results, embedded in the meaning of any particular mathematical expression.
The history of mathematics is embodied in its symbolism.
After Descartes, the machinery of manipulating equations has consisted in
procedures for transferring symbols from one side of the equal sign to the
other, and for rearranging terms until the equation takes the form of what was
to be proven. A key to this method is reversibility. The results of operations
may be taken as starting points, by assigning them symbols which can be
manipulated in the equation. The symbols for unknown numbers x, y which
satisfy particular equations are treated as if they were already known; in the
same way, any other class of expressions, including those yet to be discovered,
are represented as placeholders in the equation. The machinery is not impeded
by our ignorance of any particular fact; by the method of symbolizing whole
classes, including both past results, future results, perhaps even unattainable
or impossible results, it is possible to set in motion the procedures of manipu-
lating equations, and to come to conclusions which reveal how its terms are
related.
In one sense the symbolism is a reification. It treats items as if they were
things, by symbolizing them in a thing-like way; it gives apparent solidity to
this x, or this f(x), which is yet another temptation to treat the objects of
mathematics as if they were Platonic realities. But this reification is provisional
only, for the sake of proceeding with the technology of manipulating equations.
The symbolism belongs to an ongoing history. It points both backwards and
forwards: backwards because the most obvious referent of a symbolic place-
holder is something of the kind which has already been found on a more
concrete level. The x may be replaced with a number which solves an arithmetic
problem; the f(x) may be exemplified by some particular algebraic expression.
And because the symbolism is abstract and general, it points forward to larger
classes of mathematics: not only to all the particular unknowns which might
be substituted for a symbol, but outward toward the space of abstract possi-
bilities in a cognate family of operations. In this way the formulation of new
symbolism—which always means new systems of practice, procedures for
manipulating a collection of symbols—opens up new areas of discovery, higher
orders of mathematics to be investigated. Successive orders of symbolism thus
point not only backwards to previous work in which they are grounded, but
also onward to new kinds of problems.
Mathematics is social, then, in two successively stronger senses: Anyone
who performs mathematics even to the extent of understanding an equation
in elementary arithmetic is engaged in a form of social discourse and a network
of teachers and discoverers. And symbols and procedures which make up
mathematics reflexively embody the history of that creative network all the
866 •^ Meta-reflections