FOM: social construction?

[Charles Silver]

SUMMING UP WHAT'S QUOTED BELOW:

	Martin Davis answered a question I asked, elaborating on what
"exists" means in mathematics.  Bill Tait pretty much agrees with Davis's
explanation but suggested an improvement so that Davis's explanation does
not play into the hands of Hersh.  Davis subsequently modified his
explanation and asks Tait whether it is now ok.



M. Davis:
> >>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. 

W. Tait:
> >I am with you in spirit, Martin; but I don't think this is what you mean 
> >to say. You seem to be making the assertion of existence in mathematics 
> >amount to an assertion about us, what we do or could do---which would 
> >suit Hersh, if I understand him, but would not suit you, if I understand 
> >you (in previous communications).
> >
> >Bill Tait

M. Davis:
> Thank you Bill. Careless wording on my part. Let me try to do better:
> 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 they are
> objective in the sense that they have properties which we can determine (or
> try to determine) but which we have no power to alter.
> 
> Better?

	I want to suggest that Hersh's view should not be considered so
odious in this context.  After all, he is right (isn't he?) that we find
out what math is by observing what mathematicians actually do.  That is,
he's right that the *statements* and the *proofs* of mathematical
propositions would not exist were there no mathematicians (whether the
underlying *propositions* themselves would still "exist" is a completely
different matter [I think the ambiguity of 'theorem' has caused some of
the controversy, since 'mathematical theorem' can mean the statement of a
proposition or it can mean the proposition itself). 

	And, I think that Hersh is also right that few people have looked
carefully at what mathematicians actually *do*.  I don't think what Bill
Tait said to Martin Davis was quite right, when he said, "You seem to be
making the assertion of existence in mathematics amount to an assertion
about us, what we do or could do---which would suit Hersh...."  The
problem, as I see it, is in the "amount to". I believe Hersh is right that
we find out what math is by investigating what mathematicians would do in
certain circumstances. For example, prior to trying to figure out what
mathematicians really do, we could entertain the possibility of a bunch of
them revoking the Four Color Theorem (computer aided), or disqualifying
Wiles's proof as being too long and difficult.  We can imagine that
happening, but we know that it won't.

	What I think we find out from a Hershian (socio-culturo-...) 
investigation is that the best way to account for the activities of
mathematicians is to hypothesize underlying, immutable structures that
mathematics is *about*.  I said before that I think Hersh makes a
use-mention mistake (Randall Holmes said the very same thing) in confusing
the (socio-culturo-...) perspective used in investigating the behavior of
mathematicians with what mathematics is *about*, but what mathematics is
about is something inferred from what in fact mathematicians *do*.  That
is, the modification that Davis makes to suit Tait is to say that
mathematical truths are "objective in the sense that they have properties
which we can determine (or try to determine) but which we have no power to
alter."  I agree that this is in fact true, meaning that we find out by
investigating the activities of mathematicians that the things they call
theorems are not subject to change (unless a hole in the proof is
discovered).  Hersh can (correctly, in my view) object to Davis merely
tossing the word 'objective' in, as though all he has to do is *say* that
mathematics is objective and that's the end of it.  That is, Hersh can say
(correctly, I think) that there's no "argument" here, that merely saying
mathematics is objective doesn't make it so.

	I believe math *is* objective, but I agree with Hersh's
methodology (if not with Hersh himself) that the way we find out that math
is objective is by studying what mathematicians do.  So, I prefer Davis's
first account, even though it may seem to play into Hersh's hands. 

***************
	
	I don't want to write a separate post.  I think the chess argument
is mixed up, because I don't think a distinction has been made between the
structures underlying chess and the game itself.  I don't think it would
be too difficult for someone clever to set up a mathematical model of
chess.  Let us make the elements of the universe "pieces", and let us
create one-place predicates '_ is a pawn', '_is a knight,' etc. Then,
there will be "positions" and "moves", and so forth.  I already see that
what I'm saying here won't work.  "Pieces" seems like the wrong way to
start.  I don't want to correct it, though, and fiddle around to make it
more plausible.  I am not the "clever" person to do this, anyway. I'm sure
lots of you out there can do this much more quickly than I can.  The
underlying point is that it seems clear to me that an axiomatic theory
could be set up that would completely define the game.  And, then theorems
could be proved about this "theory".  But, the game of chess itself would
not be mathematics. Chess would be just one model (the "standard" one?) of
the resultant theory.

Charlie Silver
Smith College