[FOM] Only one proof
Vaughan Pratt
pratt at cs.stanford.edu
Sat Sep 5 04:06:09 EDT 2009
joeshipman at aol.com wrote:
> That leaves Feit/Thompson as my best example of an important theorem
> with essentially only one proof. Can anyone think of another? (It's not
> enough that there is one idea common to all proofs unless that idea is
> the only difficulty -- I'm looking for a theorem where all proofs are
> essentially similar both globally and locally.)
As we know from Gauss there are strikingly different proofs of the
Fundamental Theorem of Algebra (that every univariate polynomial with
complex coefficients has a complex root), although they all rely on
topology. (One would imagine there should also be a discrete
nontopological proof using algebraic numbers, the obstacle there seems
to be that the very existence of the algebraic numbers seems to depend
on topology.)
But what about the Fundamental Theorem of Arithmetic? All proofs I'm
aware of seem to boil down to one version or another of Euclid's GCD
algorithm. Is there a different proof?
Vaughan Pratt
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