FOM: When is a proof conclusive? Reply to Gonzales Cabillon

[jshipman@bloomberg.net]

Since published 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.  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).  Lakatos's
"refutations" identified unstated assumptions or incoherent concepts that
needed to be modified.  Proofs are more or less but never completely formalized;
greater formalization is simply a strategy for sharpening the proof.-Joe Shipman