FOM: Rota and elementary proofs

[Harvey Friedman]

Reply to Tragesser 2:23AM 10/28/99.

>        In the process of composing a philosophy of math. memorial for
>_PM_ about Rota,  I came to realize (as in Rota's _Indiscrete Thoughts_)
>that Rota had no interest in the traditional foundational schemes,
>however revamped.

I do not believe this. I do believe that he had no interest in investing
the time to obtain a research level understanding of "traditional
foundational schemes."

> He often made much about the quest of mathematicians
>for "elementary proofs" of theorems,  which he explained loosely as
>proofs that don't draw on anything more than what is contained in the
>concepts constituting the statement of the considered theorem [hardly an
>exact characterization;  though I at least can appreciate its highly
>intensional flavor].

I have worked steadily for over 30 years about this matter using
"traditional foundational schemes."

>His way of putting this was that typically deep
>theorems are first proven non-analytically (as for example drawing on
>mathematics quite apart from number theory to first prove a deep
>proposition in number theory).

For instance, I have been pushing the program of showing that all of the
famous number theoretic results of the 20th century (and earlier) can be
proved within EFA = exponential function arithmetic. I conjecture that
Mordell's Conjecture and Fermat's Last Theorem can be proved in EFA. Some
world famous number theorists are quiite interested in this conjecture. It
leads to a whole host of related investigations cast in "traditional
foundational schemes." E.g., I recently posted a sketch (#56) that EFA is
enough to prove the consistency of elementary algebra and geometry; and in
fact, a whole host of theorems about elementary algebra and geometry to the
effect that if you use elementary algera and geometry to prove a number
theoretic result, then you can eliminate the elementary algebra and
geometry in favor of elementary number theory.

When I wrote #56, I didn't explicitly discuss this way of putting the
results, but there is no problem stating them along these lines. In fact,
my responding to your posting has lead me to see that I should make a
followup posting in which I disucss this issue of "eliminability of
elementary algebra and geometry in favor of elementary number theory" in
detail. In fact, there is the much wider issue of "eliminability of various
analytic methods in favor of elementary number theory" which needs to be
addressed systematically, and which I began to address systematically in a
seminar here, but got seriously sidetracked.

All of this is in "traditional foundational schemes."

>The basic foundational drive native to
>mathematics [working toward  elementary/analytic proofs] on his view is
>fairly independent,  then,  of epistemological (e.g. certainty) and
>(broadly conceived) logical issues (e.g., coherence, consistency)
>issues.  That is,  once one give an elementary proof one can declare --
>all is in order,  something like that;  the mathematical mind is then
>foundationally satisfied.

Under this view, the burning question becomes: can elementary proofs always
be obtained, and if not, then in which interesting cases can they be
obtained, and in which interesting cases can they not be obtained?

This is exactly the sort of question that is extremely well suited to
"traditional foundational schemes." In fact, I would go further: It has not
yet been shown to be well suited to "nontraditional foundational schemes."
In fact, it would be quite interesting to see if some "nontraditional
foundational scheme" could be seen to shed any light on this at all.

>He was neutral on the matter of whether or
>not elementary proofs can always be achieved.

I take this to mean that he never claimed results along the lines that
"elementary proofs can always be achieved" or "here are examples where
elementary proofs cannot be achieved." He was fully aware of the results I
had obtained of the latter kind when he gave a speech in the mid 1980's in
Washington D.C. in celebration of an award I had just won.