FOM: Objectivity,intersubjectivity and "social construction"

[Solomon Feferman]

 Thanks to Robert Tragesser, Reuben Hersh, Julio Gonzalez Cabillon, Mic
Detlefsen and Bill Tait for mulling this over and raising some important
problems. (Perhaps there are others I have not seen yet, while I await the
next issue of my fom-digest).  A few remarks, but as with Tait, I'm not
ready to try to fill in the details, at least not here.
  First, perhaps I was a bit hasty in tagging along with Moshe Machover in
describing my view of mathematics as being "socially constructed".  These
are not words that I myself have used before in this respect, but they
are close enough to what I have said that it seemed reasonable to carry
on in kind.  What I should have realized is that nowadays this carries a
great deal of post-modernist freight which I would not want to be
associated with at all.  I would also want to disassociate myself from any
political connotations that those words might currently suggest.  
  So what did I have in mind?  Something like, mathematics is a body of
thought which is shared intersubjectively.  And while individuals may
contribute to that body of thought without sharing their ideas or proofs,
it is only with reference to it that their contributions make sense.  
  Hersh wanted to put my question--what is it about the conceptual
and verificational structure of mathematics that distinguishes it and that
makes it such a supremely objective body
of thought--on its head, by saying that mathematics may simply be defined
as *the* supremely objective part of our thought.  But that ignores other
supremely objective parts of our thought, e.g. any games, such as chess,
go, etc. governed by precise rules.  We *could* declare those to be part of
mathematics by fiat but that would be stretching it completely.  My view
is that we need at least to say something about the logical structure of
mathematics to distinguish it among all other objective parts of our
thought.
  Of course all this has been chewed over by the philosophers in one way
or another, and insofar as philosophical answers go, there is probably
nothing new to say, and perhaps Kant has already said it in the second of
the two senses of objectivity that Detlefsen explained.  What is new
though, is a body of logical knowledge and sophistication that has to be
combined suitably with an appropriate philosophical stance.  That's where
the details should come in.
  I plan to answer Gonzalez Cabillon about Wigner's wonder at the
"unreasonable effectiveness of mathematics" separately. 

Sol Feferman