From: Ben Goertzel (ben@goertzel.org)
Date: Tue Nov 08 2005 - 11:25:17 MST
Eliezer,
Your story of trying to disprove Cantor's diagonalization argument, in your
youth, really resonated with me!
I was *not* as much of a prodigy as you were -- I was just a very bright
kid. I entered university at 15 and graduated at 18; I suppose I could have
entered uni at 12 had the opportunity arisen; but I couldn't have done
university material at 8 or 10, no way, my brain just wasn't there yet.
Anyway, I remember that at some point during my teens (late teens; I was
probably 19, in fact) I spent nearly a week convinced that I had proved
mathematics inconsistent! Or, more specifically, that I had proved the
Axiom of Infinity led to logical inconsistency within Zermelo-Frankel set
theory. That was an exhilarating week...
But unfortunately (and predictably), all I had discovered was yet another
(screwy and not so elegant, really) form of Godel's Incompleteness Theorem,
somewhat related to algorithmic information theory.
Perhaps this merely proves that I'm even crazier than you: You were just
trying to disprove aleph-one, but I was nuts enough to try and disprove
aleph-zero as well!
[I still am not comfortable with the notion of the infinite in mathematics,
and especially not with aleph-one and all those nasty random entities that
exist as a group yet can never be demonstrated on an individual basis. But
unfortunately the paradoxical nature of these things doesn't seem to take
the form of a real logical inconsistency, just an incompatibility between
the commonsense notion of "is" and the mathematical notion of "is".]
I knew a real child prodigy once, fairly well -- he was a classmate of my
son's, and at age 5 he was doing trigonometry and analytic geometry (while
my son, who was gifted in math but not a prodigy, was merely having fun with
simple binary arithmetic examples and very basic algebra... my son was by
far the *second* best kid in the class in math, but this prodigy kid blew
him away). Indeed, he did not seem generally more *intellectually mature*
than my son or other very bright kids, even though he had more capability to
absorb and manipulate technical ideas. In more respects his judgment was
still that of a young child.
As member of a math dept. hiring committee, in the early 90's I I once
interviewed an ex-prodigy for a professor position. He was clearly VERY
bright, and was unusual in having well-developed research careers in two
very different areas, set theory and mathematical physics. But we ended up
hiring someone else, a number theorist who had not been a prodigy but seemed
to have more creative and important research ideas. This encounter
reinforced what the literature shows: prodigies nearly always grow into very
gifted adults, but not necessarily more gifted than some others who were not
prodigies. Yet another aspect in which brain function and development are
very mysterious!
-- Ben G
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