FOM: Re: f.o.m./TIME Magazine

[Harvey Friedman]

Reply to Silver Tue, 27 Feb 2001 17:54.

Friedman wrote:

>>The problem with the first is that the Mathematics and Philosophy
>cultures
>>have grown very far apart, with virtually no common language. The
>irony is
>>that f.o.m. could have served as the common ground, preventing the
>cultures
>>from having grown this far apart. Also, University Administrations
>have not
>>recognized the serious flaws in a Disciplinary approach to University
>life.
>
Silver wrote:

>    One problem I see is that so-called foundations seems to depend
>too much on technical stuff about set theory for philosophers to be
>interested in and/or competent at.   In fact, I'd go further.   It
>seems to me that set theory has been set up as *the* foundation, and
>competence in its highly specialized, more difficult reaches seems a
>prerequisite for philosophizing about mathematics.   Therefore,
>philosophers just can 't do it.

I would like to distinguish two levels of involvement in f.o.m.

1. One is having a clear and sophisticated understanding of the main
findings and main open issues in f.o.m. This does NOT entail any
understanding of any proofs. And this kind of understanding does not
support direct major contributions to f.o.m. However, someone armed with
this sort of understanding, could a) pass such understanding on to students
and colleagues, thereby communicating the excitement and great power and
future of f.o.m. to future generations; b) raise interesting issues and
bring alternative points of view concerning f.o.m. that have not been
properly addressed by researchers in f.o.m. which, in the hands of
researchers in f.o.m., could further the development of f.o.m. and possibly
add entirely new dimensions to f.o.m. development.

2. The second is making serious contributions to f.o.m. Here I agree with
the thrust of what Silver says in the next paragraph. Sharply put, genuine
f.o.m. research requires a combination of substantial mathematical and
philosophical power, and this combination is rarely seen in sufficient
quantities. However, I suspect that this is rare partly for ENVIRONMENTAL
reasons. Students find themselves either in a Philosophy or a Mathematics
department, where the overwhelming majority of faculty in Philosophy do not
think mathematically and do not place emphasis mathematical thought, and
the overwhelming majority of faculty in Mathematics do not think
philosophically and do not place emphasis on philosophical thought. This is
true, to a lesser extent, of even those faculty who are teaching which is
broadly construed as "logic".

However, there is another important aspect of this. We must have an
environment where appropriately gifted people find f.o.m. research an
attractive choice. There are problems at various levels regarding this.

Firstly, a mathematics student with interest in f.o.m. will normally be
sent to mathematical logicians who may have no interest and little
understanding of f.o.m., or worse still, may identify f.o.m. with
mathematical logic. To put it frankly, despite my many good friends who are
mathematical logicians, mathematical logic cannot be expected to currently
look attractive to people with philosophical inclinations, since
philosophical discussion and philosophical motivation has been nearly
expunged from the current mathematical logic literature and culture.
Mathematical logic also cannot be expected to currently look attractive to
people with standard mathematical inclinations, as it seems disconnected
from mainstream work in algebra, geometry, number theory, analysis,
etcetera. It is now more connected than it was, say, 10-30 years ago,
through relatively recent work in applied model theory, and also, to some
extent, through work in descriptive set theory and work in reverse
mathematics. It is still relatively disconnected, but I don't view this
negatively, since mathematical logic grew naturally out of f.o.m. and it
was never the intention of f.o.m. to be a part of mathematics in the sense
of algebra, geometry, number theory, analysis, etcetera - but rather a
subject with its own interdisciplinary goals and standards. But this low
level of connection does have its practical effects. I should add that I am
against prematurely forcing artificial connections.

Secondly, a philosophy student with interest in f.o.m. will normally be
sent to philosophers of mathematics. Here the problem is less severe, but
still very significant: the philosopher of mathematics may not have enough
understanding and appreciation of the mathematical aspects of f.o.m. to
orient the student properly towards f.o.m. There might be the attitude that
the more mathematical aspects of f.o.m. have less philosophical importance,
which is false. (The more mathematical aspects of mathematical logic DO
have less philosophical importance - often none at all). Thus f.o.m. may
appear to the student to be a disconnected subject with mathematical stuff
and philosophical stuff, unintegrated, and one must wear two unrelated
hats. This appearance cannot be expected to be attractive.

Thirdly, the reports that I have from attempts at joint programs in
mathematics and philosophy have not been positive. However, some new ones
seem to be emerging and may work better. But the bottom line is that
students tend to be forced to go with one side or the other. Worse still,
the job market is very likely to force students to go with one side or the
other. Of course, this is a Corollary of the attitudes and modus operandi
of the faculty and University administrations. If these attitudes change
then this "one side or the other" situation will change. This is probably
an "if and only if".

So what can be done? I mentioned two pragmatic structural alternatives.
Specifically, f.o.m.

