ries to each other when the
series, or groups of series, are represented as lines or planes. But
partly by interaction in analytic geometry, and partly in the
generalization of their own methods, both have come to recognize the
fundamental character of the relations involved in their thought, and
arithmetic, through the complex number and the algebraic unknown
quantities, has come to consider more complex serial types, while
geometry has approached the analysis of its series through interaction
with number theory. For both, the content of their entities and the
relations involved have been brought to a minimum. And this is true even
of such apparently essentially intuitional fields as projective
geometry, where entities can be substituted for directional lines and
the axioms be turned into relational postulates governing their
configurations.

Nevertheless, geometry like arithmetic, has remained true to the need
that gave it initial impulse. As in the beginning it was only a method
of dealing with a concrete situation, so in the end it is nothing but
such a method, although, as in the case of arithmetic, from ever closer
contact with the situation in question, it has been led, by refinements
that thoughtful and continual contact bring, to dissect that situation
and give heed to aspects of it which were undreamed of at the initial
moment. In a sense, then, there are no such things as mathematical
entities, as scholastic realism would conceive them. And yet,
mathematics is not dealing with unrealities, for it is everywhere
concerned with real rational types and systems where such types may be
exemplified. Or we can say in a purely practical way that mathematical
entities are constituted by their relations, but this phrase cannot here
be interpreted in the Hegelian ontological sense in which it has played
so great and so pernicious a part in contemporary philosophy. Such
metaphysical interpretation and its consequences are the basis of
paradoxical absolutisms, such as that arrived at by Professor Royce
(_World and the Individual_, Vol. II, Supplementary Essay). The peculiar
character of abstract or pure mathematics seems to be that its own
operations on a lower level constitute material which serves for the
subject-matter with which its later investigations deal. But mathematics
is, after all, not fundamentally different from the other sciences. The
concepts of all sciences alike constitute a special language peculiarly
a