consideration  of  this  point  will  serve  to  show  the  importance 
of  the  theory.  A  person  confronted  for  the  first  time  with  the 
problem  of  defining  "  ancestor  "  in  terms  of  "  parent "  would 
naturally  say  that  A  is  an  ancestor  of  Z  if,  between  A  and  Z, 
there  are  a  certain  number  of  people,  B,  C,  .  .  .,  of  whom 
B  is  a  child  of  A,  each  is  a  parent  of  the  next,  until  the  last,  who 
is  a  parent  of  Z.  But  this  definition  is  not  adequate  unless  we 
add  that  the  number  of  intermediate  terms  is  to  be  finite.  Take, 
for  example,  such  a  series  as  the  following : — 

I,          f,          J,          89    .    •    •    g>    ¥>    2?    M 

Here  we  have  first  a  series  of  negative  fractions  with  no  end, 
and  then  a  series  of  positive  fractions  with  no  beginning.  Shall 
we  say  that,  in  this  series,  —  J  is  an  ancestor  of  J  ?  It  will  be 
so  according  to  the  beginner's  definition  suggested  above,  but 
it  will  not  be  so  according  to  any  definition  which  will  give  the 
kind  of  idea  that  we  wish  to  define.  For  this  purpose,  it  is 
essential  that  the  number  of  intermediaries  should  be  finite. 
But,  as  we  saw,  "  finite  "  is  to  be  defined  by  means  of  mathe 
matical  induction,  and  it  is  simpler  to  define  the  ancestral  relation 
generally  at  once  than  to  define  it  first  only  for  the  case  of  the 
relation  of  n  to  n-f-i,  and  then  extend  it  to  other  cases.  Here, 
as  constantly  elsewhere,  generality  from  the  first,  though  it  may 


Finitude  and  Mathematical  Induction  27 

require  more  thought  at  the  start,  will  be  found  in  the  long  run 
to  economise  thought  and  increase  logical  power. 

The  use  of  mathematical  induction  in  demonstrations  was, 
in  the  past,  something  of  a  mystery.  There  seemed  no  reason 
able  doubt  that  it  was  a  valid  method  of  proof,  but  no  one  quite 
knew  why  it  was  valid.  Some  believed  it  to  be  really  a  case 
of  induction,  in  the  sense  in  which  that  word  is  used  in  logic. 
Poincare  *•  considered  it  to  be  a  principle  of  the  utmost  import 
ance,  by  means  of  which  an  infinite  number  of  syllogisms  could  be 
condensed  into  one  argument.  We  now  know  that  all  such  views 
are  mistaken,  and  that  mathematical  induction  is  a  definition, 
not  a  principle.  There  are  some  numbers  to  which  it  can  be 
applied,  and  there  are  others  (as  we  shall  see  in  Chapter  VIII.) 
to  which  it  cannot  be  applied.  We  define  the  "  natural  numbers  " 
as  those  to  which  proofs  by  mathematical  induction  can  be 
applied,  i.e.  as  those  that  possess  all  inductive  properties.  It 
follows  that  such  proofs  can  be  applied  to  the  natural  numbers, 
not  in  virtue  of  any  mysterious  intuition  or  axiom  or  principle, 
but  as  a  purely  verbal  proposition.  If  "  quadrupeds "  are 
defined  as  animals  having  four  legs,  it  will  follow  that  animals 
that  have  four  legs  are  quadrupeds ;  and  the  case  of  numbers 
that  obey  mathematical  induction  is  exactly  similar. 

We  shall  use  the  phrase  "  inductive  numbers  "  to  mean  the 
same  set  as  we  have  hitherto  spoken  of  as  the  "  natural  numbers." 
The  phrase  "  inductive  numbers  "  is  preferable  as  affording  a 
reminder  that  the  definition  of  this  set  of  numbers  is  obtained 
from  mathematical  induction. 

Mathematical  induction  affords,  more  than  anything  else, 
the  essential  characteristic  by  which  the  finite  is  distinguished 
from  the  infinite.  The  principle  of  mathematical  induction 
might  be  stated  popularly  in  some  such  form  as  "  what  can  be 
inferred  from  next  to  next  can  be  inferred  from  first  to  last." 
This  is  true  when  the  number  of  intermediate  steps  between 
first  and  last  is  finite,  not  otherwise.  Anyone  who  has  ever 
1  Science  and  Method,  chap.  iv. 


28  Introduction  to  Mathematical  Philosophy 

watched  a  goods  train  beginning  to  move  will  have  noticed  how 
the  impulse  is  communicated  with  a  jerk  from  each  truck  to 
the  next,  until  at  last  even  the  hindmost  truck  is  in  motion. 
When  the  train  is  very  long,  it  is  a  very  long  time  before  the  last 
truck  moves.  If  the  train  were  infinitely  long,  there  would  be 
an  infinite  succession  of  jerks,  and  the  time  would  never  come 
when  the  whole  train  would  be  in  motion.  Nevertheless,  if 
there  were  a  series  of  trucks  no  longer  than  the  series  of  inductive 
numbers  (which,  as  we  shall  see,  is  an  instance  of  the  smallest 
of  infinites),  every  truck  would  begin  to  move  sooner  or  later 
if  the  engine  persevered,  though  there  would  always  be  other 
trucks  further  back  which  had  not  yet  begun  to  move.  This 
image  will  help  to  elucidate  the  argument  from  next  to  next, 
and  its  connection  with  finitude.  When  we  come  to  infinite 
numbers,  where  arguments  from  mathematical  induction  will 
be  no  longer  valid,  the  properties  of  such  numbers  will  help  to 
make  clear,  by  contrast,  the  almost  unconscious  use  that  is  made 
of  mathematical  induction  where  finite  numbers  are  concerned. 


CHAPTER  IV 

THE    DEFINITION    OF    ORDER 

WE  have  now  carried  our  analysis  of  the  series  of  natural  numbers 
to  the  point  where  we  have  obtained  logical  definitions  of  the 
members  of  this  series,  of  the  whole  class  of  its  members,  and 
of  the  relation  of  a  number  to  its  immediate  successor.  We 
must  now  consider  the  serial  character  of  the  natural  numbers 
in  the  order  o,  I,  2,  3,  .  .  .  We  ordinarily  think  of  the  num 
bers  as  in  this  order,  and  it  is  an  essential  part  of  the  work 
of  analysing  our  data  to  seek  a  definition  of  "  order  "  or  "  series  " 
in  logical  terms. 

The  notion  of  order  is  one  which  has  enormous  importance 
in  mathematics.  Not  only  the  integers,  but  also  rational  frac 
tions  and  all  real  numbers  have  an  order  of  magnitude,  and 
this  is  essential  to  most  of  their  mathematical  properties.  The 
order  of  points  on  a  line  is  essential  to  geometry ;  so  is  the 
slightly  more  complicated  order  of  lines  through  a  point  in  a 
plane,  or  of  planes  through  a  line.  Dimensions,  in  geometry, 
are  a  development  of  order.  The  conception  of  a  limit,  which 
underlies  all  higher  mathematics,  is  a  serial  conception.  There 
are  parts  of  mathematics  which  do  not  depend  upon  the  notion 
of  order,  but  they  are  very  few  in  comparison  with  the  parts 
in  which  this  notion  is  involved. 

In  seeking  a  definition  of  order,  the  first  thing  to  realise  is 
that  no  set  of  terms  has  just  one  order  to  the  exclusion  of  others. 
A  set  of  terms  has  all  the  orders  of  which  it  is  capable.  Some 
times  one  order  is  so  much  more  familiar  and  natural  to  our 


30  Introduction  to  Mathematical  Philosophy 

thoughts  that  we  are  inclined  to  regard  it  as  the  order  of  that 
set  of  terms  ;  but  this  is  a  mistake.  The  natural  numbers — 
or  the  "  inductive  "  numbers,  as  we  shall  also  call  them — occur 
to  us  most  readily  in  order  of  magnitude ;  but  they  are  capable 
of  an  infinite  number  of  other  arrangements.  We  might,  for 
example,  consider  first  all  the  odd  numbers  and  then  all  the 
even  numbers  ;  or  first  I,  then  all  the  even  numbers,  then  all 
the  odd  multiples  of  3,  then  all  the  multiples  of  5  but  not  of 
2  or  3,  then  all  the  multiples  of  7  but  not  of  2  or  3  or  5,  and  so 
on  through  the  whole  series  of  primes.  When  we  say  that  we 
"  arrange "  the  numbers  in  these  various  orders,  that  is  an 
inaccurate  expression  :  what  we  really  do  is  to  turn  our  attention 
to  certain  relations  between  the  natural  numbers,  which  them 
selves  generate  such-and-such  an  arrangement.  We  can  no 
more  "  arrange  "  the  natural  numbers  than  we  can  the  starry 
heavens  ;  but  just  as  we  may  notice  among  the  fixed  stars 
either  their  order  of  brightness  or  their  distribution  in  the  sky, 
so  there  are  various  relations  among  numbers  which  may  be 
observed,  and  which  give  rise  to  various  different  orders  among 
numbers,  all  equally  legitimate.  And  what  is  true  of  numbers 
is  equally  true  of  points  on  a  line  or  of  the  moments  of  time  : 
one  order  is  more  familiar,  but  others  are  equally  valid.  We 
might,  for  example,  take  first,  on  a  line,  all  the  points  that  have 
integral  co-ordinates,  then  all  those  that  have  non-integral 
rational  co-ordinates,  then  all  those  that  have  algebraic  non- 
rational  co-ordinates,  and  so  on,  through  any  set  of  complica 
tions  we  please.  The  resulting  order  will  be  one  which  the 
points  of  the  line  certainly  have,  whether  we  choose  to  notice 
it  or  not ;  the  only  thing  that  is  arbitrary  about  the  various 
orders  of  a  set  of  terms  is  our  attention,  for  the  terms  themselves 
have  always  all  the  orders  of  which  they  are  capable. 

One  important  result  of  this  consideration  is  that  we  must 
not  look  for  the  definition  of  order  in  the  nature  of  the  set  of 
terms  to  be  ordered,  since  one  set  of  terms  has  many  orders. 
The  order  lies,  not  in  the  class  of  terms,  but  in  a  relation  among 


The  Definition  of  Order  .    3 1 

the  members  of  the  class,  in  respect  of  which  some  appear  as 
earlier  and  some  as  later.  The  fact  that  a  class  may  have  many 
orders  is  due  to  the  fact  that  there  can  be  many  relations  holding 
among  the  members  of  one  single  class.  What  properties  must 
a  relation  have  in  order  to  give  rise  to  an  order  ? 

The  essential  characteristics  of  a  relation  which  is  to  give  rise 
to  order  may  be  discovered  by  considering  that  in  respect  of 
such  a  relation  we  must  be  able  to  say,  of  any  two  terms  in 
the  class  which  is  to  be  ordered,  that  one  "  precedes  "  and  the 
other  "  follows."  Now,  in  order  that  we  may  be  able  to  use 
these  words  in  the  way  in  which  we  should  naturally  understand 
them,  we  require  that  the  ordering  relation  should  have  three 
properties  : — 

(1)  If  x  precedes  y,  y  must  not  also  precede  x.     This  is  an 
obvious  characteristic  of  the  kind  of  relations  that  lead  to  series. 
If  x  is  less  than  y,  y  is  not  also  less  than  x.     If  x  is  earlier  in 
time  than  y,  y  is  not  also  earlier  than  x.     If  x  is  to  the  left  of 
y,  y  is  not  to  the  left  of  x.     On  the  other  hand,  relations  which 
do  not  give  rise  to  series  often  do  not  have  this  property.     If 
x  is  a  brother  or  sister  of  y,  y  is  a  brother  or  sister  of  x.     If  x  is 
of  the  same  height  as  y,  y  is  of  the  same  height  as  x.     If  x  is  of  a 
different  height  from  y,  y  is  of  a  different  height  from  x.     In 
all  these  cases,  when  the  relation  holds  between  x  and  y,  it  also 
holds  between  y  and  x.     But  with  serial  relations  such  a  thing 
cannot  happen.     A  relation  having  this  first  property  is  called 
asymmetrical. 

(2)  If  x  precedes  y  and  y  precedes  z,  x  must  precede  z.     This 
may  be  illustrated  by  the  same  instances  as  before  :   less,  earlier, 
left  of.     But  as  instances  of  relations  which  do  not  have  this 
property  only  two  of  our  previous  three  instances  will  serve. 
If  x  is  brother  or  sister  of  y,  and  y  of  z,  x  may  not  be  brother 
or  sister  of  z,  since  x  and  z  may  be  the  same  person.     The  same 
applies  to  difference  of  height,  but  not  to  sameness  of  height, 
which  has  our  second  property  but  not  our  first.     The  relation 
"  father,"  on  the   other  hand,  has  our  first  property  but  not 


32  Introduction  to  Mathematical  Philosophy 

our  second.  A  relation  having  our  second  property  is  called 
transitive. 

(3)  Given  any  two  terms  of  the  class  which  is  to  be  ordered, 
there  must  be  one  which  precedes  and  the  other  which  follows. 
For  example,  of  any  two  integers,  or  fractions,  or  real  numbers, 
one  is  smaller  and  the  other  greater ;  but  of  any  two  complex 
numbers  this  is  not  true.  Of  any  two  moments  in  time,  one 
must  be  earlier  than  the  other ;  but  of  events,  which  may  be 
simultaneous,  this  cannot  be  said.  Of  two  points  on  a  line, 
one  must  be  to  the  left  of  the  other.  A  relation  having  this 
third  property  is  called  connected. 

When  a  relation  possesses  these  three  properties,  it  is  of  the 
sort  to  give  rise  to  an  order  among  the  terms  between  which  it 
holds  ;  and  wherever  an  order  exists,  some  relation  having  these 
three  properties  can  be  found  generating  it. 

Before  illustrating  this  thesis,  we  will  introduce  a  few 
definitions. 

(1)  A  relation  is  said  to  be  an  aliorelative,1  or  to  be  contained 
in  or  imply  diversity,  if  no  term  has    this   relation   to   itself. 
Thus,  for  example,  "  greater,"   "  different  in  size,"   "  brother," 
"  husband,"  "  father  "  are  aliorelatives  ;    but  "  equal,"  "  born 
of  the  same  parents,"  "  dear  friend  "  are  not. 

(2)  The  square  of  a  relation  is  that  relation  which  holds  between 
two  terms  x  and  z  when  there  is  an  intermediate  term  y  such 
that  the  given  relation  holds  between  x  and  y  and  between 
y  and  z.     Thus  "  paternal  grandfather  "  is  the  square  of  "  father," 
"  greater  by  2  "  is  the  square  of  "  greater  by  I,"  and  so  on. 

(3)  The  domain  of  a  relation  consists  of  all  those  terms  that 
have  the  relation  to  something  or  other,  and  the  converse  domain 
consists  of  all  those  terms  to  which  something  or  other  has  the 
relation.     These    words    have    been    already    defined,    but    are 
recalled  here  for  the  sake  of  the  following  definition  : — 

(4)  The  field  of  a  relation  consists  of  its  domain  and  converse 
domain  together. 

1  This  term  is  due  to  C.  S.  Peirce. 


The  Definition  of  Order  33 

(5)  One  relation  is  said  to  contain  or  be  implied  by  another  if 
it  holds  whenever  the  other  holds. 

It  will  be  seen  that  an  asymmetrical  relation  is  the  same  thing 
as  a  relation  whose  square  is  an  aliorelative.  It  often  happens 
that  a  relation  is  an  aliorelative  without  being  asymmetrical, 
though  an  asymmetrical  relation  is  always  an  aliorelative.  For 
example,  "  spouse "  is  an  aliorelative,  but  is  symmetrical, 
since  if  x  is  the  spouse  of  y,  y  is  the  spouse  of  x.  But  among 
transitive  relations,  all  aliorelatives  are  asymmetrical  as  well 
as  vice  versa. 

From  the  definitions  it  will  be  seen  that  a  transitive  relation 
is  one  which  is  implied  by  its  square,  or,  as  we  also  say,  "  con 
tains  "  its  square.  Thus  "  ancestor "  is  transitive,  because 
an  ancestor's  ancestor  is  an  ancestor ;  but  "  father "  is  not 
transitive,  because  a  father's  father  is  not  a  father.  A  transitive 
aliorelative  is  one  which  contains  its  square  and  is  contained 
in  diversity ;  or,  what  comes  to  the  same  thing,  one  whose 
square  implies  both  it  and  diversity — because,  when  a  relation 
is  transitive,  asymmetry  is  equivalent  to  being  an  aliorelative. 

A  relation  is  connected  when,  given  any  two  different  terms 
of  its  field,  the  relation  holds  between  the  first  and  the  second 
or  between  the  second  and  the  first  (not  excluding  the  possibility 
that  both  may  happen,  though  both  cannot  happen  if  the  relation 
is  asymmetrical). 

It  will  be  seen  that  the  relation  "  ancestor,"  for  example, 
is  an  aliorelative  and  transitive,  but  not  connected  ;  it  is  because 
it  is  not  connected  that  it  does  not  suffice  to  arrange  the  human 
race  in  a  series. 

The  relation  "  less  than  or  equal  to,"  among  numbers,  is 
transitive  and  connected,  but  not  asymmetrical  or  an  aliorelative. 

The  relation  "  greater  or  less  "  among  numbers  is  an  alio 
relative  and  is  connected,  but  is  not  transitive,  for  if  x  is  greater 
or  less  than  y,  and  y  is  greater  or  less  than  z,  it  may  happen 
that  x  and  z  are  the  same  number. 

Thus  the  three  properties  of  being  (i)  an  aliorelative,  (2) 

3 


34  Introduction  to  Mathematical  Philosophy 

transitive,  and  (3)  connected,  are  mutually  independent,  since 
a  relation  may  have  any  two  without  having  the  third. 

We  now  lay  down  the  following  definition  : — 

A  relation  is  serial  when  it  is  an  aliorelative,  transitive,  and 
connected ;  or,  what  is  equivalent,  when  it  is  asymmetrical, 
transitive,  and  connected. 

A  series  is  the  same  thing  as  a  serial  relation. 

It  might  have  been  thought  that  a  series  should  be  the  field 
of  a  serial  relation,  not  the  serial  relation  itself.  But  this  would 
be  an  error.  For  example, 

I,  2,  3  ;   i,  3,  2  ;   2,  3,  I  ;   2,  i,  3  ;   3,  I,  2  ;   3,  2,  I 

are  six  different  series  which  all  have  the  same  field.  If  the 
field  were  the  series,  there  could  only  be  one  series  with  a  given 
field.  What  distinguishes  the  above  six  series  is  simply  the 
different  ordering  relations  in  the  six  cases.  Given  the  ordering 
relation,  the  field  and  the  order  are  both  determinate.  Thus 
the  ordering  relation  may  be  taken  to  be  the  series,  but  the  field 
cannot  be  so  taken. 

Given  any  serial  relation,  say  P,  we  shall  say  that,  in  respect 
of  this  relation,  x  "  precedes  "  y  if  x  has  the  relation  P  to  y, 
which  we  shall  write  "  xPy  "  for  short.  The  three  characteristics 
which  P  must  have  in  order  to  be  serial  are : 

(1)  We   must    never  have  xPx,   i.e.   no  term  must  precede 

itself. 

(2)  P2  must  imply  P,  i.e.  if  x  precedes  y  and  y  precedes  z,  x  must 

precede  z. 

(3)  If  x  and  y  are  two  different  terms  in  the  field  of  P,  we  shall 

have  xPy  or  yPx,  i.e.  one  of  the  two  must  precede  the 
other. 

The  reader  can  easily  convince  himself  that,  where  these  three 
properties  are  found  in  an  ordering  relation,  the  characteristics 
we  expect  of  series  will  also  be  found,  and  vice  versa.  We  are 
therefore  justified  in  taking  the  above  as  a  definition  of  order 


The  Definition  of  Order  35 

or  series.  And  it  will  be  observed  that  the  definition  is  effected 
in  purely  logical  terms. 

Although  a  transitive  asymmetrical  connected  relation  always 
exists  wherever  there  is  a  series,  it  is  not  always  the  relation 
which  would  most  naturally  be  regarded  as  generating  the  series. 
The  natural-number  series  may  serve  as  an  illustration.  The 
relation  we  assumed  in  considering  the  natural  numbers  was 
the  relation  of  immediate  succession,  i.e.  the  relation  between 
consecutive  integers.  This  relation  is  asymmetrical,  but  not 
transitive  or  connected.  We  can,  however,  derive  from  it, 
by  the  method  of  mathematical  induction,  the  "  ancestral " 
relation  which  we  considered  in  the  preceding  chapter.  This 
relation  will  be  the  same  as  "  less  than  or  equal  to  "  among 
inductive  integers.  For  purposes  of  generating  the  series  of 
natural  numbers,  we  want  the  relation  "  less  than,"  excluding 
"  equal  to."  This  is  the  relation  oimton  when  m  is  an  ancestor 
of  n  but  not  identical  with  n,  or  (what  comes  to  the  same  thing) 
when  the  successor  of  m  is  an  ancestor  of  n  in  the  sense  in  which 
a  number  is  its  own  ancestor.  That  is  to  say,  we  shall  lay  down 
the  following  definition  : — 

An  inductive  number  m  is  said  to  be  less  than  another  number 
n  when  n  possesses  every  hereditary  property  possessed  by  the 
successor  of  m. 

It  is  easy  to  see,  and  not  difficult  to  prove,  that  the  relation 
"  less  than,"  so  defined,  is  asymmetrical,  transitive,  and  con 
nected,  and  has  the  inductive  numbers  for  its  field.  Thus  by 
means  of  this  relation  the  inductive  numbers  acquire  an  order 
in  the  sense  in  which  we  defined  the  term  "  order,"  and  this  order 
is  the  so-called  "  natural  "  order,  or  order  of  magnitude. 

The  generation  of  series  by  means  of  relations  more  or  less 
resembling  that  of  n  to  n-j-i  is  very  common.  The  series  of  the 
Kings  of  England,  for  example,  is  generated  by  relations  of  each 
to  his  successor.  This  is  probably  the  easiest  way,  where  it  is 
applicable,  of  conceiving  the  generation  of  a  series.  In  this 
method  we  pass  on  from  each  term  to  the  next,  as  long  as  there 


36  Introduction  to  Mathematical  Philosophy 

is  a  next,  or  back  to  the  one  before,  as  long  as  there  is  one  before. 
This  method  always  requires  the  generalised  form  of  mathe 
matical  induction  in  order  to  enable  us  to  define  "  earlier  "  and 
"  later  "  in  a  series  so  generated.  On  the  analogy  of  "  proper 
fractions,"  let  us  give  the  name  "  proper  posterity  of  x  with  respect 
to  R  "  to  the  class  of  those  terms  that  belong  to  the  R-posterity 
of  some  term  to  which  x  has  the  relation  R,  in  the  sense  which 
we  gave  before  to  "  posterity,"  which  includes  a  term  in  its  own 
posterity.  Reverting  to  the  fundamental  definitions,  we  find  that 
the  "  proper  posterity  "  may  be  defined  as  follows  : — 

The  "  proper  posterity  "  of  x  with  respect  to  R  consists  of 
all  terms  that  possess  every  R-hereditary  property  possessed  by 
every  term  to  which  x  has  the  relation  R. 

It  is  to  be  observed  that  this  definition  has  to  be  so  framed 
as  to  be  applicable  not  only  when  there  is  only  one  term  to  which 
x  has  the  relation  R,  but  also  in  cases  (as  e.g.  that  of  father  and 
child)  where  there  may  be  many  terms  to  which  x  has  the  relation 
R.  We  define  further  : 

A  term  x  is  a  "  proper  ancestor  "  of  y  with  respect  to  R  if  y 
belongs  to  the  proper  posterity  of  x  with  respect  to  R. 

We  shall  speak  for  short  of  "  R-posterity  "  and  "  R-ancestors  " 
when  these  terms  seem  more  convenient. 

Reverting  now  to  the  generation  of  series  by  the  relation  R 
between  consecutive  terms,  we  see  that,  if  this  method  is  to  be 
possible,  the  relation  "  proper  R-ancestor  "  must  be  an  aliorela- 
tive,  transitive,  and  connected.  Under  what  circumstances  will 
this  occur  ?  It  will  always  be  transitive  :  no  matter  what  sort 
of  relation  R  may  be,  "  R-ancestor  "  and  "  proper  R-ancestor  " 
are  always  both  transitive.  But  it  is  only  under  certain  circum 
stances  that  it  will  be  an  aliorelative  or  connected.  Consider, 
for  example,  the  relation  to  one's  left-hand  neighbour  at  a  round 
dinner-table  at  which  there  are  twelve  people.  If  we  call  this 
relation  R,  the  proper  R-posterity  of  a  person  consists  of  all  who 
can  be  reached  by  going  round  the  table  from  right  to  left.  This 
includes  everybody  at  the  table,  including  the  person  himself,  since 


The  Definition  of  Order  37 

twelve  steps  bring  us  back  to  our  starting-point.  Thus  in  such 
a  case,  though  the  relation  "  proper  R-ancestor  "  is  connected, 
and  though  R  itself  is  an  aliorelative,  we  do  not  get  a  series 
because  "  proper  R-ancestor  "  is  not  an  aliorelative.  It  is  for 
this  reason  that  we  cannot  say  that  one  person  comes  before 
another  with  respect  to  the  relation  "  right  of  "  or  to  its  ancestral 
derivative. 

The  above  was  an  instance  in  which  the  ancestral  relation  was 
connected  but  not  contained  in  diversity.  An  instance  where 
it  is  contained  in  diversity  but  not  connected  is  derived  from  the 
ordinary  sense  of  the  word  "  ancestor."  If  x  is  a  proper  ancestor 
of  y,  x  and  y  cannot  be  the  same  person  ;  but  it  is  not  true  that 
of  any  two  persons  one  must  be  an  ancestor  of  the  other. 

The  question  of  the  circumstances  under  which  series  can  be 
generated  by  ancestral  relations  derived  from  relations  of  con- 
secutiveness  is  often  important.  Some  of  the  most  important 
cases  are  the  following :  Let  R  be  a  many-one  relation,  and  let 
us  confine  our  attention  to  the  posterity  of  some  term  x.  When 
so  confined,  the  relation  "  proper  R-ancestor  "  must  be  connected ; 
therefore  all  that  remains  to  ensure  its  being  serial  is  that  it  shall 
be  contained  in  diversity.  This  is  a  generalisation  of  the  instance 
of  the  dinner-table.  Another  generalisation  consists  in  taking 
R  to  be  a  one-one  relation,  and  including  the  ancestry  of  x  as 
well  as  the  posterity.  Here  again,  the  one  condition  required 
to  secure  the  generation  of  a  series  is  that  the  relation  "  proper 
R-ancestor  "  shall  be  contained  in  diversity. 

The  generation  of  order  by  means  of  relations  of  consecutive- 
ness,  though  important  in  its  own  sphere,  is  less  general  than  the 
method  which  uses  a  transitive  relation  to  define  the  order.  It 
often  happens  in  a  series  that  there  are  an  infinite  number  of  inter 
mediate  terms  between  any  two  that  may  be  selected,  however 
near  together  these  may  be.  Take,  for  instance,  fractions  in  order 
of  magnitude.  Between  any  two  fractions  there  are  others — for 
example,  the  arithmetic  mean  of  the  two.  Consequently  there  is 
no  such  thing  as  a  pair  of  consecutive  fractions.  If  we  depended 


38  Introduction  to  Mathematical  Philosophy 

upon  consecutiveness  for  defining  order,  we  should  not  be  able 
to  define  the  order  of  magnitude  among  fractions.  But  in  fact 
the  relations  of  greater  and  less  among  fractions  do  not  demand 
generation  from  relations  of  consecutiveness,  and  the  relations 
of  greater  and  less  among  fractions  have  the  three  characteristics 
which  we  need  for  defining  serial  relations.  In  all  such  cases 
the  order  must  be  defined  by  means  of  a  transitive  relation,  since 
only  such  a  relation  is  able  to  leap  over  an  infinite  number  of 
intermediate  terms.  The  method  of  consecutiveness,  like  that 
of  counting  for  discovering  the  number  of  a  collection,  is  appro 
priate  to  the  finite  ;  it  may  even  be  extended  to  certain  infinite 
series,  namely,  those  in  which,  though  the  total  number  of  terms  is 
infinite,  the  number  of  terms  between  any  two  is  always  finite ; 
but  it  must  not  be  regarded  as  general.  Not  only  so,  but  care 
must  be  taken  to  eradicate  from  the  imagination  all  habits  of 
thought  resulting  from  supposing  it  general.  If  this  is  not  done, 
series  in  which  there  are  no  consecutive  terms  will  remain  difficult 
and  puzzling.  And  such  series  are  of  vital  importance  for  the 
understanding  of  continuity,  space,  time,  and  motion. 

There  are  many  ways  in  which  series  may  be  generated,  but 
all  depend  upon  the  finding  or  construction  of  an  asymmetrical 
transitive  connected  relation.  Some  of  these  ways  have  con 
siderable  importance.  We  may  take  as  illustrative  the  genera 
tion  of  series  by  means  of  a  three-term  relation  which  we  may 
call  "  between."  This  method  is  very  useful  in  geometry,  and 
may  serve  as  an  introduction  to  relations  having  more  than  two 
terms ;  it  is  best  introduced  in  connection  with  elementary 
geometry. 

Given  any  three  points  on  a  straight  line  in  ordinary  space, 
there  must  be  one  of  them  which  is  between  the  other  two.  This 
will  not  be  the  case  with  the  points  on  a  circle  or  any  other  closed 
curve,  because,  given  any  three  points  on  a  circle,  we  can  travel 
from  any  one  to  any  other  without  passing  through  the  third. 
In  fact,  the  notion  "  between  "  is  characteristic  of  open  series — 
or  series  in  the  strict  sense — as  opposed  to  what  may  be  called 


The  Definition  of  Order  39 

"  cyclic "  series,  where,  as  with  people  at  the  dinner-table,  a 
sufficient  journey  brings  us  back  to  our  starting-point.  This 
notion  of  "  between  "  may  be  chosen  as  the  fundamental  notion 
of  ordinary  geometry  ;  but  for  the  present  we  will  only  consider 
its  application  to  a  single  straight  line  and  to  the  ordering  of  the 
points  on  a  straight  line.1  Taking  any  two  points  #,  b,  the  line 
(ab)  consists  of  three  parts  (besides  a  and  b  themselves)  : 

(1)  Points  between  a  and  b. 

(2)  Points  x  such  that  a  is  between  x  and  b. 

(3)  Points  y  such  that  b  is  between  y  and  a. 

Thus  the  line  (ab)  can  be  defined  in  terms  of  the  relation 
"  between." 

In  order  that  this  relation  "  between  "  may  arrange  the  points 
of  the  line  in  an  order  from  left  to  right,  we  need  certain  assump 
tions,  namely,  the  following  : — 

(1)  If  anything  is  between  a  and  b,  a  and  b  are  not  identical. 

(2)  Anything  between  a  and  b  is  also  between  b  and  a. 

(3)  Anything  between  a  and  b  is  not  identical  with  a  (nor, 
consequently,  with  b,  in  virtue  of  (2)). 

(4)  If  x  is  between  a  and  b,  anything  between  a  and  x  is  also 
between  a  and  b. 

(5)  If  x  is  between  a  and  b,  and  b  is  between  x  and  y,  then  b 
is  between  a  and  y. 

(6)  If  x  and  y  are  between  a  and  b,  then  either  x  and  y  are 
identical,  or  x  is  between  a  and  y,  or  x  is  between  y  and  b. 

(7)  If  b  is  between  a  and  x  and  also  between  a  and  y,  then  either 
a:  and  y  are  identical,  or  x  is  between  £  and  y,  or  y  is  between 
b  and  #. 

These  seven  properties  are  obviously  verified  in  the  case  of  points 
on  a  straight  line  in  ordinary  space.  Any  three-term  relation 
which  verifies  them  gives  rise  to  series,  as  may  be  seen  from  the 
following  definitions.  For  the  sake  of  definiteness,  let  us  assume 

1  Cf .  Rivista  di  Matematica,  iv.  pp.  55  ft. ;  Principles  of  Mathematics,  p. 
394  (§  375). 


4-O  Introduction  to  Mathematical  Philosophy 

that  a  is  to  the  left  of  b.  Then  the  points  of  the  line  (ab)  are  (i) 
those  between  which  and  b,  a  lies — these  we  will  call  to  the  left 
of  a  ;  (2)  a  itself  ;  (3)  those  between  a  and  b  ;  (4)  b  itself  ;  (5) 
those  between  which  and  a  lies  b — these  we  will  call  to  the  right 
of  b.  We  may  now  define  generally  that  of  two  points  x,  y,  on 
the  line  (ab),  we  shall  say  that  x  is  "  to  the  left  of  "  y  in  any 
of  the  following  cases  : — 

(1)  When  x  and  y  are  both  to  the  left  of  a,  and  y  is  between 

x  and  a ; 

(2)  When  x  is  to  the  left  of  a,  and  y  is  a  or  b  or  between  a  and 

b  or  to  the  right  of  b  ; 

(3)  When  x  is  a,  and  y  is  between  a  and  b  or  is  b  or  is  to  the 

right  of  b ; 

(4)  When  x  and  y  are  both  between  a  and  £,  and  y  is  between 

#  and  b  ; 

(5)  When  x  is  between  <z  and  b,  and  y  is  3  or  to  the  right  of  b  ; 

(6)  When  x  is  £  and  y  is  to  the  right  of  b  ; 

(7)  When  x  and  y  are  both  to  the  right  of  b  and  x  is  between 

b  and  y. 

It  will  be  found  that,  from  the  seven  properties  which  we  have 
assigned  to  the  relation  "  between,"  it  can  be  deduced  that  the 
relation  "  to  the  left  of,"  as  above  defined,  is  a  serial  relation  as 
we  defined  that  term.  It  is  important  to  notice  that  nothing 
in  the  definitions  or  the  argument  depends  upon  our  meaning 
by  "  between  "  the  actual  relation  of  that  name  which  occurs  in 
empirical  space  :  any  three-term  relation  having  the  above  seven 
purely  formal  properties  will  serve  the  purpose  of  the  argument 
equally  well. 

Cyclic  order,  such  as  that  of  the  points  on  a  circle,  cannot  be 
generated  by  means  of  three-term  relations  of  "  between."  We 
need  a  relation  of  four  terms,  which  may  be  called  "  separation 
of  couples."  The  point  may  be  illustrated  by  considering  a 
journey  round  the  world.  One  may  go  from  England  to  New 
Zealand  by  way  of  Suez  or  by  way  of  San  Francisco  ;  we  cannot 


The  Definition  of  Order  41 

say  definitely  that  either  of  these  two  places  is  "  between " 
England  and  New  Zealand.  But  if  a  man  chooses  that  route 
to  go  round  the  world,  whichever  way  round  he  goes,  his  times  in 
England  and  New  Zealand  are  separated  from  each  other  by  his 
times  in  Suez  and  San  Francisco,  and  conversely.  Generalising, 
if  we  take  any  four  points  on  a  circle,  we  can  separate  them  into 
two  couples,  say  a  and  b  and  x  and  y,  such  that,  in  order  to  get 
from  a  to  b  one  must  pass  through  either  x  or  y,  and  in  order  to 
get  from  x  to  y  one  must  pass  through  either  a  or  b.  Under  these 
circumstances  we  say  that  the  couple  (a,  b)  are  "  separated  "  by 
the  couple  (x,  y).  Out  of  this  relation  a  cyclic  order  can  be  gen 
erated,  in  a  way  resembling  that  in  which  we  generated  an  open 
order  from  "  between,"  but  somewhat  more  complicated.1 

The  purpose  of  the  latter  half  of  this  chapter  has  been  to  suggest 
the  subject  which  one  may  call  "  generation  of  serial  relations." 
When  such  relations  have  been  defined,  the  generation  of  them 
from  other  relations  possessing  only  some  of  the  properties 
required  for  series  becomes  very  important,  especially  in  the 
philosophy  of  geometry  and  physics.  But  we  cannot,  within 
the  limits  of  the  present  volume,  do  more  than  make  the  reader 
aware  that  such  a  subject  exists. 

1  Cf.  Principles  of  Mathematics,  p.  205  (§  194),  and  references  there  given. 


CHAPTER  V 

KINDS    OF    RELATIONS 

A  GREAT  part  of  the  philosophy  of  mathematics  is  concerned  with 
relations,  and  many  different  kinds  of  relations  have  different 
kinds  of  uses.  It  often  happens  that  a  property  which  belongs 
to  all  relations  is  only  important  as  regards  relations  of  certain 
sorts  ;  in  these  cases  the  reader  will  not  see  the  bearing  of  the 
proposition  asserting  such  a  property  unless  he  has  in  mind  the 
sorts  of  relations  for  which  it  is  useful.  For  reasons  of  this 
description,  as  well  as  from  the  intrinsic  interest  of  the  subject, 
it  is  well  to  have  in  our  minds  a  rough  list  of  the  more 
mathematically  serviceable  varieties  of  relations. 

We  dealt  in  the  preceding  chapter  with  a  supremely  important 
class,  namely,  serial  relations.  Each  of  the  three  properties  which 
we  combined  in  defining  series — namely,  asymmetry,  transitiveness, 
and  connexity — has  its  own  importance.  We  will  begin  by  saying 
something  on  each  of  these  three. 

Asymmetry,  i.e.  the  property  of  being  incompatible  with  the 
converse,  is  a  characteristic  of  the  very  greatest  interest  and 
importance.  In  order  to  develop  its  functions,  we  will  consider 
various  examples.  The  relation  husband  is  asymmetrical,  and 
so  is  the  relation  wife  ;  i.e.  if  a  is  husband  of  b,  b  cannot  be  husband 
of  a,  and  similarly  in  the  case  of  wife.  On  the  other  hand,  the 
relation  "  spouse  "  is  symmetrical :  if  a  is  spouse  of  b,  then  b  is 
spouse  of  a.  Suppose  now  we  are  given  the  relation  spouse,  and 
we  wish  to  derive  the  relation  husband.  Husband  is  the  same  as 
male  spouse  or  spouse  of  a  female  ;  thus  the  relation  husband  can 

4* 


Kinds  of  Relations  43 

be  derived  from  spouse  either  by  limiting  the  domain  to  males 
or  by  limiting  the  converse  to  females.  We  see  from  this  instance 
that,  when  a  symmetrical  relation  is  given,  it  is  sometimes  possible, 
without  the  help  of  any  further  relation,  to  separate  it  into  two 
asymmetrical  relations.  But  the  cases  where  this  is  possible  are 
rare  and  exceptional :  they  are  cases  where  there  are  two  mutually 
exclusive  classes,  say  a  and  j3,  such  that  whenever  the  relation 
holds  between  two  terms,  one  of  the  terms  is  a  member  of  a  and 
the  other  is  a  member  of  )3 — as,  in  the  case  of  spouse,  one  term 
of  the  relation  belongs  to  the  class  of  males  and  one  to  the  class 
of  females.  In  such  a  case,  the  relation  with  its  domain  confined 
to  a  will  be  asymmetrical,  and  so  will  the  relation  with  its  domain 
confined  to  j3.  But  such  cases  are  not  of  the  sort  that  occur 
when  we  are  dealing  with  series  of  more  than  two  terms ;  for  in 
a  series,  all  terms,  except  the  first  and  last  (if  these  exist),  belong 
both  to  the  domain  and  to  the  converse  domain  of  the  generating 
relation,  so  that  a  relation  like  husband,  where  the  domain  and 
converse  domain  do  not  overlap,  is  excluded. 

The  question  how  to  construct  relations  having  some  useful 
property  by  means  of  operations  upon  relations  which  only  have 
rudiments  of  the  property  is  one  of  considerable  importance. 
Transitiveness  and  connexity  are  easily  constructed  in  many  cases 
where  the  originally  given  relation  does  not  possess  them  :  for 
example,  if  R  is  any  relation  whatever,  the  ancestral  relation 
derived  from  R  by  generalised  induction  is  transitive ;  and  if  R 
is  a  many-one  relation,  the  ancestral  relation  will  be  connected 
if  confined  to  the  posterity  of  a  given  term.  But  asymmetry  is 
a  much  more  difficult  property  to  secure  by  construction.  The 
method  by  which  we  derived  husband  from  spouse  is,  as  we  have 
seen,  not  available  in  the  most  important  cases,  such  as  greater, 
before,  to  the  right  of,  where  domain  and  converse  domain  overlap. 
In  all  these  cases,  we  can  of  course  obtain  a  symmetrical  relation 
by  adding  together  the  given  relation  and  its  converse,  but  we 
cannot  pass  back  from  this  symmetrical  relation  to  the  original 
asymmetrical  relation  except  by  the  help  of  some  asymmetrical 


44  Introauction  to  Mathematical  Philosophy 

relation.  Take,  for  example,  the  relation  greater :  the  relation 
greater  or  less — i.e.  unequal — is  symmetrical,  but  there  is  nothing 
in  this  relation  to  show  that  it  is  the  sum  of  two  asymmetrical 
relations.  Take  such  a  relation  as  "  differing  in  shape."  This 
is  not  the  sum  of  an  asymmetrical  relation  and  its  converse,  since 
shapes  do  not  form  a  single  series  ;  but  there  is  nothing  to  show 
that  it  differs  from  "  differing  in  magnitude  "  if  we  did  not  already 
know  that  magnitudes  have  relations  of  greater  and  less.  This 
illustrates  the  fundamental  character  of  asymmetry  as  a  property 
of  relations. 

From  the  point  of  view  of  the  classification  of  relations,  being 
asymmetrical  is  a  much  more  important  characteristic  than 
implying  diversity.  Asymmetrical  relations  imply  diversity, 
but  the  converse  is  not  the  case.  "  Unequal,"  for  example, 
implies  diversity,  but  is  symmetrical.  Broadly  speaking,  we 
may  say  that,  if  we  wished  as  far  as  possible  to  dispense  with 
relational  propositions  and  replace  them  by  such  as  ascribed 
predicates  to  subjects,  we  could  succeed  in  this  so  long  as  we 
confined  ourselves  to  symmetrical  relations  :  those  that  do  not 
imply  diversity,  if  they  are  transitive,  may  be  regarded  as  assert 
ing  a  common  predicate,  while  those  that  do  imply  diversity 
may  be  regarded  as  asserting  incompatible  predicates.  For 
example,  consider  the  relation  of  similarity  between  classes, 
by  means  of  which  we  defined  numbers.  This  relation  is  sym 
metrical  and  transitive  and  does  not  imply  diversity.  It  would 
be  possible,  though  less  simple  than  the  procedure  we  adopted, 
to  regard  the  number  of  a  collection  as  a  predicate  of  the  collec 
tion  :  then  two  similar  classes  will  be  two  that  have  the  same 
numerical  predicate,  while  two  that  are  not  similar  will  be  two 
that  have  different  numerical  predicates.  Such  a  method  of 
replacing  relations  by  predicates  is  formally  possible  (though 
often  very  inconvenient)  so  long  as  the  relations  concerned  are 
symmetrical ;  but  it  is  formally  impossible  when  the  relations 
are  asymmetrical,  because  both  sameness  and  difference  of  predi 
cates  are  symmetrical.  Asymmetrical  relations  are,  we  may 


Kinds  of  Relations  45 

say,  the  most  characteristically  relational  of  relations,  and  the 
most  important  to  the  philosopher  who  wishes  to  study  the 
ultimate  logical  nature  of  relations. 

Another  class  of  relations  that  is  of  the  greatest  use  is  the 
class  of  one-many  relations,  i.e.  relations  which  at  most  one 
term  can  have  to  a  given  term.  Such  are  father,  mother, 
husband  (except  in  Tibet),  square  of,  sine  of,  and  so  on.  But 
parent,  square  root,  and  so  on,  are  not  one-many.  It  is  possible, 
formally,  to  replace  all  relations  by  one-many  relations  by  means 
of  a  device.  Take  (say)  the  relation  less  among  the  inductive 
numbers.  Given  any  number  n  greater  than  I,  there  will  not 
be  only  one  number  having  the  relation  less  to  n,  but  we  can 
form  the  whole  class  of  numbers  that  are  less  than  n.  This 
is  one  class,  and  its  relation  to  n  is  not  shared  by  any  other  class. 
We  may  call  the  class  of  numbers  that  are  less  than  n  the  "  proper 
ancestry  "  of  n,  in  the  sense  in  which  we  spoke  of  ancestry  and 
posterity  in  connection  with  mathematical  induction.  Then 
"  proper  ancestry  "  is  a  one-many  relation  (one-many  will  always 
be  used  so  as  to  include  one-one),  since  each  number  determines 
a  single  class  of  numbers  as  constituting  its  proper  ancestry. 
Thus  the  relation  less  than  can  be  replaced  by  being  a  member  of 
the  proper  ancestry  of.  In  this  way  a  one-many  relation  in  which 
the  one  is  a  class,  together  with  membership  of  this  class,  can 
always  formally  replace  a  relation  which  is  not  one-many.  Peano, 
who  for  some  reason  always  instinctively  conceives  of  a  relation 
as  one-many,  deals  in  this  way  with  those  that  are  naturally 
not  so.  Reduction  to  one-many  relations  by  this  method, 
however,  though  possible  as  a  matter  of  form,  does  not  represent 
a  technical  simplification,  and  there  is  every  reason  to  think 
that  it  does  not  represent  a  philosophical  analysis,  if  only  because 
classes  must  be  regarded  as  "  logical  fictions."  We  shall  there 
fore  continue  to  regard  one-many  relations  as  a  special  kind  of 
relations. 

One-many  relations  are  involved  in  all  phrases  of  the  form 
"  the  so-and-so  of  such-and-such."     "  The  King  of  England," 


46  Introduction  to  Mathematical  Philosophy 

"  the  wife  of  Socrates,"  "  the  father  of  John  Stuart  Mill,"  and 
so  on,  all  describe  some  person  by  means  of  a  one-many  relation 
to  a  given  term.  A  person  cannot  have  more  than  one  father, 
therefore  "  the  father  of  John  Stuart  Mill "  described  some  one 
person,  even  if  we  did  not  know  whom.  There  is  much  to 
say  on  the  subject  of  descriptions,  but  for  the  present  it  is 
relations  that  we  are  concerned  with,  and  descriptions  are  only 
relevant  as  exemplifying  the  uses  of  one-many  relations.  It 
should  be  observed  that  all  mathematical  functions  result  from 
one-many  relations  :  the  logarithm  of  x,  the  cosine  of  x,  etc., 
are,  like  the  father  of  x,  terms  described  by  means  of  a  one-many 
relation  (logarithm,  cosine,  etc.)  to  a  given  term  (x).  The 
notion  of  function  need  not  be  confined  to  numbers,  or  to  the 
uses  to  which  mathematicians  have  accustomed  us  ;  it  can  be 
extended  to  all  cases  of  one-many  relations,  and  "  the  father  of  x  " 
is  just  as  legitimately  a  function  of  which  x  is  the  argument  as 
is  "  the  logarithm  of  x."  Functions  in  this  sense  are  descriptive 
functions.  As  we  shall  see  later,  there  are  functions  of  a  still 
more  general  and  more  fundamental  sort,  namely,  prepositional 
functions  ;  but  for  the  present  we  shall  confine  our  attention 
to  descriptive  functions,  i.e.  "  the  term  having  the  relation  R 
to  x,"  or,  for  short,  "  the  R  of  x"  where  R  is  any  one-many 
relation. 

It  will  be  observed  that  if  "  the  R  of  x  "  is  to  describe  a  definite 
term,  x  must  be  a  term  to  which  something  has  the  relation  R, 
and  there  must  not  be  more  than  one  term  having  the  relation 
R  to  x,  since  "  the,"  correctly  used,  must  imply  uniqueness. 
Thus  we  may  speak  of  "  the  father  of  x  "  if  x  is  any  human  being 
except  Adam  and  Eve ;  but  we  cannot  speak  of  "  the  father 
of  x  "  if  x  is  a  table  or  a  chair  or  anything  else  that  does  not 
have  a  father.  We  shall  say  that  the  R  of  x  "  exists  "  when 
there  is  just  one  term,  and  no  more,  having  the  relation  R  to  x. 
Thus  if  R  is  a  one-many  relation,  the  R  of  x  exists  whenever 
x  belongs  to  the  converse  domain  of  R,  and  not  otherwise. 
Regarding  "  the  R  of  x "  as  a  function  in  the  mathematical 


Kinds  of  Relations  47 

sense,  we  say  that  x  is  the  "  argument  "  of  the  function,  and  if 
y  is  the  term  which  has  the  relation  R  to  x,  i.e.  if  y  is  the  R  of  x, 
then  y  is  the  "  value  "  of  the  function  for  the  argument  x.  If 
R  is  a  one-many  relation,  the  range  of  possible  arguments  to 
the  function  is  the  converse  domain  of  R,  and  the  range  of  values 
is  the  domain.  Thus  the  range  of  possible  arguments  to  the 
function  "  the  father  of  x  "  is  all  who  have  fathers,  i.e.  the  con 
verse  domain  of  the  relation  father,  while  the  range  of  possible 
values  for  the  function  is  all  fathers,  i.e.  the  domain  of  the  relation. 

Many  of  the  most  important  notions  in  the  logic  of  relations 
are  descriptive  functions,  for  example :  converse,  domain,  con 
verse  domain,  field.  Other  examples  will  occur  as  we  proceed. 

Among  one-many  relations,  one-one  relations  are  a  specially 
important  class.  We  have  already  had  occasion  to  speak  of 
one-one  relations  in  connection  with  the  definition  of  number, 
but  it  is  necessary  to  be  familiar  with  them,  and  not  merely 
to  know  their  formal  definition.  Their  formal  definition  may 
be  derived  from  that  of  one-many  relations  :  they  may  be 
defined  as  one-many  relations  which  are  also  the  converses  of 
one-many  relations,  i.e.  as  relations  which  are  both  one-many 
and  many-one.  One-many  relations  may  be  defined  as  relations 
such  that,  if  x  has  the  relation  in  question  to  y,  there  is  no  other 
term  x'  which  also  has  the  relation  to  y.  Or,  again,  they  may 
be  defined  as  follows  :  Given  two  terms  x  and  x',  the  terms  to 
which  x  has  the  given  relation  and  those  to  which  x'  has  it  have 
no  member  in  common.  Or,  again,  they  may  be  defined  as 
relations  such  that  the  relative  product  of  one  of  them  and 
its  converse  implies  identity,  where  the  "  relative  product " 
of  two  relations  R  and  S  is  that  relation  which  holds  between 
x  and  2  when  there  is  an  intermediate  term  y,  such  that  x  has 
the  relation  R  to  y  and  y  has  the  relation  S  to  2.  Thus,  for 
example,  if  R  is  the  relation  of  father  to  son,  the  relative  product 
of  R  and  its  converse  will  be  the  relation  which  holds  between 
x  and  a  man  2  when  there  is  a  person  y,  such  that  x  is  the  father 
of  y  and  y  is  the  son  of  2.  It  is  obvious  that  x  and  z  must  be 


48  Introduction  to  Mathematical  Philosophy 

the  same  person.  If,  on  the  other  hand,  we  take  the  relation 
of  parent  and  child,  which  is  not  one-many,  we  can  no  longer 
argue  that,  if  x  is  a  parent  of  y  and  y  is  a  child  of  z,  x  and  z  must 
be  the  same  person,  because  one  may  be  the  father  of  y  and  the 
other  the  mother.  This  illustrates  that  it  is  characteristic  of 
one-many  relations  when  the  relative  product  of  a  relation  and 
its  converse  implies  identity.  In  the  case  of  one-one  relations 
this  happens,  and  also  the  relative  product  of  the  converse  and 
the  relation  implies  identity.  Given  a  relation  R,  it  is  convenient, 
if  x  has  the  relation  R  to  y,  to  think  of  y  as  being  reached  from 
x  by  an  "  R-step  "  or  an  "  R-vector."  In  the  same  case  x  will 
be  reached  from  y  by  a  "  backward  R-step."  Thus  we  may 
state  the  characteristic  of  one-many  relations  with  which  we 
have  been  dealing  by  saying  that  an  R-step  followed  by  a  back 
ward  R-step  must  bring  us  back  to  our  starting-point.  With 
other  relations,  this  is  by  no  means  the  case ;  for  example,  if 
R  is  the  relation  of  child  to  parent,  the  relative  product  of  R  and 
its  converse  is  the  relation  "  self  or  brother  or  sister,'*  and  if  R 
is  the  relation  of  grandchild  to  grandparent,  the  relative  product 
of  R  and  its  converse  is  "  self  or  brother  or  sister  or  first  cousin." 
It  will  be  observed  that  the  relative  product  of  two  relations 
is  not  in  general  commutative,  i.e.  the  relative  product  of  R 
and  S  is  not  in  general  the  same  relation  as  the  relative  product 
of  S  and  R.  E.g.  the  relative  product  of  parent  and  brother  is 
uncle,  but  the  relative  product  of  brother  and  parent  is  parent. 

One-one  relations  give  a  correlation  of  two  classes,  term  for 
term,  so  that  each  term  in  either  class  has  its  correlate  in  the 
other.  Such  correlations  are  simplest  to  grasp  when  the  two 
classes  have  no  members  in  common,  like  the  class  of  husbands 
and  the  class  of  wives  ;  for  in  that  case  we  know  at  once  whether 
a  term  is  to  be  considered  as  one  from  which  the  correlating 
relation  R  goes,  or  as  one  to  which  it  goes.  It  is  convenient 
to  use  the  word  referent  for  the  term  from  which  the  relation 
goes,  and  the  term  relatum  for  the  term  to  which  it  goes.  Thus 
if  x  and  y  are  husband  and  wife,  then,  with  respect  to  the  relation 


Kinas  of  Relations  49 

"  husband,"  x  is  referent  and  y  relatum,  but  with  respect  to  the 
relation  "  wife,"  y  is  referent  and  x  relatum.  We  say  that  a 
relation  and  its  converse  have  opposite  "  senses  "  ;  thus  the 
"  sense  "  of  a  relation  that  goes  from  x  to  y  is  the  opposite  of 
that  of  the  corresponding  relation  from  y  to  x.  The  fact  that  a 
relation  has  a  "  sense  "  is  fundamental,  and  is  part  of  the  reason 
why  order  can  be  generated  by  suitable  relations.  It  will  be 
observed  that  the  class  of  all  possible  referents  to  a  given  relation 
is  its  domain,  and  the  class  of  all  possible  relata  is  its  converse 
domain. 

But  it  very  often  happens  that  the  domain  and  converse 
domain  of  a  one-one  relation  overlap.  Take,  for  example, 
the  first  ten  integers  (excluding  o),  and  add  I  to  each ;  thus 
instead  of  the  first  ten  integers  we  now  have  the  integers 

2,  3,  4>  5»  6>  7>  8>  9»  I0>  «• 

These  are  the  same  as  those  we  had  before,  except  that  I  has 
been  cut  off  at  the  beginning  and  II  has  been  joined  on  at  the 
end.  There  are  still  ten  integers  :  they  are  correlated  with 
the  previous  ten  by  the  relation  of  n  to  n-{-i,  which  is  a  one-one 
relation.  Or,  again,  instead  of  adding  I  to  each  of  our  original 
ten  integers,  we  could  have  doubled  each  of  them,  thus  obtaining 
the  integers 

2,  4,  6,  8,  10,  12,  14,  16,  18,  20. 

