I think that’s more about Wolfram giving a clickbait headline to some dicking around he did in the name of “the ruliad”, a revolutionary conceptual innovation of the Wolfram Physics Project that is best studied using the Wolfram Language, brought to you by Wolfram Research.

The full ruliad—which appears at the foundations of physics, mathematics and much more—is the entangled limit of all possible computations. […] In representing all possible computations, the ruliad—like the “everything machine”—is maximally nondeterministic, so that it in effect includes all possible computational paths.

Unrelated William James quote from 1907:

The more absolutistic philosophers dwell on so high a level of abstraction that they never even try to come down. The absolute mind which they offer us, the mind that makes our universe by thinking it, might, for aught they show us to the contrary, have made any one of a million other universes just as well as this. You can deduce no single actual particular from the notion of it. It is compatible with any state of things whatever being true here below.

I study complexity theory so this is precisely my wheelhouse. I confess I did not read most of it in detail, because it does spend a ton of space working through tedious examples. This is a huge red flag for math (theoretical computer science is basically a branch of math), because if you truly have a result or idea, you need a precise statement and a mathematical proof. If you’re muddling through examples, that generally means you either don’t know what your precise statement is or you don’t have a proof. I’d say not having a precise statement is much worse, and that is what is happening here.

Wolfram here believes that he can make big progress on stuff like P vs NP by literally just going through all the Turing machines and seeing what they do. It’s the equivalent of someone saying, “Hey, I have some ideas about the Collatz conjecture! I worked out all the numbers from 1 to 30 and they all worked.” This analogy is still too generous; integers are much easier to work with than Turing machines. After all, not all Turing machines halt, and there is literally no way to decide which ones do. Even the ones that halt can take an absurd amount of time to halt (and again, how much time is literally impossible to decide). Wolfram does reference the halting problem on occasion, but quickly waves it away by saying, “in lots of particular cases … it may be easy enough to tell what’s going to happen.” That is not reassuring.

I am also doubtful that he fully understands what P and NP really are. Complexity classes like P and NP are ultimately about problems, like “find me a solution to this set of linear equations” or “figure out how to pack these boxes in a bin.” (The second one is much harder.) Only then do you consider which problems can be solved efficiently by Turing machines. Wolfram focuses on the complexity of Turing machines, but P vs NP is about the complexity of problems. We don’t care about the “arbitrary Turing machines ‘in the wild’” that have absurd runtimes, because, again, we only care about the machines that solve the problems we want to solve.

Also, for a machine to solve problems, it needs to take input. After all, a linear equation solving machine should work no matter what linear equations I give it. To have some understanding on what problems a Turing machine can solve, Wolfram would need to analyze the behavior of the machine on all (infinitely many) inputs. He doesn’t even seem to grasp the concept that a machine needs to take input; none of his examples even consider that.

Finally, here are some quibbles about some of the strange terminology he uses. He talks about “ruliology” as some kind of field of science or math, and it seems to mean the study of how systems evolve under simple rules or something. Any field of study can be summarized in this kind of way, but in the end, a field of study needs to have theories in the scientific sense or theorems in the mathematical sense, not just observations. He also talks about “computational irreducibility”, which is apparently the concept of thinking about what is the smallest Turing machine that computes a function. This doesn’t really help him with his project, but not only that, there is a legitimate subfield of complexity theory called meta-complexity that is productively investigating this idea!

If I considered this in the context of solving P vs NP, I would not disagree if someone called this crank work. I think Wolfram greatly overestimates the effectiveness of just working through a bunch of examples in comparison to having a deeper understanding of the theory. (I could make a joke about LLMs here, but I digress.)