e have nothing in the Alephs
but the symbols of certain groupings of operations expressible in
ordinary number series. And the many forms of numbers are all simply the
result of recognizing value in naming definite groups of operations of a
lower level, which may itself be a complication of processes indicated
by the simple numerical signs. To create such symbols is by no means
illegitimate and no paradox results in any forms as long as we remember
that our numbers are not things but are signs of operations that may be
performed directly upon things or upon other operations.

For example, let us consider such a symbol as sqrt{-5}. -5 signifies
the totality of a counting process carried on in an opposite sense from
that denoted by +5. To take the square root is to symbolize a number,
the totality of an operation, such that when the operation denoted by
multiplying it by itself is performed the result is 5. Consequently the
sqrt{-5} is merely the symbol of these processes combined in such a
way that the whole operation is to be considered as opposite in some
sense to that denoted by sqrt{5}. Hence, an easy method for the
representation of such imaginaries is based on the principle of analytic
geometry and a system of co-ordinates.

The nature of this last generalization of mathematics is well shown by
Mr. Whitehead in his monumental _Universal Algebra_. The work begins
with the definition of a calculus as "The art of manipulating
substitutive signs according to fixed rules, and the deduction therefrom
of true propositions" (_loc. cit._, p. 4). The deduction itself is
really a manipulation according to rules, and the truth consists
essentially in the results being actually derived from the premises
according to rule. Following Stout, substitutive signs are characterized
thus: "a word is an instrument for thinking about the meaning which it
expresses; a substitutive sign is a means of not thinking about the
meaning which it symbolizes." Mathematical symbols have, then, become
substitutive signs. But this is only possible because they were at an
early stage of their history expressive signs, and the laws which
connected them were derived from the relations of the things for which
they stood. First it became possible to forget the things in their
concreteness, and now they have become mere terms for the relations that
had been generalized between them. Consequently, the things forgotten
and the terms treated as mere elements