[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.

        Bill Taylor                  W.Taylor at math.canterbury.ac.nz
             Logic: used by mathematicians, but not talked about,
                    talked about by philosophers but not used.

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