d 'inductive' science certainly has 'pulled off' remarkable
successes in the past, but we can have no confidence that these
successes will be repeated unless there are much better reasons for
believing in its methods and initial assumptions than any which the
scientific man who is an amateur 'empiricist' in his philosophy can
offer us. We may note, in particular, that this empiricism, which has
been expounded most carefully by Pearson and Mach, coincides with
Hegelian Absolutism in leading to the denial of the truth of
mathematics. It would be a superfluous task to argue at length that,
e.g., De Moivre's theorem or Taylor's theorem is not a short-hand
formula for recording the 'routine of our perceptions'.

The general state of things at the time of which I am speaking was thus
that relations were decidedly strained between a body of philosophers
and a body of scientific men who ought at least to have met on the
common ground of a complete Agnosticism. The philosophers were, in
general, shy of Science, mainly, no doubt, because they were modest men
who knew their own limitations, but they had a way of being
condescending to Science, which naturally annoyed the scientific men.
These latter professed a theory of the structure of knowledge which the
philosophers could easily show to be grotesque, but the retort was
always ready to hand that at any rate Science seemed somehow to be
getting somewhere while Philosophy appeared to lead nowhere in
particular.

The conditions for mutual understanding have now greatly improved,
thanks mainly to the labour of mathematicians with philosophical minds
on the principles of their own science. If we admit that mathematics is
true--and it seems quite impossible to avoid the admission--we can now
see that neither the traditional Kant-Hegel doctrine nor the traditional
sensationalistic empiricism can be sound. Not to speak of inquiries
which have been actually created within our own life-time, it may fairly
be said that the whole of pure mathematics has been shown, or is on the
verge of being shown, to form a body of conclusions rigidly deduced from
a few unproved postulates which are of a purely logical character.
Descartes has proved to be right in his view that the exceptional
certainty men have always ascribed to mathematical knowledge is not due
to the supposed restriction of the science to relation of number and
magnitude--there is a good deal of pure mathematics which deals with
n