Here  we  still  have  five  of  our  previous  set  of  integers,  namely, 
2,  4,  6,  8,  10.  The  correlating  relation  in  this  case  is  the  relation 
of  a  number  to  its  double,  which  is  again  a  one-one  relation. 
Or  we  might  have  replaced  each  number  by  its  square,  thus 
obtaining  the  set 

i,  4,  9,  16,  25,  36,  49,  64,  81,  100. 

On  this  occasion  only  three  of  our  original  set  are  left,  namely, 
I,  4,  9.  Such  processes  of  correlation  may  be  varied  endlessly. 

The  most  interesting  case  of  the  above  kind  is  the  case  where 
our  one-one  relation  has  a  converse  domain  which  is  part,  but 

4 


50  Introduction  to  Mathematical  Philosophy 

not  the  whole,  of  the  domain.  If,  instead  of  confining  the  domain 
to  the  first  ten  integers,  we  had  considered  the  whole  of  the 
inductive  numbers,  the  above  instances  would  have  illustrated 
this  case.  We  may  place  the  numbers  concerned  in  two  rows, 
putting  the  correlate  directly  under  the  number  whose  correlate 
it  is.  Thus  when  the  correlator  is  the  relation  of  n  to  n-{-iy  we 
have  the  two  rows  : 

1,  2,  3,  4,  5,  ...  n  ... 

2,  3>4>  5>  6>  •  •  •  "+1  -  •  - 

When  the  correlator  is  the  relation  of  a  number  to  its  double, 
we  have  the  two  rows  : 

1,  2,  3,  4,    5,  ...  n  .  .  . 

2,  4,  6,  8,  10,  ...  2w  ... 

When  the  correlator  is  the  relation  of  a  number  to  its  square, 
the  rows  are : 

i,  2,  3,    4,    5, «... 

i,  4,  9,  1 6,  25,  .....  n2  ... 

In  all  these  cases,  all  inductive  numbers  occur  in  the  top  row, 
and  only  some  in  the  bottom  row. 

Cases  of  this  sort,  where  the  converse  domain  is  a  "  proper 
part "  of  the  domain  (i.e.  a  part  not  the  whole),  will  occupy  us 
again  when  we  come  to  deal  with  infinity.  For  the  present,  we 
wish  only  to  note  that  they  exist  and  demand  consideration. 

Another  class  of  correlations  which  are  often  important  is 
the  class  called  "  permutations,"  where  the  domain  and  converse 
domain  are  identical.  Consider,  for  example,  the  six  possible 
arrangements  of  three  letters  : 

a,     b,     c 

a,  c,     b 

b,  c,     a 

b,  a,     c 

c,  a,     b 
c,     b,     a 


Kinds  of  Relations  51 

Each  of  these  can  be  obtained  from  any  one  of  the  others  by 
means  of  a  correlation.  Take,  for  example,  the  first  and  last, 
(a,  b,  c)  and  (c,  b,  a).  Here  a  is  correlated  with  c,  b  with  itself, 
and  c  with  a.  It  is  obvious  that  the  combination  of  two  permu 
tations  is  again  a  permutation,  i.e.  the  permutations  of  a  given 
class  form  what  is  called  a  "  group." 

These  various  kinds  of  correlations  have  importance  in  various 
connections,  some  for  one  purpose,  some  for  another.  The 
general  notion  of  one-one  correlations  has  boundless  importance 
in  the  philosophy  of  mathematics,  as  we  have  partly  seen  already, 
but  shall  see  much  more  fully  as  we  proceed.  One  of  its  uses 
will  occupy  us  in  our  next  chapter. 


CHAPTER  VI 

SIMILARITY    OF   RELATIONS 

WE  saw  in  Chapter  II.  that  two  classes  have  the  same  number 
of  terms  when  they  are  "  similar,"  i.e.  when  there  is  a  one-one 
relation  whose  domain  is  the  one  class  and  whose  converse 
domain  is  the  other.  In  such  a  case  we  say  that  there  is  a 
"  one-one  correlation  "  between  the  two  classes. 

In  the  present  chapter  we  have  to  define  a  relation  between 
relations,  which  will  play  the  same  part  for  them  that  similarity 
of  classes  plays  for  classes.  We  will  call  this  relation  "  similarity 
of  relations,"  or  "  likeness  "  when  it  seems  desirable  to  use  a 
different  word  from  that  which  we  use  for  classes.  How  is 
likeness  to  be  defined  ? 

We  shall  employ  still  the  notion  of  correlation  :  we  shall 
assume  that  the  domain  of  the  one  relation  can  be  correlated 
with  the  domain  of  the  other,  and  the  converse  domain  with  the 
converse  domain  ;  but  that  is  not  enough  for  the  sort  of  resem 
blance  which  we  desire  to  have  between  our  two  relations. 
What  we  desire  is  that,  whenever  either  relation  holds  between 
two  terms,  the  other  relation  shall  hold  between  the  correlates 
of  these  two  terms.  The  easiest  example  of  the  sort  of  thing 
we  desire  is  a  map.  When  one  place  is  north  of  another,  the 
place  on  the  map  corresponding  to  the  one  is  above  the  place 
on  the  map  corresponding  to  the  other  ;  when  one  place  is  west 
of  another,  the  place  on  the  map  corresponding  to  the  one  is 
to  the  left  of  the  place  on  the  map  corresponding  to  the  other ; 
and  so  on.  The  structure  of  the  map  corresponds  with  that  of 

52 


Similarity  of  Relations  53 

the  country  of  which  it  is  a  map.  The  space-relations  in  the 
map  have  "  likeness "  to  the  space-relations  in  the  country 
mapped.  It  is  this  kind  of  connection  between  relations  that 
we  wish  to  define. 

We  may,  in  the  first  place,  profitably  introduce  a  certain 
restriction.  We  will  confine  ourselves,  in  defining  likeness,  to 
such  relations  as  have  "  fields,"  i.e.  to  such  as  permit  of  the 
formation  of  a  single  class  out  of  the  domain  and  the  converse 
domain.  This  is  not  always  the  case.  Take,  for  example, 
the  relation  "  domain,"  i.e.  the  relation  which  the  domain  of  a 
relation  has  to  the  relation.  This  relation  has  all  classes  for  its 
domain,  since  every  class  is  the  domain  of  some  relation  ;  and 
it  has  all  relations  for  its  converse  domain,  since  every  relation 
has  a  domain.  But  classes  and  relations  cannot  be  added  to 
gether  to  form  a  new  single  class,  because  they  are  of  different 
logical  "  types."  We  do  not  need  to  enter  upon  the  difficult 
doctrine  of  types,  but  it  is  well  to  know  when  we  are  abstaining 
from  entering  upon  it.  We  may  say,  without  entering  upon 
the  grounds  for  the  assertion,  that  a  relation  only  has  a  "  field  " 
when  it  is  what  we  call  "  homogeneous,"  i.e.  when  its  domain 
and  converse  domain  are  of  the  same  logical  type ;  and  as  a 
rough-and-ready  indication  of  what  we  mean  by  a  "  type," 
we  may  say  that  individuals,  classes  of  individuals,  relations 
between  individuals,  relations  between  classes,  relations  of 
classes  to  individuals,  and  so  on,  are  different  types.  Now  the 
notion  of  likeness  is  not  very  useful  as  applied  to  relations  that 
are  not  homogeneous  ;  we  shall,  therefore,  in  defining  likeness, 
simplify  our  problem  by  speaking  of  the  "  field  "  of  one  of  the 
relations  concerned.  This  somewhat  limits  the  generality  of 
our  definition,  but  the  limitation  is  not  of  any  practical  impor 
tance.  And  having  been  stated,  it  need  no  longer  be  remembered. 
We  may  define  two  relations  P  and  Q  as  "  similar,"  or  as 
having  "  likeness,"  when  there  is  a  one-one  relation  S  whose 
domain  is  the  field  of  P  and  whose  converse  domain  is  the  field 
of  Q.  and  which  is  such  that,  if  one  term  has  the  relation  P 


54  Introduction  to  Mathematical  Philosophy 

to  another,  the  correlate  of  the  one  has  the  relation  Q  to  the 
correlate  of  the  other,  and  vice  versa.  A  figure  will  make  this 

clearer.     Let  x  and  v  be  two 
x,                  P  y 

. >   .  terms   having   the   relation  P. 

Then  there  are  to  be  two  terms 
z,  w,  such  that  x  has  the  rela 
tion  S  to  z,  y  has  the  relation 
S  to  zv,  and  z  has  the  relation 

• >   •  Q  to  20.     If  this  happens  with 

z  Q  w  . 

every  pair  of  terms  such  as  x 

and  y,  and  if  the  converse  happens  with  every  pair  of  terms  such 
as  z  and  w,  it  is  clear  that  for  every  instance  in  which  the  relation 
P  holds  there  is  a  corresponding  instance  in  which  the  relation 
Q  holds,  and  vice  versa  ;  and  this  is  what  we  desire  to  secure  by 
our  definition.  We  can  eliminate  some  redundancies  in  the 
above  sketch  of  a  definition,  by  observing  that,  when  the  above 
conditions  are  realised,  the  relation  P  is  the  same  as  the  relative 
product  of  S  and  Q  and  the  converse  of  S,  i.e.  the  P-step  from 
x  to  y  may  be  replaced  by  the  succession  of  the  S-step  from 
x  to  z,  the  Q-step  from  z  to  w,  and  the  backward  S-step  from 
w  to  y.  Thus  we  may  set  up  the  following  definitions  : — 

A  relation  S  is  said  to  be  a  "  correlator  "  or  an  "  ordinal 
correlator  "  of  two  relations  P  and  Q  if  S  is  one-one,  has  the 
field  of  Q  for  its  converse  domain,  and  is  such  that  P  is  the 
relative  product  of  S  and  Q  and  the  converse  of  S. 

Two  relations  P  and  Q  are  said  to  be  "  similar,"  or  to  have 
"  likeness,"  when  there  is  at  least  one  correlator  of  P  and  Q. 

These  definitions  will  be  found  to  yield  what  we  above  decided 
to  be  necessary. 

It  will  be  found  that,  when  two  relations  are  similar,  they 
share  all  properties  which  do  not  depend  upon  the  actual  terms 
in  their  fields.  For  instance,  if  one  implies  diversity,  so  does 
the  other  ;  if  one  is  transitive,  so  is  the  other  ;  if  one  is  con 
nected,  so  is  the  other.  Hence  if  one  is  serial,  so  is  the  other. 
Again,  if  one  is  one-many  or  one-one,  the  other  is  one-many 


Similarity  of  Relations  55 

or  one-one  ;  and  so  on,  through  all  the  general  properties  of 
relations.  Even  statements  involving  the  actual  terms  of  the 
field  of  a  relation,  though  they  may  not  be  true  as  they  stand 
when  applied  to  a  similar  relation,  will  always  be  capable  of 
translation  into  statements  that  are  analogous.  We  are  led 
by  such  considerations  to  a  problem  which  has,  in  mathematical 
philosophy,  an  importance  by  no  means  adequately  recognised 
hitherto.  Our  problem  may  be  stated  as  follows  : — 

Given  some  statement  in  a  language  of  which  we  know  the 
grammar  and  the  syntax,  but  not  the  vocabulary,  what  are  the 
possible  meanings  of  such  a  statement,  and  what  are  the  mean 
ings  of  the  unknown  words  that  would  make  it  true  ? 

The  reason  that  this  question  is  important  is  that  it  represents, 
much  more  nearly  than  might  be  supposed,  the  state  of  our 
knowledge  of  nature.  We  know  that  certain  scientific  pro 
positions — which,  in  the  most  advanced  sciences,  are  expressed 
in  mathematical  symbols — are  more  or  less  true  of  the  world, 
but  we  are  very  much  at  sea  as  to  the  interpretation  to  be  put 
upon  the  terms  which  occur  in  these  propositions.  We  know 
much  more  (to  use,  for  a  moment,  an  old-fashioned  pair  of 
terms)  about  the  form  of  nature  than  about  the  matter. 
Accordingly,  what  we  really  know  when  we  enunciate  a  law 
of  nature  is  only  that  there  is  probably  some  interpretation  of 
our  terms  which  will  make  the  law  approximately  true.  Thus 
great  importance  attaches  to  the  question :  What  are  the 
possible  meanings  of  a  law  expressed  in  terms  of  which  we  do 
not  know  the  substantive  meaning,  but  only  the  grammar  and 
syntax  ?  And  this  question  is  the  one  suggested  above. 

For  the  present  we  will  ignore  the  general  question,  which 
will  occupy  us  again  at  a  later  stage;  the  subject  of  likeness 
itself  must  first  be  further  investigated. 

Owing  to  the  fact  that,  when  two  relations  are  similar,  their 
properties  are  the  same  except  when  they  depend  upon  the 
fields  being  composed  of  just  the  terms  of  which  they  are  com 
posed,  it  is  desirable  to  have  a  nomenclature  which  collects 


56  Introduction  to  Mathematical  Philosophy 

together  all  the  relations  that  are  similar  to  a  given  relation. 
Just  as  we  called  the  set  of  those  classes  that  are  similar  to  a 
given  class  the  "  number  "  of  that  class,  so  we  may  call  the  set 
of  all  those  relations  that  are  similar  to  a  given  relation  the 
"  number  "  of  that  relation.  But  in  order  to  avoid  confusion  with 
the  numbers  appropriate  to  classes,  we  will  speak,  in  this  case,  of 
a  "  relation-number."  Thus  we  have  the  following  definitions  : — 

The  "  relation-number  "  of  a  given  relation  is  the  class  of  all 
those  relations  that  are  similar  to  the  given  relation. 

"  Relation-numbers  "  are  the  set  of  all  those  classes  of  relations 
that  are  relation-numbers  of  various  relations ;  or,  what  comes  to 
the  same  thing,  a  relation  number  is  a  class  of  relations  consisting 
of  all  those  relations  that  are  similar  to  one  member  of  the  class. 

When  it  is  necessary  to  speak  of  the  numbers  of  classes  in 
a  way  which  makes  it  impossible  to  confuse  them  with  relation- 
numbers,  we  shall  call  them  "  cardinal  numbers."  Thus  cardinal 
numbers  are  the  numbers  appropriate  to  classes.  These  include 
the  ordinary  integers  of  daily  life,  and  also  certain  infinite 
numbers,  of  which  we  shall  speak  later.  When  we  speak  of 
"  numbers  "  without  qualification,  we  are  to  be  understood  as 
meaning  cardinal  numbers.  The  definition  of  a  cardinal  number, 
it  will  be  remembered,  is  as  follows  : — 

The  "  cardinal  number "  of  a  given  class  is  the  set  of  all 
those  classes  that  are  similar  to  the  given  class. 

The  most  obvious  application  of  relation-numbers  is  to  series. 
Two  series  may  be  regarded  as  equally  long  when  they  have 
the  same  relation-number.  Two  finite  series  will  have  the 
same  relation-number  when  their  fields  have  the  same  cardinal 
number  of  terms,  and  only  then — i.e.  a  series  of  (say)  15  terms 
will  have  the  same  relation-number  as  any  other  series  of  fifteen 
terms,  but  will  not  have  the  same  relation-number  as  a  series 
of  14  or  1 6  terms,  nor,  of  course,  the  same  relation-number 
as  a  relation  which  is  not  serial.  Thus,  in  the  quite  special  case 
of  finite  series,  there  is  parallelism  between  cardinal  and  relation- 
numbers.  The  relation-numbers  applicable  to  series  may  be 


Similarity  of  Relations  57 

called  "  serial  numbers  "  (what  are  commonly  called  "  ordinal 
numbers  "  are  a  sub-class  of  these) ;  thus  a  finite  serial  number 
is  determinate  when  we  know  the  cardinal  number  of  terms 
in  the  field  of  a  series  having  the  serial  number  in  question. 
If  n  is  a  finite  cardinal  number,  the  relation-number  of  a  series 
which  has  n  terms  is  called  the  "  ordinal  "  number  n.  (There 
are  also  infinite  ordinal  numbers,  but  of  them  we  shall  speak 
in  a  later  chapter.)  When  the  cardinal  number  of  terms  in 
the  field  of  a  series  is  infinite,  the  relation-number  of  the  series 
is  not  determined  merely  by  the  cardinal  number,  indeed  an 
infinite  number  of  relation-numbers  exist  for  one  infinite  cardinal 
number,  as  we  shall  see  when  we  come  to  consider  infinite  series. 
When  a  series  is  infinite,  what  we  may  call  its  "  length,"  i.e. 
its  relation-number,  may  vary  without  change  in  the  cardinal 
number  ;  but  when  a  series  is  finite,  this  cannot  happen. 

We  can  define  addition  and  multiplication  for  relation- 
numbers  as  well  as  for  cardinal  numbers,  and  a  whole  arithmetic 
of  relation-numbers  can  be  developed.  The  manner  in  which 
this  is  to  be  done  is  easily  seen  by  considering  the  case  of  series. 
Suppose,  for  example,  that  we  wish  to  define  the  sum  of  two 
non-overlapping  series  in  such  a  way  that  the  relation-number 
of  the  sum  shall  be  capable  of  being  defined  as  the  sum  of  the 
relation-numbers  of  the  two  series.  In  the  first  place,  it  is  clear 
that  there  is  an  order  involved  as  between  the  two  series  :  one 
of  them  must  be  placed  before  the  other.  Thus  if  P  and  Q 
are  the  generating  relations  of  the  two  series,  in  the  series  which 
is  their  sum  with  P  put  before  Q,  every  member  of  the  field  of 
P  will  precede  every  member  of  the  field  of  Q.  Thus  the  serial 
relation  which  is  to  be  defined  as  the  sum  of  P  and  Q  is  not 
"  P  or  Q  "  simply,  but  "  P  or  Q  or  the  relation  of  any  member 
of  the  field  of  P  to  any  member  of  the  field  of  Q."  Assuming 
that  P  and  Q  do  not  overlap,  this  relation  is  serial,  but  "  P  or  Q  " 
is  not  serial,  being  not  connected,  since  it  does  not  hold  between 
a  member  of  the  field  of  P  and  a  member  of  the  field  of  Q.  Thus 
the  sum  of  P  and  Q,  as  above  defined,  is  what  we  need  in  order 


58  Introduction  to  Mathematical  Philosophy 

to  define  the  sum  of  two  relation-numbers.  Similar  modifica 
tions  are  needed  for  products  and  powers.  The  resulting  arith 
metic  does  not  obey  the  commutative  law  :  the  sum  or  product 
of  two  relation-numbers  generally  depends  upon  the  order  in 
which  they  are  taken.  But  it  obeys  the  associative  law,  one 
form  of  the  distributive  law,  and  two  of  the  formal  laws  for 
powers,  not  only  as  applied  to  serial  numbers,  but  as  applied  to 
relation-numbers  generally.  Relation-arithmetic,  in  fact,  though 
recent,  is  a  thoroughly  respectable  branch  of  mathematics. 

It  must  not  be  supposed,  merely  because  series  afford  the 
most  obvious  application  of  the  idea  of  likeness,  that  there  are 
no  other  applications  that  are  important.  We  have  already 
mentioned  maps,  and  we  might  extend  our  thoughts  from  this 
illustration  to  geometry  generally.  If  the  system  of  relations 
by  which  a  geometry  is  applied  to  a  certain  set  of  terms  can  be 
brought  fully  into  relations  of  likeness  with  a  system  applying 
to  another  set  of  terms,  then  the  geometry  of  the  two  sets  is 
indistinguishable  from  the  mathematical  point  of  view,  i.e.  all 
the  propositions  are  the  same,  except  for  the  fact  that  they  are 
applied  in  one  case  to  one  set  of  terms  and  in  the  other  to  another. 
We  may  illustrate  this  by  the  relations  of  the  sort  that  may  be 
called  "  between,"  which  we  considered  in  Chapter  IV.  We 
there  saw  that,  provided  a  three-term  relation  has  certain  formal 
logical  properties,  it  will  give  rise  to  series,  and  may  be  called 
a  "  between-relation."  Given  any  two  points,  we  can  use  the 
between-relation  to  define  the  straight  line  determined  by  those 
two  points  ;  it  consists  of  a  and  b  together  with  all  points  x, 
such  that  the  between-relation  holds  between  the  three  points 
a,  b,  x  in  some  order  or  other.  It  has  been  shown  by  0.  Veblen 
that  we  may  regard  our  whole  space  as  the  field  of  a  three-term 
between-relation,  and  define  our  geometry  by  the  properties  we 
assign  to  our  between-relation.1  Now  likeness  is  just  as  easily 

1  This  does  not  apply  to  elliptic  space,  but  only  to  spaces  in  which 
the  straight  line  is  an  open  series.  Modern  Mathematics,  edited  by 
J.  W.  A.  Young,  pp.  3-51  (monograph  by  O.  Veblen  on  "  The  Foundations  of 
Geometry"). 


Similarity  of  Relations  59 

definable  between  three-term  relations  as  between  two-term 
relations.  If  B  and  B'  are  two  between-relations,  so  that 
"  xB(y,  z)  "  means  "  x  is  between  y  and  z  with  respect  to  B," 
we  shall  call  S  a  correlator  of  B  and  B7  if  it  has  the  field  of  B' 
for  its  converse  domain,  and  is  such  that  the  relation  B  holds 
between  three  terms  when  B'  holds  between  their  S-correlates, 
and  only  then.  And  we  shall  say  that  B  is  like  B'  when  there 
is  at  least  one  correlator  of  B  with  B'.  The  reader  can  easily 
convince  himself  that,  if  B  is  like  B'  in  this  sense,  there  can  be 
no  difference  between  the  geometry  generated  by  B  and  that 
generated  by  B'. 

It  follows  from  this  that  the  mathematician  need  not  concern 
himself  with  the  particular  being  or  intrinsic  nature  of  his  points, 
lines,  and  planes,  even  when  he  is  speculating  as  an  applied 
mathematician.  We  may  say  that  there  is  empirical  evidence 
of  the  approximate  truth  of  such  parts  of  geometry  as  are  not 
matters  of  definition.  But  there  is  no  empirical  evidence  as  to 
what  a  "  point  "  is  to  be.  It  has  to  be  something  that  as  nearly 
as  possible  satisfies  our  axioms,  but  it  does  not  have  to  be  "  very 
small "  or  "  without  parts."  Whether  or  not  it  is  those  things 
is  a  matter  of  indifference,  so  long  as  it  satisfies  the  axioms.  If 
we  can,  out  of  empirical  material,  construct  a  logical  structure, 
no  matter  how  complicated,  which  will  satisfy  our  geometrical 
axioms,  that  structure  may  legitimately  be  called  a  "  point." 
We  must  not  say  that  there  is  nothing  else  that  could  legitimately 
be  called  a  "  point  "  ;  we  must  only  say  :  "  This  object  we  have 
constructed  is  sufficient  for  the  geometer ;  it  may  be  one  of 
many  objects,  any  of  which  would  be  sufficient,  but  that  is  no 
concern  of  ours,  since  this  object  is  enough  to  vindicate  the 
empirical  truth  of  geometry,  in  so  far  as  geometry  is  not  a 
matter  of  definition."  This  is  only  an  illustration  of  the  general 
principle  that  what  matters  in  mathematics,  and  to  a  very  great 
extent  in  physical  science,  is  not  the  intrinsic  nature  of  our 
terms,  but  the  logical  nature  of  their  interrelations. 

We  may  say,  of  two  similar  relations,  that  they  have  the  same 


60  Introduction  to  Mathematical  Philosophy 

"  structure."  For  mathematical  purposes  (though  not  for  those 
of  pure  philosophy)  the  only  thing  of  importance  about  a  relation 
is  the  cases  in  which  it  holds,  not  its  intrinsic  nature.  Just  as  a 
class  may  be  defined  by  various  different  but  co-extensive  concepts 
— e.g.  "  man  "  and  "  featherless  biped," — so  two  relations  which 
are  conceptually  different  may  hold  in  the  same  set  of  instances. 
An  "  instance  "  in  which  a  relation  holds  is  to  be  conceived  as  a 
couple  of  terms,  with  an  order,  so  that  one  of  the  terms  comes 
first  and  the  other  second ;  the  couple  is  to  be,  of  course, 
such  that  its  first  term  has  the  relation  in  question  to  its  second. 
Take  (say)  the  relation  "  father  "  :  we  can  define  what  we  may 
call  the  "  extension  "  of  this  relation  as  the  class  of  all  ordered 
couples  (Xy  y)  which  are  such  that  x  is  the  father  of  y.  From 
the  mathematical  point  of  view,  the  only  thing  of  importance 
about  the  relation  "  father  "  is  that  it  defines  this  set  of  ordered 
couples.  Speaking  generally,  we  say  : 

The  "  extension  "  of  a  relation  is  the  class  of  those  ordered 
couples  (x,  y)  which  are  such  that  x  has  the  relation  in  question 
to  y. 

We  can  now  go  a  step  further  in  the  process  of  abstraction, 
and  consider  what  we  mean  by  "  structure."  Given  any  relation, 
we  can,  if  it  is  a  sufficiently  simple  one,  construct  a  map  of  it. 
For  the  sake  of  definiteness,  let  us  take  a  relation  of  which  the 
extension  is  the  following  couples  :  aby  aCy  ady  be,  ce,  dcy  dey  where 
<z,  by  Cy  dy  e  ale  five  terms,  no  matter  what.  We  may  make  a 
"  map  "  of  this  relation  by  taking  five  points 

a. >  .      on  a  plane  and  connecting  them  by  arrows, 

as  in  the  accompanying  figure.  What  is 
revealed  by  the  map  is  what  we  call  the 
"  structure  "  of  the  relation. 

It  is  clear  that  the  "  structure "   of  the 
relation  does  not  depend  upon  the  particular 
terms  that  make  up  the  field  of  the  relation. 
The  field  may  be  changed  without  changing  the  structure,  and 
the  structure  may  be  changed  without  changing  the  field — for 


Similarity  of  Relations  61 

example,  if  we  were  to  add  the  couple  ae  in  the  above  illustration 
we  should  alter  the  structure  but  not  the  field.  Two  relations 
have  the  same  "  structure,"  we  shall  say,  when  the  same  map 
will  do  for  both — or,  what  comes  to  the  same  thing,  when  either 
can  be  a  map  for  the  other  (since  every  relation  can  be  its  own 
map).  And  that,  as  a  moment's  reflection  shows,  is  the  very 
same  thing  as  what  we  have  called  "  likeness."  That  is  to  say, 
two  relations  have  the  same  structure  when  they  have  likeness, 
i./.  when  they  have  the  same  relation-number.  Thus  what  we 
defined  as  the  "  relation-number  "  is  the  very  same  thing  as  is 
obscurely  intended  by  the  word  "  structure  " — a  word  which, 
important  as  it  is,  is  never  (so  far  as  we  know)  defined  in  precise 
terms  by  those  who  use  it. 

There  has  been  a  great  deal  of  speculation  in  traditional 
philosophy  which  might  have  been  avoided  if  the  importance  of 
structure,  and  the  difficulty  of  getting  behind  it,  had  been  realised. 
For  example,  it  is  often  said  that  space  and  time  are  subjective, 
but  they  have  objective  counterparts ;  or  that  phenomena  are 
subjective,  but  are  caused  by  things  in  themselves,  which  must 
have  differences  inter  se  corresponding  with  the  differences  in 
the  phenomena  to  which  they  give  rise.  Where  such  hypotheses 
are  made,  it  is  generally  supposed  that  we  can  know  very  little 
about  the  objective  counterparts.  In  actual  fact,  however,  if 
the  hypotheses  as  stated  were  correct,  the  objective  counterparts 
would  form  a  world  having  the  same  structure  as  the  phenomenal 
world,  and  allowing  us  to  infer  from  phenomena  the  truth  of  all 
propositions  that  can  be  stated  in  abstract  terms  and  are  known 
to  be  true  of  phenomena.  If  the  phenomenal  world  has  three 
dimensions,  so  must  the  world  behind  phenomena ;  if  the  pheno 
menal  world  is  Euclidean,  so  must  the  other  be ;  and  so  on. 
In  short,  every  proposition  having  a  communicable  significance 
must  be  true  of  both  worlds  or  of  neither :  the  only  difference 
must  lie  in  just  that  essence  of  individuality  which  always  eludes 
words  and  bafHes  description,  but  which,  for  that  very  reason, 
is  irrelevant  to  science.  Now  the  only  purpose  that  philosophers 


62  Introduction  to  Mathematical  Philosophy 

have  in  view  in  condemning  phenomena  is  in  order  to  persuade 
themselves  and  others  that  the  real  world  is  very  different  from 
the  world  of  appearance.  We  can  all  sympathise  with  their  wish 
to  prove  such  a  very  desirable  proposition,  but  we  cannot  con 
gratulate  them  on  their  success.  It  is  true  that  many  of  them 
do  not  assert  objective  counterparts  to  phenomena,  and  these 
escape  from  the  above  argument.  Those  who  do  assert  counter 
parts  are,  as  a  rule,  very  reticent  on  the  subject,  probably  because 
they  feel  instinctively  that,  if  pursued,  it  will  bring  about  too 
much  of  a  rapprochement  between  the  real  and  the  phenomenal 
world.  If  they  were  to  pursue  the  topic,  they  could  hardly  avoid 
the  conclusions  which  we  have  been  suggesting.  In  such  ways, 
as  well  as  in  many  others,  the  notion  of  structure  or  relation- 
number  is  important. 


CHAPTER  VII 

RATIONAL,    REAL,    AND   COMPLEX    NUMBERS 

WE  have  now  seen  how  to  define  cardinal  numbers,  and  also 
relation-numbers,  of  which  what  are  commonly  called  ordinal 
numbers  are  a  particular  species.  It  will  be  found  that  each 
of  these  kinds  of  number  may  be  infinite  just  as  well  as  finite. 
But  neither  is  capable,  as  it  stands,  of  the  more  familiar  exten 
sions  of  the  idea  of  number,  namely,  the  extensions  to  negative, 
fractional,  irrational,  and  complex  numbers.  In  the  present 
chapter  we  shall  briefly  supply  logical  definitions  of  these  various 
extensions. 

One  of  the  mistakes  that  have  delayed  the  discovery  of  correct 
definitions  in  this  region  is  the  common  idea  that  each  extension 
of  number  included  the  previous  sorts  as  special  cases.  It  was 
thought  that,  in  dealing  with  positive  and  negative  integers,  the 
positive  integers  might  be  identified  with  the  original  signless 
integers.  Again  it  was  thought  that  a  fraction  whose  denominator 
is  I  may  be  identified  with  the  natural  number  which  is  its 
numerator.  And  the  irrational  numbers,  such  as  the  square 
root  of  2,  were  supposed  to  find  their  place  among  rational  frac 
tions,  as  being  greater  than  some  of  them  and  less  than  the  others, 
so  that  rational  and  irrational  numbers  could  be  taken  together 
as  one  class,  called  "  real  numbers."  And  when  the  idea  of 
number  was  further  extended  so  as  to  include  "  complex " 
numbers,  i.e.  numbers  involving  the  square  root  of  — I,  it  was 
thought  that  real  numbers  could  be  regarded  as  those  among 
complex  numbers  in  which  the  imaginary  part  (i.e.  the  part 

63 


64  Introduction  to  Mathematical  Philosophy 

which  was  a  multiple  of  the  square  root  of  — i)  was  zero.  All 
these  suppositions  were  erroneous,  and  must  be  discarded,  as  we 
shall  find,  if  correct  definitions  are  to  be  given. 

Let  us  begin  with  positive  and  negative  integers.  It  is  obvious 
on  a  moment's  consideration  that  +1  and  —  I  must  both  be 
relations,  and  in  fact  must  be  each  other's  converses.  The 
obvious  and  sufficient  definition  is  that  -f-i  is  the  relation  of 
tt-f  I  to  n,  and  —  I  is  the  relation  of  n  to  n-f-l.  Generally,  if  m 
is  any  inductive  number,  -\-m  will  be  the  relation  of  n-\-m  to  n 
(for  any  n),  and  — m  will  be  the  relation  of  n  to  n-\-m.  Accord 
ing  to  this  definition,  -\-m  is  a  relation  which  is  one-one  so 
long  as  n  is  a  cardinal  number  (finite  or  infinite)  and  m  is  an 
inductive  cardinal  number.  But  -\-m  is  under  no  circumstances 
capable  of  being  identified  with  my  which  is  not  a  relation,  but 
a  class  of  classes.  Indeed,  -f  m  is  every  bit  as  distinct  from  m 
as  — m  is. 

Fractions  are  more  interesting  than  positive  or  negative  integers. 
We  need  fractions  for  many  purposes,  but  perhaps  most  obviously 
for  purposes  of  measurement.  My  friend  and  collaborator  Dr 
A.  N.  Whitehead  has  developed  a  theory  of  fractions  specially 
adapted  for  their  application  to  measurement,  which  is  set  forth 
in  Principia  Mathematical  But  if  all  that  is  needed  is  to  define 
objects  having  the  required  purely  mathematical  properties,  this 
purpose  can  be  achieved  by  a  simpler  method,  which  we  shall 
here  adopt.  We  shall  define  the  fraction  m/n  as  being  that 
relation  which  holds  between  two  inductive  numbers  xt  y  when 
xn=ym.  This  definition  enables  us  to  prove  that  m/n  is  a  one- 
one  relation,  provided  neither  m  or  n  is  zero.  And  of  course  n/m 
is  the  converse  relation  to  m/n. 

From  the  above  definition  it  is  clear  that  the  fraction  m/i  is 
that  relation  between  two  integers  x  and  y  which  consists  in  the 
fact  that  x=my.  This  relation,  like  the  relation  -f-w,  is  by  no 
means  capable  of  being  identified  with  the  inductive  cardinal 
number  m%  because  a  relation  and  a  class  of  classes  are  objects 
1  Vol.  iii.  *  300  ff.,  especially  303. 


Rational)  Real,  and  Complex  Numbers  65 

of  utterly  different  kinds.1  It  will  be  seen  that  o/»  is  always  the 
same  relation,  whatever  inductive  number  n  may  be;  it  is,  in  short, 
the  relation  of  o  to  any  other  inductive  cardinal.  We  may  call 
this  the  zero  of  rational  numbers  ;  it  is  not,  of  course,  identical 
with  the  cardinal  number  o.  Conversely,  the  relation  ra/o  is 
always  the  same,  whatever  inductive  number  m  may  be.  There 
is  not  any  inductive  cardinal  to  correspond  to  m/o.  We  may  call 
it  "  the  infinity  of  rationals."  It  is  an  instance  of  the  sort  of 
infinite  that  is  traditional  in  mathematics,  and  that  is  represented 
by  "  oo ."  This  is  a  totally  different  sort  from  the  true  Cantorian 
infinite,  which  we  shall  consider  in  our  next  chapter.  The  in 
finity  of  rationals  does  not  demand,  for  its  definition  or  use,  any 
infinite  classes  or  infinite  integers.  It  is  not,  in  actual  fact,  a 
very  important  notion,  and  we  could  dispense  with  it  altogether 
if  there  were  any  object  in  doing  so.  The  Cantorian  infinite,  on 
the  other  hand,  is  of  the  greatest  and  most  fundamental  impor 
tance  ;  the  understanding  of  it  opens  the  way  to  whole  new  realms 
of  mathematics  and  philosophy. 

It  will  be  observed  that  zero  and  infinity,  alone  among  ratios, 
are  not  one-one.  Zero  is  one-many,  and  infinity  is  many-one. 

There  is  not  any  difficulty  in  defining  greater  and  less  among 
ratios  (or  fractions).  Given  two  ratios  mjn  and  p/q,  we  shall  say 
that  m/n  is  less  than  p/q  if  mq  is  less  than  pn.  There  is  no 
difficulty  in  proving  that  the  relation  "  less  than,"  so  defined,  is 
serial,  so  that  the  ratios  form  a  series  in  order  of  magnitude.  In 
this  series,  zero  is  the  smallest  term  and  infinity  is  the  largest. 
If  we  omit  zero  and  infinity  from  our  series,  there  is  no  longer 
any  smallest  or  largest  ratio  ;  it  is  obvious  that  if  m/n  is  any  ratio 
other  than  zero  and  infinity,  m/2n  is  smaller  and  2m/n  is  larger, 
though  neither  is  zero  or  infinity,  so  that  m/n  is  neither  the  smallest 

1  Of  course  in  practice  we  shall  continue  to  speak  of  a  fraction  as  (say) 
greater  or  less  than  i,  meaning  greater  or  less  than  the  ratio  i/i.  So 
long  as  it  is  understood  that  the  ratio  i/i  and  the  cardinal  number  i  are 
different,  it  is  not  necessary  to  be  always  pedantic  in  emphasising  the 
difference. 

5 


66  Introduction  to  Mathematical  Philosophy 

nor  the  largest  ratio,  and  therefore  (when  zero  and  infinity  are 
omitted)  there  is  no  smallest  or  largest,  since  m/n  was  chosen 
arbitrarily.  In  like  manner  we  can  prove  that  however  nearly 
equal  two  fractions  may  be,  there  are  always  other  fractions 
between  them.  For,  let  m/n  and  p/q  be  two  fractions,  of  which 
p/q  is  the  greater.  Then  it  is  easy  to  see  (or  to  prove)  that 
(m+p)/(n-}-q)  will  be  greater  than  m/n  and  less  than  p/q.  Thus 
the  series  of  ratios  is  one  in  which  no  two  terms  are  consecutive, 
but  there  are  always  other  terms  between  any  two.  Since  there 
are  other  terms  between  these  others,  and  so  on  ad  infinitum,  it 
is  obvious  that  there  are  an  infinite  number  of  ratios  between 
any  two,  however  nearly  equal  these  two  may  be.1  A  series 
having  the  property  that  there  are  always  other  terms  between 
any  two,  so  that  no  two  are  consecutive,  is  called  "  compact." 
Thus  the  ratios  in  order  of  magnitude  form  a  "  compact "  series. 
Such  series  have  many  important  properties,  and  it  is  important 
to  observe  that  ratios  afford  an  instance  of  a  compact  series 
generated  purely  logically,  without  any  appeal  to  space  or  time 
or  any  other  empirical  datum. 

Positive  and  negative  ratios  can  be  defined  in  a  way  analogous 
to  that  in  which  we  defined  positive  and  negative  integers. 
Having  first  defined  the  sum  of  two  ratios  m/n  and  p/q  as 
(mq+pn)/nq,  we  define  -{-p/q  as  the  relation  of  m/n-\-p/q  to  m/n, 
where  m/n  is  any  ratio ;  and  —  p/q  is  of  course  the  converse  of 
-\-p/q-  This  is  not  the  only  possible  way  of  defining  positive  and 
negative  ratios,  but  it  is  a  way  which,  for  our  purpose,  has  the 
merit  of  being  an  obvious  adaptation  of  the  way  we  adopted  in 
the  case  of  integers. 

We  come  now  to  a  more  interesting  extension  of  the  idea  of 
number,  i.e.  the  extension  to  what  are  called  "  real  "  numbers, 
which  are  the  kind  that  embrace  irrationals.  In  Chapter  I.  we 
had  occasion  to  mention  "  incommensurables  "  and  their  dis- 

1  Strictly  speaking,  this  statement,  as  well  as  those  following  to  the  end 
of  the  paragraph,  involves  what  is  called  the  "  axiom  of  infinity,"  which 
will  be  discussed  in  a  later  chapter. 


Rational,  Reaty  and  Complex  Numbers  67 

covery  by  Pythagoras.  It  was  through  them,  i.e.  through 
geometry,  that  irrational  numbers  were  first  thought  of.  A 
square  of  which  the  side  is  one  inch  long  will  have  a  diagonal  of 
which  the  length  is  the  square  root  of  2  inches.  But,  as  the 
ancients  discovered,  there  is  no  fraction  of  which  the  square  is  2. 
This  proposition  is  proved  in  the  tenth  book  of  Euclid,  which  is 
one  of  those  books  that  schoolboys  supposed  to  be  fortunately  lost 
in  the  days  when  Euclid  was  still  used  as  a  text-book.  The  proof 
is  extraordinarily  simple.  If  possible,  let  mjn  be  the  square  root 
of  2,  so  that  ra2/ft2=2,  i.e.  m2—2n2.  Thus  m2  is  an  even  number, 
and  therefore  m  must  be  an  even  number,  because  the  square  of 
an  odd  number  is  odd.  Now  if  m  is  even,  m*  must  divide  by  4, 
for  if  m=2p,  then  m2=^.p2.  Thus  we  shall  have  4£2=2«2,  where 
p  is  half  of  m.  Hence  2p2=n29  and  therefore  n/p  will  also  be  the 
square  root  of  2.  But  then  we  can  repeat  the  argument :  if 
n=2q,  pjq  will  also  be  the  square  root  of  2,  and  so  on,  through 
an  unending  series  of  numbers  that  are  each  half  of  its  predecessor. 
But  this  is  impossible ;  if  we  divide  a  number  by  2,  and  then 
halve  the  half,  and  so  on,  we  must  reach  an  odd  number  after  a 
finite  number  of  steps.  Or  we  may  put  the  argument  even  more 
simply  by  assuming  that  the  m/n  we  start  with  is  in  its  lowest 
terms  ;  in  that  case,  m  and  n  cannot  both  be  even ;  yet  we  have 
seen  that,  if  m2/n2—2,  they  must  be.  Thus  there  cannot  be  any 
fraction  m/n  whose  square  is  2. 

Thus  no  fraction  will  express  exactly  the  length  of  the  diagonal 
of  a  square  whose  side  is  one  inch  long.  This  seems  like  a 
challenge  thrown  out  by  nature  to  arithmetic.  However  the 
arithmetician  may  boast  (as  Pythagoras  did)  about  the  power 
of  numbers,  nature  seems  able  to  baffle  him  by  exhibiting  lengths 
which  no  numbers  can  estimate  in  terms  of  the  unit.  But  the 
problem  did  not  remain  in  this  geometrical  form.  As  soon  as 
algebra  was  invented,  the  same  problem  arose  as  regards  the 
solution  of  equations,  though  here  it  took  on  a  wider  form, 
since  it  also  involved  complex  numbers. 

It  is  clear  that  fractions  can  be  found  which  approach  nearer 


68  Introduction  to  Mathematical  Philosophy 

and  nearer  to  having  their  square  equal  to  2.  We  can  form  an 
ascending  series  of  fractions  all  of  which  have  their  squares 
less  than  2,  but  differing  from  2  in  their  later  members  by 
less  than  any  assigned  amount.  That  is  to  say,  suppose  I  assign 
some  small  amount  in  advance,  say  one-billionth,  it  will  be 
found  that  all  the  terms  of  our  series  after  a  certain  one,  say  the 
tenth,  have  squares  that  differ  from  2  by  less  than  this  amount. 
And  if  I  had  assigned  a  still  smaller  amount,  it  might  have  been 
necessary  to  go  further  along  the  series,  but  we  should  have 
reached  sooner  or  later  a  term  in  the  series,  say  the  twentieth, 
after  which  all  terms  would  have  had  squares  differing  from  2 
by  less  than  this  still  smaller  amount.  If  we  set  to  work  to 
extract  the  square  root  of  2  by  the  usual  arithmetical  rule,  we 
shall  obtain  an  unending  decimal  which,  taken  to  so-and-so 
many  places,  exactly  fulfils  the  above  conditions.  We  can 
equally  well  form  a  descending  series  of  fractions  whose  squares 
are  all  greater  than  2,  but  greater  by  continually  smaller  amounts 
as  we  come  to  later  terms  of  the  series,  and  differing,  sooner  or 
later,  by  less  than  any  assigned  amount.  In  this  way  we  seem 
to  be  drawing  a  cordon  round  the  square  root  of  2,  and  it  may 
seem  difficult  to  believe  that  it  can  permanently  escape  us. 
Nevertheless,  it  is  not  by  this  method  that  we  shall  actually 
reach  the  square  root  of  2. 

If  we  divide  all  ratios  into  two  classes,  according  as  their 
squares  are  less  than  2  or  not,  we  find  that,  among  those  whose 
squares  are  not  less  than  2,  all  have  their  squares  greater  than  2. 
There  is  no  maximum  to  the  ratios  whose  square  is  less  than  2, 
and  no  minimum  to  those  whose  square  is  greater  than  2.  There 
is  no  lower  limit  short  of  zero  to  the  difference  between  the 
numbers  whose  square  is  a  little  less  than  2  and  the  numbers 
whose  square  is  a  little  greater  than  2.  We  can,  in  short,  divide 
all  ratios  into  two  classes  such  that  all  the  terms  in  one  class 
are  less  than  all  in  the  other,  there  is  no  maximum  to  the  one 
class,  and  there  is  no  minimum  to  the  other.  Between  these 
two  classes,  where  V2  ought  to  be,  there  is  nothing.  Thus  our 


Rational,  Real,  and  Complex  Numbers  69 

cordon,  though  we  have  drawn  it  as  tight  as  possible,  has  been 
drawn  in  the  wrong  place,  and  has  not  caught  v  2. 

The  above  method  of  dividing  all  the  terms  of  a  series  into 
two  classes,  of  which  the  one  wholly  precedes  the  other,  was 
brought  into  prominence  by  Dedekind,1  and  is  therefore  called 
a  "  Dedekind  cut."  With  respect  to  what  happens  at  the  point 
of  section,  there  are  four  possibilities  :  (i)  there  may  be  a 
maximum  to  the  lower  section  and  a  minimum  to  the  upper 
section,  (2)  there  may  be  a  maximum  to  the  one  and  no  minimum 
to  the  other,  (3)  there  may  be  no  maximum  to  the  one,  but  a 
minimum  to  the  other,  (4)  there  may  be  neither  a  maximum  to 
the  one  nor  a  minimum  to  the  other.  Of  these  four  cases,  the 
first  is  illustrated  by  any  series  in  which  there  are  consecutive 
terms  :  in  the  series  of  integers,  for  instance,  a  lower  section 
must  end  with  some  number  n  and  the  upper  section  must 
then  begin  with  n+i.  The  second  case  will  be  illustrated 
in  the  series  of  ratios  if  we  take  as  our  lower  section  all  ratios 
up  to  and  including  I,  and  in  our  upper  section  all  ratios  greater 
than  I.  The  third  case  is  illustrated  if  we  take  for  our  lower 
section  all  ratios  less  than  I,  and  for  our  upper  section  all  ratios 
from  I  upward  (including  I  itself).  The  fourth  case,  as  we  have 
seen,  is  illustrated  if  we  put  in  our  lower  section  all  ratios  whose 
square  is  less  than  2,  and  in  our  upper  section  all  ratios  whose 
square  is  greater  than  2. 

We  may  neglect  the  first  of  our  four  cases,  since  it  only  arises 
in  series  where  there  are  consecutive  terms.  In  the  second  of 
our  four  cases,  we  say  that  the  maximum  of  the  lower  section 
is  the  lower  limit  of  the  upper  section,  or  of  any  set  of  terms 
chosen  out  of  the  upper  section  in  such  a  way  that  no  term  of 
the  upper  section  is  before  all  of  them.  In  the  third  of  our 
four  cases,  we  say  that  the  minimum  of  the  upper  section  is  the 
upper  limit  of  the  lower  section,  or  of  any  set  of  terms  chosen 
out  of  the  lower  section  in  such  a  way  that  no  term  of  the  lower 
section  is  after  all  of  them.  In  the  fourth  case,  we  say  that 
1  Stetigkeit  und  irrationale  Zahlen,  2nd  edition,  Brunswick,  1892. 


70  Introduction  to  Mathematical  Philosophy 

there  is  a  "  gap  "  :  neither  the  upper  section  nor  the  lower  has 
a  limit  or  a  last  term.  In  this  case,  we  may  also  say  that  we 
have  an  "  irrational  section,"  since  sections  of  the  series  of  ratios 
have  "  gaps  "  when  they  correspond  to  irrationals. 

What  delayed  the  true  theory  of  irrationals  was  a  mistaken 
belief  that  there  must  be  "  limits "  of  series  of  ratios.  The 
notion  of  "  limit "  is  of  the  utmost  importance,  and  before 
proceeding  further  it  will  be  well  to  define  it. 

A  term  x  is  said  to  be  an  "  upper  limit "  of  a  class  a  with 
respect  to  a  relation  P  if  (i)  a  has  no  maximum  in  P,  (2)  every 
member  of  a  which  belongs  to  the  field  of  P  precedes  x,  (3)  every 
member  of  the  field  of  P  which  precedes  x  precedes  some  member 
of  a.  (By  "  precedes  "  we  mean  "  has  the  relation  P  to.") 

This  presupposes  the  following  definition  of  a  "  maximum  "  : — 

A  term  x  is  said  to  be  a  "  maximum  "  of  a  class  a  with  respect 
to  a  relation  P  if  x  is  a  member  of  a  and  of  the  field  of  P  and  does 
not  have  the  relation  P  to  any  other  member  of  a. 

These  definitions  do  not  demand  that  the  terms  to  which 
they  are  applied  should  be  quantitative.  For  example,  given 
a  series  of  moments  of  time  arranged  by  earlier  and  later,  their 
"  maximum  "  (if  any)  will  be  the  last  of  the  moments  ;  but  if 
they  are  arranged  by  later  and  earlier,  their  "  maximum  "  (if 
any)  will  be  the  first  of  the  moments. 

The  "  minimum  "  of  a  class  with  respect  to  P  is  its  maximum 
with  respect  to  the  converse  of  P  ;  and  the  "  lower  limit  "  with 
respect  to  P  is  the  upper  limit  with  respect  to  the  converse  of  P. 

The  notions  of  limit  and  maximum  do  not  essentially  demand 
that  the  relation  in  respect  to  which  they  are  defined  should 
be  serial,  but  they  have  few  important  applications  except  to 
cases  when  the  relation  is  serial  or  quasi-serial.  A  notion  which 
is  often  important  is  the  notion  "  upper  limit  or  maximum," 
to  which  we  may  give  the  name  "  upper  boundary."  Thus  the 
"  upper  boundary  "  of  a  set  of  terms  chosen  out  of  a  series  is 
their  last  member  if  they  have  one,  but,  if  not,  it  is  the  first 
term  after  all  of  them,  if  there  is  such  a  term.  If  there  is  neither 


Rational)  Real,  and  Complex  Numbers  71 

a  maximum  nor  a  limit,  there  is  no  upper  boundary.  The 
"  lower  boundary  "  is  the  lower  limit  or  minimum. 

Reverting  to  the  four  kinds  of  Dedekind  section,  we  see  that 
in  the  case  of  the  first  three  kinds  each  section  has  a  boundary 
(upper  or  lower  as  the  case  may  be),  while  in  the  fourth  kind 
neither  has  a  boundary.  It  is  also  clear  that,  whenever  the 
lower  section  has  an  upper  boundary,  the  upper  section  has 
a  lower  boundary.  In  the  second  and  third  cases,  the  two 
boundaries  are  identical ;  in  the  first,  they  are  consecutive 
terms  of  the  series. 

A  series  is  called  "  Dedekindian  "  when  every  section  has  a 
boundary,  upper  or  lower  as  the  case  may  be. 

We  have  seen  that  the  series  of  ratios  in  order  of  magnitude 
is  not  Dedekindian. 

From  the  habit  of  being  influenced  by  spatial  imagination, 
people  have  supposed  that  series  must  have  limits  in  cases  where 
it  seems  odd  if  they  do  not.  Thus,  perceiving  that  there  was 
no  rational  limit  to  the  ratios  whose  square  is  less  than  2,  they 
allowed  themselves  to  "  postulate "  an  irrational  limit,  which 
was  to  fill  the  Dedekind  gap.  Dedekind,  in  the  above-mentioned 
work,  set  up  the  axiom  that  the  gap  must  always  be  filled,  i.e. 
that  every  section  must  have  a  boundary.  It  is  for  this  reason 
that  series  where  his  axiom  is  verified  are  called  "  Dedekindian." 
But  there  are  an  infinite  number  of  series  for  which  it  is  not 

4 

verified. 

The  method  of  "  postulating  "  what  we  want  has  many  advan 
tages  ;  they  are  the  same  as  the  advantages  of  theft  over  honest 
toil.  Let  us  leave  them  to  others  and  proceed  with  our  honest  toil. 

It  is  clear  that  an  irrational  Dedekind  cut  in  some  way  "  repre 
sents  "  an  irrational.  In  order  to  make  use  of  this,  which  to 
begin  with  is  no  more  than  a  vague  feeling,  we  must  find  some 
way  of  eliciting  from  it  a  precise  definition  ;  and  in  order  to  do 
this,  we  must  disabuse  our  minds  of  the  notion  that  an  irrational 
must  be  the  limit  of  a  set  of  ratios.  Just  as  ratios  whose  de 
nominator  is  i  are  not  identical  with  integers,  so  those  rational 


72  Introduction  to  Mathematical  Philosophy 

numbers  which  can  be  greater  or  less  than  irrationals,  or  can 
have  irrationals  as  their  limits,  must  not  be  identified  with  ratios. 
We  have  to  define  a  new  kind  of  numbers  called  "  real  numbers," 
of  which  some  will  be  rational  and  some  irrational.  Those  that 
are  rational  "  correspond  "  to  ratios,  in  the  same  kind  of  way 
in  which  the  ratio  n/i  corresponds  to  the  integer  n  ;  but  they  are 
not  the  same  as  ratios.  In  order  to  decide  what  they  are  to  be, 
let  us  observe  that  an  irrational  is  represented  by  an  irrational 
cut,  and  a  cut  is  represented  by  its  lower  section.  Let  us  confine 
ourselves  to  cuts  in  which  the  lower  section  has  no  maximum  ; 
in  this  case  we  will  call  the  lower  section  a  "  segment."  Then 
those  segments  that  correspond  to  ratios  are  those  that  consist 
of  all  ratios  less  than  the  ratio  they  correspond  to,  which  is 
their  boundary ;  while  those  that  represent  irrationals  are  those 
that  have  no  boundary.  Segments,  both  those  that  have 
boundaries  and  those  that  do  not,  are  such  that,  of  any  two 
pertaining  to  one  series,  one  must  be  part  of  the  other ;  hence 
they  can  all  be  arranged  in  a  series  by  the  relation  of  whole  and 
part.  A  series  in  which  there  are  Dedekind  gaps,  i.e.  in  which 
there  are  segments  that  have  no  boundary,  will  give  rise  to  more 
segments  than  it  has  terms,  since  each  term  will  define  a  segment 
having  that  term  for  boundary,  and  then  the  segments  without 
boundaries  will  be  extra. 

We  are  now  in  a  position  to  define  a  real  number  and  an 
irrational  number. 

A  "  real  number  "  is  a  segment  of  the  series  of  ratios  in  order 
of  magnitude. 

An  "  irrational  number  "  is  a  segment  of  the  series  of  ratios 
which  has  no  boundary. 

A  "  rational  real  number  "  is  a  segment  of  the  series  of  ratios 
which  has  a  boundary. 

Thus  a  rational  real  number  consists  of  all  ratios  less  than  a 
certain  ratio,  and  it  is  the  rational  real  number  corresponding 
to  that  ratio.  The  real  number  I,  for  instance,  is  the  class  of 
proper  fractions. 


