FOM: social construction?

[Charles Silver]

On Fri, 20 Mar 1998, Kanovei wrote:

> <Date: Fri, 20 Mar 1998 07:50:50 -0500 (EST)
> <From: Charles Silver <csilver at sophia.smith.edu>
> 
> >I'm suggesting that when
> >we say that 7 or sqrt(2) or sqrt(-1) or the real number system "exist" what
> >we (as mathematicians) mean (or at least ought to mean) is that we know how
> >to determine some of their properties as definite and have good reason to
> >think of those we can not decide as definite problems to work on. 

	I didn't write the above, Martin Davis did.  Not that I differ
with it.  I was just asking him to please say more, which he did.  I think
it's an interesting point of view, but I'm not sure I fully understand it. 

V. Kanovei:
> There has been a challenge to this point (by Steel, I think). 
> Indeed, the statement
> 
> "Sh. Holmes existed" 
> 
> obviously satisfies the definition. 

	Yes, I think John Steel objected to a "fictionalist" view of
mathematics.  It seems to me that one would want to clarify *first* the
nature of mathematical entities (apart from whether they "really exist" or
not.)  Once that has been established, presumably entities like Sherlock
Holmes would then fail to be candidates for being mathematical objects. At
that point (when the *nature of* mathematical entities has been
clarified), it seems to me, Davis's view about "existence" could be
invoked without such counter-examples from fictional realms being relevant
(though I can't tell whether Davis's view about existence adds anything or
not to what would already have been decided about the *nature of*
mathematical entities).  To me, these counter-examples seem to arise
because the question of the nature of mathematical entities is usually
intertwined with the question of their existence.  I'm suggesting we look
at the nature first and the existence second.  

	I am curious how Steel would support his Platonism if it could be
agreed beforehand exactly which objects are mathematical and which ones
are not--meaning that things like Sherlock Holmes were already
disqualified from being mathematical entities.  That is, assuming that we
all agree exactly which objects are mathematical and which ones are not,
what support can be given to the notion that, furthermore, they are "real"
in some Platonic sense?  What standard should we use to judge whether they
are in fact Platonic entities (which I think Steel wishes to claim) or
whether they exist only in our imaginations (which I think Feferman wishes
to claim), or whether they exist somewhere else.  Martin Davis's view
seems to me to have the merit of saying that there is nothing further
meant by "existence" at all (after the nature of mathematical entities has
been clarified).

Charlie Silver