FOM: intuition
Mark Steiner
marksa at vms.huji.ac.il
Sun Jan 7 22:56:20 EST 2001
In Euler's case, it wasn't just curve-fitting. The argument was mainly based
on a plausible analogy between polynomials and infinite trigonometric series,
namely that the relations between the coefficients and the roots remained true in
the infinite case. I believe Euler expanded cos(x) for this purpose (my memory
is hazy). He noticed, using another coefficient of cos(x) that the analogy would
give another result which had already been proved by Leibniz, also involving pi.
Given this kind of "higher induction", Euler's result, that the sum of the
reciprocals of the squares is pi^2/6, was inductively inevitable in my opinion,
once Euler checked the decimal expansion to 100 places. Notice that a general
analogy becomes much more plausible when it is checked for a specific case. I
believe that the kind of analogy that Polya describes and the inductive reasoning
based on it is no different than the kind that goes on in the physical
sciences--namely, it's not just "enumerative induction."
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