FOM: Conclusion on Conclusiveness

[jshipman@bloomberg.net]

The fact of the matter is (to use a favorite FOM phrase) that we never have
absolute certainty that a proof is valid.  In the case of a machine-checkable
proof there might have been a bug in the program for the machine's Pentium chip
or a cosmic ray that changed the contents of a memory location or three.  We
reduce the subjective probability we are wrong to negligible levels by various
techniques.  (For example, in the case of the 4-color theorem of Appel and
Haken, whose proof includes a computer output not realistically checkable by an
individual mathematician who is not crazy already, the proof of the program's
correctness is not difficult and many people independently wrote their own
programs which verified the result).  A randomized proof of a statement of the
form "n is prime" is just another technique for increasing our subjective
certainty re a mathematical proposition.  An interesting question is whether
this expansion of the notion of proof has any relevance for concrete statements
that are NOT totally finitary and (infeasibly) verifiable in principle.  My
guess is "no, but..."; the "but..." will have to wait for a later post.-JShipman