Mathematical Universe Hypothesis What is the current "consensus" on Max Tegmark's Mathematical Universe Hypothesis (MUH) which claims every concievable mathematical structure exists, including infinite different Universes etc.
I realize it's more metaphysics than physics and that it is not falsifiable, yet a lot of people seem to be taking a liking to it, so is it something that is plausible ?
I've yet to hear any very good objections to it other than "it's crazy", but are there any real technical problems with it?
 A: The problem with such ideas is that they are empty words. In order to say what "exist" means, you need to specify an outcome of a physical measurement which confirms the existence. For example, the Elitzur Vaidman bomb-tester is a way of giving a logical positivist meaning to the existence of counterfactual worlds within quantum mechanics. Without such a positivist meaning, the question of whether counterfactual worlds exist becomes meaningless jibberish.
The main obstacles to such an interpretation:


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*The space of mathematical structures as normally imagined to exist in ZFC is far too big for physics (this was the previous answer), but I'll try to fix that, and interpret the sapce of structures differently. The correct space of all mathematical objects is probably best given as the set of all integers and relations on the integers which encode arbitrary running histories of computer programs to finite times. This includes sequences of digits which approximate real numbers, the description of all theorems of ZFC (or any other axiomatic system), etc, so is as big as you can conceive.

*There is no obvious dynamics on this space (other than the running of computer programs)--- how do you jump from one finite graph on the integers to another? What is the relation between structures which corresponds to different experiences at different times? You have to embed physics into this structure somehow.

*There is no quantum dynamics on these structures: the basic structures of mathematics differ from the basic structures of physics in that physics deals with quantum mechanics. These types of "mathematical existence" a-priori methods sometimes produce discrete models with probability, like cellular automata, but they don't give rise to quantum mechanics. You need a way for quantum mechanics to emerge from a non-quantum probabilistic substructure, and this is not known to be possible, and might be impossible.


