FOM: naive or brainwashed?

[Charles Silver]

Reuben Hersh:
> "What determines what math is?"
> Do you think this is a clearly formulated question?
> 
> I gave a criterion that distinguishes math from other 
> fields of academic study.  The fact that mathematicians can check
> each other and come to agreement on their conclusions is
> an essential, non-ignorable feature that makes
> mathematics what it is.

	Maybe I can put my point better by saying that I think you make a
mistake similar to a "use-mention error".  For example, you say (p. 236,
_What...?_) "By recognizing mathematics as the study of certain
social-cultural-historic objects, humanism connects philosophy of
mathematics to the rest of philosophy."  There is certainly a lot to
disagree with in this single sentence, but I'll just stick to the
"use-mention error". 

	As part of your investigation into the nature of mathematics, you
have looked at mathematics from a social, cultural, and historic
perspective.  That does not mean that <*mathematics itself*> is the study
of "social-cultural-historic objects".  Your own methodology for studying
the subject is distinct from what the subject itself deals with.  To my
mind, you fudge over this elementary distinction, and thus equivocate when
you talk about what mathematics really is.  

	I do not have an objection to your using this perspective
("social-cultural-historic") in analyzing mathematics.  I think it's a
useful one, and I think other investigators into the nature of mathematics
have not spent as much time as you have in looking at the trappings of
mathematics and mathematicians.  But, what mathematics is *about* is
distinct from these trappings and can only be inferred from them.

	Let us now try to figure out what mathematics is really about (and
let us also try hard *not* to confuse the nature of our investigation with
the nature of what we are investigating).  I am on shaky ground here,
because I do not have the knowledge you and many others on this list have
about mathematics.  But, I'll push on, anyway. The kind of view I find
most attractive is contained in Feferman's theses.  But, I don't care for
his statement that mathematical objects exist "only" in the imagination,
because I think we should postpone existence questions until later.  For
one thing, what "to exist" means here is itself questionable, at least to
me. 

	At any rate, if we look at the kinds of things mathematicians
*think* they're talking about, I believe we will find certain structures. 
Maybe this is already something about which many people will disagree. 
So, I'll write '#structures' to indicate this is only a hypothesis. 
Without going any further, let's say that these #structures may be just in
the minds of mathematicians (their "imaginations"), or in the world, or in
some other place(s).  You recently rebuked Randall Holmes for asserting
(without argument, you said) that these #structures were part of the
world.  I think there is lots of room here for your insights about how
mathematicians talk about these #structures (or whatever they are).  Maybe
you'll find through your social-cultural-historic investigation that we
shouldn't have been so fast as to say there is a single thing that
mathematics is about. So, maybe even '#structures' was too fast, maybe
there are multiple things (which perhaps resemble each other in certain
ways).  All I'm saying here is that there is lots of room for you to use
your particular methodology to uncover the nature of mathematical objects.

	It seems to me to be an entirely separate question whether these
objects have any "real" existence or not.  I think one's answer to this
depends on wider and difficult views about existence itself.  In an
exchange awhile back, Feferman asserted that he did not think it was
necessarily the case that "X exists" simply because our best theories
about the world use X to explain things.  (This was in the context of a
discussion of Quine's naturalism.  Unfortunately, due to various kinds of
insensitivities exhibited here [please note that I am saying that there
are several kinds, not just one], Feferman cannot defend himself or even
say that I misunderstood him.) At any rate, my point is that uncovering
the nature of mathematical objects still leaves open the question of their
"real" existence. 

	This is about as far as I can go, given my ignorance about a
number of things about which I wish I knew more. 

Charlie Silver
Smith College