Nice blog post. I wasn’t aware of those comments by Yann LeCunn and Murray Gell-Mann, but it’s reassuring to know there are some experts who have been wondering about this “flaw” in Kolmogorov complexity as well.

I wouldn’t go so far as to say Kolmogorov complexity is useless as an objective complexity measure however. The invariance theorem does provide a truly universal and absolute measure of algorithmic complexity — but it’s the complexity between two things rather than of one thing. You can think of U and V as “representatives” of any two partial recursive functions u(x) and v(x) capable of universal computation. The constant c(u, v) is interesting then because it is a natural number that depends only on the two abstract functions themselves and not the specific Turing machines that compute the functions.

What does that mean philosophically? I’m not sure. It might mean that the notion of absolute complexity for a finite string isn’t a coherent concept, i.e., complexity is fundamentally a property of the relationship between things rather than of a thing.

Yeah, we may have to take seriously that the notion of complexity for a finite string (or system) has no reasonable definition.