FOM: Consensus vs Indubitability

[JoeShipman@aol.com]

In a message dated 9/18/98 6:21:11 PM Eastern Daylight Time,
fallis at u.arizona.edu writes:

<< Why does our evidence that the probabilistic proof is reliable have to be a
 conventional mathematical proof?  Why can't it be probabilistic proofs all
 the way down - as long as the probabilistic proofs allow for consensus?
  >>

I find it very hard to imagine how this might work without some core layer of
non-probabilistic logical foundations, but this is sufficiently intriguing
that I would like to see you explicate the possibility of "probabilistic
proofs all the way down" further. 

Is your article "The epistemic status of probabilistic proof" available on
line?
I agree with you that statements confirmed probabilistically should not be
disqualified fromm "Annals of Mathematics"; but what makes you suppose that
they are?  Does anyone know of any examples of probabilistic proofs that have
appeared in major journals or been rejected from them because they were
probabilistic?  I can't think of any interesting candidates for probabilistic
proof that are not delta_0 (i.e. previously reduced to a finite problem),
though occasionally a delta_0 result is sufficiently interesting to be
publishable (nonexistence of finite projective planes of order n for some n, a
value for a Ramsey number, the existence of a finite simple group of a
particular order).  Can you give some additional examples of probabilistic
proofs besides the ones Steve Cook mentioned (Miller/Rabin, Berlekamp,
Schwartz)? -- Joe Shipman