[FOM] Only one proof
MELVYN NATHANSON
MELVYN.NATHANSON at lehman.cuny.edu
Sat Sep 12 12:37:13 EDT 2009
Vaughn Pratt writes, "the Fundamental Theorem of Arithmetic is not the sort of
deep number-theoretic fact that serious mathematicians prior to the 20th
century would have felt the need to revisit. It's just too elementary."
This is a curious statement. To me, the Fundamental Theorem of Arithmetic is
one of the deepest facts in mathematics, and it is mistake to believe that
theorems with elementary proofs cannot be deep, or that the adjectives "deep"
and "elementary" cannot be applied simultaneously to the same mathematical
statement. There is some discussion of this in my two recent notes,
"Desperately seeking mathematical truth," Notices of the American
Mathematical Society, August 2008, and "Desperately seeking mathematical
proof," The Mathematical Intelligencer, Volume. 31, Number 2, 2009.
Nor is it clear to me what is the significance of the question that began this
string of emails: Why does it matter if a theorem has one or more essentially
different proofs? The brain is a frail instrument. Sometimes we are lucky or
clever enough to find two different proofs of a theorem, and sometimes not.
You cannot measure the depth or importance of a theorem by the number of its
proofs.
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