FOM: FundamentalConcepts/Analysis

[Martin Davis]

At 02:31 AM 1/18/98 +0100, Harvey Friedman wrote:

>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. Examples: Cantor's work on trigonometric series, led him to
>>transfinite iteration of the process of forming the derived set of a set of
>>points and thus to developing set theory. G\"odel's fundamental
>>contributions occurred in the context of very specific problems that had
>>been set by Hilbert.
>
>>Frege is the one contributor to fom I can think of who really did proceed by
>>setting forth to analyze concepts (logical reasoning, cardinality)
>
>The point of this posting is to disagree with the spirit - if not the
>letter - of this writing of Martin Davis. OK, maybe it's a bit rare, but
>there is a lot to do of this kind, and I have have done some - even
>recently. The key phrase is: RARELY IS IT A MATTER OF AN INVESTIGATOR
>SETTING OUT TO ANALYZE SOME FUNDAMENTAL CONCEPT.
>
>1. Frege is obviously an example like Martin says. And this is a great big
>giant example. This reminds me of discussion I have had with many
>physicists over the years about foundational work in physics and Albert
>Einstein. Most physicists think that foundational and philosophical
>reflection on physics does not lead to important advances in physics. But
>when I mention Einstein and his thought experiments, they immediately say -
>"OK, you're right. But that's the exception." Some exception. I have
>trouble when the exceptions are virtually the greatest and most admired and
>influential examples.

Einstein is an excellent example of what I mean (although of course strictly
speaking it's not fom). Of course, Einstein's brilliant insights resulted
from thinking in a profound and fundamental way about such matters as
simultaneity. But what HE SET OUT TO DO was to try to solve a foundational
problem: the contradiction between Newtonian physics and Maxwellian
electromagnetism. It was in the process of thinking this through that he was
led to rethink fundamental concepts. (And of course it certainly helped that
he had read Ernst Mach.) But note my emphasis: it wasn't that he decided
that simultaneity had not been sufficiently studied on a really fundamental
level and proceed from that. He solved a crucial puzzle: an apparent
contradiction between the two key physical theories of his day.


>2. What about Frege's "system", which was inconsistent? OK, it didn't work.
>But what about Russell's type theory and various fixes? What about
>Zermelo's set theory? What about Frankel's replacement axiom? What about
>Turing's model of computation? What about Aristotle's syllogisms? And what
>you say about Cantor is misleading. OK, a case can be made that Cantor's
>work on trigonometric series led him to develop set theory. Even here, I am
>a little bit skeptical - who says he wouldn't have developed it anyways,
>and that was just a spark, replaceable by other sparks? But I don't want to
>labor that speculation particularly. What I do want to labor is that once
>Cantor got going, he was obviously propelled by the subject itself.
>Including analyzing concepts such as equinumerous and infinite and finite.

Let's see; this is a loaded paragraph - where to begin. I already mentioned
Frege. His contributions were brilliant and they were motivated as far as I
can see by seeking to provide a deep analysis of fundamental concepts. So,
agreed, one for your side. As far as the inconsistency goes, George Boolos
and others have largely rehabilitated his system. Frege would likely have
seen it himself if providing a DEFINITION of the cardinal numbers that was
philosophically adequate hadn't seemed to him to be so crucial. [What
suffices, in place of Frege's attempt to provide such a definition, is the
second order "Hume's Principle" which asserts, in effect (that is allowing
for the fact that Frege didn't talk set-speak), that sets in one-one
correspondence have equal cardinal numbers.] So if I wanted to be a real
debator, I could even argue that if Frege had been less eager to provide an
analysis for a particular fundamental concept, he could have saved his
system and not died a bitter man convinced his life had been a failure.

Russell, Zermelo: Come on Harvey, you know better. It's a commonplace that
they were trying to figure out to avoid the paradoxes, to resolve a
contradiction, just like Einstein. Replacement? Needed to solve a definite
problem: how to get past V-sub-(omega+omega). Turing's model of computation?
He was trying to solve a problem: the Entscheidungsproblem. He didn't start
out trying to analyze computability; it occurred to him (a brilliant
thought) that maybe there was no algorithm and maybe that could even be
proved. But in that case, algorithm needed to be defined. Aristotle? You may
well be right there - I just don't know enough about it; perhaps some of the
philosophers on the list could comment. As for what you say about Cantor:
the history of how he came to think about set theory and how slowly he moved
from analysis to general set theory is perfectly clear. Whatever does it
mean to say that maybe he would have done it anyway? I'm talking about what
HAS happened. When he was much younger, I heard an important philospher give
a lecture in which he talked about the possibility of the ancient Greeks
having developed modern logic. I thought this was silly then, and I still
do. It's hard enough in the history of ideas to understand what DID happen
without wandering off into what might have happened.

>3. OK, with Godel you have a case. His great work normally is not normally
>considered to involve axiomatizations or conceptual analysis of fundamental
>notions. But I'm not so sure. There is the case of the constructible sets
>and also of ordinal definability. A good case can be made that
>constructible sets is a profound analysis of a more concrete notion of set
>than the usual informal notion. And if you read his Princeton Bicentennial
>lecture in the Godel volume, you see that ordinal definability was
>explicitly proposed by him as an analysis of a notion of absolute
>definability which is language independent. So you're not really correct
>even about Godel.

>I would even go so far as to say that he very likely did a lot of private
>fundamental analysis of the liar paradox and related matters, and so was in
>a position to get the incompleteness theorems much more easily than his
>contemporaries. And his incompleteness theorems are infinitely inspiring -
>which is a good deal more inspiring than any of his likely private
>fundamental analysis of the liar paradox. Does this count? In fact, I have
>a suspicion that Godel may have been perpetually engaged in analyses of
>fundamental concepts all his career, and was only really satisfied with the
>consequences of that analysis for f.o.m.!

