[FOM] What produces certainty in mathematical proofs?
martin at eipye.com
Fri Sep 14 01:02:09 EDT 2007
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. I had a colleague who refuses to accept
any non-constructive proof I don't have an
example at hand, but I suspect that there are
non-constructive proofs that are fully
predicative as that notion is usually explicated.
Going in the other direction, here in Berkeley
there are logicians who accept very large
cardinals without a qualm, whereas another
equally fine logician has devoted a great effort
in attempting to prove the inconsistency of the
existence of measurable cardinals with ZFC.
Harvey Friedman has emphasized the progression of
means of proof from the most basic to the highest
regions of the transfinite with a decision to
draw the line at at particular level a matter of
personal idiosyncrasy rather than the result of philosophical acumen.
There are two further points I'd like to make.
One is that, in the last analysis mathematics is
a social activity, and as time goes on, methods
that achieve desired results without leading to
catastrophe will just gain acceptance.
The other is that we know by Gödel that however
far we go out in this progression there will be
true propositions, even Pi-0-1 sentences, that
require that one proceed further up the
transfinite staircase in order to prove them.
Harvey's work gives tantalizing examples, but so
far there is no example of a true Pi-0-1 sentence
that number theorists have longed to prove but
which requires set theoretic, or even highly
transfinite, methods for its proof. Gödel himself
once suggested that the Riemann Hypothesis could be such a proposition.
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
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