ce which
Socrates rejected for its supposed want of utility; but perhaps he had
another ground in reserve to justify his humorous prejudice. He may have
felt that such a science, if admitted, would endanger his thesis about
the identity of virtue and knowledge.

[Sidenote: Quantity submits easily to dialectical treatment.]

Mathematical method has been the envy of philosophers, perplexed and
encumbered as they are with the whole mystery of existence, and they
have attempted at times to emulate mathematical cogency. Now the
lucidity and certainty found in mathematics are not inherent in its
specific character as the science of number or dimension; they belong to
dialectic as a whole which is essentially elucidation. The effort to
explain meanings is in most cases abortive because these meanings melt
in our hands--a defeat which Hegel would fain have consecrated, together
with all other evils, into necessity and law. But the merit of
mathematics is that it is so much less Hegelian than life; that it holds
its own while it advances, and never allows itself to misrepresent its
original intent. In all it finds to say about the triangle it never
comes to maintain that the triangle is really a square. The privilege of
mathematics is simply to have offered the mind, for dialectical
treatment, a material to which dialectical treatment could be honestly
applied. This material consists in certain general aspects of
sensation--its extensity, its pulsation, its distribution into related
parts. The wakefulness that originally makes these abstractions is able
to keep them clear, and to elaborate them infinitely without
contradicting their essence.

For this reason it is always a false step in mathematical science, a
step over its brink into the abyss beyond, when we try to reduce its
elements to anything not essentially sensible. Intuition must continue
to furnish the subject of discourse, the axioms, and the ultimate
criteria and sanctions. Calculation and transmutation can never make
their own counters or the medium in which they move. So that space,
number, continuity, and every other elementary intuition remains at
bottom opaque--opaque, that is, to mathematical science; for it is no
paradox, but an obvious necessity, that the data of a logical operation
should not be producible by its workings. Reason would have nothing to
do if it had no irrational materials. Saint Augustine's rhetoric
accordingly covered--as so often with hi