700 RENAISSANCES

difference of essence, but as a gradational difference of emphasis, habitus,
or penchant.1

We can, for example, agree with Spengler in holding that the pre-
dominant penchant that gives a particular culture its distinctive style
may impart to all the men and women who have been brought up in the
atmosphere of that culture, whatever the native psychic orientation of
each individual may happen to be, a uniform inclination either towards
or away from the mathematical, scientific, and technological approach
to life, or, short of that, may at least incline them, within the bounds of
this broad field of activity in the realm of Non-Human Nature, to
address themselves to one branch of Mathematics, Science, or Tech-
nology rather than to another. In this sense and within these limits it
may be true, as Spengler contends,2 that the 'Apollinean' spirit of an
Hellenic upbringing would foster astatic-minded Weltanschauung which,
in the province of Mathematics, would find its most congenial expres-
sion in Geometry, whereas the 'Faustian* spirit of a Western upbringing
would foster a dynamic-minded Weltanschauung which, within the same
mathematical field of intellectual activity, would find its most congenial
expression in Algebra and the Calculus. But of course, in assenting to
this more judiciously formulated version of Spengler's proposition, we
are admitting nothing beyond the indisputable facts that there are diverse
schools or fashions of education (in the widest meaning of that word)
and that the uniform impress of any such educational habitus is bound
to leave its mark on all individual human beings whose fate it may be to
be put through that particular cultural mill, even when the penchant of
the society into which a particular individual happens to have been born
is at variance with that individual's native personal bent. In admitting
this much, we are not committing ourselves to the nonsense implicit in
the pontifical version of Spengler's thesis, in which the philosopher-
hierophant goes so far as to assert that 'there is not and cannot be any
such thing as Number-in-itself' and that 'the notion of a universally
valid Science ... is an illusion'.

It would, indeed, be as fantastic to suggest that Geometry and the
Calculus are diverse, alternative, and incompatible systems of Mathe-
matics as it would be reasonable to say that these are different aspects of
one identical object of mathematical study that can properly be called
'Number-m-Itself. We may go on to observe that the several provinces
of this realm of Mathematical Science have been opened up at different
times and places by divers members of a single mathematical fraternity
whose choices of their particular fields of mathematical research have
been always influenced, and sometimes virtually determined, by a mental
penchant or habitus imparted to the individual mathematician by his
social milieu. In going thus far, however, we must be careful to steer
clear of the nonsense that Spengler makes of his own thesis when he
propounds it in its extreme form; for Spengler is manifestly flying in
the face of the facts when he suggests that a pioneer in the realm of
Mathematics cannot occupy the whole kingdom simultaneously, but

i Sec Spengler, op. cit., vol. i, p. 156, quoted in this Study in III. iii. 383-4.
a Ibid., pp. 380-3, quoted in III. iii. 388-9,