Rational,  Real,  and  Complex  Numb  en  73 

In  the  cases  in  which  we  naturally  supposed  that  an  irrational 
must  be  the  limit  of  a  set  of  ratios,  the  truth  is  that  it  is  the  limit 
of  the  corresponding  set  of  rational  real  numbers  in  the  series 
of  segments  ordered  by  whole  and  part.  For  example,  ^/^  is 
the  upper  limit  of  all  those  segments  of  the  series  of  ratios  that 
correspond  to  ratios  whose  square  is  less  than  2.  More  simply 
still,  \/2  is  the  segment  consisting  of  all  those  ratios  whose  square 
is  less  than  2. 

It  is  easy  to  prove  that  the  series  of  segments  of  any  series 
is  Dedekindian.  For,  given  any  set  of  segments,  their  boundary 
will  be  their  logical  sum,  i.e.  the  class  of  all  those  terms  that 
belong  to  at  least  one  segment  of  the  set.1 

The  above  definition  of  real  numbers  is  an  example  of  "  con 
struction  "  as  against  "  postulation,"  of  which  we  had  another 
example  in  the  definition  of  cardinal  numbers.  The  great 
advantage  of  this  method  is  that  it  requires  no  new  assumptions, 
but  enables  us  to  proceed  deductively  from  the  original  apparatus 
of  logic. 

There  is  no  difficulty  in  defining  addition  and  multiplication 
for  real  numbers  as  above  defined.  Given  two  real  numbers 
\L  and  v,  each  being  a  class  of  ratios,  take  any  member  of  JJL  and 
any  member  of  v  and  add  them  together  according  to  the  rule 
for  the  addition  of  ratios.  Form  the  class  of  all  such  sums 
obtainable  by  varying  the  selected  members  of  p  and  v.  This 
gives  a  new  class  of  ratios,  and  it  is  easy  to  prove  that  this  new 
class  is  a  segment  of  the  series  of  ratios.  We  define  it  as  the 
sum. of  p  and  v.  We  may  state  the  definition  more  shortly  as 
follows  : — 

The  arithmetical  sum  of  two  real  numbers  is  the  class  of  the 
arithmetical  sums  of  a  member  of  the  one  and  a  member  of  the 
other  chosen  in  all  possible  ways. 

1  For  a  fuller  treatment  of  the  subject  of  segments  and  Dedekindian 
relations,  see  Principia  Mathematical,  vol.  ii.  *  210-214.  For  a  fuller 
treatment  of  real  numbers,  see  ibid.,  vol.  iii.  *  310  ff.,  and  Principles  of 
Mathematics,  chaps,  xxxiii.  and  xxxiv. 


74  Introduction  to  Mathematical  Philosophy 

We  can  define  the  arithmetical  product  of  two  real  numbers 
in  exactly  the  same  way,  by  multiplying  a  member  of  the  one  by 
a  member  of  the  other  in  all  possible  ways.  The  class  of  ratios 
thus  generated  is  defined  as  the  product  of  the  two  real  numbers. 
(In  all  such  definitions,  the  series  of  ratios  is  to  be  defined  as 
excluding  o  and  infinity.) 

There  is  no  difficulty  in  extending  our  definitions  to  positive 
and  negative  real  numbers  and  their  addition  and  multiplication. 

It  remains  to  give  the  definition  of  complex  numbers. 

Complex  numbers,  though  capable  of  a  geometrical  interpreta 
tion,  are  not  demanded  by  geometry  in  the  same  imperative  way 
in  which  irrationals  are  demanded.  A  "  complex  "  number  means 
a  number  involving  the  square  root  of  a  negative  number,  whether 
integral,  fractional,  or  real.  Since  the  square  of  a  negative 
number  is  positive,  a  number  whose  square  is  to  be  negative  has 
to  be  a  new  sort  of  number.  Using  the  letter  i  for  the  square 
root  of  —I,  any  number  involving  the  square  root  of  a  negative 
number  can  be  expressed  in  the  form  x-\-yi,  where  x  and  y  are 
real.  The  part  yi  is  called  the  "  imaginary  "  part  of  this  number, 
x  being  the  "  real "  part.  (The  reason  for  the  phrase  "  real 
numbers  "  is  that  they  are  contrasted  with  such  as  are  "  ima 
ginary.")  Complex  numbers  have  been  for  a  long  time  habitually 
used  by  mathematicians,  in  spite  of  the  absence  of  any  precise 
definition.  It  has  been  simply  assumed  that  they  would  obey 
the  usual  arithmetical  rules,  and  on  this  assumption  their  employ 
ment  has  been  found  profitable.  They  are  required  less  for 
geometry  than  for  algebra  and  analysis.  We  desire,  for  example, 
to  be  able  to  say  that  every  quadratic  equation  has  two  roots, 
and  every  cubic  equation  has  three,  and  so  on.  But  if  we  are 
confined  to  real  numbers,  such  an  equation  as  #2-|-i=o  has  no 
roots,  and  such  an  equation  as  x^—i—o  has  only  one.  Every 
generalisation  of  number  has  first  presented  itself  as  needed  for 
some  simple  problem  :  negative  numbers  were  needed  in  order 
that  subtraction  might  be  always  possible,  since  otherwise  a— b 
would  be  meaningless  if  a  were  less  than  b  ;  fractions  were  needed 


Rational)  Real,  and  Complex  Numbers  75 

in  order  that  division  might  be  always  possible  ;  and  complex 
numbers  are  needed  in  order  that  extraction  of  roots  and  solu 
tion  of  equations  may  be  always  possible.  But  extensions  of 
number  are  not  created  by  the  mere  need  for  them  :  they  are 
created  by  the  definition,  and  it  is  to  the  definition  of  complex 
numbers  that  we  must  now  turn  our  attention. 

A  complex  number  may  be  regarded  and  defined  as  simply  an 
ordered  couple  of  real  numbers.  Here,  as  elsewhere,  many 
definitions  are  possible.  All  that  is  necessary  is  that  the  defini 
tions  adopted  shall  lead  to  certain  properties.  In  the  case  of 
complex  numbers,  if  they  are  defined  as  ordered  couples  of  real 
numbers,  we  secure  at  once  some  of  the  properties  required, 
namely,  that  two  real  numbers  are  required  to  determine  a  com 
plex  number,  and  that  among  these  we  can  distinguish  a  first 
and  a  second,  and  that  two  complex  numbers  are  only  identical 
when  the  first  real  number  involved  in  the  one  is  equal  to  the 
first  involved  in  the  other,  and  the  second  to  the  second.  What 
is  needed  further  can  be  secured  by  defining  the  rules  of  addition 
and  multiplication.  We  are  to  have 


Thus  we  shall  define  that,  given  two  ordered  couples  of  real 
numbers,  (#,  y)  and  (#',  y'),  their  sum  is  to  be  the  couple  (x+xr, 
y+y')>  and  their  product  is  to  be  the  couple  (xxf—  yy',  xy'-\-x'y). 
By  these  definitions  we  shall  secure  that  our  ordered  couples 
shall  have  the  properties  we  desire.  For  example,  take  the 
product  of  the  two  couples  (o,  y)  and  (o,  y').  This  will,  by  the 
above  rule,  be  the  couple  (—  yy',  o).  Thus  the  square  of  the 
couple  (o,  i)  will  be  the  couple  (  —  I,  o).  Now  those  couples  in 
which  the  second  term  is  o  are  those  which,  according  to  the  usual 
nomenclature,  have  their  imaginary  part  zero  ;  in  the  notation 
x-\-  yi,  they  are  x+oi,  which  it  is  natural  to  write  simply  x.  Just 
as  it  is  natural  (but  erroneous)  to  identify  ratios  whose  de 
nominator  is  unity  with  integers,  so  it  is  natural  (but  erroneous) 


j6  Introduction  to  Mathematical  Philosophy 

to  identify  complex  numbers  whose  imaginary  part  is  zero  with 
real  numbers.  Although  this  is  an  error  in  theory,  it  is  a  con 
venience  in  practice  ;  "  x-}-oi  "  may  be  replaced  simply  by  "  x  " 
and  "  o-\-yi  "  by  "  yi,"  provided  we  remember  that  the  "  x  "  is 
not  really  a  real  number,  but  a  special  case  of  a  complex  number. 
And  when  y  is  I,  "  yi"  may  of  course  be  replaced  by  "  *."  Thus 
the  couple  (o,  l)  is  represented  by  *,  and  the  couple  (—1,  o)  is 
represented  by  —  I.  Now  our  rules  of  multiplication  make  the 
square  of  (o,  l)  equal  to  (—1,  o),  i.e.  the  square  of  i  is  —  i.  This 
is  what  we  desired  to  secure.  Thus  our  definitions  serve  all 
necessary  purposes. 

It  is  easy  to  give  a  geometrical  interpretation  of  complex 
numbers  in  the  geometry  of  the  plane.  This  subject  was  agree 
ably  expounded  by  W.  K.  Clifford  in  his  Common  Sense  of  the 
Exact  Sciences,  a  book  of  great  merit,  but  written  before  the 
importance  of  purely  logical  definitions  had  been  realised. 

Complex  numbers  of  a  higher  order,  though  much  less  useful 
and  important  than  those  what  we  have  been  defining,  have 
certain  uses  that  are  not  without  importance  in  geometry,  as 
may  be  seen,  for  example,  in  Dr  Whitehead's  Universal  Algebra. 
The  definition  of  complex  numbers  of  order  n  is  obtained  by  an 
obvious  extension  of  the  definition  we  have  given.  We  define  a 
complex  number  of  order  n  as  a  one-many  relation  whose  domain 
consists  of  certain  real  numbers  and  whose  converse  domain 
consists  of  the  integers  from  I  to  n.1  This  is  what  would  ordi 
narily  be  indicated  by  the  notation  (xl9  x2,  #3,  .  .  .  xn),  where  the 
suffixes  denote  correlation  with  the  integers  used  as  suffixes,  and 
the  correlation  is  one-many,  not  necessarily  one-one,  because  xr 
and  xa  may  be  equal  when  r  and  s  are  not  equal.  The  above 
definition,  with  a  suitable  rule  of  multiplication,  will  serve  all 
purposes  for  which  complex  numbers  of  higher  orders  are  needed. 

We  have  now  completed  our  review  of  those  extensions  of 
number  which  do  not  involve  infinity.  The  application  of  number 
to  infinite  collections  must  be  our  next  topic. 

1  Cf .  Principles  of  Mathematics,  §  360,  p.  379. 


CHAPTER  VIII 

INFINITE    CARDINAL    NUMBERS 

THE  definition  of  cardinal  numbers  which  we  gave  in  Chapter  II. 
was  applied  in  Chapter  III.  to  finite  numbers,  i.e.  to  the  ordinary 
natural  numbers.  To  these  we  gave  the  name  "  inductive 
numbers,"  because  we  found  that  they  are  to  be  defined  as 
numbers  which  obey  mathematical  induction  starting  from  o. 
But  we  have  not  yet  considered  collections  which  do  not  have  an 
inductive  number  of  terms,  nor  have  we  inquired  whether  such 
collections  can  be  said  to  have  a  number  at  all.  This  is  an 
ancient  problem,  which  has  been  solved  in  our  own  day,  chiefly 
by  Georg  Cantor.  In  the  present  chapter  we  shall  attempt  to 
explain  the  theory  of  transfinite  or  infinite  cardinal  numbers  as 
it  results  from  a  combination  of  his  discoveries  with  those  of 
Frege  on  the  logical  theory  of  numbers. 

It  cannot  be  said  to  be  certain  that  there  are  in  fact  any  infinite 
collections  in  the  world.  The  assumption  that  there  are  is  what 
we  call  the  "  axiom  of  infinity."  Although  various  ways  suggest 
themselves  by  which  we  might  hope  to  prove  this  axiom,  there 
is  reason  to  fear  that  they  are  all  fallacious,  and  that  there  is  no 
conclusive  logical  reason  for  believing  it  to  be  true.  At  the  same 
time,  there  is  certainly  no  logical  reason  against  infinite  collections, 
and  we  are  therefore  justified,  in  logic,  in  investigating  the  hypo 
thesis  that  there  are  such  collections.  The  practical  form  of  this 
hypothesis,  for  our  present  purposes,  is  the  assumption  that,  if 
n  is  any  inductive  number,  n  is  not  equal  to  w-j-i.  Various 
subtleties  arise  in  identifying  this  form  of  our  assumption  with 

77 


7  8  Introduction  to  Mathematical  Philosophy 

the  form  that  asserts  the  existence  of  infinite  collections  ;  but 
we  will  leave  these  out  of  account  until,  in  a  later  chapter,  we 
come  to  consider  the  axiom  of  infinity  on  its  own  account.  For 
the  present  we  shall  merely  assume  that,  if  n  is  an  inductive 
number,  n  is  not  equal  to  n-\-i.  This  is  involved  in  Peano's 
assumption  that  no  two  inductive  numbers  have  the  same  suc 
cessor  ;  for,  if  n=n-}-i,  then  n—  I  and  n  have  the  same  successor, 
namely  n.  Thus  we  are  assuming  nothing  that  was  not  involved 
in  Peano's  primitive  propositions. 

Let  us  now  consider  the  collection  of  the  inductive  numbers 
themselves.  This  is  a  perfectly  well-defined  class.  In  the  first 
place,  a  cardinal  number  is  a  set  of  classes  which  are  all  similar 
to  each  other  and  are  not  similar  to  anything  except  each  other. 
We  then  define  as  the  "  inductive  numbers "  those  among 
cardinals  which  belong  to  the  posterity  of  o  with  respect  to  the 
relation  of  n  to  w-f-i,  *•<?•  those  which  possess  every  property 
possessed  by  o  and  by  the  successors  of  possessors,  meaning  by 
the  "successor"  of  n  the  number  n-\-\.  Thus  the  class  of 
"  inductive  numbers "  is  perfectly  definite.  By  our  general 
definition  of  cardinal  numbers,  the  number  of  terms  in  the  class 
of  inductive  numbers  is  to  be  defined  as  "  all  those  classes  that 
are  similar  to  the  class  of  inductive  numbers  " — i.e.  this  set  of 
classes  is  the  number  of  the  inductive  numbers  according  to  our 
definitions. 

Now  it  is  easy  to  see  that  this  number  is  not  one  of  the  inductive 
numbers.  If  n  is  any  inductive  number,  the  number  of  numbers 
from  o  to  n  (both  included)  is  n-\-i  ;  therefore  the  total  number 
of  inductive  numbers  is  greater  than  n,  no  matter  which  of  the 
inductive  numbers  n  may  be.  If  we  arrange  the  inductive 
numbers  in  a  series  in  order  of  magnitude,  this  series  has  no  last 
term ;  but  if  n  is  an  inductive  number,  every  series  whose  field 
has  n  terms  has  a  last  term,  as  it  is  easy  to  prove.  Such  differences 
might  be  multiplied  a<L  lib.  Thus  the  number  of  inductive 
numbers  is  a  new  number,  different  from  all  of  them,  not  possess 
ing  all  inductive  properties.  It  may  happen  that  o  has  a  certain 


Infinite  Cardinal  Numbers  79 

property,  and  that  if  n  has  it  so  has  w+i,  and  yet  that  this  new 
number  does  not  have  it.  The  difficulties  that  so  long  delayed 
the  theory  of  infinite  numbers  were  largely  due  to  the  fact  that 
some,  at  least,  of  the  inductive  properties  were  wrongly  judged 
to  be  such  as  must  belong  to  all  numbers  ;  indeed  it  was  thought 
that  they  could  not  be  denied  without  contradiction.  The  first 
step  in  understanding  infinite  numbers  consists  in  realising  the 
mistakenness  of  this  view. 

The  most  noteworthy  and  astonishing  difference  between  an 
inductive  number  and  this  new  number  is  that  this  new  number 
is  unchanged  by  adding  I  or  subtracting  I  or  doubling  or  halving 
or  any  of  a  number  of  other  operations  which  we  think  of  as 
necessarily  making  a  number  larger  or  smaller.  The  fact  of  being 
not  altered  by  the  addition  of  I  is  used  by  Cantor  for  the  defini 
tion  of  what  he  calls  "  transfinite  "  cardinal  numbers ;  but  for 
various  reasons,  some  of  which  will  appear  as  we  proceed,  it  is 
better  to  define  an  infinite  cardinal  number  as  one  which  does 
not  possess  all  inductive  properties,  i.e.  simply  as  one  which  is 
not  an  inductive  number.  Nevertheless,  the  property  of  being 
unchanged  by  the  addition  of  I  is  a  very  important  one,  and  we 
must  dwell  on  it  for  a  time. 

To  say  that  a  class  has  a  number  which  is  not  altered  by  the 
addition  of  I  is  the  same  thing  as  to  say  that,  if  we  take  a  term  x 
which  does  not  belong  to  the  class,  we  can  find  a  one-one  relation 
whose  domain  is  the  class  and  whose  converse  domain  is  obtained 
by  adding  x  to  the  class.  For  in  that  case,  the  class  is  similar 
to  the  sum  of  itself  and  the  term  x,  i.e.  to  a  class  having  one  extra 
term ;  so  that  it  has  the  same  number  as  a  class  with  one  extra 
term,  so  that  if  n  is  this  number,  n=n-\-\.  In  this  case,  we  shall 
also  have  n—n—  I,  i.e.  there  will  be  one-one  relations  whose 
domains  consist  of  the  whole  class  and  whose  converse  domains 
consist  of  just  one  term  short  of  the  whole  class.  It  can  be  shown 
that  the  cases  in  which  this  happens  are  the  same  as  the  apparently 
more  general  cases  in  which  some  part  (short  of  the  whole)  can  be 
put  into  one-one  relation  with  the  whole.  When  this  can  be  done, 


8o  Introduction  to  Mathematical  Philosophy 

the  correlator  by  which  it  is  done  may  be  said  to  "  reflect  "  the 
whole  class  into  a  part  of  itself ;  for  this  reason,  such  classes  will 
be  called  "  reflexive."  Thus  : 

A  "  reflexive  "  class  is  one  which  is  similar  to  a  proper  part 
of  itself.  (A  "  proper  part  "  is  a  part  short  of  the  whole.) 

A  "  reflexive  "  cardinal  number  is  the  cardinal  number  of  a 
reflexive  class. 

We  have  now  to  consider  this  property  of  reflexiveness. 

One  of  the  most  striking  instances  of  a  "  reflexion  "  is  Royce's 
illustration  of  the  map  :  he  imagines  it  decided  to  make  a  map 
of  England  upon  a  part  of  the  surface  of  England.  A  map,  if 
it  is  accurate,  has  a  perfect  one-one  correspondence  with  its 
original ;  thus  our  map,  which  is  part,  is  in  one-one  relation  with 
the  whole,  and  must  contain  the  same  number  of  points  as  the 
whole,  which  must  therefore  be  a  reflexive  number.  Royce  is 
interested  in  the  fact  that  the  map,  if  it  is  correct,  must  contain 
a  map  of  the  map,  which  must  in  turn  contain  a  map  of  the  map 
of  the  map,  and  so  on  ad  infinitum.  This  point  is  interesting, 
but  need  not  occupy  us  at  this  moment.  In  fact,  we  shall  do 
well  to  pass  from  picturesque  illustrations  to  such  as  are  more 
completely  definite,  and  for  this  purpose  we  cannot  do  better 
than  consider  the  number-series  itself. 

The  relation  of  n  to  w-f-i,  confined  to  inductive  numbers,  is 
one-one,  has  the  whole  of  the  inductive  numbers  for  its  domain, 
and  all  except  o  for  its  converse  domain.  Thus  the  whole  class 
of  inductive  numbers  is  similar  to  what  the  same  class  becomes 
when  we  omit  o.  Consequently  it  is  a  "  reflexive  "  class  according 
to  the  definition,  and  the  number  of  its  terms  is  a  "  reflexive  " 
number.  Again,  the  relation  of  n  to  2n,  confined  to  inductive 
numbers,  is  one-one,  has  the  whole  of  the  inductive  numbers  for 
its  domain,  and  the  even  inductive  numbers  alone  for  its  converse 
domain.  Hence  the  total  number  of  inductive  numbers  is  the 
same  as  the  number  of  even  inductive  numbers.  This  property 
was  used  by  Leibniz  (and  many  others)  as  a  proof  that  infinite 
numbers  are  impossible ;  it  was  thought  self-contradictory  that 


Infinite  Cardinal  Numbers  81 

"  the  part  should  be  equal  to  the  whole."  But  this  is  one  of  those 
phrases  that  depend  for  their  plausibility  upon  an  unperceived 
vagueness  :  the  word  "  equal  "  has  many  meanings,  but  if  it  is 
taken  to  mean  what  we  have  called  "  similar,"  there  is  no  contra 
diction,  since  an  infinite  collection  can  perfectly  well  have  parts 
similar  to  itself.  Those  who  regard  this  as  impossible  have, 
unconsciously  as  a  rule,  attributed  to  numbers  in  general  pro 
perties  which  can  only  be  proved  by  mathematical  induction, 
and  which  only  their  familiarity  makes  us  regard,  mistakenly, 
as  true  beyond  the  region  of  the  finite. 

Whenever  we  can  "  reflect "  a  class  into  a  part  of  itself,  the 
same  relation  will  necessarily  reflect  that  part  into  a  smaller 
part,  and  so  on  ad  infinitum.  For  example,  we  can  reflect, 
as  we  have  just  seen,  all  the  inductive  numbers  into  the  even 
numbers  ;  we  can,  by  the  same  relation  (that  of  n  to  2n)  reflect 
the  even  numbers  into  the  multiples  of  4,  these  into  the  multiples 
of  8,  and  so  on.  This  is  an  abstract  analogue  to  Royce's  problem 
of  the  map.  The  even  numbers  are  a  "  map  "  of  all  the  inductive 
numbers  ;  the  multiples  of  4  are  a  map  of  the  map  ;  the  multiples 
of  8  are  a  map  of  the  map  of  the  map  ;  and  so  on.  If  we  had 
applied  the  same  process  to  the  relation  of  «  to  w+l,"  our  "  map  " 
would  have  consisted  of  all  the  inductive  numbers  except  o ; 
the  map  of  the  map  would  have  consisted  of  all  from  2  onward, 
the  map  of  the  map  of  the  map  of  all  from  3  onward  ;  and  so  on. 
The  chief  use  of  such  illustrations  is  in  order  to  become  familiar 
with  the  idea  of  reflexive  classes,  so  that  apparently  paradoxical 
arithmetical  propositions  can  be  readily  translated  into  the 
language  of  reflexions  and  classes,  in  which  the  air  of  paradox 
is  much  less. 

It  will  be  useful  to  give  a  definition  of  the  number  which  is 
that  of  the  inductive  cardinals.  For  this  purpose  we  will 
first  define  the  kind  of  series  exemplified  by  the  inductive  cardinals 
in  order  of  magnitude.  The  kind  of  series  which  is  called  a 
"  progression  "  has  already  been  considered  in  Chapter  I.  It  is  a 
series  which  can  be  generated  by  a  relation  of  consecutiveness  : 

6 


82  Introduction  to  Mathematical  Philosophy 

every  member  of  the  series  is  to  have  a  successor,  but  there  is 
to  be  just  one  which  has  no  predecessor,  and  every  member  of 
the  series  is  to  be  in  the  posterity  of  this  term  with  respect  to 
the  relation  "  immediate  predecessor."  These  characteristics 
may  be  summed  up  in  the  following  definition  : —  * 

A  "  progession  "  is  a  one-one  relation  such  that  there  is  just 
one  term  belonging  to  the  domain  but  not  to  the  converse  domain, 
and  the  domain  is  identical  with  the  posterity  of  this  one  term. 

It  is  easy  to  see  that  a  progression,  so  defined,  satisfies  Peano's 
five  axioms.  The  term  belonging  to  the  domain  but  not  to  the 
converse  domain  will  be  what  he  calls  "  o  "  ;  the  term  to  which 
a  term  has  the  one-one  relation  will  be  the  "  successor  "  of  the 
term ;  and  the  domain  of  the  one-one  relation  will  be  what 
he  calls  "  number."  Taking  his  five  axioms  in  turn,  we  have 
the  following  translations  : — 

(1)  "  o  is  a  number  "  becomes  :  "  The  member  of  the  domain 
which  is  not  a  member  of  the  converse  domain  is  a  member  of 
the  domain."     This  is   equivalent  to  the  existence  of  such  a 
member,  which  is  given  in  our  definition.     We  will  call  this 
member  "  the  first  term." 

(2)  "  The  successor  of  any  number  is  a  number  "  becomes  : 
"  The  term  to  which  a  given  member  of  the  domain  has  the  rela 
tion  in  question  is  again  a  member  of  the  domain."     This  is 
proved  as  follows  :    By  the  definition,   every  member  of  the 
domain  is  a  member  of  the  posterity  of  the  first  term  ;    hence 
the  successor  of  a  member  of  the  domain  must  be  a  member  of 
the  posterity  of  the  first  term  (because  the  posterity  of  a  term 
always  contains  its  own  successors,  by  the  general  definition  of 
posterity),  and  therefore  a  member  of  the  domain,  because  by 
the  definition  the  posterity  of  the  first  term  is  the  same  as  the 
domain. 

(3)  "  No   two  numbers   have   the  same   successor."     This  is 
only  to  say  that  the  relation  is  one-many,  which  it  is  by  definition 
(being  one-one). 

1  Cf.  Pnncipia  Mathematica,  vol.  ii.  #   123. 


Infinite  Cardinal  Numbers  83 

(4)  "  o  is  not  the  successor  of  any  number  "  becomes  :    "  The 
first  term  is  not  a  member  of  the  converse  domain,"  which  is 
again  an  immediate  result  of  the  definition. 

(5)  This  is  mathematical  induction,  and  becomes  :    "  Every 
member  of  the  domain  belongs  to  the  posterity  of  the  first  term," 
which  was  part  of  our  definition. 

Thus  progressions  as  we  have  defined  them  have  the  five 
formal  properties  from  which  Peano  deduces  arithmetic.  It  is 
easy  to  show  that  two  progessions  are  "  similar  "  in  the  sense 
defined  for  similarity  of  relations  in  Chapter  VI.  We  can,  of 
course,  derive  a  relation  which  is  serial  from  the  one-one  relation 
by  which  we  define  a  progression :  the  method  used  is  that 
explained  in  Chapter  IV.,  and  the  relation  is  that  of  a  term  to 
a  member  of  its  proper  posterity  with  respect  to  the  original 
one-one  relation. 

Two  transitive  asymmetrical  relations  which  generate  pro 
gressions  are  similar,  for  the  same  reasons  for  which  the  cor 
responding  one-one  relations  are  similar.  The  class  of  all  such 
transitive  generators  of  progressions  is  a  "  serial  number  "  in 
the  sense  of  Chapter  VI.;  it  is  in  fact  the  smallest  of  infinite 
serial  numbers,  the  number  to  which  Cantor  has  given  the  name 
o>,  by  which  he  has  made  it  famous. 

But  we  are  concerned,  for  the  moment,  with  cardinal  numbers. 
Since  two  progressions  are  similar  relations,  it  follows  that  their 
domains  (or  their  fields,  which  are  the  same  as  their  domains) 
are  similar  classes.  The  domains  of  progressions  form  a  cardinal 
number,  since  every  class  which  is  similar  to  the  domain  of  a 
progression  is  easily  shown  to  be  itself  the  domain  of  a  progression. 
This  cardinal  number  is  the  smallest  of  the  infinite  cardinal 
numbers  ;  it  is  the  one  to  which  Cantor  has  appropriated  the 
Hebrew  Aleph  with  the  suffix  o,  to  distinguish  it  from  larger 
infinite  cardinals,  which  have  other  suffixes.  Thus  the  name  of 
the  smallest  of  infinite  cardinals  is  N0. 

To  say  that  a  class  has  N0  terms  is  the  same  thing  as  to  say 
that  it  is  a  member  of  N0,  and  this  is  the  same  thing  as  to  say 


84  Introduction  to  Mathematical  Philosophy 

that  the  members  of  the  class  can  be  arranged  in  a  progression. 
It  is  obvious  that  any  progression  remains  a  progression  if  we 
omit  a  finite  number  of  terms  from  it,  or  every  other  term,  or 
all  except  every  tenth  term  or  every  hundredth  term.  These 
methods  of  thinning  out  a  progression  do  not  make  it  cease  to 
be  a  progression,  and  therefore  do  not  diminish  the  number  of 
its  terms,  which  remains  N0.  In  fact,  any  selection  from  a  pro 
gression  is  a  progression  if  it  has  no  last  term,  however  sparsely 
it  may  be  distributed.  Take  (say)  inductive  numbers  of  the  form 
nw,  or  nnW.  Such  numbers  grow  very  rare  in  the  higher  parts 
of  the  number  series,  and  yet  there  are  just  as  many  of  them  as 
there  are  inductive  numbers  altogether,  namely,  N0. 

Conversely,  we  can  add  terms  to  the  inductive  numbers  without 
increasing  their  number.  Take,  for  example,  ratios.  One 
might  be  inclined  to  think  that  there  must  be  many  more  ratios 
than  integers,  since  ratios  whose  denominator  is  I  correspond 
to  the  integers,  and  seem  to  be  only  an  infinitesimal  proportion 
of  ratios.  But  in  actual  fact  the  number  of  ratios  (or  fractions) 
is  exactly  the  same  as  the  number  of  inductive  numbers,  namely, 
N0.  This  is  easily  seen  by  arranging  ratios  in  a  series  on  the 
following  plan  :  If  the  sum  of  numerator  and  denominator  in 
one  is  less  than  in  the  other,  put  the  one  before  the  other ;  if 
the  sum  is  equal  in  the  two,  put  first  the  one  with  the  smaller 
numerator.  This  gives  us  the  series 

i,  1/2,  2,  1/3,  3,  1/4,  2/3,  3/2,  4,  1/5,  .  .  . 

This  series  is  a  progression,  and  all  ratios  occur  in  it  sooner  or 
later.  Hence  we  can  arrange  all  ratios  in  a  progression,  and 
their  number  is  therefore  N0. 

It  is  not  the  case,  however,  that  all  infinite  collections  have 
N0  terms.  The  number  of  real  numbers,  for  example,  is  greater 
than  N0 ;  it  is,  in  fact,  2^°,  and  it  is  not  hard  to  prove  that  2n 
is  greater  than  n  even  when  n  is  infinite.  The  easiest  way  of 
proving  this  is  to  prove,  first,  that  if  a  class  has  n  members,  it 
contains  2n  sub-classes — in  other  words,  that  there  are  2n  ways 


Infinite  Carainal  Numbers  85 

of  selecting  some  of  its  members  (including  the  extreme  cases 
where  we  select  all  or  none) ;  and  secondly,  that  the  number  of 
sub-classes  contained  in  a  class  is  always  greater  than  the  number 
of  members  of  the  class.  Of  these  two  propositions,  the  first 
is  familiar  in  the  case  of  finite  numbers,  and  is  not  hard  to  extend 
to  infinite  numbers.  The  proof  of  the  second  is  so  simple  and 
so  instructive  that  we  shall  give  it : 

In  the  first  place,  it  is  clear  that  the  number  of  sub-classes 
of  a  given  class  (say  a)  is  at  least  as  great  as  the  number  of 
members,  since  each  member  constitutes  a  sub-class,  and  we  thus 
have  a  correlation  of  all  the  members  with  some  of  the  sub 
classes.  Hence  it  follows  that,  if  the  number  of  sub-classes  is 
not  equal  to  the  number  of  members,  it  must  be  greater.  Now 
it  is  easy  to  prove  that  the  number  is  not  equal,  by  showing  that, 
given  any  one-one  relation  whose  domain  is  the  members  and 
whose  converse  domain  is  contained  among  the  set  of  sub 
classes,  there  must  be  at  least  one  sub-class  not  belonging  to 
the  converse  domain.  The  proof  is  as  follows  :  x  When  a  one- 
one  correlation  R  is  established  between  all  the  members  of  a 
and  some  of  the  sub-classes,  it  may  happen  that  a  given  member 
x  is  correlated  with  a  sub-class  of  which  it  is  a  member ;  or, 
again,  it  may  happen  that  x  is  correlated  with  a  sub-class  of 
which  it  is  not  a  member.  Let  us  form  the  whole  class,  )3  say, 
of  those  members  x  which  are  correlated  with  sub-classes  of  which 
they  are  not  members.  This  is  a  sub-class  of  a,  and  it  is  not 
correlated  with  any  member  of  a.  For,  taking  first  the  members 
of  ]3,  each  of  them  is  (by  the  definition  of  )8)  correlated  with 
some  sub-class  of  which  it  is  not  a  member,  and  is  therefore  not 
correlated  with  j3.  Taking  next  the  terms  which  are  not  members 
of  jS,  each  of  them  (by  the  definition  of  j3)  is  correlated  with 
some  sub-class  of  which  it  is  a  member,  and  therefore  again 
is  not  correlated  with  j8.  Thus  no  member  of  a  is  correlated 
with  )3.  Since  R  was  any  one-one  correlation  of  all  members 

1  This  proof  is  taken  from  Cantor,  with  some  simplifications  :  see 
Jahresbericht  der  deutschen  Mathematiker-Vereinigung,  i.  (1892),  p.  77. 


86  Introduction  to  Mathematical  Philosophy 

with  some  sub-classes,  it  follows  that  there  is  no  correlation 
of  all  members  with  all  sub-classes.  It  does  not  matter  to  the 
proof  if  j3  has  no  members  :  all  that  happens  in  that  case  is  that 
the  sub-class  which  is  shown  to  be  omitted  is  the  null-class. 
Hence  in  any  case  the  number  of  sub-classes  is  not  equal  to  the 
number  of  members,  and  therefore,  by  what  was  said  earlier, 
it  is  greater.  Combining  this  with  the  proposition  that,  if  n  is 
the  number  of  members,  2n  is  the  number  of  sub-classes,  we  have 
the  theorem  that  2n  is  always  greater  than  n,  even  when  n  is 
infinite. 

It  follows  from  this  proposition  that  there  is  no  maximum 
to  the  infinite  cardinal  numbers.  However  great  an  infinite 
number  n  may  be,  2n  will  be  still  greater.  The  arithmetic  of 
infinite  numbers  is  somewhat  surprising  until  one  becomes 
accustomed  to  it.  We  have,  for  example, 


N0-fw=N0,  where  n  is  any  inductive  number, 
«o2=»o- 

(This  follows  from  the  case  of  the  ratios,  for,  since  a  ratio  is 
determined  by  a  pair  of  inductive  numbers,  it  is  easy  to  see  that 
the  number  of  ratios  is  the  square  of  the  number  of  inductive 
numbers,  i.e.  it  is  N02  ;  but  we  saw  that  it  is  also 


N0«=N0>  where  n  is  any  inductive  number. 
(This  follows  from      N02=No  by  induction  ;  for  if  NO"=NO, 
then  N0«+i=N02=N0.) 

But  2^0  >N0. 

In  fact,  as  we  shall  see  later,  2^°  is  a  very  important  number, 
namely,  the  number  of  terms  in  a  series  which  has  "  continuity  " 
in  the  sense  in  which  this  word  is  used  by  Cantor.  Assuming 
space  and  time  to  be  continuous  in  this  sense  (as  we  commonly 
do  in  analytical  geometry  and  kinematics),  this  will  be  the 
number  of  points  in  space  or  of  instants  in  time  ;  it  will  also  be 
the  number  of  points  in  any  finite  portion  of  space,  whether 


Infinite  Cardinal  Numbers  87 

line,  area,  or  volume.     After  N0,  2^  is  the  most  important  and 
interesting  of  infinite  cardinal  numbers. 

Although  addition  and  multiplication  are  always  possible 
with  infinite  cardinals,  subtraction  and  division  no  longer  give 
definite  results,  and  cannot  therefore  be  employed  as  they  are 
employed  in  elementary  arithmetic.  Take  subtraction  to  begin 
with  :  so  long  as  the  number  subtracted  is  finite,  all  goes  well  ; 
if  the  other  number  is  reflexive,  it  remains  unchanged.  Thus 
N0— n=&0,  if  n  is  finite;  so  far,  subtraction  gives  a  perfectly 
definite  result.  But  it  is  otherwise  when  we  subtract  N0  from 
itself;  we  may  then  get  any  result,  from  o  up  to  N0.  This  is 
easily  seen  by  examples.  From  the  inductive ,  numbers,  take 
away  the  following  collections  of  N0  terms  : — 

(1)  All  the  inductive  numbers — remainder,  zero. 

(2)  All  the  inductive  numbers  from  n  onwards— remainder, 
the  numbers  from  o  to  n—  I,  numbering  n  terms  in  all. 

(3)  All  the  odd  numbers — remainder,  all  the  even  numbers, 
numbering  N0  terms. 

All  these  are  different  ways  of  subtracting  N0  from  N0,  and 
all  give  different  results. 

As  regards  division,  very  similar  results  follow  from  the  fact 
that  N0  is  unchanged  when  multiplied  by  2  or  3  or  any  finite 
number  n  or  by  N0.  It  follows  that  N0  divided  by  N0  may  have 
any  value  from  I  up  to  N0. 

From  the  ambiguity  of  subtraction  and  division  it  results 
that  negative  numbers  and  ratios  cannot  be  extended  to  infinite 
numbers.  Addition,  multiplication,  and  exponentiation  proceed 
quite  satisfactorily,  but  the  inverse  operations — subtraction, 
division,  and  extraction  of  roots — are  ambiguous,  and  the  notions 
that  depend  upon  them  fail  when  infinite  numbers  are  concerned. 

The  characteristic  by  which  we  defined  finitude  was  mathe 
matical  induction,  i.e.  we  defined  a  number  as  finite  when  it 
obeys  mathematical  induction  starting  from  o,  and  a  class  as 
finite  when  its  number  is  finite.  This  definition  yields  the  sort 
of  result  that  a  definition  ought  to  yield,  namely,  that  the  finite 


88  Introduction  to  Mathematical  Philosophy 

numbers  are  those  that  occur  in  the  ordinary  number-series 
o,  i,  2,  3,  ...  But  in  the  present  chapter,  the  infinite  num 
bers  we  have  discussed  have  not  merely  been  non-inductive : 
they  have  also  been  reflexive.  Cantor  used  reflexiveness  as  the 
definition  of  the  infinite,  and  believes  that  it  is  equivalent  to 
non-inductiveness  ;  that  is  to  say,  he  believes  that  every  class 
and  every  cardinal  is  either  inductive  or  reflexive.  This  may  be 
true,  and  may  very  possibly  be  capable  of  proof  ;  but  the  proofs 
hitherto  offered  by  Cantor  and  others  (including  the  present 
author  in  former  days)  are  fallacious,  for  reasons  which  will  be 
explained  when  we  come  to  consider  the  "  multiplicative  axiom." 
At  present,  it  is  not  known  whether  there  are  classes  and  cardinals 
which  are  neither  reflexive  nor  inductive.  If  n  were  such  a 
cardinal,  we  should  not  have  n—n-\-iy  but  n  would  not  be  one 
of  the  "  natural  numbers,"  and  would  be  lacking  in  some  of  the 
inductive  properties.  All  known  infinite  classes  and  cardinals 
are  reflexive ;  but  for  the  present  it  is  well  to  preserve  an  open 
mind  as  to  whether  there  are  instances,  hitherto  unknown,  of 
classes  and  cardinals  which  are  neither  reflexive  nor  inductive. 
Meanwhile,  we  adopt  the  following  definitions  : — 

A.  finite  class  or  cardinal  is  one  which  is  inductive. 

An  infinite  class  or  cardinal  is  one  which  is  not  inductive. 
All  refiexive  classes  and  cardinals  are  infinite  ;  but  it  is  not  known 
at  present  whether  all  infinite  classes  and  cardinals  are  reflexive. 
We  shall  return  to  this  subject  in  Chapter  XII. 


CHAPTER  IX 

INFINITE    SERIES   AND    ORDINALS 

AN  "  infinite  series  "  may  be  defined  as  a  series  of  which  the  field 
is  an  infinite  class.  We  have  already  had  occasion  to  consider 
one  kind  of  infinite  series,  namely,  progressions.  In  this  chapter 
we  shall  consider  the  subject  more  generally. 

The  most  noteworthy  characteristic  of  an  infinite  series  is 
that  its  serial  number  can  be  altered  by  merely  re-arranging 
its  terms.  In  this  respect  there  is  a  certain  oppositeness  between 
cardinal  and  serial  numbers.  It  is  possible  to  keep  the  cardinal 
number  of  a  reflexive  class  unchanged  in  spite  of  adding  terms 
to  it ;  on  the  other  hand,  it  is  possible  to  change  the  serial 
number  of  a  series  without  adding  or  taking  away  any  terms, 
by  mere  re-arrangement.  At  the  same  time,  in  the  case  of  any 
infinite  series  it  is  also  possible,  as  with  cardinals,  to  add  terms 
without  altering  the  serial  number :  everything  depends  upon 
the  way  in  which  they  are  added. 

In  order  to  make  matters  clear,  it  will  be  best  to  begin  with 
examples.  Let  us  first  consider  various  different  kinds  of  series 
which  can  be  made  out  of  the  inductive  numbers  arranged  on 
various  plans.  We  start  with  the  series 


3, 


which,  as  we  have  already  seen,  represents  the  smallest  of  in 
finite  serial  numbers,  the  sort  that  Cantor  calls  co.  Let  us 
proceed  to  thin  out  this  series  by  repeatedly  performing  the 

89 


90  Introduction  to  Mathematical  Philosophy 

operation  of  removing  to  the  end  the  first  even  number  that 
occurs.  We  thus  obtain  in  succession  the  various  series  : 

i,  3,  4>  5,  •  •  •  w>  •  •  •  2> 

i,  3,  5>  6>  -      •  n+l>  •  •  •  2>4> 

i,  3>  5,  7>  •  •  •  w+2>  •  '     2>  4>  6> 

and  so  on.  If  we  imagine  this  process  carried  on  as  long  as 
possible,  we  finally  reach  the  series 

i,  3,5,  7,  .  .  .  2n+i,         .  2,4,6,8,         ..  2n, 
in  which  we  have  first  all  the  odd  numbers  and  then  all  the  even 
numbers. 

The  serial  numbers  of  these  various  series  are  w+i,  co+2, 
w~l~3>  •  •  2aj-  Each  of  these  numbers  is  "  greater"  than  any 
of  its  predecessors,  in  the  following  sense  : — 

One  serial  number  is  said  to  be  "  greater  "  than  another  if 
any  series  having  the  first  number  contains  a  part  having  the 
second  number,  but  no  series  having  the  second  number  contains 
a  part  having  the  first  number. 
If  we  compare  the  two  series 

i,  2,  3,  4,  .      .  n,  .  . 

i,  3,4,5,  .  .  ,  «+i,  .  .     2, 

we  see  that  the  first  is  similar  to  the  part  of  the  second  which 
omits  the  last  term,  namely,  the  number  2,  but  the  second  is 
not  similar  to  any  part  of  the  first.  (This  is  obvious,  but  is 
easily  demonstrated.)  Thus  the  second  series  has  a  greater 
serial  number  than  the  first,  according  to  the  definition — i.e. 
CD+I  is  greater  than  o>.  But  if  we  add  a  term  at  the  beginning 
of  a  progression  instead  of  the  end,  we  still  have  a  progression. 
Thus  I  -\-a>—o}.  Thus  i-fo>  is  not  equal  to  co+i.  This  is 
characteristic  of  relation-arithmetic  generally  :  if  p,  and  v  are 
two  relation-numbers,  the  general  rule  is  that  p+v  is  not  equal 
to  v-\-p,.  The  case  of  finite  ordinals,  in  which  there  is  equality, 
is  quite  exceptional. 

The  series  we  finally  reached  just  now  consisted  of  first  all  the 
odd  numbers  and  then  all  the  even  numbers,  and  its  serial 


Infinite  Series  and  Ordinals  91 

number  is  2o>.  This  number  is  greater  than  o>  or  a)-{-n9  where 
n  is  finite.  It  is  to  be  observed  that,  in  accordance  with  the 
general  definition  of  order,  each  of  these  arrangements  of  integers 
is  to  be  regarded  as  resulting  from  some  definite  relation.  E.g. 
the  qne  which  merely  removes  2  to  the  end  will  be  defined  by 
the  following  relation  :  "  x  and  y  are  finite  integers,  and  either 
y  is  2  and  x  is  not  2,  or  neither  is  2  and  x  is  less  than  y."  The 
one  which  puts  first  all  the  odd  numbers  and  then  all  the  even 
ones  will  be  defined  by  :  "  x  and  y  are  finite  integers,  and  either 
x  is  odd  and  y  is  even  or  x  is  less  than  y  and  both  are  odd  or  both 
are  even."  We  shall  not  trouble,  as  a  rule,  to  give  these  formulae 
in  future  ;  but  the  fact  that  they  could  be  given  is  essential. 

The  number  which  we  have  called  2o>,  namely,  the  number  of 
a  series  consisting  of  two  progressions,  is  sometimes  called  a>  .2. 
Multiplication,  like  addition,  depends  upon  the  order  of  the 
factors  :  a  progression  of  couples  gives  a  series  such  as 


which  is  itself  a  progression  ;  but  a  couple  of  progressions  gives 
a  series  which  is  twice  as  long  as  a  progression.  It  is  therefore 
necessary  to  distinguish  between  2cu  and  to  .  2.  Usage  is  variable  ; 
we  shall  use  2o>  for  a  couple  of  progressions  and  a>  .  2  for  a  pro 
gression  of  couples,  and  this  decision  of  course  governs  our 
general  interpretation  of  "  a  .  )3  "  when  a  and  j3  are  relation- 
numbers  :  "  a  .  j3  "  will  have  to  stand  for  a  suitably  constructed 
sum  of  a  relations  each  having  jS  terms. 

We  can  proceed  indefinitely  with  the  process  of  thinning 
out  the  inductive  numbers.  For  example,  we  can  place  first 
the  odd  numbers,  then  their  doubles,  then  the  doubles  of  these, 
and  so  on.  We  thus  obtain  the  series 

•»  3»  5t  7»  •,•  •  ;  2>   6»  I0>  H>  •  •  •  ;  4>  I2>  20>  28,  .  .  ; 

8,  24,  40,  56,  .  .  ., 

of  which  the  number  is  o>2,  since  it  is  a  progression  of  progressions  . 
Any  one  of  the  progressions  in  this  new  series  can  of  course  be 


92  Introduction  to  Mathematical  Philosophy 

thinned  out  as  we  thinned  out  our  original  progression.  We  can 
proceed  to  o>3,  o>4,  .  „  .  cow,  and  so  on ;  however  far  we  have  gone, 
we  can  always  go  further. 

The  series  of  all  the  ordinals  that  can  be  obtained  in  this  way, 
i.e.  all  that  can  be  obtained  by  thinning  out  a  progression,  is 
itself  longer  than  any  series  that  can  be  obtained  by  re-arranging 
the  terms  of  a  progression.  (This  is  not  difficult  to  prove.) 
The  cardinal  number  of  the  class  of  such  ordinals  can  be  shown 
to  be  greater  than  N0 ;  it  is  the  number  which  Cantor  calls 
Nj.  The  ordinal  number  of  the  series  of  all  ordinals  that  can 
be  made  out  of  an  N0,  taken  in  order  of  magnitude,  is  called  o)v 
Thus  a  series  whose  ordinal  number  is  coj  has  a  field  whose 
cardinal  number  is  Nj. 

We  can  proceed  from  cox  and  Nx  to  co2  and  N2  by  a  process 
exactly  analogous  to  that  by  which  we  advanced  from  w  and  N0 
to  o>!  and  Mx.  And  there  is  nothing  to  prevent  us  from  advancing 
indefinitely  in  this  way  to  new  cardinals  and  new  ordinals.  It 
is  not  known  whether  2^°  is  equal  to  any  of  the  cardinals  in  the 
series  of  Alephs.  It  is  not  even  known  whether  it  is  comparable 
with  them  in  magnitude ;  for  aught  we  know,  it  may  be  neither 
equal  to  nor  greater  nor  less  than  any  one  of  the  Alephs.  This 
question  is  connected  with  the  multiplicative  axiom,  of  which 
we  shall  treat  later. 

All  the  series  we  have  been  considering  so  far  in  this  chapter 
have  been  what  is  called  "well-ordered."  A  well-ordered 
series  is  one  which  has  a  beginning,  and  has  consecutive  terms, 
and  has  a  term  next  after  any  selection  of  its  terms,  provided 
there  are  any  terms  after  the  selection.  This  excludes,  on  the 
one  hand,  compact  series,  in  which  there  are  terms  between 
any  two,  and  on  the  other  hand  series  which  have  no  beginning, 
or  in  which  there  are  subordinate  parts  having  no  beginning. 
The  series  of  negative  integers  in  order  of  magnitude,  having 
no  beginning,  but  ending  with  —  I,  is  not  well-ordered;  but 
taken  in  the  reverse  order,  beginning  with  — I,  it  is  well-ordered, 
being  in  fact  a  progression.  The  definition  is  : 


Infinite  Series  and  Ordinals  93 

A  "  well-ordered "  series  is  one  in  which  every  sub-class 
(except,  of  course,  the  null-class)  has  a  first  term. 

An  "  ordinal "  number  means  the  relation-number  of  a  well- 
ordered  series.  It  is  thus  a  species  of  serial  number. 

Among  well-ordered  series,  a  generalised  form  of  mathematical 
induction  applies.  A  property  may  be  said  to  be  "  transfinitely 
hereditary "  if,  when  it  belongs  to  a  certain  selection  of  the 
terms  in  a  series,  it  belongs  to  their  immediate  successor  pro 
vided  they  have  one.  In  a  well-ordered  series,  a  transfinitely 
hereditary  property  belonging  to  the  first  term  of  the  series 
belongs  to  the  whole  series.  This  makes  it  possible  to  prove 
many  propositions  concerning  well-ordered  series  which  are  not 
true  of  all  series. 

It  is  easy  to  arrange  the  inductive  numbers  in  series  which 
are  not  well-ordered,  and  even  to  arrange  them  in  compact 
series.  For  example,  we  can  adopt  the  following  plan  :  consider 
the  decimals  from  *i  (inclusive)  to  I  (exclusive),  arranged  in  order 
of  magnitude.  These  form  a  compact  series  ;  between  any 
two  there  are  always  an  infinite  number  of  others.  Now  omit 
the  dot  at  the  beginning  of  each,  and  we  have  a  compact  series 
consisting  of  all  finite  integers  except  such  as  divide  by  10.  If 
we  wish  to  include  those  that  divide  by  10,  there  is  no  difficulty  ; 
instead  of  starting  with  *i,  we  will  include  all  decimals  less  than 
I,  but  when  we  remove  the  dot,  we  will  transfer  to  the  right  any 
o's  that  occur  at  the  beginning  of  our  decimal.  Omitting  these, 
and  returning  to  the  ones  that  have  no  o's  at  the  beginning, 
we  can  state  the  rule  for  the  arrangement  of  our  integers  as 
follows  :  Of  two  integers  that  do  not  begin  with  the  same  digit, 
the  one  that  begins  with  the  smaller  digit  comes  first.  Of  two 
that  do  begin  with  the  same  digit,  but  differ  at  the  second  digit, 
the  one  with  the  smaller  second  digit  comes  first,  but  first  of  all 
the  one  with  no  second  digit ;  and  so  on.  Generally,  if  two 
integers  agree  as  regards  the  first  n  digits,  but  not  as  regards 
the  (n-f-i)**,  that  one  comes  first  which  has  either  no  (n+i)th 
digit  or  a  smaller  one  than  the  other.  This  rule  of  arrangement, 


94  Introduction  to  Mathematical  Philosophy 

as  the  reader  can  easily  convince  himself,  gives  rise  to  a  compact 
series  containing  all  the  integers  not  divisible  by  10  ;  and, 
as  we  saw,  there  is  no  difficulty  about  including  those 
that  are  divisible  by  10.  It  follows  from  this  example  that 
it  is  possible  to  construct  compact  series  having  N0  terms. 
In  fact,  we  have  already  seen  that  there  are  N0  ratios,  and 
ratios  in  order  of  magnitude  form  a  compact  series  ;  thus 
we  have  here  another  example.  We  shall  resume  this  topic 
in  the  next  chapter. 

Of  the  usual  formal  laws  of  addition,  multiplication,  and  ex 
ponentiation,  all  are  obeyed  by  transfinite  cardinals,  but  only 
some  are  obeyed  by  transfinite  ordinals,  and  those  that  are  obeyed 
by  them  are  obeyed  by  all  relation-numbers.  By  the  "  usual 
formal  laws  "  we  mean  the  following  :  — 

I.  The  commutative  law  : 

a+jS=j8+a     and     aX0=j8xa. 
II.  The  associative  law  : 

(a+jS)+y=a+(j3-hy)     and     (aXjS)Xy=aX  (£xy). 
III.  The  distributive  law  : 


When  the  commutative  law  does  not  hold,  the  above  form 
of  the  distributive  law  must  be  distinguished  from 


As  we  shall  see  immediately,  one  form  may  be  true  and  the 
other  false. 

IV.  The  laws  of  exponentiation  : 


All  these  laws  hold  for  cardinals,  whether  finite  or  infinite, 
and  {QI  finite  ordinals.  But  when  we  come  to  infinite  ordinals, 
or  indeed  to  relation-numbers  in  general,  some  hold  and  some 
do  not.  The  commutative  law  does  not  hold  ;  the  associative 
law  does  hold  ;  the  distributive  law  (adopting  the  convention 


Infinite  Series  and  Ordinals  95 

we  have  adopted  above  as  regards  the  order  of  the  factors  in  a 
product)  holds  in  the  form 


but  not  in  the  form 

the  exponential  laws 

a?  . 
still  hold,  but  not  the  law 


which  is  obviously  connected  with  the  commutative  law  for 
multiplication. 

The  definitions  of  multiplication  and  exponentiation  that 
are  assumed  in  the  above  propositions  are  somewhat  complicated. 
The  reader  who  wishes  to  know  what  they  are  and  how  the 
above  laws  are  proved  must  consult  the  second  volume  of 
Principia  Mathematics  *  172-176. 

Ordinal  transfinite  arithmetic  was  developed  by  Cantor  at 
an  earlier  stage  than  cardinal  transfinite  arithmetic,  because  it 
has  various  technical  mathematical  uses  which  led  him  to  it. 
But  from  the  point  of  view  of  the  philosophy  of  mathematics 
it  is  less  important  and  less  fundamental  than  the  theory  of 
transfinite  cardinals.  Cardinals  are  essentially  simpler  than 
ordinals,  and  it  is  a  curious  historical  accident  that  they  first 
appeared  as  an  abstraction  from  the  latter,  and  only  gradually 
came  to  be  studied  on  their  own  account.  This  does  not  apply 
to  Frege's  work,  in  which  cardinals,  finite  and  transfinite,  were 
treated  in  complete  independence  of  ordinals  ;  but  it  was 
Cantor's  work  that  made  the  world  aware  of  the  subject,  while 
Frege's  remained  almost  unknown,  probably  in  the  main  on 
account  of  the  difficulty  of  his  symbolism.  And  mathematicians, 
like  other  people,  have  more  difficulty  in  understanding  and 
using  notions  which  are  comparatively  "  simple  "  in  the  logical 
sense  than  in  manipulating  more  complex  notions  which  are 


96  Introduction  to  Mathematical  Philosophy 

more  akin  to  their  ordinary  practice.  For  these  reasons,  it  was 
only  gradually  that  the  true  importance  of  cardinals  in  mathe 
matical  philosophy  was  recognised.  The  importance  of  ordinals, 
though  by  no  means  small,  is  distinctly  less  than  that  of  cardinals, 
and  is  very  largely  merged  in  that  of  the  more  general  conception 
of  relation-numbers. 


