lusion flowing from those
premises, he will have enlarged his knowledge of his art and discovered
a congenial good. He will have made progress in the Socratic science of
knowing his own intent.

[Sidenote: Double status of mathematics.]

Mathematics, for all its applications in nature, is a part of ideal
philosophy. It is logic applied to certain simple intuitions. These
intuitions and many of their developments happen to appear in that
efficacious and self-sustaining moiety of being which we call material;
so that mathematics is _per accidens_ the dialectical study of nature's
efficacious form. Its use and application in the world rather hide its
dialectical principle. Mathematics owes its public success to the happy
choice of a simple and widely diffused subject-matter; it owes its inner
cogency, however, to its ideality and the merely adventitious
application it has to existence. Mathematics has come to seem the type
of good logic because it is an illustration of logic in a sphere so
highly abstract in idea and so pervasive in sense as to be at once
manageable and useful.

The delights and triumphs of mathematics ought, therefore, to be a great
encouragement to ideal philosophy. If in a comparatively uninteresting
field attention can find so many treasures of harmony and order, what
beauties might it not discover in interpreting faithfully ideas nobler
than extension and number, concretions closer to man's spiritual life?
But unfortunately the logic of values is subject to voluntary and
involuntary confusions of so discouraging a nature that the flight of
dialectic in that direction has never been long and, even when short,
often disastrous. What is needed, as the example of mathematics shows,
is a steadfast intent and an adventurous inquiry. It would not occur to
a geometer to ask with trepidation what difference it would make to the
Pythagorean proposition if the hypothenuse were said to be wise and
good. Yet metaphysicians, confounding dialectic with physics and
thereby corrupting both, will discuss for ever the difference it makes
to substance whether you call it matter or God. Nevertheless, no
decorative epithets can give substance any other attributes than those
which it has; that is, other than the actual appearances that substance
is needed to support. Similarly, neither mathematicians nor astronomers
are exercised by the question whether [Greek: pi] created the ring of
Saturn; yet naturalists and logician