phy? We find it happening across the board in the generation of 1900: Husserl,
Russell, later the Vienna Circle. Mathematical logic was a meta-territory
arising from the confluence of axiomatic programs in algebra, in geometry, and
in the arithmetization of analysis. Having been formulated on an appropriately
general level, it rose above mathematics like a balloon slipping its moorings
and floating away. Although it arose out of disputes over the techniques of
mathematical construction and proof, the axiomatic programs become unnec-
essary for mathematicians’ ordinary work of discovering and proving algo-
rithms; it became a goal displacement of the sort familiar in the sociology of
organizations, an end in itself pursued by its own specialized community. Soon
disputes and factions among meta-mathematicians were generating their own
dynamics, no longer dependent on stimulation from ordinary mathematical
work. Meta-mathematics now converged with philosophical turf, and thus
with the community of philosophers. For philosophy is precisely the attention
space of most generalized issues; the episode of meta-mathematics recapitu-
lates the way philosophies have emerged throughout history, as the disputes
of substantive areas (nature cosmologies of the early Greeks, salvation tech-
niques in India, theological training for Christians) take on an attention focus
of their own.
In this case logic got a tremendous boost in attention when it shifted from
a technical concern within mathematics to an activity of philosophers. For
philosophers occupy the space where claims are made about knowledge in
general, and thus about the general-purpose role of all intellectuals; major
controversies on this turf are hard to ignore. It was for this reason that
mathematicians hostile to philosophy, such as Russell and Wittgenstein, ended
up as philosophers, that is, in the attention space where they were most
successful and where they could avoid the fate of Frege and the early Peirce.
The explicit emergence of meta-mathematics and of the logicist movement
within philosophy occurred in a few years at the turn of the century. Russell
was the shock center from which emanated the second wave of the mathemati-
cal foundations crisis, stimulating the competing programs of Hilbert’s formal-
ism (1904) and Brouwer’s intuitionism (1907). Already in exchanges starting
in 1897 and continuing through 1906 (Coffa, 1991: 129–133), the mathema-
tician Poincaré had taken issue with Russell’s approach, raising objections in
the direction that would become intuitionism. Russell soon managed to escalate
the crisis. In an appendix to Principles of Mathematics (1903), Russell draws
attention to Frege’s logic and points out a contradiction in the notion of sets
which are not members of themselves. (Is this set a member of itself? If yes,
no; if no, yes.) Since Frege’s extensional logic depends on translating all
predicates (i.e., concepts or intentions) into sets, and thus on forming sets of
elements of any kind whatever, Russell’s paradox was a blow at the heart of712 •^ Intellectual Communities: Western Paths