FOM: Mathematical Certainty: reply to Silver
Charles Silver
csilver at sophia.smith.edu
Wed Dec 16 04:57:03 EST 1998
On Tue, 15 Dec 1998, Joe Shipman wrote:
> Charles Silver wrote:
>
> > 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.
> >
JS:
> I agree with you of course, but Hersh doesn't; I am not defending
> Hersh's position, just talking about what arguments can be used against
> it.
>
CS:
> > 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.
JS:
> Yes; but if no counterexamples can be found Hersh can maintain that this
> is a distinction without a difference! The point is that a
> counterexample would show that his notion of mathematical truth did not
> entail a property of mathematical truth that we would all agree on
> (namely incorrigibility) and therefore could not be correct; without
> such a counterexample he is free to redefine what mathematicians are
> "really" doing.
No, Hersh can't avail himself of this sort of reasoning, because
he's attempting an analysis of meaning. But, I don't want to keep
repeating this mantra. Perhaps I can come up with an example or two. Ok.
try this (if this doesn't quite work, please make up one yourself that
works better). In most neighborhoods, one sees a fair number of pet dogs.
At least I do. I'm supposing that we have some idea what it means to *be*
a dog. Something to do with the animal's internal structure, what other
creatures it can mate with, whatever. Here's an excellent test for
"dogness" that has nothing to do with what it really means to be a dog:
small animal that wags its tail in a regular fashion. All the dogs around
here, even the ones that bark and charge at you, wag their tails back and
forth, back and forth. I've seen nothing else that does this. I suppose
cats *could* do this--I'm guessing--but they happen not to (at least not
around here, anyway). The point is that if Hersh were to avail himself of
such a test, he'd be in the position of having to say that tail-wagging
*constitutes* dogness, even though looking for a wagging tail is just a
simple way to pick dogs out (around here, anyway) and has nothing to do
with what makes them dogs (I'm supposing this, anyway; there *could* be
some DNA connection). Maybe someone can arrive at a far better example.
Charlie
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