[FOM] Only one proof

[Timothy Y. Chow]

Vaughan Pratt wrote:
>I find it hard to believe Fermat or Euler would have considered this a 
>"new proof," and I would be shocked if Gauss had.  It's simply the 
>distillation to its essence of the only way it's ever been argued.

I'm not sure it's productive to debate this point further without agreeing 
on a more careful definition of what makes two proofs different.  The 
"scope test" mentioned by Joe Shipman sounds like an interesting idea and 
I will be interested to see the results of that analysis.

"Distillation to its essence" is a useful concept for trying to understand 
a piece of mathematics, but it doesn't strike me as a very robust 
criterion if one's goal is to classify proofs.  Presumably, the more 
powerful your intellect, the more everything seems trivial, and the more 
insignificant the differences between two arguments appear to be.  Take 
the four-color theorem, for example.  Is the Appel-Haken-Koch proof 
"essentially the same" as the Robertson et al. proof?  Yes and no.  
They're both based on finding a set of unavoidable, reducible
configurations, which is some sense is the crucial idea.  However, 
Robertson et al. did prove some important lemmas along the way that 
greatly simplified the search for such a set.  But a sufficiently powerful
mind might be able to see instantly that such sets of configurations exist 
and might regard the differences between the two proofs as inessential.  A 
somewhat more objective criterion for deciding equality of proofs seems 
desirable.

Tim