creativity and sometimes ending up far from the starting place. The original
thrust of Frege’s method was fruitful in all these senses. The takeoff came at
just the time when Russell was promoting his famous paradox about higher-
order sets. Now the movement had not merely the straightforward path of
carrying out the formalist program in mathematics, but also a branch con-
cerned with the underlying problems of set theory. Frege’s Platonism, initially
an innocuous position within mathematics, opened all sorts of controversies
when it was expanded into militant claims about truth and meaningfulness.
And the assumption that logic is the foundational language of mathematics,
when widened into a program for all of language, turned into an almost polar
opposite by the end of Wittgenstein’s tortuous career. The logical formalists
provide one of the strongest proofs that a philosophical movement lives not
on its solutions but on its problems.
The Social Invention of Higher Mathematics
Mathematics by the time of Frege had become very different from that familiar
to Kant. In the 1700s the field consisted mostly of analysis, exploring the
branches of Leibnizian calculus and their applications in physical science. By
around 1780, the belief had become widespread among leading mathemati-
cians that mathematics had exhausted itself, that there was little left to dis-
cover.^1 Unexpectedly, the following century was the most flamboyant in the
history of the field, proliferating new areas and opening the realms of abstract
higher mathematics.
The sudden expansion of creativity arose from shifts in the social bases of
mathematics. Competition for recognition increased with a large expansion in
the numbers of mathematicians. The older bases for full-time professional
mathematicians had consisted of the official academies of science, notably
Paris, along with Berlin, St. Petersburg, and a few others. The foundation of
the École Polytechnique in 1794 introduced continuity of training for a highly
selected group of students while providing teaching positions for the most
creative. In Germany, the new public school system widened the selection net
to pick up penurious students of potential talent (such as Gauss, a mathemati-
cal parallel to Fichte in this respect). The university reform extended to mathe-
matics the emphasis on innovative research, as well as giving a distinctive slant
toward pure knowledge apart from practical application. The process of dis-
ciplinary differentiation split math from physics and astronomy, encouraging
the tendency to abstraction.
Along with these organizational changes came journals devoted largely or
wholly to mathematics, beginning with the journal of the École Polytechnique
in 1794, and the private journals of Crelle in Germany in 1826 and Liouville
The Post-revolutionary Condition^ •^697