FOM: my non-constructive proof
ayanm at yahoo.com
Sat Jun 10 05:09:28 EDT 2000
From: Martin Davis <martin at eipye.com>
To: Fred Richman <richman at fau.edu>; FOM <fom at math.psu.edu>
Sent: Friday, June 09, 2000 11:37 AM
Subject: Re: FOM: my non-constructive proof
> Of course the construction of an example provides additional information;
> one needn't be a constructivist to enjoy constructive proofs.
I seem to disagree with this sweeping statement. For example one can
"construct" a non-measurable set or Banach-Taraski paradox, (one sphere
transformed into two spheres) assuming axiom of choice. I doubt if these
examples provide any additional information. Do you think these are
constructive proofs? If not (well, I am assuming here) then we need to draw
some line between the "actual" constructive proof and "constructive" proof.
How do you think to achieve that?
> "challenge" has to do with whether a constructivist would really doubt or
> deny the truth of my conclusion.
Technically he can't deny the truth of the conclusion but of course he can
doubt. Suppose one proves a theorem for the abelian group, further suppose
there is no known counterexample to that for non-abelian groups. How will
one see the theorem in context of groups? I guess, constructive
mathematicians look at a classical theorem in the same way, unless he can
find some weak counterexample to that.
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