Max Tegmark's Mathematical Universe Hypothesis is extremely compelling - it's the only idea I've ever seen that comes close to the holy grail of theories of everything "Of course! How could it possibly be any other way!". And yet it seems unsatisfying in a way, perhaps because it's conclusion is (when you think about it for a bit) completely obvious. Of course the universe is a mathematical structure containing self-aware substructures - what else could it be?
Also, if it is true then it leaves a lot of questions unanswered:
- What is the mathematical structure that describes our universe?
- What is the mathematical definition of a self-aware substructure (SAS)?
- If any mathematical (or even computable, or finite) structure that contains SASs exists in the same way for those SASs that our universe exists for us, why is our universe as simple and explicable as it is (Tegmark calls this the "measure problem").
My feeling is (as I've said on this blog before) that it's extremely likely that our universe is in some sense the simplest possible universe that supports SASs (i.e. the "measure" punishes complexity to the maximum possible extent). I have no a priori justification for this, though - it just seems to me to be the most likely explanation for the third point above. While it may seem unnecessary to have three generations of leptons and quarks, I strongly believe that when we have a more complete theory of physics we'll discover that they are completely indispensible - a universe like ours but with only a single generation of these particles would either be more complex (mathematically speaking) or make it impossible for SASs to exist. I suppose it is possible however, that when we do find the theory of everything (and/or a theory of SASs) we'll be able to think of a simpler structure in which SASs are possible.
The other thing about MUH is that I'm not convinced that it really does make any predictions at all, because it seems like whatever we discover about our universe with respect to the other possible universes in the Level IV multiverse can be made consistent with MUH by an appropriate choice of measure.