From: epfaith@twinkies.berkeley.edu (Edward Paul Faith)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 11 Apr 1992 07:36:21 GMT
Organization: U.C. Berkeley Math. Department.
NNTP-Posting-Host: twinkies.berkeley.edu

  This article is 120 lines long.

  In article <1992Apr10.212950.27333@mp.cs.niu.edu>, rickert@mp.cs.niu.edu (Neil
Rickert) writes:

>  There are good reasons to say that the nature of biology (and its relation
>to its foundations) is different from the nature of mathematics.  But the
>argument to separate mathematics from linguistics is much less persuasive.
>Like mathematics, language is an invention of man.  Just as mathematics is
>used to study the foundations of mathematics, so also language is used to
>study the basis of language.

  As you probably noticed with a smile, I found that I could not actually argue
(in article <s4rpbINNe04@agate.berkeley.edu> ) that linguistics was much different
from math.  Perhaps it is not.

  Mathematics seems to be a phenomenon that occurs in language, and without
language, and for example the referential properties of language, I cannot
conceive of mathematics.  Mathematicians need to say things about all spheres.
"No sphere (of two dimensions, like the surface of the earth) may be given a
smooth vector field which is nowhere zero".  How could mathematicians even begin
to say such things without being able to refer to all spheres?  They would have
to produce actual spheres, and not only that, they would have to produce all of
them!  Thus mathematics is very much a linguistic phenomenon.

  Linguists have run up against one big problem, the problem of getting an adequate
theory of semantics.  I would submit that this very problem, if ever resolved,
would also resolve many late-night worries of some mathematicians.

>  The majority of mathematicians probably view "foundations" as just another
>field of math.  Indeed they probably view it as a branch of math in which
>they have little interest.  A typical attitude might be "I'm sure glad
>somebody is doing it, because it is probably important, but it is not very
>interesting to me."

  A healthy attitude to have, when one is trying to sleep at night. And as a
member of the majority, an utterly distasteful thing to consider myself as, I
share this opinion.  I admit that many results of this branch may
be fascinating and may be applicable to all branches.  But then, most
branches of math assist each other somewhat, with an algebraist using a
topological theorem here, a topologist using an algebraic theorem there, and so
forth.

  Mathematics is self-referential to a high degree.  The theorem that "No sphere
may be given a smooth vector field which is nowhere zero" is arguably a theorem
about the geometry. One may interpret it as the statement that there can be no
mathematical construct for such a field because then we could construct wild
things from this, things which both do and do not have a certain property.  In
this way, mathematicians are constantly making statements with wider and wider
implications for mathematics. Mathematics about mathematics (such as
unprovability theorems) are taken in stride by all and considered to be valuable
contributions, but are in most occurrences hardly considered to be foundational.
Yet as far as I know foundational mathematicians do essentially this sort of
thing, only in great generality.  Perhaps you know better than I.  This is why I
view foundational mathematics as not foundational, but simply broad.  For
example theorems about symbol manipulation have very broad application, since so
much of math involves symbol-crunching, but are nevertheless theorems, just like
the theorem about the spheres (called the fixed-point theorem, by the way).

  In my message (ID given above), I wrote, "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."

Neil replied (may I call you that, please call me Edward):
> While this is a common view, perhaps the preponderant view of mathematics,
>it is not universally accepted.  It was not nearly as common a view 50 years
>ago.  It quite possibly may cease to be the common view at some future time.

  I think that this question is simply one of definitions.  I do not usually think
of math (when I do it) as simple symbol-manipulation.  However, I can do so,
even though it would make me a much less motivated mathematician. You may or may
not include extra meaning in what you call math, as you like. I choose the
restrictive definition to stress the sharp difference between different aspects
of what I do when I do math.  This one aspect, the symbol-manipulation plus
formal English statements, is ultimately the only aspect that we all share in
common.  The rest, for me, is in my own private world, and in my power to
build bridges that don't fall down.:-)

>  Some would claim that calculus is also loaded with meaning and utility
>within mathematics itself, and thus that the value of calculus is inherent
>in the math.

  It depends on what we mean.  I agree with what you say, but I change my
definitions to do so.  I do no physics, yet find calculus meaningful.
Moreover, as I said above, mathematics is highly self-referential.

>  Many mathematicians are not bothered at all by this.  When mathematicians
>work in foundations their work is usually respected, but is treated as being
>in a somewhat exotic branch of math.  When philosophers talk about
>"extramathematical meaning" they are largely ignored my most mathematicians
>who tend to have a low opinion of philosophers, particularly of philosophers
>who work in the philosophy of mathematics yet don't know much mathematics
>themselves.

  Poor Wittgenstein.  It's true, Godel was needed to get us off our duffs.
However, I find suspicious the notion that sophisticated math gets us closer to
the bottom of things.  And yet, Imre Lakatos, in his book _Proofs and
Refutations_, uses the development of ever-more sophisticated theorems as the
motivation and backdrop for his main points.  Highly recommended. (He uses the
Galilean or Socratic conversation approach, which is always a delight.)

>  Without some concern for foundations, it is not at all certain that
>arithmetic can be trusted either.  There is no difficulty adding 2+2,
>since you can use your fingers.  And when you get over 10 you can take
>off your socks.  But if you try to add two numbers, each equal to the
>number of fundamental particles in the universe, you have to start
>confronting the question of whether such an operation has any meaning.

  But I doubt whether the field of foundations, as a purely mathematical field,
can ever help us.  As far as I know, "proof by induction", which is at the root
of the question of very large integers, has never been supported mathematically
except by induction itself, and I don't think it ever will, since even the
theorems of the foundationalists are finite objects, and cannot be said to touch
infinity any more than induction can.

>  Perhaps the philosophy of mathematics is too important to be trusted
>to the philosophers.

  Hear, hear!

  By the way, is this message too long?  Philosophy tends to be wordy,