FOM: F.O.M. and foundations

[Harvey Friedman]

This is a reply to Shoenfield 11:55AM 9/1/98. I intend to reply to the
other thread represented by the Shoenfield/Simpson interchange (Con(ZFC))
in a later posting.

>     We disagree on the truth-value of the statement "fom is mathematics";
>but this is less a disagreement over fom than a disagreement over the
>criteria for saying something is mathematics.

You have not addressed my claim that practically noone in (core) statistics
or computer science says or thinks that their subject is a branch of
mathematics, either. Many classifications consider mathematics, statistics,
and computer science "mathematical science."

You have also not addressed my claim that the extreme disadvantage to
calling fom or mathematical logic a branch of mathematics is that it looks
quite weak as a branch of mathematics compared to, say, number theory,
geometry, analysis, algebra, etcetera, especially in terms of interactions
with other branches of mathematics. And interactions is perhaps the most
highly valued criterion being used by mathematicians to evaluate branches.

>      My original point was that the really important results in fom are
>mathematical; that is, they consist of definitions, theorems, and proofs,
>all meeting the established standards of rigor in mathematics.

In your original posting, you omitted "definitions." Also, f.o.m. at its
highest level, seeks to understand mathematical proof and construction,
etcetera, and so inevitably requires a very refined judgement as to what
are the appropriate definitions. At the moment, not very much is known in
the way of a theory of what constitutes a good or important or incisive or
elegant definition. And in the case of f.o.m., the most common definitions
are the introduction of formalisms, including especially formal systems.

(This is, of course, one of many counterexamples to what you are arguing
about with Simpson in the other thread. All mathematicians are confronted
with this, and none I know have any intention of providing a formal
criteria for a good definition.)

In fact, a case can be made that an unusually large number of the most
important contributions to f.o.m. have been the correct definitions,
starting with Frege. This makes f.o.m. a lot more like the rest of
theoretical science and engineering than, say, number theory. This is a
major reason why it is counterproductive to refer to f.o.m. as a branch of
mathematics. Doing so also does not reflect on the special relationship
that the foundations of a field has to the field itself.

Shoenfield quotes me:

>>     >Some intuitive ideas may not yield to such analysis, but still may
>>be essential to consider.   On doesn't simply pretend that the concepts
>>don't exist.

>     Sounds good; but what should one do?   It is no use to just assert
>very strongly that the concept is important and that those who do not
>agree are obtuse.   Perhaps one should just put the concept aside until
>another day, as one usually does with problems one cannot solve.

Almost every mathematician every day - in fact, every intellectual - is
using such intuitive ideas every day. This is because, e.g., every
mathematician must decide what is worth working on, and what is worth
publishing, which inevitably involves such criteria as "is this elegant?"
"is this nontrivial?" "is this deep" "is this memorable" "is this a simpler
proof" "does this explain the situation" "is this really new" "does this
get to the heart of the matter" "is this interesting" "is this fundamental"
"is this way of doing it understandable" "is this proof better than that
one" "is this a direct proof"

These generic intuitive ideas also multiply when considering special
topics. E.g., "is this algebraic" "is this geometric" "is this
combinatorial".

Your word "obtuse" has an analog, as I have indicated previously, in the
case of color and pitch. A person is called "color blind" or "tone deaf."
There are also people who prefer snickers candy bars to 3-star French
restaurants (of course, the former is less expensive, which is a
preference).

To a large extent, the importancce of one's work depends on how good one's
intuition is regarding such matters. There is a severe limit to the level
of acheivment of people who have poor intuitions along these lines, but who
can construct complicated technical arguments.

>      Harvey disagrees with my statement that there are no significant
>results on foundations in general, but I do not find his remarks on this
>convincing.

In a unified approach to the foundations of subjects, basic constructions
in f.o.m. such as propositional calculus, predicate calculus, structures,
etcetera, certainly are fundamental. For instance, in the myriads of papers
on foundations of computer science, such things are now taken as given,
both in theoretical and applied computer science. These things are
routinely taught to students in computer science departments.

Perhaps I don't understand the question "is there any significant results
on foundations in general." Or: are there "any significant results on
mathematics in general?" What does "in general" mean in these question?

>The problem is that in each field, the substantial work on foundations is
>done by researchers in that field; and this is natural, since one cannot
>say anything substantial about the foundations of the field without a good
>understanding of the work that has been done in that field.

This is ambiguous. The statement "a lot can be said about f.o.m. without a
*good* understanding of mathematics" is true or false or unknown depending
on what *good* means.

Experts in fields are notoriously incompetent and notoriously uninterested
in doing foundational work of the appropriate kind, and I am convinced that
the appropriate foundational work will be done by people who are either
originally from other fields, or who are not working in the mainstream of
that field.

The current state of foundations of fields is such that the initial
advances in appropriate foundations will likely involve only consideration
of comparatively well understood and classical parts of the fields. Even
though f.o.m. is very highly developed and impressive, this is still true
of f.o.m. vs. mathematics. On the other hand, foundational work involves an
incomparably more penetrating understanding of the basic material. In fact,
it should involve a major reworking of the basic material with new
emphases.

Today, computer science is the one area where foundational work of roughly
the kind we are talking about goes on all over the map. It is of course
very unsettled and rapidly evolving. Much of the stuff is so new and
fertile that people can get into this kind of foundational work from areas
outside computer science often with very little difficulty. They do not
have to be steeped in years of conventional developments.

> The result
>is that the work on foundations will be done using the tools of the
>particular subject.

This is false or misleading as indicated above by the computer science
example. And Frege's fundamental work on foundations hardly relies on
serious tools of mathematics.

>I think Harvey's dream of departments of foundations
>appearing in academia and taking over thw work of foundations of
>particular fields could only happen in an alternate universe.

It can be said that an intellectual revolution is the creation of an
alternative universe. If you shared my vision and ideas for doing this, you
would be working on it.

>     Harvey seems to have no strong objections to my analysis of the
>achievements of reverse mathematics.   He says I should amplify my
>statement that the next step should be to prove mathematical theorems
>about these concepts (the five basic theories).  Well, the theorm which
>says that these theories are linear ordered has the right form, but has
>two disadvantages: the proof is trivial and uniformative, and the
>foundational significance of the theorem is not clear.

And what is your formal definition of "foundational significance?" Do you
need a formal definition here, as you often indicate?

>I am sure Harvey
>agrees that a theorem which people would agree explains this phenomena of
>linear ordering would be a jewel of reverse mathematics.

I emphasized this already in my ICM talk in 1974, and every day since then,
and many days before then. An appropriate explanation would be a major
jewel of f.o.m. But if this is the standard for judging the importance of
reverse mathematics, then it is obvious that almost nothing important has
been done in logic in general.

>     I am sorry that I must disagree with Harvey's statement that it is
>inconceivable that the idea of reverse mathematics is not a permanent part
>of fom.   Improbable perhaps; but not inconceivable.

>If the researchers
>in reverse mathematics II do not obtain results at least as interesting as
>those in reverse mathematics I, the subject will disappear from fom.

This is false. There are hundreds of interesting situations at the heart of
classical mathematics to be explored with reverse math I, and a successful
analysis of these situations is of obvious permanent value - simply as a
consequence of the robustness of classical mathematics and RCA_0. You can't
erase a robust claslsification of classical objects. I don't know of a
single case of this.

>Researchers may continue their work, but they will be a lonely crowd.

Godel was more lonely. With Simpson's overdue book coming out, reverse math
will grow in terms of the number of people.