There is no relation between this type of metaphysics and the program of physics, which is defined by logical positivist questions one can ask of nature.
A: The MUH is related to but separate from Multiverse theories. Multiverse theories are becoming quite respectable because of the String Landscape, eternal inflation, anthropic coincidences, Everett manyworlds, all of which point towards multiverses. MUH is more controversial, however given a broad-enough definition of "mathematics" (all non-contradictory structures) it can almost be seen as tautological; of course the universe is a mathematical structures. Those who object to the MUH are essentially dualists: they must believe that existing things have properties that are describable mathematically (this seems uncontroversial give the success of mathematical physics), but must also have properties (essential thingyness! existenceness!) that are indescribable. This approaches mysticism.
A: It's just mathematical Platonism. That's the mainstream view among mathematicians, but among physicists, it provokes the 'ick' factor, if you know what I mean.
All joking aside, do you really believe nonmeasurable subsets exist? A well-ordering of the real exists? That the power set of an infinite set containing all possible conceivable and inconceivable combinations of subsets exists?
Cantor, Russell and the like had to grapple with paradoxes involving the set of all sets. The set of all sets which aren't elements of themselves? That's sheer nonsense. The cardinality of the power set of the set of everything is greater than the cardinality of the set of everything? Modern day mathematical Platonists have gotten more sophisticated and only study the class of all mathematical structures. What is the difference between a set and a class? 
A: In the space of all mathematical structures, there exists artificial intelligence programs running on abstract Turing machines. The MUH hypothesis is often combined with the anthropic principle. So why aren't we such an abstract artificial intelligence program? Why bother to include the entire universe and the tedious cumbersome process of evolution and fine tuning of the laws of physics as well? It all boils down to relative probabilities, but just what are the relative probabilities?
A: Let me elaborate upon Jonathan's answer because I believe it is worth replying to. Suppose there is some mathematical universe of all possible mathematical structures. Already, "all" and "possible" have to be defined. Each mathematical structure is a whole in itself, but a structure also has to be made up of or at least contain parts. Otherwise, it can't be a structure. A single structure by itself is insufficient to describe our universe because our universe is not some undifferentiated formless singular entity. To say a given mathematical structure is our universe, that is to say, "is" and not "corresponds to" or "describes", is incomplete. Subentities within our universe exist. Many of them. They have "existenceness". MUH says these subentities are also mathematical structures in their own right because they exist, but they have to be a different mathematical structure from our universe as a whole. To say our universe is a solution to some laws of physics is to say the laws of physics exists as a mathematical structure, and there exists some set of multiverses containing all possible solutions to these laws of physics. Relations between structures have to exist, and by existing, these relations must also be mathematical structures in their own right. A tangled web of relations between structures, which is also a structure, has to exist. The many worlds interpretation posits that the wave function of the universe exists as a mathematical structure, but also that each branch exists as a mathematical structure, including the branch we find ourselves in. My point is, selecting a single structure and calling it our reality is insufficient to describe our reality.
"Exist" comes from a Latin root meaning "to stand out". To exist is to stand out from the other potential structures. A common analogy is the light analogy. If a beam of light shines upon a structure, it stands out and by virtue of standing out, exists. The light can be traced back to the Source from which the light of the world shines. The Source shines its light on one particular choice of laws of physics among all possible ones. Among all possible solutions to these laws, it shines upon one particular solution. Among all the branches making up the multiverse, it shines on our branch, a branch containing conscious observers. To shine is to "collapse the wave function". Within that branch, it shines on the conscious observer.
The light and the Source lies outside the mathematical universe. Anything which exists has to be more than just a mathematical structure of relations. It has "suchness" or "essence" coming from the Source. It is "like this" and not "like that".
A: The mathematical universe of Tegmark is Platonic Heaven. As mentioned by some of the other posters, axiom of choice infinite sets, non-measurable sets and well-orderings of the reals all exist in this Platonic Heaven. Unfortunately, there is absolutely zero empirical evidence that such a Platonic Heaven actually exists. All the "evidence" comes from mathematical formulas and mathematical proofs written in a mixture of formulae and natural language. There is a school of thought --- formalism --- claiming that only the formulae, perhaps in the form of first order logic statements or something similar exist. Symbols supposedly acting as signifiers of a signified which has no existence, pointing to a supposed Platonic Heaven described by ZFC with no real referent. A map about a territory which doesn't exist. It is from these symbols manipulated according to some formal rules that all the evidence of this Heaven comes from, but the symbols are all there is. Unless a concrete instatiation of this Platonic Heaven can be created in accordance with the laws of physics, it doesn't exist. See the wikipedia article http://en.wikipedia.org/wiki/Instantiation_principle.
It is obvious where the intuition of Platonic Heaven came from. It comes from the metaphor of completed infinity in opposition to potential infinity. Empirical observations can only support potential infinity, that is, a process which can go on forever without end with a clear prescription for getting to the next step from each current step. Going from potential infinity to completed infinity is an unwarranted metaphysical leap of faith from Becoming to Being, and Platonic Heaven can only exist as Being. It's too bad that potential infinity has a hard time incorporating the axiom of choice in general when there are infinitely many choices to be made. Cantor's diagonalization argument clearly demonstrates that the power set of an infinite set can never exist as a potential infinity.  
In this regard, the philosopher Aristotle demonstrated a lot more insight than Plato.
A: Tegmark, Hut and Alford wrote an article at http://arxiv.org/abs/physics/0510188 . Penrose had the idea of a causal loop between matter, mind and math. The authors don't like causal loops and tried to cut them at various points. Tegmark is the Platonist who believes math is fundamental and the ground of being. 
Is that really so? Math rests upon a substrate of matter. Just as Landauer showed information has to have a physical substrate, so does math. Immaterial disembodied mathematical structures don't exist. Platonic ideal forms not made up of matter?
A: According to the metaphysical Church-Turing thesis, all that exists has to be Turing computable. There should also be a unity oneness to all that exists.
So, dare I say, maybe all that exists is ONE universal computer dovetailing over all possible programs a la Schmidhuber, or ONE quantum computer running a superposition of all possible quantum programs.
