FOM: Re: When is a proof conclusive? Reply to GonzaleZ Cabillon

[Julio Gonzalez Cabillon]

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  [Published or unpublished mathematics is always very incompletely formalized]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 consensus--they 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