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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?

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I didn't vote to close, but I do think this question is much too vague and/or broad. Have you thought about these things at all yourself? Is there something that bothers you in particular? – Danu Feb 10 '14 at 16:24
Yes I have. What bothers me is whether there could be different notions of mathematics underlying these hypotheses - yet are these notions in any way connected? – vonjd Feb 10 '14 at 16:26
I didn't vote to close, but I did flag for migration to because I don't see what this has to do with physics. – Michiel Feb 10 '14 at 16:41
@Michiel: Well, Tegmark is a physicist (cosmologist) at MIT and the notion here is that at the core of physics lies mathematics in a very literal sense. – vonjd Feb 10 '14 at 16:43
I'm voting to reopen this because I think there is a real question buried here, but as it stands I think this is a very poor question. You need to make clear exactly what the two 'hypotheses' are, so the question is intelligible to someone who hasn't read those two huge books. You also need to rephrase very precisely what the question actually is. If you're thinking "I'd like someone to explain X to me" it's fine, but threads along the lines of "I'd like to participate in a discussion about X" are not really for this site. Keep it to the point! – Emilio Pisanty Feb 12 '14 at 9:46
up vote 4 down vote accepted

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.

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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.

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