results of counting: adding gives rules or shortcuts as to what will happen if
one counts first one group, then another, then counts them both, and so forth.
Elementary algebra generalizes the results from various kinds of arithmetic
problems. There is a generalizing, and reflexive, chain from one form of mathe-
matics to another, from the operations of counting up through the study of
operations upon operations which are raised to intricate degrees in abstract
math. At each level mathematics investigates and classifies operations. It makes
operations equivalent to one another by treating them as equivalent, by sub-
jecting them to a systematic set of higher-order operations. We make the num-
bers of a counting system equivalent by imposing the conventions of adding
and subtracting them. For mathematics, there are no problems of mixing apples
and oranges; the mathematician coins a new concept of that which is equivalent
among them. Nor need that equivalent be a “natural kind,” a concept in rebus
(e.g., fruit), but just that equivalence which is given by the operation imposed
on them. If counting consists in making a series of gestures which thereby
constitutes something as a series, arithmetic consists in making gestures toward
number operations, elementary algebra in making gestures toward arithmetic
operations, higher algebra in making gestures toward elementary algebraic
operations which treat them as equivalent.
These gestures are made in common with others in the community of math-
ematicians. One becomes a member by adopting its conventions of communi-
cation. The social structure of mathematics is pyramid-shaped. Across the base,
there is the huge community of those who use the conventions of counting and
of arithmetic. Building step-wise upon this are the communities of increasingly
specialized and esoteric mathematicians, networks which have taken the lower-
level communicative operations, the lower-level conventions, as topics for
abstraction and reflexive generalization.
Mathematical objects are real in the same sense that human communication
is real. It is the reality of activities of real human beings, carried out in time,
located in space. And it is the doubly strong, obdurate reality of the social: of
widespread conventions of discourse, which is to say activities carried out in
common, and which constitute a community out of just those persons who
adopt these conventional operations. We may even say it is triply strong, since
the network of mathematicians is that which has grown up around the central
activity of constructing techniques of building meta-operations which take as
content the previous operations of the community.
The long-standing view of mathematics as a realm of Platonic ideals is
mistaken. Some Greeks philosophers and mathematicians argued that the
objects of mathematics must be ideal because the truths they proved about
geometric figures referred to ideal circles and lines, not to the imperfectly drawn
lines on the sand.^7 Others have argued for the ideality of mathematics by using868 •^ Meta-reflections