An introduction to mathematics, by A. N. Whitehead.

16 INTRODUCTION TO MATHEMATICS
is of vital importance, both to philosophy
and to mathematics; for by it the notion of
infinity is introduced. Perhaps it required
the introduction of the arabic numerals, by
which the use of letters as standing for definite numbers has been completely discarded
in mathematics, in order to suggest to mathematicians the technical convenience of the
use of letters for the ideas of any number
and some number. The Romans would have
stated the number of the year in which this
is written in the form MDCCCCX, whereas
we write it 1910, thus leaving the letters for
the other usage. But this is merely a speculation. After the rise of algebra the differential calculus was invented by Newton and
Leibniz, and then a pause in the progress
of the philosophy of mathematical thought
occurred so far as these notions are concerned; and it was not till within the last
few years that it has been realized how fundamental any and some are to the very
nature of mathematics, with the result of
opening out still further subjects for mathematical exploration.
Let us now make some simple algebraic
statements, with the object of understanding
exactly how these fundamental ideas occur.
(1) For any number x, x+2 = 2+x;
(2) For some number x, x+2 =3;
(3) For some number x, x +2 > 3.