between genealogies of research equipment and the argumentative network of
scientific intellectuals. Mathematical rapid-discovery science adds a third net-
work, the lineage of techniques for manipulating formal symbols representing
classes of communicative operations. Mathematics does not provide a magical
eye by which we see the transcendently existing objects behind the phenomenal
surface of experience, the invisible entities of scientific theories. Mathematics
is connected to the other two networks in the phenomenal world of experience.
On one side, the measurements given by the research equipment are made
into mathematical realities because humans use them as markers, in the same
sense that the primitive mathematical operation of counting is the social
procedure of gesturing toward (and thereby setting equivalent) items of expe-
rience. As Searle (1992) would say, there is no homunculus in the research
equipment; it is the human mathematicians who use equipment as extensions
of their own capacity for making gestures. These are gestures simultaneously
toward the non-human world and toward the social community, which has
built up a repertoire of reliable methods of transforming one set of symbolic
gestures into another.
On the other side, the genealogy of mathematical techniques connects to
the network of scientific intellectuals, which constructs the meaningful objects
and arguments which make up the humanly familiar contents of science.
Equipment genealogies produce phenomena in the world of experience; scien-
tific intellectuals turn these phenomena into interpretations, useful for winning
arguments and setting off the network onto yet further topics of investigation.
The “invisible” world of scientific entities comes from the intellectuals, not
directly from the equipment. Mathematical technique becomes important for
scientists because it enables them to give an especially obdurate character to
at least parts of their arguments; but this is the obdurate reality of certain
chains of reflexive communicative operations, which it has been the business
of mathematicians to investigate. The crystal-hard social reality of mathematics
gives backbone to the socially negotiated arguments of scientific coalitions.
Mathematics is a bridge: it shares with the scientific network the character
of being social; it shares with equipment genealogies the character of being a
lineage of techniques. Since the lineage of mathematical technique is a lineage
of discoveries about processes in time-space reality (i.e., about mathematical
operations), it meshes well with equipment-generated phenomena which are
multi-dimensional processes, whose form cannot be interpreted on the low
levels of abstraction and reflexivity encountered in ordinary noun-adjective-
verb grammar or ordinary arithmetic. (That is why the investigation of higher
algebras—quaternions, vectors, matrices—was so fruitful for the development
of modern physics.) Here again we find the social reality of mathematics874 •^ Meta-reflections