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

156 INTRODUCTION TO MATHEMATICS
whole difference between the older and the
newer mathematics lies in the fact that vague
half-metaphorical terms like "gradually" are
no longer tolerated in its exact statements.
Modern mathematics will only admit statements and definitions and arguments which
exclusively employ the few simple ideas about
number and magnitude and variables on
which the science is founded. Of two numbers one can be greater or less than the
other; and one can be such and such a multiple of the other; but there is no relation of
"graduality" between two numbers, and
hence the term is inadmissible. Now this
may seem at first sight to be great pedantry.
To this charge there are two answers. In
the first place, during the first half of the
nineteenth century it was found by some
great mathematicians, especially Abel in
Sweden, and Weierstrass in Germany, that
large parts of mathematics as enunciated in
the old happy-go-lucky manner were simply
wrong. Macaulay in his essay on Bacon
contrasts the certainty of mathematics with
the uncertainty of philosophy; and by way
of a rhetorical example he says, "There has
been no reaction against Taylor's theorem."
He could not have chosen a worse example.
For, without having made an examination of
English text-books on mathematics contemporary with the publication of this essay, the