Discussion about this post

User's avatar
Mark's avatar

I think the notion of limitation here obscures more than it clarifies. There are plenty of more intuitively limited things that are simpler than less limited things. The Game of Life, for example, is a simpler kind of dynamical system than idealized Newtonian gravitation, even though there's a kind of "speed limit" in the former that doesn't exist in the latter.

I'd speculate that what you have in mind in most paradigmatic examples of "pathological limitation" which decreases simplicity is taking some fixed description of a system, and then tacking on an extraneous detail declaring some specific case/configuration (hitherto unremarkable) to be forbidden. For instance, Newtonian gravitation plus the weird addendum that everything must be moving below 100 mph at all times. There's an argument to be made that most of the time this will increase Kolmogorov complexity, hence makes things more complex.

But if we have two extraordinarily different theories, neither of which is a close restriction of the other, this observation seems inapplicable. Instead, we need to zoom out and do our best to compare (our approximations of) Kolmogorov complexity overall directly. And I'd argue that when we do this, none of the attributes you propose come out looking very simple.

For example, "knowing every true thing" is extremely complicated. If we discovered a highly mysterious alien computer somewhere that correctly answered some hard math questions (e.g., proof/disproof of the Riemann Hypothesis or P = NP), the hypothesis that it could correctly answer *any* math question (including all the undecidable ones!) would nevertheless remain very complex. Because it would involve postulating that the fundamental laws of physics which underly the computer's physical operations are uncomputable: for every true undecidable mathematical statement, there'd have to be something like an extra law stating atoms have to move a certain way in such-and-such a situation to reflect that statement's truth. Otherwise, how would the alien computer always answer in the right way? By contrast, a computable set of laws on which the computer is maybe superintelligent but still mathematically non-omniscient, and constrained to only being able to compute things these laws allow (unless it answers randomly and gets lucky), should should be regarded as simpler.

It gets worse when you broaden "can answer any mathematical question correctly" to "can answer any question correctly." Because now there are a bunch of semantic paradoxes your theory about the alien computer has to circumvent, such as "Hey Alien Computer, will you answer negatively to this question?" So instead of "can answer any question," you have to sharpen it to "can answer any question in [restricted class of questions that avoids paradox]." But spelling out that restricted class *precisely* is hard. It may also be uncomputable, i.e., there might not be a reasonable decision procedure to identify whether a question is in that class or not. So this hypothesis looks really bad! Better to go with a "limited" universe.

Expand full comment
Hochreiter's avatar

He saith unto them, But whom say ye that I am?

And Simon Peter answered and said, "Suppose you have a mind and just don’t place any limits on it. I claim that what you get is God! Because he is unlimited there is no limit on what he knows or can do. But if he knows everything then he knows the moral facts, and thus understands at the deepest conceivable level why he should follow them. You might worry—isn’t he limited in various ways. For instance, presumably his love of evil is limited. But this misses the point. It’s true that there are various properties in which he’s limited—but he’s unlimited qua mind."

And Jesus answered and said unto him, "What?"

Expand full comment
10 more comments...

No posts