foundations

[Harvey Friedman]

*temporarily, please readdress replies to friedman at math.ohio-state.edu
instead of using the reply key*

[for a quick punch line in this long essay, which I intended to be only a
single paragraph, see item 11 near the end]

I am unboundedly excited about the responses from Lou, Anand, Dave, Steve,
and indirectly John Baldwin and John Burgess about foundations of
mathematics. I have been restraining myself from putting in my own two cents
again as I am committed to giving Colloq at Math and Phil at Stanford next
week and am under pressure to prepare stuff right now. I feel that I gain
some sort of perspective by dealing with multiple communities (departments),
and I want to share that with you when I get back and have the time to
explain this in depth. 

If you entertain the idea that Simpson and {Lou, Anand, Dave} are talking
past each other, you don't know what "talking past each other is"; witness
mathematicians and philosophers in the same room!! By comparison, you and
Steve are in complete agreement.

However, I can't resist saying just a tiny tiny bit right now, before I
really get serious about all of the points raised by your very thoughtful
little essays. So I will restrict myself here to talk only about why I am
now terribly excited about classical foundations of mathematics in the sense
practiced by Kurt Godel. Below I will use FOM to mean the foundations of
mathematics in this sense.

1. FOM is and also is not a branch of mathematics. It is a branch of
mathematics when people doing it are looking for jobs in mathematics
departments, or have to be classified in academia. It is a branch of
mathematics if it is to be classified in terms of the methodology - which is
largely proving theorems rigorously. It is not a branch of mathematics is
the sense that mathematics is motivated ultimately by quantity, shape,
motion, symmetry, etcetera. It is not even a branch of mathematics in the
sense that applied mathematics is a branch of mathematics, with its (applied
mathematics) applications to subjects outside mathematics such as science
and engineering. 

2. Since FOM is not a branch of mathematics, it is understandable that many
mathematicians such as Lou, Anand, Dave, would be less interested in FOM
than in many branches of mathematics such as algebra, geometry, number
theory, etcetera, or mathematics generally. It's understandable - but not
inevitable. Of course, I would like to think that Lou, Anand, Dave could
become more interested in FOM than in mathematics itself, under appropriate
circumstances. In fact, I would like to think that all mathematicians could
become more interested in FOM than in mathematics itself, and that is even
more ambitious. One of the driving forces behind me getting this e-mail
going is to learn more about why Lou, Anand, Dave, etc. find mathematics
more interesting than FOM, and also to explore alternative kinds of
"foundations of mathematics" that they have in mind. I leave open the
possibility that there are alternative kinds of "foundations of mathematics"
that I could become at least as interested in as I am currently in FOM.

3. Once we have cleared the air by agreeing that FOM is not a branch of
mathematics, we can stop trying to evaluate it as a branch of mathematics.
This is as out of place as evaluating statistics as a branch of mathematics,
or evaluating physics as a branch of mathematics. For instance one can say
that "statistics is not an impressive branch of mathematics like number
theory in that it doesn't have the deep interplay with so many other
branches of mathematics with deep theorems such as Faltings and Wiles." Or
one can say that "physics is not an impressive branch of mathematics because
of the comparative lack of rigorous methods." Or for that matter
"mathematics is not an impressive branch of science because of its general
lack of experimental confirmation." Blah, blah, blah.       

It seems to me that some of what Lou, Dave, Anand are saying appears to be a
bit like evaluating FOM as a branch of mathematics.

4. Another point I think is worth making is that Lou, Dave, and Anand do
sometimes have a tendency to too quickly trust the judgment of
mathematicians. They know full well the virtually pathological resistance of
nearly all of the mathematicians they have come into contact with to learn
what predicate calculus is, despite its by now obvious use in proving some
things (in, e.g., algebraic and semi-algeraic geometry) in the space of a
microsecond when it normally takes a great deal of work without it. Of
course, I myself rather like to complain about the general lack of
intellectual breadth and sophistication represented by this aversion, since
I emphasize the great general intellectual importance of predicate calculus
in the history of ideas. But the applied model theorists are seeing another
side to this, and this has been going on for a very long time. I know full
well that there are serious moves to rectify this situation, perhaps on both
sides, in connection with the upcoming events early next year at MSRI, and I
wish everyone the best of luck in this regard.

