FOM: indubitability

[M. Randall Holmes]

Reuben Hersh said:


There is a  practically insignificant but philosophically crucial 
difference between indubitability and high standards of rigor.

Go back to Plato.  Why does he claim that knowledge of Ideas is superior
to knowledge of the visible world?  Because, says Plato, knowledge of
Ideas is certain, indubitable, whereas knowledge of the visible world is
always open to doubt.  He thinks knowledge of the visible world  may
approach a high degree of certitude, but it can never be *absolutely*
certain.

My reply is:

High standards of rigor are meaningless (one cannot understand why
they are desirable) except as an approximation to a standard of proof
which, if achieved, would guarantee the correctness of the results
proved (mod the correctness of axioms, of course).

Our practice witnesses this.  When we find out that a proof is
invalid, we look for a mistake in it.  We do make mistakes -- that is
evidence that we are fallible, not that the logic of our proofs is
fallible.

Correct proofs are indubitable -- but, due to the limits of our power
to survey a proof, we may doubt that any given purported proof is
actually a correct proof (this does _not_ cast doubt on the
indubitability of proofs which are correct). Attention to rigor is
understandable under these conditions; attention to rigor would be
neurotic if there were no standard of (indubitable) correct proof to
which to adhere.

Sincerely, M. Randall Holmes

holmes at math.idbsu.edu or mrh29 at dpmms.cam.ac.uk
http://diamond.idbsu.edu/~holmes

Boise State University and the University of Cambridge
must be held harmless for any silly thing I may say.

"And God posted an angel with a flaming sword at the gates 
of Cantor's paradise, that the slow-witted 
and the deliberately obtuse might not glimpse 
the wonders therein."  (Holmes, with apologies to Hilbert)