and was especially close to Gödel. It was with some sense of showdown that
Brouwer was invited to lecture at Vienna in March 1928. Wittgenstein sud-
denly ended his long intellectual withdrawal and began talking about philoso-
phy again immediately after hearing Brouwer’s lectures, although he at first
had to be coaxed by his Vienna Circle acquaintances to attend (Wang, 1987:
81; Monk, 1990: 249). That summer Hilbert, in a celebrated address to the
International Congress of Mathematicians, challenged the field to solve four
basic problems of the consistency and completeness of analysis, number theory,
and logic. Gödel immediately chose these problems for his dissertation; by
1930 he had solved them all.
The conflicting philosophical schools also became tied to the mathematical
foundations battle. Husserl was invoked by the intuitionists, especially as he
moved in the 1920s to emphasize the pre-formalized life-world, within which
geometry was seen to originate (Heyting, 1983; Coffa, 1991: 253–255). The
growing sense of antagonism between phenomenologists and logical positivists
became overlaid upon the rivalry between the mathematical camps. Tarski,
who like Husserl was descended from the Brentano school, adopted from
Husserl a hierarchy of semantic categories, which played the same role as
Russell’s theory of types but in a more guarded fashion by generalizing from
logic to languages. Tarski visited the Vienna mathematics department during
February 1930, with a galvanizing effect on Carnap.
Even more of a Trojan horse was Wittgenstein. Originally the heir apparent
to the Frege-Russell program, by 1930 he was explicitly turning away in a
direction that increasingly resembled that of the mathematical intuitionists. His
strong personal influence led to a growing split within the Circle. In 1930
Gödel was challenging Wittgenstein in the Circle on philosophical issues: How
does one distinguish the allowably meaningless statements which constitute
Wittgenstein’s own higher-order clarifications from the meaninglessness of
metaphysics, which it is our business to destroy (Coffa, 1991: 272)? Soon
everything clicked in new directions for philosophers and mathematicians alike.
In the summer of 1930 Gödel, not yet 25 years old, conveyed to the Circle his
proof that any logical system capable of generating arithmetic contains propo-
sitions undecidable within the system.
The creativity of Gödel and of the later Wittgenstein spun off from much
the same point, and from their mutual disagreements. This was a clash between
two highly sponsored group favorites, an older and a younger. Gödel was
sponsored by his mathematical mentors in the Vienna Circle as the bright
young student capable of solving the most central problems; through them, his
results were immediately and widely publicized.
More fireworks exploded at this moment when the Circle’s conflicts became
most intense. Within the next year Popper, a peripheral member of the Vienna
The Post-revolutionary Condition^ •^727