Re: Article in Scientific American

T.H. Ray <thray123@xxxxxxx> wrote:
Hmmm. I don't know. To the extent that not
disprovable
means not not provable, is it not equivalent to
provable?

Only if disprovable meant "not provable". It
doesn't: disprovable means
that there exists a proof for the negation. If you
want a single word
for "not provable", then unprovable will work.

--
Chris Smith

I am not quite sold. Montgomery-Smith's context was
that of computers checking computers. You said to M-S:

2. The probability of a proof being correct increases each time a
qualified person reads it. The probability of a computer program being
correct doesn't increase because someone runs it.

Montogomery-Smith replied:

Yes, but the probability of the computer program being correct does
increase it if it is run on different computers with different
compilers, operating systems, etc.<<

That sure sounds like not not provable to me. I mean,
a proposition not provable is trivally unproved, but
not necessarily proved to be unprovable. A proposition
proved disprovable is so by the convention of agreement,
whether among mathematicians, or among computer proofs.

It's a sticky question particularly for us pencil and
paper mathematicians, who would like to think that a
sound existence proof is sufficient to prove that an
unproved proposition is not unprovable. A program written
to check such a proof has to try and prove
that the proposition is _dis_provable. That is, if such
an algorithm were to agree among computer
proofs, it could only do so by proving the proposition.
That is, the computer proof (if the pencil and paper proof
were true)would have to agree with the proof that the
proposition is not unprovable, by proving the proposition,
or else it could neither verify nor disprove the pencil
and paper proof.

Tautology -- if the pencil and paper existence proof is
true, the proposition is not unprovable; if the computer
proof is true, the proposition is not not unprovable,
and neither is its negation. What has one proved?

Yes, I know some would eject existence proofs entirely.
I am not ready to do that, and neither are you, or
"disprovable" would not mean "there exists ..."

Tom
.