From: rickert@mp.cs.niu.edu (Neil Rickert)
Subject: Re: Sets in mathematics
Organization: Northern Illinois University
Date: Sat, 11 Apr 1992 16:56:04 GMT

In article <s64tlINNip0@agate.berkeley.edu> epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:
>  In article <1992Apr10.212950.27333@mp.cs.niu.edu>, rickert@mp.cs.niu.edu (Neil
>Rickert) writes:
>  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.

 I venture to suggest that most mathematicians, when they have late night
worries, are not thinking about anything closely related to this.  More
likely they are just stuck on a crucial step in their attempted proof.

>  Mathematics is self-referential to a high degree.

 Many human activities are self-referential.  It is not at all clear that
mathematics should be singled out for this.

>                                                   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.

  Perhaps "foundational mathematics" is not really anything special in
comparison with other mathematics.  It set out to be special, but Goedel's
incompleteness theorem can be interpreted as demonstrating that this special
status was unattainable.

>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).
                                           ^^^^^^^^^^^^^^^^^^^
 Gee.  I always thought it was called the hairy billiard ball theorem.

 When counting my thumbs, I can ponder the question,

	What is the number 2?

	  Is it really an ordered pair   # a complex number
	  of Dedekind cuts	         # real numbers
	  of equivalence classes	 # rational numbers
	  of ordered pairs		 # fractions
	  of ordered pairs		 # integers
	  of sets			 # natural numbers
	  of sets			 # the basic elements of ZF

 If I really had to think about numbers in this way, counting my thumbs would
be a real struggle.

 One way of looking at foundations is to consider it the construction of a
model.  As long as we can embed our mathematics in this model (by some
sort of isomorphism) there can be no inconsistency in our mathematics unless
there is a corresponding inconsistency in ZF.

 This leaves unanswered, and perhaps therefore a question of philosophy,
whether the isomorphism is the identity (i.e. a number really is the monster
described above) or not.  But to most mathematicians this question is of
no importance.  The presumed existence of an isomorphism is enough to
consider foundations a possibly useful branch of mathematics.  With
Goedel's results, the ultimate goal of foundations -- to prove the
consistency of mathematics -- cannot be reached.  But as long as other
useful results come from foundations, the field has validity.  The
relatively small size of the set of axioms needed in ZF is perhaps already
sufficient justification.  But it is really the need to avoid problems
such as Russell's paradox which motivate most mathematicians in their views
of foundations.

>  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.

  Let me expand a little on my comment that this is perhaps the preponderant
view.  I think many mathematicians would state this as their view if asked,
although it might depend on who asked them, and how the question were
phrased.  But it does not seem to be the view they use when they do their
research.  They tend, instead, to behave as if they believe there is
some intrinsic, perhaps undefineable, meaning in mathematics.

  Alas, in all too many text books the idea that mathematics is empty
symbol manipulation with no intrinsic meaning is taken too far, with the
result that many would-be mathematicians are turned away from mathematics
because they find this view unappealing.

  My personal view is that mathematics does have intrinsic meaning, value
and purpose.  However the view that mathematics is empty symbol manipulation
is a valid (even if incorrect) interpretation.  It is important that this
interpretation remain valid, for this is what makes mathematics developed
in one context applicable in a completely different context.  In particular
the validity of the proof of a theorem must be based on its correctness
under the interpretation of mathematics as empty symbol manipulation.
In contrast, the importance of the theorem is judged in terms of its
relevance to the intrinsic meaning of the mathematics within some context.

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