[FOM] Certainty in mathematical proofs
Arnon Avron
aa at tau.ac.il
Fri Oct 19 06:45:11 EDT 2007
On September 14 Martin Davis reacted on FOM to one of my posting.
His reaction triggered me to write a far too long reply.
Following the good advice of Martin and other editors of FOM,
I divided my original reply into parts, and I'll
post each of them separately. This is the first of them.
Martin Davis Wrote:
> Arnon Avron in recent posts has explained that
> only proofs that meet criteria of predicativity
> rise in his view to the level of full
> acceptability. He acknowledges the interest of
> proofs that go beyond this bound, but insists
> that they can not provide him with full
> confidence in the truth of the theorem proved.
> Now, as long as this is expressed as a personal
> feeling, no one can reasonably dispute the
> matter. But it should be realized that capable
> mathematicians who think about foundational
> matters are all over the map with respect to
> where they draw the line of full acceptability of
> a proof.
So?
If pushed with determination, than there will
be no escape from the claim that certainty is
at the end a personal matter, and that one
can never be certain about what is certain.
If we leave the issue at that, then there is no
point in discussing FOM. The only choice would
be: everyone for himself/herself, and that's it.
Well I, for one, do think that it makes sense to
discuss and make research on FOM - and *for the
original goal*: to provide secure and certain
foundations for large parts of mathematics,
and to do so on the basis of *objective* criteria,
which goes beyond someone's personal feelings and judgements.
Yes, it might indeed be impossible to objectively draw
the *exact* line between the absolutely certain
mathematical propositions, and those which are
less-than-absolutely-certain. Indeed, I do not pretend
that I know where the exact line is. But even without exact line
there are obvious cases of the two sorts. Thus the fact
that there are infinitely many primes was absolutely
certain at the time of Euclid, it is still
absolutely certain today, and so it will remain forever.
The same applies (since 1931) to Godel's incompleteness theorems.
In contrast, it should be obvious that
neither GCH nor its negation will ever be
recognized as absolutely certain. The only way to change
the status of a proposition to "absolutely certain" is by
*proving* it on the basis of absolutely certain axioms,
using absolutely certain methods of proofs. We know
that GCH can be decided only on the basis of *new* axioms, and
by the nature of absolute certainty, no proposition
that was doubtful in the past can suddenly become
absolutely certain to the point it would deserve full
acceptance as a certain *axiom* (which does not requires a proof).
Once we recognize that there are some absolutely certain
mathematical propositions and some that are not (Of course,
there will always be strange people who deny it.
The only thing one can do about it is to ignore them),
it is important to find out (as far as possible)
what is the extent and applicability of those who are.
It is not necessary to find the
exact line (if such exists). It suffices to determine
a large body of mathematics that has absolutely
secure foundations on one hand, and supports the
most important parts of classical mathematics on the other.
This is what the predicativist program is all about,
and I think that a very strong
case can be made that not only has it succeeded,
but also that it is the only current approach
that does work, and does achieve the primary goal
of FOM described above.
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