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*To*: fom@math.psu.edu*Subject*: FOM: NYC logic conference and panel discussion*From*: Stephen G Simpson <simpson@math.psu.edu>*Date*: Sun, 5 Dec 1999 19:58:03 -0500 (EST)*Cc*: jhamkins@gc.cuny.edu, rkobb@cunyvm.cuny.edu, richard@llaic.u-clermont1.fr, schopra@gc.cuny.edu, rparikh@gc.cuny.edu, aheller@gc.cuny.edu, rcowen@gis.net, cherlin@math.rutgers.edu, cwood@mail.wesleyan.edu*Organization*: Department of Mathematics, Pennsylvania State University*Reply-To*: simpson@math.psu.edu*Sender*: owner-fom@math.psu.edu

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/

**Follow-Ups**:**Stephen G Simpson**- FOM: NYC logic conference and panel discussion**Harvey Friedman**- FOM: NYC panel discussion/formalization

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