[FOM] Re: Mathematical Experiments
Bill Taylor
W.Taylor at math.canterbury.ac.nz
Fri Jun 27 01:24:24 EDT 2003
Don Fallis wrote:
> As you suggest, there is an important distinction between proofs-2 and
> probabilistic primality tests. ... However, it is not clear to me
> why this is an *epistemically important* distinction.
Proof doesn't have all that much significance to epistemic matters.
After all, epistemology is concerned with how we know things, which to
a large extent is how we *find them out*. This has almost nothing to do
with proof, but with trial & error, guesswork, examples, diagrams,
hard work, inspiration, and who knows what else.
Proof is for *hygiene*, not discovery, as (was it?) Hilbert observed.
> We never know *for sure* that there are no mistakes in a long proof-2.
Nor do we ever know almost anything else, *for sure*.
> A proof-2 (just like a probabilistic primality test) can only provide
> defeasible evidence that a mathematical claim is true
True, probabilistic proof and regular (published) proof(2) are both forms
of *evidence*; but evidence of quite different types. The former is
merely evidence of a certain *claim*, perhaps quite strong evidence;
whereas the latter is evidence of both the claim *and* the existence
of another thing - the implied proof-1 object. Significantly different!
>> mistakes are *mistakes* ... Their possible presence is a very real
>> ... concern, but NOT a philosophical one. At least, not math-philosophical
>
> The claim that mistakes are not a philosophical concern would sound
> strange to most epistemologists [and] reliabilists.
No doubt; but you ignored my last addendum - not of great concern
to the philosophy OF MATH. Naturally it is to philosophy in general.
> I guess that I don't understand how you are using the term
> *validation* here. If we do not see it, how can a proof-1 provide
> *evidence* that a mathematical claim is true?
Admitedly validation is not a precise term. But it certainly means
a lot more than mere evidence. There is an element of "certification",
in the sense of authority rather than evidence; even a positivistic
element of *definition* of truth, at least for statements beyond mere N.
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Bill Taylor W.Taylor at math.canterbury.ac.nz
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Logic: used by mathematicians, but not talked about,
talked about by philosophers but not used.
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