FOM: Reply to Friedman's ReplyToDavis

[Martin Davis]

>This is a reply to Davis, 11/7/97.
>
>In a way, I agree and welcome this posting. However, the choice of words
>does reveal an undercurrent here that I take some issue with. In
>particular, the posting contains at least the suggestion that mainline
>f.o.m. may not be as fruitful and active as I know it to be.

Sorry. I meant no such suggestion. I was taking issue with what I took to be
Steve Simpson's approach via singling out the appropriate *concepts* for
foundational investigations. I believe much of the discussion resulting from
that stance to have been sterile and at times uncomfortably rude.

>
>>Something I miss in many of the postings here is the realization that
>>foundational work tends to be in response to specific foundational problems.
>
>The problem of "what (kind of) interesting mathematical information can
>only be obtained by expanding the usual axioms for mathematics" is very
>mainline f.o.m., clearly discussed in various ways by Kurt Godel. However,
>I'm not sure that it is "in response to a specific foundational problem" in
>the sense that you mean later in your e-mail. What do you think?

It surely is. I mentioned in an earlier posting what I think of as a key
foundational problem in our time: how to make sense of the \Pi_1 statements
implied by large cardinal axioms. 


>Another very mainline foundational problem: "what kind of coherent
>justification can one give for the generation of the large cardinal axioms"
>or for that matter ".... for higher set theory?"
>
>Yet another very mainline foundational problem: "what is so special or
>perhaps canoncial about the currently studied formal systems of f.o.m.?"
>
>There are many more, and at a later time I want to make a systematic
>posting of them. These are all topics of great promise and current activity
>at various levels. They are not dead subjects in any way shape or form.

I agree completely.

>>There are various things in mathematical practice that lead mathematicians
>>beyond what they are able to justify in terms of their contemporary
>>understanding. When this happens, a foundational problem arises. In the
>>happiest instances, foundational work will not only provide a satisfactory
>>explanation, but also will provide new directions for mathematical research.
>
>I agree that there are fruitful interpretations and reinterpretations of
>foundational developments along these lines. However, what specifically do
>you propose as a foundational problem that fits this mold now?

I don't have a really good contemporary example. But one would arise if some
important question were decided by using a new set theoretic axiom. That
would make the rpoblem of justifying such an axiom clearly important for
mainstream mathematics.

>>Analogy is another force leading mathematicians beyond what they are able to
>>justify. (This has great contemporary relevance for set theory: the infinite
>>by analogy with the finite, the huge infinite by analogy with the countably
>>infinite.)
>
>Now this is something that is currently being adressed in f.o.m. E.g.,
>there are transfer principles from discrete to transfinite that generate
>some large cardinals. This seems to fit into your mold.

Yes.

>>The obvious success of infinitesimal methods, used freely by engineers and
>>physicists long after mathematicians had insisted they were illegitimate,
>>was an example of a foundational problem not resolved until Abraham Robinson
>>explained it.
>
>Frankly, I have not been a big fan of the f.o.m. aspect of nonstandard
>analysis - at least in its present form.

I always felt there was something mysterious about the "construction" of the
real numbers. In, for example, Dedekind's version, one does technical
fussing about the rational cuts (does the rational point belong in the left
or right class), fusses about how to deal with negative cuts, and how to
define the arithmetic operations, and then, lo and behold, we have an
ordered field. Nonstandard analysis clarifes this. In an enlargement, we
have Q* (where the rationals are Q) and then the reals are seen as the
quotient ring of the finite elements of Q* modulo the (maximal) ideal of the
infinitesimals. The completion of the rationals to form the reals is then
seen as one example of the use of enlargements for sustaining limit
operations, and the field properties as seen as just an instance of the
(foundational!) transfer principle.

>
>I personally get a lot out of considering what is truly "foundational" -
>although not in the argumentative sense of some of the FOM interchanges. I
>use such considerations in choosing focused topics such as those mentioned
>at the beginning of this e-mail. Some of the mainline f.o.m. is obviously
>of timeless importance - especially, "what (kind of) interesting
>mathematical information can only be obtained by expanding the usual axioms
>for mathematics?"
>
>I have to admit that many people don't think that the particular arguments
>over the FOM about whether or not some specific topics are foundational or
>not is fruitful. Is that what you are criticizing?
>

Yes.

>And it would be nice if you could list some foundational problems "needing
>to be addressed." Do you think that
>
>"the free use of algebraic manipulation of
>operators in solving ordinary differential equations (see George Boole's
>classical text for masterful use of these techniques without a shred of
>justification), the Heaviside operator calculus, the Dirac delta function"
>
>are still timely topics for f.o.m.?
>

No, of course not. But they do lend historical perspective.

I do appreciate the thoughtful reply.

Martin Davis