FOM: Riis:What is the motivation behind the Kreiselian school?

[Harvey Friedman]

**note: there are now 276 distinguised subscribers to this e-mail list on
foundations of mathematics!!**

This is the first of a two part reply to Riis, 6:02PM 1/5/98, What is the
motivation behind the Kreiselian school?

I want to go into detail into this important posting. Some of the people on
fom might wonder why I spend so much effort with the dialog with Lou. The
answer is that Lou represents a caricature of a bias among core
mathematicians against FOM, which needs to be analyzed, amplified,
scrutinized, and answered systematically and publicly for the good of FOM.
And Lou represents a tiny but significant minority of mathematical
logicians in this regard. Lou and this significant minority are no doubt
very talented mathematicians who have significant acheivments, and should
not be ignored. So I have rejected the strong suggestion by some colleagues
on the fom that carrying on this debate with Lou is a counterproductive
waste of time.

In this first part of the reply to Riis, I want to summarize my own view on
these matters. This summary goes off in a somewhat different direction than
my ealier summarizes of views on the fom.

1. There is a reasonably well defined concept of mainstream mathematics, or
core mathematics, which is intellectually challenging and richly rewarding
for its practitioners. And there is substantial interaction between the
participants at a deep and practical level, even if their work concentrates
on quite different aspects of core mathematics.

2. In particular, the structures considered, and the kind of problems
addressed, have characteristic features which are instantly recognizable to
the practitioners and, to some extent, to informed outsiders. This enables
the practitioners to instantly sense whether a piece of mathematics, or
mathematical thought, is admissible as core mathematics.

3. There is a heavy emphasis on the way a problem is solved, often more so
than the problem itself. The effective use of substantial machinery is
particularly highly valued. In many instances, very famous old problems,
generally considered as lying in core mathematics, get solved - yet the
excitement generated and rewards given to the solver are surprisingly
limited. At least surprising to an outsider. This is a function of the way
that problem was solved. Such considerations lead one to model core
mathematics as more of an art/sport than a subject or science.

4. The interconnections and synergies and collaborations between core
mathematicians are at an incomparably more active and intense level than
those between noncore mathematicians and core mathematicians. In
particular, FOM is obviously not core mathematics. Mathematical logic,
which is a technical spinoff of FOM, is also obviously not core
mathematics. However, there are now a growing body of applications of
mathematical logic in core mathematics. These applications include   a)
different proofs of known core mathematics;  b) simpler proofs of known
core mathematics;  c) solutions to problems in core mathematics where the
logic is later removed;  d) solutions to problems in core mathematics where
the logic has not (yet) been removed.

5. Nevertheless, this growing success in applications is still much too
limited to elevate mathematical logic to the status of even "near" core
mathematics. And it is unclear how far it will go. Or whether it will have
a substantial effect on employability of mathematical logicians as a whole
or in part.

6. Because of the structure of Universities and related reasons,
mathematical logic and FOM find themselves housed in mathematics
departments, and consequently judged as mathematics. This causes a great
deal of misunderstanding. The judgement of mathematical logic and FOM as
mathematics inevitably fails to properly take into account the
distinguishing features of FOM that I have been discussing on the fom. This
is one reason why it is important to discuss these distinguishing features.
In this respect, the judgements by mathematicians are very uneven. Some
don't take these distinguishing features into account at all, whereas
others take them into account to some extent, and still others take them
into account very negatively! In any case, such matters - namely how to
properly judge mathematical logic and FOM - are not seriously (let alone
systematically) discussed among mathematicians.

7. There is much precedence for areas of mathematical science to be
originally housed in mathematics departments, where they suffered greatly
as a consequence of not being core mathematics. I'm thinking of computer
science and statistics. But this is, by all accounts, also true today of
various kinds of applied mathematics. I have heard many applied
mathematicians state that their approach to intellectual life also differs
greatly from that of core mathematicians, and claim to suffer from this
fact. They claim to be grossly misunderstood. They point to the fact that
they are better appreciated and valued outside mathematics. They often talk
of moving outside mathematics departments, and some of them actually do to
varying extents - e.g., joint appointments.

