To develop the complete history of arithmetic and geometry would be a
task quite beyond the limits of this paper, and of the writer's
knowledge. In arithmetic we were able to observe a stage in which
spontaneous behavior led to the invention of number names and methods of
counting. Then, by certain speculative and "play" impulses, there arose
elementary arithmetical problems which began to be of interest in
themselves. Geometry here also comes into consideration, and, in
connection with positional number symbols, begin those interactions
between arithmetic and geometry that result in the forms of our
contemporary mathematics. The complex quantities represented by number
symbols are no longer merely the necessary results of analyzing
commercial relations or practical measurements, and geometry is no
longer directly based upon the intuitively given line, point, and plane.
If number relations are to be expressed in terms of empirical spatial
positions, it is necessary to construct many imaginary surfaces, as is
done by Riemann in his theory of functions, a construction representing
the type of imagination which Poincare has called the intuitional in
contradistinction to the logical (_Value of Science_, Ch. I). And
geometry has not only been led to the construction of many non-Euclidian
spaces, but has even, with Peano and his school, been freed from the
bonds of any necessary spatial interpretation whatsoever.

To trace in concrete detail the attainment of modern refinements of
number theory would likewise exhibit nothing new in the building up of
mathematical intelligence. We should find, here, a process carried out
without thought of the consequences, there, an analogy suggesting an
operation that might lead us beyond a difficulty that had blocked
progress; here, a play interest leading to a combination of symbols out
of which a new idea has sprung; there, a painstaking and methodical
effort to overcome a difficulty recognized from the start. It is rather
for us now to ask what it is that has been attained by these means, to
inquire finally what are those things called "number" and "line" in the
broad sense in which the terms are now used.

In so far as the cardinal number at least is concerned, the answer
generally accepted by Dedekind, Peano, Russell, and such writers is
this: the number is a "class of similar classes" (Whitehead and Russell,
_Prin. Math._, Vol. II, p. 4). To the interpretation of this answer, Mr.
R