CHAPTER  X 

LIMITS    AND    CONTINUITY 

THE  conception  of  a  "  limit "  is  one  of  which  the  importance  in 
mathematics  has  been  found  continually  greater  than  had  been 
thought.  The  whole  of  the  differential  and  integral  calculus, 
indeed  practically  everything  in  higher  mathematics,  depends 
upon  limits.  Formerly,  it  was  supposed  that  infinitesimals  were 
involved  in  the  foundations  of  these  subjects,  but  Weierstrass 
showed  that  this  is  an  error  :  wherever  infinitesimals  were  thought 
to  occur,  what  really  occurs  is  a  set  of  finite  quantities  having 
zero  for  their  lower  limit.  It  used  to  be  thought  that  "  limit " 
was  an  essentially  quantitative  notion,  namely,  the  notion  of  a 
quantity  to  which  others  approached  nearer  and  nearer,  so  that 
among  those  others  there  would  be  some  differing  by  less  than  any 
assigned  quantity.  But  in  fact  the  notion  of  "  limit "  is  a  purely 
ordinal  notion,  not  involving  quantity  at  all  (except  by  accident 
when  the  series  concerned  happens  to  be  quantitative).  A  given 
point  on  a  line  may  be  the  limit  of  a  set  of  points  on  the  line, 
without  its  being  necessary  to  bring  in  co-ordinates  or  measure 
ment  or  anything  quantitative.  The  cardinal  number  N0  is  the 
limit  (in  the  order  of  magnitude)  of  the  cardinal  numbers  I,  2, 
3,  ...«,...,  although  the  numerical  difference  between  NO 
and  a  finite  cardinal  is  constant  and  infinite  :  from  a  quantitative 
point  of  view,  finite  numbers  get  no  nearer  to  N0  as  they  grow 
larger.  What  makes  NO  the  limit  of  the  finite  numbers  is  the 
fact  that,  in  the  series,  it  comes  immediately  after  them,  which 
is  an  ordinal  fact,  not  a  quantitative  fact. 

97  7 


98  Introduction  to  Mathematical  Philosophy 

There  are  various  forms  of  the  notion  of  "  limit,"  of  in 
creasing  complexity.  The  simplest  and  most  fundamental  form, 
from  which  the  rest  are  derived,  has  been  already  defined,  but 
we  will  here  repeat  the  definitions  which  lead  to  it,  in  a  general 
form  in  which  they  do  not  demand  that  the  relation  concerned 
shall  be  serial.  The  definitions  are  as  follows  : — 

The  "  minima  "  of  a  class  a  with  respect  to  a  relation  P  are 
those  members  of  a  and  the  field  of  P  (if  any)  to  which  no  member 
of  a  has  the  relation  P. 

The  "  maxima  "  with  respect  to  P  are  the  minima  with  respect 
to  the  converse  of  P. 

The  "  sequents  "  of  a  class  a  with  respect  to  a  relation  P  are 
the  minima  of  the  "  successors  "  of  a,  and  the  "  successors  "  of 
a  are  those  members  of  the  field  of  P  to  which  every  member  of 
the  common  part  of  a  and  the  field  of  P  has  the  relation  P. 

The  "  precedents  "  with  respect  to  P  are  the  sequents  with 
respect  to  the  converse  of  P. 

The  "  upper  limits  "  of  a  with  respect  to  P  are  the  sequents 
provided  a  has  no  maximum  ;  but  if  a  has  a  maximum,  it  has  no 
upper  limits. 

The  "  lower  limits  "  with  respect  to  P  are  the  upper  limits  with 
respect  to  the  converse  of  P. 

Whenever  P  has  connexity,  a  class  can  have  at  most  one 
maximum,  one  minimum,  one  sequent,  etc.  Thus,  in  the  cases 
we  are  concerned  with  in  practice,  we  can  speak  of  "  the  limit  " 
(if  any). 

When  P  is  a  serial  relation,  we  can  greatly  simplify  the  above 
definition  of  a  limit.  We  can,  in  that  case,  define  first  the 
"  boundary  "  of  a  class  a,  i.e.  its  limits  or  maximum,  and  then 
proceed  to  distinguish  the  case  where  the  boundary  is  the  limit 
from  the  case  where  it  is  a  maximum.  For  this  purpose  it  is 
best  to  use  the  notion  of  "  segment." 

We  will  speak  of  the  "  segment  of  P  defined  by  a  class  a  "  as 
all  those  terms  that  have  the  relation  P  to  some  one  or  more  of 
the  members  of  a.  This  will  be  a  segment  in  the  sense  defined 


Limits  and  Continuity  99 

in  Chapter  VII. ;  indeed^  every  segment  in  the  sense  there  denned 
is  the  segment  defined  by  some  class  a.  If  P  is  serial,  the 
segment  defined  by  a  consists  of  all  the  terms  that  precede 
some  term  or  other  of  a.  If  a  has  a  maximum,  the  segment  will 
be  all  the  predecessors  of  the  maximum.  But  if  a  has  no 
maximum,  every  member  of  a  precedes  some  other  member  of 
a,  and  the  whole  of  a  is  therefore  included  in  the  segment  defined 
by  a.  Take,  for  example,  the  class  consisting  of  the  fractions 

i    i,    I,    if,    •  •  •» 

i.e.  of  all  fractions  of  the  form  I  —  —  for  different  finite  values 

2" 

of  n.  This  series  of  fractions  has  no  maximum,  and  it  is  clear 
that  the  segment  which  it  defines  (in  the  whole  series  of  fractions 
in  order  of  magnitude)  is  the  class  of  all  proper  fractions.  Or, 
again,  consider  the  prime  numbers,  considered  as  a  selection  from 
the  cardinals  (finite  and  infinite)  in  order  of  magnitude.  In  this 
case  the  segment  defined  consists  of  all  finite  integers. 

Assuming  that  P  is  serial,  the  "  boundary  "  of  a  class  a  will  be 
the  term  x  (if  it  exists)  whose  predecessors  are  the  segment 
defined  by  a. 

A  "  maximum  "  of  a  is  a  boundary  which  is  a  member  of  a. 

An  " upper  limit"  of  a  is  a  boundary  which  is  not  a  member  of  cu 

If  a  class  has  no  boundary,  it  has  neither  maximum  nor  limit. 
This  is  the  case  of  an  "  irrational "  Dedekind  cut,  or  of  what  is 
called  a  "  gap." 

Thus  the  "  upper  limit  "  of  a  set  of  terms  a  with  respect  to  a 
series  P  is  that  term  x  (if  it  exists)  which  comes  after  all  the  a's, 
but  is  such  that  every  earlier  term  comes  before  some  of  the  a's. 

We  may  define  all  the  "  upper  limiting-points  "  of  a  set  of 
terms  j3  as  all  those  that  are  the  upper  limits  of  sets  of  terms 
chosen  out  of  j8.  We  shall,  of  course,  have  to  distinguish  upper 
limiting-points  from  lower  limiting-points.  If  we  consider,  for 
example,  the  series  of  ordinal  numbers  : 

I,  2,  3,  ...  CO,  CO-f  I,  .   .   .  2CO,  2CO-H,  ...  3^0,  ...  CD2,  ...  CO3,  ..., 


ioo  Introduction  to  Mathematical  Philosophy 

the  upper  limiting-points  of  the  field  of  this  series  are  those  that 
have  no  immediate  predecessors,  i.e. 

I,  CO,  2CO,  3&>>    •    •    •    &J2>  to>2-\-O),    .    ,    .    2CO2,    *    .    .    CO3    »    .    . 

The  upper  limiting-points  of  the  field  of  this  new  series  will  be 
I,  co2,  2co2,  ...  co3,  co3+co2  .  .  . 

On  the  other  hand,  the  series  of  ordinals — and  indeed  every  well- 
ordered  series — has  no  lower  limiting-points,  because  there  are 
no  terms  except  the  last  that  have  no  immediate  successors.  But 
if  we  consider  such  a  series  as  the  series  of  ratios,  every  member 
of  this  series  is  both  an  upper  and  a  lower  limiting-point  for 
suitably  chosen  sets.  If  we  consider  the  series  of  real  numbers, 
and  select  out  of  it  the  rational  real  numbers,  this  set  (the 
rationals)  will  have  all  the  real  numbers  as  upper  and  lower 
limiting-points.  The  limiting-points  of  a  set  are  called  its  "  first 
derivative,"  and  the  limiting-points  of  the  first  derivative  are 
called  the  second  derivative,  and  so  on. 

With  regard  to  limits,  we  may  distinguish  various  grades  of 
what  may  be  called  "  continuity  "  in  a  series.  The  word  "  con 
tinuity  "  had  been  used  for  a  long  time,  but  had  remained  without 
any  precise  definition  until  the  time  of  Dedekind  and  Cantor. 
Each  of  these  two  men  gave  a  precise  significance  to  the  term, 
but  Cantor's  definition  is  narrower  than  Dedekind's  :  a  series 
which  has  Cantorian  continuity  must  have  Dedekindian  con 
tinuity,  but  the  converse  does  not  hold. 

The  first  definition  that  would  naturally  occur  to  a  man  seeking 
a  precise  meaning  for  the  continuity  of  series  would  be  to  define 
it  as  consisting  in  what  we  have  called  "  compactness,"  i.e.  in  the 
fact  that  between  any  two  terms  of  the  series  there  are  others. 
But  this  would  be  an  inadequate  definition,  because  of  the 
existence  of  "  gaps  "  in  series  such  as  the  series  of  ratios.  We 
saw  in  Chapter  VII.  that  there  are  innumerable  ways  in  which 
the  series  of  ratios  can  be  divided  into  two  parts,  of  which  one 
wholly  precedes  the  other,  and  of  which  the  first  has  no  last  term, 


Limits  and  Continuity  101 

while  the  second  has  no  first  term.  Such  a  state  of  affairs  seems 
contrary  to  the  vague  feeling  we  have  as  to  what  should  character 
ise  "  continuity,"  and,  what  is  more,  it  shows  that  the  series  of 
ratios  is  not  the  sort  of  series  that  is  needed  for  many  mathematical 
purposes.  Take  geometry,  for  example  :  we  wish  to  be  able  to 
say  that  when  two  straight  lines  cross  each  other  they  have  a 
point  in  common,  but  if  the  series  of  points  on  a  line  were  similar 
to  the  series  of  ratios,  the  two  lines  might  cross  in  a  "  gap  "  and 
have  no  point  in  common.  This  is  a  crude  example,  but  many 
others  might  be  given  to  show  that  compactness  is  inadequate  as 
a  mathematical  definition  of  continuity. 

It  was  the  needs  of  geometry,  as  much  as  anything,  that  led 
to  the  definition  of  "  Dedekindian  "  continuity.  It  will  be  re 
membered  that  we  defined  a  series  as  Dedekindian  when  every 
sub-class  of  the  field  has  a  boundary.  (It  is  sufficient  to  assume 
that  there  is  always  an  upper  boundary,  or  that  there  is  always 
a  lower  boundary.  If  one  of  these  is  assumed,  the  other  can  be 
deduced.)  That  is  to  say,  a  series  is  Dedekindian  when  there 
are  no  gaps.  The  absence  of  gaps  may  arise  either  through 
terms  having  successors,  or  through  the  existence  of  limits  in  the 
absence  of  maxima.  Thus  a  finite  series  or  a  well-ordered  series 
is  Dedekindian,  and  so  is  the  series  of  real  numbers.  The  former 
sort  of  Dedekindian  series  is  excluded  by  assuming  that  our 
series  is  compact ;  in  that  case  our  series  must  have  a  property 
which  may,  for  many  purposes,  be  fittingly  called  continuity. 
Thus  we  are  led  to  the  definition  : 

A  series  has  "  Dedekindian  continuity  "  when  it  is  Dedekindian 
and  compact. 

But  this  definition  is  still  too  wide  for  many  purposes.  Suppose, 
for  example,  that  we  desire  to  be  able  to  assign  such  properties 
to  geometrical  space  as  shall  make  it  certain  that  every  point 
can  be  specified  by  means  of  co-ordinates  which  are  real  numbers  : 
this  is  not  insured  by  Dedekindian  continuity  alone.  We  want 
to  be  sure  that  every  point  which  cannot  be  specified  by  rational 
co-ordinates  can  be  specified  as  the  limit  of  a  progression  of  points 


IO2  Introduction  to  Mathematical  Philosophy 

whose  co-ordinates  are  rational,  and  this  is  a  further  property 
which  our  definition  does  not  enable  us  to  deduce. 

We  are  thus  led  to  a  closer  investigation  of  series  with  respect 
to  limits.  This  investigation  was  made  by  Cantor  and  formed 
the  basis  of  his  definition  of  continuity,  although,  in  its  simplest 
form,  this  definition  somewhat  conceals  the  considerations  which 
have  given  rise  to  it.  We  shall,  therefore,  first  travel  through 
some  of  Cantor's  conceptions  in  this  subject  before  giving  his 
definition  of  continuity. 

Cantor  defines  a  series  as  "  perfect "  when  all  its  points  are 
limiting-points  and  all  its  limiting-points  belong  to  it.  But  this 
definition  does  not  express  quite  accurately  what  he  means. 
There  is  no  correction  required  so  far  as  concerns  the  property 
that  all  its  points  are  to  be  limiting-points  ;  this  is  a  property 
belonging  to  compact  series,  and  to  no  others  if  all  points  are  to 
be  upper  limiting-  or  all  lower  limiting-points.  But  if  it  is  only 
assumed  that  they  are  limiting-points  one  way,  without  specify 
ing  which,  there  will  be  other  series  that  will  have  the  property 
in  question — for  example,  the  series  of  decimals  in  which  a  decimal 
ending  in  a  recurring  9  is  distinguished  from  the  corresponding 
terminating  decimal  and  placed  immediately  before  it.  Such  a 
series  is  very  nearly  compact,  but  has  exceptional  terms  which 
are  consecutive,  and  of  which  the  first  has  no  immediate  prede 
cessor,  while  the  second  has  no  immediate  successor.  Apart  from 
such  series,  the  series  in  which  every  point  is  a  limiting-point 
are  compact  series  ;  and  this  holds  without  qualification  if  it  is 
specified  that  every  point  is  to  be  an  upper  limiting-point  (or 
that  every  point  is  to  be  a  lower  limiting-point). 

Although  Cantor  does  not  explicitly  consider  the  matter,  we 
must  distinguish  different  kinds  of  limiting-points  according  to 
the  nature  of  the  smallest  sub-series  by  which  they  can  be  defined. 
Cantor  assumes  that  they  are  to  be  defined  by  progressions,  or 
by  regressions  (which  are  the  converses  of  progressions).  When 
every  member  of  our  series  is  the  limit  of  a  progression  or  regres 
sion,  Cantor  calls  our  series  "  condensed  in  itself  "  (insichdicht). 


Limits  and  Continuity  103 

We  come  now  to  the  second  property  by  which  perfection  was 
to  be  defined,  namely,  the  property  which  Cantor  calls  that  of 
being  "  closed  "  (abgescblosseri).  This,  as  we  saw,  was  first  defined 
as  consisting  in  the  fact  that  all  the  limiting-points  of  a  series 
belong  to  it.  But  this  only  has  any  effective  significance  if  our 
series  is  given  as  contained  in  some  other  larger  series  (as  is  the 
case,  e.g.,  with  a  selection  of  real  numbers),  and  limiting-points 
are  taken  in  relation  to  the  larger  series.  Otherwise,  if  a  series 
is  considered  simply  on  its  own  account,  it  cannot  fail  to  contain 
its  limiting-points.  What  Cantor  means  is  not  exactly  what 
he  says ;  indeed,  on  other  occasions  he  says  something  rather 
different,  which  is  what  he  means.  What  he  really  means  is  that 
every  subordinate  series  which  is  of  the  sort  that  might  be  ex 
pected  to  have  a  limit  does  have  a  limit  within  the  given  series  ; 
i.e.  every  subordinate  series  which  has  no  maximum  has  a  limit, 
i.e.  every  subordinate  series  has  a  boundary.  But  Cantor  does 
not  state  this  for  every  subordinate  series,  but  only  for  progres 
sions  and  regressions.  (It  is  not  clear  how  far  he  recognises  that 
this  is  a  limitation.)  Thus,  finally,  we  find  that  the  definition  we 
want  is  the  following  : — 

A  series  is  said  to  be  "  closed  "  (abgescblossen)  when  every  pro 
gression  or  regression  contained  in  the  series  has  a  limit  in  the 
series. 

We  then  have  the  further  definition  : — 

A  series  is  "  perfect "  when  it  is  condensed  in  itself  and  closed, 
i.e.  when  every  term  is  the  limit  of  a  progression  or  regression, 
and  every  progression  or  regression  contained  in  the  series  has  a 
limit  in  the  series. 

In  seeking  a  definition  of  continuity,  what  Cantor  has  in  mind 
is  the  search  for  a  definition  which  shall  apply  to  the  series  of 
real  numbers  and  to  any  series  similar  to  that,  but  to  no  others. 
For  this  purpose  we  have  to  add  a  further  property.  Among 
the  real  numbers  some  are  rational,  some  are  irrational ;  although 
the  number  of  irrationals  is  greater  than  the  number  of  rationals, 
yet  there  are  rationals  between  any  two  real  numbers,  however 


IO4  Introduction  to  Mathematical  Philosophy 

little  the  two  may  differ.  The  number  of  rationals,  as  we  saw, 
is  >S0.  This  gives  a  further  property  which  suffices  to  characterise 
continuity  completely,  namely,  the  property  of  containing  a  class 
of  N0  members  in  such  a  way  that  some  of  this  class  occur 
between  any  two  terms  of  our  series,  however  near  together. 
This  property,  added  to  perfection,  suffices  to  define  a  class  of 
series  which  are  all  similar  and  are  in  fact  a  serial  number.  This 
class  Cantor  defines  as  that  of  continuous  series. 

We  may  slightly  simplify  his  definition.  To  begin  with, 
we  say  : 

A  "  median  class  "  of  a  series  is  a  sub-class  of  the  field  such 
that  members  of  it  are  to  be  found  between  any  two  terms  of 
the  series. 

Thus  the  rationals  are  a  median  class  in  the  series  of  real 
numbers.  It  is  obvious  that  there  cannot  be  median  classes 
except  in  compact  series. 

We  then  find  that  Cantor's  definition  is  equivalent  to  the 
following : — 

A  series  is  "  continuous  "  when  (i)  it  is  Dedekindian,  (2)  it 
contains  a  median  class  having  N0  terms. 

To  avoid  confusion,  we  shall  speak  of  this  kind  as  "  Cantorian 
continuity."  It  will  be  seen  that  it  implies  Dedekindian  con 
tinuity,  but  the  converse  is  not  the  case.  All  series  having 
Cantorian  continuity  are  similar,  but  not  all  series  having 
Dedekindian  continuity. 

The  notions  of  limit  and  continuity  which  we  have  been  defining 
must  not  be  confounded  with  the  notions  of  the  limit  of  a  function 
for  approaches  to  a  given  argument,  or  the  continuity  of  a  function 
in  the  neighbourhood  of  a  given  argument.  These  are  different 
notions,  very  important,  but  derivative  from  the  above  and  more 
complicated.  The  continuity  of  motion  (if  motion  is  continuous) 
is  an  instance  of  the  continuity  of  a  function  ;  on  the  other  hand, 
the  continuity  of  space  and  time  (if  they  are  continuous)  is  an 
instance  of  the  continuity  of  series,  or  (to  speak  more  cautiously) 
of  a  kind  of  continuity  which  can,  by  sufficient  mathematical 


Limits  and  Continuity  105 

manipulation,  be  reduced  to  the  continuity  of  series.  In  view 
of  the  fundamental  importance  of  motion  in  applied  mathe 
matics,  as  well  as  for  other  reasons,  it  will  be  well  to  deal 
briefly  with  the  notions  of  limits  and  continuity  as  applied 
to  functions ;  but  this  subject  will  be  best  reserved  for  a 
separate  chapter. 

The  definitions  of  continuity  which  we  have  been  considering, 
namely,  those  of  Dedekind  and  Cantor,  do  not  correspond  very 
closely  to  the  vague  idea  which  is  associated  with  the  word  in 
the  mind  of  the  man  in  the  street  or  the  philosopher.  They 
conceive  continuity  rather  as  absence  of  separateness,  the  sort 
of  general  obliteration  of  distinctions  which  characterises  a  thick 
fog.  A  fog  gives  an  impression  of  vastness  without  definite 
multiplicity  or  division.  It  is  this  sort  of  thing  that  a  meta 
physician  means  by  "  continuity,"  declaring  it,  very  truly, 
to  be  characteristic  of  his  mental  life  and  of  that  of  children 
and  animals. 

The  general  idea  vaguely  indicated  by  the  word  "  continuity  " 
when  so  employed,  or  by  the  word  "  flux,"  is  one  which  is  certainly 
quite  different  from  that  which  we  have  been  defining.  Take, 
for  example,  the  series  of  real  numbers.  Each  is  what  it  is, 
quite  definitely  and  uncompromisingly ;  it  does  not  pass  over 
by  imperceptible  degrees  into  another ;  it  is  a  hard,  separate 
unit,  and  its  distance  from  every  other  unit  is  finite,  though 
it  can  be  made  less  than  any  given  finite  amount  assigned  in 
advance.  The  question  of  the  relation  between  the  kind  of 
continuity  existing  among  the  real  numbers  and  the  kind  ex 
hibited,  e.g.  by  what  we  see  at  a  given  time,  is  a  difficult  and 
intricate  one.  It  is  not  to  be  maintained  that  the  two  kinds 
are  simply  identical,  but  it  may,  I  think,  be  very  well  main 
tained  that  the  mathematical  conception  which  we  have  been 
considering  in  this  chapter  gives  the  abstract  logical  scheme  to 
which  it  must  be  possible  to  bring  empirical  material  by  suitable 
manipulation,  if  that  material  is  to  be  called  "  continuous  " 
in  any  precisely  definable  sense.  It  would  be  quite  impossible 


106  Introduction  to  Mathematical  Philosophy 

to  justify  this  thesis  within  the  limits  of  the  present  volume. 
The  reader  who  is  interested  may  read  an  attempt  to  justify 
it  as  regards  time  in  particular  by  the  present  author  in  the 
Monist  for  1914-5,  as  well  as  in  parts  of  Our  Knowledge  of  the 
External  World.  With  these  indications,  we  must  leave  this 
problem,  interesting  as  it  is,  in  order  to  return  to  topics  more 
closely  connected  with  mathematics. 


CHAPTER  XI 

LIMITS    AND    CONTINUITY    OF    FUNCTIONS 

IN  this  chapter  we  shall  be  concerned  with  the  definition  of  the 
limit  of  a  function  (if  any)  as  the  argument  approaches  a  given 
value,  and  also  with  the  definition  of  what  is  meant  by  a  "  con 
tinuous  function."  Both  of  these  ideas  are  somewhat  technical, 
and  would  hardly  demand  treatment  in  a  mere  introduction 
to  mathematical  philosophy  but  for  the  fact  that,  especially 
through  the  so-called  infinitesimal  calculus,  wrong  views  upon 
our  present  topics  have  become  so  firmly  embedded  in  the  minds 
of  professional  philosophers  that  a  prolonged  and  considerable 
effort  is  required  for  their  uprooting.  It  has  been  thought 
ever  since  the  time  of  Leibniz  that  the  differential  and  integral 
calculus  required  infinitesimal  quantities.  Mathematicians 
(especially  Weierstrass)  proved  that  this  is  an  error ;  but  errors 
incorporated,  e.g.  in  what  Hegel  has  to  say  about  mathematics, 
die  hard,  and  philosophers  have  tended  to  ignore  the  work  of 
such  men  as  Weierstrass. 

Limits  and  continuity  of  functions,  in  works  on  ordinary 
mathematics,  are  defined  in  terms  involving  number.  This  is 
not  essential,  as  Dr  Whitehead  has  shown.1  We  will,  however, 
begin  with  the  definitions  in  the  text-books,  and  proceed  after 
wards  to  show  how  these  definitions  can  be  generalised  so  as  to 
apply  to  series  in  general,  and  not  only  to  such  as  are  numerical 
or  numerically  measurable. 

Let  us  consider  any  ordinary  mathematical  function  fx9  where 

1  See  Principia  Mathematica,  vol.  ii.  *  230-234. 
107 


io8  Introduction  to  Mathematical  Philosophy 

x  and/*  are  both  real  numbers,  and  fx  is  one-valued — i.e.  when 
x  is  given,  there  is  only  one  value  that/*  can  have.  We  call  x 
the  "  argument,"  and/*  the  "  value  for  the  argument  *."  When 
a  function  is  what  we  call  "  continuous,"  the  rough  idea  for  which 
we  are  seeking  a  precise  definition  is  that  small  differences  in  * 
shall  correspond  to  small  differences  in/*,  and  if  we  make  the 
differences  in  *  small  enough,  we  can  make  the  differences  in 
/*  fall  below  any  assigned  amount.  We  do  not  want,  if  a  function 
is  to  be  continuous,  that  there  shall  be  sudden  jumps,  so  that, 
for  some  value  of  *,  any  change,  however  small,  will  make  a 
change  in/*  which  exceeds  some  assigned  finite  amount.  The 
ordinary  simple  functions  of  mathematics  have  this  property  : 
it  belongs,  for  example,  to  *2,  *3,  .  .  .  log  *,  sin  *,  and  so  on. 
But  it  is  not  at  all  difficult  to  define  discontinuous  functions. 
Take,  as  a  non-mathematical  example,  "  the  place  of  birth  of 
the  youngest  person  living  at  time  t"  This  is  a  function  of  t ; 
its  value  is  constant  from  the  time  of  one  person's  birth  to  the 
time  of  the  next  birth,  and  then  the  value  changes  suddenly 
from  one  birthplace  to  the  other.  An  analogous  mathematical 
example  would  be  "  the  integer  next  below  *,"  where  x  is  a  real 
number.  This  function  remains  constant  from  one  integer  to 
the  next,  and  then  gives  a  sudden  jump.  The  actual  fact  is 
that,  though  continuous  functions  are  more  familiar,  they  are 
the  exceptions  :  there  are  infinitely  more  discontinuous  functions 
than  continuous  ones. 

Many  functions  are  discontinuous  for  one  or  several  values  of 
the  variable,  but  continuous  for  all  other  values.  Take  as  an 
example  sin  I/*.  The  function  sin  6  passes  through  all  values 
from  — I  to  I  every  time  that  6  passes  from  — 77/2  to  77/2,  or  from 
77/2  to  377/2,  or  generally  from  (2w— 1)77/2  to  (2n-\- 1)77/2,  where 
n  is  any  integer.  Now  if  we  consider  I/*  when  *  is  very  small, 
we  see  that  as  *  diminishes  I/*  grows  faster  and  faster,  so  that 
it  passes  more  and  more  quickly  through  the  cycle  of  values  from 
one  multiple  of  77/2  to  another  as  *  becomes  smaller  and  smaller. 
Consequently  sin  i/x  passes  more  and  more  quickly  from  — I 


Limits  and  Continuity  of  Functions  109 

to  i  and  back  again,  as  x  grows  smaller.  In  fact,  if  we  take 
any  interval  containing  o,  say  the  interval  from  — e  to  -fe  where 
e  is  some  very  small  number,  sin  i/x  will  go  through  an  infioite 
number  of  oscillations  in  this  interval,  and  we  cannot  diminish 
the  oscillations  by  making  the  interval  smaller.  Thus  round 
about  the  argument  o  the  function  is  discontinuous.  It  is  easy 
to  manufacture  functions  which  are  discontinuous  in  several 
places,  or  in  N0  places,  or  everywhere.  Examples  will  be  found 
in  any  book  on  the  theory  of  functions  of  a  real  variable. 

Proceeding  now  to  seek  a  precise  definition  of  what  is  meant 
by  saying  that  a  function  is  continuous  for  a  given  argument, 
when  argument  and  value  are  both  real  numbers,  let  us  first 
define  a  "  neighbourhood  "  of  a  number  x  as  all  the  numbers 
from  x — c  to  #-|-e,  where  e  is  some  number  which,  in  important 
cases,  will  be  very  small.  It  is  clear  that  continuity  at  a  given 
point  has  to  do  with  what  happens  in  any  neighbourhood  of  that 
point,  however  small. 

What  we  desire  is  this  :  If  a  is  the  argument  for  which  we  wish 
our  function  to  be  continuous,  let  us  first  define  a  neighbourhood 
(a  say)  containing  the  value /#  which  the  function  has  for  the 
argument  a  ;  we  desire  that,  if  we  take  a  sufficiently  small 
neighbourhood  containing  a,  all  values  for  arguments  throughout 
this  neighbourhood  shall  be  contained  in  the  neighbourhood  a, 
no  matter  how  small  we  may  have  made  a.  That  is  to  say,  if 
we  decree  that  our  function  is  not  to  differ  from/rf  by  more  than 
some  very  tiny  amount,  we  can  always  find  a  stretch  of  real 
numbers,  having  a  in  the  middle  of  it,  such  that  throughout 
this  stretch  fx  will  not  differ  f rom  fa  by  more  than  the  pre 
scribed  tiny  amount.  And  this  is  to  remain  true  whatever 
tiny  amount  we  may  select.  Hence  we  are  led  to  the  following 
definition : — 

The  function  f(x)  is  said  to  be  "  continuous  "  for  the  argu 
ment  a  if,  for  every  positive  number  CT,  different  from  o,  but  as 
small  as  we  please,  there  exists  a  positive  number  e,  different 
from  o,  such  that,  for  all  values  of  8  which  are  numerically 


no  Introduction  to  Mathematical  Philosophy 

less1  than  e,  the  difference  /(#+§)—/(#)  is  numerically  less 
than  a. 

In  this  definition,  a  first  defines  a  neighbourhood  of  /(#), 
namely,  the  neighbourhood  from/(tf)  —  a  to/(tf)-j-cr.  The  defini 
tion  then  proceeds  to  say  that  we  can  (by  means  of  e)  define  a 
neighbourhood,  namely,  that  from  #— e  to  a-\-e,  such  that,  for 
all  arguments  within  this  neighbourhood,  the  value  of  the  function 
lies  within  the  neighbourhood  horn  f (a)  —  a  tof(a)+cr.  If  this 
can  be  done,  however  cr  may  be  chosen,  the  function  is  "  con 
tinuous  "  for  the  argument  a. 

So  far  we  have  not  defined  the  "  limit "  of  a  function  for  a 
given  argument.  If  we  had  done  so,  we  could  have  defined  the 
continuity  of  a  function  differently  :  a  function  is  continuous 
at  a  point  where  its  value  is  the  same  as  the  limit  of  its  value  for 
approaches  either  from  above  or  from  below.  But  it  is  only 
the  exceptionally  "  tame  "  function  that  has  a  definite  limit  as 
the  argument  approaches  a  given  point.  The  general  rule  is 
that  a  function  oscillates,  and  that,  given  any  neighbourhood 
of  a  given  argument,  however  small,  a  whole  stretch  of  values 
will  occur  for  arguments  within  this  neighbourhood.  As  this 
is  the  general  rule,  let  us  consider  it  first. 

Let  us  consider  what  may  happen  as  the  argument  approaches 
some  value  a  from  below.  That  is  to  say,  we  wish  to  consider 
what  happens  for  arguments  contained  in  the  interval  from 
a — e  to  a,  where  e  is  some  number  which,  in  important  cases, 
will  be  very  small. 

The  values  of  the  function  for  arguments  from  a— e  to  a  (a 
excluded)  will  be  a  set  of  real  numbers  which  will  define  a  certain 
section  of  the  set  of  real  numbers,  namely,  the  section  consisting 
of  those  numbers  that  are  not  greater  than  all  the  values  for 
arguments  from  a— e  to  a.  Given  any  number  in  this  section, 
there  are  values  at  least  as  great  as  this  number  for  arguments 
between  a— e  and  #,  i.e.  for  arguments  that  fall  very  little  short 

1  A  number  is  said  to  be  "  numerically  less  "  than  e  when  it  lies  between 
— e  and  +e. 


Limits  and  Continuity  of  Functions  in 

of  a  (if  c  is  very  small).  Let  us  take  all  possible  e's  and  all 
possible  corresponding  sections.  The  common  part  of  all  these 
sections  we  will  call  the  "  ultimate  section  "  as  the  argument 
approaches  a.  To  say  that  a  number  z  belongs  to  the  ultimate 
section  is  to  say  that,  however  small  we  may  make  e,  there  are 
arguments  between  a— e  and  a  for  which  the  value  of  the  function 
is  not  less  than  z. 

We  may  apply  exactly  the  same  process  to  upper  sections, 
i.e.  to  sections  that  go  from  some  point  up  to  the  top,  instead  of 
from  the  bottom  up  to  some  point.  Here  we  take  those  numbers 
that  are  not  less  than  all  the  values  for  arguments  from  a— e 
to  a ;  this  defines  an  upper  section  which  will  vary  as  e  varies. 
Taking  the  common  part  of  all  such  sections  for  all  possible  e's, 
we  obtain  the  "  ultimate  upper  section."  To  say  that  a  number 
z  belongs  to  the  ultimate  upper  section  is  to  say  that,  however 
small  we  make  e,  there  are  arguments  between  a — e  and  a  for 
which  the  value  of  the  function  is  not  greater  than  z. 

If  a  term  z  belongs  both  to  the  ultimate  section  and  to  the 
ultimate  upper  section,  we  shall  say  that  it  belongs  to  the 
"  ultimate  oscillation."  We  may  illustrate  the  matter  by  con 
sidering  once  more  the  function  sin  i/x  as  x  approaches  the 
value  o.  We  shall  assume,  in  order  to  fit  in  with  the  above 
definitions,  that  this  value  is  approached  from  below. 

Let  us  begin  with  the  "  ultimate  section."  Between  —  e 
and  o,  whatever  e  may  be,  the  function  will  assume  the  value 
I  for  certain  arguments,  but  will  never  assume  any  greater  value. 
Hence  the  ultimate  section  consists  of  all  real  numbers,  positive 
and  negative,  up  to  and  including  I  ;  i.e.  it  consists  of  all  negative 
numbers  together  with  o,  together  with  the  positive  numbers 
up  to  and  including  I. 

Similarly  the  "  ultimate  upper  section  "  consists  of  all  positive 
numbers  together  with  o,  together  with  the  negative  numbers 
down  to  and  including  — I. 

Thus  the  "  ultimate  oscillation  "  consists  of  all  real  numbers 
from  —I  to  i,  both  included. 


112  Introduction  to  Mathematical  Philosophy 

We  may  say  generally  that  the  "  ultimate  oscillation "  of 
a  function  as  the  argument  approaches  a  from  below  consists 
of  all  those  numbers  x  which  are  such  that,  however  near  we 
come  to  ay  we  shall  still  find  values  as  great  as  x  and  values  as 
small  as  x. 

The  ultimate  oscillation  may  contain  no  terms,  or  one  term, 
or  many  terms.  In  the  first  two  cases  the  function  has  a  definite 
limit  for  approaches  from  below.  If  the  ultimate  oscillation 
has  one  term,  this  is  fairly  obvious.  It  is  equally  true  if  it  has 
none ;  for  it  is  not  difficult  to  prove  that,  if  the  ultimate  oscilla 
tion  is  null,  the  boundary  of  the  ultimate  section  is  the  same  as 
that  of  the  ultimate  upper  section,  and  may  be  defined  as  the 
limit  of  the  function  for  approaches  from  below.  But  if  the 
ultimate  oscillation  has  many  terms,  there  is  no  definite  limit  to 
the  function  for  approaches  from  below.  In  this  case  we  can 
take  the  lower  and  upper  boundaries  of  the  ultimate  oscillation 
(i.e.  the  lower  boundary  of  the  ultimate  upper  section  and  the 
upper  boundary  of  the  ultimate  section)  as  the  lower  and  upper 
limits  of  its  "  ultimate "  values  for  approaches  from  below. 
Similarly  we  obtain  lower  and  upper  limits  of  the  "  ultimate  " 
values  for  approaches  from  above.  Thus  we  have,  in  the  general 
case,/owr  limits  to  a  function  for  approaches  to  a  given  argument. 
The  limit  for  a  given  argument  a  only  exists  when  all  these  four 
are  equal,  and  is  then  their  common  value.  If  it  is  also  the 
value  for  the  argument  a,  the  function  is  continuous  for  this 
argument.  This  may  be  taken  as  defining  continuity :  it  is 
equivalent  to  our  former  definition. 

We  can  define  the  limit  of  a  function  for  a  given  argument 
(if  it  exists)  without  passing  through  the  ultimate  oscillation 
and  the  four  limits  of  the  general  case.  The  definition  proceeds, 
in  that  case,  just  as  the  earlier  definition  of  continuity  proceeded. 
Let  us  define  the  limit  for  approaches  from  below.  If  there  is  to 
be  a  definite  limit  for  approaches  to  a  from  below,  it  is  necessary 
and  sufficient  that,  given  any  small  number  cr,  two  values  for 
arguments  sufficiently  near  to  a  (but  both  less  than  a)  will  differ 


Limits  and  Continuity  of  Functions  113 

by  less  than  cr ;  i.e.  if  e  is  sufficiently  small,  and  our  arguments 
both  lie  between  a— e  and  a  (a  excluded),  then  the  difference 
between  the  values  for  these  arguments  will  be  less  than  cr. 
This  is  to  hold  for  any  cr,  however  small ;  in  that  case  the 
function  has  a  limit  for  approaches  from  below.  Similarly 
we  define  the  case  when  there  is  a  limit  for  approaches  from 
above.  These  two  limits,  even  when  both  exist,  need  not  be 
identical ;  and  if  they  are  identical,  they  still  need  not  be  identical 
with  the  value  for  the  argument  a.  It  is  only  in  this  last  case 
that  we  call  the  function  continuous  for  the  argument  a. 

A  function  is  called  "  continuous "  (without  qualification) 
when  it  is  continuous  for  every  argument. 

Another  slightly  different  method  of  reaching  the  definition 
of  continuity  is  the  following  : — 

Let  us  say  that  a  function  "  ultimately  converges  into  a 
class  a  "  if  there  is  some  real  number  such  that,  for  this  argument 
and  all  arguments  greater  than  this,  the  value  of  the  function 
is  a  member  of  the  class  a.  Similarly  we  shall  say  that  a  function 
"  converges  into  a  as  the  argument  approaches  x  from  below  " 
if  there  is  some  argument  y  less  than  x  such  that  throughout 
the  interval  from  y  (included)  to  x  (excluded)  the  function  has 
values  which  are  members  of  a.  We  may  now  say  that  a 
function  is  continuous  for  the  argument  a,  for  which  it  has  the 
value  fa,  if  it  satisfies  four  conditions,  namely  : — 

(1)  Given  any  real  number  less  than  /#,  the  function  con 
verges   into   the   successors   of   this   number   as   the   argument 
approaches  a  from  below  ; 

(2)  Given  any  real  number  greater  than  /£,  the  function  con 
verges  into  the  predecessors  of  this  number  as  the  argument 
approaches  a  from  below  ; 

(3)  and  (4)  Similar  conditions  for  approaches  to  a  from  above. 
The  advantages  of  this  form  of  definition  is  that  it  analyses 

the  conditions  of  continuity  into  four,  derived  from  considering 
arguments  and  values  respectively  greater  or  less  than  the 
argument  and  value  for  which  continuity  is  to  be  defined. 

8 


H4  Introduction  to  Mathematical  Philosophy 

We  may  now  generalise  our  definitions  so  as  to  apply  to  series 
which  are  not  numerical  or  known  to  be  numerically  measurable. 
The  case  of  motion  is  a  convenient  one  to  bear  in  mind.  There 
is  a  story  by  H.  G.  Wells  which  will  illustrate,  from  the  case  of 
motion,  the  difference  between  the  limit  of  a  function  for  a  given 
argument  and  its  value  for  the  same  argument.  The  hero  of 
the  story,  who  possessed,  without  his  knowledge,  the  power  of 
realising  his  wishes,  was  being  attacked  by  a  policeman,  but  on 

ejaculating  "Go  to "  he  found  that  the  policeman  disappeared. 

If  f(t)  was  the  policeman's  position  at  time  t,  and  t0  the  moment 
of  the  ejaculation,  the  limit  of  the  policeman's  positions  as  t 
approached  to  t0  from  below  would  be  in  contact  with  the  hero, 
whereas  the  value  for  the  argument  t0  was  — .  But  such  occur 
rences  are  supposed  to  be  rare  in  the  real  world,  and  it  is  assumed, 
though  without  adequate  evidence,  that  all  motions  are  continu 
ous,  i.e.  that,  given  any  body,  if /(*)  is  its  position  at  time  t,f(t) 
is  a  continuous  function  of  t.  It  is  the  meaning  of  "  continuity  " 
involved  in  such  statements  which  we  now  wish  to  define  as 
simply  as  possible. 

The  definitions  given  for  the  case  of  functions  where  argument 
and  value  are  real  numbers  can  readily  be  adapted  for  more 
general  use. 

Let  P  and  Q  be  two  relations,  which  it  is  well  to  imagine 
serial,  though  it  is  not  necessary  to  our  definitions  that  they 
should  be  so.  Let  R  be  a  one-many  relation  whose  domain 
is  contained  in  the  field  of  P,  while  its  converse  domain  is  con 
tained  in  the  field  of  Q.  Then  R  is  (in  a  generalised  sense)  a 
function,  whose  arguments  belong  to  the  field  of  Q,  while  its 
values  belong  to  the  field  of  P.  Suppose,  for  example,  that  we 
are  dealing  with  a  particle  moving  on  a  line :  let  Q  be  the  time- 
series,  P  the  series  of  points  on  our  line  from  left  to  right,  R  the 
relation  of  the  position  of  our  particle  on  the  line  at  time  a  to 
the  time  a,  so  that  "  the  R  of  a  "  is  its  position  at  time  a.  This 
illustration  may  be  borne  in  mind  throughout  our  definitions. 

We  shall  say  that  the  function  R  is  continuous  for  the  argument 


Limits  and  Continuity  of  Functions  115 

a  if,  given  any  interval  a  on  the  P-series  containing  the  value 
of  the  function  for  the  argument  #,  there  is  an  interval  on  the 
Q-series  containing  a  not  as  an  end-point  and  such  that,  through 
out  this  interval,  the  function  has  values  which  are  members 
of  a.  (We  mean  by  an  "  interval  "  all  the  terms  between  any 
two  ;  i.e.  if  x  and  y  are  two  members  of  the  field  of  P,  and  x  has 
the  relation  P  to  y,  we  shall  mean  by  the  "  P-interval  x  to  y  " 
all  terms  z  such  that  x  has  the  relation  P  to  x  and  z  has  the  rela 
tion  P  to  y — together,  when  so  stated,  with  x  or  y  themselves.) 

We  can  easily  define  the  "  ultimate  section  "  and  the  "  ulti 
mate  oscillation."  To  define  the  "  ultimate  section "  for 
approaches  to  the  argument  a  from  below,  take  any  argument 
y  which  precedes  a  (i.e.  has  the  relation  Q  to  a),  take  the  values 
of  the  function  for  all  arguments  up  to  and  including  y,  and 
form  the  section  of  P  defined  by  these  values,  i.e.  those  members 
of  the  P-series  which  are  earlier  than  or  identical  with  some  of 
these  values.  Form  all  such  sections  for  all  y's  that  precede  a, 
and  take  their  common  part ;  this  will  be  the  ultimate  section. 
The  ultimate  upper  section  and  the  ultimate  oscillation  are  then 
defined  exactly  as  in  the  previous  case. 

The  adaptation  of  the  definition  of  convergence  and  the 
resulting  alternative  definition  of  continuity  offers  no  difficulty 
of  any  kind. 

We  say  that  a  function  R  is  "  ultimately  Q-convergent  into 
a  "  if  there  is  a  member  y  of  the  converse  domain  of  R  and  the 
field  of  Q  such  that  the  value  of  the  function  for  the  argument 
y  and  for  any  argument  to  which  y  has  the  relation  Q  is  a  member 
of  a.  We  say  that  R  "  Q-converges  into  a  as  the  argument 
approaches  a  given  argument  a  "  if  there  is  a  term  y  having 
the  relation  Q  to  a  and  belonging  to  the  converse  domain  of  R 
and  such  that  the  value  of  the  function  for  any  argument  in  the 
Q-interval  from  y  (inclusive)  to  a  (exclusive)  belongs  to  a. 

Of  the  four  conditions  that  a  function  must  fulfil  in  order 
to  be  continuous  for  the  argument  a,  the  first  is,  putting  b  for 
the  value  for  the  argument  a  : 


1 1 6  Introduction  to  Mathematical  Philosophy 

Given  any  term  having  the  relation  P  to  b,  R  Q-converges 
into  the  successors  of  b  (with  respect  to  P)  as  the  argument 
approaches  a  from  below. 

The  second  condition  is  obtained  by  replacing  P  by  its 
converse ;  the  third  and  fourth  are  obtained  from  the  first  and 
second  by  replacing  Q  by  its  converse. 

There  is  thus  nothing,  in  the  notions  of  the  limit  of  a  function 
or  the  continuity  of  a  function,  that  essentially  involves  number. 
Both  can  be  defined  generally,  and  many  propositions  about 
them  can  be  proved  for  any  two  series  (one  being  the  argument- 
series  and  the  other  the  value-series).  It  will  be  seen  that  the 
definitions  do  not  involve  infinitesimals.  They  involve  infinite 
classes  of  intervals,  growing  smaller  without  any  limit  short  of 
zero,  but  they  do  not  involve  any  intervals  that  are  not  finite. 
This  is  analogous  to  the  fact  that  if  a  line  an  inch  long  be  halved, 
then  halved  again,  and  so  on  indefinitely,  we  never  reach  infini 
tesimals  in  this  'way  :  after  n  bisections,  the  length  of  our  bit  is 

—  of  an  inch ;   and  this  is  finite  whatever  finite  number  n  may 

2n 

be.  The  process  of  successive  bisection  does  not  lead  to 
divisions  whose  ordinal  number  is  infinite,  since  it  is  essentially 
a  one-by-one  process.  Thus  infinitesimals  are  not  to  be  reached 
in  this  way.  Confusions  on  such  topics  have  had  much  to  do 
with  the  difficulties  which  have  been  found  in  the  discussion  of 
infinity  and  continuity. 


CHAPTER  XII 

SELECTIONS    AND   THE   MULTIPLICATIVE    AXIOM 

IN  this  chapter  we  have  to  consider  an  axiom  which  can  be 
enunciated,  but  not  proved,  in  terms  of  logic,  and  which  is  con 
venient,  though  not  indispensable,  in  certain  portions  of  mathe 
matics.  It  is  convenient,  in  the  sense  that  many  interesting 
propositions,  which  it  seems  natural  to  suppose  true,  cannot 
be  proved  without  its  help  ;  but  it  is  not  indispensable,  because 
even  without  those  propositions  the  subjects  in  which  they 
occur  still  exist,  though  in  a  somewhat  mutilated  form. 

Before  enunciating  the  multiplicative  axiom,  we  must  first 
explain  the  theory  of  selections,  and  the  definition  of  multi 
plication  when  the  number  of  factors  may  be  infinite. 

In  defining  the  arithmetical  operations,  the  only  correct  pro 
cedure  is  to  construct  an  actual  class  (or  relation,  in  the  case 
of  relation-numbers)  having  the  required  number  of  terms. 
This  sometimes  demands  a  certain  amount  of  ingenuity,  but 
it  is  essential  in  order  to  prove  the  existence  of  the  number 
defined.  Take,  as  the  simplest  example,  the  case  of  addition. 
Suppose  we  are  given  a  cardinal  number  ^,  and  a  class  a  which 
has  fji  terms.  How  shall  we  define  ju.+/z  ?  For  this  purpose 
we  must  have  two  classes  having  //,  terms,  and  they  must  not 
overlap.  We  can  construct  such  classes  from  a  in  various  ways, 
of  which  the  following  is  perhaps  the  simplest :  Form  first  all 
the  ordered  couples  whose  first  term  is  a  class  consisting  of  a 
single  member  of  a,  and  whose  second  term  is  the  null-class ; 
then,  secondly,  form  all  the  ordered  couples  whose  first  term  is 

117 


n8  Introduction  to  Mathematical  Philosophy 

the  null-class  and  whose  second  term  is  a  class  consisting  of  a 
single  member  of  a.  These  two  classes  of  couples  have  no 
member  in  common,  and  the  logical  sum  of  the  two  classes  will 
have  /z-f/*-  terms.  Exactly  analogously  we  can  define  p,-\-v, 
given  that  /z,  is  the  number  of  some  class  a  and  v  is  the  number 
of  some  class  j3. 

Such  definitions,  as  a  rule,  are  merely  a  question  of  a  suitable 
technical  device.  But  in  the  case  of  multiplication,  where  the 
number  of  factors  may  be  infinite,  important  problems  arise  out 
of  the  definition. 

Multiplication  when  the  number  of  factors  is  finite  offers  no 
difficulty.  Given  two  classes  a  and  j8,  of  which  the  first  has 
ju,  terms  and  the  second  v  terms,  we  can  define  fix  v  as  the  number 
of  ordered  couples  that  can  be  formed  by  choosing  the  first  term 
out  of  a  and  the  second  out  of  ]3.  It  will  be  seen  that  this  de 
finition  does  not  require  that  a  and  j3  should  not  overlap  ;  it 
even  remains  adequate  when  a  and  jS  are  identical.  For  example, 
let  a  be  the  class  whose  members  are  xl9  #2,  #3.  Then  the  class 
which  is  used  to  define  the  product  /x  X  p,  is  the  class  of  couples  : 

(*i,  *i),  (*i,  *a)>  (*i>  *B)  5  (**>  *i)»  (*2>  *2)>  (*2>  *a)  >  (*3>  *i), 
(#3>  *s)»  (*s>  *a)« 

This  definition  remains  applicable  when  \i  or  v  or  both  are 
infinite,  and  it  can  be  extended  step  by  step  to  three  or  four  or 
any  finite  number  of  factors.  No  difficulty  arises  as  regards 
this  definition,  except  that  it  cannot  be  extended  to  an  infinite 
number  of  factors. 

The  problem  of  multiplication  when  the  number  of  factors 
may  be  infinite  arises  in  this  way  :  Suppose  we  have  a  class  K 
consisting  of  classes  ;  suppose  the  number  of  terms  in  each  of 
these  classes  is  given.  How  shall  we  define  the  product  of  all 
these  numbers  ?  If  we  can  frame  our  definition  generally,  it 
will  be  applicable  whether  K  is  finite  or  infinite.  It  is  to  be 
observed  that  the  problem  is  to  be  able  to  deal  with  the  case 
when  K  is  infinite,  not  with  the  case  when  its  members  are.  If 


Selections  and  the  Multiplicative  Axiom  119 

K  is  not  infinite,  the  method  defined  above  is  just  as  applicable 
when  its  members  are  infinite  as  when  they  are  finite.  It  is 
the  case  when  K  is  infinite,  even  though  its  members  may  be 
finite,  that  we  have  to  find  a  way  of  dealing  with. 

The  following  method  of  defining  multiplication  generally  is 
due  to  Dr  Whitehead.  It  is  explained  and  treated  at  length  in 
Principia  Mathematics*,  vol.  i.  *  80  ff.,  and  vol.  ii.  *  114. 

Let  us  suppose  to  begin  with  that  K  is  a  class  of  classes  no  two 
of  which  overlap — say  the  constituencies  in  a  country  where 
there  is  no  plural  voting,  each  constituency  being  considered 
as  a  class  of  voters.  Let  us  now  set  to  work  to  choose  one  term 
out  of  each  class  to  be  its  representative,  as  constituencies  do 
when  they  elect  members  of  Parliament,  assuming  that  by  law 
each  constituency  has  to  elect  a  man  who  is  a  voter  in  that 
constituency.  We  thus  arrive  at  a  class  of  representatives,  who 
make  up  our  Parliament,  one  being  selected  out  of  each  con 
stituency.  How  many  different  possible  ways  of  choosing  a 
Parliament  are  there  ?  Each  constituency  can  select  any  one 
of  its  voters,  and  therefore  if  there  are  p  voters  in  a  constituency, 
it  can  make  JLC  choices.  The  choices  of  the  different  constituencies 
are  independent ;  thus  it  is  obvious  that,  when  the  total  number 
of  constituencies  is  finite,  the  number  of  possible  Parliaments 
is  obtained  by  multiplying  together  the  numbers  of  voters  in  the 
various  constituencies.  When  we  do  not  know  whether  the 
number  of  constituencies  is  finite  or  infinite,  we  may  take  the 
number  of  possible  Parliaments  as  defining  the  product  of  the 
numbers  of  the  separate  constituencies.  This  is  the  method 
by  which  infinite  products  are  defined.  We  must  now  drop  our 
illustration,  and  proceed  to  exact  statements. 

Let  K  be  a  class  of  classes,  and  let  us  assume  to  begin  with  that 
no  two  members  of  ic  overlap,  i.e.  that  if  a  and  j3  are  two  different 
members  of  K,  then  no  member  of  the  one  is  a  member  of  the 
other.  We  shall  call  a  class  a  "  selection  "  from  K  when  it  con 
sists  of  just  one  term  from  each  member  of  K  ;  i.e.  p,  is  a  "  selec 
tion  "  from  K  if  every  member  of  JJL  belongs  to  some  member 


I2O  Introduction  to  Mathematical  Philosophy 

of  K,  and  if  a  be  any  member  of  K,  /i  and  a  have  exactly  one  term 
in  common.  The  class  of  all  "  selections  "  from  K  we  shall  call 
the  "  multiplicative  class  "  of  K.  The  number  of  terms  in  the 
multiplicative  class  of  /c,  i.e.  the  number  of  possible  selections 
from  K,  is  defined  as  the  product  of  the  numbers  of  the  members 
of  K.  This  definition  is  equally  applicable  whether  K  is  finite 
or  infinite. 

Before  we  can  be  wholly  satisfied  with  these  definitions,  we 
must  remove  the  restriction  that  no  two  members  of  K  are  to 
overlap.  For  this  purpose,  instead  of  defining  first  a  class 
called  a  "  selection,"  we  will  define  first  a  relation  which  we  will 
call  a  "  selector."  A  relation  R  will  be  called  a  "  selector " 
from  K  if,  from  every  member  of  /c,  it  picks  out  one  term  as  the 
representative  of  that  member,  i.e.  if,  given  any  member  a  of  /c, 
there  is  just  one  term  x  which  is  a  member  of  a  and  has  the 
relation  R  to  a ;  and  this  is  to  be  all  that  R  does.  The  formal 
definition  is  : 

A  "  selector  "  from  a  class  of  classes  K  is  a  one-many  relation, 
having  K  for  its  converse  domain,  and  such  that,  if  x  has  the 
relation  to  a,  then  x  is  a  member  of  a. 

If  R  is  a  selector  from  /c,  and  a  is  a  member  of  K,  and  x  is  the 
term  which  has  the  relation  R  to  a,  we  call  x  the  "  representative  " 
of  a  in  respect  of  the  relation  R. 