8. The question of where FOM and its technical spinoff, mathematical logic,
should be housed in a University is a critical issue that is not going to
be resolved quickly. At present, Universities are not equipped to handle
joint appointments very well, particularly at the junior level. And
mathematics, philosophy, and computer science departments seem to be moving
farther apart in intellectual outlook. So for the forseeable future, the
answer is: mathematical logic and FOM will be housed in mathematics
departments, where they will be largely misunderstood, and be forced to
swallow a relatively marginal status.

9. But there is a far superior solution to this problem in the long run.
The area called Foundational Studies will emerge, which cuts across all
disciplines in the University, with a unified approach to intellectual
life. This is where the realization of the Leibnizian unification vision
will take place. And FOM will have the leading role in Foundational
Studies, being the most well developed part of it, and serving as a model
of what can be acheived, and how it can be acheived. The quality of the
Foundational Studies School in a University will be the most important
determining factor in the status and quality of that University, both in
terms of scholarly activity and in terms of educational effectiveness.
Foundational Studies is also the place where new ways to organize and unify
knowledge are developed which radically change the effectiveness of
education.

10. All subjects, including FOM, core mathematics, theoretical physics,
etcetera, have to be ultimately judged in the general intellectual arena.
The biggest drawback of Universities today is the utter lack of presence of
an organized general intellectual arena. Instead, one has merely a
hodgepodge of Departments representing totally different approaches to
intellectual life, with little or no substantive interaction. In
particular, little or no communication and mutual understanding of these
myriad different approaches to intellectual life actually takes place.

11. The importance of contemporary core mathematics, as well as
contemporary FOM, needs to be argued. Taking the importance of core
mathematics for granted is no more appropriate than taking the importance
of contemporary FOM for granted. I think that some of my colleagues on this
list think that since FOM = foundations of mathematics, it is obvious that
the importance of contemporary FOM is tied to the importance of
contemporary mathematics, and should be judged primarily by reference to
what it does for contemporary mathematics - whose importance is to be taken
for granted without explanation. This is wrong on two separate counts.

12. Firstly, this is like saying that one judges contemporary theoretical
physics according to how it helps us to build airplanes; no - how it helps
us to build contemporary airplanes, not the airplanes of yesteryears. Well,
not much of contemporary theoretical physics actually helps us with the
latest designs. One actually goes out and tests designs, and performs
computer simulations, etcetera, and doesn't worry too much about string
theory or quark theory. Or, for that matter, cohomology.

13. No doubt, one can "expect" and "hope" that in the long run, major
revelations in theoretical physics will help us to build then contemporary
airplanes. But it would be absurd and counterproductive to prematurely
expect this of theoretical physics. Similarly, it would be absurd and
counterproductive to prematurely expect large cardinals to be necessarily
used in order to solve basic open problems of number theory.

14. Secondly, FOM is really best thought of as: foundations of mathematical
thought. This is a much better name. This makes it clear that FOM is not to
be thought of strictly as: foundations of contemporary mathematics. In
fact, let me interweave this point in the context of the arguments on fom.

15. It is clear that Lou and also even more explicitly Pillay want to argue
that FOM should be judged by what it says about contemporary core
mathematics and not about, say, classical mathematics. Also, it is clear
from the Riis posting that Kreisel and Macintyre also want to judge it this
way. And, of course, those who seek to judge it this way judge it
negatively.

Well, I think this is a mild case of the theoretical physics/new airplanes
situation in 12 above. FOM hasn't even scratched the surface of saying what
it can say about very clasical mathematics. And thinking of FOM as FOMT =
foundations of mathematical thought, emphasizes these facts:

a) an enormous amount of mathematical thought is going on as we speak;
b) the overwhelming majority of it is not done by professional mathematicians;
c) the majority of it is not even done by academics;
d) the majority of it that is done by academics is not done by professional
mathematicians;
e) almost all of it is within very classical settings.

16. With these considerations in mind, I officially propose a name change
of FOM to FOMT.