FOM: Lakatos as logic chopper
Robert S Tragesser
RTragesser at compuserve.com
Mon Dec 15 09:06:39 EST 1997
I was greatly hoping that Jeff's question would have inspired some
philosophical discussion of Lakatos rather than bibiography, so how abot
this:
In his review of Lakatos, Sol Feferman worried about the
implicature underlying Lakatosian "proof criticism" with the implication
that no mathematical proof is really conclusive. Perhaps this is what is
atypical in Lakatos' examples, that they catch mathematics in moments of
problems in local foundations.
Are there conclusive proofs? Are there informal conclusive
proofs?
(In part to think there is one must have a sensitivity to what
Kreisel called doubtful doubts -- see Leibniz on Descartes' method of doubt
-- the Leibniz had Descartes very wrong).
(BTW: Does anyone know what Hilbert meant by "proof criticism"?)
That's one question.
Here's a comment about the right sort of (non logic chopping) kind
of proof criticism, inspired by Polya:
Mathematics students need to be taught how to think about and milk
ideas and understanding from proofs rather than the merely base and
sophomoric skill of skepticism. The need the skill of a higher kind of
"proof criticism", that can appreciate proofs beyond the exigencies of
mere vailidity. For example, one gives the students the usual RAA proof
for the irrationality of the square root of two and asks them for homework
to work the same sort of game on irrationality of square root of three,
five. . ., giving the obvious hint of what they'd have to work out.
Okay, most can do the routine and be absolutely untouched by it.
Curiously it is the best and the worst students who have the right
response. The good student will look for some lemma that will enable them
to avoid working out a construction for each case. The poor student just
won't get it, won't understand. Proper pedagogy would be to get the
student to see, well this proof does the job, but you should really hate
it, or want something better, one that let's you see what is going on.
By careful positive proof criticism, out loud, with the students, one
can wind one's way to the lemma that if a postivie integer does not have an
kth root among the positive integers, it doesn't have one among the
rationals. TGhen if one proves this lemma in a sufficiently gentle way
from the fundamental theorem of arithmetic, light-bulbs glow in students
eyes.
robert tragesser
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