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Monday, May 14, 2012

Mathematics, Logic, and Faith

Filed under: General,Geometry — m759 @ 4:44 PM

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.

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