What are some points to refute mathematical universe hypothesis? How can the concept of math cause the emergence of universes just because there is a possibility of it? And how does it determine the size,features of the universe? It doesn't seem right in some ways. Some living beings have consciousness which math itself doesn't have. Math cannot feel or think or doesn't have free will to do as it wants so how can living beings have consciousness when math itself doesn't have it and but somehow gave rise to a universe with consciousness? It doesn't seem correct or necessary.
My suggestion is don't call it a category error, or problematize his math. He can easily find refuge within the mysteries and idiosyncratic views of consciousness. Most arguments against MUH attack something else for this reason, even though he's written much of current math won't be much more than a conscious illusion. I have to point out too Pigliuicci's objections came before printed release and the chapters Tegmark brings up in his response are largely about consciousness.
With such a large pre-established gap between objective and subjective reality (see the Maudlin quote below), in the very same move of claiming a category error, detractors are hardly refuting much at all. I find these attacks weak because Tegmark is equally writing about subjective experience. Calling a category error is just giving Tegmark more line to connect two disparate realms--we know there's a huge gap. "How pulses of water in pipes might give rise to toothaches is indeed entirely incomprehensible, but no less so than how electro-chemical impulses along neurons can." (Maudlin).
For these reasons, don't levy a category error critique. If the gap to explaining subjective experience is so great, this critique is unoriginal and washes right off. Extend him the same mighty courtesy we've extended to other philosophers who dare write on what consciousness is. He is trying the same near-impossible task.
Refute it for being too unspecific or nigh vacuous. Psychologists doing the heavy lifting in the future, his main hope for explaining all the illusions be brings up, is not specific enough.
My two cents ...
Sticking me neck out ... what's left of it anyway ... but if any mathematization of reality ... the number crunching therein ... involves 1) Zero, division by zero to be precise and/or 2) Infinity, calculations involving it, it spells trouble for The Mathematical Universe Hypothesis.
That said, there are pretty neat workarounds like, for instance, The Holographic Universe which states that what's inside a black hole (, infinity, mathematically inaccessible) is on its surface (mathematically accessible) or something like that and per Numberphile (you tube channel), we can substitute (-1/12) for infinity for some physcis calculations. So much for infinity.
Too, what about matter, they're considered to be constituted of of point particles - mass(ive) objects with zero volume - and what may we say about their density (mass/volume = mass/0)?
Just a repeat of Hippasus of Metapontum vs. the Pythagoreans. Hippasus discovered the square root of 2 is irrational and that, I'm told, messed up Pythagoras' doctrine, MUH version 1.0.
Like many other hypotheses, the MUH cannot be refuted because it is not open to experimental testing. If I told you, for example, that the Universe is the energy of Kaskri Rama, you would not be able to refute my hypothesis, because it is an unsubstantiated point of view that has no experimental consequences and it is sufficiently vague that I can adapt it at will to dodge any logical objections you might raise.
That said, if the vague and unsubstantiated nature of the MUH is not enough for you to write it off as manifest nonsense, there are certain other grounds that might lead you to find it intellectually unpalatable. For instance, one might raise the following objections...
Mathematics rests upon a set of foundational axioms, and the axioms themselves are in some respects arbitrary. By adopting different sets of axioms we can arrive at different mathematical results, all of which are equally valid in a certain sense, so one problem is that 'mathematics' is very ill-defined.
The reality around us is very complicated, and markedly different from mathematical entities with which we are very familiar. For example, as far as we know, we never encounter perfect circles, perfect cubes, perfectly parallel lines and so on; we don't encounter objects that can be split into simple fractions (eg a third of an atom). It seems odd, therefore, that in a mathematical universe we don't encounter some of the mathematical entities we consider to be among the most basic. Conversely, there are morse esoteric aspects of mathematics that we don't encounter in reality. For example, our spacetime seems to be limited to 4 dimensions, rather than 57, say. So the question then is why does our Universe consist of a particular subset of mathematical entities? You might dodge these questions by declaring that there are infinitely many Universes in which perfect circles etc are encountered, and ours is just one at random, but in what sense is that of any value?
Clearly, too, there is a difference between abstract mathematical objects and physical ones, so how is that to be understood? What causes certain mathematical entities to be reified and others not?
Indeed, why limit the universe to mathematics? Why not generalise it to a Conceptual Universe Hypothesis, in which there is a multiverse containing all concepts, whether mathematical or not?
In short, the MUH might be an interesting collection of ideas, but since has no predictive powers, a cynic might be inclined to file it in the drawer labelled 'So what?'.