FOM: intuition
Alexander R. Pruss
pruss+ at pitt.edu
Sun Jan 7 11:38:56 EST 2001
On Sun, 7 Jan 2001, Robert Black wrote:
> 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.
Well, I agree that the evidence raised the probability of the correctness
of the result. But did it raise it above 1/2? Did it make it more likely
than the alternatives? After all, the numerical data was equally
compatible with the hypothesis that the sum was (pi^2+2^(-117))/6.
The problem here is this. As applied to concrete reality, the basic
problem with induction is, essentially, the curve-fitting problem.
Infinitely many curves fit the known data but give different results at
unknown data points. In practice, we tend to choose the simplest
curve/hypothesis. How we're justified in doing this is, of course, the
big question. Probably any epistemic solution will have to say that
nature just has a certain kind of simplicity (and perhaps add something
about how we are justified in assuming this kind of simplicity--the only
two promising lines I know are the theistic and axiarchic, but that's a
different matter).
The parallel in mathematics would be a belief in the simplicity of
solutions to problems. Now, it's obvious that there are problems whose
solutions are abitrarily complicated (e.g., infinite sums that sum to
values like (pi^2+2^(-117))/6). So, the claim presumably would have to be
something like: "The solutions to simple problems tend to be simple."
Without something like this being true, induction in mathematics cannot be
justified. But is something like this true? If so, it begs for a
definition of simplicity and then it begs for a precise statement under
which it could become a theorem... I'm still puzzled.
Alex
--
Alexander R. Pruss || e-mail: pruss+ at pitt.edu
Graduate Student || home page: http://www.pitt.edu/~pruss
Department of Philosophy || alternate e-mail address: pruss at member.ams.org
University of Pittsburgh || Erdos number: 4
Pittsburgh, PA 15260 ||
U.S.A. ||
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