In Our Mathematical Universe, Max Tegmark sets out, for the layman, his argument for the notorious Level IV Multiverse. Okay, he also gets to the other three multiverse levels, too.

So, he starts out discussing cosmology, both its history and his personal involvement in it. In some ways this was the most interesting part of the book. It goes over early modern theories about the origin and nature of the universe, their motivation, and why they were ultimately abandoned. It also discusses the Cosmic Background in detail, how it was analyzed, the attempts to map it, and his involvement in using information theory to map it more effectively.

It then goes into the best layman's explanation of inflation that I have ever read. It is clear and concise and easy to follow. He segues neatly from there into what he calls the Level 1 Multiverse. This (and the third) are the least controversial of his multiverses. This is accepted by, basically, everyone who accepts inflation. It suggests that if the universe is infinite, then everything possible happens infinitely many times and there are infinitely many versions of you reading this, and tiny variations on that. Given that he mentions Nick Bostrom several times in the end of the book and also discusses his view of our place in the universe, I'm surprised he didn't mention Bostrom's Infinite Ethics.

He then transitions from there neatly into the Level 2 multiverse. This gets the shortest treatment of any of the. I can't really blame him since it relies on the String Theory landscape and String Theory is hard. His Level 2 multiverse treats physical constants as stable phases of empty space that crystallize into different values in various pocket-universes within the inflationary field. This section is notable for an intuitively appealing argument for parallel universes in terms of a sun fine-tuned for life and whether we should expect to find multiple stars as a result and if we hadn't seen independent evidence for them. It also argues that the anthropic principle is perfectly valid and any responsible reasoner should draw on it.

The treatment of the Level 3 multiverse is completely satisfying. He explains the Copenhagen interpretation and the problems with it and offers Everett's many worlds interpretation as a counterpoint. He also takes a brief excursion into the question of why we don't see quantum effects on the macro scale and offers decoherence as the explanation, demolishing quantum consciousness in the process. I was troubled by the obvious similarity between what we would expect to find in the Level 3 multiverse and the Level 1 multiverse (the same laws but different histories), and I was happy to see he united them with his Cosmological Interpretation, which gives probabilities as fractions of worlds where we should expect to see one outcome or the other in the Level 1 multiverse.

We then get to the Level IV multiverse in which all mathematical
objects exist. As you are likely aware, I
have
quite a bit of sympathy for this idea. Tegmark bases his
reasoning on the unrealistic effectiveness of mathematics in science
and the question of why *some* mathematical structures
but *not others* should have mathematical reality. This is
the *most* controversial part of the book. How persuasive
you'll find it depends, basically, on your feelings about
parsimony. Since we have one physically instantiated mathematical
structure, adding more (even if ‘more’ is ultimately ‘all of them’)
feels more parsimonious to me than postulating some selection
mechanism that seems as if it would require its own explanation.

He addresses some potential problems, like the problems with computability of real-valued functions and Gödelian trouble arising from infinity. He ultimately rejects the continuum, mostly (it seems) motivated by a desire to solve the measure problem, but provisionally keeps infinity. This bothers me a bit, since it gets back to his question of why some mathematical objects should be privileged over others. His assertion that we will find that uncountable infinity disappears as an inconsistent illusion is not really backed up by anything. I guess you could be an Intuitionist.

He then goes on a few interesting digressions about living in
infinite multiverses covering things like the Self Sample
Assumption, the way the world ends, the likelihood of alien life,
how we give the world meaning, existential risk, and how we should
indoctrinate people into thinking scientifically. It's a mixed bag,
the bits on reasoning, meaning, and alien life are quite good. The
bit about science education is sort of brief and vague, though at
least keeps on the side of truth as the ultimate advantage in
argument. The existential threat bit is reasonable. Finding out he
was an adviser to MIRI makes me like him *slightly* less, not
because I don't agree completely with their AI risk argument (I
don't, but I don't have any problem with them making it), but
because the MIRI people seem way more into secrecy and aristocracy
than anyone I'd want to deal with would be.

I quite enjoyed this book and recommend it to anyone. Even if you reject his Level IV multiverse outright, it should at least be enjoyable as a fantasy, and the rest of the book should be enjoyable to anyone who doesn't indulge in ridiculous moral panics at the thought of any theory implying multiple cosmoi.