A  "  selection  "  from  K  will  now  be  defined  as  the  domain  of  a 
selector  ;  and  the  multiplicative  class,  as  before,  will  be  the  class 
of  selections. 

But  when  the  members  of  K  overlap,  there  may  be  more  selectors 
than  selections,  since  a  term  x  which  belongs  to  two  classes  a 
and  j8  may  be  selected  once  to  represent  a  and  once  to  represent  j3, 
giving  rise  to  different  selectors  in  the  two  cases,  but  to  the  same 
selection.  For  purposes  of  defining  multiplication,  it  is  the 
selectors  we  require  rather  than  the  selections.  Thus  we  define  : 

"  The  product  of  the  numbers  of  the  members  of  a  class  of 
classes  K  "  is  the  number  of  selectors  from  /c. 

We  can  define  exponentiation  by  an  adaptation  of  the  above 


Selections  and  the  Multiplicative  Axiom  121 

plan.  We  might,  of  course,  define  /A"  as  the  number  of  selectors 
from  v  classes,  each  of  which  has  ju,  terms.  But  there  are 
objections  to  this  definition,  derived  from  the  fact  that  the 
multiplicative  axiom  (of  which  we  shall  speak  shortly)  is  unneces 
sarily  involved  if  it  is  adopted.  We  adopt  instead  the  following 
construction  : — 

Let  a  be  a  class  having  (JL  terms,  and  j3  a  class  having  v  terms. 

Let  y  be  a  member  of  j3,  and  form  the  class  of  all  ordered 
couples  that  have  y  for  their  second  term  and  a  member  of  a  for 
their  first  term.  There  will  be  p  such  couples  for  a  given  y,  since 
any  member  of  a  may  be  chosen  for  the  first  term,  and  a  has  /z 
members.  If  we  now  form  all  the  classes  of  this  sort  that  result 
from  varying  y,  we  obtain  altogether  v  classes,  since  y  may  be 
any  member  of  j8,  and  j8  has  v  members.  These  v  classes  are  each 
of  them  a  class  of  couples,  namely,  all  the  couples  that  can  be 
formed  of  a  variable  member  of  a  and  a  fixed  member  of  j8.  We 
define  \LV  as  the  number  of  selectors  from  the  class  consisting  of 
these  v  classes.  Or  we  may  equally  well  define  ju,"  as  the  number  of 
selections,  for,  since  our  classes  of  couples  are  mutually  exclusive, 
the  number  of  selectors  is  the  same  as  the  number  of  selections. 
A  selection  from  our  class  of  classes  will  be  a  set  of  ordered  couples, 
of  which  there  will  be  exactly  one  having  any  given  member  of  jS 
for  its  second  term,  and  the  first  term  may  be  any  member  of  a. 
Thus  ju,"  is  defined  by  the  selectors  from  a  certain  set  of  v  classes 
each  having  p,  terms,  but  the  set  is  one  having  a  certain  structure 
and  a  more  manageable  composition  than  is  the  case  in  general. 
The  relevance  of  this  to  the  multiplicative  axiom  will  appear 
shortly. 

What  applies  to  exponentiation  applies  also  to  the  product  of 
two  cardinals.  We  might  define  "jz.Xi'"  as  the  sum  of  the 
numbers  of  v  classes  each  having  JJL  terms,  but  we  prefer  to  define 
it  as  the  number  of  ordered  couples  to  be  formed  consisting  of  a 
member  of  a  followed  by  a  member  of  j5,  where  a  has  ^  terms 
and  j8  has  v  terms.  This  definition,  also,  is  designed  to  evade  the 
necessity  of  assuming  the  multiplicative  axiom. 


122  Introduction  to  Mathematical  Philosophy 

With  our  definitions,  we  can  prove  the  usual  formal  laws  of 
multiplication  and  exponentiation.  But  there  is  one  thing  we 
cannot  prove  :  we  cannot  prove  that  a  product  is  only  zero  when 
one  of  its  factors  is  zero.  We  can  prove  this  when  the  number 
of  factors  is  finite,  but  not  when  it  is  infinite.  In  other  words, 
we  cannot  prove  that,  given  a  class  of  classes  none  of  which  is 
null,  there  must  be  selectors  from  them ;  or  that,  given  a  class 
of  mutually  exclusive  classes,  there  must  be  at  least  one  class 
consisting  of  one  term  out  of  each  of  the  given  classes.  These 
things  cannot  be  proved  ;  and  although,  at  first  sight,  they  seem 
obviously  true,  yet  reflection  brings  gradually  increasing  doubt, 
until  at  last  we  become  content  to  register  the  assumption  and 
its  consequences,  as  we  register  the  axiom  of  parallels,  without 
assuming  that  we  can  know  whether  it  is  true  or  false.  The 
assumption,  loosely  worded,  is  that  selectors  and  selections  exist 
when  we  should  expect  them.  There  are  many  equivalent  ways 
of  stating  it  precisely.  We  may  begin  with  the  following  : — 

"  Given  any  class  of  mutually  exclusive  classes,  of  which  none 
is  null,  there  is  at  least  one  class  which  has  exactly  one  term  in 
common  with  each  of  the  given  classes." 

This  proposition  we  will  call  the  "  multiplicative  axiom."  1 
We  will  first  give  various  equivalent  forms  of  the  proposition, 
and  then  consider  certain  ways  in  which  its  truth  or  falsehood 
is  of  interest  to  mathematics. 

The  multiplicative  axiom  is  equivalent  to  the  proposition  that 
a  product  is  only  zero  when  at  least  one  of  its  factors  is  zero  ; 
i.e.  that,  if  any  number  of  cardinal  numbers  be  multiplied  together, 
the  result  cannot  be  o  unless  one  of  the  numbers  concerned  is  o. 

The  multiplicative  axiom  is  equivalent  to  the  proposition  that, 
if  R  be  any  relation,  and  K  any  class  contained  in  the  converse 
domain  of  R,  then  there  is  at  least  one  one-many  relation  implying 
R  and  having  K  for  its  converse  domain. 

The  multiplicative  axiom  is  equivalent  to  the  assumption  that 
if  a  be  any  class,  and  K  all  the  sub-classes  of  a  with  the  exception 

1  See  Principia  Mathematica,  vol.  i.  *  88.     Also  vol.  iii.  *  257-258. 


Selections  and  the  Multiplicative  Axiom  123 

of  the  null-class,  then  there  is  at  least  one  selector  from  K.  This 
is  the  form  in  which  the  axiom  was  first  brought  to  the  notice  of 
the  learned  world  by  Zermelo,  in  his  "  Beweis,  dass  jede  Menge 
wohlgeordnet  werden  kann."  1  Zermelo  regards  the  axiom  as  an 
unquestionable  truth.  It  must  be  confessed  that,  until  he  made 
it  explicit,  mathematicians  had  used  it  without  a  qualm  ;  but  it 
would  seem  that  they  had  done  so  unconsciously.  And  the  credit 
due  to  Zermelo  for  having  made  it  explicit  is  entirely  independent 
of  the  question  whether  it  is  true  or  false. 

The  multiplicative  axiom  has  been  shown  by  Zermelo,  in  the 
above-mentioned  proof,  to  be  equivalent  to  the  proposition  that 
every  class  can  be  well-ordered,  i.e.  can  be  arranged  in  a  series  in 
which  every  sub-class  has  a  first  term  (except,  of  course,  the  null- 
class).  The  full  proof  of  this  proposition  is  difficult,  but  it  is  not 
difficult  to  see  the  general  principle  upon  which  it  proceeds.  It 
uses  the  form  which  we  call  "  Zermelo's  axiom,"  i.e.  it  assumes 
that,  given  any  class  a,  there  is  at  least  one  one-many  relation  R 
whose  converse  domain  consists  of  all  existent  sub-classes  of  a 
and  which  is  such  that,  if  x  has  the  relation  R  to  f ,  then  x  is  a 
member  of  f .  Such  a  relation  picks  out  a  "  representative  " 
from  each  sub-class  ;  of  course,  it  will  often  happen  that  two 
sub-classes  have  the  same  representative.  What  Zermelo  does, 
in  effect,  is  to  count  off  the  members  of  a,  one  by  one,  by  means 
of  R  and  transfinite  induction.  We  put  first  the  representative 
of  a;  call  it  xr  Then  take  the  representative  of  the  class  consisting 
of  all  of  a  except  x1 ;  call  it  x2.  It  must  be  different  from  xl9 
because  every  representative  is  a  member  of  its  class,  and  x±  is 
shut  out  from  this  class.  Proceed  similarly  to  take  away  X2,  and 
let  #3  be  the  representative  of  what  is  left.  In  this  way  we  first 
obtain  a  progression  x^  X2,  .  .  .  xm  .  .  .,  assuming  that  a  is  not 
finite.  We  then  take  away  the  whole  progression  ;  let  #w  be  the 
representative  of  what  is  left  of  a.  In  this  way  we  can  go  on 
until  nothing  is  left.  The  successive  representatives  will  form  a 

1  Mathematische  Annalen,  vol.  lix.  pp.  514-6.  In  this  form  we  shall 
speak  of  it  as  Zermelo's  axiom. 


124  Introduction  to  Mathematical  Philosophy 

well-ordered  series  containing  all  the  members  of  a.  (The  above 
is,  of  course,  only  a  hint  of  the  general  lines  of  the  proof.)  This 
proposition  is  called  "  Zermelo's  theorem." 

The  multiplicative  axiom  is  also  equivalent  to  the  assumption 
that  of  any  two  cardinals  which  are  not  equal,  one  must  be  the 
greater.  If  the  axiom  is  false,  there  will  be  cardinals  p  and  v 
such  that  ju-  is  neither  less  than,  equal  to,  nor  greater  than  v.  We 
have  seen  that  Nj  and  2No  possibly  form  an  instance  of  such  a  pair. 

Many  other  forms  of  the  axiom  might  be  given,  but  the  above 
are  the  most  important  of  the  forms  known  at  present.  As  to 
the  truth  or  falsehood  of  the  axiom  in  any  of  its  forms,  nothing 
is  known  at  present. 

The  propositions  that  depend  upon  the  axiom,  without  being 
known  to  be  equivalent  to  it,  are  numerous  and  important.  Take 
first  the  connection  of  addition  and  multiplication.  We  naturally 
think  that  the  sum  of  v  mutually  exclusive  classes,  each  having 
jit  terms,  must  have  p,Xv  terms.  When  v  is  finite,  this  can  be 
proved.  But  when  v  is  infinite,  it  cannot  be  proved  without  the 
multiplicative  axiom,  except  where,  owing  to  some  special  cir 
cumstance,  the  existence  of  certain  selectors  can  be  proved.  The 
way  the  multiplicative  axiom  enters  in  is  as  follows  :  Suppose 
we  have  two  sets  of  v  mutually  exclusive  classes,  each  having  ^ 
terms,  and  we  wish  to  prove  that  the  sum  of  one  set  has  as  many 
terms  as  the  sum  of  the  other.  In  order  to  prove  this,  we  must 
establish  a  one-one  relation.  Now,  since  there  are  in  each  case 
v  classes,  there  is  some  one-one  relation  between  the  two  sets  of 
classes  ;  but  what  we  want  is  a  one-one  relation  between  their 
terms.  Let  us  consider  some  one-one  relation  S  between  the 
classes.  Then  if  K  and  A  are  the  two  sets  of  classes,  and  a  is  some 
member  of  K,  there  will  be  a  member  j3  of  A  which  will  be  the 
correlate  of  a  with  respect  to  S.  Now  a  and  j3  each  have  /x  terms, 
and  are  therefore  similar.  There  are,  accordingly,  one-one  cor 
relations  of  a  and  jS.  The  trouble  is  that  there  are  so  many.  In 
order  to  obtain  a  one-one  correlation  of  the  sum  of  K  with  the 
sum  of  A,  we  have  to  pick  out  one  selection  from  a  set  of  classes 


Selections  and  the  Multiplicative  Axiom  125 

of  correlators,  one  class  of  the  set  being  all  the  one-one  correlators 
of  a  with  j3.  If  K  and  A  are  infinite,  we  cannot  in  general  know 
that  such  a  selection  exists,  unless  we  can  know  that  the  multi 
plicative  axiom  is  true.  Hence  we  cannot  establish  the  usual 
kind  of  connection  between  addition  and  multiplication. 

This  fact  has  various  curious  consequences.  To  begin  with, 
we  know  that  N02=N0x«0=No-  ^  *s  commonly  inferred  from 
this  that  the  sum  of  N0  classes  each  having  N0  members  must 
itself  have  N0  members,  but  this  inference  is  fallacious,  since  we 
do  not  know  that  the  number  of  terms  in  such  a  sum  is  N0  X  N0» 
nor  consequently  that  it  is  N0.  This  has  a  bearing  upon  the  theory 
of  transfinite  ordinals.  It  is  easy  to  prove  that  an  ordinal  which 
has  NO  predecessors  must  be  one  of  what  Cantor  calls  the  "  second 
class,"  i.e.  such  that  a  series  having  this  ordinal  number  will  have 
N0  terms  in  its  field.  It  is  also  easy  to  see  that,  if  we  take  any 
progression  of  ordinals  of  the  second  class,  the  predecessors  of 
their  limit  form  at  most  the  sum  of  N0  classes  each  having  N0 
terms.  It  is  inferred  thence — fallaciously,  unless  the  multi 
plicative  axiom  is  true — that  the  predecessors  of  the  limit  are  N0 
in  number,  and  therefore  that  the  limit  is  a  number  of  the  "  second 
class."  That  is  to  say,  it  is  supposed  to  be  proved  that  any  pro 
gression  of  ordinals  of  the  second  class  has  a  limit  which  is  again 
an  ordinal  of  the  second  class.  This  proposition,  with  the  corol 
lary  that  a}±  (the  smallest  ordinal  of  the  third  class)  is  not  the 
limit  of  any  progression,  is  involved  in  most  of  the  recognised 
theory  of  ordinals  of  the  second  class.  In  view  of  the  way  in 
which  the  multiplicative  axiom  is  involved,  the  proposition  and 
its  corollary  cannot  be  regarded  as  proved.  They  may  be  true, 
or  they  may  not.  All  that  can  be  said  at  present  is  that  we  do 
not  know.  Thus  the  greater  part  of  the  theory  of  ordinals  of 
the  second  class  must  be  regarded  as  unproved. 

Another  illustration  may  help  to  make  the  point  clearer.  We 
know  that  2XN0=N0.  Hence  we  might  suppose  that  the  sum 
of  N0  pairs  must  have  N0  terms.  But  this,  though  we  can  prove 
that  it  is  sometimes  the  case,  cannot  be  proved  to  happen  always 


126  Introduction  to  Mathematical  Philosophy 

unless  we  assume  the  multiplicative  axiom.  This  is  illustrated 
by  the  millionaire  who  bought  a  pair  of  socks  whenever  he  bought 
a  pair  of  boots,  and  never  at  any  other  time,  and  who  had  such 
a  passion  for  buying  both  that  at  last  he  had  N0  pairs  of  boots 
and  NO  pairs  of  socks.  The  problem  is  :  How  many  boots  had 
he,  and  how  many  socks  ?  One  would  naturally  suppose  that 
he  had  twice  as  many  boots  and  twice  as  many  socks  as  he  had 
pairs  of  each,  and  that  therefore  he  had  N0  of  each,  since  that 
number  is  not  increased  by  doubling.  But  this  is  an  instance  of 
the  difficulty,  already  noted,  of  connecting  the  sum  of  v  classes 
each  having  p  terms  with  fjiXv.  Sometimes  this  can  be  done, 
sometimes  it  cannot.  In  our  case  it  can  be  done  with  the  boots, 
but  not  with  the  socks,  except  by  some  very  artificial  device. 
The  reason  for  the  difference  is  this  :  Among  boots  we  can  dis 
tinguish  right  and  left,  and  therefore  we  can  make  a  selection  of 
one  out  of  each  pair,  namely,  we  can  choose  all  the  right  boots  or 
all  the  left  boots ;  but  with  socks  no  such  principle  of  selection 
suggests  itself,  and  we  cannot  be  sure,  unless  we  assume  the 
multiplicative  axiom,  that  there  is  any  class  consisting  of  one 
sock  out  of  each  pair.  Hence  the  problem. 

We  may  put  the  matter  in  another  way.  To  prove  that  a 
class  has  N0  terms,  it  is  necessary  and  sufficient  to  find  some  way 
of  arranging  its  terms  in  a  progression.  There  is  no  difficulty  in 
doing  this  with  the  boots.  The  pairs  are  given  as  forming  an  NO, 
and  therefore  as  the  field  of  a  progression.  Within  each  pair, 
take  the  left  boot  first  and  the  right  second,  keeping  the  order 
of  the  pair  unchanged ;  in  this  way  we  obtain  a  progression  of 
all  the  boots.  But  with  the  socks  we  shall  have  to  choose  arbi 
trarily,  with  each  pair,  which  to  put  first ;  and  an  infinite  number 
of  arbitrary  choices  is  an  impossibility.  Unless  we  can  find  a 
rule  for  selecting,  i.e.  a  relation  which  is  a  selector,  we  do  not  know 
that  a  selection  is  even  theoretically  possible.  Of  course,  in  the 
case  of  objects  in  space,  like  socks,  we  always  can  find  some 
principle  of  selection.  For  example,  take  the  centres  of  mass 
of  the  socks  :  there  will  be  points  p  in  space  such  that,  with  any 


Selections  and  the  Multiplicative  Axiom  127 

pair,  the  centres  of  mass  of  the  two  socks  are  not  both  at  exactly 
the  same  distance  from  p  ;  thus  we  can  choose,  from  each  pair, 
that  sock  which  has  its  centre  of  mass  nearer  to  p.  But  there  is 
no  theoretical  reason  why  a  method  of  selection  such  as  this 
should  always  be  possible,  and  the  case  of  the  socks,  with  a  little 
goodwill  on  the  part  of  the  reader,  may  serve  to  show  how  a 
selection  might  be  impossible. 

It  is  to  be  observed  that,  if  it  were  impossible  to  select  one  out 
of  each  pair  of  socks,  it  would  follow  that  the  socks  could  not  be 
arranged  in  a  progression,  and  therefore  that  there  were  not  N0 
of  them.  This  case  illustrates  that,  if  fj,  is  an  infinite  number, 
one  set  of  p  pairs  may  not  contain  the  same  number  of  terms  as 
another  set  of  p,  pairs  ;  for,  given  N0  pairs  of  boots,  there  are 
certainly  N0  boots,  but  we  cannot  be  sure  of  this  in  the  case  of 
the  socks  unless  we  assume  the  multiplicative  axiom  or  fall  back 
upon  some  fortuitous  geometrical  method  of  selection  such  as 
the  above. 

Another  important  problem  involving  the  multiplicative 
axiom  is  the  relation  of  reflexiveness  to  non-inductiveness.  It 
will  be  remembered  that  in  Chapter  VIII.  we  pointed  out  that  a 
reflexive  number  must  be  non-inductive,  but  that  the  converse 
(so  far  as  is  known  at  present)  can  only  be  proved  if  we  assume 
the  multiplicative  axiom.  The  way  in  which  this  comes  about 
is  as  follows  : — 

It  is  easy  to  prove  that  a  reflexive  class  is  one  which  contains 
sub-classes  having  N0  terms.  (The  class  may,  of  course,  itself 
have  N0  terms.)  Thus  we  have  to  prove,  if  we  can,  that,  given 
any  non-inductive  class,  it  is  possible  to  choose  a  progression 
out  of  its  terms.  Now  there  is  no  difficulty  in  showing  that 
a  non-inductive  class  must  contain  more  terms  than  any  inductive 
class,  or,  what  comes  to  the  same  thing,  that  if  a  is  a  non-induc 
tive  class  and  v  is  any  inductive  number,  there  are  sub-classes 
of  a  that  have  v  terms.  Thus  we  can  form  sets  of  finite  sub 
classes  of  a  :  First  one  class  having  no  terms,  then  classes  having 
I  term  (as  many  as  there  are  members  of  a),  then  classes  having 


128  Introduction  to  Mathematical  Philosophy 

2  terms,  and  so  on.  We  thus  get  a  progression  of  sets  of  sub 
classes,  each  set  consisting  of  all  those  that  have  a  certain  given 
finite  number  of  terms.  So  far  we  have  not  used  the  multiplica 
tive  axiom,  but  we  have  only  proved  that  the  number  of  collec 
tions  of  sub-classes  of  a  is  a  reflexive  number,  i.e.  that,  if  p  is 
the  number  of  members  of  a,  so  that  2*  is  the  number  of  sub 
classes  of  a  and  22^  is  the  number  of  collections  of  sub-classes, 
then,  provided  JLC  is  not  inductive,  22f*  must  be  reflexive.  But 
this  is  a  long  way  from  what  we  set  out  to  prove. 

In  order  to  advance  beyond  this  point,  we  must  employ  the 
multiplicative  axiom.  From  each  set  of  sub-classes  let  us 
choose  out  one,  omitting  the  sub-class  consisting  of  the  null- 
class  alone.  That  is  to  say,  we  select  one  sub-class  containing 
one  term,  04,  say ;  one  containing  two  terms,  a2,  say  ;  one  con 
taining  three,  a3,  say ;  and  so  on.  (We  can  do  this  if  the  multipli 
cative  axiom  is  assumed ;  otherwise,  we  do  not  know  whether 
we  can  always  do  it  or  not.)  We  have  now  a  progression 
ai»  a2>  as>  •  •  •  °f  sub-classes  of  a,  instead  of  a  progression  of 
collections  of  sub-classes  ;  thus  we  are  one  step  nearer  to  our 
goal.  We  now  know  that,  assuming  the  multiplicative  axiom, 
if  ju,  is  a  non-inductive  number,  2*  must  be  a  reflexive  number. 

The  next  step  is  to  notice  that,  although  we  cannot  be  sure 
that  new  members  of  a  come  in  at  any  one  specified  stage  in  the 
progression  ax,  a2,  a3,  .  .  .  we  can  be  sure  that  new  members 
keep  on  coming  in  from  time  to  time.  Let  us  illustrate. 
The  class  c^,  which  consists  of  one  term,  is  a  new  beginning; 
let  the  one  term  be  xv  The  class  a2,  consisting  of  two  terms, 
may  or  may  not  contain  x1 ;  if  it  does,  it  introduces  one  new 
term  ;  and  if  it  does  not,  it  must  introduce  two  new  terms,  say 
#2,  xz.  In  this  case  it  is  possible  that  a3  consists  of  xl9  #2,  xst 
and  so  introduces  no  new  terms,  but  in  that  case  a4  must  introduce 
a  new  term.  The  first  v  classes  aly  a2,  a3,  .  .  .  av  contain,  at 
the  very  most,  1+2+3+  •  •  •  +"  terms,  i.e.  j/(v+i)/2  terms; 
thus  it  would  be  possible,  if  there  were  no  repetitions  in  the 
first  v  classes,  to  go  on  with  only  repetitions  from  the 


Selections  and  the  Multiplicative  Axiom  129 

class  to  the  v(v+i)/2th  class.  But  by  that  time  the  old  terms 
would  no  longer  be  sufficiently  numerous  to  form  a  next  class 
with  the  right  number  of  members,  i.e.  v(i/-|-i)/2-[-i,  therefore 
new  terms  must  come  in  at  this  point  if  not  sooner.  It 
follows  that,  if  we  omit  from  our  progression  04,  a2,  a3,  ,  .  ,  all 
those  classes  that  are  composed  entirely  of  members  that  have 
occurred  in  previous  classes,  we  shall  still  have  a  progression. 
Let  our  new  progression  be  called  fil9  j82,  j83.  .  .  .  (We  shall 
have  a>i=pi  and  a2=j32,  because  ax  and  a2  must  introduce  new 
terms.  We  may  or  may  not  have  a3=j83,  but,  speaking  generally, 
p^  will  be  a,,  where  v  is  some  number  greater  than  p ;  i.e.  the 
j8's  are  some  of  the  a's.)  Now  these  jS's  are  such  that  any  one 
of  them,  say  jS^,  contains  members  which  have  not  occurred  in 
any  of  the  previous  j8's.  Let  y^  be  the  part  of  /^  which  consists 
of  new  members.  Thus  we  get  a  new  progression  yl9  y2,  y3,  .  .  . 
(Again  y5  will  be  identical  with  j8j  and  with  c^  ;  if  a2  does  not 
contain  the  one  member  of  al9  we  shall  have  y2=j32=a2,  but  if 
a2  does  contain  this  one  member,  y2  will  consist  of  the  other 
member  of  a2.)  This  new  progression  of  y's  consists  of  mutually 
exclusive  classes.  Hence  a  selection  from  them  will  be  a  pro 
gression  ;  i.e.  if  xl  is  the  member  of  yl9  x2  is  a  member  of  ya,  xs 
is  a  member  of  ys,  and  so  on  ;  then  xl9  #2,  #3,  .  .  .  is  a  progression, 
and  is  a  sub-class  of  a.  Assuming  the  multiplicative  axiom, 
such  a  selection  can  be  made.  Thus  by  twice  using  this  axiom 
we  can  prove  that,  if  the  axiom  is  true,  every  non-inductive 
cardinal  must  be  reflexive.  This  could  also  be  deduced  from 
Zermelo's  theorem,  that,  if  the  axiom  is  true,  every  class  can  be 
well  ordered  ;  for  a  well-ordered  series  must  have  either  a  finite 
or  a  reflexive  number  of  terms  in  its  field. 

There  is  one  advantage  in  the  above  direct  argument,  as 
against  deduction  from  Zermelo's  theorem,  that  the  above 
argument  does  not  demand  the  universal  truth  of  the  multi 
plicative  axiom,  but  only  its  truth  as  applied  to  a  set  of  N0  classes. 
It  may  happen  that  the  axiom  holds  for  N0  classes,  though  not 
for  larger  numbers  of  classes.  For  this  reason  it  is  better,  when 

9 


130  Introduction  to  Mathematical  Philosophy 

it  is  possible,  to  content  ourselves  with  the  more  restricted 
assumption.  The  assumption  made  in  the  above  direct  argu 
ment  is  that  a  product  of  N0  factors  is  never  zero  unless  one  of 
the  factors  is  zero.  We  may  state  this  assumption  in  the  form  : 
"  N0  is  a  multipliable  number,"  where  a  number  v  is  defined  as 
"  multipliable  "  when  a  product  of  v  factors  is  never  zero  unless 
one  of  the  factors  is  zero.  We  can  prove  that  a  finite  number  is 
always  multipliable,  but  we  cannot  prove  that  any  infinite  number 
is  so.  The  multiplicative  axiom  is  equivalent  to  the  assumption 
that  all  cardinal  numbers  are  multipliable.  But  in  order  to 
identify  the  reflexive  with  the  non-inductive,  or  to  deal  with  the 
problem  of  the  boots  and  socks,  or  to  show  that  any  progression 
of  numbers  of  the  second  class  is  of  the  second  class,  we  only 
need  the  very  much  smaller  assumption  that  N0  is  multipliable. 

It  is  not  improbable  that  there  is  much  to  be  discovered 
in  regard  to  the  topics  discussed  in  the  present  chapter.  Cases 
may  be  found  where  propositions  which  seem  to  involve  the 
multiplicative  axiom  can  be  proved  without  it.  It  is  conceivable 
that  the  multiplicative  axiom  in  its  general  form  may  be  shown 
to  be  false.  From  this  point  of  view,  Zermelo's  theorem  offers 
the  best  hope :  the  continuum  or  some  still  more  dense  series 
might  be  proved  to  be  incapable  of  having  its  terms  well  ordered, 
which  would  prove  the  multiplicative  axiom  false,  in  virtue  of 
Zermelo's  theorem.  But  so  far,  no  method  of  obtaining  such 
results  has  been  discovered,  and  the  subject  remains  wrapped  in 
obscurity. 


CHAPTER  XIII 

THE    AXIOM    OF    INFINITY    AND    LOGICAL   TYPES 

THE  axiom  of  infinity  is  an  assumption  which  may  be  enunciated 
as  follows  : — 

"  If  n  be  any  inductive  cardinal  number,  there  is  at  least  one 
class  of  individuals  having  n  terms." 

If  this  is  true,  it  follows,  of  course,  that  there  are  many  classes 
of  individuals  having  n  terms,  and  that  the  total  number  of 
individuals  in  the  world  is  not  an  inductive  number.  For,  by 
the  axiom,  there  is  at  least  one  class  having  n-f- 1  terms,  from  which 
it  follows  that  there  are  many  classes  of  n  terms  and  that  n  is 
not  the  number  of  individuals  in  the  world.  Since  n  is  any 
inductive  number,  it  follows  that  the  number  of  individuals 
in  the  world  must  (if  our  axiom  be  true)  exceed  any  inductive 
number.  In  view  of  what  we  found  in  the  preceding  chapter, 
about  the  possibility  of  cardinals  which  are  neither  inductive 
nor  reflexive,  we  cannot  infer  from  our  axiom  that  there  are  at 
least  N0  individuals,  unless  we  assume  the  multiplicative  axiom. 
But  we  do  know  that  there  are  at  least  N0  classes  of  classes, 
since  the  inductive  cardinals  are  classes  of  classes,  and  form  a 
progression  if  our  axiom  is  true.  The  way  in  which  the  need 
for  this  axiom  arises  may  be  explained  as  follows  : — One  of 
Peano's  assumptions  is  that  no  two  inductive  cardinals  have  the 
same  successor,  i.e.  that  we  shall  not  have  ra-f  !=«-{- 1  unless 
m=n,  if  m  and  n  are  inductive  cardinals.  In  Chapter  VIII.  we 
had  occasion  to  use  what  is  virtually  the  same  as  the  above 
assumption  of  Peano's,  namely,  that,  if  n  is  an  inductive  cardinal, 


132  Introduction  to  Mathematical  Philosophy 

n  is  not  equal  to  w-f-i.  It  might  be  thought  that  this  could  be 
proved.  We  can  prove  that,  if  a  is  an  inductive  class,  and  n 
is  the  number  of  members  of  a,  then  n  is  not  equal  to  «+i. 
This  proposition  is  easily  proved  by  induction,  and  might  be 
thought  to  imply  the  other.  But  in  fact  it  does  not,  since  there 
might  be  no  such  class  as  a.  What  it  does  imply  is  this  :  If 
n  is  an  inductive  cardinal  such  that  there  is  at  least  one  class 
having  n  members,  then  n  is  not  equal  to  n-\-i.  The  axiom  of 
infinity  assures  us  (whether  truly  or  falsely)  that  there  are  classes 
having  n  members,  and  thus  enables  us  to  assert  that  n  is  not 
equal  to  »+i.  But  without  this  axiom  we  should  be  left  with 
the  possibility  that  n  and  n-\-i  might  both  be  the  null-class. 

Let  us  illustrate  this  possibility  by  an  example :  Suppose 
there  were  exactly  nine  individuals  in  the  world.  (As  to  what 
is  meant  by  the  word  "  individual,"  I  must  ask  the  reader  to 
be  patient.)  Then  the  inductive  cardinals  from  o  up  to  9  would 
be  such  as  we  expect,  but  10  (defined  as  9+1)  would  be  the 
null-class.  It  will  be  remembered  that  n-\-i  may  be  defined  as 
follows  :  tt-j-  I  is  the  collection  of  all  those  classes  which  have  a 
term  x  such  that,  when  x  is  taken  away,  there  remains  a  class 
of  n  terms.  Now  applying  this  definition,  we  see  that,  in  the 
case  supposed,  9+1  is  a  class  consisting  of  no  classes,  i.e.  it  is 
the  null-class.  The  same  will  be  true  of  9+2,  or  generally  of 
9+w,  unless  n  is  zero.  Thus  10  and  all  subsequent  inductive 
cardinals  will  all  be  identical,  since  they  will  all  be  the  null-class. 
In  such  a  case  the  inductive  cardinals  will  not  form  a  progression, 
nor  will  it  be  true  that  no  two  have  the  same  successor,  for  9 
and  10  will  both  be  succeeded  by  the  null-class  (10  being  itself 
the  null-class).  It  is  in  order  to  prevent  such  arithmetical 
catastrophes  that  we  require  the  axiom  of  infinity. 

As  a  matter  of  fact,  so  long  as  we  are  content  with  the  arith 
metic  of  finite  integers,  and  do  not  introduce  either  infinite 
integers  or  infinite  classes  or  series  of  finite  integers  or  ratios, 
it  is  possible  to  obtain  all  desired  results  without  the  axiom  of 
infinity.  That  is  to  say,  we  can  deal  with  the  addition,  multi- 


The  Axiom  of  Infinity  and  Logical  Types         133 

plication,  and  exponentiation  of  finite  integers  and  of  ratios, 
but  we  cannot  deal  with  infinite  integers  or  with  irrationals. 
Thus  the  theory  of  the  transfinite  and  the  theory  of  real  numbers 
fails  us.  How  these  various  results  come  about  must  now  be 
explained. 

Assuming  that  the  number  of  individuals  in  the  world  is  n, 
the  number  of  classes  of  individuals  will  be  2n.  This  is  in  virtue 
of  the  general  proposition  mentioned  in  Chapter  VIII.  that  the 
number  of  classes  contained  in  a  class  which  has  n  members 
is  2n.  Now  2n  is  always  greater  than  n.  Hence  the  number 
of  classes  in  the  world  is  greater  than  the  number  of  individuals. 
If,  now,  we  suppose  the  number  of  individuals  to  be  9,  as  we  did 
just  now,  the  number  of  classes  will  be  29,  i.e.  512.  Thus  if  we 
take  our  numbers  as  being  applied  to  the  counting  of  classes 
instead  of  to  the  counting  of  individuals,  our  arithmetic  will 
be  normal  until  we  reach  512  :  the  first  number  to  be  null  will 
be  513.  And  if  we  advance  to  classes  of  classes  we  shall  do  still 
better  :  the  number  of  them  will  be  2512,  a  number  which  is  so 
large  as  to  stagger  imagination,  since  it  has  about  153  digits. 
And  if  we  advance  to  classes  of  classes  of  classes,  we  shall  obtain 
a  number  represented  by  2  raised  to  a  power  which  has  about 
153  digits  ;  the  number  of  digits  in  this  number  will  be  about 
three  times  io152.  In  a  time  of  paper  shortage  it  is  undesirable 
to  write  out  this  number,  and  if  we  want  larger  ones  we  can 
obtain  them  by  travelling  further  along  the  logical  hierarchy. 
In  this  way  any  assigned  inductive  cardinal  can  be  made  to 
find  its  place  among  numbers  which  are  not  null,  merely  by 
travelling  along  the  hierarchy  for  a  sufficient  distance.1 

As  regards  ratios,  we  have  a  very  similar  state  of  affairs. 
If  a  ratio  p,/v  is  to  have  the  expected  properties,  there  must 
be  enough  objects  of  whatever  sort  is  being  counted  to  insure 
that  the  null-class  does  not  suddenly  obtrude  itself.  But  this 
can  be  insured,  for  any  given  ratio  JJL/V,  without  the  axiom  of 

1  On  this  subject  see  Principia  Mathematica,  vol.  ii.  *  120  ff.  On  the 
corresponding  problems  as  regards  ratio,  see  ibid.,  vol.  iii.  *  303  ff. 


134  Introduction  to  Mathematical  Philosophy 

infinity,  by  merely  travelling  up  the  hierarchy  a  sufficient  distance. 
If  we  cannot  succeed  by  counting  individuals,  we  can  try  counting 
classes  of  individuals  ;  if  we  still  do  not  succeed,  we  can  try 
classes  of  classes,  and  so  on.  Ultimately,  however  few  indi 
viduals  there  may  be  in  the  world,  we  shall  reach  a  stage  where 
there  are  many  more  than  /x  objects,  whatever  inductive  number 
p  may  be.  Even  if  there  were  no  individuals  at  all,  this  would 
still  be  true,  for  there  would  then  be  one  class,  namely,  the  null- 
class,  2  classes  of  classes  (namely,  the  null-class  of  classes  and  the 
class  whose  only  member  is  the  null-class  of  individuals),  4  classes 
of  classes  of  classes,  16  at  the  next  stage,  65,536  at  the  next 
stage,  and  so  on.  Thus  no  such  assumption  as  the  axiom  of 
infinity  is  required  in  order  to  reach  any  given  ratio  or  any  given 
inductive  cardinal. 

It  is  when  we  wish  to  deal  with  the  whole  class  or  series  of 
inductive  cardinals  or  of  ratios  that  the  axiom  is  required.  We 
need  the  whole  class  of  inductive  cardinals  in  order  to  establish 
the  existence  of  N0,  and  the  whole  series  in  order  to  establish 
the  existence  of  progressions  :  for  these  results,  it  is  necessary 
that  we  should  be  able  to  make  a  single  class  or  series  in  which 
no  inductive  cardinal  is  null.  We  need  the  whole  series  of  ratios 
in  order  of  magnitude  in  order  to  define  real  numbers  as  segments  : 
this  definition  will  not  give  the  desired  result  unless  the  series 
of  ratios  is  compact,  which  it  cannot  be  if  the  total  number  of 
ratios,  at  the  stage  concerned,  is  finite. 

It  would  be  natural  to  suppose  —  as  I  supposed  myself  in  former 
days  —  that,  by  means  of  constructions  such  as  we  have  been 
considering,  the  axiom  of  infinity  could  be  proved.  It  may  be 
said  :  Let  us  assume  that  the  number  of  individuals  is  n,  where 
n  may  be  o  without  spoiling  our  argument  ;  then  if  we  form  the 
complete  set  of  individuals,  classes,  classes  of  classes,  etc.,  all 
taken  together,  the  number  of  terms  in  our  whole  set  will  be 


which  is  N0.     Thus  taking  all  kinds  of  objects  together,  and  not 


The  Axiom  of  Infinity  and  Logical  Types         135 

confining  ourselves  to  objects  of  any  one  type,  we  shall  certainly 
obtain  an  infinite  class,  and  shall  therefore  not  need  the  axiom 
of  infinity.  So  it  might  be  said. 

Now,  before  going  into  this  argument,  the  first  thing  to  observe 
is  that  there  is  an  air  of  hocus-pocus  about  it :  something  reminds 
one  of  the  conjurer  who  brings  things  out  of  the  hat.  The  man 
who  has  lent  his  hat  is  quite  sure  there  wasn't  a  live  rabbit  in  it 
before,  but  he  is  at  a  loss  to  say  how  the  rabbit  got  there.  So 
the  reader,  if  he  has  a  robust  sense  of  reality,  will  feel  convinced 
that  it  is  impossible  to  manufacture  an  infinite  collection  out  of 
a  finite  collection  of  individuals,  though  he  may  be  unable  to 
say  where  the  flaw  is  in  the  above  construction.  It  would  be  a 
mistake  to  lay  too  much  stress  on  such  feelings  of  hocus-pocus  ; 
like  other  emotions,  they  may  easily  lead  us  astray.  But  they 
afford  a  prima  facie  ground  for  scrutinising  very  closely  any 
argument  which  arouses  them.  And  when  the  above  argument 
is  scrutinised  it  will,  in  my  opinion,  be  found  to  be  fallacious, 
though  the  fallacy  is  a  subtle  one  and  by  no  means  easy  to  avoid 
consistently. 

The  fallacy  involved  is  the  fallacy  which  may  be  called  "  con 
fusion  of  types."  To  explain  the  subject  of  "  types  "  fully  would 
require  a  whole  volume  ;  moreover,  it  is  the  purpose  of  this  book 
to  avoid  those  parts  of  the  subjects  which  are  still  obscure  and 
controversial,  isolating,  for  the  convenience  of  beginners,  those 
parts  which  can  be  accepted  as  embodying  mathematically  ascer 
tained  truths.  Now  the  theory  of  types  emphatically  does  not 
belong  to  the  finished  and  certain  part  of  our  subject :  much  of 
this  theory  is  still  inchoate,  confused,  and  obscure.  But  the  need 
of  some  doctrine  of  types  is  less  doubtful  than  the  precise  form 
the  doctrine  should  take ;  and  in  connection  with  the  axiom  of 
infinity  it  is  particular'y  easy  to  see  the  necessity  of  some  such 
doctrine. 

This  necessity  results,  for  example,  from  the  "  contradiction  of 
the  greatest  cardinal."  We  saw  in  Chapter  VIII.  that  the  number 
of  classes  contained  in  a  given  class  is  always  greater  than  the 


136  Introduction  to  Mathematical  Philosophy 

number  of  members  of  the  class,  and  we  inferred  that  there  is 
no  greatest  cardinal  number.  But  if  we  could,  as  we  suggested 
a  moment  ago,  add  together  into  one  class  the  individuals,  classes 
of  individuals,  classes  of  classes  of  individuals,  etc.,  we  should 
obtain  a  class  of  which  its  own  sub-classes  would  be  members. 
The  class  "consisting  of  all  objects  that  can  be  counted,  of  whatever 
sort,  must,  if  there  be  such  a  class,  have  a  cardinal  number  which 
is  the  greatest  possible.  Since  all  its  sub-classes  will  be  members 
of  it,  there  cannot  be  more  of  them  than  there  are  members. 
Hence  we  arrive  at  a  contradiction. 

When  I  first  came  upon  this  contradiction,  in  the  year  1901, 
I  attempted  to  discover  some  flaw  in  Cantor's  proof  that  there  is 
no  greatest  cardinal,  which  we  gave  in  Chapter  VIII.  Apply 
ing  this  proof  to  the  supposed  class  of  all  imaginable  objects, 
I  was  led  to  a  new  and  simpler  contradiction,  namely,  the 
following : — 

The  comprehensive  class  we  are  considering,  which  is  to  embrace 
everything,  must  embrace  itself  as  one  of  its  members.  In  other 
words,  if  there  is  such  a  thing  as  "  everything,"  then  "  every 
thing  "  is  something,  and  is  a  member  of  the  class  "  everything." 
But  normally  a  class  is  not  a  member  of  itself.  Mankind,  for 
example,  is  not  a  man.  Form  now  the  assemblage  of  all  classes 
which  are  not  members  of  themselves.  This  is  a  class  :  is  it  a 
member  of  itself  or  not  ?  If  it  is,  it  is  one  of  those  classes  that 
are  not  members  of  themselves,  i.e.  it  is  not  a  member  of  itself. 
If  it  is  not,  it  is  not  one  of  those  classes  that  are  not  members  of 
themselves,  i.e.  it  is  a  member  of  itself.  Thus  of  the  two  hypo 
theses — that  it  is,  and  that  it  is  not,  a  member  of  itself — each 
implies  its  contradictory.  This  is  a  contradiction. 

There  is  no  difficulty  in  manufacturing  similar  contradictions 
ad  lib.  The  solution  of  such  contradictions  by  the  theory  of 
types  is  set  forth  fully  in  Principia  Mathematical  and  also,  more 
briefly,  in  articles  by  the  present  author  in  the  American  Journal 

1  Vol.  i.,  Introduction,  chap,  ii.,  #  12  and  *  20;  vol  ii.,  Prefatory 
Statement. 


The  Axiom  of  Infinity  and  Logical  Types         137 

of  Mathematics  1  and  in  the  Revue  de  Metaphysique  et  de  Morale? 
For  the  present  an  outline  of  the  solution  must  suffice. 

The  fallacy  consists  in  the  formation  of  what  we  may  call 
"  impure  "  classes,  i.e.  classes  which  are  not  pure  as  to  "  type." 
As  we  shall  see  in  a  later  chapter,  classes  are  logical  fictions,  and 
a  statement  which  appears  to  be  about  a  class  will  only  be  signi 
ficant  if  it  is  capable  of  translation  into  a  form  in  which  no  mention 
is  made  of  the  class.  This  places  a  limitation  upon  the  ways  in 
which  what  are  nominally,  though  not  really,  names  for  classes 
can  occur  significantly  :  a  sentence  or  set  of  symbols  in  which 
such  pseudo-names  occur  in  wrong  ways  is  not  false,  but  strictly 
devoid  of  meaning.  The  supposition  that  a  class  is,  or  that  it 
is  not,  a  member  of  itself  is  meaningless  in  just  this  way.  And 
more  generally,  to  suppose  that  one  class  of  individuals  is  a 
member,  or  is  not  a  member,  of  another  class  of  individuals 
will  be  to  suppose  nonsense  ;  and  to  construct  symbolically  any 
class  whose  members  are  not  all  of  the  same  grade  in  the  logical 
hierarchy  is  to  use  symbols  in  a  way  which  makes  them  no 
longer  symbolise  anything. 

Thus  if  there  are  n  individuals  in  the  world,  and  2n  classes  of 
individuals,  we  cannot  form  a  new  class,  consisting  of  both 
individuals  and  classes  and  having  w-f-2n  members.  In  this  way 
the  attempt  to  escape  from  the  need  for  the  axiom  of  infinity 
breaks  down.  I  do  not  pretend  to  have  explained  the  doctrine 
of  types,  or  done  more  than  indicate,  in  rough  outline,  why  there 
is  need  of  such  a  doctrine.  I  have  aimed  only  at  saying  just 
so  much  as  was  required  in  order  to  show  that  we  cannot  'prove 
the  existence  of  infinite  numbers  and  classes  by  such  conjurer's 
methods  as  we  have  been  examining.  There  remain,  however, 
certain  other  possible  methods  which  must  be  considered. 

Various  arguments  professing  to  prove  the  existence  of  infinite 
classes  are  given  in  the  Principles  of  Mathematics,  §  339  (p.  357). 

1  "  Mathematical  Logic  as  based  on  the  Theory  of  Types,"  vol.  xxx., 
1908,  pp.  222-262. 

"  Les  paradoxes  de  la  logique,"  1906,  pp.  627-650. 


138  Introduction  to  Mathematical  Philosophy 

In  so  far  as  these  arguments  assume  that,  if  n  is  an  inductive 
cardinal,  n  is  not  equal  to  n-\-i,  they  have  been  already  dealt 
with.  There  is  an  argument,  suggested  by  a  passage  in  Plato's 
ParmfnidlSy  to  the  effect  that,  if  there  is  such  a  number  as  I, 
then  I  has  being  ;  but  I  is  not  identical  with  being,  and  therefore 
I  and  being  are  two,  and  therefore  there  is  such  a  number  as  2, 
and  2  together  with  I  and  being  gives  a  class  of  three  terms,  and 
so  on.  This  argument  is  fallacious,  partly  because  "  being  "  is 
not  a  term  having  any  definite  meaning,  and  still  more  because, 
if  a  definite  meaning  were  invented  for  it,  it  would  be  found  that 
numbers  do  not  have  being — they  are,  in  fact,  what  are  called 
"  logical  fictions,''  as  we  shall  see  when  we  come  to  consider 
the  definition  of  classes. 

The  argument  that  the  number  of  numbers  from  o  to  n  (both 
inclusive)  is  n-\-i  depends  upon  the  assumption  that  up  to  and 
including  n  no  number  is  equal  to  its  successor,  which,  as  we  have 
seen,  will  not  be  always  true  if  the  axiom  of  infinity  is  false.  It 
must  be  understood  that  the  equation  n=n-\-i,  which  might  be 
true  for  a  finite  n\in  exceeded  the  total  number  of  individuals 
in  the  world,  is  quite  different  from  the  same  equation  as  applied 
to  a  reflexive  number.  As  applied  to  a  reflexive  number,  it 
means  that,  given  a  class  of  n  terms,  this  class  is  "  similar  "  to 
that  obtained  by  adding  another  term.  But  as  applied  to  a 
number  which  is  too  great  for  the  actual  world,  it  merely  means 
that  there  is  no  class  of  n  individuals,  and  no  class  of  n-\-\  indi 
viduals  ;  it  does  not  mean  that,  if  we  mount  the  hierarchy  of 
types  sufficiently  far  to  secure  the  existence  of  a  class  of  n  terms, 
we  shall  then  find  this  class  "  similar  "  to  one  of  n-\- 1  terms,  for 
if  n  is  inductive  this  will  not  be  the  case,  quite  independently  of 
the  truth  or  falsehood  of  the  axiom  of  infinity. 

There  is  an  argument  employed  by  both  Bolzano  1  and  Dede- 
kind  2  to  prove  the  existence  of  reflexive  classes.  The  argument, 
in  brief,  is  this  :  An  object  is  not  identical  with  the  idea  of  the 

1  Bolzano,  Paradoxien  des  Unendlichen,  13. 

1  Dedekind,  Was  sind  und  was  sollen  die  Zahlen  ?    No.  66. 


The  Axiom  of  Infinity  and  Logical  Types         139 

object,  but  there  is  (at  least  in  the  realm  of  being)  an  idea  of  any 
object.  The  relation  of  an  object  to  the  idea  of  it  is  one-one,  and 
ideas  are  only  some  among  objects.  Hence  the  relation  "  idea 
of  "  constitutes  a  reflexion  of  the  whole  class  of  objects  into  a 
part  of  itself,  namely,  into  that  part  which  consists  of  ideas. 
Accordingly,  the  class  of  objects  and  the  class  of  ideas  are  both 
infinite.  This  argument  is  interesting,  not  only  on  its  own 
account,  but  because  the  mistakes  in  it  (or  what  I  judge  to  be 
mistakes)  are  of  a  kind  which  it  is  instructive  to  note.  The 
main  error  consists  in  assuming  that  there  is  an  idea  of  every 
object.  It  is,  of  course,  exceedingly  difficult  to  decide  what  is 
meant  by  an  "  idea  "  ;  but  let  us  assume  that  we  know.  We  are 
then  to  suppose  that,  starting  (say)  with  Socrates,  there  is  the 
idea  of  Socrates,  and  so  on  ad  inf.  Now  it  is  plain  that  this  is  not 
the  case  in  the  sense  that  all  these  ideas  have  actual  empirical 
existence  in  people's  minds.  Beyond  the  third  or  fourth  stage 
they  become  mythical.  If  the  argument  is  to  be  upheld,  the 
"  ideas  "  intended  must  be  Platonic  ideas  laid  up  in  heaven,  for 
certainly  they  are  not  on  earth.  But  then  it  at  once  becomes 
doubtful  whether  there  are  such  ideas.  If  we  are  to  know  that 
there  are,  it  must  be  on  the  basis  of  some  logical  theory,  proving 
that  it  is  necessary  to  a  thing  that  there  should  be  an  idea  of  it. 
We  certainly  cannot  obtain  this  result  empirically,  or  apply  it, 
as  Dedekind  does,  to  "  meine  Gedankenwelt " — the  world  of  my 
thoughts. 

If  we  were  concerned  to  examine  fully  the  relation  of  idea  and 
object,  we  should  have  to  enter  upon  a  number  of  psychological 
and  logical  inquiries,  which  are  not  relevant  to  our  main  purpose. 
But  a  few  further  points  should  be  noted.  If  "  idea  "  is  to  be 
understood  logically,  it  may  be  identical  with  the  object,  or  it 
may  stand  for  a  description  (in  the  sense  to  be  explained  in  a 
subsequent  chapter).  In  the  former  case  the  argument  fails, 
because  it  was  essential  to  the  proof  of  reflexiveness  that  object 
and  idea  should  be  distinct.  In  the  second  case  the  argument 
also  fails,  because  the  relation  of  object  and  description  is  not 


140  Introduction  to  Mathematical  Philosophy 

one-one  :  there  are  innumerable  correct  descriptions  of  any  given 
object.  Socrates  (e.g)  may  be  described  as  "  the  master  of 
Plato,"  or  as  "  the  philosopher  who  drank  the  hemlock,"  or  as 
"  the  husband  of  Xantippe."  If — to  take  up  the  remaining 
hypothesis — "  idea  "  is  to  be  interpreted  psychologically,  it  must 
be  maintained  that  there  is  not  any  one  definite  psychological 
entity  which  could  be  called  the  idea  of  the  object :  there  are  in 
numerable  beliefs  and  attitudes,  each  of  which  could  be  called  an 
idea  of  the  object  in  the  sense  in  which  we  might  say  "  my  idea 
of  Socrates  is  quite  different  from  yours,"  but  there  is  not  any 
central  entity  (except  Socrates  himself)  to  bind  together  various 
"  ideas  of  Socrates,"  and  thus  there  is  not  any  such  one-one  rela 
tion  of  idea  and  object  as  the  argument  supposes.  Nor,  of  course, 
as  we  have  already  noted,  is  it  true  psychologically  that  there  are 
ideas  (in  however  extended  a  sense)  of  more  than  a  tiny  proportion 
of  the  things  in  the  world.  For  all  these  reasons,  the  above 
argument  in  favour  of  the  logical  existence  of  reflexive  classes 
must  be  rejected. 

It  might  be  thought  that,  whatever  may  be  said  of  logical 
arguments,  the  empirical  arguments  derivable  from  space  and 
time,  the  diversity  of  colours,  etc.,  are  quite  sufficient  to  prove 
the  actual  existence  of  an  infinite  number  of  particulars.  I  do 
not  believe  this.  We  have  no  reason  except  prejudice  for  believ 
ing  in  the  infinite  extent  of  space  and  time,  at  any  rate  in  the  sense 
in  which  space  and  time  are  physical  facts,  not  mathematical 
fictions.  We  naturally  regard  space  and  time  as  continuous,  or, 
at  least,  as  compact ;  but  this  again  is  mainly  prejudice.  The 
theory  of  "  quanta  "  in  physics,  whether  true  or  false,  illustrates 
the  fact  that  physics  can  never  afford  proof  of  continuity,  though 
it  might  quite  possibly  afford  disproof.  The  senses  are  not 
sufficiently  exact  to  distinguish  between  continuous  motion  and 
rapid  discrete  succession,  as  anyone  may  discover  in  a  cinema. 
A  world  in  which  all  motion  consisted  of  a  series  of  small  finite 
jerks  would  be  empirically  indistinguishable  from  one  in  which 
motion  was  continuous.  It  would  take  up  too  much  space  to 


The  Axiom  of  Infinity  and  Logical  Types          141 

defend  these  theses  adequately  ;  for  the  present  I  am  merely 
suggesting  them  for  the  reader's  consideration.  If  they  are  valid, 
it  follows  that  there  is  no  empirical  reason  for  believing  the 
number  of  particulars  in  the  world  to  be  infinite,  and  that  there 
never  can  be ;  also  that  there  is  at  present  no  empirical  reason 
to  believe  the  number  to  be  finite,  though  it  is  theoretically 
conceivable  that  some  day  there  might  be  evidence  pointing, 
though  not  conclusively,  in  that  direction. 

From  the  fact  that  the  infinite  is  not  self-contradictory,  but  is 
also  not  demonstrable  logically,  we  must  conclude  that  nothing 
can  be  known  a  priori  as  to  whether  the  number  of  things 
in  the  world  is  finite  or  infinite.  The  conclusion  is,  therefore, 
to  adopt  a  Leibnizian  phraseology,  that  some  of  the  possible 
worlds  are  finite,  some  infinite,  and  we  have  no  means  of 
knowing  to  which  of  these  two  kinds  our  actual  world  belongs. 
The  axiom  of  infinity  will  be  true  in  some  possible  worlds 
and  false  in  others  ;  whether  it  is  true  or  false  in  this  world, 
we  cannot  tell. 

Throughout  this  chapter  the  synonyms  "  individual "  and 
"  particular  "  have  been  used  without  explanation.  It  would  be 
impossible  to  explain  them  adequately  without  a  longer  disquisi 
tion  on  the  theory  of  types  than  would  be  appropriate  to  the 
present  work,  but  a  few  words  before  we  leave  this  topic  may 
do  something  to  diminish  the  obscurity  which  would  otherwise 
envelop  the  meaning  of  these  words. 

