FOM: Methodological Purity & Elementary

[Robert S Tragesser]

        Michael Detlefsen's suggestion that "methodologically pure" and
"elementary" may be at root the same idea is very suggestive indeed.   We
are getting an exciting convergence of phenomena:

[1]  Inspired by Detlefsen, re-reading Rota,  I notice that as an
alternative (albeit negative) characterization of elementary proofs is that
they don't apply external techniques,  but only techniques somehow germane
to the subject presumably as determined by the terms of the theorem at
issue.
[2]  Detlefesen's connection with Aristotle's Organon is extremely
important.   For Aristotle each subject matter had its own principles or
causes.   It was for Aristotle a serious fallacy to transport techniques
germane to on subject matter to another subject matter.   He called the
fallacy(if I may butcher the Greek): metabasis eis allo genos).   
        Aristotle is often represented as differing from Plato most
importantly at this juncture.  Plato,  but not Aristotle [under this
representation of the difference],  believed that there was an
over-arching,  unifying science [dialectic].
        Paolo Mancosu's recent PHILOSOPHY OF MATHEMATICS AND MATHEMATICAL
PRACTICE IN THE SEVENTEENTH CENTURY powerfully draws attention to the role
of this Aristotelian outlook at the birth scene of modern mathematics.
        Amos Funkenstein's amazing THEOLOGY AND THE SCIENTIFIC IMAGINATION
FROM THE MIDDLE AGES TO THE SEVENTEENTH CENTURY (but actually he works his
way through to Kant) tracks the importance of overcoming the sense of
fallacy attaching to metabasis in the formation of modern scientific and
mathematical thought.
        Felix Klein in this Elementary Mathematics from an Advanced
Standpoint finds their to be two equally energetic tendencies in
mathematical work,  which he calls Plan A and Plan B.   Plan A is
Aristotelian (the development of separate subject matters),  whereas Plan B
is Platonistic(unifying).
        As Detlefsen and I in effect conjointly observe(putting out
observations together),  elementary proof,  methodological purity,  
germane rather than external technique,  analyticity,  local foundations, 
and self-containedness all seem to be different perspectives on the same
thing.

[2]   Rota has given us a kind of hermeneutics of the mathematical practice
of seeking elementary proofs.   Note that with many practices,   not all
who seek for an elementary proof may be all that clear about what the point
of finding elementary proofs is,  so Rota has refreshed the deeper-layer
motivation behind the practice,  and I think that the tie of quest for
elementary proofs to analyticity,  trivialy,  self-containedness,  local
coundations,  avoidance of the fallacy of metabasis eis allo genos,  the
Aristotelian sense that each subject matter has its own causes, 
principles,  reasons,  techniques,  logic come together to prepare for a
philosophical,  mathematical,  and historical sharpening of the substance
and terrific importance of the very idea of elementary (or self-contained)
proof.

[3]  The failure of the existence of nonelementary proofs for important
theorems would go a ways toward vindicating Plato and (the gods help us)
Schelling,  that we are one world after all (you know the melody).

[4] An alternative outlook:  Elementary proofs as ethnic cleansing--the
down side of purity of method:.    Sprinkled throughout the literature
involving elementary proofs are comments like "to the horror of the
algebraists,  topological methods seem unavoidable. . ."   This points to a
purely territorial motivation for elementary proof--but at the same time it
seems to indicate a faith in the autonomy of mathematical genuses (though
ot course "purity of blood" mythos).
                        rbrt tragesser