FOM: Cabillon's ? abt: ADEQUATE PROOF&LAKATOS
Robert S Tragesser
RTragesser at compuserve.com
Mon Dec 15 18:26:15 EST 1997
I raised the question whether Lakatos' method of
proof/criticism/analysis, the method of proofs and refutations,
as central to mathematical performance was not rather misleading.
(I am just putting some of the critical points made by Sol
Feferman out there; albeit I am stating them more loosely than
Sol did.)
First, it seems to suggest that "adequate proofs" are hard to come by But
they are everywhere. Adequte proof? "There, that's proved. We don't
have to worry about finding a proof for it. But we might like a more
transparent or elementary or revealing or. . .proof than this.
Second, it is over skeptical, overly logic chopping.
Third, there are adequate imperfectly formal proofs, where
I use formal in Leibniz's sense: every aspect of the proofis covered
by a principle given in advance [so "formal proof" need not mean
"derivation in a meaning-vacated formal arithmetic".
Fourth, Lakatos' method is more appropriate for rational arguments in
general than mathematical proof.
Fifth, by concentrating on a logic-chopers conception of validity,
Lakatos
coompletely overlooks what is important in proofs -- the ideas in them,
and what can be learned (about the theorem and its possible generalizations
or possible more instructive lemmas)from thinking about them
and trying to understand them.
Sixth, in conection with the last, perhaps the greatest sin of
mathematical
pedagogy is just presenting proofs, emphasizing only their validity, and
no dwelling on them, thinking them out, instructively varying them.
robert tragesser
them
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