In  an  ordinary  statement  we  can  distinguish  a  verb,  expressing 
an  attribute  or  relation,  from  the  substantives  which  express  the 
subject  of  the  attribute  or  the  terms  of  the  relation.  "  Caesar 
lived  "  ascribes  an  attribute  to  Caesar ;  "  Brutus  killed  Caesar  " 
expresses  a  relation  between  Brutus  and  Caesar.  Using  the  word 
"subject"  in  a  generalised  sense,  we  may  call  both  Brutus  and 
Caesar  subjects  of  this  proposition  :  the  fact  that  Brutus  is  gram 
matically  subject  and  Caesar  object  is  logically  irrelevant,  since 
the  same  occurrence  may  be  expressed  in  the  words  "  Caesar  was 
killed  by  Brutus,"  where  Caesar  is  the  grammatical  subject. 


142  Introduction  to  Mathematical  Philosophy 

Thus  in  the  simpler  sort  of  proposition  we  shall  have  an  attribute 
or  relation  holding  of  or  between  one,  two  or  more  "  subjects  " 
in  the  extended  sense.  (A  relation  may  have  more  than  two 
terms  :  e.g.  "  A  gives  B  to  C  "  is  a  relation  of  three  terms.)  Now 
it  often  happens  that,  on  a  closer  scrutiny,  the  apparent  subjects 
are  found  to  be  not  really  subjects,  but  to  be  capable  of  analysis  ; 
the  only  result  of  this,  however,  is  that  new  subjects  take  their 
places.  It  also  happens  that  the  verb  may  grammatically  be 
made  subject :  e.g.  we  may  say,  "  Killing  is  a  relation  which 
holds  between  Brutus  and  Caesar."  But  in  such  cases  the 
grammar  is  misleading,  and  in  a  straightforward  statement, 
following  the  rules  that  should  guide  philosophical  grammar, 
Brutus  and  Cssar  will  appear  as  the  subjects  and  killing 
as  the  verb. 

We  are  thus  led  to  the  conception  of  terms  which,  when  they 
occur  in  propositions,  can  only  occur  as  subjects,  and  never  in 
any  other  way.  This  is  part  of  the  old  scholastic  definition 
of  substance ;  but  persistence  through  time,  which  belonged  to 
that  notion,  forms  no  part  of  the  notion  with  which  we  are  con 
cerned.  We  shall  define  "  proper  names  "  as  those  terms  which 
can  only  occur  as  subjects  in  propositions  (using  "  subject " 
in  the  extended  sense  just  explained).  We  shall  further  define 
"  individuals "  or  "  particulars "  as  the  objects  that  can  be 
named  by  proper  names.  (It  would  be  better  to  define  them 
directly,  rather  than  by  means  of  the  kind  of  symbols  by  which 
they  are  symbolised ;  but  in  order  to  do  that  we  should  have 
to  plunge  deeper  into  metaphysics  than  is  desirable  here.)  It 
is,  of  course,  possible  that  there  is  an  endless  regress  :  that 
whatever  appears  as  a  particular  is  really,  on  closer  scrutiny, 
a  class  or  some  kind  of  complex.  If  this  be  the  case,  the  axiom 
of  infinity  must  of  course  be  true.  But  if  it  be  not  the  case, 
it  must  be  theoretically  possible  for  analysis  to  reach  ultimate 
subjects,  and  it  is  these  that  give  the  meaning  of  "  particulars  " 
or  "  individuals."  It  is  to  the  number  of  these  that  the  axiom 
of  infinity  is  assumed  to  apply.  If  it  is  true  of  them,  it  is  true 


The  Axiom  of  Infinity  and  Logical  Types          143 

of  classes  of  them,  and  classes  of  classes  of  them,  and  so  on ; 
similarly  if  it  is  false  of  them,  it  is  false  throughout  this  hierarchy. 
Hence  it  is  natural  to  enunciate  the  axiom  concerning  them  rather 
than  concerning  any  other  stage  in  the  hierarchy.  But  whether 
the  axiom  is  true  or  false,  there  seems  no  known  method  of 
discovering. 


CHAPTER  XIV 

INCOMPATIBILITY    AND   THE   THEORY    OF    DEDUCTION 

WE  have  now  explored,  somewhat  hastily  it  is  true,  that  part 
of  the  philosophy  of  mathematics  which  does  not  demand  a 
critical  examination  of  the  idea  of  class.  In  the  preceding 
chapter,  however,  we  found  ourselves  confronted  by  problems 
which  make  such  an  examination  imperative.  Before  we  can 
undertake  it,  we  must  consider  certain  other  parts  of  the  philos 
ophy  of  mathematics,  which  we  have  hitherto  ignored.  In  a 
synthetic  treatment,  the  parts  which  we  shall  now  be  concerned 
with  come  first :  they  are  more  fundamental  than  anything 
that  we  have  discussed  hitherto.  Three  topics  will  concern  us 
before  we  reach  the  theory  of  classes,  namely  :  (i)  the  theory 
of  deduction,  (2)  prepositional  functions,  (3)  descriptions.  Of 
these,  the  third  is  not  logically  presupposed  in  the  theory  of 
classes,  but  it  is  a  simpler  example  of  the  kind  of  theory  that 
is  needed  in  dealing  with  classes.  It  is  the  first  topic,  the  theory 
of  deduction,  that  will  concern  us  in  the  present  chapter. 

Mathematics  is  a  deductive  science  :  starting  from  certain 
premisses,  it  arrives,  by  a  strict  process  of  deduction,  at  the 
various  theorems  which  constitute  it.  It  is  true  that,  in  the  past, 
mathematical  deductions  were  often  greatly  lacking  in  rigour ; 
it  is  true  also  that  perfect  rigour  is  a  scarcely  attainable  ideal. 
Nevertheless,  in  so  far  as  rigour  is  lacking  in  a  mathematical 
proof,  the  proof  is  defective  ;  it  is  no  defence  to  urge  that  common 
sense  shows  the  result  to  be  correct,  for  if  we  were  to  rely  upon 
that,  it  would  be  better  to  dispense  with  argument  altogether, 

144 


Incompatibility  and  the  Theory  of  Deduction        145 

rather  than  bring  fallacy  to  the  rescue  of  common  sense.  No 
appeal  to  common  sense,  or  "  intuition,"  or  anything  except  strict 
deductive  logic,  ought  to  be  needed  in  mathematics  after  the 
premisses  have  been  laid  down. 

Kant,  having  observed  that  the  geometers  of  his  day  could 
not  prove  their  theorems  by  unaided  argument,  but  required 
an  appeal  to  the  figure,  invented  a  theory  of  mathematical 
reasoning  according  to  which  the  inference  is  never  strictly 
logical,  but  always  requires  the  support  of  what  is  called 
"  intuition."  The  whole  trend  of  modern  mathematics,  with 
its  increased  pursuit  of  rigour,  has  been  against  this  Kantian 
theory.  The  things  in  the  mathematics  of  Kant's  day  which 
cannot  be  proved,  cannot  be  known — for  example,  the  axiom  of 
parallels.  What  can  be  known,  in  mathematics  and  by  mathe 
matical  methods,  is  what  can  be  deduced  from  pure  logic.  What 
else  is  to  belong  to  human  knowledge  must  be  ascertained  other 
wise — empirically,  through  the  senses  or  through  experience  in 
some  form,  but  not  a  priori.  The  positive  grounds  for  this 
thesis  are  to  be  found  in  Principia  Mathematica,  passim ;  a 
controversial  defence  of  it  is  given  in  the  Principles  of  Mathe 
matics.  We  cannot  here  do  more  than  refer  the  reader  to  those 
works,  since  the  subject  is  too  vast  for  hasty  treatment.  Mean 
while,  we  shall  assume  that  all  mathematics  is  deductive,  and 
proceed  to  inquire  as  to  what  is  involved  in  deduction. 

In  deduction,  we  have  one  or  more  propositions  called  pre 
misses,  from  which  we  infer  a  proposition  called  the  conclusion. 
For  our  purposes,  it  will  be  convenient,  when  there  are  originally 
several  premisses,  to  amalgamate  them  into  a  single  proposition, 
so  as  to  be  able  to  speak  of  the  premiss  as  well  as  of  the  con 
clusion.  Thus  we  may  regard  deduction  as  a  process  by  which 
we  pass  from  knowledge  of  a  certain  proposition,  the  premiss, 
to  knowledge  of  a  certain  other  proposition,  the  conclusion. 
But  we  shall  not  regard  such  a  process  as  logical  deduction  unless 
it  is  correct,  i.e.  unless  there  is  such  a  relation  between  premiss 
and  conclusion  that  we  have  a  right  to  believe  the  conclusion 

10 


146  Introduction  to  Mathematical  Philosophy 

if  we  know  the  premiss  to  be  true.  It  is  this  relation  that  is 
chiefly  of  interest  in  the  logical  theory  of  deduction. 

In  order  to  be  able  validly  to  infer  the  truth  of  a  proposition, 
we  must  know  that  some  other  proposition  is  true,  and  that 
there  is  between  the  two  a  relation  of  the  sort  called  "implication," 
i.e.  that  (as  we  say)  the  premiss  "  implies  "  the  conclusion.  (We 
shall  define  this  relation  shortly.)  Or  we  may  know  that  a  certain 
other  proposition  is  false,  and  that  there  is  a  relation  between 
the  two  of  the  sort  called  "  disjunction,"  expressed  by  "  p  or  ^,"  1 
so  that  the  knowledge  that  the  one  is  false  allows  us  to  infer 
that  the  other  is  true.  Again,  what  we  wish  to  infer  may  be 
the  falsehood  of  some  proposition,  not  its  truth.  This  may  be 
inferred  from  the  truth  of  another  proposition,  provided  we  know 
that  the  two  are  "  incompatible,"  i.e.  that  if  one  is  true,  the  other 
is  false.  It  may  also  be  inferred  from  the  falsehood  of  another 
proposition,  in  just  the  same  circumstances  in  which  the  truth 
of  the  other  might  have  been  inferred  from  the  truth  of  the  one  ; 
i.e.  from  the  falsehood  of  p  we  may  infer  the  falsehood  of  q,  when 
q  implies  p.  All  these  four  are  cases  of  inference.  When  our 
minds  are  fixed  upon  inference,  it  seems  natural  to  take  "  impli 
cation  "  as  the  primitive  fundamental  relation,  since  this  is  the 
relation  which  must  hold  between  p  and  q  if  we  are  to  be  able 
to  infer  the  truth  of  q  from  the  truth  of  p.  But  for  technical 
reasons  this  is  not  the  best  primitive  idea  to  choose.  Before 
proceeding  to  primitive  ideas  and  definitions,  let  us  consider 
further  the  various  functions  of  propositions  suggested  by  the 
above-mentioned  relations  of  propositions. 

The  simplest  of  such  functions  is  the  negative,  "  not-^>." 
This  is  that  function  of  p  which  is  true  when  p  is  false,  and  false 
when  p  is  true.  It  is  convenient  to  speak  of  the  truth  of  a  pro 
position,  or  its  falsehood,  as  its  "  truth-value  "  2  ;  i.e.  truth  is 
the  "  truth-value  "  of  a  true  proposition,  and  falsehood  of  a  false 
one.  Thus  not-£  has  the  opposite  truth-value  to  p. 

1  We  shall  use  the  letters  p,  q,  r,  s,  t  to  denote  variable  propositions. 

2  This  term  is  due  to  Frege. 


Incompatibility  and  the  Theory  of  Deduction        147 

We  may  take  next  disjunction,  "  p  or  <?."  This  is  a  function 
whose  truth-value  is  truth  when  p  is  true  and  also  when  q  is  true, 
but  is  falsehood  when  both  p  and  q  are  false. 

Next  we  may  take  conjunction,  "  p  and  q"  This  has  truth 
for  its  truth-value  when  p  and  q  are  both  true  ;  otherwise  it 
has  falsehood  for  its  truth-value. 

Take  next  incompatibility,  i.e.  "  p  and  q  are  not  both  true." 
This  is  the  negation  of  conjunction  ;  it  is  also  the  disjunction 
of  the  negations  of  p  and  q,  i.e.  it  is  "  not-/)  or  not-y."  Its  truth- 
value  is  truth  when  p  is  false  and  likewise  when  q  is  false  ;  its 
truth-value  is  falsehood  when  p  and  q  are  both  true. 

Last  take  implication,  i.e.  "  p  implies  q,"  or  "  if  p,  then  <?." 
This  is  to  be  understood  in  the  widest  sense  that  will  allow  us 
to  infer  the  truth  of  q  if  we  know  the  truth  of  p.  Thus  we  inter 
pret  it  as  meaning  :  "  Unless  p  is  false,  q  is  true,"  or  "  either 
p  is  false  or  q  is  true."  (The  fact  that  "  implies  "  is  capable 
of  other  meanings  does  not  concern  us  ;  this  is  the  meaning  which 
is  convenient  for  us.)  That  is  to  say,  "  p  implies  q  "  is  to  mean 
"  not-/>  or  q  "  :  its  truth-value  is  to  be  truth  if  p  is  false,  likewise 
if  q  is  true,  and  is  to  be  falsehood  if  p  is  true  and  q  is  false. 

We  have  thus  five  functions:  negation,  disjunction,  conjunction, 
incompatibility,  and  implication.  We  might  have  added  others, 
for  example,  joint  falsehood,  "  not-p  and  not-^,"  but  the  above 
five  will  suffice.  Negation  differs  from  the  other  four  in  being 
a  function  of  one  proposition,  whereas  the  others  are  functions 
of  two.  But  all  five  agree  in  this,  that  their  truth-value  depends 
only  upon  that  of  the  propositions  which  are  their  arguments. 
Given  the  truth  or  falsehood  of  p,  or  of  p  and  q  (as  the  case  may 
be),  we  are  given  the  truth  or  falsehood  of  the  negation,  disjunc 
tion,  conjunction,  incompatibility,  or  implication.  A  function  of 
propositions  which  has  this  property  is  called  a  "  truth-function." 

The  whole  meaning  of  a  truth-function  is  exhausted  by  the 
statement  of  the  circumstances  under  which  it  is  true  or  false. 
"  Not-/),"  for  example,  is  simply  that  function  of  p  which  is  true 
when  p  is  false,  and  false  when  p  is  true  :  there  is  no  further 


148  Introduction  to  Mathematical  Philosophy 

meaning  to  be  assigned  to  it.  The  same  applies  to  "  p  or  q  " 
and  the  rest.  It  follows  that  two  truth-functions  which  have 
the  same  truth-value  for  all  values  of  the  argument  are  indis 
tinguishable.  For  example,  "  p  and  q "  is  the  negation  of 
"  not-/)  or  not-^  "  and  vice  versa  ;  thus  either  of  these  may  be 
defined  as  the  negation  of  the  other.  There  is  no  further  meaning 
in  a  truth-function  over  and  above  the  conditions  under  which 
it  is  true  or  false. 

It  is  clear  that  the  above  five  truth-functions  are  not  all  inde 
pendent.  We  can  define  some  of  them  in  terms  of  others.  There 
is  no  great  difficulty  in  reducing  the  number  to  two ;  the  two 
chosen  in  Principia  Mathematica  are  negation  and  disjunction. 
Implication  is  then  defined  as  "  not-/)  or  q  "  ;  incompatibility 
as  "  not-/>  or  not-q  "  ;  conjunction  as  the  negation  of  incompati 
bility.  But  it  has  been  shown  by  Sheffer  *  that  we  can  be  content 
with  one  primitive  idea  for  all  five,  and  by  Nicod  2  that  this  enables 
us  to  reduce  the  primitive  propositions  required  in  the  theory 
of  deduction  to  two  non-formal  principles  and  one  formal  one. 
For  this  purpose,  we  may  take  as  our  one  indefinable  either 
incompatibility  or  joint  falsehood.  We  will  choose  the  former. 

Our  primitive  idea,  now,  is  a  certain  truth-function  called 
"  incompatibility,"  which  we  will  denote  by  p/q.  Negation 
can  be  at  once  defined  as  the  incompatibility  of  a  proposition 
with  itself,  i.e.  "  not-/)  "  is  defined  as  "  />//>."  Disjunction  is 
the  incompatibility  of  not-/)  and  not-<?,  i.e.  it  is  (p/p)\(q/q). 
Implication  is  the  incompatibility  of  p  and  not-^,  i.e.  p\(q/q)> 
Conjunction  is  the  negation  of  incompatibility,  i.e.  it  is  (p/q)  \ 
(p/q)'  Thus  all  our  four  other  functions  are  defined  in  terms 
of  incompatibility. 

It  is  obvious  that  there  is  no  limit  to  the  manufacture  of  truth- 
functions,  either  by  introducing  more  arguments  or  by  repeating 
arguments.  What  we  are  concerned  with  is  the  connection  of 
this  subject  with  inference. 

1  Trans.  Am.  Math.  Soc.,  vol.  xiv.  pp.  481-488. 

2  Proc.  Camb.  Phil.  Soc.,  vol.  xix.,  i.,  January  1917. 


Incompatibility  and  the  Theory  of  Deduction        149 

If  we  know  that  p  is  true  and  that  p  implies  q,  we  can  proceed 
to  assert  q.  There  is  always  unavoidably  something  psycho 
logical  about  inference  :  inference  is  a  method  by  which  we  arrive 
at  new  knowledge,  and  what  is  not  psychological  about  it  is  the 
relation  which  allows  us  to  infer  correctly  ;  but  the  actual  passage 
from  the  assertion  of  p  to  the  assertion  of  q  is  a  psychological 
process,  and  we  must  not  seek  to  represent  it  in  purely  logical 
terms. 

In  mathematical  practice,  when  we  infer,  we  have  always 
some  expression  containing  variable  propositions,  say  p  and  qy 
which  is  known,  in  virtue  of  its  form,  to  be  true  for  all  values 
of  p  and  q  ;  we  have  also  some  other  expression,  part  of  the  former, 
which  is  also  known  to  be  true  for  all  values  of  p  and  q ;  and  in 
virtue  of  the  principles  of  inference,  we  are  able  to  drop  this  part 
of  our  original  expression,  and  assert  what  is  left.  This  somewhat 
abstract  account  may  be  made  clearer  by  a  few  examples. 

Let  us  assume  that  we  know  the  five  formal  principles  of 
deduction  enumerated  in  Principia  Matbematica.  (M.  Nicod  has 
reduced  these  to  one,  but  as  it  is  a  complicated  proposition, 
we  will  begin  with  the  five.)  These  five  propositions  are  as 
follows  : — 

(1)  "  p  or  p  "  implies  p — i.e.  if  either  p  is  true  or  p  is  true, 
then  p  is  true. 

(2)  q  implies  "  p  or  q  " — i.e.  the  disjunction  "  p  or  q  "  is  true 
when  one  of  its  alternatives  is  true. 

(3)  "  p  or  q  "  implies  "  q  or  />."     This  would  not  be  required 
if  we  had  a  theoretically  more  perfect  notation,  since  in  the 
conception  of  disjunction  there  is  no  order  involved,  so  that 
"  p  or  q  "  and  "  q  or  p  "  should  be  identical.     But  since  our 
symbols,  in  any  convenient  form,  inevitably  introduce  an  order, 
we  need  suitable  assumptions   for  showing  that   the  order  is 
irrelevant. 

(4)  If  either  p  is  true  or  "  q  or  r  "  is  true,  then  either  q  is  true 
or  "  p  or  r "  is  true.     (The  twist  in  this  proposition  serves  to 
increase  its  deductive  power.) 


150  Introduction  to  Mathematica    Philosophy 

(5)  If  q  implies  r,  then  "  p  or  q  "  implies  "  p  or  r." 
These  are  the  formal  principles  of  deduction  employed  in 
Principia  Mathematica.  A  formal  principle  of  deduction  has  a 
double  use,  and  it  is  in  order  to  make  this  clear  that  we  have 
cited  the  above  five  propositions.  It  has  a  use  as  the  premiss 
of  an  inference,  and  a  use  as  establishing  the  fact  that  the  pre 
miss  implies  the  conclusion.  In  the  schema  of  an  inference 
we  have  a  proposition  p,  and  a  proposition  "  p  implies  £,"  from 
which  we  infer  q.  Now  when  we  are  concerned  with  the  princi 
ples  of  deduction,  our  apparatus  of  primitive  propositions  has 
to  yield  both  the  p  and  the  "  p  implies  q  "  of  our  inferences. 
That  is  to  say,  our  rules  of  deduction  are  to  be  used,  not  only  as 
rules,  which  is  their  use  for  establishing  "  p  implies  q"  but  also 
as  substantive  premisses,  i.e.  as  the  p  of  our  schema.  Suppose, 
for  example,  we  wish  to  prove  that  if  p  implies  q,  then  if  q 
implies  r  it  follows  that  p  implies  r.  We  have  here  a  relation  of 
three  propositions  which  state  implications.  Put 

pi=p  implies  q,  p2=q  implies  r,  and  p3=p  implies  r. 

Then  we  have  to  prove  that  p±  implies  that  pz  implies  ps.  Now 
take  the  fifth  of  our  above  principles,  substitute  not-/>  for  p, 
and  remember  that  "  not-/)  or  q  "  is  by  definition  the  same  as 
"  p  implies  q."  Thus  our  fifth  principle  yields  : 

"  If  q  implies  r,  then  '  p  implies  q  '  implies  '  p  implies  r,'  ' 
i.e.    "  p2  implies    that   p^   implies  p3."     Call  this  propo 
sition  A. 

But  the  fourth  of  our  principles,  when  we  substitute  not-/), 
not-£,  for  p  and  qy  and  remember  the  definition  of  implication, 
becomes  : 

"  If  p  implies  that  q  implies  r,  then  q  implies  that  p  implies  r." 

Writing  p2  in  place  of  p,  p±  in  place  of  q,  and  p3  in  place  of  ry  this 
becomes  : 

"  If  pz  implies  that  p1  implies  />3,  then  pl  implies  that  p2  implies 
1>"     Call  this  B. 


Incompatibility  and  the  Theory  of  Deduction        1 5 1 

Now  we  proved  by  means  of  our  fifth  principle  that 

"  p2  implies  that  p^  implies  p3"  which  was  what  we  called  A. 

Thus  we  have  here  an  instance  of  the  schema  of  inference, 
since  A  represents  the  p  of  our  scheme,  and  B  represents  the 
"  p  implies  q."  Hence  we  arrive  at  q,  namely, 

"  pl  implies  that  pz  implies  p3" 

which  was  the  proposition  to  be  proved.  In  this  proof,  the 
adaptation  of  our  fifth  principle,  which  yields  A,  occurs  as  a 
substantive  premiss  ;  while  the  adaptation  of  our  fourth  principle, 
which  yields  B,  is  used  to  give  the  form  of  the  inference.  The 
formal  and  material  employments  of  premisses  in  the  theory 
of  deduction  are  closely  intertwined,  and  it  is  not  very  important 
to  keep  them  separated,  provided  we  realise  that  they  are  in 
theory  distinct. 

The  earliest  method  of  arriving  at  new  results  from  a  premiss 
is  one  which  is  illustrated  in  the  above  deduction,  but  which 
itself  can  hardly  be  called  deduction.  The  primitive  propositions, 
whatever  they  may  be,  are  to  be  regarded  as  asserted  for  all 
possible  values  of  the  variable  propositions  p,  q,  r  which  occur 
in  them.  We  may  therefore  substitute  for  (say)  p  any  expression 
whose  value  is  always  a  proposition,  e.g.  not-p,  "  s  implies  t," 
and  so  on.  By  means  of  such  substitutions  we  really  obtain 
sets  of  special  cases  of  our  original  proposition,  but  from  a  prac 
tical  point  of  view  we  obtain  what  are  virtually  new  propositions. 
The  legitimacy  of  substitutions  of  this  kind  has  to  be  insured  by 
means  of  a  non-formal  principle  of  inference.1 

We  may  now  state  the  one  formal  principle  of  inference  to 
which  M.  Nicod  has  reduced  the  five  given  above.  For  this 
purpose  we  will  first  show  how  certain  truth-functions  can  be 
defined  in  terms  of  incompatibility.  We  saw  already  that 

p  |  (q/q)  means  "  p  implies  q" 

1  No  such  principle  is  enunciated  in  Pnncipia  Mathematics,  or  in  M. 
Nicod's  article  mentioned  above.  But  this  would  seem  to  be  an  omission, 


152  Introduction  to  Mathematical  Philosophy 

We  now  observe  that 

p  |  (q/r)  means  "  p  implies  both  q  and  r" 

For  this  expression  means  "  p  is  incompatible  with  the  incom 
patibility  of  q  and  r,"  i.e.  "  p  implies  that  q  and  r  are  not  incom 
patible,"  i.e.  "  p  implies  that  q  and  r  are  both  true  "  —  for,  as 
we  saw,  the  conjunction  of  q  and  r  is  the  negation  of  their 
incompatibility. 

Observe  next  that  t  \  (t/t)  means  "  t  implies  itself."  This  is  a 
particular  case  of  p  \  (q/q). 

Let  us  write  p  for  the  negation  of  p  ;  thus  p/s  will  mean  the 
negation  of  p/s,  i.e.  it  will  mean  the  conjunction  of  p  and  s.  It 
follows  that 

<V?)f?A 

expresses  the  incompatibility  of  s/q  with  the  conjunction  of 
p  and  s  ;  in  other  words,  it  states  that  if  p  and  s  are  both  true, 
s/q  is  false,  i.e.  s  and  q  are  both  true  ;  in  still  simpler  words, 
it  states  that  p  and  s  jointly  imply  s  and  q  jointly. 

Now,  put  P=p  |  (q/r), 


Q=(s/q)\p/s. 
Then  M.  Nicod's  sole  formal  principle  of  deduction  is 

Pk/Q, 

in  other  words,  P  implies  both  TT  and  Q. 

He  employs  in  addition  one  non-formal  principle  belonging 
to  the  theory  of  types  (which  need  not  concern  us),  and  one 
corresponding  to  the  principle  that,  given  p,  and  given  that 
p  implies  q,  we  can  assert  q.  This  principle  is  : 

"If  p  |  (r/q)  is  true,  and  p  is  true,  then  q  is  true."  From 
this  apparatus  the  whole  theory  of  deduction  follows,  except 
in  so  far  as  we  are  concerned  with  deduction  from  or  to  the 
existence  or  the  universal  truth  of  "  prepositional  functions," 
which  we  shall  consider  in  the  next  chapter. 

There  is?  if  I  am  not  mistaken,  a  certain  confusion  in  the 


Incompatibility  and  the  Theory  of  Deauction        153 

minds  of  some  authors  as  to  the  relation,  between  propositions, 
in  virtue  of  which  an  inference  is  valid.  In  order  that  it  may 
be  valid  to  infer  q  from  />,  it  is  only  necessary  that  p  should  be 
true  and  that  the  proposition  "  not-/>  or  q  "  should  be  true. 
Whenever  this  is  the  case,  it  is  clear  that  q  must  be  true.  But 
inference  will  only  in  fact  take  place  when  the  proposition  "  not-/> 
or  q  "  is  known  otherwise  than  through  knowledge  of  not-/)  or 
knowledge  of  q.  Whenever  p  is  false,  "  not-/>  or  q  "  is  true, 
but  is  useless  for  inference,  which  requires  that  p  should  be  true. 
Whenever  q  is  already  known  to  be  true,  "  not-/)  or  q  "  is  of 
course  also  known  to  be  true,  but  is  again  useless  for  inference, 
since  q  is  already  known,  and  therefore  does  not  need  to  be 
inferred.  In  fact,  inference  only  arises  when  "  not-/)  or  q " 
can  be  known  without  our  knowing  already  which  of  the  two 
alternatives  it  is  that  makes  the  disjunction  true.  Now,  the 
circumstances  under  which  this  occurs  are  those  in  which  certain 
relations  of  form  exist  between  p  and  q.  For  example,  we  know 
that  if  r  implies  the  negation  of  s,  then  s  implies  the  negation 
of  r .  Between  "  r  implies  not-5  "  and  "  s  implies  not-r  "  there 
is  a  formal  relation  which  enables  us  to  know  that  the  first  implies 
the  second,  without  having  first  to  know  that  the  first  is  false 
or  to  know  that  the  second  is  true.  It  is  under  such  circum 
stances  that  the  relation  of  implication  is  practically  useful  for 
drawing  inferences. 

But  this  formal  relation  is  only  required  in  order  that  we  may 
be  able  to  know  that  either  the  premiss  is  false  or  the  conclusion 
is  true.  It  is  the  truth  of  "  not-/)  or  q  "  that  is  required  for 
the  validity  of  the  inference  ;  what  is  required  further  is  only 
required  for  the  practical  feasibility  of  the  inference.  Professor 
C.  I.  Lewis  *  has  especially  studied  the  narrower,  formal  relation 
which  we  may  call  "  formal  deducibility."  He  urges  that  the 
wider  relation,  that  expressed  by  "  not-/>  or  q"  should  not  be 
called  "  implication."  That  is,  however,  a  matter  of  words. 

1  See  Mind,  vol.  xxi.,  1912,  pp.  522-531 ;  and  vol.  xxiii.,  1914,  pp. 
240-247. 


154  Introduction  to  Mathematical  Philosophy 

Provided  our  use  of  words  is  consistent,  it  matters  little  how  we 
define  them.  The  essential  point  of  difference  between  the 
theory  which  I  advocate  and  the  theory  advocated  by  Professor 
Lewis  is  this  :  He  maintains  that,  when  one  proposition  q  is 
"  formally  deducible "  from  another  p,  the  relation  which  we 
perceive  between  them  is  one  which  he  calls  "  strict  implication," 
which  is  not  the  relation  expressed  by  "  not-p  or  q  "  but  a  narrower 
relation,  holding  only  when  there  are  certain  formal  connections 
between  p  and  q.  I  maintain  that,  whether  or  not  there  be 
such  a  relation  as  he  speaks  of,  it  is  in  any  case  one  that  mathe 
matics  does  not  need,  and  therefore  one  that,  on  general  grounds 
of  economy,  ought  not  to  be  admitted  into  our  apparatus  of 
fundamental  notions ;  that,  whenever  the  relation  of  "  formal 
deducibility  "  holds  between  two  propositions,  it  is  the  case  that 
we  can  see  that  either  the  first  is  false  or  the  second  true,  and  that 
nothing  beyond  this  fact  is  necessary  to  be  admitted  into  our 
premisses ;  and  that,  finally,  the  reasons  of  detail  which  Professor 
Lewis  adduces  against  the  view  which  I  advocate  can  all  be  met 
in  detail,  and  depend  for  their  plausibility  upon  a  covert  and 
unconscious  assumption  of  the  point  of  view  which  I  reject. 
I  conclude,  therefore,  that  there  is  no  need  to  admit  as  a  funda 
mental  notion  any  form  of  implication  not  expressible  as  a 
truth-function. 


CHAPTER  XV 

PROPOSITIONAL   FUNCTIONS 

WHEN,  in  the  preceding  chapter,  we  were  discussing  propositions, 
we  did  not  attempt  to  give  a  definition  of  the  word  "  proposition." 
But  although  the  word  cannot  be  formally  defined,  it  is  necessary 
to  say  something  as  to  its  meaning,  in  order  to  avoid  the  very 
common  confusion  with  "  prepositional  functions,"  which  are  to 
be  the  topic  of  the  present  chapter. 

We  mean  by  a  "  proposition  "  primarily  a  form  of  words  which 
expresses  what  is  either  true  or  false.  I  say  "  primarily," 
because  I  do  not  wish  to  exclude  other  than  verbal  symbols,  or 
even  mere  thoughts  if  they  have  a  symbolic  character.  But  I 
think  the  word  "  proposition  "  should  be  limited  to  what  may, 
in  some  sense,  be  called  "  symbols,"  and  further  to  such  symbols 
as  give  expression  to  truth  and  falsehood.  Thus  "  two  and  two 
are  four  "  and  "  two  and  two  are  five  "  will  be  propositions, 
and  so  will  "  Socrates  is  a  man  "  and  "  Socrates  is  not  a  man." 
The  statement :  "  Whatever  numbers  a  and  b  may  be, 
a*+2ab+b2"  is  a  proposition  ;  but  the  bare  formula  " 
a?-\-2ab-\-b2  "  alone  is  not,  since  it  asserts  nothing  definite  unless 
we  are  further  told,  or  led  to  suppose,  that  a  and  b  are  to  have 
all  possible  values,  or  are  to  have  such-and-such  values.  The 
former  of  these  is  tacitly  assumed,  as  a  rule,  in  the  enunciation 
of  mathematical  formulae,  which  thus  become  propositions ; 
but  if  no  such  assumption  were  made,  they  would  be  "  preposi 
tional  functions."  A  "  prepositional  function,"  in  fact,  is  an 
expression  containing  one  or  more  undetermined  constituents, 


156  Introduction  to  Mathematical  Philosophy 

such  that,  when  values  are  assigned  to  these  constituents,  the 
expression  becomes  a  proposition.  In  other  words,  it  is  a  function 
whose  values  are  propositions.  But  this  latter  definition  must 
be  used  with  caution.  A  descriptive  function,  e.g.  "  the  hardest 
proposition  in  A's  mathematical  treatise,"  will  not  be  a  pro- 
positional  function,  although  its  values  are  propositions.  But  in 
such  a  case  the  propositions  are  only  described  :  in  a  proposi- 
tional  function,  the  values  must  actually  enunciate  propositions. 

Examples  of  prepositional  functions  are  easy  to  give :  "  x 
is  human  "  is  a  prepositional  function  ;  so  long  as  x  remains 
undetermined,  it  is  neither  true  nor  false,  but  when  a  value 
is  assigned  to  x  it  becomes  a  true  or  false  proposition.  Any 
mathematical  equation  is  a  prepositional  function.  So  long  as 
the  variables  have  no  definite  value,  the  equation  is  merely  an 
expression  awaiting  determination  in  order  to  become  a  true  or 
false  proposition.  If  it  is  an  equation  containing  one  variable, 
it  becomes  true  when  the  variable  is  made  equal  to  a  root 
of  the  equation,  otherwise  it  becomes  false ;  but  if  it  is  an 
"  identity  "  it  will  be  true  when  the  variable  is  any  number. 
The  equation  to  a  curve  in  a  plane  or  to  a  surface  in  space  is  a 
propositional  function,  true  for  values  of  the  co-ordinates  belong 
ing  to  points  on  the  curve  or  surface,  false  for  other  values. 
Expressions  of  traditional  logic  such  as  "  all  A  is  B  "  are  pro- 
positional  functions  :  A  and  B  have  to  be  determined  as  definite 
classes  before  such  expressions  become  true  or  false. 

The  notion  of  "  cases  "  or  "  instances  "  depends  upon  pro- 
positional  functions.  Consider,  for  example,  the  kind  of  process 
suggested  by  what  is  called  "  generalisation,"  and  let  us  take 
some  very  primitive  example,  say,  "  lightning  is  followed  by 
thunder."  We  have  a  number  of  "  instances "  of  this,  i.e.  a 
number  of  propositions  such  as  :  "  this  is  a  flash  of  lightning 
and  is  followed  by  thunder."  What  are  these  occurrences 
"  instances "  of  ?  They  are  instances  of  the  propositional 
function  :  "  If  x  is  a  flash  of  lightning,  x  is  followed  by  thunder." 
The  process  of  generalisation  (with  whose  validity  we  are  fortun- 


Prepositional  Functions  157 

ately  not  concerned)  consists  in  passing  from  a  number  of  such 
instances  to  the  universal  truth  of  the  prepositional  function  : 
"  If  x  is  a  flash  of  lightning,  x  is  followed  by  thunder."  It  will 
be  found  that,  in  an  analogous  way,  prepositional  functions 
are  always  involved  whenever  we  talk  of  instances  or  cases  or 
examples. 

We  do  not  need  to  ask,  or  attempt  to  answer,  the  question  : 
"  What  is  a  prepositional  function  ?  "  A  prepositional  function 
standing  all  alone  may  be  taken  to  be  a  mere  schema,  a  mere 
shell,  an  empty  receptacle  for  meaning,  not  something  already 
significant.  We  are  concerned  with  prepositional  functions, 
broadly  speaking,  in  two  ways  :  first,  as  involved  in  the  notions 
"  true  in  all  cases  "  and  "  true  in  some  cases  "  ;  secondly,  as 
involved  in  the  theory  of  classes  and  relations.  The  second  of 
these  topics  we  will  postpone  to  a  later  chapter  ;  the  first  must 
occupy  us  now. 

When  we  say  that  something  is  "  always  true  "  or  "  true  in 
all  cases,"  it  is  clear  that  the  "  something  "  involved  cannot  be 
a  proposition.  A  proposition  is  just  true  or  false,  and  there 
is  an  end  of  the  matter.  There  are  no  instances  or  cases  of 
"  Socrates  is  a  man  "  or  "  Napoleon  died  at  St  Helena."  These 
are  propositions,  and  it  would  be  meaningless  to  speak  of  their 
being  true  "  in  all  cases."  This  phrase  is  only  applicable  to 
prepositional  functions.  Take,  for  example,  the  sort  of  thing 
that  is  often  said  when  causation  is  being  discussed.  (We  are 
net  concerned  with  the  truth  or  falsehood  of  what  is  said,  but 
only  with  its  logical  analysis.)  We  are  told  that  A  is,  in  every 
instance,  followed  by  B.  Now  if  there  are  "  instances  "  of  A, 
A  must  be  some  general  concept  of  which  it  is  significant  to  say 
"  #!  is  A,"  "  x2  is  A,"  "  #3  is  A,"  and  so  on,  where  xl9  x2,  x3  are 
particulars  which  are  not  identical  one  with  another.  This 
applies,  e.g.,  to  our  previous  case  of  lightning.  We  say  that 
lightning  (A)  is  followed  by  thunder  (B).  But  the  separate 
flashes  are  particulars,  not  identical,  but  sharing  the  common 
property  of  being  lightning.  The  only  way  of  expressing  a 


158  Introduction  to  Mathematical  Philosophy 

common  property  generally  is  to  say  that  a  common  property 
of  a  number  of  objects  is  a  prepositional  function  which  becomes 
true  when  any  one  of  these  objects  is  taken  as  the  value  of  the 
variable.  In  this  case  all  the  objects  are  "  instances  "  of  the 
truth  of  the  prepositional  function — for  a  prepositional  function, 
though  it  cannot  itself  be  true  or  false,  is  true  in  certain  instances 
and  false  in  certain  others,  unless  it  is  "  always  true  "  or  "  always 
false."  When,  to  return  to  our  example,  we  say  that  A  is  in 
every  instance  followed  by  B,  we  mean  that,  whatever  x  may  be, 
if  x  is  an  A,  it  is  followed  by  a  B  ;  that  is,  we  are  asserting  that 
a  certain  propositional  function  is  "  always  true." 

Sentences  involving  such  words  as  "  all,"  "  every,"  "  a," 
"  the,"  "  some  "  require  propositional  functions  for  their  inter 
pretation.  The  way  in  which  propositional  functions  occur 
can  be  explained  by  means  of  two  of  the  above  words,  namely, 
"  all  "  and  "  some." 

There  are,  in  the  last  analysis,  only  two  things  that  can  be 
done  with  a  propositional  function  :  one  is  to  assert  that  it  is 
true  in  all  cases,  the  other  to  assert  that  it  is  true  in  at  least  one 
case,  or  in  some  cases  (as  we  shall  say,  assuming  that  there  is 
to  be  no  necessary  implication  of  a  plurality  of  cases).  All  the 
other  uses  of  propositional  functions  can  be  reduced  to  these  two. 
When  we  say  that  a  propositional  function  is  true  "  in  all  cases," 
or  "  always  "  (as  we  shall  also  say,  without  any  temporal  sugges 
tion),  we  mean  that  all  its  values  are  true.  If  "  fa  "  is  the 
function,  and  a  is  the  right  sort  of  object  to  be  an  argument  to 
"  fa,"  then  (f>a  is  to  be  true,  however  a  may  have  been  chosen. 
For  example,  "  if  a  is  human,  a  is  mortal  "  is  true  whether  a 
is  human  or  not ;  in  fact,  every  proposition  of  this  form  is  true. 
Thus  the  propositional  function  "  if  x  is  human,  x  is  mortal " 
is  "  always  true,"  or  "  true  in  all  cases."  Or,  again,  the  state 
ment  "  there  are  no  unicorns  "  is  the  same  as  the  statement 
"  the  propositional  function  *  x  is  not  a  unicorn '  is  true  in  all 
cases."  The  assertions  in  the  preceding  chapter  about  pro 
positions,  e.g.  "  '  p  or  q  '  implies  *  q  or  p,  "  are  really  assertions 


Prepositional  Functions  159 

that  certain  prepositional  functions  are  true  in  all  cases.  We  do 
not  assert  the  above  principle,  for  example,  as  being  true  only 
of  this  or  that  particular  p  or  q,  but  as  being  true  of  any  p  or  q 
concerning  which  it  can  be  made  significantly.  The  condition 
that  a  function  is  to  be  significant  for  a  given  argument  is  the  same 
as  the  condition  that  it  shall  have  a  value  for  that  argument, 
either  true  or  false.  The  study  of  the  conditions  of  significance 
belongs  to  the  doctrine  of  types,  which  we  shall  not  pursue 
beyond  the  sketch  given  in  the  preceding  chapter. 

Not  only  the  principles  of  deduction,  but  all  the  primitive 
propositions  of  logic,  consist  of  assertions  that  certain  preposi 
tional  functions  are  always  true.  If  this  were  not  the  case,  they 
would  have  to  mention  particular  things  or  concepts — Socrates, 
or  redness,  or  east  and  west,  or  what  not, — and  clearly  it  is  not 
the  province  of  logic  to  make  assertions  which  are  true  concerning 
one  such  thing  or  concept  but  not  concerning  another.  It  is 
part  of  the  definition  of  logic  (but  not  the  whole  of  its  definition) 
that  all  its  propositions  are  completely  general,  i.e.  they  all 
consist  of  the  assertion  that  some  propositional  function  con 
taining  no  constant  terms  is  always  true.  We  shall  return  in 
our  final  chapter  to  the  discussion  of  propositional  functions 
containing  no  constant  terms.  For  the  present  we  will  proceed 
to  the  other  thing  that  is  to  be  done  with  a  propositional  function, 
namely,  the  assertion  that  it  is  "  sometimes  true,"  i.e.  true  in  at 
least  one  instance. 

When  we  say  "  there  are  men,"  that  means  that  the  pro- 
positional  function  "  x  is  a  man  "  is  sometimes  true.  When  we 
say  "  some  men  are  Greeks,"  that  means  that  the  propositional 
function  "  x  is  a  man  and  a  Greek  "  is  sometimes  true.  When  we 
say  "  cannibals  still  exist  in  Africa,"  that  means  that  the  pro- 
positional  function  "  x  is  a  cannibal  now  in  Africa  "  is  sometimes 
true,  i.e.  is  true  for  some  values  of  x.  To  say  "  there  are  at  least 
n  individuals  in  the  world "  is  to  say  that  the  propositional 
function  "  a  is  a  class  of  individuals  and  a  member  of  the  cardinal 
number  n  "  is  sometimes  true,  or,  as  we  may  say,  is  true  for  certain 


160  Introduction  to  Mathematical  Philosophy 

values  of  a.  This  form  of  expression  is  more  convenient  when  it 
is  necessary  to  indicate  which  is  the  variable  constituent  which 
we  are  taking  as  the  argument  to  our  prepositional  function. 
For  example,  the  above  prepositional  function,  which  we  may 
shorten  to  "  a,  is  a  class  of  n  individuals,"  contains  two  variables, 
a  and  n.  The  axiom  of  infinity,  in  the  language  of  prepositional 
functions,  is  :  "  The  prepositional  function  *  if  n  is  an  inductive 
number,  it  is  true  for  some  values  of  a  that  a  is  a  class  of  n  indi 
viduals  '  is  true  for  all  possible  values  of  «."  Here  there  is  a 
subordinate  function,  "  a  is  a  class  of  n  individuals,"  which  is 
said  to  be,  in  respect  of  a,  sometimes  true ;  and  the  assertion 
that  this  happens  if  n  is  an  inductive  number  is  said  to  be,  in 
respect  of  »,  always  true. 

The  statement  that  a  function  fa  is  always  true  is  the  negation 
of  the  statement  that  not- fa  is  sometimes  true,  and  the  state 
ment  that  fa  is  sometimes  true  is  the  negation  of  the  state 
ment  that  Tiot-fa  is  always  true.  Thus  the  statement  "  all 
men  are  mortals  "  is  the  negation  of  the  statement  that  the 
function  "  x  is  an  immortal  man  "  is  sometimes  true.  And  the 
statement  "  there  are  unicorns "  is  the  negation  of  the  state 
ment  that  the  function  "  x  is  not  a  unicorn  "  is  always  true.1 
We  say  that  fa  is  "  never  true  "  or  "  always  false  "  if  not-fa  is 
always  true.  We  can,  if  we  choose,  take  one  of  the  pair  "  always," 
"  sometimes  "  as  a  primitive  idea,  and  define  the  other  by  means 
of  the  one  and  negation.  Thus  if  we  choose  "  sometimes  "  as 
our  primitive  idea,  we  can  define  :  "  '  (f>x  is  always  true '  is  to 
mean  *  it  is  false  that  not- fa  is  sometimes  true.' "  2  But  for 
reasons  connected  with  the  theory  of  types  it  seems  more  correct 
to  take  both  "  always  "  and  "  sometimes  "  as  primitive  ideas, 
and  define  by  their  means  the  negation  of  propositions  in  which 
they  occur.  That  is  to  say,  assuming  that  we  have  already 

1  The    method    of    deduction    is    given    in    Principia    Mathematica, 
vol.  i.  *  9. 

2  For  linguistic  reasons,  to  avoid  suggesting  either  the  plural  or  the 
singular,  it  is  often  convenient  to  say  "  yx  is  not  always  false  "  rather 
than  "  cpx  sometimes  "  or  "  <px  is  sometimes  true." 


Prepositional  Functions  161 

defined  (or  adopted  as  a  primitive  idea)  the  negation  of  pro 
positions  of  the  type  to  which  x  belongs,  we  define  :  "  The 
negation  of  '  </>x  always  '  is  *  not-0#  sometimes  ' ;  and  the  nega 
tion  of  '  (j>x  sometimes  '  is  *  not-<£#  always.'  '  In  like  manner 
we  can  re-define  disjunction  and  the  other  truth-functions, 
as  applied  to  propositions  containing  apparent  variables,  in 
terms  of  the .  definitions  and  primitive  ideas  for  propositions 
containing  no  apparent  variables.  Propositions  containing  no 
apparent  variables  are  called  "  elementary  propositions."  From 
these  we  can  mount  up  step  by  step,  using  such  methods  as  have 
just  been  indicated,  to  the  theory  of  truth-functions  as  applied 
to  propositions  containing  one,  two,  three  .  .  .  variables,  or  any 
number  up  to  n,  where  n  is  any  assigned  finite  number. 

The  forms  which  are  taken  as  simplest  in  traditional  formal 
logic  are  really  far  from  being  so,  and  all  involve  the  assertion 
of  all  values  or  some  values  of  a  compound  prepositional  function. 
Take,  to  begin  with,  "  all  S  is  P."  We  will  take  it  that  S  is 
defined  by  a  prepositional  function  </>x,  and  P  by  a  prepositional 
function  i/jx.  E.g.,  if  S  is  men,  <j)X  will  be  "  x  is  human  "  ;  if  P  is 
mortals,  t/jx  will  be  "  there  is  a  time  at  which  x  dies."  Then 
"  all  S  is  P  "  means  :  "  '  <f>x  implies  i/jx '  is  always  true."  It  is 
to  be  observed  that  "  all  S  is  P  "  does  not  apply  only  to  those 
terms  that  actually  are  S's  ;  it  says  something  equally  about 
terms  which  are  not  S's.  Suppose  we  come  across  an  x  of  which 
we  do  not  know  whether  it  is  an  S  or  not ;  still,  our  statement 
"  all  S  is  P  "  tells  us  something  about  x,  namely,  that  if  x  is  an  S, 
then  x  is  a  P.  And  this  is  every  bit  as  true  when  x  is  not  an  S  as 
when  x  is  an  S.  If  it  were  not  equally  true  in  both  cases,  the 
reductio  ad  absurdum  would  not  be  a  valid  method  ;  for  the 
essence  of  this  method  consists  in  using  implications  in  cases 
where  (as  it  afterwards  turns  out)  the  hypothesis  is  false.  We  may 
put  the  matter  another  way.  In  order  to  understand  "  all  S  is  P," 
it  is  not  necessary  to  be  able  to  enumerate  what  terms  are  S's  ; 
provided  we  know  what  is  meant  by  being  an  S  and  what  by 
being  a  P,  we  can  understand  completely  what  is  actually  affirmed 

II 


1 62  Introduction  to  Mathematical  Philosophy 

by  "  all  S  is  P,"  however  little  we  may  know  of  actual  instances 
of  either.  This  shows  that  it  is  not  merely  the  actual  terms  that 
are  S's  that  are  relevant  in  the  statement  "  all  S  is  P,"  but  all  the 
terms  concerning  which  the  supposition  that  they  are  S's  is 
significant,  i.e.  all  the  terms  that  are  S's,  together  with  all  the 
terms  that  are  not  S's — i.e.  the  whole  of  the  appropriate  logical 
"  type."  What  applies  to  statements  about  all  applies  also  to 
statements  about  some.  "  There  are  men,"  e.g.,  means  that 
"  x  is  human  "  is  true  for  some  values  of  x.  Here  all  values  of  x 
(i.e.  all  values  for  which  "  x  is  human  "  is  significant,  whether 
true  or  false)  are  relevant,  and  not  only  those  that  in  fact  are 
human.  (This  becomes  obvious  if  we  consider  how  we  could 
prove  such  a  statement  to  be  false.)  Every  assertion  about 
"  all "  or  "  some  "  thus  involves  not  only  the  arguments  that 
make  a  certain  function  true,  but  all  that  make  it  significant, 
i.e.  all  for  which  it  has  a  value  at  all,  whether  true  or  false. 

We  may  now  proceed  with  our  interpretation  of  the  traditional 
forms  of  the  old-fashioned  formal  logic.  We  assume  that  S 
is  those  terms  x  for  which  fa  is  true,  and  P  is  those  for  which  fa 
is  true.  (As  we  shall  see  in  a  later  chapter,  all  classes  are  derived 
in  this  way  from  prepositional  functions.)  Then  : 

"  All  S  is  P  "  means  "  '  fa  implies  fa  '  is  always  true." 
"  Some  S  is  P  "  means  "  *  fa  and  fa '  is  sometimes  true." 
"  No  S  is  P  "  means  "  '  fa  implies  not-fa  '  is  always  true." 
"  Some  S  is  not  P  "  means  " '  fa  and  not-fa '  is  sometimes 
true." 

It  will  be  observed  that  the  propositional  functions  which  are 
here  asserted  for  all  or  some  values  are  not  fa  and  fa  them 
selves,  but  truth-functions  of  fa  and  fa  for  the  same  argument 
x.  The  easiest  way  to  conceive  of  the  sort  of  thing  that  is 
intended  is  to  start  not  from  fa  and  fa  in  general,  but  from 
(j>a  and  ipa,  where  a  is  some  constant.  Suppose  we  are  consider 
ing  all  "  men  are  mortal  "  :  we  will  begin  with 

"  If  Socrates  is  human,  Socrates  is  mortal," 


Prepositional  Functions  163 

and  then  we  will  regard  "  Socrates  "  as  replaced  by  a  variable  x 
wherever  "  Socrates  "  occurs.  The  object  to  be  secured  is  that, 
although  x  remains  a  variable,  without  any  definite  value,  yet 
it  is  to  have  the  same  value  in  "  fa  "  as  in  "  fa  "  when  we  are 
asserting  that  "  fa  implies  fa  "  is  always  true.  This  requires 
that  we  shall  start  with  a  function  whose  values  are  such  as 
"  cf>a  implies  $a"  rather  than  with  two  separate  functions  fa 
and  fa ;  for  if  we  start  with  two  separate  functions  we  can 
never  secure  that  the  x,  while  remaining  undetermined,  shall 
have  the  same  value  in  both. 

For  brevity  we  say  "  fa  always  implies  iftx "  when  we 
mean  that  "  (j>x  implies  fa "  is  always  true.  Propositions 
of  the  form  "  fa  always  implies  fa "  are  called  "  formal 
implications  "  ;  this  name  is  given  equally  if  there  are  several 
variables. 

The  above  definitions  show  how  far  removed  from  the  simplest 
forms  are  such  propositions  as  "  all  S  is  P,"  with  which  tradi 
tional  logic  begins.  It  is  typical  of  the  lack  of  analysis  involved 
that  traditional  logic  treats  "all  S  is  P  "  as  a  proposition  of 
the  same  form  as  "  x  is  P  " — e.g.,  it  treats  "  all  men  are  mortal  " 
as  of  the  same  form  as  "  Socrates  is  mortal."  As  we  have  just 
seen,  the  first  is  of  the  form  "  fa  always  implies  fa"  while  the 
second  is  of  the  form  "  fa"  The  emphatic  separation  of  these 
two  forms,  which  was  effected  by  Peano  and  Frege,  was  a  very 
vital  advance  in  symbolic  logic. 

It  will  be  seen  that  "  all  S  is  P  "  and  "  no  S  is  P  "  do  not 
really  differ  in  form,  except  by  the  substitution  of  not-iffx  for  fa, 
and  that  the  same  applies  to  "  some  S  is  P  "  and  "  some  S  is 
not  P."  It  should  also  be  observed  that  the  traditional  rules 
of  conversion  are  faulty,  if  we  adopt  the  view,  which  is  the  only 
technically  tolerable  one,  that  such  propositions  as  "  all  S  is  P  " 
do  not  involve  the  "  existence  "  of  S's,  i.e.  do  not  require  that 
there  should  be  terms  which  are  S's.  The  above  definitions 
lead  to  the  result  that,  if  fa  is  always  false,  i.e.  if  there  are  no 
S's,  then  "  all  S  is  P  "  and  «  no  S  is  P  "  will  both  be  true,  what- 


164  Introduction  to  Mathematical  Philosophy 

ever  P  may  be.  For,  according  to  the  definition  in  the  last 
chapter,  "  fa  implies  fa "  means  "  not- fa  or  fa"  which  is 
always  true  if  not-<£#  is  always  true.  At  the  first  moment, 
this  result  might  lead  the  reader  to  desire  different  definitions, 
but  a  little  practical  experience  soon  shows  that  any  different 
definitions  would  be  inconvenient  and  would  conceal  the  important 
ideas.  The  proposition  "  fa  always  implies  fa,  and  fa 
is  sometimes  true  "  is  essentially  composite,  and  it  would  be 
very  awkward  to  give  this  as  the  definition  of  "all  S  is  P," 
for  then  we  should  have  no  language  left  for  "  fa  always  implies 
fa,"  which  is  needed  a  hundred  times  for  once  that  the  other  is 
needed.  But,  with  our  definitions,  "  all  S  is  P  "  does  not  imply 
"  some  S  is  P,"  since  the  first  allows  the  non-existence  of  S  and 
the  second  does  not;  thus  conversion  per  accidens  becomes 
invalid,  and  some  moods  of  the  syllogism  are  fallacious,  e.g. 
Darapti :  "  All  M  is  S,  all  M  is  P,  therefore  some  S  is  P,"  which 
fails  if  there  is  no  M. 

