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The Mathematical universe hypothesis, mainly by Max Tegmark and A new Kind of Science, mainly by Stephen Wolfram both claim (as least as I understand it) that at its innermost core reality is mathematics.
Can this statement be made more precise, i.e. what is the exact relationship between these two hypotheses?
I think that Wolfram is arguing that the study of cellular automata and perhaps similar computational systems could serve as an organizational principle, providing a coherent framework to look at different problem (just like the more familiar frameworks provided by physics and chemistry). This explains the title of his new book, A new kind of Science (i.e. the study of the above-mentioned structures).
On the other hand, Tegmark argues that our Universe is one big mathematical structure. This may be difficult to wrap your head around, but it would mean that we are just mathematical structures that are complex enough to be self-aware and do everything we do. I assume this would not have any observational consequences (as we cannot proof that something cannot be described by mathematics, exactly because we need mathematics to prove anything) and is therefore purely speculative.
As you can see, Wolfram is calling for a new framework to conceptualize and study problems, while Tegmark is positing a theory of the Universe. In my opinion, these are two completely different things. Disclaimer: I have not read the book by Wolfram, nor was I previously familiar with Tegmark's proposal.
It's my impression that Wolfram/ANKOS approaches the issue from a much more technical, detailed angle, while Tegmark is arguing the case on general principles, without concern for the specific implementation. Mind you, my impression of Tegmarks work is only based on online references, not his actual writing (I've read ANKOS, though).
For me, Wolfram/ANKOS is more interesting because of the more practical approach, but Tegmark has one important element that Wolfram fumbles; Tegmark makes the reasonable assumption that if the universe is a MUH or CUH, it must also be completely abstract. Wolfram mentions this possibility only briefly in his book, but then goes on to say he doesn't believe this is the case.
Wolfram claims that the universe at its core is described by some Turing universal computations (it doesn't matter what specific form it takes, such as cellular automata, tag systems, Peano arithmetic etc)as all Turing universal computations are equivalent). He also claims that mathematical descriptions are a special case of general computations where you can predict a future state without having to compute all intermediate states. But mathematics includes, at least in theory (but not generally in practice), any computation, because Peano arithmetic is Turing Universal. Tegmark goes a step further and states that any computational system (or computable mathematical structure) exists as a universe. Wolfram, instead, tries to find out which one is the "real one", that is, the one describing our physical universe. Tegmark limits his universes to Godel complete ones. However, in my opinion, this leaves out many non-computable mathematical structures that are pretty much "likely to exist" (whatever that means), such as the models of ZFC + large cardinal axioms.