5. We can now concentrate on what the aims and goals are of FOM and see
whether or not I can state them in so compelling a fashion that you
eventually become terribly excited about it - maybe more excited about it
than mathematics. At a later time I want to carefully address the
differences between the topics put forth as "foundational" by Lou, Dave, and
Anand, and FOM; as well as the differences between these topics put forth by
you as "foundational" and the normal use of the word "foundational" as used
in foundations of physics, mechanics, biology, philosophy, psychology,
computer science, economics, law, etcetera. 

At the risk of digression, here is a hint of something that is not strictly
FOM but is clearly foundational in the normal sense of the word: give a
systematic treatment of the concepts of natural number and the continuum
that doesn't rely on set theory. In general, give an appropriately elegant
and powerful non set theoretic foundation for mathematics, or at least
foundation for significant parts of mathematics. Even better, do it in such
a way that one is not subject to the usual negative results of Godel. Or
alternatively, prove that this cannot be done in that a tiny minimum number
of indespensable mathematical facts quoted and used every day in some
appropriate sense force you into the negative results of Godel. These
alternatives are not really incompatible.   

6. It is generally acknowledged that FOM had a golden age from late 1800's
through the 1930's, with Kurt Godel dominating this development in the
1930's. And that a substantial portion of the greatest mathematicians of the
early part of the Century were profoundly interested in FOM, and put their
own views on some aspects of it in writing. E.g., Weyl, Hilbert, Brouwer,
Poincare, von Neumann, etcetera. And to remind you of the temperment of
these people, Poincare wrote a large book called Foundations of Science,
although it was not really successful. There was tremendous interest in such
topics as the elimination of transcendental methods in number theory and the
use of impredicative methods in analysis. And not just because they thought
that such methods might lead to inconsistency. If my memory is right,
Poincare and Weyl suggested that impredicative methods were unlikely to ever
result in meaningful theorems - that, as we say in modern terms, a
predicatively meaningful theorem of any interest should have a predicatively
acceptable proof. As all of you know, this has been strongly and
convincingly refuted already by the early 1980's. But maybe I should say:
this wasn't refuted until the early 1980's.

There is no doubt that some people in philosophy and other subjects, and
some mathematical logicians, would not only be convinced that this is a
triumph of FOM, but that already, from what I said, FOM looks more
interesting than mathematics. Here we have the following ingredients:

a) mathematicians generally, including the greatest of them, were profoundly
interested in the use of transcendental methods in mathematics, especially
their eliminability and the extent to which they can be used to prove
nontranscendental theorems;

b) any result to the effect that certain transcendental machinery cannot be
eliminated in the proof of an important nontranscendental result was clearly
regarded at that time as an unimaginable achievment that would profoundly
change their views as to the nature of mathematics;

c) after many decades, results of this kind were acheived, and continued to
be acheived, using substantial mathematical machinery from the technical
development of mathematical logic.  

Of course, I am talking about Kruskal's theorem and the graph minor theorem.
E.g., Carol Wood reports the great interest of Thurston in these two
theorems, something I have known about from conversations with Thurston many
years ago.

These are the most convincing results to date which establish the absolute
necessity of machinery. This moves an issue of classical concern into the
realm of the scientific and the quantitative. 

7. Let us reflect on this from two points of view. First as a mathematician.
Well, these results are not mathematics in the sense discussed above of
being directly about numbers and shapes, etcetera. So what's the excitement?
Moreover, it's not likely to help the mathematician prove any theorems in
mathematics. And the mathematician will not have his papers rejected because
he has used impredicative methods.

