Our Mathematical Universe
I have not read Max Tegmark’s new book, but the argument that mathematics is the ultimate reality of the universe is a strong one that has been around for a long time. I would agree that mathematics is an irreplaceable tool for understanding the universe, and for understanding knowledge, but mathematics alone is not sufficient to derive the actual universe which we experience.
In my view, mathematics can only be an emergent property of representation and therefore ephiphenomenal. The underlying (and overarching) phenomena of presence or presentation is fundamentally aesthetic and consists of sensory-motor experiences. This is not a biocentric view as inorganic matter is also, by my understanding, a tokenization of aesthetic experiences. The universe is a significance-building machine, where significance is the temporal super-saturation and transcendence of sensory qualities.
As far as comparing mathematics to computation, mathematics seems to be a broader category which would embrace ideas which computation cannot, such as irrational numbers and geometric forms. While computation can be used to drive a sensory experience in which geometric forms are inscribed visually or sculpted tangibly, those outputs are irrelevant to the computation itself and are desirable to us purely for aesthetic reasons.
Computation is, however, closer to empirical realism than other kinds of mathematics, since it is rooted in digital interactions which can be reproduced and re-presented in any solid-body/persistent-position form-function. If there is no discrete fundamental unit which is subject to reliable inspection (which is an experiential and aesthetic property that is generally overlooked ) then computation cannot be initiated or preserved.
I get into this a bit here: https://multisenserealism.com/2013/06/06/mathmatical-musings/
Mathematics requires a mind and a brain while computation requires only a brain substitute. By this I mean that the sense of computation is a low level sensory-motor interaction through which higher level interactions can be transported from one location to another. This transportation offers the opportunity for reconstruction only if the receiver has the appropriate frame of reference to imitate the sender’s intents. We use a computer to listen to music or watch a video, but in the absence of human receivers, there would only product would be disconnected instants of acoustic or optical activity.
Mathematics similarly owes its universality to its exploitation of a low level ‘common sense’ which depends on similarly overlooked assumptions about the validity of conceptual realism. Mathematics depends on sanity in the intellectual and logical sense. It presumes an aesthetic minimalism. Where computation can be more clearly seen to depend on concrete mechanisms of read/write/erase, storage, pattern recognition, loops, etc., mathematics seeks a more anesthetic representation – as Baudrillard might have said, a simulacra: A representation without any presentation. In my understanding, mathematics can be thought of as an ultimate reality only in the sense that all of our intellectual models of reality can be rendered in mathematical ways.
Unfortunately, most people conflate the idea of reality with the experience of it, and have developed a misplaced authority for “information” as the progenitor of physics and awareness. This is, in my view, almost correct, but actually upside down as information can only ride on top of an aesthetic exchange of experiences, which involves public to private extractions of significance and private to public export of entropy. Information, by itself, has never done anything. No byte of data will ever feel anything, be anything, want anything, go anywhere, etc. Mathematics deals in figures, which have no form or function but represent forms and functions. What figures cannot represent is presence itself. There is no substitute for experience, and that is why it is experience which is the ultimate reality – the absolute and authentic substrate of the universe is a unique agenda of aesthetics, not a generic consequence of configured figures.
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Multisense Realism 
If Max Tegmark thinks that “mathematics is the ultimate reality of the universe,” alls I gotta say is wow.
Math describes what, where, and when. It does not explain how, and, most importantly, why. Never has, never will.
I’d like Tegmark to answer the question about what a number, in fact, is. Or, alternatively, tell me where numbers come from.
Sheesh. The truth is that I am the numbers themselves. I am the mathematician and the math in One.
Yes, the who and the why are two questions that math-centric worldviews can only try to un-ask by making them seem irrelevant. It’s like an ambiguous image in that when someone focuses on the vase in the middle, they can’t see the two faces on either side in the negative space. Focusing on math gives us an idea of what, where, and how (aka matter, space, and energy btw) that is so clear and compelling that we can convince ourselves that the who, when, and why (sense, time, motive) can only be unreal.