[FOM] Re: Mathematical Experiments

[Don Fallis]

Quoting Bill Taylor <W.Taylor at math.canterbury.ac.nz>:
> Proof-1 is a strictly formal object - a series of strings in a
> language,
> following certain well-known rules.
>
> Proof-2 is a far more informal object. It is the published version of
> a proof-1,
> containing many informal short-cuts such as diagrams and explanations
> in natural
> language.

As you suggest, there is an important distinction between proofs-2 and
probabilistic primality tests.  If there are no mistakes in it, a
proof-2 (unlike a probabilistic primality test) can be expanded into a
proof-1.  However, it is not clear to me why this is an *epistemically
important* distinction.  We never know *for sure* that there are no
mistakes in a long proof-2.  Thus, we do not know for sure that it can
be expanded into a proof-1.  A proof-2 (just like a probabilistic
primality test) can only provide defeasible evidence that a
mathematical claim is true and that a proof-1 of this claim exists.

See pages 180 to 185 of
http://links.jstor.org/sici?sici=0022-362X%28199704%2994%3A4%3C165%3ATESOPP%3E2.0.CO%3B2-B
for a more detailed discussion of these issues.

Finally, here are two minor points:

> But again, mistakes are *mistakes*, not math.  Their possible
> presence is
> a very real academic and social concern, but NOT a philosophical one.
> At least, not a math-philosophical one.

The claim that mistakes are not a philosophical concern would sound
strange to most epistemologists.  It would sound especially strange to
reliabilists.

> This proof-1 a validation of the truth of FLT (even though
> we never
> see it).

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?

take care,
don

Don Fallis
School of Information Resources
University of Arizona
http://www.sir.arizona.edu/faculty/fallis.html