[FOM] What produces certainty in mathematical proofs?

[Martin Davis]

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.

Martin



                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com