From: epfaith@twinkies.berkeley.edu (Edward Paul Faith)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 11 Apr 1992 10:22:39 GMT
Organization: U.C. Berkeley Math. Department.
NNTP-Posting-Host: twinkies.berkeley.edu

In article <15196@optima.cs.arizona.edu>, gudeman@cs.arizona.edu (David Gudeman)
replies to something I wrote:

]]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.

His reply:

]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.  Hilbert wanted to formalize
all of mathematics into a single calculus (so I recall), and Godel showed that
this was impossible, since no matter what calculus we develop, if it is
sufficiently complex, we will be able to prove things about it, without being
able to carry out the proofs within that calculus.  However, these proofs, while
not realizable within the calculus, are nevertheless mathematical proofs. This
is why mathematics cannot be completely formalized.

My careless wording has led you to believe that I agree with Hilbert.  Well, I
don't, as you can plainly see.  In truth, mathematics uses not a formal system,
but an enormously sophisticated symbolic system, namely language.

So the problem of whether or not mathematics is vacuous becomes the problem of
semantics, the problem of the meaning of language.  Just as I thought [read my
reply to <1992Apr10.212950.27333@mp.cs.niu.edu>, rickert@mp.cs.niu.edu (Neil
Rickert)]. Two fields of philosophical inquiry collapse into one.

]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_.

Unlike grammar, which as a science can be wrong, and even as a judgement of
right and wrong often comes into conflict with itself (grammar books disagree
sometimes as to correct crammar) mathematics holds a very special place in
language, because its theorems cannot be disproved by typical experience.  For
example, the real world is curved and so some day a geometer might discover that
the three angles of triangles do not really add up to exactly pi.  This would be
a revolutionary discovery in a world without Einstein.  However, it would not
invalidate Euclidean geometry.  Rather, it would invalidate (or modify) physics.
I suggest that mathematical statements are absolutely certain in a way that
most statements are not.

Let us return to Godel's proof.  Suppose somebody constructs within a certain
calculus a proof that Godel has proven impossible to construct.  What do we
conclude?  We conclude that the person must have made an error.  Or, if this
turns out to be not the case, then we conclude that the actual practice of
following the calculus is for some reason different from Godel's (and our)
inductive understanding of it.  We return to the problem of infinity and
induction, mentioned in the abovementioned reply to Neil Rickert.

We have another option.  We may simply say that the person cannot have
followed the calculus, because by definition to follow the calculus means to
follow Godel's and our inductive understanding of it.  In other words, we say
that the existence of the impossible proof shows that there are different
calculi that appear to follow the same rules, but actually don't.

We could do this.  However, it would be a real pain in the neck to people who
had thought that it said things about real mathematical systems, and not just
hypothetical inductively defined systems.  His theorem would no longer be a
theorem about mathematics, and it would no longer obstruct the Hilbertian
program of formalizing mathematics completely.  How do we know that his theorem
does in fact apply?  I consider this a question of foundations. 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.

So now, "mathematical foundations", the math field, can be about math, but the
bottom line is that it cannot resolve many of those problems which are considered by
philosophers and by myself and by the writer whose identity I do not
recall to comprise the foundations of mathematics.

]"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?  Grammatically, 
you are saying the latter.  I'm not sure what you mean, then. If you
are saying that I am ignoring fundamental issues, then I will not defend
myself.  I have just finished a thought experiment in which math fails
to tell us true things about the physical world, and even about mathematics,
until we understand it properly.  Euclidean geometry has failed for
two thousand years to tell us true things about the real world, because
the real world is not Euclidean.  And yet, all is forgiven.  We have found
a new role for it in our lives.  So I am not sure that mathematics tells us
(precisely) anything true about the world.  Perhaps the world is ever so
slightly logically inconsistent.

How does the physical world manage to be so mathematical?  Is that really a
topic that the mathematical theory of foundations can address?  I think that it
is not.  I may have ignored this topic as you say, but now I am not ignoring it,
and still I say that this foundational question cannot be answered by any
mathematics. Whatever we find using mathematical theories, the question will
remain as to why we have thus found anything out about the real world, or even
about mathematics.

]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.  It's nothing new. Indeed, hidden assumptions are much on the mind
of any mathematician.  The many failed proofs of Fermat's last theorem have all
been shown to rely on hidden assumptions.  Ordinary, everyday mathematicians
exposed these hidden assumptions as a matter of course.  You have just stated in
effect that foundational theorists are not doing anything that is not being done
Xref: sparky sci.philosophy.tech:2086 sci.physics:6689