Descartes was the network member who reaped the first great fame for this
revolution. He set out the general methods for what henceforth became the
standard forms for setting up equations and symbolizing unknowns and the
rules for manipulating them. His algebraic geometry unified all the field of
mathematics into a single conceptual realm; it also brought to the center of
attention a meta-space on which points, lines, squares, and dimensions higher
than those of the physical universe could all be combined in a single process
of mathematical manipulation. Descartes made the intellectual world aware of
a meta-level of abstraction, expounding not only his particular results but also
a general theory of equations, and pointed to a higher dimension of mathe-
matics. Most important, he trumpeted it abroad that the mathematical com-
munity now possessed a method for producing a stream of new and reliable
results, a machine for making discoveries.
Exploring the turf opened up by this discovery-making machine made up
the first wave of modern mathematics. Around 1800 a second wave of mathe-
matical abstraction and discovery took off. This time it proceeded by height-
ening reflexiveness about what mathematicians themselves were doing, a grow-
ing self-consciousness which led up to the realization that mathematics itself
is a study not of things but of its own operations. This wave of higher
mathematics was set off by an organizational shift, the emergence of academic
positions, networks, and publications in which mathematics could be pursued
independently of the physical sciences and their applications. The machinery
of mathematics began to be scrutinized in its own right, apart from the results
it gave for generalizations about the world of nature. The practical methods
of the previous period, when the various territories of calculus were explored,
now began to be criticized for their lack of rigor, for operations such as
discarding terms and dividing by zero, and for assuming concepts such as series
which are never seen to converge.
Rigor became a new standard for the competitive game, now played on its
own abstract turf. Conceptually sloppy methods could no longer be justified
merely because they brought good results in physics or engineering; at the same
time, new standards of proof were held up in the competitive challenges be-
tween one mathematician and another. This purposely involuted game turned
out to be a success; rather than being a needless refinement, rigor opened up
a vast new realm of abstract mathematics to be explored. The discovery-mak-
ing machinery took off again, this time on a higher plane. The method of
axiomatization, carefully laying out the rules of operations, was applied not
only to geometry (where a set of restricted, physically commonsensical princi-
ples had been standard since Euclid), but also to algebra, and then to still more
abstract realms such as the theory of groups.
The march into this realm was made exciting by the recurrent experience
of discovering paradoxes: patterns which seemed absurd from the point of view
Sequence and Branch in the Social Production of Ideas^ •^849