The  notion  of  "  existence  "  has  several  forms,  one  of  which 
will  occupy  us  in  the  next  chapter ;  but  the  fundamental  form 
is  that  which  is  derived  immediately  from  the  notion  of  "  some 
times  true."  We  say  that  an  argument  a  (<  satisfies  "  a  function 
fa  if  <f>a  is  true  ;  this  is  the  same  sense  in  which  the  roots  of  an 
equation  are  said  to  satisfy  the  equation.  Now  if  fa  is  sometimes 
true,  we  may  say  there  are  #'s  for  which  it  is  true,  or  we  may  say 
"  arguments  satisfying  fa  exist"  This  is  the  fundamental  mean 
ing  of  the  word  "  existence."  Other  meanings  are  either  derived 
from  this,  or  embody  mere  confusion  of  thought.  We  may 
correctly  say  "  men  exist,"  meaning  that  "  x  is  a  man  "  is  some 
times  true.  But  if  we  make  a  pseudo-syllogism  :  "  Men  exist, 
Socrates  is  a  man,  therefore  Socrates  exists,"  we  are  talking 
nonsense,  since  "  Socrates  "  is  not,  like  "  men,"  merely  an  un 
determined  argument  to  a  given  prepositional  function.  The 
fallacy  is  closely  analogous  to  that  of  the  argument :  "  Men  are 
numerous,  Socrates  is  a  man,  therefore  Socrates  is  numerous." 
In  this  case  it  is  obvious  that  the  conclusion  is  nonsensical,  but 


Prepositional  Functions  165 

in  the  case  of  existence  it  is  not  obvious,  for  reasons  which  will 
appear  more  fully  in  the  next  chapter.  For  the  present  let  us 
merely  note  the  fact  that,  though  it  is  correct  to  say  "  men  exist," 
it  is  incorrect,  or  rather  meaningless,  to  ascribe  existence  to  a 
given  particular  x  who  happens  to  be  a  man.  Generally,  "  terms 
satisfying  fa  exist"  means  "fa  is  sometimes  true";  but  "a 
exists  "  (where  a  is  a  term  satisfying  fa)  is  a  mere  noise  or  shape, 
devoid  of  significance.  It  will  be  found  that  by  bearing  in  mind 
this  simple  fallacy  we  can  solve  many  ancient  philosophical 
puzzles  concerning  the  meaning  of  existence. 

Another  set  of  notions  as  to  which  philosophy  has  allowed 
itself  to  fall  into  hopeless  confusions  through  not  sufficiently 
separating  propositions  and  prepositional  functions  are  the 
notions  of  "  modality "  :  necessary,  possible,  and  impossible. 
(Sometimes  contingent  or  assertoric  is  used  instead  of  possible) 
The  traditional  view  was  that,  among  true  propositions,  some 
were  necessary,  while  others  were  merely  contingent  or  assertoric  ; 
while  among  false  propositions  some  were  impossible,  namely, 
those  whose  contradictories  were  necessary,  while  others  merely 
happened  not  to  be  true.  In  fact,  however,  there  was  never 
any  clear  account  of  what  was  added  to  truth  by  the  conception 
of  necessity.  In  the  case  of  prepositional  functions,  the  three 
fold  division  is  obvious.  If  "  fa  "  is  an  undetermined  value  of  a 
certain  prepositional  function,  it  will  be  necessary  if  the  function 
is  always  true,  possible  if  it  is  sometimes  true,  and  impossible  if 
it  is  never  true.  This  sort  of  situation  arises  in  regard  to  prob 
ability,  for  example.  Suppose  a  ball  x  is  drawn  from  a  bag 
which  contains  a  number  of  balls  :  if  all  the  balls  are  white, 
"  x  is  white  "  is  necessary ;  if  some  are  white,  it  is  possible ; 
if  none,  it  is  impossible.  Here  all  that  is  known  about  x  is  that 
it  satisfies  a  certain  prepositional  function,  namely,  "  x  was  a 
ball  in  the  bag."  This  is  a  situation  which  is  general  in  prob 
ability  problems  and  not  uncommon  in  practical  life — e.g.  when 
a  person  calls  of  whom  we  know  nothing  except  that  he  brings 
a  letter  of  introduction  from  our  friend  so-and-so.  In  all  such 


1 66  Introduction  to  Mathematical  Philosophy 

cases,  as  in  regard  to  modality  in  general,  the  prepositional 
function  is  relevant.  For  clear  thinking,  in  many  very  diverse 
directions,  the  habit  of  keeping  prepositional  functions  sharply 
separated  from  propositions  is  of  the  utmost  importance,  and 
the  failure  to  do  so  in  the  past  has  been  a  disgrace  to 
philosophy. 


CHAPTER  XVI 

DESCRIPTIONS 

WE  dealt  in  the  preceding  chapter  with  the  words  all  and  some ; 
in  this  chapter  we  shall  consider  the  word  the  in  the  singular, 
and  in  the  next  chapter  we  shall  consider  the  word  the  in  the 
plural.  It  may  be  thought  excessive  to  devote  two  chapters 
to  one  word,  but  to  the  philosophical  mathematician  it  is  a 
word  of  very  great  importance  :  like  Browning's  Grammarian 
with  the  enclitic  Se,  I  would  give  the  doctrine  of  this  word  if  I 
were  "  dead  from  the  waist  down  "  and  not  merely  in  a  prison. 

We  have  already  had  occasion  to  mention  "  descriptive 
functions,"  i.e.  such  expressions  as  "  the  father  of  x  "  or  "  the  sine 
of  x"  These  are  to  be  defined  by  first  defining  "  descriptions." 

A  "  description  "  may  be  of  two  sorts,  definite  and  indefinite 
(or  ambiguous).  An  indefinite  description  is  a  phrase  of  the 
form  "  a  so-and-so,"  and  a  definite  description  is  a  phrase  of 
the  form  "  the  so-and-so  "  (in  the  singular).  Let  us  begin  with 
the  former. 

"  Who  did  you  meet  ?  "  "I  met  a  man."  "  That  is  a  very 
indefinite  description."  We  are  therefore  not  departing  from 
usage  in  our  terminology.  Our  question  is  :  What  do  I  really 
assert  when  I  assert  "  I  met  a  man  "  ?  Let  us  assume,  for  the 
moment,  that  my  assertion  is  true,  and  that  in  fact  I  met  Jones. 
It  is  clear  that  what  I  assert  is  not  "  I  met  Jones."  I  may  say 
"  I  met  a  man,  but  it  was  not  Jones  "  ;  in  that  case,  though  I  lie, 
I  do  not  contradict  myself,  as  I  should  do  if  when  I  say  I  met  a 

167 


1 68  Introduction  to  Mathematical  Philosophy 

man  I  really  mean  that  I  met  Jones.  It  is  clear  also  that  the 
person  to  whom  I  am  speaking  can  understand  what  I  say,  even 
if  he  is  a  foreigner  and  has  never  heard  of  Jones. 

But  we  may  go  further  :  not  only  Jones,  but  no  actual  man, 
enters  into  my  statement.  This  becomes  obvious  when  the  state 
ment  is  false,  since  then  there  is  no  more  reason  why  Jones 
should  be  supposed  to  enter  into  the  proposition  than  why  any 
one  else  should.  Indeed  the  statement  would  remain  significant, 
though  it  could  not  possibly  be  true,  even  if  there  were  no  man 
at  all.  "  I  met  a  unicorn "  or  "  I  met  a  sea-serpent "  is  a 
perfectly  significant  assertion,  if  we  know  what  it  would  be  to 
be  a  unicorn  or  a  sea-serpent,  i.e.  what  is  the  definition  of  these 
fabulous  monsters.  Thus  it  is  only  what  we  may  call  the  concept 
that  enters  into  the  proposition.  In  the  case  of  "  unicorn," 
for  example,  there  is  only  the  concept :  there  is  not  also,  some 
where  among  the  shades,  something  unreal  which  may  be  called 
"  a  unicorn."  Therefore,  since  it  is  significant  (though  false) 
to  say  "  I  met  a  unicorn,"  it  is  clear  that  this  proposition,  rightly 
analysed,  does  not  contain  a  constituent  "  a  unicorn,"  though 
it  does  contain  the  concept  "  unicorn." 

The  question  of  "  unreality,"  which  confronts  us  at  this 
point,  is  a  very  important  one.  Misled  by  grammar,  the  great 
majority  of  those  logicians  who  have  dealt  with  this  question 
have  dealt  with  it  on  mistaken  lines.  They  have  regarded 
grammatical  form  as  a  surer  guide  in  analysis  than,  in  fact, 
it  is.  And  they  have  not  known  what  differences  in  gram 
matical  form  are  important.  "  I  met  Jones  "  and  "  I  met  a 
man  "  would  count  traditionally  as  propositions  of  the  same  form, 
but  in  actual  fact  they  are  of  quite  different  forms  :  the  first 
names  an  actual  person,  Jones ;  while  the  second  involves  a 
prepositional  function,  and  becomes,  when  made  explicit :  "  The 
function  '  I  met  x  and  x  is  human '  is  sometimes  true."  (It 
will  be  remembered  that  we  adopted  the  convention  of  using 
"  sometimes  "  as  not  implying  more  than  once.)  This  proposi 
tion  is  obviously  not  of  the  form  "  I  met  x,"  which  accounts 


Descriptions  1 69 

for  the  existence  of  the  proposition  "  I  met  a  unicorn  "  in  spite 
of  the  fact  that  there  is  no  such  thing  as  "  a  unicorn." 

For  want  of  the  apparatus  of  prepositional  functions,  many 
logicians  have  been  driven  to  the  conclusion  that  there  are 
unreal  objects.  It  is  argued,  e.g.  by  Meinong,1  that  we  can 
speak  about  "  the  golden  mountain,"  "  the  round  square," 
and  so  on ;  we  can  make  true  propositions  of  which  these  are 
the  subjects ;  hence  they  must  have  some  kind  of  logical  being, 
since  otherwise  the  propositions  in  which  they  occur  would  be 
meaningless.  In  such  theories,  it  seems  to  me,  there  is  a  failure 
of  that  feeling  for  reality  which  ought  to  be  preserved  even  in 
the  most  abstract  studies.  Logic,  I  should  maintain,  must  no 
more  admit  a  unicorn  than  zoology  can ;  for  logic  is  concerned 
with  the  real  world  just  as  truly  as  zoology,  though  with  its 
more  abstract  and  general  features.  To  say  that  unicorns  have 
an  existence  in  heraldry,  or  in  literature,  or  in  imagination, 
is  a  most  pitiful  and  paltry  evasion.  What  exists  in  heraldry 
is  not  an  animal,  rrlade  of  flesh  and  blood,  moving  and  breathing 
of  its  own  initiative.  What  exists  is  a  picture,  or  a  description 
in  words.  Similarly,  to  maintain  that  Hamlet,  for  example, 
exists  in  his  own  world,  namely,  in  the  world  of  Shakespeare's 
imagination,  just  as  truly  as  (say)  Napoleon  existed  in  the 
ordinary  world,  is  to  say  something  deliberately  confusing,  or 
else  confused  to  a  degree  which  is  scarcely  credible.  There  is 
only  one  world,  the  "  real "  world  :  Shakespeare's  imagination 
is  part  of  it,  and  the  thoughts  that  he  had  in  writing  Hamlet 
are  real.  So  are  the  thoughts  that  we  have  in  reading  the  play. 
But  it  is  of  the  very  essence  of  fiction  that  only  the  thoughts, 
feelings,  etc.,  in  Shakespeare  and  his  readers  are  real,  and  that 
there  is  not,  in  addition  to  them,  an  objective  Hamlet.  When 
you  have  taken  account  of  all  the  feelings  roused  by  Napoleon 
in  writers  and  readers  of  history,  you  have  not  touched  the  actual 
man  ;  but  in  the  case  of  Hamlet  you  have  come  to  the  end  of 
him.  If  no  one  thought  about  Hamlet,  there  would  be  nothing 
1  Untersuchungen  zur  Gegenstandstheorie  und  Psychologic,  1904. 


170  Introduction  to  Mathematical  Philosophy 

left  of  him  ;  if  no  one  had  thought  about  Napoleon,  he  would 
have  soon  seen  to  it  that  some  one  did.  The  sense  of  reality  is 
vital  in  logic,  and  whoever  juggles  with  it  by  pretending  that 
Hamlet  has  another  kind  of  reality  is  doing  a  disservice  to 
thought.  A  robust  sense  of  reality  is  very  necessary  in  framing 
a  correct  analysis  of  propositions  about  unicorns,  golden  moun 
tains,  round  squares,  and  other  such  pseudo-objects. 

In  obedience  to  the  feeling  of  reality,  we  shall  insist  that, 
in  the  analysis  of  propositions,  nothing  "  unreal "  is  to  be 
admitted.  But,  after  all,  if  there  is  nothing  unreal,  how,  it 
may  be  asked,  could  we  admit  anything  unreal  ?  The  reply 
is  that,  in  dealing  with  propositions,  we  are  dealing  in  the  first 
instance  with  symbols,  and  if  we  attribute  significance  to  groups 
of  symbols  which  have  no  significance,  we  shall  fall  into  the 
error  of  admitting  unrealities,  in  the  only  sense  in  which  this  is 
possible,  namely,  as  objects  described.  In  the  proposition 
"  I  met  a  unicorn,"  the  whole  four  words  together  make  a  signi 
ficant  proposition,  and  the  word  "  unicorn  "  by  itself  is  significant, 
in  just  the  same  sense  as  the  word  "  man."  But  the  two  words 
"  a  unicorn  "  do  not  form  a  subordinate  group  having  a  meaning 
of  its  own.  Thus  if  we  falsely  attribute  meaning  to  these  two 
words,  we  find  ourselves  saddled  with  "  a  unicorn,"  and  with 
the  problem  how  there  can  be  such  a  thing  in  a  world  where 
there  are  no  unicorns.  "  A  unicorn  "  is  an  indefinite  descrip 
tion  which  describes  nothing.  It  is  not  an  indefinite  description 
which  describes  something  unreal.  Such  a  proposition  as 
"  x  is  unreal "  only  has  meaning  when  "  x  "  is  a  description, 
definite  or  indefinite  ;  in  that  case  the  proposition  will  be  true 
if  "  x  "  is  a  description  which  describes  nothing.  But  whether 
the  description  "  x  "  describes  something  or  describes  nothing, 
it  is  in  any  case  not  a  constituent  of  the  proposition  in  which  it 
occurs  ;  like  "  a  unicorn  "  just  now,  it  is  not  a  subordinate  group 
having  a  meaning  of  its  own.  All  this  results  from  the  fact  that, 
when  "  x  "  is  a  description,  "  x  is  unreal  "  or  "  x  does  not  exist " 
is  not  nonsense,  but  is  always  significant  and  sometimes  true. 


Descriptions  171 

We  may  now  proceed  to  define  generally  the  meaning  of 
propositions  which  contain  ambiguous  descriptions.  Suppose 
we  wish  to  make  some  statement  about  "  a  so-and-so,"  where 
"so-and-so's"  are  those  objects  that  have  a  certain  property 
<£,  i.e.  those  objects  x  for  which  the  prepositional  function  (fax  is 
true.  (E.g.  if  we  take  "  a  man  "  as  our  instance  of  "  a  so-and-so," 
t/)X  will  be  "  x  is  human.")  Let  us  now  wish  to  assert  the  property 
ifj  of  "  a  so-and-so,"  i.e.  we  wish  to  assert  that  "  a  so-and-so  "  has 
that  property  which  x  has  when  i/jx  is  true.  (E.g.  in  the  case 
of  "  I  met  a  man,"  ifix  will  be  "  I  met  #.")  Now  the  proposition 
that  "  a  so-and-so  "  has  the  property  ift  is  not  a  proposition  of 
the  form  "  0#."  If  it  were,  "  a  so-and-so  "  would  have  to  be 
identical  with  x  for  a  suitable  x ;  and  although  (in  a  sense)  this 
may  be  true  in  some  cases,  it  is  certainly  not  true  in  such  a  case 
as  "  a  unicorn."  It  is  just  this  fact,  that  the  statement  that  a 
so-and-so  has  the  property  ijj  is  not  of  the  form  ifrx,  which  makes 
it  possible  for  "  a  so-and-so  "  to  be,  in  a  certain  clearly  definable 
sense,  "  unreal."  The  definition  is  as  follows  : — 

The  statement  that  "  an  object  having  the  property  ^  has 

the  property  ift " 
means  : 

"  The  joint  assertion  of  <f>x  and  i/ix  is  not  always  false." 

So  far  as  logic  goes,  this  is  the  same  proposition  as  might 
be  expressed  by  "  some  <£'s  are  ^'s  "  ;  but  rhetorically  there  is 
a  difference,  because  in  the  one  case  there  is  a  suggestion  of 
singularity,  and  in  the  other  case  of  plurality.  This,  however, 
is  not  the  important  point.  The  important  point  is  that,  when 
rightly  analysed,  propositions  verbally  about  "  a  so-and-so " 
are  found  to  contain  no  constituent  represented  by  this  phrase. 
And  that  is  why  such  propositions  can  be  significant  even  when 
there  is  no  such  thing  as  a  so-and-so. 

The  definition  of  existence,  as  applied  to  ambiguous  descrip 
tions,  results  from  what  was  said  at  the  end  of  the  preceding 
chapter.  We  say  that  "  men  exist  "  or  "  a  man  exists  "  if  the 


172  Introduction  to  Mathematical  Philosophy 

prepositional  function  "  x  is  human  "  is  sometimes  true  ;  and 
generally  "  a  so-and-so  "  exists  if  "  x  is  so-and-so  "  is  sometimes 
true.  We  may  put  this  in  other  language.  The  proposition 
"  Socrates  is  a  man  "  is  no  doubt  equivalent  to  "  Socrates  is 
human,"  but  it  is  not  the  very  same  proposition.  The  is  of 
"  Socrates  is  human "  expresses  the  relation  of  subject  and 
predicate  ;  the  is  of  "  Socrates  is  a  man  "  expresses  identity. 
It  is  a  disgrace  to  the  human  race  that  it  has  chosen  to  employ 
the  same  word  "  is  "  for  these  two  entirely  different  ideas — a 
disgrace  which  a  symbolic  logical  language  of  course  remedies. 
The  identity  in  "  Socrates  is  a  man  "  is  identity  between  an 
object  named  (accepting  "  Socrates "  as  a  name,  subject  to 
qualifications  explained  later)  and  an  object  ambiguously 
described.  An  object  ambiguously  described  will  "  exist  "  when 
at  least  one  such  proposition  is  true,  i.e.  when  there  is  at  least 
one  true  proposition  of  the  form  "  x  is  a  so-and-so,"  where  "  x  " 
is  a  name.  It  is  characteristic  of  ambiguous  (as  opposed  to 
definite)  descriptions  that  there  may  be  any  number  of  true 
propositions  of  the  above  form — Socrates  is  a  man,  Plato  is  a 
man,  etc.  Thus  "  a  man  exists "  follows  from  Socrates,  or 
Plato,  or  anyone  else.  With  definite  descriptions,  on  the  other 
hand,  the  corresponding  form  of  proposition,  namely,  "  x  is  the 
so-and-so  "  (where  "  x  "  is  a  name),  can  only  be  true  for  one 
value  of  x  at  most.  This  brings  us  to  the  subject  of  definite 
descriptions,  which  are  to  be  defined  in  a  way  analogous  to 
that  employed  for  ambiguous  descriptions,  but  rather  more 
complicated. 

We  come  now  to  the  main  subject  of  the  present  chapter, 
namely,  the  definition  of  the  word  the  (in  the  singular).  One 
very  important  point  about  the  definition  of  "  a  so-and-so " 
applies  equally  to  "  the  so-and-so  "  ;  the  definition  to  be  sought 
is  a  definition  of  propositions  in  which  this  phrase  occurs,  not  a 
definition  of  the  phrase  itself  in  isolation.  In  the  case  of  "  a 
so-and-so,"  this  is  fairly  obvious  :  no  one  could  suppose  that 
"  a  man  "  was  a  definite  object,  which  could  be  defined  by  itself. 


Descriptions  173 

Socrates  is  a  man,  Plato  is  a  man,  Aristotle  is  a  man,  but  we 
cannot  infer  that  "  a  man "  means  the  same  as  "  Socrates  " 
means  and  also  the  same  as  "  Plato  "  means  and  also  the  same 
as  "  Aristotle  "  means,  since  these  three  names  have  different 
meanings.  Nevertheless,  when  we  have  enumerated  all  the 
men  in  the  world,  there  is  nothing  left  of  which  we  can  say, 
"  This  is  a  man,  and  not  only  so,  but  it  is  the  '  a  man,'  the  quintes 
sential  entity  that  is  just  an  indefinite  man  without  being  any 
body  in  particular."  It  is  of  course  quite  clear  that  whatever 
there  is  in  the  world  is  definite  :  if  it  is  a  man  it  is  one  definite 
man  and  not  any  other.  Thus  there  cannot  be  such  an  entity 
as  "  a  man  "  to  be  found  in  the  world,  as  opposed  to  specific 
man.  And  accordingly  it  is  natural  that  we  do  not  define  "  a 
man  "  itself,  but  only  the  propositions  in  which  it  occurs. 

In  the  case  of  "  the  so-and-so  "  this  is  equally  true,  though 
at  first  sight  less  obvious.  We  may  demonstrate  that  this  must 
be  the  case,  by  a  consideration  of  the  difference  between  a  name 
and  a  definite  description.  Take  the  proposition,  "  Scott  is  the 
author  of  Waverley"  We  have  here  a  name,  "  Scott,"  and  a 
description,  "  the  author  of  Waverley"  which  are  asserted  to 
apply  to  the  same  person.  The  distinction  between  a  name  and 
all  other  symbols  may  be  explained  as  follows  : — 

A  name  is  a  simple  symbol  whose  meaning  is  something  that 
can  only  occur  as  subject,  i.e.  something  of  the  kind  that,  in 
Chapter  XIII.,  we  defined  as  an  "  individual "  or  a  "  particular." 
And  a  "  simple  "  symbol  is  one  which  has  no  parts  that  are 
symbols.  Thus  "  Scott "  is  a  simple  symbol,  because,  though  it 
has  parts  (namely,  separate  letters),  these  parts  are  not  symbols. 
On  the  other  hand,  "  the  author  of  Waverley  "  is  not  a  simple 
symbol,  because  the  separate  words  that  compose  the  phrase 
are  parts  which  are  symbols.  If,  as  may  be  the  case,  whatever 
seems  to  be  an  "  individual  "  is  really  capable  of  further  analysis, 
we  shall  have  to  content  ourselves  with  what  may  be  called 
"  relative  individuals,"  which  will  be  terms  that,  throughout 
the  context  in  question,  are  never  analysed  and  never  occur 


174  Introduction  to  Mathematical  Philosophy 

otherwise  than  as  subjects.  And  in  that  case  we  shall  have 
correspondingly  to  content  ourselves  with  "  relative  names." 
From  the  standpoint  of  our  present  problem,  namely,  the  defini 
tion  of  descriptions,  this  problem,  whether  these  are  absolute 
names  or  only  relative  names,  may  be  ignored,  since  it  con 
cerns  different  stages  in  the  hierarchy  of  "  types,"  whereas  we 
have  to  compare  such  couples  as  "  Scott "  and  "  the  author  of 
Waverley"  which  both  apply  to  the  same  object,  and  do  not 
raise  the  problem  of  types.  We  may,  therefore,  for  the  moment, 
treat  names  as  capable  of  being  absolute ;  nothing  that  we  shall 
have  to  say  will  depend  upon  this  assumption,  but  the  wording 
may  be  a  little  shortened  by  it. 

We  have,  then,  two  things  to  compare  :  (i)  a  name,  which 
is  a  simple  symbol,  directly  designating  an  individual  which 
is  its  meaning,  and  having  this  meaning  in  its  own  right,  in 
dependently  of  the  meanings  of  all  other  words  ;  (2)  a  description, 
which  consists  of  several  words,  whose  meanings  are  already 
fixed,  and  from  which  results  whatever  is  to  be  taken  as  the 
"  meaning  "  of  the  description. 

A  proposition  containing  a  description  is  not  identical  with 
what  that  proposition  becomes  when  a  name  is  substituted, 
even  if  the  name  names  the  same  object  as  the  description 
describes.  "  Scott  is  the  author  of  Waverley  "  is  obviously  a 
different  proposition  from  "  Scott  is  Scott "  :  the  first  is  a  fact 
in  literary  history,  the  second  a  trivial  truism.  And  if  we  put 
anyone  other  than  Scott  in  place  of  "  the  author  of  Waverley" 
our  proposition  would  become  false,  and  would  therefore  certainly 
no  longer  be  the  same  proposition.  But,  it  may  be  said,  our 
proposition  is  essentially  of  the  same  form  as  (say)  "  Scott  is 
Sir  Walter,"  in  which  two  names  are  said  to  apply  to  the  same 
person.  The  reply  is  that,  if  "  Scott  is  Sir  Walter  "  really  means 
"  the  person  named  e  Scott '  is  the  person  named  '  Sir  Walter,' ' 
then  the  names  are  being  used  as  descriptions  :  i.e.  the  individual, 
instead  of  being  named,  is  being  described  as  the  person  having 
that  name.  This  is  a  way  in  which  names  are  frequently  used 


Descriptions  175 

in  practice,  and  there  will,  as  a  rule,  be  nothing  in  the  phraseology 
to  show  whether  they  are  being  used  in  this  way  or  as  names. 
When  a  name  is  used  directly,  merely  to  indicate  what  we  are 
speaking  about,  it  is  no  part  of  the  fact  asserted,  or  of  the  falsehood 
if  our  assertion  happens  to  be  false  :  it  is  merely  part  of  the 
symbolism  by  which  we  express  our  thought.  What  we  want 
to  express  is  something  which  might  (for  example)  be  translated 
into  a  foreign  language ;  it  is  something  for  which  the  actual 
words  are  a  vehicle,  but  of  which  they  are  no  part.  On  the  other 
hand,  when  we  make  a  proposition  about  "  the  person  called 
'  Scott,'  "  the  actual  name  "  Scott "  enters  into  what  we  are 
asserting,  and  not  merely  into  the  language  used  in  making  the 
assertion.  Our  proposition  will  now  be  a  different  one  if  we 
substitute  "  the  person  called  '  Sir  Walter.' '  But  so  long  as 
we  are  using  names  as  names,  whether  we  say  "  Scott  "  or  whether 
we  say  "  Sir  Walter  "  is  as  irrelevant  to  what  we  are  asserting 
as  whether  we  speak  English  or  French.  Thus  so  long  as  names 
are  used  as  names,  "  Scott  is  Sir  Walter  "  is  the  same  trivial 
proposition  as  "  Scott  is  Scott."  This  completes  the  proof  that 
"  Scott  is  the  author  of  Waverley  "  is  not  the  same  proposition 
as  results  from  substituting  a  name  for  "  the  author  of  Waverley" 
no  matter  what  name  may  be  substituted. 

When  we  use  a  variable,  and  speak  of  a  propositional  function, 
(/>x  say,  the  process  of  applying  general  statements  about  x  to 
particular  cases  will  consist  in  substituting  a  name  for  the  letter 
"  x"  assuming  that  ^  is  a  function  which  has  individuals  for  its 
arguments.  Suppose,  for  example,  that  <j>x  is  "  always  true  "  ; 
let  it  be,  say,  the  "  law  of  identity,"  x=x.  Then  we  may  sub 
stitute  for  "  x  "  any  name  we  choose,  and  we  shall  obtain  a  true 
proposition.  Assuming  for  the  moment  that  "  Socrates," 
"  Plato,"  and  "  Aristotle  "  are  names  (a  very  rash  assumption), 
we  can  infer  from  the  law  of  identity  that  Socrates  is  Socrates, 
Plato  is  Plato,  and  Aristotle  is  Aristotle.  But  we  shall  commit 
a  fallacy  if  we  attempt  to  infer,  without  further  premisses,  that 
the  author  of  Waverley  is  the  author  of  Waverley.  This  results 


176  Introduction  to  Mathematical  Philosophy 

from  what  we  have  just  proved,  that,  if  we  substitute  a  name  for 
"  the  author  of  Waverley "  in  a  proposition,  the  proposition 
we  obtain  is  a  different  one.  That  is  to  say,  applying  the  result 
to  our  present  case  :  If  "  x  "  is  a  name,  "  x=x  "  is  not  the  same 
proposition  as  "  the  author  of  Waverley  is  the  author  of  Waverley" 
no  matter  what  name  "  x  "  may  be.  Thus  from  the  fact  that 
all  propositions  of  the  form  "  x=x  "  are  true  we  cannot  infer, 
without  more  ado,  that  the  author  of  Waverley  is  the  author  of 
Waverley.  In  fact,  propositions  of  the  form  "  the  so-and-so 
is  the  so-and-so  "  are  not  always  true  :  it  is  necessary  that  the 
so-and-so  should  exist  (a  term  which  will  be  explained  shortly). 
It  is  false  that  the  present  King  of  France  is  the  present  King  of 
France,  or  that  the  round  square  is  the  round  square.  When  we 
substitute  a  description  for  a  name,  prepositional  functions 
which  are  "  always  true  "  may  become  false,  if  the  description 
describes  nothing.  There  is  no  mystery  in  this  as  soon  as  we 
realise  (what  was  proved  in  the  preceding  paragraph)  that  when 
we  substitute  a  description  the  result  is  not  a  value  of  the 
propositional  function  in  question. 

We  are  now  in  a  position  to  define  propositions  in  which  a 
definite  description  occurs.  The  only  thing  that  distinguishes 
"  the  so-and-so "  from  "  a  so-and-so "  is  the  implication  of 
uniqueness.  We  cannot  speak  of  "  the  inhabitant  of  London," 
because  inhabiting  London  is  an  attribute  which  is  not  unique. 
We  cannot  speak  about  "  the  present  King  of  France,"  because 
there  is  none ;  but  we  can  speak  about  "  the  present  King  of 
England."  Thus  propositions  about  "  the  so-and-so "  always 
imply  the  corresponding  propositions  about  "  a  so-and-so," 
with  the  addendum  that  there  is  not  more  than  one  so-and-so. 
Such  a  proposition  as  "  Scott  is  the  author  of  Waverley  "  could 
not  be  true  if  Waverley  had  never  been  written,  or  if  several 
people  had  written  it ;  and  no  more  could  any  other  proposition 
resulting  from  a  propositional  function  x  by  the  substitution 
of  "  the  author  of  Waverley  "  for  "  x."  We  may  say  that  "  the 
author  of  Waverley  "  means  "  the  value  of  x  for  which  (  x  wrote 


Descriptions  177 

Waverley '  is  true."  Thus  the  proposition  "  the  author  of 
Waverley  was  Scotch,"  for  example,  involves  : 

(1)  "  x  wrote  Waverley  "  is  not  always  false  ; 

(2)  "  if  x  and  y  wrote  Waverley,  x  and  y  are  identical  "  is 

always  true  ; 

(3)  "  if  x  wrote  Waverley,  x  was  Scotch  "  is  always  true. 

These  three  propositions,  translated  into  ordinary  language, 
state  : 

(1)  at  least  one  person  wrote  Waverley  ; 

(2)  at  most  one  person  wrote  Waverley  ; 

(3)  whoever  wrote  Waverley  was  Scotch. 

All  these  three  are  implied  by  "  the  author  of  Waverley  was 
Scotch."  Conversely,  the  three  together  (but  no  two  of  them) 
imply  that  the  author  of  Waverley  was  Scotch.  Hence  the 
three  together  may  be  taken  as  defining  what  is  meant  by  the 
proposition  "  the  author  of  Waverley  was  Scotch." 

We  may  somewhat  simplify  these  three  propositions.  The 
first  and  second  together  are  equivalent  to  :  "  There  is  a  term 
c  such  that  '  x  wrote  Waverley  '  is  true  when  x  is  c  and  is  false 
when  x  is  not  c ."  In  other  words,  "  There  is  a  term  c  such  that 
*  x  wrote  Waverley  '  is  always  equivalent  to  *  x  is  c.9  "  (Two 
propositions  are  "  equivalent "  when  both  are  true  or  both  are 
false.)  We  have  here,  to  begin  with,  two  functions  of  x,  "  x 
wrote  Waverley  "  and  "  x  is  r,"  and  we  form  a  function  of  c  by 
considering  the  equivalence  of  these  two  functions  of  x  for  all 
values  of  x  ;  we  then  proceed  to  assert  that  the  resulting  function 
of  c  is  "  sometimes  true,"  i.e.  that  it  is  true  for  at  least  one  value 
of  c.  (It  obviously  cannot  be  true  for  more  than  one  value  of  c .) 
These  two  conditions  together  are  defined  as  giving  the  meaning 
of  "  the  author  of  Waverley  exists." 

We  may  now  define  "  the  term  satisfying  the  function  <f>x 
exists."  This  is  the  general  form  of  which  the  above  is  a  par 
ticular  case.  "  The  author  of  Waverley  "  is  "  the  term  satisfying 
the  function  '  x  wrote  Waverley'  "  And  "  the  so-and-so  "  will 

12 


178  Introduction  to  Mathematical  Philosophy 

always  involve  reference  to  some  prepositional  function,  namely, 
that  which  defines  the  property  that  makes  a  thing  a  so-and-so. 
Our  definition  is  as  follows  : — 

"  The  term  satisfying  the  function  fa  exists  "  means  : 
"  There  is  a  term  c  such  that  fa  is  always  equivalent  to  '  x  is  c?  ' 
In  order  to  define   "  the   author  of  Waverley  was   Scotch," 
we  have  still  to  take  account  of  the  third  of  our  three  proposi 
tions,  namely,   "  Whoever  wrote  Waverley  was  Scotch."     This 
will  be  satisfied  by  merely  adding  that  the  c  in  question  is  to 
be  Scotch.     Thus  "  the  author  of  Waverley  was  Scotch  "  is  : 

"  There  is  a  term  c  such  that  (i)  *  x  wrote  Waverley  9  is  always 
equivalent  to  '  x  is  cj  (2)  c  is  Scotch." 

And  generally :  "  the  term  satisfying  </>x  satisfies  fa "  is 
defined  as  meaning : 

"  There  is  a  term  c  such  that  (i)  <{>x  is  always  equivalent  to 
'  x  is  c,9  (2)  ific  is  true." 

This  is  the  definition  of  propositions  in  which  descriptions  occur. 
It  is  possible  to  have  much  knowledge  concerning  a  term 
described,  i.e.  to  know  many  propositions  concerning  "  the  so- 
and-so,"  without  actually  knowing  what  the  so-and-so  is,  i.e. 
without  knowing  any  proposition  of  the  form  "  x  is  the  so-and-so," 
where  "  x  "  is  a  name.  In  a  detective  story  propositions  about 
"  the  man  who  did  the  deed  "  are  accumulated,  in  the  hope 
that  ultimately  they  will  suffice  to  demonstrate  that  it  was 
A  who  did  the  deed.  We  may  even  go  so  far  as  to  say  that, 
in  all  such  knowledge  as  can  be  expressed  in  words — with  the 
exception  of  "  this "  and  "  that "  and  a  few  other  words  of 
which  the  meaning  varies  on  different  occasions — no  names, 
in  the  strict  sense,  occur,  but  what  seem  like  names  are  really 
descriptions.  We  may  inquire  significantly  whether  Homer 
existed,  which  we  could  not  do  if  "  Homer  "  were  a  name.  The 
proposition  "  the  so-and-so  exists "  is  significant,  whether 
true  or  false  ;  but  if  a  is  the  so-and-so  (where  "  a  "  is  a  name), 
the  words  "  a  exists  "  are  meaningless.  It  is  only  of  descriptions 


Descriptions  179 

— definite  or  indefinite — that  existence  can  be  significantly 
asserted  ;  for,  if  "a  "  is  a  name,  it  must  name  something  :  what 
does  not  name  anything  is  not  a  name,  and  therefore,  if  intended 
to  be  a  name,  is  a  symbol  devoid  of  meaning,  whereas  a  descrip 
tion,  like  "  the  present  King  of  France,"  does  not  become  in 
capable  of  occurring  significantly  merely  on  the  ground  that  it 
describes  nothing,  the  reason  being  that  it  is  a  complex  symbol, 
of  which  the  meaning  is  derived  from  that  of  its  constituent 
symbols.  And  so,  when  we  ask  whether  Homer  existed,  we  are 
using  the  word  "  Homer  "  as  an  abbreviated  description  :  we 
may  replace  it  by  (say)  "  the  author  of  the  Iliad  and  the  Odyssey" 
The  same  considerations  apply  to  almost  all  uses  of  what  look 
like  proper  names. 

When  descriptions  occur  in  propositions,  it  is  necessary  to 
distinguish  what  may  be  called  "  primary  "  and  "  secondary  " 
occurrences.  The  abstract  distinction  is  as  follows.  A  descrip 
tion  has  a  "  primary "  occurrence  when  the  proposition  in 
which  it  occurs  results  from  substituting  the  description  for 
"  x "  in  some  prepositional  function  (/>x ;  a  description  has  a 
"  secondary "  occurrence  when  the  result  of  substituting  the 
description  for  x  in  <j>x  gives  only  part  of  the  proposition  con 
cerned.  An  instance  will  make  this  clearer.  Consider  "  the 
present  King  of  France  is  bald."  Here  "  the  present  King  of 
France  "  has  a  primary  occurrence,  and  the  proposition  is  false. 
Every  proposition  in  which  a  description  which  describes  nothing 
has  a  primary  occurrence  is  false.  But  now  consider  "  the 
present  King  of  France  is  not  bald."  This  is  ambiguous.  If 
we  are  first  to  take  "  x  is  bald,"  then  substitute  "  the  present 
King  of  France  "  for  "  x"  and  then  deny  the  result,  the  occurrence 
of  "  the  present  King  of  France  "  is  secondary  and  our  proposition 
is  true  ;  but  if  we  are  to  take  "  x  is  not  bald  "  and  substitute 
"  the  present  King  of  France "  for  "  x"  then  "  the  present 
King  of  France  "  has  a  primary  occurrence  and  the  proposition 
is  false.  Confusion  of  primary  and  secondary  occurrences  is  a 
ready  source  of  fallacies  where  descriptions  are  concerned. 


i8o  Introduction  to  Mathematical  Philosophy 

Descriptions  occur  in  mathematics  chiefly  in  the  form  of 
descriptive  functions,  i.e.  "  the  term  having  the  relation  R  to 
y,"  or  "  the  R  of  y  "  as  we  may  say,  on  the  analogy  of  "  the 
father  of  y  "  and  similar  phrases.  To  say  "  the  father  of  y  is 
rich,"  for  example,  is  to  say  that  the  following  prepositional 
function  of  c  :  "  c  is  rich,  and  '  x  begat  y  '  is  always  equivalent 
to  '  x  is  cj  "  is  "  sometimes  true,"  i.e.  is  true  for  at  least  one 
value  of  c.  It  obviously  cannot  be  true  for  more  than  one 
value. 

The  theory  of  descriptions,  briefly  outlined  in  the  present 
chapter,  is  of  the  utmost  importance  both  in  logic  and  in  theory 
of  knowledge.  But  for  purposes  of  mathematics,  the  more 
philosophical  parts  of  the  theory  are  not  essential,  and  have 
therefore  been  omitted  in  the  above  account,  which  has  confined 
itself  to  the  barest  mathematical  requisites. 


CHAPTER  XVIi 

CLASSES 

IN  the  present  chapter  we  shall  be  concerned  with  the  in  the 
plural :  the  inhabitants  of  London,  the  sons  of  rich  men,  and 
so  on.  In  other  words,  we  shall  be  concerned  with  classes.  We 
saw  in  Chapter  II.  that  a  cardinal  number  is  to  be  defined  as  a 
class  of  classes,  and  in  Chapter  III.  that  the  number  I  is  to  be 
defined  as  the  class  of  all  unit  classes,  i.e.  of  all  that  have  just 
one  member,  as  we  should  say  but  for  the  vicious  circle.  Of 
course,  when  the  number  I  is  defined  as  the  class  of  all  unit 
classes,  "  unit  classes  "  must  be  defined  so  as  not  to  assume 
that  we  know  what  is  meant  by  "  one  "  ;  in  fact,  they  are  defined 
in  a  way  closely  analogous  to  that  used  for  descriptions,  namely  : 
A  class  a  is  said  to  be  a  "  unit  "  class  if  the  prepositional  function 
"  *  x  is  an  a '  is  always  equivalent  to  '  x  is  c  9  "  (regarded  as  a 
function  of  c)  is  not  always  false,  i.e.,  in  more  ordinary  language, 
if  there  is  a  term  c  such  that  x  will  be  a  member  of  a  when  x  is  c 
but  not  otherwise.  This  gives  us  a  definition  of  a  unit  class  if  we 
already  know  what  a  class  is  in  general.  Hitherto  we  have,  in 
dealing  with  arithmetic,  treated  "  class  "  as  a  primitive  idea. 
But,  for  the  reasons  set  forth  in  Chapter  XIII.,  if  for  no  others, 
we  cannot  accept  "  class  "  as  a  primitive  idea.  We  must  seek  a 
definition  on  the  same  lines  as  the  definition  of  descriptions, 
i.e.  a  definition  which  will  assign  a  meaning  to  propositions  in 
whose  verbal  or  symbolic  expression  words  or  symbols  apparently 
representing  classes  occur,  but  which  will  assign  a  meaning  that 
altogether  eliminates  all  mention  of  classes  from  a  right  analysis 

181 


1 82  Introduction  to  Mathematical  Philosophy 

of  such  propositions.  We  shall  then  be  able  to  say  that  the 
symbols  for  classes  are  mere  conveniences,  not  representing 
objects  called  "  classes,"  and  that  classes  are  in  fact,  like  descrip 
tions,  logical  fictions,  or  (as  we  say)  "  incomplete  symbols." 

The  theory  of  classes  is  less  complete  than  the  theory  of  descrip 
tions,  and  there  are  reasons  (which  we  shall  give  in  outline) 
for  regarding  the  definition  of  classes  that  will  be  suggested  as 
not  finally  satisfactory.  Some  further  subtlety  appears  to  be 
required ;  but  the  reasons  for  regarding  the  definition  which 
will  be  offered  as  being  approximately  correct  and  on  the  right 
lines  are  overwhelming. 

The  first  thing  is  to  realise  why  classes  cannot  be  regarded 
as  part  of  the  ultimate  furniture  of  the  world.  It  is  difficult 
to  explain  precisely  what  one  means  by  this  statement,  but  one 
consequence  which  it  implies  may  be  used  to  elucidate  its  meaning. 
If  we  had  a  complete  symbolic  language,  with  a  definition  for 
everything  definable,  and  an  undefined  symbol  for  everything 
indefinable,  the  undefined  symbols  in  this  language  would  repre 
sent  symbolically  what  I  mean  by  "  the  ultimate  furniture  of 
the  world."  I  am  maintaining  that  no  symbols  either  for  "  class  " 
in  general  or  for  particular  classes  would  be  included  in  this 
apparatus  of  undefined  symbols.  On  the  other  hand,  all  the 
particular  things  there  are  in  the  world  would  have  to  have 
names  which  would  be  included  among  undefined  symbols. 
We  might  try  to  avoid  this  conclusion  by  the  use  of  descriptions. 
Take  (say)  "  the  last  thing  Cassar  saw  before  he  died."  This 
is  a  description  of  some  particular ;  we  might  use  it  as  (in  one 
perfectly  legitimate  sense)  a  definition  of  that  particular.  But 
if  "  a  "  is  a  name  for  the  same  particular,  a  proposition  in  which 
"  a  "  occurs  is  not  (as  we  saw  in  the  preceding  chapter)  identical 
with  what  this  proposition  becomes  when  for  "  a  "  we  substitute 
"  the  last  thing  Caesar  saw  before  he  died."  If  our  language 
does  not  contain  the  name  "  a"  or  some  other  name  for  the  same 
particular,  we  shall  have  no  means  of  expressing  the  proposition 
which  we  expressed  by  means  of  "  a  "  as  opposed  to  the  one  that 


Classes  183 

we  expressed  by  means  of  the  description.  Thus  descriptions 
would  not  enable  a  perfect  language  to  dispense  with  names  for 
all  particulars.  In  this  respect,  we  are  maintaining,  classes 
differ  from  particulars,  and  need  not  be  represented  by  undefined 
symbols.  Our  first  business  is  to  give  the  reasons  for  this  opinion. 

We  have  already  seen  that  classes  cannot  be  regarded  as  a 
species  of  individuals,  on  account  of  the  contradiction  about 
classes  which  are  not  members  of  themselves  (explained  in 
Chapter  XIIL),  and  because  we  can  prove  that  the  number  of 
classes  is  greater  than  the  number  of  individuals. 

We  cannot  take  classes  in  the  pure  extensional  way  as  simply 
heaps  or  conglomerations.  If  we  were  to  attempt  to  do  that, 
we  should  find  it  impossible  to  understand  how  there  can  be  such 
a  class  as  the  null-class,  which  has  no  members  at  all  and  cannot 
be  regarded  as  a  "  heap  "  ;  we  should  also  find  it  very  hard  to 
understand  how  it  comes  about  that  a  class  which  has  only  one 
member  is  not  identical  with  that  one  member.  I  do  not  mean 
to  assert,  or  to  deny,  that  there  are  such  entities  as  "  heaps." 
As  a  mathematical  logician,  I  am  not  called  upon  to  have  an 
opinion  on  this  point.  All  that  I  am  maintaining  is  that,  if  there 
are  such  things  as  heaps,  we  cannot  identify  them  with  the  classes 
composed  of  their  constituents. 

We  shall  come  much  nearer  to  a  satisfactory  theory  if  we 
try  to  identify  classes  with  prepositional  functions.  Every 
class,  as  we  explained  in  Chapter  II.,  is  defined  by  some  pro- 
positional  function  which  is  true  of  the  members  of  the  class 
and  false  of  other  things.  But  if  a  class  can  be  defined  by  one 
prepositional  function,  it  can  equally  well  be  defined  by  any 
other  which  is  true  whenever  the  first  is  true  and  false  when 
ever  the  first  is  false.  For  this  reason  the  class  cannot  be  identi 
fied  with  any  one  such  prepositional  function  rather  than  with 
any  other — and  given  a  prepositional  function,  there  are  always 
many  others  which  are  true  when  it  is  true  and  false  when  it  is 
false.  We  say  that  two  prepositional  functions  are  "  formally 
equivalent  "  when  this  happens.  Two  propositions  are  "  equiva- 


184  Introduction  to  Mathematical  Philosophy 

lent "  when  both  are  true  or  both  false ;  two  prepositional 
functions  <f>x,  ifjx  are  "  formally  equivalent "  when  <frx  is  always 
equivalent  to  iftx.  It  is  the  fact  that  there  are  other  functions 
formally  equivalent  to  a  given  function  that  makes  it  impossible 
to  identify  a  class  with  a  function  ;  for  we  wish  classes  to  be  such 
that  no  two  distinct  classes  have  exactly  the  same  members, 
and  therefore  two  formally  equivalent  functions  will  have  to 
determine  the  same  class. 

When  we  have  decided  that  classes  cannot  be  things  of  the 
same  sort  as  their  members,  that  they  cannot  be  just  heaps  or 
aggregates,  and  also  that  they  cannot  be  identified  with  pro- 
positional  functions,  it  becomes  very  difficult  to  see  what  they 
can  be,  if  they  are  to  be  more  than  symbolic  fictions.  And  if 
we  can  find  any  way  of  dealing  with  them  as  symbolic  fictions, 
we  increase  the  logical  security  of  our  position,  since  we  avoid 
the  need  of  assuming  that  there  are  classes  without  being  com 
pelled  to  make  the  opposite  assumption  that  there  are  no  classes. 
We  merely  abstain  from  both  assumptions.  This  is  an  example 
of  Occam's  razor,  namely,  "  entities  are  not  to  be  multiplied 
without  necessity."  But  when  we  refuse  to  assert  that  there 
are  classes,  we  must  not  be  supposed  to  be  asserting  dogmatically 
that  there  are  none.  We  are  merely  agnostic  as  regards  them  : 
like  Laplace,  we  can  say,  "  je  n'ai  pas  besoin  de  cette  hypotbese." 

Let  us  set  forth  the  conditions  that  a  symbol  must  fulfil  if 
it  is  to  serve  as  a  class.  I  think  the  following  conditions  will 
be  found  necessary  and  sufficient : — 

(i)  Every  prepositional  function  must  determine  a  class, 
consisting  of  those  arguments  for  which  the  function  is  true. 
Given  any  proposition  (true  or  false),  say  about  Socrates,  we 
can  imagine  Socrates  replaced  by  Plato  or  Aristotle  or  a  gorilla 
or  the  man  in  the  moon  or  any  other  individual  in  the  world. 
In  general,  some  of  these  substitutions  will  give  a  true  proposition 
and  some  a  false  one.  The  class  determined  will  consist  of  all 
those  substitutions  that  give  a  true  one.  Of  course,  we  have 
still  to  decide  what  we  mean  by  "  all  those  which,  etc."  All  that 


Classes  1 8  5 

we  are  observing  at  present  is  that  a  class  is  rendered  determinate 
by  a  prepositional  function,  and  that  every  propositional  function 
determines  an  appropriate  class. 

(2)  Two    formally    equivalent    propositional    functions    must 
determine  the  same  class,  and  two  which  are  not  formally  equiva 
lent  must  determine  different  classes.     That  is,  a  class  is  deter 
mined  by  its  membership,  and  no  two  different  classes  can  have 
the  same  membership.     (If  a  class  is  determined  by  a  function 
<f>x,  we  say  that  a  is  a  "  member  "  of  the  class  if  c/>a  is  true.) 

(3)  We  must  find  some  way  of  defining  not  only  classes,  but 
classes  of  classes.     We  saw  in  Chapter  II.  that  cardinal  numbers 
are  to  be  defined  as  classes  of  classes.     The  ordinary  phrase 
of   elementary   mathematics,    "  The   combinations   of  n   things 
m  at  a  time  "  represents  a  class  of  classes,  namely,  the  class  of 
all  classes  of  m  terms  that  can  be  selected  out  of  a  given  class 
of  n  terms.     Without  some  symbolic  method  of  dealing  with 
classes  of  classes,  mathematical  logic  would  break  down. 

(4)  It  must  under  all  circumstances  be  meaningless  (not  false) 
to  suppose  a  class  a  member  of  itself  or  not  a  member  of  itself. 
This    results    from    the    contradiction    which    we    discussed    in 
Chapter  XIII. 

(5)  Lastly — and  this  is  the  condition  which  is  most  difficult 
of  fulfilment, — it  must  be  possible  to  make  propositions  about 
all  the  classes  that  are  composed  of  individuals,  or  about  all  the 
classes  that  are  composed  of  objects  of  any  one  logical  "  type." 
If  this  were  not  the  case,  many  uses  of  classes  would  go  astray 
— for  example,  mathematical  induction.     In  defining  the  posterity 
of  a  given  term,  we  need  to  be  able  to  say  that  a  member  of  the 
posterity  belongs  to  all  hereditary  classes  to  which  the  given 
term  belongs,  and  this  requires  the  sort  of  totality  that  is  in 
question.     The  reason  there  is  a  difficulty  about  this  condition 
is  that  it  can  be  proved  to  be  impossible  to  speak  of  all  the  pro- 
positional  functions  that  can  have  arguments  of  a  given  type. 

We  will,  to  begin  with,  ignore  this  last  condition  and  the 
problems   which   it   raises.     The   first   two   conditions   may   be 


1 86  Introduction  to  Mathematical  Philosophy 

taken  together.  They  state  that  there  is  to  be  one  class,  no 
more  and  no  less,  for  each  group  of  formally  equivalent  pro- 
positional  functions  ;  e.g.  the  class  of  men  is  to  be  the  same  as 
that  of  featherless  bipeds  or  rational  animals  or  Yahoos  or  what 
ever  other  characteristic  may  be  preferred  for  defining  a  human 
being.  Now,  when  we  say  that  two  formally  equivalent  pro- 
positional  functions  may  be  not  identical,  although  they  define 
the  same  class,  we  may  prove  the  truth  of  the  assertion  by  point 
ing  out  that  a  statement  may  be  true  of  the  one  function  and 
false  of  the  other ;  e.g.  "  I  believe  that  all  men  are  mortal " 
may  be  true,  while  "  I  believe  that  all  rational  animals  are 
mortal "  may  be  false,  since  I  may  believe  falsely  that  the 
Phoenix  is  an  immortal  rational  animal.  Thus  we  are  led  to 
consider  statements  about  functions,  or  (more  correctly)  functions 
of  functions. 

Some  of  the  things  that  may  be  said  about  a  function  may 
be  regarded  as  said  about  the  class  defined  by  the  function, 
whereas  others  cannot.  The  statement  "  all  men  are  mortal " 
involves  the  functions  "  x  is  human  "  and  "  x  is  mortal  "  ;  or, 
if  we  choose,  we  can  say  that  it  involves  the  classes  men  and 
mortals.  We  can  interpret  the  statement  in  either  way,  because 
its  truth-value  is  unchanged  if  we  substitute  for  "  x  is  human  " 
or  for  "  x  is  mortal "  any  formally  equivalent  function.  But, 
as  we  have  just  seen,  the  statement  "  I  believe  that  all  men  are 
mortal "  cannot  be  regarded  as  being  about  the  class  determined 
by  either  function,  because  its  truth-value  may  be  changed 
by  the  substitution  of  a  formally  equivalent  function  (which 
leaves  the  class  unchanged).  We  will  call  a  statement  involving 
a  function  <frx  an  "  extensional  "  function  of  the  function  <£#,  if 
it  is  like  "  all  men  are  mortal,"  i.e.  if  its  truth-value  is  unchanged 
by  the  substitution  of  any  formally  equivalent  function  ;  and 
when  a  function  of  a  function  is  not  extensional,  we  will  call  it 
"  intensional,"  so  that  "  I  believe  that  all  men  are  mortal " 
is  an  intensional  function  of  "  x  is  human  "  or  "  x  is  mortal." 
Thus  extensional  functions  of  a  function  x  may,  for  practical 


Classes  187 

purposes,  be  regarded  as  functions  of  the  class  determined  by 
x,  while  intensional  functions  cannot  be  so  regarded. 

It  is  to  be  observed  that  all  the  specific  functions  of  functions 
that  we  have  occasion  to  introduce  in  mathematical  logic  are 
extensional.  Thus,  for  example,  the  two  fundamental  functions 
of  functions  are  :  "  </>x  is  always  true  "  and  "  <j>x  is  sometimes 
true."  Each  of  these  has  its  truth-value  unchanged  if  any 
formally  equivalent  function  is  substituted  for  </>x.  In  the 
language  of  classes,  if  a  is  the  class  determined  by  </>x9  "  (f>x  is 
always  true  "  is  equivalent  to  "  everything  is  a  member  of  a," 
and  "  </>x  is  sometimes  true  "  is  equivalent  to  "  a  has  members  " 
or  (better)  "  a  has  at  least  one  member."  Take,  again,  the 
condition,  dealt  with  in  the  preceding  chapter,  for  the  existence 
of  "  the  term  satisfying  <£#."  The  condition  is  that  there  is  a 
term  c  such  that  $x  is  always  equivalent  to  "  x  is  c"  This 
is  obviously  extensional.  It  is  equivalent  to  the  assertion 
that  the  class  defined  by  the  function  (f>x  is  a  unit  class,  i.e.  a 
class  having  one  member;  in  other  words,  a  class  which  is  a 
member  of  I. 

