FOM: Re: Consensus vs Indubitability

[Reuben Hersh]

The question is what you mean by rigorous proof.

Maybe you mean formal proof as defined in logic books.

Or maybe you mean rigorous proofs at the level of detail
or "rigor" actually practised by representative mathematicians,
or if you like, the leading mathematicians, in one or another
field of mathematics.

I have been at pains to point out that these two meanings are
not identical. If by rigor you mean the first interpretation,
than nearly all published mathematics lacks rigor.
\
If you mean the second, then it inevitably carries with it
a certain possibility of error or misunderstanding.

It is possible to claim that the two are reallly the same.
"Any proof in the second sense can be filled in to be a proof
in the first sense."  That claim certainly has not been and cannot
be proved rigorously.  I discussed all this at greater length
in my book.

It's not that I am saying rigorous proof is unnecessary.  It
is desirable, and approachable, but not in practise fully attainable
in the bulk of mathematical research and publication.  So our
certainty in our results is diminished a little bit.

If it's only a little bit, you can say, So what?

The significance is not in  practical mathematicasl work.
It's in our conception of the essential nature of mathematics, its status
ontologically and epistemologically.