FOM: FundamentalConcepts/Analysis

[Harvey Friedman]

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.

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.

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.!

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.

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.

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.

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.

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.

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.

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.

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.....