FOM: FundamentalConcepts/Analysis

[Harvey Friedman]

This is a reply to Martin Davis, 10:02PM, 1/18/98.

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.

The capitalization and *'s are mine for extreme emphasis. This is what I
was responding to. In your 1/18/98, MartinD writes:

>But what HE SET OUT TO DO was to try to solve a ***FOUNDATIONAL
>PROBLEM***: the contradiction between Newtonian physics and Maxwellian
>electromagnetism.

You were referring to Einstein. Again the capitalization and *'s are mine
for extreme emphasis.
More 1/18/98:

>I never said that analysis of fundamental concepts wasn't a crucial element
>of past work in fom; I'm sorry to have been so misunderstood. My point is
>only that really important work has tended to begin by trying to resolve a
>**********foundational********** PROBLEM.

I put 20 *'s there for extreme emphasis because you had already capitalized
"problem" for your emphasis. More from 1/18/98:

>Who on the fom list
>would deny the importance of deep analyses of fundamental concepts? My point
>was that in the great majority of cases such analyses are carried out in the
>course of solving a ***DEFINITE FOUNDATIONAL PROBLEM*** rather than
>initiated on
>their own.

And more from 1/18/98:

>I count this one on my side. The ***PROBLEM*** of how to resolve the
>contradiction
>between the cumulative hierarchy seeming to go on forever and the desire to
>have a coherent general concept of set, the paradoxes again (as you
>emphasized in a post addressed to Lou), is perhaps the most important
>***PROBLEM*** in fom today. Can you honestly say that you had none of this
>in mind
>when you did this work?

(NOTE: Steve will have a fit over your "incorrect" use of fom which should
have been f.o.m.)

And even more from 1/18/98:

>This would certainly resolve a ***REAL FOUNDATIONAL PROBLEM*** if it could be
>carried off. (I thought about this myself some years ago.)

This was in response to my bringing up Church's thesis.

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.

in these 6 quotes.

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.

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?