by the dominance of physical practices, embodied in material equipment (one
might call this materialist constructivism). It is possible that a different line of
research equipment or a different line of tinkering could produce repeatable
results as well; and this might be combined with a different lineage of idea
interpretations from the intellectual community. Because research discoveries
are driven by recombining equipment genealogies, a fan-shaped pattern of
discovery paths is hypothetically possible, although some of these are not
followed up because of the social focus of attention. Different sciences might
be historically constructed from the same point in time. It is the social process
of seizing on the lineage of equipment that advances most quickly on the
research front which cuts off some of these directions and exalts one of them.
Our formula for high-consensus science becomes: competitive philosophical
networks plus empiricism carried out with a fast-moving genealogy of research
technologies.
mathematics becomes a discovery-making machine
There is an alternative route to rapid-discovery science. Another key to the
scientific revolution was not laboratory equipment but mathematics. Coperni-
cus overthrew geocentric astronomy not with new observations but by mathe-
matically simplifying old data. The two routes may coincide; many aspects of
the scientific revolution of the 1600s and 1700s were carried out not only by
experiment but also by formulating quantitative principles for the results. But
the two routes were not identical. A mathematical revolution preceded the
takeoff of scientific research by two or three generations; the upsurge in the
number of noteworthy mathematicians in Europe started in the 1490s,^12 and
the first big advances began around 1520–1550 with Ferro, Cardan, and
Tartaglia in the general solution of higher-order algebraic equations, leading
to the expansion of new mathematical fields with Viète.
According to a familiar line of argument, mathematization of the world-
view produced modern science. The difficulty is that traditional mathematical
science, such as astronomy among the Greeks, Chinese, or Indians, does not
have the characteristics of consensus and rapid discovery which are central to
modern science. Mathematics in general is not sufficient to bring about con-
sensus-making, fast-moving science; only a particular kind of mathematics
provides the key.
What kind of mathematics can this be? The mathematical revolution un-
folds when mathematics itself becomes a research technology. That is to say,
technology is a set of embodied practices which bring about reliable, repeatable
results. Such techniques, although not consisting in a complex physical appa-
ratus, nevertheless are material: they consist in methods for writing equations
on wax or paper, or placing sticks on a counting board, following procedures
538 • (^) Intellectual Communities: Western Paths