[FOM] FOM currents

[Harvey Friedman]

There has recently been a large increase in the number of postings from
various thinkers, most of whom do not work in f.o.m. or mathematical logic.

I think that it is useful to classify the main themes of these postings into
two groups. Such a broad classification, of course, cannot do justice to the
many subtleties involved.

1. A preoccupation with issues in the philosophy of language, confusing them
with issues in the foundations of mathematics.

It would help the FOM list by thinkers starting with the nature and
structure of mathematical thought - taking into account important features
that are second nature to competent precollege students. This is much more
appropriate than starting with features of natural language, and forcing
them on mathematical thought.

Issues such as "can they in fact be forced on mathematical thought" or "can
we prove that they cannot be forced on mathematical thought" are of course
the domain of f.o.m. professionals.

2. Claims that f.o.m. should be pursuing substantially different directions
than it does now. 

There are all sorts of new directions for f.o.m. known to many f.o.m.
professionals. Some of these emerge from time to time - but generally only
when something significant is done with them. The overwhelming majority of
ideas for expansion of f.o.m. lead nowhere, or at least lead nowhere for
very substantial periods of time.

But many of these suggested topics that have recently appeared here are not
aimed at the basic features of mathematical thought that are of the highest
general intellectual interest. Instead these suggested topics are aimed at
much more special features of contemporary mathematical thought, that are of
interest mainly to specialists in certain areas of mathematics, and are
difficult to properly explain to the kind of wide audience that f.o.m. has
been spectacularly successful in addressing. Furthermore, and most
importantly: there were no indications provided of substantial lines of
attack of a foundational nature.

I further divide these posters (writers) into three OVERLAPPING groups.

2a. Those who make these suggestions gently, without any visible contempt
for contemporary f.o.m.

2b. Whose who make these suggestions with varying degrees and shades of
disrespect and contempt for contemporary f.o.m.

2c. Those who wish to deny the glaring distinction between mathematical
thought of a foundational nature, and "normal" mathematical thought.

I make the following points.

A. F.o.m. is such a subtlety difficult matter, that it only really came into
being in the late 1800's and early 1900's, despite an enormously successful
development of mathematics over thousands of years, and a tremendous
development from the time of the calculus onwards.

B. At the present time, f.o.m. has barely scratched the surface of the
surface of the most obviously profound, significant, and deep issues about
the nature of mathematical thought, of obvious general intellectual
interest. 

C. Over the last 100 years, there has been an entirely orderly and
appropriate expansion of features of mathematical thought addressed by
f.o.m. in an effective, and sometimes spectacular, fashion.

D. This orderly expansion of features addressed by f.o.m. takes place in
tandem with the gradual expansion of knowledge, experience, insights, and
tools developed in f.o.m. and mathematical logic.

E. One must be aware of the minimal prospects for doing serious foundational
work concerning more specialized and focused matters, before the proper
groundwork is laid with matters of a more basic nature.

F. Consequently, armchair suggestions as to what f.o.m. should instead be
doing are unimpressive unless accompanied by some striking insights that
would indicate prospects for success - e.g., how generally to proceed.

G. I have not seen any even unstriking insight that would indicate how,
generally, to proceed, on any foundational issue, by thinkers in category 2,
in their postings. 

H. If I ever see any even unstriking insight, stated in clear nontechnical
terms, indicating how, generally, to proceed, on any foundational issue
related to mathematics, whether or not it is on the "orderly schedule of
f.o.m." as referred to in C,D above, then I, for one, will pay serious
attention to it.  

I. For H, it does not count to merely give some reference to the literature
of such an insight. It should still be encapsulated in some effective way
right here on the FOM list, in order to save time and effort.

J. I believe that it is highly unlikely that any given foundational issue of
general intellectual interest, concerning mathematical thought, can be dealt
with in a truly profound and effective way - meeting f.o.m. standards!! -
without substantial use of current f.o.m. perspectives, developments, and
methods. 

K. In those unusual exceptions to J, it is virtually certain that the
foundational development would then be greatly expanded through use of
current f.o.m. perspectives, developments, and methods.

Some additional comments:

The FOM is a great resource that has not nearly reached its potential. There
are many students across the world who subscribe and/or read the FOM
Archies. 

Because of their lack of experience, students can not be expected to be able
to tell what is of value on the FOM.

Consequently, the vicarious thrill that may be associated with making grand
declarations of a negative nature one is not prepared to defend, should be
avoided, especially by those with no significant relevant track record.

Harvey Friedman