FOM: Rota, "natural Foundations", Elementary Proof.
Robert Tragesser
RTragesser at compuserve.com
Tue Oct 26 22:35:08 EDT 1999
FOMers:
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. 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]. 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). 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. He was neutral on the matter of whether or
not elementary proofs can always be achieved.
Any one care to comment?
Robert Tragesser
West(running)brook, Connecticut
(A sealapped place where things go strangely. . .like lobsters, crabs
and other submarines, which preponderate.)
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