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*To*: fom@math.psu.edu*Subject*: Re: FOM: NYC logic conference and panel discussion*From*: Sam Buss <sbuss@herbrand.ucsd.edu>*Date*: Mon, 6 Dec 1999 16:51:38 -0800 (PST)*Sender*: owner-fom@math.psu.edu

This is in reply to Steve's comments on the panel discussion at the recent NYC Logic Conference on "The Role of Set Theory and its Alternatives in the Foundations of Mathematics". In particular, Steve asked my to clarify my own comments (made as a member of the audience). I made two comments, one on foundations of mathematics and one on the possible future computerized theorem provers; but I will only discuss the first topic below. My basic assertion was that the real foundations of mathematics is first-order logic, or the use of rigorous reasoning and mathematical rigor that can be formalized in first logic. This is in contrast to the usual point of view that set theory is the foundations of mathematics. Remark: I should make it clear that when I say "first-order logic" I allow more generally any logic that can be recursively axiomatized, including multi-sorted logic and even second-order logic, but I am restricting to classical logics. That is to say, I am saying the real foundations of mathematics is the use of the kind of rigor and reasoning that is formalized by first-order logic, regardless of whether this rigor is applied to first-order systems or to systems which have stronger expressive power. To bolster this (perhaps surprising?) assertion, one can make several observations: 1. Nearly every mathematicians is quite comfortable with set notation and informal set formation; however, few mathematicians are familiar with the formalization of mathematics in ZFC. For instance, how many mathematicians pay attention to when they use the axiom of choice? How many know the difference between the dependent axiom of choice versus the full axiom of choice? How many know what the axiom of replacement is? Furthermore, is there any point to mathematicians watching out for these issues? Shouldn't they just take the axiom of choice as being obviously true? (Unless they are being careful about constructivity, and even then the axiom of choice is generally accepted as true.) It is also rare for mathematical theorems to depend on large cardinal axioms. (Some people, notably Harvey Friedman, are trying to change this situation.) As additional evidence of the cultural fact that mathematicians no longer use set theory essentially, note that (in the US at least), set theory and mathematical logic are not commonly taught in either the graduate or undergraduate curriculum. I have served on the graduate student admissions committee here at UCSD and know from experience that almost no prospective graduate students have studied logic or set theory as undergraduates (well under 5%, I'd estimate). 2. The bulk of mathematics can be done in second-order logic rather than in set theory --- however to really do mathematics in second-order logic would be very awkward. Take as an example the study of Banach spaces. These are abstract spaces providing a very general framework for theorems in analysis, differential equations, etc. Nearly every interesting example of a Banach space is definable in higher-logic over the reals. However, it is far better to formalize Banach spaces in a general setting with no reference to higher-order logic over the reals. This is generally done in an informal set-theoretic setting, but in practice it is done (a) by formulating first-order axioms about Banach spaces and functions on Banach spaces, and (b) taking the reals as given. 3. The theory of the reals provides a good example of how both set theory and first-order logic are used in foundations for mathematics. Of course the traditional foundations of the reals uses set theory: Dedekind cuts or Cauchy sequences with the construction resting ultimately on the definition of integers. The integers themselves can be defined in terms of sets (the von Neumann integers). However, it is also possible to formalize the integers in first-order logic (Peano arithmetic). And in fact, most mathematical reasoning about first-order theorems about the integers is done more in the style of Peano arithmetic, rather than in the style of set theory. Thus, it is common to think of induction as a fundamental principle rather than a derived principle. Also, it is extremely rare for a mathematician to write an assertion like "7 \in 9" which depends on the formalization of von Neumann integers. 4. One of the distinguishing features of mathematics is the use of proof and of mathematically rigorous reasoning. Especially noteworthy is the fact that mathematicians will nearly always agree on whether a given asserted theorem has been correctly proved. When there are agreements on whether a proof is correct, mathematicians try to resolve their disagreement by breaking their argument into smaller steps, down to the level of first-order logic if necessary (but not down to the level of set theory.) Lasting disagreements on whether a proof is correct are rare; on the contrary, they can usually be resolved to *everyone's* satisfaction. I believe that this fact is due directly to the fact that mathematicians are reasoning with methods that are *formalizable in principle* in a first-order system. This makes the notion of mathematical rigor a robust and objective concept. 5. In defense of set theory, I should mention that it provides the best general "ontology" for mathematics. We have good intuitive reasons for believing in the "existence" of mathematical concepts such as the integers, the reals, functions on the reals, etc. Set theory has the advantage that it provides a single framework for a Platonic view of *all* of mathematics. In other words, set theory may not be optimal foundation for common concepts such as the integers or the reals (since we have clear direct intuitions of these concepts that do not require the complexity of set theory), but set theory has the generality to handle *all* mathematical constructions. This is a remarkable fact (and one that Haim Gaifman made very clearly in his introductory remarks at the panel discussion). Nonetheless, one can consider a thought-experiment: suppose that the next generation were to stop thinking about sets for their foundations. Would this stop the study of mathematics? Would this necessarily fundamentally alter the nature of mathematics? (Setting aside the areas of the mathematics, such as set theory, that study sets directly.) I think that, as a thought-experiment, the answers to these questions is "No." [Please note I am *not* predicting the demise of set theory, I am only conducting a thought-experiment! Set theory will clearly be around at least until some clearly better alternative approach is found, and such an alternative would obviously be a *major* innovation. I suspect that any possible alternative would build heavily on the highly successful example of set theory.] One the other hand, it is very difficult to conceive of mathematics being carried out without the use of the type of rigor inherent in first-order logical reasoning. ------ Some time ago there was a lot of fom discussion on the nature of mathematics and "What is Mathematics, Really". I'd like to offer my own definition: "Mathematics is the study of objects and constructions, or of aspects of objects and constructions, which are capable of being fully and completely defined. A defining characteristic of mathematics is that once mathematical objects are sufficiently well-specified then mathematical reasoning can be carried out with a robust and objective standard of rigor." (Please note that "objects" is intended to include non-physical objects!) --- Sam Buss I am supposed to say something about my background: My job title is: Professor, Mathematics and Computer Science, UCSD My research intesests include proof theory and theoretical computer science.

**Follow-Ups**:**Stephen G Simpson**- FOM: speculations on the future of f.o.m.**Stephen G Simpson**- FOM: defining ``mathematics''

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