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

FUNCTIONS 157
assumption is a fairly safe one that Taylor's
theorem was enunciated and proved wrongly
in every one of them. Accordingly, the
anxious precision of modern mathematics is
necessary for accuracy. In the second place
it is necessary for research. It makes for
clearness of thought, and thence for boldness
of thought and for fertility in trying new
combinations of ideas. When the initial
statements are vague and slipshod, at every
subsequent stage of thought, common sense
has to step in to limit applications and to
explain meanings. Now in creative thought
common sense is a bad master. Its sole
criterion for judgment is that the new ideas
shall look like the old ones.- In other words
it can only act by suppressing originality.
In working our way towards the precise
definition of continuity (as applied to functions) let us consider more closely the statement that there is no relation of "graduality"
between numbers. It may be asked, Cannot
one number be only slightly greater than
another number, or in other words, cannot
the difference between the two numbers be
small? The whole point is that in the abstract, apart from some arbitrarily assumed
application, there is no such thing as a great
or a small number. A million miles is a
small number of miles for an astronomer
investigating the fixed stars, but a million