dent, mathematicians began to split into rival movements for shoring up
foundations.
By the 1860s, above all in Germany, recognition had dawned that a large
part of mathematics consists in investigating arbitrary and abstract concepts
without physical referents or even conventional geometric representation on
paper. The result was a movement toward rebuilding the various branches of
mathematics on numbers, arithmetizing analysis and geometry. Numbers, too,
the last bastion of realist interpretation, came under scrutiny in the 1870s and
1880s by Weierstrass, Dedekind, Cantor, and Frege. When Cantor in 1879–
1884 demonstrated the existence of transfinite numbers—successive orders of
infinity—the movement of rigorization broke into open scandal. Kronecker,
the powerful journal editor in Berlin, opposed publication of papers which
created unnatural monstrosities, which he attributed to the methodological
fallacy of using derivations going beyond finite series of steps. For Kronecker,
only the natural numbers really exist; in defending a conservative position in
this respect, he was provoked into formulating a radical program to reconstruct
all of mathematics, which after 1900 became the intuitionist program. Cantor,
who foreshadowed the formalists, was clearly aware of the power struggle
taking place over the nature and organization of mathematics. In 1883 he
argued that the distinctiveness of mathematics as a field is its freedom to create
its own concepts without regard for reality (Kline, 1972: 1031). To back up
his position, he pushed for the separation of German mathematicians from the
Gesellschaft Deutscher Naturforscher und Ärzte, and became the founding
president of the Deutsche Mathematiker-Vereinigung in 1891, and organizer
of the first International Congress of Mathematicians in 1897, a stronghold of
the formalist movement (Dauben, 1979; Collins and Restivo, 1983). The
formalists represent the tendency of autonomous specialization in mathematics
at its most extreme.Frege’s Anti-psychologistic Logic
It was in this context that Frege developed his project to found arithmetic on
logic. In the process he had to create a usable logic. It was hardly a matter of
supporting one discipline on a more prestigeful one; mathematics was much
more advanced, and the prestige of logic dates largely from the adoption of
Frege’s system.
Logic was in the air, but the strongest winds were blowing from another
direction. Just at this time the Neo-Kantians were intruding as usual into other
specialties. Cohen and the Marburg school (starting in 1874) made a program
out of investigating the logics of the various disciplines; the regulatory logic of
jurisprudence, for instance, is ethics; other fields—mathematics, education, so-
700 •^ Intellectual Communities: Western Paths