FOM: FundamentalConcepts/Analysis

[Harvey Friedman]

This is a reply to MartinD 1/19/98 12:07PM.

You know, MartinD, I can't tell if there is any substantial disagreement
between us about these matters we have been writing about to each other.
However, I am still interest in finding out what, if any, is the
disagreement.

As I said in 10:11AM 1/19/98, I originally reacted to your statement of
5:23PM 1/15/98:

>In my judgement this [Weyl's work on Riemann surfaces] is not atypical of
>advances in fom.
>Rarely is it a
>matter of an investigator setting out to analyze some fundamental concept.
>Rather in the process of working on a ***TECHNICAL PROBLEM***, the
>investigator is
>forced to dig deeper, and results may be obtained that compel new
>understandings.

I was moved to reply because of your phrase "technical problem" (emphasis
mine). In you later posting, you instead used the following terminology 2-6:

1. technical problem.
2. foundational problem.
3. foundational problem.
4. definite foundational problem.
5. problem.
6. real foundational problem.

For me, there is an extreme difference between 1 and 2-6 (except that 5 is
quite indefinite). And one of the reasons I am writing again about this is
that you have not retracted the use of the word "technical", or
alternatively, explained your use of "technical". Indeed, I had expected
that you would have retracted your use of the word "technical" in your
1/19/98 posting.

>I understand Steve to be proposing to DEFINE f.o.m. as consisting
>of of investigations in a fundamental manner of what he takes to be truly
>basic mathematical concepts. Without really disputing this (and generally
>agreeing about what belongs on the list), I felt that putting it that way
>does not do justice to the history.

This does some justice to some history. But I think that one can
systematically and deliberately engage in such fundamental investigations
fruitfully despite any patterns of any kind in history. I for one feel very
comfortable doing a systematic investigation of basic mathematical concepts
in a fundamental manner, and expect to get more seriously and
systematically involved in the future than I have been in the past. As you
know, most of my work has been in more classical f.o.m.

In particular, I strongly believe in raising the quality of our
understanding of basic matters even in contexts where there is no pressing
"need" for raigin the quality. It's simply intrinsically important. Quality
is important even when it is not a crisis. I don't feel constrained by any
history. E.g., I don't sense any pressing need by the vast majority of
people who do classical mechanics and applications to have any conceptually
coherent formulation of classical mechanics and its applications - at least
none that they acknowledge. Yet even this part of physical science is, by
my standards, a disgusting pig pen, conceptually. I would like to clean
this up.

>Again and again, progress is made in
>deeper analysis of fundamental concepts when such analysis is necessary in
>working on a problem.

The kind of "problems" that are relevant to this context are very very
special indeed. Just random even famous problems are in general totally
irrelevant. ANd by default, given a fundamental mathematical concept C,
there is always the "problem" - what does C really mean? Can we formlaize
C? Can we formalize C in a manner that is intellectually illumiinating?
That creates new subjects? If that's what you mean by a "problem", then
what you say is unassailable.

On the other hand, if you are saying: well, let's continue the study of the
hemi-demi-semi-femi-zions of the demi-quasi-remnons in order to solve the
humpty-dumpty-dalory conjecture of Mr. Silly-Billy Willy, since nobody's
been able to do it for over 10 years, and there is a stumbling block, and
there is grant money in it, and "let a thousand flowers bloom" becuase
that's where the unexpected important discoveries come from - well, you've
got a major disagreement with me of the largest proportion.

>Often the problem is foundational in nature: a
>contradiction, or ambiguity in existing methods and approaches calls for
>deeper analysis. Investigators begin with such a problem AND IN THE PROCESS
>OF RESOLVING IT are led to a deeper analysis of some concept. This has been
>true throughout the history of mathematics. The sophisticated Eudoxus theory
>of proportion (which almost anticipates Dedekind cuts) incorporated into
>Euclid was a response to the problem of incommensurability. My previous post
>contained a number of examples.

Now that's more like it!!. However, these are not "technical" problems in
my understanding of the word. If you are saying that we should concentrate
on such "foundational" problems rather than "technical" problems, then we
agree completely.

>It's "OK" to do whatever. We're free people. It's
>certainly neither "necessary" nor IMHO desirable in the least "to ignore
>foundational and philosophical issues when conducting research in logic". If
>the history has a moral at all, it is just the opposite: LOOK FOR
>FOUNDATIONAL PROBLEMS AT THE EVER SHIFTING EDGE OF THE DOMAIN WHERE
>MATHEMATICAL CLARITY HAS BEEN ACHIEVED. It is not that philosophical issues
>should be "ignored" but just the opposite. It is researchers who have the
>imagination to think in terms of deeper analysis of basic concepts and the
>ability to carry through such fundamental analyses that will mmake progress.

Now you're talking!! Great!!!

>Used? Who me? Harvey, there is a flavor of orthodoxy about this that bothers
>me.

I wouldn't be writing this way if it were just us, but there are over 280
looking at this, and, frankly, a lot of them have barely heard of you and
me, and certainly a minority of them know anything about what you or I
think.

>I've heard it before in very different contexts.

Are you referring to my orthodoxy or other's stating some similar orthodoxy?

>I've said what I think
>as clearly as I know how.

Well, you could explain the original use of the word "technical" or disavow it.

>Anyone who's read my postings and lots that I've
>written over the years will surely know that, while I admire applications of
>logic to other parts of mathematics (I hate that word "core") deeply, I also
>have the greatest admiration for the powerful applications to f.o.m. Others
>are free to "use" what I've written any way they please.

I personally don't assume that anybody has read anything I have ever
written, or even more than a fraction of my postings.

>What are you afraid of?

Where did that come from? I'm always afraid that genuine f.o.m. will not
survive the present intellectually backward period we are going through.
Hence the fom. Yes, MartinD, I know you are one of the good guys, and you
have my respect.