Suppose, for example, that you had a computer program that defined a synthetic biology and a synthetic gene system. And this computer program ran through an artificial evolution that simulated 10^44 organisms (which is one estimate for the number of creatures who have ever lived on Earth), and subjected them to some kind of competition. And finally, after some predefined number of generations, it selected the highest-scoring organism, and printed it out. In an intuitive sense, you would (expect to) say that the best organisms on each round were getting more complicated, their biology more intricate, as the artificial biosystem evolved. But from the standpoint of Kolmogorov complexity, the final organism, even after a trillion years, is no more “complex” than the program code it takes to specify the tournament and the criterion of competition. The organism that wins is implicit in the specification of the tournament, so it can’t be any more “complex” than that.
and later: …then this intrauniversal civilization—and everything it’s ever learned by theory or experiment over the last up-arrow years—is said to contain 400 bits of complexity, or however long the original program was.
“You say it as if it were a good thing.”
But this seems, on the contrary, to indicate that Kolmogorov complexity might not be a good candidate for the formalized notion of intuitive complexity, including the one used by Occam’s Razor. The fact that it often can be anti-intuitive doesn’t necessarily mean that it’s our intuition of what is simple and what is complicated that must change; the mere fact that the notion of Kolmogorov complexity is itself very simple and easy to play with mathematically doesn’t by itself entitle it to anything.
There exists a proof of Fermat’s Last Theorem with very small Kolmogorov complexity, much smaller than, say, the size of this post of yours in characters, or the Kolmogorov complexity of some well-known formal proof of a basic theorem in calculus. Does that mean that this proof of FLT is a much simpler object than these others? It does if you insist on interpreting “simple” to mean “small Kolmogorov complexity”, but why must you so insist? If you show this proof to a mathematician, he or she won’t say “oh, this is very simple, simpler than basic calculus proofs.” If you then tell the mathematician that in fact this very complex proof was obtained by running this very short Turing machine, they’ll just shrug in response—so what? As a piece of mathematics, the proof is still evidently extremely complicated. Maybe your notion of complexity just isn’t so goo d at capturing that.
Suppose, for example, that you had a computer program that defined a synthetic biology and a synthetic gene system. And this computer program ran through an artificial evolution that simulated 10^44 organisms (which is one estimate for the number of creatures who have ever lived on Earth), and subjected them to some kind of competition. And finally, after some predefined number of generations, it selected the highest-scoring organism, and printed it out. In an intuitive sense, you would (expect to) say that the best organisms on each round were getting more complicated, their biology more intricate, as the artificial biosystem evolved. But from the standpoint of Kolmogorov complexity, the final organism, even after a trillion years, is no more “complex” than the program code it takes to specify the tournament and the criterion of competition. The organism that wins is implicit in the specification of the tournament, so it can’t be any more “complex” than that.
and later: …then this intrauniversal civilization—and everything it’s ever learned by theory or experiment over the last up-arrow years—is said to contain 400 bits of complexity, or however long the original program was.
“You say it as if it were a good thing.”
But this seems, on the contrary, to indicate that Kolmogorov complexity might not be a good candidate for the formalized notion of intuitive complexity, including the one used by Occam’s Razor. The fact that it often can be anti-intuitive doesn’t necessarily mean that it’s our intuition of what is simple and what is complicated that must change; the mere fact that the notion of Kolmogorov complexity is itself very simple and easy to play with mathematically doesn’t by itself entitle it to anything.
There exists a proof of Fermat’s Last Theorem with very small Kolmogorov complexity, much smaller than, say, the size of this post of yours in characters, or the Kolmogorov complexity of some well-known formal proof of a basic theorem in calculus. Does that mean that this proof of FLT is a much simpler object than these others? It does if you insist on interpreting “simple” to mean “small Kolmogorov complexity”, but why must you so insist? If you show this proof to a mathematician, he or she won’t say “oh, this is very simple, simpler than basic calculus proofs.” If you then tell the mathematician that in fact this very complex proof was obtained by running this very short Turing machine, they’ll just shrug in response—so what? As a piece of mathematics, the proof is still evidently extremely complicated. Maybe your notion of complexity just isn’t so goo d at capturing that.