Chaitin's paper is at the same time much better and much worse than I
was expecting.
I was expecting bad philosophy which relied heavily on resemblances to
make its inference. This is bad philosophy in the same way that
Hofstadter's GEB is bad philosophy: he draws your attention to a
number of interesting things that have a certain resemblance, and then
goes on to suggest that this resemblance is more than superficial.
(Yes, look, they ALL have STRANGE LOOPS; therefore, er, therefore
they're all very interesting.) [Good books may nonetheless contain
bad philosophy.]
Chaitin's paper relies heavily on an equation he's cooked up which
takes parameters. Pop in the parameters and you get equations which
either have a finite number of solutions or an infinite number of
solutions; you can't tell which, of course, without solving the
halting problem. Accepting as an axiom that one of them has a finite
number of solutions (or an infinite number of solutions, your pick)
doesn't help you to derive the solutions to any of the others, at all.
Their solutions are formally independent, just as it is independent
whether, when you flip a coin, the next toss will be heads or tails.
No amount of reasoning will yield the next result.
Yet, interestingly, you can also show that in half of the cases, the
equations will have a finite number of solutions, and in the other
half, they'll have an infinite number of solutions. Thus, he can make
a provable statistical claim about these things, despite the fact that
he can't formally derive any one of them on the basis of the others.
That, he says, is just like the statistical result that half your coin
flips will be heads.
He suggests a new course of research: that we should check out these
arithmetical coin tosses, yielding a branch of statistical arithmetic
just as we have statistical mechanics.
The relationship to epistemology, though, looks rather weak to me.
-unless you love someone-
-nothing else makes any sense-
e.e. cummings
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