# FOM: the role of informal concepts in evaluating f.o.m. research

Stephen G Simpson simpson at math.psu.edu
Mon Aug 31 22:40:52 EDT 1998

```Joseph Shoenfield 31 Aug 1998 11:32:27 writes:
> Let me repeat why I think vague intuitive concepts are unsuitable
> for fom.  I think the important results in fom are mathematical;
> that is, they consist of definitions, theorems, and proofs which
> are precise and rigourous is the sense of mathematics.  Thus we can
> use intuitive concepts only by replacing them be precise concepts,
> or, as Harvey prefers to say, by analysing them in a rigorous
> manner.

OK.  For the sake of argument, let me temporarily concede your point
that all of the important results in f.o.m. are rigorous mathematical
results.  Nevertheless, I would still maintain that informal or
intuitive concepts are relevant and even essential to the process of
*evaluating and formulating the interest and importance of* results in
f.o.m.

This is quite consistent with the way things are usually done in
mathematics.  For example, if you read reviews or survey papers in
mathematics, you will find evaluative statements along the lines of
"theorem X is important because it shows that phenomenon Y can be
understood from a purely algebraic standpoint", or "theorem X is the
most important discovery in subject Z in the last 20 years".  And
these statements will be backed up by detailed mathematical
explanations.  Note however that all of the key concepts in these
statements are informal and intuitive, i.e. vague, to use your
pejorative terminology.  I'm talking about concepts like "important",
"understood", "purely algebraic standpoint", "phenomenon Y", "subject
Z", etc.  Do you think we ought to dismiss these thoughtful
evaluations without a hearing, simply because informal or intuitive
concepts appear in them?

I said that Harvey's independent statements are qualitatively
different from Con(ZFC), because they are of a finite combinatorial
nature.  If you dismiss this out of hand on the grounds that "finite
combinatorial" is an informal concept, then I submit you are missing
out on something.

> I disagree totally with your statement that in normal mathematical
> usage, the term "combinatorial" already has a component of
> understandibility.

Maybe I didn't say what I meant.  What I meant was that, in normal
mathematical usage, the term "combinatorial" includes a component of
"understandable in combinatorial terms".  Just as "geometrical"
includes a component of "understandable in geometrical terms".  For
example, if we speak of a geometrical explanation of some phenomenon,
what we mean is that the explanation makes use of geometrical
intuition and is understandable in geometrical terms.  This kind of
talk is informal and intuitive, but it is nevertheless reasonably
precise and scientifically valid.

> You say I could find this out by trying to convince the editors of
> the Journal of Combinatory Theorem that statements like ConZFC are
> combinatorial and therefore would fit perfectly in their journal.
> I believe they woud reply that ConZFC may be combinatorial but that
> it does not fit in their journal.

They might reply that way out of politeness.  However, they would
privately be thinking something along the lines of: "Doesn't this dude
have a clue as to what combinatorics is all about?"

> I will agree that in some sense, Harvey's principal is an
> understandable f.c.p and ConZFC is not.

OK, good, I'm glad you agree that there is at least some sort of
difference between Harvey's independent statements and Con(ZFC).

> Whether this is serious progess in f.o.m. depends on whether this
> notion of understandable f.c.p. is a significant notion for f.o.m.

The notion of f.c.p. (or understandable f.c.p. in your terminology) is
one of those intuitive, informal notions which, although non-rigorous,
are of tremendous significance *for evaluating progess* in f.o.m.

> ConZFC is pi-0-1 and Harvey's principle is not;

Wait a minute.  Some of Harvey's independent statements *are* Pi^0_1.

> This is what I meant in a earlier communication, where I suggested
> that Harvey's result could lead to a sort of analogue of the
> Martin-Steel theorem.

I'm sorry; I didn't follow that remark.  What is the Martin-Steel
theorem, and what analogue did you have in mind?

> strong meanings of approval or disapproval without indicating in
> any way the reasons for approval or disapproval.  A good example is
> the golden in golden opportunity, which also seems to have upset
> Martin a litle.

When I spoke of a "golden opportunity", I did indeed want to convey my
approval or positive evaluation of the opportunity in question.  And
I'm pretty sure I explained my reasons for that evaluation.  I'm sorry
if the word "golden" caused Martin to get upset, but I feel that there
was no real need for him to get upset.  After all, he himself has
excellent judgement and has seized many golden opportunities when they
presented themselves.

-- Steve

```