FOM: indubitability (or "certainty")
rhersh at math.math.unm.edu
Fri Oct 2 13:04:14 EDT 1998
You say one cannot have a usable or worthwhile notion of partial
or incomplete rigor or proof without having in advance a notion
of perfect rigor or certainty.
I don't think this necessity has ever been demonstrated.
Certainly it works the other way--*if* one could have a meaningful,
valid criterion or notion or test of absolute certainty or perfect
proof or rigor, then indeed that would be useful in dealing with
imperfect or incomplete proofs.
However, it may be that the search for perfect or absolute certainty or
proof or rigor is actually made by successively refining and making more
satisfactory our standards of incomplete proof.
History does show successive refinements and strengthening of proof
(betweenness in geometry, uniform convergence in analysis, and others
in set theory which you are better qualified than I to tell about.
You can say that those advances were made possible on the basis of
an existing notion of absolute certainty. I would say that our
notion of absolute certainty was developed in the course of more
careful and critical incomplete proofs.
I emphasize that "incomplete proofs" means virtually all proofs
as presented to seminars or colloquia or printed in journals and treatises
or broadcast on the world wide web. Harvey recently gave examples
of typical ways in which gaps are consciously left in published
proofs. Complete proofs, for most problems
and theorems of any substance, would be far too long and tedious to be
read or published.
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