From: rickert@mp.cs.niu.edu (Neil Rickert)
Subject: Re: Sets in mathematics
Organization: Northern Illinois University
Date: Fri, 10 Apr 1992 21:29:50 GMT

In article <s4rpbINNe04@agate.berkeley.edu> epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:
>One might think that the study of foundations is a sort of science,
>like biology or linguistics.  This would justify saying that for the
>usefulness and validity of "foundations" it does not matter whether or
>not mathematics was already around long before its foundations were
>examined, since living beings were around long before biology was
>studied, and languagage was around long before there was a
>linguistics.
>
>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.

  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.

>What keeps "foundations" from being just another field of math, as
>opposed to being a commentary on all math?

  The majority of mathematician 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."

>                                            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.

  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.

>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?

  Some would deny that there is any "extramathematical meaning" in
foundations.

>To be more specific, calculus has clear application.  It is loaded
>with meaning and utility.  This is not inherent in the math, but
>rather is a gift from sciences like physics.

  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.

>But the extramathematical meaning of "foundations" is philosophically
>much more worrisome, not least because it gives primacy to a certain
>field of mathematics which on the face of it (just looking over the
>theorems from a symbolic point of view) does not seem any more
>inherently meaningful than other fields like topology and arithmetic.

  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.

>Indeed, if I wished to assure myself of the validity of "foundations",
>I would probably found it on arithmetic, which I trust more.

  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.

>                        However, the real sticky questions of the
>meaning of math and so forth are in the end purely philosophical, and
>it is to the philosophers that we must turn (this includes lay
>philosophers).  Thus in my view "foundations" has most to offer as a
>field of pure philosophy.

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

-- 
=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=
  Neil W. Rickert, Computer Science               <rickert@cs.niu.edu>
  Northern Illinois Univ.
  DeKalb, IL 60115                                   +1-815-753-6940