From: gudeman@cs.arizona.edu (David Gudeman)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 11 Apr 92 20:00:22 GMT
Organization: U of Arizona CS Dept, Tucson

It seems that you have misunderstood my position and I'm beginning to
suspect that I have misunderstood yours.  Sometimes people can be led
astray by their reaction to key words and phrases.  I'll try to be
more circumspect...

In article  <s6elfINNjd1@agate.berkeley.edu> Edward Paul Faith writes:
]In article <15196@optima.cs.arizona.edu>, gudeman@cs.arizona.edu (David Gudeman)
]][about the formalist view of mathematics]
]]Godel's incompleteness theorems showed that this view entails
]]that there are true statements about mathematics that we can know are
]]true but that we cannot prove mathematically.

]Godel's theorems prove that within a certain calculus one could not
]construct a formal proof of certain facts about that calculus.

It seems that I was too terse.  In the following, a calculus is a set
of meaningless formulas along with a set of rules for manipulating
these formulas.  If mathematics is, in fact, no more than empty
formulas, then there is some sense in which all of mathematics is
nothing but a single calculus, as I will try to demonstrate.  First,
note that if mathematics is only the manipulation of empty formulas,
then this manipulation must constitute, at least within a specific
subfield of mathematics, a calculus.  This is so because proofs
consist of various transformations on formulas, and if you are going
to rule out any semantic meaning to the formulas, then there is no
other possible justification for the transformations except that they
are a part of a calculus.

Now take the calculi associated with all subfields of mathematics and
join them together with appropriate substitutions of symbols so that
no rule that was intended to go with subfield A now applies to a
formula of subfield B.  This, I claim, is all there is to mathematics,
given the hypothesis that the formulas of mathematics are meaningless.
You get a Hilbert-style single unifying language of mathematics merely
as a consequence of assuming that mathematics is meaningless.  Given
this, then Godel's theorems show that there are things true of
mathematics that can be proven true, but that cannot be proven true
within mathematics.

]When I mean foundations in this paragraph, I mean foundations, and
]not that branch of math calling itself such.  Indeed, such a branch
]runs up against precisely the same problem.  So it can never be used
]to resolve that problem.  Someone mentioned a few articles ago that
]foundations was interesting to him, and proceeded to list some
]foundational problems.  These problems were of the sort that the
]mathematical field calling itself by that name cannot address,
]because it runs up against the same problems.

This is where I think you misunderstood me.  When I say foundations,
I'm not refering to any particular theory of foundations, and
especially not set theory, which I consider a failed attempt at
foundations.  I believe the article you refer to was written by me,
and that gave a fairly accurate description of what I consider
foundations.  Proofs that various mathematics can be modeled in set
theory is not foundations, since sets are no more fundamental than
other objects of mathematics.  If that was your original point, then I
can agree with it.

]]"Just a bunch of handwaving" more accurately describes your statement
]]that mathematics is no more than empty symbol manipulation, while
]]ignoring fundamental issues such as how empty symbol manipulation can
]]manage to tell us true things about the physical world and how we know
]]that symbol manipulations are going to tell us true things.
]
]Excuse my confusion, but are you saying that I am ignoring fundamental
]issues, or that the quote ignores fundamental issues?

I don't know whether you are ignoring the fundamental issues or not,
but the quote does.  At least to me, it looks like a direct and
unsupported statement of the formalist philosophy of mathematics.
Formalism may be common among scientists who aren't aware of the all
the problems it brings up, but it is not common among those who have
actually studied the issue.

Certainly, formalism has a beguiling simplicity to it that makes it
immediately plausible when it is first thought of, but it just doesn't
stand up to a close inspection.  As Neil Rickert has pointed out, most
mathematicians who _claim_ to believe this view actually do
mathematics as though they thought the symbols had meaning.  I'm
inclined to be more interested in the actual _practice_ of mathematics
than in the (rather naive) philosophy of its practitioners.  (It is
unfortunate that any statement that imputes any ignorance is taken as
derrogatory.  I really it don't mean it that way, I'm just stating a
fact.  I personally think it is silly to expect anyone to know
everything.)

]]Foundations on
]]the other hand tries to begin with fundamental intuitions and
]]construct mathematics without hidden assumptions. 

]I submit that all mathematicians since Euclid have tried to begin
]with fundamental intuitions and to construct their mathematics
]without hidden assumptions...

True enough.  The point is that mathematics is rife with hidden
assumptions --or it was before the early work in foundational studies.
The current state of affairs is that mathematics is rife with
ill-founded assumptions that are at least recognized.  The purpose of
foundational theories is to find ways to support these assumptions
that are already being used.  But as I said, I do not mean to suggest
that I think the current trends are doing a good job of it.
-- 
					David Gudeman
gudeman@cs.arizona.edu
noao!arizona!gudeman
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