FOM: FundamentalConcepts/Analysis (Do Harvey & I really disagree?)

[Martin Davis]

At 09:23 AM 1/20/98 +0100, Harvey Friedman wrote:
>
>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.

For what it's worth, I think that, what disagreement there is, is POLITICAL
rather than mathematical, f.o.m.-ical, or philosophical. You see enemies out
there who have to be defeated so that the OBJECTIVE (Steve's word) TRUTH can
prevail. You are fighting the good fight against the philistine
anti-intellectual atmosphere you (correctly - in my judgement) see all
around us. My belief is that this is a mistake. We could only hope to FORCE
the truth on people if we had a coercive apparatus at our disposal, and this
bloody century has seen quite enough of that! The only real way to fight the
Zeitgeist is the way G\"odel did. Produce unimpeachable TECHNICAL results
which serve to refute the dominant current of ideas. And my God, you of all
people are doing more of that than anyone else. So keep at it, keep telling
us about it, and stop tilting at windmills. 

> 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.

OK. I surrender. I used the word just to mean: involving concepts and
methods which only trained experts know. There is nothing to prevent some
foundational work from being highly technical. The best of it is.

>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.

Sure. It was even one of Hilbert's problems. Have you considered the
possibility that there may be an intrinsic reason for the "pigpen" because
the real world isn't classical? In any case, more power to you! 

History does suggest, and simple reason confirms, that the emergence of
paradox helps to point the way. 

>
>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.

I'm too ignorant about so much of current mathematical research to be able
to make serious judgements about the ultimate value of much of it. But if by
some chance you're talking about FLT, I find what happened there really
exciting although not relevant to f.o.m. (A famous problem, not very
interesting in itself, became approachable because of deep mathematical
advances of general interest.) 

>>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.

As indicated, I don't see the contradiction. 

>>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.

"it" [that is "orthodoxy"]: think for example of "dialectical materialism"
and the claim to possess the key to what is "objectively" true. (I didn't
mean to be obscure - just didn't want to go too far afield.)
>
>
>>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.
>

You certainly have mine. Remember: "This too shall pass!" 

Martin