FOM: NYC logic conference and panel discussion

[Stephen G Simpson]

Recently I attended a logic conference at the Graduate Center of the
City University of New York, in midtown Manhattan.  The conference was
very well attended.  Apparently a lot of people took advantage of the
opportunity to ``spend a weekend in the city''.

A typeset version of the transparencies for my talk on

     Forcing With Trees and Conservation Results for WKL_0

is now available on the web at
<http://www.math.psu.edu/simpson/nyclc/>.  There is also a link to the
official conference web page, including the program and abstracts,
photographs taken at the conference, etc.

A special event at the conference was a lively and wide-ranging panel
discussion on

         The Role of Set Theory and its Alternatives 
            in the Foundations of Mathematics

The moderator of the panel discussion was Rohit Parikh.  The panelists
were Haim Gaifman, myself, Alex Heller, and Harvey Friedman.  The
entire discussion was videotaped.  Perhaps Joel Hamkins can provide
information about how to obtain the videotape.

Here on the FOM list, I would like to initiate a discussion of the NYC
logic conference and particularly the panel discussion.  Let me start
by giving some of my impressions of the panel discussion as I remember
it.  I know that my account below is woefully inadequate, but others
may fill in the gaps.

In his opening remarks, Gaifman set the tone by giving an overview of
the subject.  One of Gaifman's memorable points was an analogy between
the ZFC formalism and the gold in Fort Knox.  Gaifman's said that,
even if nobody ever formalizes mathematical practice in ZFC, it is
possible in principle to do so, and this is the currently accepted
basis of mathematical rigor, just as gold backing is the basis of a
sound currency.

[ I would interject that this analogy may not be perfect.  For
instance, what aspect of f.o.m. would be analogous to the fractional
reserve system?  What would be analogous to fiat money? ]

In my opening remarks, I started off with my usual definition of
f.o.m. as the study of the most basic mathematical concepts (number,
shape, set, function, algorithm, mathematical definition, mathematical
proof, mathematical axiom, ...) and the logical structure of
mathematics, with an eye to the unity of human knowledge.  I then
tried to make a few additional points:

  1. Opinions about which mathematical concepts are truly basic have
     changed over time and undergone various revolutions.  Examples:
     analytic geometry, arithmetization of analysis, the set-theoretic
     revolution.

  2. Set-theoretical foundations (ZFC) is the currently reigning
     foundational orthodoxy, but there are heretical views, including:
     intutionism, constructivism, categorical foundations based on the
     free topos, lambda calculus, predicativity, predicative
     reductionism, finitistic reductionism, ultrafinitism.

  3. Right now, the FOM list is the place to be for on-line discussion
     of f.o.m. issues.

  4. One key issue in f.o.m. is the choice of appropriate axioms for
     mathematics.  Recent research has revealed a lot.

     a. Reverse mathematics yields precise data on the role of
        specific set-existence axioms in core mathematics
        (algebra-analysis-topology).

     b. Friedman's recent work on finite combinatorial statements
        requiring large cardinal axioms to prove them is of great
        importance.

After that, Alex Heller made his opening remarks.  Alex is primarily
an algebraic topologist with a strong interest in category theory.  He
made a plea for tolerance and a broad view of foundations.

I know that this summary of Alex's remarks is far from adequate.
Perhaps Alex would care to elaborate his points here in the FOM forum.

Harvey Friedman focused his opening remarks on a provocative,
long-range conjecture that he has recently formulated.  The thrust of
the conjecture is that we are going to discover a completely
coding-free and base-theory-free way of calibrating the strength of
mathematical statements and groups of statements.  I know that this
summary does not do justice to Harvey's vision, so I call on Harvey to
elaborate here on FOM.

In the audience discussion, many interesting issues were raised.  

Samir Chopra raised the question of necessary conditions for an
adequate foundational scheme, alternative ideas of basic mathematical
concepts, etc.  I tried to point out that important concepts in the
standard mathematics curriculum such as Riemannian manifolds, Lebesgue
measure, etc etc, are not ``basic'' in the relevant sense, because
they are standardly defined in terms of more basic concepts.  I
suggested that, for a concept to be considered truly basic, it may
even be desirable for it to have pre-mathematical content.  For
instance, the concept of a *set* (of marbles or whatever) can be
explained to a child who knows no mathematics, but the same cannot be
said for the concept of a *category*.

Alex Heller pointed out that he himself is not an advocate of
``categorical foundations'', but some people such as MacLane and
Lawvere have put forth such ideas, and Lawvere has even tried to teach
category theory to young children.  

[ I was unaware of these pedagogical experiments of Lawvere.  Can
anyone give a reference? ]

Another interesting question, raised by Robert Cowen I think, was that
of completely formalized proofs of reasonable length for standard
mathematical theorems.  If such proofs do not exist, what is the point
of formalization in ZFC?  Harvey said that the existence of such
proofs is an open question, but he conjectures that such proofs exist,
and he has high hopes for theorem-proving technology such as the Mizar
project.  Gaifman said that the existence of short, completely
formalized proofs is not essential in order for the predicate calculus
and ZFC to play their accustomed foundational role.  All we need to
know is that complete formalization is possible in principle.  I said
that the 20th century idea of mathematical rigor is very important and
wonderful and is closely related to (indeed grew up hand in hand with)
the idea of formalization of mathematical arguments in the predicate
calculus.  Sam Buss spoke up in favor of my position and Harvey's, I
think.  (Sam, could you please elaborate?)

The next day after the panel discussion, Gregory Cherlin in private
conversation raised an objection to my point about Riemannian
manifolds, Lebesgue measure, etc.  According to Cherlin, the
development of these and many other mathematical concepts ought to be
viewed as foundational work.  When I pressed him, he admitted that he
is entertaining the following proposition: All high-level conceptual
work in mathematics ought to be considered part of f.o.m. in the best
sense.  I was also able to get Cherlin to admit that the
Frege-Hilbert-G"odel line is foundational in a different sense of the
word ``foundational''.  But according to Cherlin, core mathematicians
view this ``traditional f.o.m.'' line as dull, passe, uninteresting,
etc.  I said that this is a mistake on the part of the core
mathematicians, as witness their shock over the fact G"odel and Turing
are the only two mathematicians on the Time Magazine list of great
20th century thinkers.

Basically I think Cherlin is going back to the position of denying the
interest of f.o.m.  This was of course the view taken by Cherlin's
fellow applied model theorists (van den Dries et al) in the early days
of FOM.

Anyway, I hope that people such as Parikh, Gaifman, Heller, Friedman,
Chopra, Cowen, Buss, Cherlin, et al will join in the discussion here
on the FOM list.

-- Steve Simpson

Name: Stephen G. Simpson
Position: Professor of Mathematics
Institution: Penn State University
Research interest: foundations of mathematics
More information: http://www.math.psu.edu/simpson/