# [FOM] Model theory and foundations IV

Stephen G Simpson simpson at math.psu.edu
Thu Jul 17 12:06:19 EDT 2003

```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.

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

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