From the NY Times philosophy column "The Stone"
yesterday at 5 PM—
Timothy Williamson, Wykeham Professor of Logic at Oxford,
claims that all the theorems of mathematics
"… are ultimately derived from a few simple axioms
by chains of logical reasoning, some of them
hundreds of pages long…."
Williamson gives as an example recent (1986-1995)
work on Fermat's conjecture.
He does not, however, cite any axioms or "chains of
logical reasoning" in support of his claim that
a proof of Fermat's conjecture can be so derived.
Here is a chain of reasoning that forms a crucial part
of recent arguments for the truth of Fermat's conjecture—
K. A. Ribet, "On modular representations of Gal(Q̄/Q)
arising from modular forms," Invent. Math. 100 (1990), 431-476.
Whether this chain of reasoning is in fact logical is no easy question.
It is not the sort of argument easily reduced to a series of purely
logical symbol-strings that could be checked by a computer.
Few mathematicians, even now, can follow each step
in the longer chain of reasoning that led to a June 1993 claim
that Fermat's conjecture is true.
Williamson is not a mathematician, and his view of
Fermat's conjecture as a proven fact is clearly based
not on logic, but on faith.