DetlefsenFOM: The Aristotelian ideal and reverse mathematics

[Robert S Tragesser]

Summaryt:  It is highly implausible that the Aristotelian ideal can be
realized in its orginal terms (whatever exactly they might be).  It is
surely a good historical exercise to think about what those original terms
might be,  as Defletsen is gamely attempting.   But there is no point in
blindly imposing them on reverse mathematics.  The logic needed for
mathematics is just too powerful to realize that ideal in its original
terms,  even if there were some point to doing so (it is curious that
Detlefsen didn't say what the point would be).   The important question is
to determine the measure in which a particular mathematical genos is
self-founding,  whether the proofs of all its theorems can be "elementary
proofs."   Is any mathematical genos genuinely self-contained, 
"elementary",  composed entirley of "trivial" truths,  dominated by
analyticity?  Or must "transcendental methods" always be evoked
(=transcendental = techniques from another genos),  so that unity rather
than separateness is characteristic of mathematics.
        

        It is important to be more cautious than Detlefsen is being in
invoking and framing the terms of the Aristotelian ideal.   For one thing, 
the elaborate and failed attempts to characterize and realize the ideal in
mathematics in the Late and Post Renaissance,  as exposed in Mancosu, 
should serve as a serious warning that it is highly problematic.   For
another,  recent attempts to specify in the context of a logic adequate to
mathematics the class of proofs which are in some strong sense explanatory
(not to mention the wider and more dramatic failure to do so in the
philosophy of science) suggest that one can't even get of the ground. (The
very fact that any mathematical domain can be represented in very many very
different ways militates vigorously against the possibility of specifying a
priviledged explanatory frame for any domain!)
        It was exactly because Aristotle's logic was a term logic which
rather limited vehicles of deductive inference that he could imagine the
possibility of specifying which deductive inference structures are
causal/explantory.   And more importantly,  recall that it was a necessary
condition for an "axiom" or fundamental "definition" that it not be
deducible.  How was one to recognize when by backward deduction [which
essentially would involve inferences which were not causal!] one got to an
axiom?   The crude idea [that is,  the idea which would have to be
explicated rigorously] wsas that one got to an AisB for which there were, 
and could be,  no "middle terms" which would allow one to
explantorily/causally deduce AisB from something more fundamental (from
anything else at all, in fact).   One had to be able to recognize that the
concrete term logic of the genos at issue made (explanatory) proofs of the
proposed axioms impossible (without committing metabasis and going outside
the terms of the genos.
        THE ONLY IMPORTANT AND DO-ABLE ASPECT OF THE ARISTOTELIAN IDEAL: 
setting things up,  perhaps in the context of RevMath,  to see which
mathematical subjects can be generated by "elementary" rather than
"transcendental" methods (where transcendental means,  not necessarily
infinitary,  but rather OTHER,  that is,  a transcendental technique is one
which is borrowed from another genos,  as for example geometry to algebra
or algebra to geometry.   
        THE REALLY DEEP ASSUMPTION IN ARISTOTLE -- whose truth is most
importantly as issue in the context of our mathematics -- IN THE CONTEXT OF
THE QUEST FOR ITS FOUNDATIONS,  IS WHETHER OR NOT WHAT TECHNIQUES ARE
PROPER OR NATURAL TO A GENOS IS WELL-DETERMINED,  THAT IS,  WHETHER OR NOT
A PROOF IS ELEMENTARY.
                        Robert Tragesser