From: gudeman@cs.arizona.edu (David Gudeman)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 11 Apr 92 06:52:12 GMT
Organization: U of Arizona CS Dept, Tucson

In article  <s4rpbINNe04@agate.berkeley.edu> Edward Paul Faith writes:
]
]One might think that the study of foundations is a sort of science,
]like biology or linguistics...

It is.

]But the study of foundations is different from biology and
]linguistics, in that it is itself a field of mathematics, and
]mathematics is what it claims to study.  Biology, in contrast, is not
]a living being, at least not in the usual sense.

So how is this relevant?

]What keeps "foundations" from being just another field of math, as
]opposed to being a commentary on all math?  As mathematicians like to
]say, math is defined by being not about anything.  Mathematical
]statements are ultimately empty, mere symbol manipulation, until the
]user imbues them with some meaning, some extramathematical use.  But
]this imbuing is most unmathematical.

If you were similarly confused about the nature of language and
thought that documents are no more than collections of meaningless
symbols, then you might ask a similar question about grammar books:
"What keeps a grammar book from being just another document as opposed
to being a commentary on all documents?"  Obviously there is no direct
answer to this question because it is ill-posed.  A grammar book _is_
just another document and is _also_ a commentary on all documents.

Likewise, foundations _is_ just another field of math and is _also_ a
commentary on all math.  What makes foundations a commentary on all
math is the same thing that makes a grammer book a commentary on all
documents, namely _that's what it is about_.

The comments of any given mathematician may or may not be relevant,
depending on how much that particular mathematican knows about
foundations.  There are many brilliant mathematicans who would agree
with you and who are wrong because they haven't studied the subject.
You just cannot expect, with a few minutes in introspection, to arrive
at answers that have eluded brilliant minds for centuries.

In particular, the theory you offer above is unsupportable (By the
way, my favorite statement of this view goes something like
"Mathematicians get their certainty of being right by not saying what
they are talking about." -- anyone know who said that?).  This view
was first expressed and defended by Hilbert, who gave up the theory
after 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.

]Strictly as a field of mathematics, then, the study of foundations is
]as vacuous as any field of mathematics.  What then gives it its
]extramathematical meaning?  And is this meaning something that a
]self-respecting mathematician should trust, or is it just a bunch of
]handwaving?

"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.
(Incidentally, Frege showed that Hilbert's view was "just a bunch of
handwaving" long before anyone ever heard of Godel.)  Foundations on
the other hand tries to begin with fundamental intuitions and
construct mathematics without hidden assumptions.  Yes, hidden
assumptions and wrong intuitions have always crept in, but that does
not invalidate the science, it just shows that the science is very
hard.

]"The study of foundations" involves not only math, but also
]philosophy.  I have ignored the latter aspect, since the worldly
]sciences do not generally seem to need philosophy to back them up, and
]I have been trying to show how different "foundations" is from worldly
]sciences like biology.

You have not ignored philosophy, you have been making (outdated)
philosophical claims all through your article.  And no science can get
along without philosophy to back it up.  You just don't see the
philosophy in most modern sciences because the philosophical questions
have been mostly answered in those sciences (which is why they are now
considered sciences, and no longer philosphy).  Actually though, there
are still unanswered questions in the foundations of all sciences, and
until those questions get answered the conclusions drawn in the
sciences will be more-or-less suspect.
-- 
					David Gudeman
gudeman@cs.arizona.edu
noao!arizona!gudeman
Newsgroups: sci.philosophy.tech