FOM: (mercifully brief) reply to Hersh's reply
holmes at catseye.idbsu.edu
Mon Mar 23 13:04:40 EST 1998
To talk about real mathematics in the proper ideal eternal
immaterial insubstantial sense, we must evidently rise out of
or above our own being (as humans) which is physical, material
and social, cultural. We, physical, material, social, cultural
beings must contact and interact with and in some Godelian sense
actually perceive the transcendental abstract reality of real
We actually interact with structure of the kind that I think
mathematics is about all the time in everyday life. There's nothing
especially mysterious about it. Moreover, people (even those who
profess to be materialists) talk about "abstractions" all the time,
and appear to know what they are talking about :-)
On any kind of qualification of "being", see below.
In other words, you can be religious or materialistic,
but not half and half.
The alternative to "materialist" is not "religious". One can be a realist
(as opposed to a nominalist) without being religious in the least.
I think that it is very clear that existence comes in different
senses or levels.
And I think that it is very clear that it does _not_ :-) Things either
are or they aren't, as it were (in the final analysis). Reference of
ostensibly naming terms does come in different flavors: for example,
on my own recent account the expression "the number 7" does not refer
to any specific object; it refers to an object in an (implicitly
understood) model of arithmetic (or set theory, if one wants to use it
to count elements of a set). The unqualified existential assertion I
would be making is "there are models of arithmetic (or set theory)";
in any such model a "number 7" can be found. Similar phenomena of
context dependent reference are found in ordinary language, of course.
Then if we have math being real, and we want to understand how that
can be, we have five choices:
2) mental (individual subjectivity)
3) social (intersubjective)
5) it's a non-problem, doesn't interest me, leave it to the philosophers.
No one has advocated 2. I have tried to deal with 1. 4 and 5 are
obviously tenable. It seems to me both involve a certain amount
of self-deception and wishful thinking.
I'm not a dualist; I recognize alternatives 1 and 2 as variations of
the same thing (leaving out the word "subjective"). I regard 3 as
incoherent (we can't really "make things up", even by mutual
agreement; nor can we make them up all on our own, thus my stricture
on the word "subjective" under alternative 2). The definition of 4
doesn't resonate with me; "transcendental" is probably (from Hersh) a
term of abuse; in any event I would not use it. "abstract" (used of
universals) presumes a certain philosophical view of their nature
which I probably don't hold. But universals (properties, relations)
can live under heading 4, I suppose.
Like Hersh, I really don't think 5 is an option.
Please note that in my recent posting outlining a proposal for the
nature of the reference of mathematical statements, the only
requirement I placed on the objects whose actual existence is needed
to underpin the reference of mathematical assertions is that there be
infinitely many of them (this requirement may be met in the physical
universe, which is, so far as anyone can tell, infinite in extent).
There were no assumptions about "transcendental", "immaterial", etc.
qualities; in fact, no assumptions about the individual character of
these objects were needed at all. I specifically noted that they might
be physical. Certain second-order assumptions might lead to one
assuming that the number of these objects was too large to fit in
the physical universe we are familiar with, but such assumptions were
not necessary to my account.
Since I admitted second-order quantification, I suppose I'm reifying
properties of and relations on these objects. Universals are not
generally regarded as material objects, it is true. It is also true
that actual reference to universals is a _universal_ feature of daily
speech of ordinary people. The materialist (nominalist) position on
this would be, I suppose, that statements involving references to
universals can always be rephrased so as to avoid such references; no
adequate proposal has actually been given for eliminating such
references (Quine thought about it and concluded that it was
impossible or at least much harder than nominalists claim), and they
proliferate as much in the speech of materialists as in the speech of
the rest of us. Things one cannot avoid talking about have a strong
claim to being real.
This is longer than I hoped when I wrote the subject line :-)
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes
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