FOM: elementary proofs, terminology, etc.
Detlefsen.1 at nd.edu
Fri Dec 12 09:01:16 EST 1997
The history of mathematics and foundational provides a term that at least
comes close to what some in this discussion have in mind with they speak of
elementary proofs. It comes from the German term 'methoden Reinheit'
(which, translated into an English adjectival form would be
'methodologically pure'). Hilbert, for one, used the term in his
foundations of geometry. The idea goes back to Aristotle's Psterior
Analytics. It appears in Leibniz' thought and also, prominently, in
Bolzano's. Frege, too, pursued the ideal in his work on the foundations of
The idea is roughly this: there is an objectively real ordering both
notions and truths, including those of the mathematical variety. The
mathematician's most important tasks are to discover and to pursue those
orderings in her definitions and proofs. In that way, but only in that way,
will her proofs be of the proper explanatory character (in Leibniz' term,
be proofs 'par excellence'), because only then will they expose the
objective metaphyical 'bloodline' of a truth, or trace it back to the
ur-truths which objectively 'cause' or 'ground' it.
Hilbert liberalized this idea in his foundational thinking to this extent.
Famously, he didn't advocate using only 'methodologically pure' proofs. Not
even the angels could persuade us to give up the heuristically useful (or
perhaps even psychologically necessary) impure methods (at least not the
ones Hilbert classified as impure). Nonetheless, he believed, it could be
insisted that theorems proved by means of 'impure' proofs either also be
provable by means of pure proofs or that they at least not conflict with
what can be proved by such. In a way, his view suggests a kind of
'psychological' replacement for Aristotle's metaphysical hierarchy: or,
better, he sees a hierarchy of psychological necessity or convenience (the
basis for the introduction of his so-called 'ideal' elements and proofs),
does not rule out the idea of a metaphysical hierarchy (requiring real
proofs of real theorems) and sees the two as being potentially at odds with
one another. (In this, he can be seen to follow Kant, of course.) He then
takes the foundational task to center on that of showing this potential
conflict not actually to ensue.
Clearly, there are commitments in (all variants of) this picture that are
imperspicuous and problematic. (E.g. the notions of objective metaphysical
or psyhcological orderings of terms or notions and of truths.) The question
for us, it seems to me, is this: Is there some articulable and defensible
variant of these ideas which justifies pursuit of 'pure' or 'elementary'
proofs, and which implies that there will be some deficiency in our
knowledge either of them or of other related things if we do not have them?
I hope this helps ... both for those seeking a term to express their ideas
of elementarity of proof, and for those wondering about the deeper question
of the justifiability of demanding 'pure' proofs.
To Neil: I hope before too long to get the time to say something on your
question of the place of 'logical' axioms and the consequences of Church's
Theorem in another posting. I think it's a good question.
Department of Philosophy
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