e concretion in nature is
never legislated about nor so much as thought of except possibly when,
under warrant of sense, it is chosen to illustrate the concept
investigated dialectically. It does not even occur to a man to ask if
the sun's circle can be squared, for every one understands that the sun
is circular only in so far as it conforms to the circle's ideal nature;
which is as if Socrates and his interlocutors had clearly understood
that the _virtue_ of courage in an intemperate villain meant only
whatever in his mood or action was rational and truly desirable, and had
then said that courage, so understood, was identical with wisdom or with
the truly rational and desirable rule of life.

[Sidenote: The fact that mathematics applies to existence is empirical.]

The applicability of mathematics is not vouched for by mathematics but
by sense, and its application in some distant part of nature is not
vouched for by mathematics but by inductive arguments about nature's
uniformity, or by the character which the notion, "a distant part of
nature," already possesses. Inapplicable mathematics, we are told, is
perfectly thinkable, and systematic deductions, in themselves valid, may
be made from concepts which contravene the facts of perception. We may
suspect, perhaps, that even these concepts are framed by analogy out of
suggestions found in sense, so that some symbolic relevance or
proportion is kept, even in these dislocated speculations, to the matter
of experience. It is like a new mythology; the purely fictitious idea
has a certain parallelism and affinity to nature and moves in a human
and familiar way. Both data and method are drawn from applicable
science, elements of which even myth, whether poetic or mathematical,
may illustrate by a sort of variant or fantastic reduplication.

The great glory of mathematics, like that of virtue, is to be useful
while remaining free. Number and measure furnish an inexhaustible
subject-matter which the mind can dominate and develop dialectically as
it is the mind's inherent office to develop ideas. At the same time
number and measure are the grammar of sense; and the more this inner
logic is cultivated and refined the greater subtlety and sweep can be
given to human perception. Astronomy on the one hand and mechanical arts
on the other are fruits of mathematics by which its worth is made known
even to the layman, although the born mathematician would not need the
sanction of suc