FOM: Mathematical Certainty: reply to Silver

[Charles Silver]

On Tue, 15 Dec 1998, Joe Shipman wrote:

> Charles Silver wrote:
> 
> > On Fri, 11 Dec 1998, Joe Shipman wrote:
> > > If my statement is correct Reuben Hersh will be somewhat vindicated.

CS:
> >         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.

JS:
> Hersh is PARTIALLY vindicated, because a sociological criterion turns out to
> be sufficient for a statement's incorrigibility.  

	This criterion may be right in all cases that we know of, but it
is still possible for it to come out wrong.  This establishes that
agreement and correctness are distinct.  


> 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....

	I think you are doing something very different from what Hersh
wanted to do.  Hersh wanted to capture the *meaning* of mathematical
truth.  For him, agreement of a certain sort simply *is* mathematical
truth.  I don't think you are claiming that your criteria capture the
*meaning* of mathematical truth.  The very fact that you are asking
whether anyone knows any counterexamples shows that the concepts
"mathematically true" and "satisfy the criteria" are distinct.

Charlie