In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the Ultimate Ensemble, is a speculative "theory of everything" (TOE) proposed by the theoretical physicist, Max Tegmark.This seems silly to me. I am more interested in what I call the weak mathematical universe hypothesis (WMUH). It merely says that there exists at least one structure that exists both mathematically and physically.
Tegmark's sole postulate is: All structures that exist mathematically also exist physically. That is, in the sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world". The hypothesis suggests that worlds corresponding to different sets of initial conditions, physical constants, or altogether different equations should be considered equally real.
For example, if we had a faithful mathematical representation (model) of an electron, then that would qualify. We do not. By faithful, I mean that some set of numbers, formulas, and other mathematical entities fully capture all aspects of the electron.
Tegmark seems to assume that our universe has a perfect mathematical representation, so he thinks that related models could correspond to other universes. But the statement that our own universe can be so mathematized requires a huge leap of faith. As far as I know, there is no known way to do it, and no good reason to believe that it is possible.
If an electron were a (classical) particle, then it could be faithfully represented by its position, velocity, mass, electric charge, and maybe its spin (angular momentum). This is impossible for a quantum particle, because of the uncertainty principle.
So here is the crux of my argument. If you believe in an external reality independent of humans, then you must also believe in what I call the MUH: that our physical reality is a mathematical structure. In other words, we all live in a gigantic mathematical object – one that is more elaborate than a dodecahedron, ... Everything in our world is purely mathematical – including you.I say that we should start with the weak MUH hypothesis that something in our world is purely mathematical. I do believe in an external reality independent of humans, but I don't see any reason to believe in the weak MUH. I prefer to reject it, until someone shows how it can be true.
The belief that physical reality has an objective mathematical structure has been known as hidden variable theory for most of the 20th century. It has been a complete failure, and the consensus is that these theories are wrong.
Tegmark also confusingly calls his view an “extreme shut-up-and-calculate approach to physics”. This terminology refers to using quantum mechanics to predict experiments, without having a realistic model of the underlying physics. The "shut up" is an admonition to ignore the underlying physics as long as you can calculate answers. So the shut-up-and-calculate approach is directly contrary to the mathematical universe hypothesis, because it disavows any need for the physical universe to be a mathematical structure.
I think that Tegmark's MUH is at the core of many misunderstandings of modern physics. The WMUH is implicitly assumed by most physicists, in spite of decades of evidence to the contrary. It is time to recognize it as the implausible hypothesis that it is, and to only use it when it is explicitly stated as a dubious assumption.
When physicists are presented evidence against the WMUH, they will often despair that reality does not exist, or that nature is incomprehensible, or some such nonsense. None of that follows. We just have to reject Tegmark's extreme views.