the entire system. This was no isolated incident; within a year, Zermelo made
a sensation with his proof of a more generalized paradox.^18 Zermelo’s axiom
of choice set off much criticism by mathematicians, especially helping to
provoke the formation of the opposing intuitionists, including once again the
anti-logicist Poincaré.
Paradoxes and controversy do not derail logicism but launch it. Although
Frege, emotionally exhausted by lack of support, declared the bankruptcy of
his set theory approach, Russell forged ahead immediately to re-ground the
program on a theory of types. There is an ontological hierarchy of individuals,
sets, sets of sets, and so on; statements about membership in sets are mean-
ingful only between adjacent levels (Wedberg, 1984: 134–135). Far from
abjuring The Principles of Mathematics, Russell developed its sketch into a
fully formalized system. He enlisted his old mathematics teacher, and by 1910
he and Whitehead in Principia Mathematica had derived in detail a portion of
elementary arithmetic going considerably further than Frege. Logicians were
more interested in the Principia than were mathematicians. A full-scale move-
ment of logicist philosophers was springing up; and their favorite topics, in
a well-worn pattern of intellectual life, were connected not with extending
Russell’s method along the path he had marked, but with re-digging the
foundations at just the points where he indicated there was trouble. The logicist
program claimed to build mathematics from the simplest possible starting
point, reducing assumptions to the most unassailable and obvious premises.
But the theory of types, like Zermelo’s axiom of choice, was neither obvious
nor simple. Replacing such devices became the central puzzle around which
both meta-mathematics and logical formalist philosophies grew.
As Russell came into his own wielding the weapons of logicism, his stance
toward philosophy was that it is a history of conceptual mistakes. Hazy
metaphysical systems can be cleared away by using the tools which had worked
so well in mathematics, just as Dedekind had cleared up Zeno’s paradoxes of
motion and modern rigor had given the final answer to Berkeley’s objections
to the calculus. Unlike the German mathematicians, who generally had no
animus against philosophy, Russell took this militant stance because he was
simultaneously engaged in two battles: for set theory in mathematics and
against Idealism in philosophy. Idealism, as we have seen, was everywhere the
philosophy of the transition from the old religious university to secularism. In
Germany the transition was two generations past, but in England it had taken
place in the generation of Russell’s teachers. Russell himself along with his
friends was an Idealist into the 1890s. Russell’s break with Idealism was
especially pointed because Idealists too tended to take logic as their turf. Bear
in mind that logic, before the revolution worked by Frege and Russell, was
widely considered an ancient and stagnant field, a soft territory tempting in-
The Post-revolutionary Condition^ •^713