Given  a  function  of  a  function  which  may  or  may  not  be 
extensional,  we  can  always  derive  from  it  a  connected  and 
certainly  extensional  function  of  the  same  function,  by  the 
following  plan  :  Let  our  original  function  of  a  function  be  one 
which  attributes  to  <j>x  the  property  f\  then  consider  the  asser 
tion  "  there  is  a  function  having  the  property  /  and  formally 
equivalent  to  <£#."  This  is  an  extensional  function  of  <f>x ;  it 
is  true  when  our  original  statement  is  true,  and  it  is  formally 
equivalent  to  the  original  function  of  </>x  if  this  original  function 
is  extensional ;  but  when  the  original  function  is  intensional, 
the  new  one  is  more  often  true  than  the  old  one.  For  example, 
consider  again  "  I  believe  that  all  men  are  mortal,"  regarded 
as  a  function  of  "  x  is  human."  The  derived  extensional  function 
is  :  "  There  is  a  function  formally  equivalent  to  *  x  is  human  ' 
and  such  that  I  believe  that  whatever  satisfies  it  is  mortal." 
This  remains  true  when  we  substitute  "  x  is  a  rational  animal  " 


1 88  Introduction  to  Mathematical  Philosophy 

for  "  x  is  human,"  even  if  I  believe  falsely  that  the  Phoenix  is 
rational  and  immortal. 

We  give  the  name  of  "  derived  extensional  function  "  to  the 
function  constructed  as  above,  namely,  to  the  function  :  "  There 
is  a  function  having  the  property  /  and  formally  equivalent  to 
$x,"  where  the  original  function  was  "  the  function  j>x  has 
the  property/." 

We  may  regard  the  derived  extensional  function  as  having 
for  its  argument  the  class  determined  by  the  function  <f>x,  and 
as  asserting/  of  this  class.  This  may  be  taken  as  the  definition 
of  a  proposition  about  a  class.  I.e.  we  may  define : 

To  assert  that  "  the  class  determined  by  the  function  <f>x 
has  the  property/"  is  to  assert  that  <j>x  satisfies  the  extensional 
function  derived  from/. 

This  gives  a  meaning  to  any  statement  about  a  class  which 
can  be  made  significantly  about  a  function ;  and  it  will  be 
found  that  technically  it  yields  the  results  which  are  required 
in  order  to  make  a  theory  symbolically  satisfactory.1 

What  we  have  said  just  now  as  regards  the  definition  of 
classes  is  sufficient  to  satisfy  our  first  four  conditions.  The 
way  in  which  it  secures  the  third  and  fourth,  namely,  the  possi 
bility  of  classes  of  classes,  and  the  impossibility  of  a  class  being 
or  not  being  a  member  of  itself,  is  somewhat  technical ;  it  is 
explained  in  Principia  Mathematics  but  may  be  taken  for 
granted  here.  It  results  that,  but  for  our  fifth  condition,  we 
might  regard  our  task  as  completed.  But  this  condition— at 
once  the  most  important  and  the  most  difficult — is  not  fulfilled 
in  virtue  of  anything  we  have  said  as  yet.  The  difficulty  is 
connected  with  the  theory  of  types,  and  must  be  briefly  discussed.2 

We  saw  in  Chapter  XIII.  that  there  is  a  hierarchy  of  logical 
types,  and  that  it  is  a  fallacy  to  allow  an  object  belonging  to 
one  of  these  to  be  substituted  for  an  object  belonging  to  another. 

1  See  Principia  Mathematica,  vol.  i.  pp.  75-84  and  *  20. 

2  The  reader  who  desires  a  fuller  discussion  should  consult  Principia 
Mathematica,  Introduction,  chap,  ii.;  also  *  12. 


Classes  189 

Now  it  is  not  difficult  to  show  that  the  various  functions  which 
can  take  a  given  object  a  as  argument  are  not  all  of  one  type. 
Let  us  call  them  all  ^-functions.  We  may  take  first  those  among 
them  which  do  not  involve  reference  to  any  collection  of  functions  ; 
these  we  will  call  "  predicative  ^-functions."  If  we  now  proceed 
to  functions  involving  reference  to  the  totality  of  predicative 
^-functions,  we  shall  incur  a  fallacy  if  we  regard  these  as  of  the 
same  type  as  the  predicative  ^-functions.  Take  such  an  every 
day  statement  as  "  a  is  a  typical  Frenchman."  How  shall 
we  define  a  "  typical  "  Frenchman  ?  We  may  define  him  as 
one  "  possessing  all  qualities  that  are  possessed  by  most  French 
men."  But  unless  we  confine  "  all  qualities  "  to  such  as  do  not 
involve  a  reference  to  any  totality  of  qualities,  we  shall  have  to 
observe  that  most  Frenchmen  are  not  typical  in  the  above  sense, 
and  therefore  the  definition  shows  that  to  be  not  typical  is 
essential  to  a  typical  Frenchman.  This  is  not  a  logical  contra 
diction,  since  there  is  no  reason  why  there  should  be  any  typical 
Frenchmen;  but  it  illustrates  the  need  for  separating  off 
qualities  that  involve  reference  to  a  totality  of  qualities  from 
those  that  do  not. 

Whenever,  by  statements  about  "  all  "  or  "  some "  of  the 
values  that  a  variable  can  significantly  take,  we  generate  a 
new  object,  this  new  object  must  not  be  among  the  values  which 
our  previous  variable  could  take,  since,  if  it  were,  the  totality 
of  values  over  which  the  variable  could  range  would  only  be 
definable  in  terms  of  itself,  and  we  should  be  involved  in  a  vicious 
circle.  For  example,  if  I  say  "Napoleon  had  all  the  qualities 
that  make  a  great  general,"  I  must  define  "  qualities  "  in  such  a 
way  that  it  will  not  include  what  I  am  now  saying,  i.e.  "  having 
all  the  qualities  that  make  a  great  general  "  must  not  be  itself  a 
quality  in  the  sense  supposed.  This  is  fairly  obvious,  and  is 
the  principle  which  leads  to  the  theory  of  types  by  which  vicious- 
circle  paradoxes  are  avoided.  As  applied  to  ^-functions,  we 
may  suppose  that  "  qualities  "  is  to  mean  "  predicative  functions." 
Then  when  I  say  "  Napoleon  had  all  the  qualities,  etc.,"  I  mean 


190  Introduction  to  Mathematical  Philosophy 

"  Napoleon  satisfied  all  the  predicative  functions,  etc."  This 
statement  attributes  a  property  to  Napoleon,  but  not  a  pre 
dicative  property ;  thus  we  escape  the  vicious  circle.  But 
wherever  "  all  functions  which  "  occurs,  the  functions  in  question 
must  be  limited  to  one  type  if  a  vicious  circle  is  to  be  avoided ; 
and,  as  Napoleon  and  the  typical  Frenchman  have  shown,  the 
type  is  not  rendered  determinate  by  that  of  the  argument.  It 
would  require  a  much  fuller  discussion  to  set  forth  this  point 
fully,  but  what  has  been  said  may  suffice  to  make  it  clear  that 
the  functions  which  can  take  a  given  argument  are  of  an  infinite 
series  of  types.  We  could,  by  various  technical  devices,  con 
struct  a  variable  which  would  run  through  the  first  n  of  these 
types,  where  n  is  finite,  but  we  cannot  construct  a  variable  which 
will  run  through  them  all,  and,  if  we  could,  that  mere  fact  would 
at  once  generate  a  new  type  of  function  with  the  same  arguments, 
and  would  set  the  whole  process  going  again. 

We  call  predicative  ^-functions  the  first  type  of  ^-functions  ; 
^-functions  involving  reference  to  the  totality  of  the  first  type 
we  call  the  second,  type  ;  and  so  on.  No  variable  ^-function 
can  run  through  all  these  different  types  :  it  must  stop  short  at 
some  definite  one. 

These  considerations  are  relevant  to  our  definition  of  the 
derived  extensional  function.  We  there  spoke  of  "  a  function 
formally  equivalent  to  fa"  It  is  necessary  to  decide  upon 
the  type  of  our  function.  Any  decision  will  do,  but  some  decision 
is  unavoidable.  Let  us  call  the  supposed  formally  equivalent 
function  0.  Then  ^  appears  as  a  variable,  and  must  be  of 
some  determinate  type.  All  that  we  know  necessarily  about 
the  type  of  (/>  is  that  it  takes  arguments  of  a  given  type — that 
it  is  (say)  an  ^-function.  But  this,  as  we  have  just  seen,  does 
not  determine  its  type.  If  we  are  to  be  able  (as  our  fifth  requisite 
demands)  to  deal  with  all  classes  whose  members  are  of  the  same 
type  as  a,  we  must  be  able  to  define  all  such  classes  by  means  of 
functions  of  some  one  type ;  that  is  to  say,  there  must  be  some 
type  of  ^-function,  say  the  nih9  such  that  any  ^-function  is  formally 


Classes  191 

equivalent  to  some  ^-function  of  the  nth  type.  If  this  is  the  case, 
then  any  extensional  function  which  holds  of  all  ^-functions 
of  the  nth  type  will  hold  of  any  ^-function  whatever.  It  is  chiefly 
as  a  technical  means  of  embodying  an  assumption  leading  to 
this  result  that  classes  are  useful.  The  assumption  is  called  the 
"  axiom  of  reducibility,"  and  may  be  stated  as  follows  : — 

"  There  is  a  type  (r  say)  of  ^-functions  such  that,  given  any 
tf-f unction,  it  is  formally  equivalent  to  some  function  of  the  type 
in  question." 

If  this  axiom  is  assumed,  we  use  functions  of  this  type  in 
defining  our  associated  extensional  function.  Statements  about 
all  ^-classes  (i.e.  all  classes  defined  by  ^-functions)  can  be  reduced 
to  statements  about  all  ^-functions  of  the  type  r.  So  long  as 
only  extensional  functions  of  functions  are  involved,  this  gives 
us  in  practice  results  which  would  otherwise  have  required  the 
impossible  notion  of  "  all  ^-functions."  One  particular  region 
where  this  is  vital  is  mathematical  induction. 

The  axiom  of  reducibility  involves  all  that  is  really  essential 
in  the  theory  of  classes.  It  is  therefore  worth  while  to  ask 
whether  there  is  any  reason  to  suppose  it  true. 

This  axiom,  like  the  multiplicative  axiom  and  the  axiom 
of  infinity,  is  necessary  for  certain  results,  but  not  for  the  bare 
existence  of  deductive  reasoning.  The  theory  of  deduction, 
as  explained  in  Chapter  XIV.,  and  the  laws  for  propositions 
involving  "  all  "  and  "  some,"  are  of  the  very  texture  of  mathe 
matical  reasoning :  without  them,  or  something  like  them, 
we  should  not  merely  not  obtain  the  same  results,  but  we  should 
not  obtain  any  results  at  all.  We  cannot  use  them  as  hypo 
theses,  and  deduce  hypothetical  consequences,  for  they  are 
rules  of  deduction  as  well  as  premisses.  They  must  be  absolutely 
true,  or  else  what  we  deduce  according  to  them  does  not  even 
follow  from  the  premisses.  On  the  other  hand,  the  axiom  of 
reducibility,  like  our  two  previous  mathematical  axioms,  could 
perfectly  well  be  stated  as  an  hypothesis  whenever  it  is  used, 
instead  of  being  assumed  to  be  actually  true.  We  can  deduce 


192  Introduction  to  Mathematical  Philosophy 

its  consequences  hypothetically ;  we  can  also  deduce  the  con 
sequences  of  supposing  it  false.  It  is  therefore  only  convenient, 
not  necessary.  And  in  view  of  the  complication  of  the  theory 
of  types,  and  of  the  uncertainty  of  all  except  its  most  general 
principles,  it  is  impossible  as  yet  to  say  whether  there  may 
not  be  some  way  of  dispensing  with  the  axiom  of  reducibility 
altogether.  However,  assuming  the  correctness  of  the  theory 
outlined  above,  what  can  we  say  as  to  the  truth  or  falsehood  of 
the  axiom  ? 

The  axiom,  we  may  observe,  is  a  generalised  form  of  Leibniz's 
identity  of  indiscernibles.  Leibniz  assumed,  as  a  logical  principle, 
that  two  different  subjects  must  differ  as  to  predicates.  Now 
predicates  are  only  some  among  what  we  called  "  predicative 
functions,"  which  will  include  also  relations  to  given  terms, 
and  various  properties  not  to  be  reckoned  as  predicates.  Thus 
Leibniz's  assumption  is  a  much  stricter  and  narrower  one  than 
ours.  (Not,  of  course,  according  to  his  logic,  which  regarded 
all  propositions  as  reducible  to  the  subject-predicate  form.) 
But  there  is  no  good  reason  for  believing  his  form,  so  far  as  I  can 
see.  There  might  quite  well,  as  a  matter  of  abstract  logical 
possibility,  be  two  things  which  had  exactly  the  same  predicates, 
in  the  narrow  sense  in  which  we  have  been  using  the  word  "  pre 
dicate."  How  does  our  axiom  look  when  we  pass  beyond  pre 
dicates  in  this  narrow  sense  ?  In  the  actual  world  there  seems 
no  way  of  doubting  its  empirical  truth  as  regards  particulars, 
owing  to  spatio-temporal  differentiation  :  no  two  particulars 
have  exactly  the  same  spatial  and  temporal  relations  to  all  other 
particulars.  But  this  is,  as  it  were,  an  accident,  a  fact  about 
the  world  in  which  we  happen  to  find  ourselves.  Pure  logic, 
and  pure  mathematics  (which  is  the  same  thing),  aims  at  being 
true,  in  Leibnizian  phraseology,  in  all  possible  worlds,  not  only 
in  this  higgledy-piggledy  job-lot  of  a  world  in  which  chance  has 
imprisoned  us.  There  is  a  certain  lordliness  which  the  logician 
should  preserve  :  he  must  not  condescend  to  derive  arguments 
from  the  things  he  sees  about  him. 


Classes  193 

Viewed  from  this  strictly  logical  point  of  view,  I  do  not  see 
any  reason  to  believe  that  the  axiom  of  reducibility  is  logically 
necessary,  which  is  what  would  be  meant  by  saying  that  it  is 
true  in  all  possible  worlds.  The  admission  of  this  axiom  into 
a  system  of  logic  is  therefore  a  defect,  even  if  the  axiom  is  empir 
ically  true.  It  is  for  this  reason  that  the  theory  of  classes  cannot 
be  regarded  as  being  as  complete  as  the  theory  of  descriptions. 
There  is  need  of  further  work  on  the  theory  of  types,  in  the  hope 
of  arriving  at  a  doctrine  of  classes  which  does  not  require  such  a 
dubious  assumption.  But  it  is  reasonable  to  regard  the  theory 
outlined  in  the  present  chapter  as  right  in  its  main  lines,  i.e.  in 
its  reduction  of  propositions  nominally  about  classes  to  pro 
positions  about  their  defining  functions.  The  avoidance  of 
classes  as  entities  by  this  method  must,  it  would  seem,  be  sound 
in  principle,  however  the  detail  may  still  require  adjustment. 
It  is  because  this  seems  indubitable  that  we  have  included  the 
theory  of  classes,  in  spite  of  our  desire  to  exclude,  as  far  as  possible, 
whatever  seemed  open  to  serious  doubt. 

The  theory  of  classes,  as  above  outlined,  reduces  itself  to  one 
axiom  and  one  definition.  For  the  sake  of  definiteness,  we  will 
here  repeat  them.  The  axiom  is  : 

Ther  e  is  a  type  r  such  that  if  $  is  a  function  which  can  take  a 
given  object  a  as  argument,  then  there  is  a  Junction  $  of  the  type 
r  which  is  formally  equivalent  to  <j>. 

The  definition  is  : 

If  </)  i s  a  function  which  can  take  a  given  object  a  as  argument, 
and  r  the  type  mentioned  in  the  above  axiom,  then  to  say  that 
the  class  determined  by  <j>  has  the  property  f  is  to  say  that  there 
is  a  function  of  type  T,  formally  equivalent  to  <£,  and  having  the 
property  f. 


CHAPTER  XVIII 

MATHEMATICS   AND    LOGIC 

MATHEMATICS  and  logic,  historically  speaking,  have  been  entirely 
distinct  studies.  Mathematics  has  been  connected  with  science, 
logic  with  Greek.  But  both  have  developed  in  modern  times  : 
logic  has  become  more  mathematical  and  mathematics  has 
become  more  logical.  The  consequence  is  that  it  has  now  become 
wholly  impossible  to  draw  a  line  between  the  two ;  in  fact,  the 
two  are  one.  They  differ  as  boy  and  man :  logic  is  the  youth 
of  mathematics  and  mathematics  is  the  manhood  of  logic.  This 
view  is  resented  by  logicians  who,  having  spent  their  time  in 
the  study  of  classical  texts,  are  incapable  of  following  a  piece 
of  symbolic  reasoning,  and  by  mathematicians  who  have  learnt 
a  technique  without  troubling  to  inquire  into  its  meaning  or 
justification.  Both  types  are  now  fortunately  growing  rarer. 
So  much  of  modern  mathematical  work  is  obviously  on  the 
border-line  of  logic,  so  much  of  modern  logic  is  symbolic  and 
formal,  that  the  very  close  relationship  of  logic  and  mathematics 
has  become  obvious  to  every  instructed  student.  The  proof 
of  their  identity  is,  of  course,  a  matter  of  detail :  starting  with 
premisses  which  would  be  universally  admitted  to  belong  to 
logic,  and  arriving  by  deduction  at  results  which  as  obviously 
belong  to  mathematics,  we  find  that  there  is  no  point  at  which 
a  sharp  line  can  be  drawn,  with  logic  to  the  left  and  mathe 
matics  to  the  right.  If  there  are  still  those  who  do  not  admit 
the  identity  of  logic  and  mathematics,  we  may  challenge  them 
to  indicate  at  what  point,  in  the  successive  definitions  and 

194 


Mathematics  and  Logic  195 

deductions  of  Principia  Maihematica,  they  consider  that  logic 
ends  and  mathematics  begins.  It  will  then  be  obvious  that  any 
answer  must  be  quite  arbitrary. 

In  the  earlier  chapters  of  this  book,  starting  from  the  natural 
numbers,  we  have  first  defined  "  cardinal  number  "  and  shown 
how  to  generalise  the  conception  of  number,  and  have  then 
analysed  the  conceptions  involved  in  the  definition,  until  we  found 
ourselves  dealing  with  the  fundamentals  of  logic.  In  a  synthetic, 
deductive  treatment  these  fundamentals  come  first,  and  the 
natural  numbers  are  only  reached  after  a  long  journey.  Such 
treatment,  though  formally  more  correct  than  that  which  we 
have  adopted,  is  more  difficult  for  the  reader,  because  the  ultimate 
logical  concepts  and  propositions  with  which  it  starts  are  remote 
and  unfamiliar  as  compared  with  the  natural  numbers.  Also 
they  represent  the  present  frontier  of  knowledge,  beyond  which 
is  the  still  unknown ;  and  the  dominion  of  knowledge  over  them 
is  not  as  yet  very  secure. 

It  used  to  be  said  that  mathematics  is  the  science  of  "  quantity." 
"  Quantity "  is  a  vague  word,  but  for  the  sake  of  argument 
we  may  replace  it  by  the  word  "  number."  The  statement 
that  mathematics  is  the  science  of  number  would  be  untrue 
in  two  different  ways.  On  the  one  hand,  there  are  recognised 
branches  of  mathematics  which  have  nothing  to  do  with  number 
— all  geometry  that  does  not  use  co-ordinates  or  measurement, 
for  example :  projective  and  descriptive  geometry,  down  to 
the  point  at  which  co-ordinates  are  introduced,  does  not  have 
to  do  with  number,  or  even  with  quantity  in  the  sense  of  greater 
and  less.  On  the  other  hand,  through  the  definition  of  cardinals, 
through  the  theory  of  induction  and  ancestral  relations,  through 
the  general  theory  of  series,  and  through  the  definitions  of  the 
arithmetical  operations,  it  has  become  possible  to  generalise  much 
that  used  to  be  proved  only  in  connection  with  numbers.  The 
result  is  that  what  was  formerly  the  single  study  of  Arithmetic 
has  now  become  divided  into  numbers  of  separate  studies,  no 
one  of  which  is  specially  concerned  with  numbers.  The  most 


196  Introduction  to  Mathematical  Philosophy 

elementary  properties  of  numbers  are  concerned  with  one-one 
relations,  and  similarity  between  classes.  Addition  is  concerned 
with  the  construction  of  mutually  exclusive  classes  respectively 
similar  to  a  set  of  classes  which  are  not  known  to  be  mutually 
exclusive.  Multiplication  is  merged  in  the  theory  of  "  selec 
tions,"  i.e.  of  a  certain  kind  of  one-many  relations.  Finitude 
is  merged  in  the  general  study  of  ancestral  relations,  which  yields 
the  whole  theory  of  mathematical  induction.  The  ordinal 
properties  of  the  various  kinds  of  number-series,  and  the  elements 
of  the  theory  of  continuity  of  functions  and  the  limits  of  functions, 
can  be  generalised  so  as  no  longer  to  involve  any  essential  reference 
to  numbers.  It  is  a  principle,  in  all  formal  reasoning,  to  generalise 
to  the  utmost,  since  we  thereby  secure  that  a  given  process  of 
deduction  shall  have  more  widely  applicable  results ;  we  are, 
therefore,  in  thus  generalising  the  reasoning  of  arithmetic, 
merely  following  a  precept  which  is  universally  admitted  in 
mathematics.  And  in  thus  generalising  we  have,  in  effect, 
created  a  set  of  new  deductive  systems,  in  which  traditional 
arithmetic  is  at  once  dissolved  and  enlarged ;  but  whether  any 
one  of  these  new  deductive  systems — for  example,  the  theory  of 
selections — is  to  be  said  to  belong  to  logic  or  to  arithmetic  is 
entirely  arbitrary,  and  incapable  of  being  decided  rationally. 

We  are  thus  brought  face  to  face  with  the  question  :  What 
is  this  subject,  which  may  be  called  indifferently  either  mathe 
matics  or  logic  ?  Is  there  any  way  in  which  we  can  define  it  ? 

Certain  characteristics  of  the  subject  are  clear.  To  begin 
with,  we  do  not,  in  this  subject,  deal  with  particular  things  or 
particular  properties  :  we  deal  formally  with  what  can  be  said 
about  any  thing  or  any  property.  We  are  prepared  to  say  that 
one  and  one  are  two,  but  not  that  Socrates  and  Plato  are  two, 
because,  in  our  capacity  of  logicians  or  pure  mathematicians, 
we  have  never  heard  of  Socrates  and  Plato.  A  world  in  which 
there  were  no  such  individuals  would  still  be  a  world  in  which 
one  and  one  are  two.  It  is  not  open  to  us,  as  pure  mathematicians 
or  logicians,  to  mention  anything  at  all,  because,  if  we  do  so, 


Mathematics  and  Logic  197 

we  introduce  something  irrelevant  and  not  formal.  We  may 
make  this  clear  by  applying  it  to  the  case  of  the  syllogism. 
Traditional  logic  says  :  "  All  men  are  mortal,  Socrates  is  a  man, 
therefore  Socrates  is  mortal."  Now  it  is  clear  that  what  we 
mean  to  assert,  to  begin  with,  is  only  that  the  premisses  imply 
the  conclusion,  not  that  premisses  and  conclusion  are  actually 
true ;  even  the  most  traditional  logic  points  out  that  the  actual 
truth  of  the  premisses  is  irrelevant  to  logic.  Thus  the  first 
change  to  be  made  in  the  above  traditional  syllogism  is  to  state 
it  in  the  form  :  "  If  all  men  are  mortal  and  Socrates  is  a  man, 
then  Socrates  is  mortal."  We  may  now  observe  that  it  is  intended 
to  convey  that  this  argument  is  valid  in  virtue  of  its  form,  not 
in  virtue  of  the  particular  terms  occurring  in  it.  If  we  had 
omitted  "  Socrates  is  a  man  "  from  our  premisses,  we  should 
have  had  a  non-formal  argument,  only  admissible  because 
Socrates  is  in  fact  a  man  ;  in  that  case  we  could  not  have  general 
ised  the  argument.  But  when,  as  above,  the  argument  is  formal, 
nothing  depends  upon  the  terms  that  occur  in  it.  Thus  we  may 
substitute  a  for  men,  j8  for  mortals,  and  x  for  Socrates,  where 
«  and  j3  are  any  classes  whatever,  and  x  is  any  individual.  We 
then  arrive  at  the  statement :  "  No  matter  what  possible  values 
x  and  a  and  j3  may  have,  if  all  a's  are  j8's  and  x  is  an  a,  then  x 
is  a  j8  "  ;  in  other  words,  "  the  prepositional  function  '  if  all  a's 
are  ]8  and  x  is  an  a,  then  x  is  a  j8 '  is  always  true."  Here  at  last 
we  have  a  proposition  of  logic — the  one  which  is  only  suggested  by 
the  traditional  statement  about  Socrates  and  men  and  mortals. 

It  is  clear  that,  if  formal  reasoning  is  what  we  are  aiming  at, 
we  shall  always  arrive  ultimately  at  statements  like  the  above, 
in  which  no  actual  things  or  properties  are  mentioned ;  this 
will  happen  through  the  mere  desire  not  to  waste  our  time  proving 
in  a  particular  case  what  can  be  proved  generally.  It  would  be 
ridiculous  to  go  through  a  long  argument  about  Socrates,  and  then 
go  through  precisely  the  same  argument  again  about  Plato.  If 
our  argument  is  one  (say)  which  holds  of  all  men,  we  shall  prove 
it  concerning  "  x"  with  the  hypothesis  "  if  x  is  a  man."  With 


198  Introduction  to  Mathematical  Philosophy 

this  hypothesis,  the  argument  will  retain  its  hypothetical  validity 
even  when  x  is  not  a  man.  But  now  we  shall  find  that  our  argu 
ment  would  still  be  valid  if,  instead  of  supposing  x  to  be  a  man, 
we  were  to  suppose  him  to  be  a  monkey  or  a  goose  or  a  Prime 
Minister.  We  shall  therefore  not  waste  our  time  taking  as  our 
premiss  "  x  is  a  man  "  but  shall  take  "  x  is  an  a,"  where  a  is  any 
class  of  individuals,  or  "  (f>x "  where  </)  is  any  prepositional 
function  of  some  assigned  type.  Thus  the  absence  of  all  mention 
of  particular  things  or  properties  in  logic  or  pure  mathematics 
is  a  necessary  result  of  the  fact  that  this  study  is,  as  we  say, 
"  purely  formal." 

At  this  point  we  find  ourselves  faced  with  a  problem  which 
is  easier  to  state  than  to  solve.  The  problem  is  :  "  What  are 
the  constituents  of  a  logical  proposition  ?  "  I  do  not  know  the 
answer,  but  I  propose  to  explain  how  the  problem  arises. 

Take  (say)  the  proposition  "  Socrates  was  before  Aristotle." 
Here  it  seems  obvious  that  we  have  a  relation  between  two  terms, 
and  that  the  constituents  of  the  proposition  (as  well  as  of  the 
corresponding  fact)  are  simply  the  two  terms  and  the  relation, 
i.e.  Socrates,  Aristotle,  and  before.  (I  ignore  the  fact  that 
Socrates  and  Aristotle  are  not  simple ;  also  the  fact  that  what 
appear  to  be  their  names  are  really  truncated  descriptions. 
Neither  of  these  facts  is  relevant  to  the  present  issue.)  We  may 
represent  the  general  form  of  such  propositions  by  "  x  R  y," 
which  may  be  read  "  x  has  the  relation  R  to  y."  This  general 
form  may  occur  in  logical  propositions,  but  no  particular  instance 
of  it  can  occur.  Are  we  to  infer  that  the  general  form  itself  is  a 
constituent  of  such  logical  propositions  ? 

Given  a  proposition,  such  as  "  Socrates  is  before  Aristotle," 
we  have  certain  constituents  and  also  a  certain  form.  But  the 
form  is  not  itself  a  new  constituent ;  if  it  were,  we  should  need  a 
new  form  to  embrace  both  it  and  the  other  constituents.  We 
can,  in  fact,  turn  all  the  constituents  of  a  proposition  into 
variables,  while  keeping  the  form  unchanged.  This  is  what  we 
do  when  we  use  such  a  schema  as  "  x  R  y,"  which  stands  for  any 


Mathematics  and  Logic  199 

one  of  a  certain  class  of  propositions,  namely,  those  asserting 
relations  between  two  terms.  We  can  proceed  to  general  asser 
tions,  such  as  "  x  R  y  is  sometimes  true  " — i.e.  there  are  cases 
where  dual  relations  hold.  This  assertion  will  belong  to  logic 
(or  mathematics)  in  the  sense  in  which  we  are  using  the  word. 
But  in  this  assertion  we  do  not  mention  any  particular  things 
or  particular  relations  ;  no  particular  things  or  relations  can 
ever  enter  into  a  proposition  of  pure  logic.  We  are  left  with  pure 
forms  as  the  only  possible  constituents  of  logical  propositions. 

I  do  not  wish  to  assert  positively  that  pure  forms — e.g.  the 
form  "  x  R  y  " — do  actually  enter  into  propositions  of  the  kind 
we  are  considering.  The  question  of  the  analysis  of  such  pro 
positions  is  a  difficult  one,  with  conflicting  considerations  on  the 
one  side  and  on  the  other.  We  cannot  embark  upon  this  question 
now,  but  we  may  accept,  as  a  first  approximation,  the  view 
that  forms  are  what  enter  into  logical  propositions  as  their 
constituents.  And  we  may  explain  (though  not  formally  define) 
what  we  mean  by  the  "  form  "  of  a  proposition  as  follows  : — 

The  "  form  "  of  a  proposition  is  that,  in  it,  that  remains  un 
changed  when  every  constituent  of  the  proposition  is  replaced 
by  another. 

Thus  "  Socrates  is  earlier  than  Aristotle  "  has  the  same  form 
as  "  Napoleon  is  greater  than  Wellington,"  though  every  con 
stituent  of  the  two  propositions  is  different. 

We  may  thus  lay  down,  as  a  necessary  (though  not  sufficient) 
characteristic  of  logical  or  mathematical  propositions,  that  they 
are  to  be  such  as  can  be  obtained  from  a  proposition  containing 
no  variables  (i.e.  no  such  words  as  all,  some,  a,  the,  etc.)  by  turning 
every  constituent  into  a  variable  and  asserting  that  the  result 
is  always  true  or  sometimes  true,  or  that  it  is  always  true  in 
respect  of  some  of  the  variables  that  the  result  is  sometimes  true 
in  respect  of  the  others,  or  any  variant  of  these  forms.  And 
another  way  of  stating  the  same  thing  is  to  say  that  logic  (or 
mathematics)  is  concerned  only  with  forms,  and  is  concerned 
with  them  only  in  the  way  of  stating  that  they  are  always  or 


2oo  Introduction  to  Mathematical  Philosophy 

sometimes  true — with  all  the  permutations  of  "  always  "  and 
"  sometimes  "  that  may  occur. 

There  are  in  every  language  some  words  whose  sole  function  is 
to  indicate  form.  These  words,  broadly  speaking,  are  commonest 
in  languages  having  fewest  inflections.  Take  "  Socrates  is 
human."  Here  "  is  "  is  not  a  constituent  of  the  proposition, 
but  merely  indicates  the  subject-predicate  form.  Similarly 
in  "  Socrates  is  earlier  than  Aristotle,"  "  is "  and  "  than " 
merely  indicate  form  ;  the  proposition  is  the  same  as  "  Socrates 
precedes  Aristotle,"  in  which  these  words  have  disappeared 
and  the  form  is  otherwise  indicated.  Form,  as  a  rule,  can  be 
indicated  otherwise  than  by  specific  words  :  the  order  of  the 
words  can  do  most  of  what  is  wanted.  But  this  principle 
must  not  be  pressed.  For  example,  it  is  difficult  to  see  how  we 
could  conveniently  express  molecular  forms  of  propositions 
(i.e.  what  we  call  "  truth-functions  ")  without  any  word  at  all. 
We  saw  in  Chapter  XIV.  that  one  word  or  symbol  is  enough  for 
this  purpose,  namely,  a  word  or  symbol  expressing  incompati 
bility.  But  without  even  one  we  should  find  ourselves  in  diffi 
culties.  This,  however,  is  not  the  point  that  is  important  for 
our  present  purpose.  What  is  important  for  us  is  to  observe 
that  form  may  be  the  one  concern  of  a  general  proposition, 
even  when  no  word  or  symbol  in  that  proposition  designates 
the  form.  If  we  wish  to  speak  about  the  form  itself,  we  must 
have  a  word  for  it ;  but  if,  as  in  mathematics,  we  wish  to  speak 
about  all  propositions  that  have  the  form,  a  word  for  the  form 
will  usually  be  found  not  indispensable  ;  probably  in  theory  it 
is  never  indispensable. 

Assuming — as  I  think  we  may — that  the  forms  of  propositions 
can  be  represented  by  the  forms  of  the  propositions  in  which 
they  are  expressed  without  any  special  word  for  forms,  we  should 
arrive  at  a  language  in  which  everything  formal  belonged  to 
syntax  and  not  to  vocabulary.  In  such  a  language  we  could 
express  all  the  propositions  of  mathematics  even  if  we  did  not 
know  one  single  word  of  the  language.  The  language  of  mathe- 


Mathematics  and  Logic  201 

matical  logic,  if  it  were  perfected,  would  be  such  a  language. 
We  should  have  symbols  for  variables,  such  as  "  x  "  and  "  R  " 
and  "  y,"  arranged  in  various  ways  ;  and  the  way  of  arrange 
ment  would  indicate  that  something  was  being  said  to  be  true  of 
all  values  or  some  values  of  the  variables.  We  should  not  need 
to  know  any  words,  because  they  would  only  be  needed  for  giving 
values  to  the  variables,  which  is  the  business  of  the  applied 
mathematician,  not  of  the  pure  mathematician  or  logician. 
It  is  one  of  the  marks  of  a  proposition  of  logic  that,  given  a 
suitable  language,  such  a  proposition  can  be  asserted  in  such  a 
language  by  a  person  who  knows  the  syntax  without  knowing 
a  single  word  of  the  vocabulary. 

But,  after  all,  there  are  words  that  express  form,  such  as  "  is  " 
and  "  than."  And  in  every  symbolism  hitherto  invented  for 
mathematical  logic  there  are  symbols  having  constant  formal 
meanings.  We  may  take  as  an  example  the  symbol  for  in 
compatibility  which  is  employed  in  building  up  truth-functions. 
Such  words  or  symbols  may  occur  in  logic.  The  question  is  : 
How  are  we  to  define  them  ? 

Such  words  or  symbols  express  what  are  called  "  logical 
constants."  Logical  constants  may  be  defined  exactly  as 
we  denned  forms  ;  in  fact,  they  are  in  essence  the  same  thing. 
A  fundamental  logical  constant  will  be  that  which  is  in  common 
among  a  number  of  propositions,  any  one  of  which  can  result 
from  any  other  by  substitution  of  terms  one  for  another.  For 
example,  "  Napoleon  is  greater  than  Wellington  "  results  from 
"  Socrates  is  earlier  than  Aristotle "  by  the  substitution  of 
"Napoleon"  for  "Socrates,"  "Wellington"  for  "Aristotle," 
and  "  greater  "  for  "  earlier."  Some  propositions  can  be  obtained 
in  this  way  from  the  prototype  "  Socrates  is  earlier  than  Aris 
totle  "  and  some  cannot ;  those  that  can  are  those  that  are  of 
the  form  "  x  R  y,"  i.e.  express  dual  relations.  We  cannot  obtain 
from  the  above  prototype  by  term-for-term  substitution  such 
propositions  as  "  Socrates  is  human  "  or  "  the  Athenians  gave 
the  hemlock  to  Socrates,"  because  the  first  is  of  the  subject- 


2O2  Introduction  to  Mathematical  Philosophy 

predicate  form  and  the  second  expresses  a  three-term  relation. 
If  we  are  to  have  any  words  in  our  pure  logical  language,  they 
must  be  such  as  express  "  logical  constants,"  and  "  logical 
constants  "  will  always  either  be,  or  be  derived  from,  what  is  in 
common  among  a  group  of  propositions  derivable  from  each 
other,  in  the  above  manner,  by  term-for-term  substitution.  And 
this  which  is  in  common  is  what  we  call  "  form." 

In  this  sense  all  the  "  constants  "  that  occur  in  pure  mathe 
matics  are  logical  constants.  The  number  I,  for  example,  is 
derivative  from  propositions  of  the  form :  "  There  is  a  term  c 
such  that  (f>x  is  true  when,  and  only  when,  x  is  c"  This  is  a 
function  of  ^,  and  various  different  propositions  result  from 
giving  different  values  to  <£.  We  may  (with  a  little  omission 
of  intermediate  steps  not  relevant  to  our  present  purpose)  take 
the  above  function  of  <f>  as  what  is  meant  by  "  the  class  deter 
mined  by  ^  is  a  unit  class  "  or  "  the  class  determined  by  <j>  is  a 
member  of  I  "  (i  being  a  class  of  classes).  In  this  way,  proposi 
tions  in  which  I  occurs  acquire  a  meaning  which  is  derived  from 
a  certain  constant  logical  form.  And  the  same  will  be  found 
to  be  the  case  with  all  mathematical  constants  :  all  are  logical 
constants,  or  symbolic  abbreviations  whose  full  use  in  a  proper 
context  is  defined  by  means  of  logical  constants. 

But  although  all  logical  (or  mathematical)  propositions  can 
be  expressed  wholly  in  terms  of  logical  constants  together  with 
variables,  it  is  not  the  case  that,  conversely,  all  propositions 
that  can  be  expressed  in  this  way  are  logical.  We  have  found 
so  far  a  necessary  but  not  a  sufficient  criterion  of  mathematical 
propositions.  We  have  sufficiently  defined  the  character  of  the 
primitive  ideas  in  terms  of  which  all  the  ideas  of  mathematics 
can  be  defined,  but  not  of  the  primitive  propositions  from  which 
all  the  propositions  of  mathematics  can  be  deduced.  This  is  a 
more  difficult  matter,  as  to  which  it  is  not  yet  known  what  the 
full  answer  is. 

We  may  take  the  axiom  of  infinity  as  an  example  of  a  pro 
position  which,  though  it  can  be  enunciated  in  logical  terms, 


Mathematics  and  Logic  203 

cannot  be  asserted  by  logic  to  be  true.  All  the  propositions  of 
logic  have  a  characteristic  which  used  to  be  expressed  by  saying 
that  they  were  analytic,  or  that  their  contradictories  were  self- 
contradictory.  This  mode  of  statement,  however,  is  not  satis 
factory.  The  law  of  contradiction  is  merely  one  among  logical 
propositions ;  it  has  no  special  pre-eminence ;  and  the  proof 
that  the  contradictory  of  some  proposition  is  self-contradictory 
is  likely  to  require  other  principles  of  deduction  besides  the 
law  of  contradiction.  Nevertheless,  the  characteristic  of  logical 
propositions  that  we  are  in  search  of  is  the  one  which  was  felt, 
and  intended  to  be  defined,  by  those  who  said  that  it  consisted 
in  deducibility  from  the  law  of  contradiction.  This  character 
istic,  which,  for  the  moment,  we  may  call  tautology,  obviously 
does  not  belong  to  the  assertion  that  the  number  of  individuals 
in  the  universe  is  «,  whatever  number  n  may  be.  But  for  the 
diversity  of  types,  it  would  be  possible  to  prove  logically  that 
there  are  classes  of  n  terms,  where  n  is  any  finite  integer  ;  or  even 
that  there  are  classes  of  N0  terms.  But,  owing  to  types,  such 
proofs,  as  we  saw  in  Chapter  XIII.,  are  fallacious.  We  are  left 
to  empirical  observation  to  determine  whether  there  are  as  many 
as  n  individuals  in  the  world.  Among  "  possible "  worlds, 
in  the  Leibnizian  sense,  there  will  be  worlds  having  one,  two, 
three,  .  .  .  individuals.  There  does  not  even  seem  any  logical 
necessity  why  there  should  be  even  one  individual 1 — why,  in 
fact,  there  should  be  any  world  at  all.  The  ontological  proof 
of  the  existence  of  God,  if  it  were  valid,  would  establish  the 
logical  necessity  of  at  least  one  individual.  But  it  is  generally 
recognised  as  invalid,  and  in  fact  rests  upon  a  mistaken  view  of 
existence — i.e.  it  fails  to  realise  that  existence  can  only  be  asserted 
of  something  described,  not  of  something  named,  so  that  it  is 
meaningless  to  argue  from  "  this  is  the  so-and-so  "  and  "  the 
so-and-so  exists  "  to  "  this  exists."  If  we  reject  the  ontological 

1  The  primitive  propositions  in  Principia  Mathematica  are  such  as  to 
allow  the  inference  that  at  least  one  individual  exists.  But  I  now  view 
this  as  a  defect  in  logical  purity. 


204  Introduction  to  Mathematical  Philosophy 

argument,  we  seem  driven  to  conclude  that  the  existence  of  a 
world  is  an  accident — i.e.  it  is  not  logically  necessary.  If  that 
be  so,  no  principle  of  logic  can  assert  "  existence  "  except  under 
a  hypothesis,  i.e.  none  can  be  of  the  form  "  the  prepositional 
function  so-and-so  is  sometimes  true."  Propositions  of  this 
form,  when  they  occur  in  logic,  will  have  to  occur  as  hypotheses 
or  consequences  of  hypotheses,  not  as  complete  asserted  pro 
positions.  The  complete  asserted  propositions  of  logic  will  all 
be  such  as  affirm  that  some  prepositional  function  is  always  true. 
For  example,  it  is  always  true  that  if  p  implies  q  and  q  implies 
r  then  p  implies  r,  or  that,  if  all  a's  are  jS's  and  x  is  an  a  then 
x  is  a  ]8.  Such  propositions  may  occur  in  logic,  and  their  truth 
is  independent  of  the  existence  of  the  universe.  We  may  lay 
it  down  that,  if  there  were  no  universe,  all  general  propositions 
would  be  true ;  for  the  contradictory  of  a  general  proposition 
(as  we  saw  in  Chapter  XV.)  is  a  proposition  asserting  existence, 
and  would  therefore  always  be  false  if  no  universe  existed. 

Logical  propositions  are  such  as  can  be  known  a  'priori,  without 
study  of  the  actual  world.  We  only  know  from  a  study  of 
empirical  facts  that  Socrates  is  a  man,  but  we  know  the  correct 
ness  of  the  syllogism  in  its  abstract  form  (i.e.  when  it  is  stated 
in  terms  of  variables)  without  needing  any  appeal  to  experience. 
This  is  a  characteristic,  not  of  logical  propositions  in  themselves, 
but  of  the  way  in  which  we  know  them.  It  has,  however,  a 
bearing  upon  the  question  what  their  nature  may  be,  since  there 
are  some  kinds  of  propositions  which  it  would  be  very  difficult 
to  suppose  we  could  know  without  experience. 

It  is  clear  that  the  definition  of  "  logic  "  or  "  mathematics  " 
must  be  sought  by  trying  to  give  a  new  definition  of  the  old 
notion  of  "  analytic  "  propositions.  Although  we  can  no  longer 
be  satisfied  to  define  logical  propositions  as  those  that  follow 
from  the  law  of  contradiction,  we  can  and  must  still  admit  that 
they  are  a  wholly  different  class  of  propositions  from  those  that 
we  come  to  know  empirically.  They  all  have  the  characteristic 
which,  a  moment  ago,  we  agreed  to  call  "  tautology."  This, 


Mathematics  and  Logic  205 

combined  with  the  fact  that  they  can  be  expressed  wholly  in  terms 
of  variables  and  logical  constants  (a  logical  constant  being  some 
thing  which  remains  constant  in  a  proposition  even  when  all 
its  constituents  are  changed) — will  give  the  definition  of  logic 
or  pure  mathematics.  For  the  moment,  I  do  not  know  how  to 
define  "  tautology."  1  It  would  be  easy  to  offer  a  definition 
which  might  seem  satisfactory  for  a  while  ;  but  I  know  of  none 
that  I  feel  to  be  satisfactory,  in  spite  of  feeling  thoroughly 
familiar  with  the  characteristic  of  which  a  definition  is  wanted. 
At  this  point,  therefore,  for  the  moment,  we  reach  the  frontier 
of  knowledge  on  our  backward  journey  into  the  logical  founda 
tions  of  mathematics. 

We  have  now  come  to  an  end  of  our  somewhat  summary  intro 
duction  to  mathematical  philosophy.  It  is  impossible  to  convey 
adequately  the  ideas  that  are  concerned  in  this  subject  so  long 
as  we  abstain  from  the  use  of  logical  symbols.  Since  ordinary 
language  has  no  words  that  naturally  express  exactly  what  we 
wish  to  express,  it  is  necessary,  so  long  as  we  adhere  to  ordinary 
language,  to  strain  words  into  unusual  meanings  ;  and  the  reader 
is  sure,  after  a  time  if  not  at  first,  to  lapse  into  attaching  the  usual 
meanings  to  words,  thus  arriving  at  wrong  notions  as  to  what  is 
intended  to  be  said.  Moreover,  ordinary  grammar  and  syntax 
is  extraordinarily  misleading.  This  is  the  case,  e.g.,  as  regards 
numbers ;  "  ten  men "  is  grammatically  the  same  form  as 
"  white  men,"  so  that  10  might  be  thought  to  be  an  adjective 
qualifying  "  men."  It  is  the  case,  again,  wherever  propositional 
functions  are  involved,  and  in  particular  as  regards  existence  and 
descriptions.  Because  language  is  misleading,  as  well  as  because 
it  is  diffuse  and  inexact  when  applied  to  logic  (for  which  it  was 
never  intended),  logical  symbolism  is  absolutely  necessary  to 
any  exact  or  thorough  treatment  of  our  subject.  Those  readers, 

1  The  importance  of  "  tautology  "  for  a  definition  of  mathematics  was 
pointed  out  to  me  by  my  former  pupil  Ludwig  Wittgenstein,  who  was 
working  on  the  problem.  I  do  not  know  whether  he  has  solved  it,  or  even 
whether  he  is  alive  or  dead. 


206  Introduction  to  Mathematical  Philosophy 

therefore,  who  wish  to  acquire  a  mastery  of  the  principles  of 
mathematics,  will,  it  is  to  be  hoped,  not  shrink  from  the  labour 
of  mastering  the  symbols — a  labour  which  is,  in  fact,  much  less 
than  might  be  thought.  As  the  above  hasty  survey  must  have 
made  evident,  there  are  innumerable  unsolved  problems  in  the 
subject,  and  much  work  needs  to  be  done.  If  any  student  is 
led  into  a  serious  study  of  mathematical  logic  by  this  little 
book,  it  will  have  served  the  chief  purpose  for  which  it  has  been 
written. 


INDEX 


Aggregates,  12. 

Alephs,  83,  92,  97,  125. 

Aliorelatives,  32. 

All,  158  &. 

Analysis,  4. 

Ancestors,  25,  33. 

Argument  of  a  function,  47,  108. 

Arithmetising  of  mathematics,  4. 

Associative  law,  58,  94. 

Axioms,  i. 

Between,  38  ff.,  58. 
Bolzano,  138  n. 
Boots  and  socks,  126. 
Boundary,  70,  98,  99. 

Cantor,  Georg,  77,  79,  85  «.,  86,  89, 
95,  102,  136. 

Classes,  12,  137,  181  ff. ;  reflexive,  80, 
127,  138  ;  similar,  15,  16. 

Clifford,  W.  K.,  76. 

Collections,  infinite,  13. 

Commutative  law,  58,  94. 

Conjunction,  147. 

Consecutiveness,  37,  38,  81. 

Constants,  202. 

Construction,  method  of,  73. 

Continuity,  86,  97  ff. ;  Cantorian,  102 
ff. ;  Dedekindian,  101 ;  in  philos 
ophy,  105  ;  of  functions,  106  ff. 

Contradictions,  135  ff. 

Convergence,  115. 

Converse,  16,  32,  49. 

Correlators,  54. 

Counterparts,  objective,  61. 

Counting,  14,  16. 

Dedekind,  69,  99,  138  n. 
Deduction,  144  ff. 

Definition,  3  ;    extensional  and  inten 
sion  al,  12. 
Derivatives,  100. 
Descriptions,  139,  144,  167  ff. 
Dimensions,  29. 
Disjunction,  147. 
Distributive  law,  58,  94. 
Diversity,  87. 
Domain,  16,  32,  49. 


Equivalence,  183. 
Euclid,  67. 

Existence,  164,  171,  177. 
Exponentiation,  94,  120. 
Extension  of  a  relation,  60. 

Fictions,  logical,  14  n.,  45,  137. 

Field  of  a  relation,  32,  53. 

Finite,  27. 

Flux,  105. 

Form,  198. 

Fractions,  37,  64. 

Frege,  7,  10,  25  n.,  77,  95,  146  «. 

Functions,  46  ;    descriptive,  46,   180  ; 

intensional    and    extensional,    186  ; 

predicative,  189  ;    prepositional,  46, 

144,  155  ff. 

Gap,  Dedekindian,  70  ff.,  99. 

Generalisation,  156. 

Geometry,  29,  59,  67,  74,  100,  145  ; 

analytical,  4,  86. 
Greater  and  less,  65,  90. 

Hegel,  107. 

Hereditary  properties,  21. 

Implication,  146,  153  ;  formal,  163. 

Incommensurables,  4,  66. 

Incompatibility,  147  ff.,  200. 

Incomplete  symbols,  182. 

Indiscernibles,  192. 

Individuals,  132,  141,  173. 

Induction,  mathematical,  20  ff.,  87,  93, 
185. 

Inductive  properties,  21. 

Inference,  148  ff. 

Infinite,  28  ;  of  ratipnals,  65  ;  Can 
torian,  65  ;  of  cardinals,  77  ff.  ;  and 
series  and  ordinals,  89  ff. 

Infinity,  axiom  of,  66  n.,  77,  131  ff., 
202. 

Instances,  156. 

Integers,  positive  and  negative,  64. 

Intervals,  115. 

Intuition,  145. 

Irrationals,  66,  72. 


307 


208  Introduction  to  Mathematical  Philosophy 


Kant,  145. 

Leibniz,  80,  107,  192. 

Lewis,  C.  I.,  153,  154. 

Likeness,  52. 

Limit,29,69ff.,97ff.;  of  functions,  1 06  ff. 

Limiting  points,  99. 

Logic,  159,  169,  194  ff.  ;  mathematical, 

v,  201,  206. 
Logicising  of  mathematics,  7. 

Maps,  52,  60  ff.,  80. 
Mathematics,  194  ff. 
Maximum,  70,  98. 
Median  class,  104. 
Meinong,  169. 
Method,  vi. 
Minimum,  70,  98. 
Modality,  165. 
Multiplication,  118  ff. 
Multiplicative  axiom,  92,  117  ff. 

Names,  173,  182. 

Necessity,  165. 

Neighbourhood,  109. 

Nicod,  148,  149,  151  H. 

Null-class,  23,  132. 

Number,  cardinal,  10  ff.,  56,  77  ff.,  95  ; 
complex,  74  ff. ;  finite,  20  ff.  ;  in 
ductive,  27,  78,  131  ;  infinite,  77  ff.  ; 
irrational,  66,  72  ;  maximum  ?  135  ; 
multipliable,  130  ;  natural,  2  ff.,  22  ; 
non-inductive,  88,  127 ;  real,  66,  72, 
84  ;  reflexive,  80,  127  ;  relation,  56, 
94  ;  serial,  57- 

Occam,  184. 

Occurrences,  primary  and  secondary, 

179- 

Ontological  proof,  203. 
Order,  29  ff.  ;  cyclic,  40. 
Oscillation,  ultimate,  in. 

Parmenides,  138. 
Particulars,  140  ff.,  173. 
Peano,  5  ff.,  23,  24,  78,  81,  131,  163. 
Peirce,  32  n. 
Permutations,  50. 
Philosophy,  mathematical,  v,  i. 
Plato,  138. 
Plurality,  10. 
Poincare,  27. 
Points,  59. 

Posterity,  32  ff. ;  proper,  36. 
Postulates,  71,  73. 
Precedent,  98. 
Premisses  of  arithmetic,  5. 
Primitive  ideas  and  propositions,  5,  202. 
Progressions,  8,  81  ff. 
Propositions,  155  ;   analytic,  204  ;  ele 
mentary,  161. 
Pythagoras,  4,  67. 


Quantity,  97,  195. 

Ratios,  64,  71,  84,  133. 

Reducibility,  axiom  of,  191. 

Referent,  48. 

Relation  numbers,  56  ff. 

Relations,  asymmetrical,  31,  42  ;  con 
nected,  32 ;  many-one,  15  ;  one- 
many,  15,  45;  one-one,  15,  47,  79  J 
reflexive,  16 ;  serial,  34 ;  similar, 
52  ff  ;  squares  of,  32  ;  symmetrical, 
16,  44  ;  transitive,  16,  32. 

Relatum,  48. 

Representatives,  120. 

Rigour,  144. 

Royce,  80. 

Section,  Dedekindian,  69  ff. ;  ultimate, 
in. 

Segments,  72,  98. 

Selections,  117  ff. 

Sequent,  98. 

Series,  29  ff.  ;  closed,  103  ;  compact, 
66,  93,  100 ;  condensed  in  itself, 
102 ;  Dedekindian,  71,  73,  101  ; 
generation  of,  41  ;  infinite,  89  ff.  ; 
perfect,  102,  103  ;  well-ordered,  92, 
123. 

Sheffer,  148. 

Similarity,  of  classes,  15  ff.  ;  of  rela 
tions,  52  ff.,  83. 

Some,  158  ff. 

Space,  61,  86,  140. 

Structure,  60  ff. 

Sub-classes,  84  ff. 

Subjects,  142. 

Subtraction,  87. 

Successor  of  a  number,  23,  35. 

Syllogism,  197. 

Tautology,  203,  205. 

The,  167,  172  ff. 

Time,  61,  86,  140. 

Truth-function,  147. 

Truth-value,  146. 

Types,  logical,  53,  135  ff.,  185,  i&8. 

Unreality,  168. 

Value  of  a  function,  47,  108. 
Variables,  10,  161,  199. 
Veblen,  58. 
Verbs,  141. 

Weierstrass,  97,  107. 
Wells,  H.  G.,  114. 
Whitehead,  64,  76,  107,  119. 
Wittgenstein,  205  n. 

Zermelo,  123,  129. 
Zero,  65. 


PRINTED  IN  GREAT  BRITAIN  BT  NEILL  AND  CO.,  LTD.,  EDINBURGH. 


UA  y  .  KO 

snc 

Russe 1 1 ,  Bertrand, 

1872-1970. 
Introduction  to 

mathematical  phi losophy 
AIC-3633  (mbab)