[FOM] Model theory and foundations IV

[Stephen G Simpson]

This is a follow-up to John Baldwin's sequence of postings

   Foundations and Model Theory I, II, III, IV 

of Wed, 09 Jul 2003 21:08:16 -0500 and immediately afterward.

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1. What is f.o.m.?

 My little 1996 essay on *foundations of mathematics* (abbreviated
 f.o.m.) is still on-line at

   http://www.math.psu.edu/simpson/hierarchy.html

 And here is an even shorter formulation, from the preface to my book,
 Subsystems of Second Order Arithmetic.

   *Foundations of mathematics* is the study of the most basic
   concepts and logical structure of mathematics, with an eye to the
   unity of human knowledge.  Among the most basic mathematical
   concepts are: number, shape, set, function, algorithm, mathematical
   axiom, mathematical definition, mathematical proof.  Typical
   questions in foundations of mathematics are: What is a number?
   What is a shape?  What is a set?  What is a function?  What is an
   algorithm?  What is a mathematical axiom?  What is a mathematical
   definition?  What is a mathematical proof?  What are the most basic
   concepts of mathematics?  What is the logical structure of
   mathematics?  What are the appropriate axioms for numbers?  What
   are the appropriate axioms for shapes?  What are the appropriate
   axioms for sets?  What are the appropriate axioms for functions?
   Etc., etc.

   Obviously foundations of mathematics is a subject which is of the
   greatest mathematical and philosophical importance.  Beyond this,
   foundations of mathematics is a rich subject with a long history,
   going back to Aristotle and Euclid and continuing in the hands of
   outstanding modern figures such as Descartes, Cauchy, Weierstrass,
   Dedekind, Peano, Frege, Russell, Cantor, Hilbert, Brouwer, Weyl,
   von Neumann, Skolem, Tarski, Heyting, and Goedel.  An excellent
   reference for the modern era in foundations of mathematics is van
   Heijenoort [vanh].

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2. Why the hostility to f.o.m.?

 A theme of Baldwin's postings is the idea, widespread among
 contemporary model theorists, that f.o.m. is allegedly old-fashioned
 and futile, while applied model theory (i.e., model theory applied to
 algebra) is more exciting and cogent.

 My opinion is that f.o.m. is a good subject, and applied model theory
 is also a good subject, but they are good in different ways.

 Baldwin begins by rashly asserting that an "important discovery" of
 20th century mathematics is "the futility of trying to find a general
 foundations of mathematics".  I have no idea what "discovery" Baldwin
 is referring to.  Several people have already objected to Baldwin's
 assertion, for various reasons.

 To me, the overwhelming fact here is the huge success of ZFC
 (Zermelo/Fraenkel set theory with the Axiom of Choice) as a
 convenient and widely accepted framework for formalizing most or all
 of rigorous mathematics.  One manifestation of this success is that
 most or all textbooks of rigorous mathematics for beginners begin
 with a chapter on set theory, and build everything on this
 foundation.  The simple, flexible framework consisting of set theory
 plus classical logic is of great benefit as a common standard of
 rigor, tacitly subscribed by most or all mathematicians, thus largely
 sparing mathematics the methodological disputes which are widespread
 in other branches of science.  If this doesn't qualify as "a general
 foundations of mathematics", what would qualify?  I can't imagine how
 Baldwin hoped to brush away this most dramatic foundational success
 story of the 20th century.

 Baldwin then backpedals and says that what he really meant was that
 "the foundational enterprise has separated itself from much of what
 mathematicians do".

 One could mistake this revised formulation for the usual jab at
 mathematical logic (abbreviated m.l.), criticizing m.l. as being not
 only isolated from core mathematics, but also subdivided into many
 mutually isolated subbranches.  By "core mathematics" I mean branches
 of mathematics such as algebra, analysis, number theory, geometry,
 combinatorics, and differential equations, which are dominant in
 university mathematics departments.  Among the allegedly mutually
 isolated subbranches of m.l. are: forcing, inner models, recursively
 enumerable sets and degrees, ordinal notations in proof theory, the
 lambda calculus, models of arithmetic, etc.

 If we read Baldwin's remark as a criticism of m.l., then it has a
 certain amount of merit from the core mathematics viewpoint, though
 much less so from the viewpoints of other fields such as philosophy
 of mathematics, foundations of computer science, and f.o.m. itself.

