blending seamlessly with the non-human natural reality of equipment-gener-
ated phenomena.
The core social network in science remains that of the human intellectuals.
Science, to be successful, ultimately issues in words and imagery. A purely
mathematical enterprise such as string theory does not acquire full, socially
accepted “reality” until there is a verbal interpretation of its main points, which
translates them into the familiar noun-like “entities” which count as para-
mount reality in ordinary language. But here we should note that mathematics
too is embedded in words.^11 This reminds us that “mathematics” is two
networks in one, a genealogy of techniques and a human network which both
knows how to work the techniques and engages in the usual intellectual contest
of setting arguments for one another. Verbal discourse is the most encompass-
ing frame, the home ground of intellectual life. If mathematics is an important
bridge among the human and non-human networks which constitute science,
it is because mathematicians are hybrids sharing all the traits of humanness,
from verbal discourse to their own special form of formal reflexiveness.
Mathematics is simultaneously empirical and conceptual. It encompasses
both the observation of experience in time and space, which is always particu-
lar and situationally located. But it deals with the universal and general, indeed
with patterns which are irrefutably found among universal concepts, because
its topic is the pure generality of human communicative operations. These are
the activities of making things equivalent, of transposing them for one another.
The topic is universal because it comprises the operations of treating things as
universals. It is simultaneously empirical, arising within experience and appli-
cable to experience, because doing mathematics is an activity taking place in
time and within a social network. The universal features of mathematics are
empirically discovered, by the work of mathematicians investigating various
systems of operations. The topic of mathematics is their system of communi-
cative conventions. What they discover about this is objective, obdurate reality.
If we say it is socially constructed, it is an empirical investigation of the
obdurate qualities of social construction. It is so real because it is so thoroughly
socially constructed.
Why Should Intellectual Networks Undermine Themselves?
A social constructivist theory of the intellectual world affirms several inescap-
able realities: other people and their intercommunication, the time-space world,
and human-sized material things with our own bodies among them. Mathe-
matics makes discoveries about the obdurate realities of intellectual operations,
the nested chains of gestures by which we designate equivalences and transform
Epilogue: Sociological Realism^ •^875