>>1. As a joint enterprise between Mathematics and Philosophy, facilitated by
>University Administrations.
>>
>>2. As the leading component of a new Academic structure - Foundational
>Studies.

But at an intellectual level, what can be done? A viable intellectual
movement would go a long way towards facilitating 1 or 2 above.

Perhaps the single clearest intellectual suggestion is:

**the development of philosophically critical expositions of mathematical
subjects, and specifically a philosophically critical exposition of f.o.m.**

Personally, I am not satisfied with any philosophically critical treatment
I have seen of any mathematical subject, including f.o.m. Because of the
lack of such philosophically critical expositions, it seems natural that
students will normally develop at most one side of their
mathematical/philosophical thinking. This is particularly natural as they
also are being advised by faculty who also have developed at most one side
of their mathematical/philosophical thinking. However, sufficiently clear
and powerful philosophically critical treatments of mathematical and
scientific topics should go a long way to fix the current stifling
environment.

Yet philosophically critical treatments of mathematical and scientific
topics has not been a major thrust in either the Mathematical or
Philosophical communities.

>So, there are a bunch of
>specialists--you are one--who do set theory, and a very few of these
>specialists--you are one again--happen to have strong philosophical in
>terests.   But, for example, stuff about large cardinals is
>inaccessible to virtually all philosophers (i.e., those philosophers
>who inhabit philosophy departments).

A philosophically critical treatment of f.o.m. would definitely be readily
understandable to at least the analytic philosophy community, and to many
other communities. Such a treatment need not contain any detailed proofs of
anything.

Such a treatment must contain a philosophically critical treatment of set
theory, as set theory has a special role in f.o.m. One could call this
macroscopic global foundations of mathematics. Macroscopic because of the
high strength of the axioms. Global because one is founding all of
mathematical thought. (Obviously set theory is not sensitive to many
subtleties of mathematical thought, but that is a wholly different matter).

But such a treatment must also contain microscopic global foundations of
mathematics, and also various local foundations of mathematics. By the
latter, I mean, say, the elementary theory of algebra and geometry which is
completely axiomatized through Tarski. Microscopic global foundations of
mathematics is presented in terms of relatively weak systems such as those
in reverse mathematics, or weak fragments of Peano Arithmetic, etc.

If the relevant Philosophers find the usual treatments of set theory
difficult to make sense of, then they will likely also find the usual
treatments of reverse mathematics and tame model theory (with the
elementary theory of algebra and geometry as a prime example) difficult.

This problem should be solvable through the appearance of philosophically
critical treatments of f.o.m.

I want to emphasize that any foundational scheme that accomplishes the
crucial things that set theoretic foundations accomplishes is going to be
at least as difficult as set theory to appropriately understand.

>    About your (and Steve's) concern that math. logic is not
>necessarily foundations: part of math logic is set theory, and set
>theory features prominently in foundational talk.   Therefore, it's
>natural for people to think math logic is foundations.   That is,
>foundations seems to be set theory, which is also part of math. logic.
>(I'm not saying this is a terrific argument, just a natural conclusion
>to draw.)

An absurd argument which should be fully offset by the following declaration:

**mathematical logic is the study of mathematical structures that arose out
of foundations of mathematics, studied for their own sake, or for their
applications to mathematics, independently of their connection with
foundations  of mathematics or related philosophical topics. Sometimes such
connections exist, but that is generally an accidental byproduct of these
studies for their own sake.**

Even though set theory is the preferred global foundation for mathematics,
a particular result in set theory may have anywhere from no to crucial
significance for foundations of mathematics.

>    I personally think reverse mathematics seems foundational, but
>several mathematicians I've spoken to seem not to hold it in high
>regard.   I don't know what exactly this implies, but I'm speculating
>that they don't consider it either to be "interesting" math logic or
>they think it has no foundational interest.

If you are talking about mathematicians outisde mathematical logic and
f.o.m., then reverse mathematics has to be explained in just the right way
so that they know what the fundamental points are. I have had great success
in communicating this to some of the most famous mathematicians on the
planet. This happened yet again during my recent visit to Rutgers for a
MAMLS meeting.

If you are talking about mathematical logicians then nearly the entire
recursion theory community now openly recognizes the importance of reverse
mathematics, since most of the most well known recursion theorists write
papers in reverse mathematics from time to time and mention it in some of
their talks. And nearly the entire proof theory community openly recognizes
the importance of reverse mathematics. In model theory, some luminaries
definitely see the importance, and others now appear quite open to it.
Reverse mathematics can also be recast to some extent in terms of
interpretability, and when recast this way, it probably interests more
model theorists. The set theory community is so engaged, by its very
nature, in systems that extend ZFC, that somehow reverse mathematics seems
too remote. They generally do not even think about reverse mathematics
enough to have a distinctly negative opinion, lumping it together with all
metamathematical investigations based on "only" fragments of ZFC - at least
that is my impression.