 However, such a reading is problematic, because Baldwin speaks not of
 m.l. but of "the foundational enterprise".  It appears that Baldwin
 is trying to sweep under the rug the voluminous contemporary work
 that is specifically in f.o.m., not just m.l., which interacts
 directly with core mathematics in a variety of ways.  Why is Baldwin
 doing this?

 Naturally I am thinking partly of Reverse Mathematics.  Some of
 Baldwin's model-theoretic colleagues have verbally expressed disdain
 for Reverse Mathematics, and Angus Macintyre has done so in print.
 What is behind this hostility?  I have conjectures about this ....

 As a less-publicized example of recent f.o.m. research that interacts
 directly with core mathematics, let me mention Ulrich Kohlenbach's
 work in "proof mining", leading to new bounds and uniformities in
 various branches of analysis such as approximation theory and metric
 fixed point theory.  This work of Kohlenbach is simultaneously good
 f.o.m. and good core mathematics.  Perhaps Baldwin was not aware of
 Kohlenbach, but why does he deliberately downplay this *kind* of
 research?

 It is noteworthy that, when for a few years in the 1990's an applied
 model theorist was in charge of the US government's program of
 funding for university research in mathematical logic (officially
 known as the Foundations Program within the Division of Mathematical
 Sciences of NSF), f.o.m. fared particularly badly.

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3. Foundations of algebraic groups?

 Baldwin repeats a claim made in the glorious early months of the FOM
 list, that model theory applied to algebra is "foundational" in a way
 that puts traditional f.o.m. to shame.  Why?  He gives three
 examples: (a) Chow's Theorem characterizing analytic manifolds in
 complex projective n-space, (b) the Hrushovski-Weil characterization
 of algebraic groups over an algebraically closed field, and (c) a new
 proof of a theorem of Borel and Tits.  (This is Armand Borel, the
 algebraist, not his father, Emile Borel, the set theorist.)

 I don't see that any of these excellent core mathematical results
 have "foundational character", except perhaps in a very broad sense
 of establishing or otherwise dealing with some non-obvious
 relationship between logical notions and mathematical notions.

 For example, Hrushovski-Weil is an elegant, illuminating theorem, but
 it is not foundational in the standard sense of the term, because it
 does not provide a way of expounding or formalizing a subject from
 the ground up.  In textbooks of algebraic groups, we will never see
 the following:

   Definition.  Let C be the field of complex numbers.  An algebraic
   group over C is a group which is first-order definable over C.

 Instead we will always see the usual definition of algebraic groups,
 in terms of algebraic varieties, while first-order definability will
 be relegated to a footnote or an appendix, if mentioned at all.

 And even if core mathematicians someday learn about first-order
 definability (this would be wonderful!), they will still continue to
 prefer the usual definition of algebraic groups, in terms of
 algebraic varieties, because it parallels the usual definitions of
 other kinds of groups, in terms of other kinds of varieties or
 manifolds.

 On the other hand, core mathematics textbooks will always contain
 definitions like this:

   Definition.  A group is defined to be a *set* with a binary
   operation satisfying the following axioms ....

   Definition.  A variety consists of a *set* together with a system
   of overlapping charts ....

   Definition.  A field is a *set* with two binary operations ....

 Thus sets will continue to play the foundational role.

 Or, is Baldwin claiming that this commonplace impression is
 superficial, concealing a deeper reality?  If so, what is the deeper
 reality?

 Baldwin says: "The foundational character is in the analysis of the
 FUNDAMENTAL CONCEPTS OF THE SUBJECT UNDER STUDY -which are groups,
 group morphisms etc. not sets and orderings."  (The capital letters
 are Baldwin's.)  I don't understand this.  It seems illogical,
 because a group is first of all a *set*.

 Baldwin recommends a model theory book by Poizat which he says will
 "give a feel" for what he calls "the foundational character" of
 Hrushovski-Weil, etc.  Is "the foundational character" so esoteric
 that it cannot be adequately explained here, on the FOM list?

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4. Final remark.

 It goes without saying that the kind of activity represented by
 Hrushovski-Weil etc will always be of interest to a much smaller
 circle of people -- smaller by 5 or 6 orders of magnitude -- than the
 truly foundational work of, e.g., Goedel and Turing.  In Time
 Magazine's list of the 20 most influential thinkers of the 20th
 century in all disciplines, Goedel and Turing appeared, but no other
 mathematicians.  And this is entirely appropriate.  After all, how
 many people know about algebraic groups?

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 Stephen G. Simpson
 Professor of Mathematics
 Pennsylvania State University