Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

134 MATHEMATICAL PHILOSOPHY
mathematics given by Benjamin Peirce as "the science
which draws necessary conclusions"; for the theorems
of a doctrinal function are necessary consequences of its
postulates in the sense that the former just are the implicates of the latter.
It accords with the judgment of Pieri that pure
mathematics is a "hypothetico-deductive" science; for the
postulates of a doctrinal function appear in the rôle of
hypotheses and the theorems in that of conclusions logically deduced.
It accords with the exquisite penetrating saying of
William Benjamin Smith that pure mathematics is "the
universal art apodictic"; for the logical validity of a
propositional function as such is completely independent
of any and all particular subject-matters, whether of our
world or of any other that may be conceivable or possible, and the logical coherence of the theorems and postulates of such a function is apodictically certain.
It accords with the seemingly shallow but really profound saying of Henri Poincaré that mathematics is "the
giving of the same name to different things"; for, despite
the confusion thus arising, a doctrinal function and its
various values are commonly given a single name, which
is usually that of a specially important or familiar one of
the values.
It accords well with the saying of an eminent jurist
that "mathematics is the attempt to seize hold of God
where the hair is shortest"; for the pure forms of thought
present clean-shaven aspects-they are "bald as the bare
mountain tops are bald, with a baldness that is sublime,"
and the discourse of a Gauss or a Lagrange is naturally
less:'woolly" than that of a Cicero or a Justinian or a
Coke or a Montesquieu or a Blackstone.