FOM: FundamentalConcepts/Analysis (reply to Harvey's reply)

[Martin Davis]

At 10:11 AM 1/19/98 +0100, Harvey Friedman wrote:

>The original quote that caused me to get into this matter was this:
>
>>>Martin Davis wrote on 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.
>

>So I think the source of any disagreement between us may reside in your
>original use of the term ***TECHNICAL PROBLEM***. To see if there really is
>any disagreement at all between us, please clarify what you mean by:
>
>1. technical problem.
>2. foundational problem.
>3. foundational problem.
>4. definite foundational problem.
>5. problem.
>6. real foundational problem.
>

Homework! I don't really want to go back to my old postings. Here is what is
involved: 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. Again and again, progress is made in
deeper analysis of fundamental concepts when such analysis is necessary in
working on a problem. 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.

>
>I want to talk about the motivation for me responding to your original
>1/15/98. Since you used the phrase ***TECHNICAL PROBLEM*** I felt that you
>might be saying or at least implying something like this:
>
>it is OK, and maybe even necessary to ignore foundational and philosophical
>issues when conducting research in logic; instead, follow the ***TECHNICAL
>PROBLEMS***, for truly significant work in f.o.m. comes unexpectedly out of
>there.

As Huck Finn would have said: "That's a stretcher!" I neither said nor meant
any such thing. In conversation,  Post once complained that in his work on
CH, G\"odel had left behind the truly fundamental (Post's word) issues
opened by the incompleteness theorem. (To avoid misunderstanding, I'm not
endorsing this statement - just reporting.) Then he added, "But you have to
work on what you have to work on." [paraphrase by me - not claimed to be an
exact quote] Please! 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.

>You can be guaranteed that many people who make the awful mistake of
>treating mathematical logic as if it were only an imbred technical subject
>divorced from the great issues of f.o.m. will read your original 1/15/98
>that way. So will those like Lou who treat mathematical logic as if it were
>only an interesting tool for core mathematics. Do you want to be used in
>this way?
>
>
Used? Who me? Harvey, there is a flavor of orthodoxy about this that bothers
me. I've heard it before in very different contexts. I've said what I think
as clearly as I know how. 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. What are you afraid of?

Martin