Robert.Black at nottingham.ac.uk
Sun Jan 7 10:45:45 EST 2001
Alexander Pruss raises an interesting question about our confidence that
mistakes in proofs are unlikely to matter, to which Martin Davis's remarks
about robustness are surely a partial answer.
Pruss however seems to assume that inductive reasoning can't work in
mathematics. That just has to be false. It seems to me obvious that
inductive reasoning works in mathematics, and if according to some theory
of inductive reasoning it shouldn't work in mathematics, that just shows
that it's a false theory of inductive reasoning.
A classic example, first pointed out, I think, by Polya, and since used, if
I remember correctly, by both Mark Steiner and Hilary Putnam, was Euler's
justified confidence about the sum of a certain infinite series. He had a
hopelessly inadequate proof that the series summed to (something like - I
forget the details) (pi^2)/6. But it was a rapidly converging series, and
just doing the arithmetic of the first few terms made it obvious that that
was indeed where it was going. It would be mad to suggest that this
evidence didn't raise the probability of the correctness of the result.
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