Second as a mathematical intellectual. This is the level at which Hilbert,
Godel, Brouwer, Weyl, Poincare, von Neumann, etcetera, were operating at. It
rather throroughly and profoundly refutes an extremely attractive and
plausible looking view as to the nature of "real" mathematics. Is there any
doubt that if these people were active during the 1980's that they might
well be deeply moved by such results - in a way that they would not be moved
by any number of important results in mathematics proper? 

8. FOM has been subject to the same general features that any great new
theoretical subject in science or engineering does - that after an initial
monumental thrust, reality sets in, and the inevitable gap between the great
new theory and the phenomena being theorized about becomes apparent. There
is a search for major expansions of the theory to become "more realistic" or
"more sensitive to the actual phenomena" or "further developed to encompass
more phenomena." The future impact of the subject depends on how and to what
extent such gaps are bridged. 

For quite some time after the 1930's, it was quite unclear just how and to
what extent such gaps would be bridged for FOM. I think it is now clear that
because of the developments mentioned above, as well as Reverse Mathematics,
and also some other developments I have hinted at in earlier e-mail and on
the phone with some of you, that FOM is very much on track.    

Godel had explicitly complained about the lack of achievement of
mathematical logic, and speculates that the future will be different. Now,
however, many of the most optimistic speculations of Kurt Godel in his
writings have been acheived or at least now look acheivable. 

9. Now I want you to imagine the following further acheivment. The
production of a coherent series of utterly transparent and highly valued
discrete/finite theorems representing diverse areas of mathematics that can
only be proved by going well beyond the usual axioms for mathematics as
represented by the large cardinal axioms. I will be making some status
reports on this goal in the weeks and months ahead. 

But let us now hypothesize that this has already happened (not yet). As a
mathematician, this is certainly something that pretty much has to be paid
attention to, since it threatens to be relevant, and one should at least be
aware of it in order to be on the lookout for possible missed opportunities.
But as a mathematician, the same can be said for a large number of
contemporary developments in mathematics. One always has to be on the lookout. 

However, as a mathematical intellectual, this is quite different. It
overhauls some of the most cherished beliefs about the nature of mathematics
- that the rules of the game are fixed while the challenge is one of
creativity and ingenuity within a fixed framework - and creates an at least
temporary crisis in understanding not unlike big events like relativity and
quantum mechanics in physics. It cannot be shunted aside with the
explanation that "one doesn't care about such generality and pathology." 

10. Some of you may now be so moved that you feel that such a (as yet)
hypothetical development would rank far higher than contemporary mathematics
as an intellectual acheivment, and can only be compared with the greatest
intellectual events of all time. Some of you may not go nearly that far. A
few of you may still put on your strictly mathematical hat and say - so this
kind of thing happens from time to time in core mathematics, and thus it is
simply ordinarily really good stuff. Aside from those few in the third
category, the important issue of contention may be: to what extent is this
goal realistic in light of the existing results? This is a matter that I
hope I can address in the near future. 

11. So now you know in some detail one of the biggest thrills for me in FOM.
The prospect of acheiving something that can only be compared with the
greatest intellectual events of all time. I simply don't see how to do this in
mathematics. Do you? 

There are other thrills of FOM for me that I will talk about later. Here is
one more thrill: its place in foundations generally. Lou writes: "I agree
Foundations (of mathematics) represents a coherent and precise style of
thinking ..." 

I claim something further: that Foundations generally; i.e., Foundational
Studies (Foundations of Subjects) represents a coherent and precise style of
thinking. And that the crown jewel at this time, the paradigm case, is FOM.
It is the example to emulate, to learn from, to work towards elsewhere in
other myriad contexts. Indeed, I fully intended - in fact promised myself -
to move on to Foundations generally. However, developments in FOM have
literally forced me to stay put - at least for the near future. - HMF