FOM: Rota, "natural Foundations", Elementary Proof.

[Robert Tragesser]

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.)