> It's the simplest hypothesis to date that I know of. Therefore, I assign at least a non trivial probability to it.

I don't think it's simple at all. What is the role of "mathematics" in the conjecture - what is the word supposed to mean, and what about this new definition of mathematics results in the creation of physical realities? It seems unrelated to the discipline I know of by that name, which at its most general, is an approach to defining, analyzing, and using formal models. Essentially Tegmark is saying "all formal models (of a universe) must have a concrete realization", but why, and what is the point of introducing formal models into the picture - what role do they play?

It comes down to this comment of yours:

> we try to express the laws of physics with pure mathematics since Newton, if not earlier. I think it at least indicates a hope that reality may be accurately described by pure math. Provided we can, good luck trying distinguishing that, and "being an incarnation of math." No way we can test it from within.

To test a claim, the claim first has to be stated coherently. Mathematics is an approach we use to describe and model things, including the universe. It simply isn't some sort of creative force existing independently of the minds that can contemplate it. So when someone says "the Big Bang and cosmological evolution of the universe arose out of pure geometry", they are speaking incoherent nonsense, because "pure geometry" is simply not the sort of entity that can produce such an effect.

For this to make sense, someone would have to describe the nature of this creative force that they're calling pure geometry, and then the only connection to what we normally call geometry is that ordinary geometry would be a way of describing the effects of that creative force.

Geometry or mathematics are approaches to modeling, and are neither physical phenomena themselves nor the cause of those phenomena. As soon as someone claims that "it" is the cause of phenomena, they have committed an equivocation fallacy and begun talking about some other "it" which they haven't defined.

> You mean, current evidence speaks again the level IV multiverse hypothesis? Or something else? Anyway, please name three.

I was responding more to your characterization than to Tegmark's actual definition: if "every mathematical structure just exists" and "the simpler ones ... have greater weight", then it should have observable consequences, but we don't observe such consequences. For example, we might expect things to be constructed from pure Platonic solids, we might expect the subatomic realm to be less fuzzy, etc.

> With enough double and triple checking, computer-verified proofs… we can be rather sure we did find the simplest explanation.

Actually I don't agree with that. Finding simplicity does not always lend itself to formal process. But I'm saying that although Occam's Razor can remind us that a better explanation could be simpler than the one we currently have, it does not tell us that this is the case.

> I was just saying that when I hear someone claiming something that I don't find very surprising, I generally take that as serious evidence that the claim is true.

We differ on that. The fact that something is not surprising is not evidence. Many claims that are not surprising turn out to be false. To be more likely to believe something that seems unsurprising to you implies that you're choosing beliefs based on personal bias.

> Quantum mechanics has to do with complex amplitudes, whose square determine the Born statistics. Plus, the wave function as we know it is deterministic.

None of this contradicts what I was saying. Those complex amplitudes are complex to deal with superposition, and once that is taken into account, probability densities in QM are described perfectly by probability theory, so I don't know in what sense you mean that the relation is "tenuous". This has very little to do with one's position on interpretations, it's there in the math whether you like it or not.