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

GENERALIZATIONS OF NUMBERS 83
general equation of the same form as x +1 = 3.
This equation is x+a =b, and its solution is
x=b-a. Here our difficulties become acute;
for this form can only be used for the numerical interpretation so long as b is greater than
a, and we cannot say without qualification
that a and b may be any constants. In other
words we have introduced a limitation on
the variability of the "constants" a and b,
which we must drag like a chain throughout
all our reasoning. Really prolonged mathematical investigations would be impossible
under such conditions. Every equation
would at least be buried under a pile of limitations. But if we now interpret our symbols
as "operations," all limitation vanishes like
magic. The equation x+1 =3 gives x = +2,
the equation x+3 = 1 gives x = -2, the equation x+a =b gives x =b-a which is an operation of addition or subtraction as the case
may be. We need never decide whether b-a
represents the operation of addition or of
subtraction, for the rules of procedure with
the symbols are the same in either case.
It does not fall within the plan of this work
to write a detailed chapter of elementary
algebra. Our object is merely to make plain
the fundamental ideas which guide the formation of the science. Accordingly we do not
further explain the detailed rules by which
the "positive and negative numbers" are