FOM: Mathematical Certainty: reply to Silver

[Joe Shipman]

Charles Silver wrote:

> On Fri, 11 Dec 1998, Joe Shipman wrote:
>
> > If my statement is correct Reuben Hersh will be somewhat vindicated.
>
>         I don't think so.  Reuben Hersh wrote of mathematical "truth", not
> "certainty".  'Certainty' is a funny word because it seems sometimes to
> pertain to the mental state of one who is certain and sometimes to pertain
> to the thing about which one is certain.  I think you may be using it in
> both senses.  Using your notion, I believe it is always *possible* for a
> statement to be "mathematically certain" yet false.  But, according to H.,
> agreement alone establishes its truth, because that's all that truth is.
> I think this is a very big difference.
>
> Charlie

Hersh is PARTIALLY vindicated, because a sociological criterion turns out to
be sufficient for a statement's incorrigibility.  In other words, even if you
don't accept a Platonic notion of absolute truth of mathematical statements,
it is still the case that a properly proven theorem will never be "overturned"
so that it does no harm to speak of it as "true".  The problem with Hersh's
sociological methods of evaluation is that mathematicians may be WRONG even
when a theorem is widely accepted.  My strong version of his criterion (the
problem must have already been considered very important to guarantee the
proof will be well-scrutinized, the proof must be readable by one person,  and
no serious workers in the field say the proof is unconvincing for at least 5
years) appears to have no counterexamples in the 20th century; such a
counterexample would refute Hersh's thesis and the absence of one therefore
partially vindicates his stance (though his stance is still unsatisfactory in
failing to provide an explanation for the existence of such incorrigible
results).

-- Joe Shipman