FOM: What is FOMT?

[Harvey Friedman]

I want to begin weighing in on the question: what is foundations of
mathematics? As I indicated earlier on the fom, I am now fond of talking
instead about the potentially wider topic, FOMT = foundaitons of
mathematical thought. The benefit of talking about FOMT instead of FOM is
that it reflects the undisputable fact that mathematical thought is much
broader than the practice of mathematics - either professional pure or
professional applied mathematics. And as I have said earlier, most
mathematical thought is done by nonprofessional mathematicians. And I would
even go so far as to say that most original mathematical thought is
probably done by nonprofessional mathematicians, in their particular
extramathematical settings where mathematics is used in a variety of ways.
But of course, virtually all original mathematics is done by professional
mathematicians. So there is some serious distinction here.

When pondering the difficulties in getting a good clear handle on what FOMT
is, I realized that there are two quite different kinds of FOMT. And it is
very easy to combine them inadvertently. I think they should be separated.

1. There is FOMT as a special case of FOX, where X is potentially virtually
any subject - or at least a wide range of subjects. In fact, X may be, more
generally, a mode of thinking or mode of human activity. In this kind of
FOMT, mathematics and mathematical thought are treated as just another
subject and way of thinking or intellectual activity, like many others;
e.g., statistics and statistical thought, or probability and probabilistic
thought, or mechanics and "mechanics thought" (although that's an awkward
term), or music and musical thought, etcetera. Here aspects that are
relevant to any X are addressed systematically.

E.g., the emphasize is on conceptually clear systematic accounts of what is
asserted in the subject, how findings are verified or proved, etcetera. Or
more generally, (e.g., in music), a reference frame for the evaluation of
activity. In all cases, this is singularly reflective in a way that is
uncharacteristic of practitioners. It involves a thorough systematic
conceptual analysis into primitive notions, some of which may not be at all
obvious from the normal development of the field or the normal practice of
the activity. It is expected to include dramatic findings that arise out of
such systematic analysis - e.g., impossibility results concerning what can
be proved or established or acheived.

2. There is FOMT as foundational mathematics. This constitutes the actual
development and/or redevelopment of the subject matter that is driven by
connections with the rest of human intellectual activity, general
intellectual interest, and fundamental motivation. A principal aim is to
use this singular mode of development/redevelopment to give new treatments
of the subject matter that is far more accessible than what is done by
practioners. There is to be no reliance on special features and talents of
practitioners.

Of course, there are overlapping considerations between 1 and 2 and plenty
of cross currents. But, generally, 1 will emphasize such matters as:

a. mathematical proof
b. mathematical definition
c. mathematical modelling
d. mathematical existence
e. mathematical algorithm/computation
f. axioms and rules for mathematics
g. consistency
h. independence results
i. other impossibility results; e.g., noncomputability
j. features of proofs; e.g., elementary, constructive, length
k. alternative notions of proof; e.g., using computers, Monte Carlo
l. analysis of actual proofs and possible proofs of actual propositions.

And, generally, 2 will emphasize such matters as:

i. development/redevelopment of geometry and topology driven by the
concepts of shape and motion.
ii. development/redevelopment of analysis/differential/partial differential
equaltions that support quantitative descriptions of the physical world.

In each case, a proper treatment will depend on developments of secondary
subject matter that plays an essential intellectual role in the development
of the primary subject matter. E.g., linear algebra and group theory are
primarily secondary but intellectually essential.

In light of the richness of FOMT in all of these senses, it is more
rewarding and timely to be developing FOMT rather than reflecting on what
it is. But a certain amount of this clarification is definitely worth
doing. But a truly definitive treatment of what FOMT is or should be is
certainly going to have to wait until it is in a much more highly developed
state than it is now.