empiricism as a foil; numbers are not the things we observe in the world, since
it is by numbers that we can enumerate things. Both lines of argument make
the same mistake (and so does that which holds that mathematics arises by
induction from experience of things): assuming that reality must consist either
in substantive things or in self-subsistent ideas. But mathematical concepts are
neither; they are emblems for actions, for operations of mathematical dis-
course. Universals and ideals are activities of social discourse; they are as real
as that discourse, which is to say as real as the ordinary human-sized world
of action. There is no need to assign them to another world.
Another mistake is to interpret mathematics as consisting in tautologies.
The identity between those items on opposite sides of the equal sign in a
mathematical equation is not the same kind of identity as that established
by giving something a name; it is not the empty tautology exemplified by
explaining “gravity” as “the propensity to fall.” Mathematical equivalence and
verbal tautology are embedded in different language games, in different systems
of operations. The arbitrary tautologies of ordinary language lead nowhere;
mathematical procedure is a discovery-making machine. The machinery of
mathematical equations operates in multiple directions, as Frege noted in
distinguishing sense and reference. The equivalence-making conventions of
mathematics open up successive classes of abstract operations whose properties
can be investigated. Conventions are arbitrary, but discovery in mathematics
consists in exploring the patterns opened up by adopting various kinds of
conventions. Mathematics is a special field of empirical discovery, insofar as
“empirical” means investigation of experience in time; it is the experience of
the mathematical network of investigating what is implied in the symbolic
conventions it adopts.
The theories that mathematics must be a transcendent realm of Platonic
objects, or at least the a priori truths inherent in tautologies, are attractive
because they help explain the feeling that mathematics is certain, that its results
are as high and irrefutable a level of truth as humans can attain. This certainty
can be explained by the special social character of mathematical networks.
Because the contents of mathematics are chained together over time, from the
most rarefied abstractions back to the ordinary operations of counting, the
edifice is tied together at an awesome level of tightness. It is not simply that
results are lazily passed along from one generation to the next, as a long-stand-
ing traditional paradigm which no one bothers to question. On the contrary,
the linkage is deep and inescapable, since the topics of successively more
abstract mathematics have been the underlying patterns of the operations of
previous mathematics. Mathematics embodies its history, in its procedures for
using symbolism, to a degree found in no other field. The most naive practi-
tioner produces the same results as everyone else, because anyone who learns
Epilogue: Sociological Realism^ •^869