[FOM] Am I a Platonist?

[Martin Davis]

On Mon, 27 Oct 2003, replying to my earlier posting, Harvey Friedman wrote:

 > Reply to Davis. A very nicely written and thought provoking posting.

Thank you.
...........................
 >> I regard this objectivity as a hard-won result
 >
 >This I find confusing. It would seem that, according to you, we also have no
 >ability to alter the objectivity of this all. So how can it be a "hard-won
 >result"?

It's not the general fact of objectivity, but the specific matters about 
which we gain objective knowledge. Just as the General Theory of Relativity 
was "hard-won" although we have no ability to change how our universe is 
put together.
...

 >Do you mean then that this objectivity became only clear to us from
 >mathematical practice? Otherwise, although it certainly exists (the
 >objectivity), and we may never have discovered it without hard work??

Yes, exactly.

 >> In the 17th century Torricelli stunned his contemporaries by flouting the
 >> Aristotelian doctrine by exhibiting a geometric body that was infinite in
 > >extent, but had a finite volume. This discovery of what was OBJECTIVELY
 > >the case about the relation between the finite and the infinite was not 
the
 > >result of metaphysical speculation, but the direct result of the reach of
 > >our mathematical powers.
 >
 >I do think that this can be attacked as an oversimplification. There is the
 >question of the meaning of volume - rather than any objective situation. It
 >could be said that Torricelli succeeded in showing merely that it was a good
 >idea to extend the notion of volume to more objects. Lebesgue also did that
 >later.

I think this comment misses the point in a number of directions. 
Torricelli's work occurred in a pre-calculus environment when there was no 
general notion of volume to "extend". People like Cavalieri and Torricelli 
(following the example of Archemedes) worked specific examples using a 
Euclidean framework. In modern terms the solid Torricelli considered can be 
characterized as formed by rotating the curve y = 1/x, x>=1 about the 
Y-axis. He showed that this infinite solid had the same volume as a certain 
finite cylindrical solid. (See Paolo Mancosu's fascinating monograph, 
"Philosophy of Mathematics and Mathematical Practice in the Seventeenth 
Century" Oxford 1996 where Torricelli's proof is given in detail.) The 
reason that the result (which today is a homework problem in a freshman 
calculus class) was so shocking to contemporaries was because it violated 
classical ideas going back to Anaximander and codified by Aristotle about 
the unapproachability of the infinite.

The comparison with Lebesgue is particularly misplaced in my opinion. When 
I used to sit on qualifying exam boards, one of my favorite questions was: 
"Why do analysts like to use the Lebesgue integral?" The wrong answer was: 
"Because there are Lebesgue integrable functions that are not Riemann 
integrable." Of course, analysts never felt the inability to integrate the 
characteristic function of the irrationals as a limitation.  It was the 
limit theorems enjoyed by the Lebesgue integral that accounts for its 
popularity,

 >>When Cantor began his study of transfinite
 >> numbers, it was not at all clear that this could be carried out in a
 >> satisfactory manner, and of course many refused (and continue to 
refuse) to
 >> accept it. But the work of the set theorists continue to develop this
 >> domain, and even when assumptions that transcend ZFC are used, the
 >> objective nature of their results seems compelling.
 >
 >This may seem compelling to you, but not to many people. I don't take sides.

And yet, your own beautiful work (which I advertise whenever I can) in 
which you find finite combinatorial consequences of strongly infinitary 
principles will only gain general currency to the extent that this 
objective nature is accepted.

......................

 >The big difference between physics and mathematics that has to be confronted
 >is as follows.
 >
 >When Einstein's GR is confirmed over and over again, in various settings, it
 >is clear that it could have been horribly disconfirmed - to the point of
 >refutation, by these experiments.

This is true. And  yet, large cardinal axioms could still be "horribly 
disconfirmed - to the point of
refutation" by someone proving their inconsistency with ZFC.

Martin