[FOM] Re: Mathematical Experiments
fallis at email.arizona.edu
Wed Jun 25 10:44:57 EDT 2003
Quoting Bill Taylor <W.Taylor at math.canterbury.ac.nz>:
> Proof-1 is a strictly formal object - a series of strings in a
> 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
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
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
> 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?
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