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It seems obvious that the pragmatic account of all this routine of
phenomenal knowledge is accurate. Truth here is a relation, not
of our ideas to non-human realities, but of conceptual parts of our
experience to sensational parts. Those thoughts are true which
guide us to BENEFICIAL INTERACTION with sensible particulars as they
occur, whether they copy these in advance or not.

From the frequency of copying in the knowledge of phenomenal fact,
copying has been supposed to be the essence of truth in
matters rational also. Geometry and logic, it has been supposed,
must copy archetypal thoughts in the Creator. But in these abstract
spheres there is no need of assuming archetypes. The mind is free to
carve so many figures out of space, to make so many numerical
collections, to frame so many classes and series, and it can analyze
and compare so endlessly, that the very superabundance of the
resulting ideas makes us doubt the ‘objective’ pre-existence of
their models. It would be plainly wrong to suppose a God whose
thought consecrated rectangular but not polar co-ordinates, or
Jevons’s notation but not Boole’s. Yet if, on the other hand, we
assume God to have thought in advance of every POSSIBLE flight of
human fancy in these directions, his mind becomes too much like
a Hindoo idol with three heads, eight arms and six breasts, too much
made up of superfoetation and redundancy for us to wish to copy it,
and the whole notion of copying tends to evaporate from these
sciences. Their objects can be better interpreted as being created
step by step by men, as fast as they successively conceive them.

If now it be asked how, if triangles, squares, square roots, genera,
and the like, are but improvised human ‘artefacts,’ their
properties and relations can be so promptly known to be ‘eternal,’
the humanistic answer is easy. If triangles and genera are of our
own production we can keep them invariant. We can make them
‘timeless’ by expressly decreeing that on THE THINGS WE MEAN time
shall exert no altering effect, that they are intentionally and it
may be fictitiously abstracted from every corrupting real associate
and condition. But relations between invariant objects will
themselves be invariant. Such relations cannot be happenings, for by
hypothesis nothing shall happen to the objects. I have tried to
show in the last chapter of my Principles of Psychology [Footnote:
Vol. ii, pp. 641 ff.] that they can only be relations of comparison.
No one so far seems to have noticed my suggestion, and I am too
ignorant of the development of mathematics to feel very confident of
my own view. But if it were correct it would solve the difficulty
perfectly. Relations of comparison are matters of direct inspection.
As soon as mental objects are mentally compared, they are perceived
to be either like or unlike. But once the same, always the same,
once different, always different, under these timeless conditions.
Which is as much as to say that truths concerning these man-made
objects are necessary and eternal. We can change our conclusions
only by changing our data first.

The whole fabric of the a priori sciences can thus be treated as a
man-made product. As Locke long ago pointed out, these sciences have
no immediate connection with fact. Only IF a fact can be humanized
by being identified with any of these ideal objects, is what
was true of the objects now true also of the facts. The truth itself
meanwhile was originally a copy of nothing; it was only a relation
directly perceived to obtain between two artificial mental
things. [Footnote: Mental things which are realities of course
within the mental world.]