discourse, in the very saying I display a statement in a discourse. If I fall back
on asserting that mathematics must be transcendent because it can be translated
from one language to another, I rest my claim on the existence of translations,
operations which connect several discourses together; this not only does not
escape from discourse but adds yet another kind of discourse.^5
Mathematics has a social reality in that it is inescapably a discourse within
a social community. This might seem a minimal kind of reality. We should not
assume, however, that social discourse has no objective, obdurate quality, the
kind of strong constraint that answers the concept of “truth.” To show why
mathematical discourse has this quality, we must explore the distinctive char-
acter of mathematical networks.
Mathematical networks are historically linked backwards to previous
mathematicians. This is so not merely in the genealogical sense typical of all
creative intellectuals, that the central network of famous creators in one
generation typically spawns the next generation of discoverers. Mathemati-
cians are distinctively focused on their history, insofar as the main path of
mathematical discovery is to make a topic out of the methods used in a
previous level of mathematics, to formulate a symbolism which makes explicit
some operations previously tacitly assumed, and to explore the implications of
the higher order of abstract symbolism. Algebra generalizes the rules of arith-
metic, formulating methods whereby entire classes of arithmetical problems
can be solved. Successively higher levels of algebra produce general rules about
the solvability of various kinds of algebraic equations. Similar sequences have
occurred in analysis, number theory, geometry, and their various hybrids.
Over the course of such sequences, new concepts are created which abstract
and summarize whole classes of previous work. The conventional algebraic
symbols for unknowns, x, y, stand for any number whatsoever; at a higher
level, the function sign f(x) stands for entire expressions of whatever form.
Functions of functions abstract still further; so do groups, rings, fields, and so
forth. It matters not if what is abstracted is taken to be a number, an unknown,
or an operation; on a higher level, the operations of conventional arithmetic
are abstracted into a class of operations, which may be selected and elaborated
in various ways to give rise to alternative arithmetics, alternative algebras—in
short, the pathway to higher mathematics.
Mathematics is the most historical of disciplines in the sense that its central
topic is digging backwards into what is taken for granted in its previous work.
Algebra not only implies arithmetic; and higher levels of algebra, of analysis,
and so forth, not only imply the previously investigated lower levels of abstrac-
tion in these fields. The symbolism of mathematics at any point in its history
refers to the kinds of operations which were carried out in the earlier mathe-
matics. It is impossible to escape from the historical accumulation of previous
Epilogue: Sociological Realism^ •^865