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. As in the examples above. It is in the PROCESS of
working on such problems that the great innovators in fom find themselves
led to resort to such analyses. G\"odel's own account of how he was led to
the incompleteness theorem is quite different from what you suggest. He was
working on a problem: specifically Hilbert's second problem which G\"odel
understood as the problem of the consistency of second order arithmetic. I
discuss this at length in a review of Dawson's wonderful recent biography of
G\"odel for Philosophica Mathematics (to appear soon). Here's an edited
quote from my review:

"The task he set himself was to prove the consistency of A [second order
arithmetic] on the assumption that first order arithmetic (which, as usual,
we refer to as PA) is consistent; in other words he wished to show that
second order arithmetic is consistent relative to first order arithmetic. He
proposed to do this by constructing a model of A within PA. This amounted to
finding a relation eps that is definable in PA such that the comprehension
principle
            (En)(Ax)[ x eps n iff phi(x)]
holds for each formula phi of A. G\"odel saw that if phi(x) is replaced by
[phi(x) is provable in A], such an eps would be readily obtainable. On the
other hand he had also seen that the notion of truth in A was not definable
in A and  hence certainly not in PA. Hence, the notions provable formula of
A and true formula of A could not be equivalent. So, assuming that only true
formulas could be provable, there would have to be true formulas of A not
provable in A. This is how G\"odel discovered his incompleteness theorem.
However, he realized that as thus formulated, his result would not be
persuasive to [his positivistically inclined colleagues] for whom any notion
of mathematical truth other than provability was simply meaningless. In its
published form, no traces of the means of discovery were present. Instead
the exposition was meticulous and detailed so as to convince any careful
reader."  

Constructible sets? Now he was working on another PROBLEM: CH (Hilbert's
first problem). [It's remarkable how much of G\"odel's early work was a
specific reponse to problems Hilbert had posed.]

I think you're right about ordinal definability. It seems G\"odel was struck
by the success of the Church-Turing analysis ("a kind of miracle") and
simply wanted to see whether a similar success could be had for
"definability". Another for your side! [By the way Post had very much the
same idea and came up, independently with the same notion.]

>Aristotle, Frege, Einstein, Godel, Russell, Zermelo, Frankel, Turing - what
>do you think of the quality of the exceptions, MartinD? I think these
>people are pretty well known.

Did I really do such I poor job of explaining myself? 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. I think I've made that case with your examples. Except for
Aristotle these are people whose work I've studied with fascination since I
was a boy, and they are very much the people I had in mind when I wrote my
earlier piece.

>4. One of the things I have done recently is try to say something new about
>the murky concept of "predicates and functions on absolutely everything." I
>picked a test problem for such a theory. What sentences in predicate
>calculus have a model whose domain is absolutely everything? I gave a
>plausible set of elegant and fundamental axioms about "absolutely
>everything" which formally determines the answer to this problem. And I
>show that my solution is the best possible solution in some particular
>sense. Does this count? The fundamental axioms imply, e.g., that there is
>no linear ordering on absolutely everything. This is on my website.

I think this is a beautiful result, and certainly counts as one where
analysis of a concept came first.

>5. Another thing I have done recently is to reaxiomatize set theory in a
>new way based on the idea of two (or more) interacting worlds. The
>axiomatizations are very simple, and seem to have general philosophical
>meaning. Simple natural extensions of them reaxiomatize various large
>cardinal axioms. They give a new kind of uniform treatment of large
>cardinals which, at the very least, states them in much simpler and
>uncluttered terms, and holds out the promise of a new fundamental
>philosophical theory that may reveal what is really behind them. Does this
>count? This is also on my website.

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?


>6. Finally, there is my work on transfer principles. Everybody has the
>feeling that in some way, one should be able to pass from the truly
>innocent principles of hereditarily finite set theory to transfinite set
>theory. After all, from a formal point of view, only the axiom of infinity
>is new. But to pull this off would require a careful analysis of what kind
>of statements transfer. There is some concept of "transferable" statement.
>I am optimistic about this program, but I already had a lot of success with
>a related program. I worked out very simple classes of statements about
>functions of several variables on N, which "should" transfer to functions
>of several variables on On = class of ordinal numbers. "Should" means: if I
>take the assertion that these statements do transfer, then I get
>axiomatizations of various large cardinal axioms ranging from Mahlo
>cardinals of finite order through Ramsey cardinals incompatible with V = L.
>Does this count? This is also on my website.

Ditto.

>7. I still believe that there is a stunning analysis of the concept of
>computable function which would constitue a true heralded proof of Church's
>Thesis.

This would certainly resolve a real foundational problem if it could be
carried off. (I thought about this myself some years ago.)

>8. I also believe that there is a stunning new solution to Russell's
>Paradox which will provide a totally new theory of predication, whose
>development will be as intriguing, new, and satisfying, as Cantor's
>development of set theory. It will be more general than set theory.

See above.

>9. I also still believe that the last word has not been written on the
>concept of "predicative" proof. That there is an analysis which is much
>more convincing than even Sol's.

One for you.

>10. And much much more that I have up my sleeve that I have not worked out
>- I'm holding back from you. E.g., related to constructive proof. Related
>to measures of how trivial a proof is. Related to "what is a natural
>axiom?" And the development of "pictorial set theory" and neo-relativism.
>Etc. Lots of work to do.....
>
>
The very best of luck!
Martin