way back to its earliest links; self-reflection on its own prior operations is itself
the edifice of higher mathematics.
Let us underline a further aspect which demonstrates that mathematics is
thoroughly social. The topic of mathematics is operations, not things. It is not
a field which examines what kinds of things exist in the world, or somehow
in a world beyond the world. Return to the ground zero of mathematics,
numbers. Because a number may be treated as a noun in a sentence, it is easy
to assume it is a thing. But the primitive of number is simply counting; it
consists in making gestures, verbal or otherwise, toward something while
telling off a sequence, “1, 2, 3.. .” The answer to “How many?” is the
number that one stops with when one’s gesturing to each in turn is complete.
Numbers are primordially the activity or operation of numbering.
In this respect, numbers are like other symbols which make up human dis-
course. Their universality comes from their universal use, not from any char-
acter of the objects on which they are used. Numbering is a process of dividing
and pointing. It may be applied to anything: to material objects which may
have obvious separations among them, but also to things whose outlines are
vague and shifting (clouds, for instance), or indeed to “things” which are not
things at all, which may be operations or abstractions or imaginations. Num-
bering is the operation of making items equivalent by counting them off; they
become items because they are treated as such. This does not mean that
numbers are illusions. They are real, as operations carried out by human
beings, activities carried out in time and spatial location. They are also gener-
alizable, transferable from one situation to another, because they are operations
which we can apply over and over again. Their generality comes from their
being operations of human discourse.
The operations of mathematics are social, from the primitive level of count-
ing on up. It is not merely that we learn to count by being taught by others,
and that the skill of counting is extremely widespread in most societies. The
principles of the sociology of mind apply. Counting can be an overt social
activity: I count these things in front of us, I invite you to count them as well,
or to accept my count because if you follow the same procedure, you will come
to the same conclusion.^6 Since conceptual thinking is internalized from external
discourse, and takes its meaning only because it implies an external audience,
counting which I do for myself alone is also an operation within a social frame.
And the conclusion reached earlier may be repeated in this instance: counting
produces universals because it takes a universal stance, the stance of anyone
at all who follows this convention of discourse.
What has been said about counting may be said again about each of the
successively more abstract forms of mathematics. Arithmetic generalizes the
Epilogue: Sociological Realism^ •^867