FOM: "The unreasonable effectiveness of mathematics"

[Martin Davis]

At 11:27 AM 1/26/98 -0800, Solomon Feferman wrote:

>I guess I have to add one more thing.  What the conservation result in
>question shows is that only very weak closure conditions on the real
>numbers are needed for the various results in analysis that are derived in
>the extension.  Thus they do not support assumption that the "full set of
>real numbers" in their usual set-theoretical conception is actually out
>there.  But then, what is out there of this character, if anything? 
>

This discussion was in the context of Wigner's question rather than the
Quine/Putnam indispensibility argument (which, as it happens, I never found
convincing). I think that faced with your system (assuming Wigner agreed
that all the mathematics needed for physics was there), he would be led to
reply that a version of real numbers satisfying at least your weak closure
conditions could be "actually out there". At least I don't see how the
conservative extension result rules out that possibility.

Martin