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digest 2001-02-09 #001.txt

11:17 PM 2/8/01 -0800
From: "Society for Literature & Science" 
Daily SLS Email Digest

-> Call for Applicants
     by "Miller, Wayne" 

----------------------------------------------------------------------

Date: 8 Feb 2001 09:21:05 -0800
From: "Miller, Wayne" 
Subject: Call for Applicants

CALL FOR APPLICANTS

Direct inquiries to Colin McLarty cxm7@po.cwru.edu
 or see Seminar website
                       
www.cwru.edu/artsci/phil/events/summerseminar1.html

 NEH Summer Seminar in History & Philosophy of Recent Mathematics
 June 25-Aug 3, 2001
 Stipend, $3700 per participant

 PROOFS AND REFUTATIONS IN MATHEMATICS TODAY
 Directors, Colin McLarty & David Corfield

GENERAL

We are looking for a diverse group of scholars from many disciplinary
backgrounds and perspectives interested in exploring the relations
between
proof and intuition in mathematical practice. NO mathematical training
is
required, though the group will include talented mathematicians. We look
for an interest in the issues and
problems posed by mathematics.

We will examine how mathematicians perceive and represent the broadly
important goals of math at tension points within the on-going
practice--not
a technical issue but a matter of values and meaning.

Applications should be mailed by March 1 (or not much later--e-mail me
soon
about any delay)

SPECIFICS

Some mathematicians and philosophers claim mathematics deals only in
formal
logic because there is no intuition of abstract objects. Others say
mathematical rigor must be based on pure intuition of intellectual
objects.
Nearly all mathematicians agree rigor and intuition are two poles of
mathematical experience, but still argue about the relations between
them.

Historical emphasis will be from Poincare to the present. Of particular
interest in recent math is a debate provoked by Jaffe & Quinn's article
"Theoretical mathematics: towards a cultural synthesis of mathematics
and
theoretical physics" (Bulletin of the Amer. Math. Soc.) These
mathematicians emphasize how current developments in mathematical
physics,
especially, rely on intuition on geometric subjects where no one yet can
give proofs; so there is now a serious practical need to re-evaluate and
better articulate the relations between proof, speculation, and
intuition.

The Bulletin of the American Mathematical Society not only published the
paper (July 1993) but solicited responses from more than a dozen leading
mathematicians known for their interest in these conceptual issues.
These
were published (April 1994) along with rejoinders from Jaffe and Quinn.
Among the respondents were 4 Fields Medalists, the closest thing to a
Nobel
Prize in mathematics, and that statistic understates the prestige of the
group.

Great mathematicians have often written philosophic views of the
subject.
But the Bulletin debate is the most sustained, engaged philosophical
debate
ever among such a large group of leaders.

The essays in the debate are very accessible overall as the issue is not
technical. Fascinatingly, the mathematics that most provoked Jaffe and
Quinn is closely descended from some 19th century topology which the
philosopher Lakatos took as case study in his classic 1976 book Proofs
and
Refutations. Lakatos shows how certain intuitively appealing theorems
evolved through a series of purported proofs which kept meeting
refutations, sometimes by counterexample and sometimes just by logical
critique. The dialectic forced constant revision of concepts, often
productive for further math. That book is very accessible, assuming no
specialized mathematics nor expertise on the philosophic issues, yet
giving
sophisticated insights into both.

Lakatos's model of progress has been very popular among mathematicians,
and
mathematics educators have used Lakatos's model of dialogue. In
philosophy
it has probably had more impact in philosophy of science, as for example
on
Feyerabend, than on specialists in the philosophy of mathematics. Many
people read Lakatos as concluding that there are no finally rigorous
proofs
in mathematics; and each existing proof will eventually be refuted.

Seminar co-director David Corfield, from King's College London, has
established a more penetrating reading. He shows that Lakatos did
believe
there are perfectly rigorous proofs in mathematics, but only in fields
where the intuitive ideas have been exhausted and replaced by lifeless
formalizations. According to Lakatos, every creative insightful proof
will
meet refutation but later routine, modified versions using artificially
formalized concepts need not. David's dissertation and later talks have
related this view to the history of 20th century topology, exploring
strengths in Lakatos's ideas, but also showing how Lakatos underrated
the
creative power of formalization itself. David and I met through years of
E-mail correspondence on these ideas. We have drawn on his physics
background and especially his study of topological methods in recent
physics.

As for myself, along with philosophical work on the foundations of
mathematics and the nature of 20th century math, I have been actively
involved in the mathematical subject of "category theory" for nearly 30
years. I have participated in the rise of several branches of research
in
that field, watching new ideas arise and seeing new intuitions develop
into
proofs. This past year I have had extensive conversations with Saunders
Mac
Lane about how to formalize some of his ideas in set theory. Mac Lane is
among the mathematicians in the AMS Bulletin debate. A key theme in his
long career has been that successful, clear formalization of the central
ideas in any field is a great step towards further creativity leading to
inconceivable new developments in algebra, topology, and geometry not
the
end of creativity as Lakatos described.

The historian Asprey and the philosopher Kitcher have urged looking at
actually controversial issues in working mathematics. This is valuable
not
only for specialist research, or for any adjudicating role philosophers
might try to take, but most of all because controversy arises where
purely
technical considerations fail. It arises around the deep issues of what
mathematics is about. Which moves will advance the broadly important
goals?
Which merely advance an idiosyncratic program?

So they raise the question of what are the broadly important goals. And
that is what non-specialists, and certainly humanists, most need to know
about mathematics. We do not so much need an answer to the question,
which
can surely never be final, but an appreciation for the ways of answering
and the grounds for each. In this Bulletin debate a group of great
mathematicians show their ways and give their grounds.

David and I are looking forward to working with a vibrant group of
scholars!

 Colin McLarty
 Dept of Philosophy
 Case Western Reserve University
 cxm7@po.cwru.edu
 216-368-2632

 FMI please call or email me or see our website
 www.cwru.edu/artsci/phil/events/summerseminar1.html

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