FOM: Re: When is a proof conclusive? Reply to GonzaleZ Cabillon
Julio Gonzalez Cabillon
jgc at adinet.com.uy
Tue Dec 16 09:26:54 EST 1997
Joe Shipman writes:

 Since published mathematics is always very incompletely formalized,

[Published or unpublished mathematics is always very incompletely formalized]
 in practice
 the proper definition of proof is simply "a completely convincing argument".
 If it is not completely convincing it is not conclusive.
According to the "proper definition", if the argument is not completely
convincing the argument is not a proof. This leaves the question answered,
viz, what's the difference between a "proof", and a "conclusive proof"?
 defining both within the same theory T, and using the same metalanguage.
 Note how relative this
 definition is  it leaves unsaid who is supposed to be convinced. One could
 say that if any highly competent professional mathematician in the relevant area
 who has put in the appropriate effort is not convinced, then the proof needs to
 be sharpened (either by adding intermediate steps or identifying an unaccepted
 assumption A and recasting the theorem T as "A>T"). This highly pragmatic
 definition WORKS because mathematicians operate by consensusthey keep at it
 until they are sure there is or is not a proof (after a reasonable period of
 time there was a clear consensus Wiles's original proof had a gap, and when he
 fixed it this was also accepted widely within a few months).
...
Agreed.
Julio Gonzalez Cabillon
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