Book review: Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, by Max Tegmark.

His most important claim is the radical Platonist view that all well-defined mathematical structures exist, therefore most physics is the study of which of those we inhabit. His arguments are more tempting than any others I’ve seen for this view, but I’m left with plenty of doubt.

He points to ways that we can imagine this hypothesis being testable, such as via the fine-tuning of fundamental constants. But he doesn’t provide a good reason to think that those tests will distinguish his hypothesis from other popular approaches, as it’s easy to imagine that we’ll never find situations where they make different predictions.

The most valuable parts of the book involve the claim that the multiverse is spatially infinite. He mostly talks as if that’s likely to be true, but his explanations caused me to lower my probability estimate for that claim.

He gets that infinity by claiming that inflation continues in places for infinite time, and then claiming there are reference frames for which that infinite time is located in a spatial rather than a time direction. I have a vague intuition why that second step might be right (but I’m fairly sure he left something important out of the explanation).

For the infinite time part, I’m stuck with relying on argument from authority, without much evidence that the relevant authorities have much confidence in the claim.

Toward the end of the book he mentions reasons to doubt infinities in physics theories – it’s easy to find examples where we model substances such as air as infinitely divisible, when we know that at some levels of detail atomic theory is more accurate. The eternal inflation theory depends on an infinitely expandable space which we can easily imagine is only an approximation. Plus, when physicists explicitly ask whether the universe will last forever, they don’t seem very confident. I’m also tempted to say that the measure problem (i.e. the absence of a way to say some events are more likely than others if they all happen an infinite number of times) is a reason to doubt infinities, but I don’t have much confidence that reality obeys my desire for it to be comprehensible.

I’m disappointed by his claim that we can get good evidence that we’re not Boltzmann brains. He wants us to test our memories, because if I am a Boltzmann brain I’ll probably have a bunch of absurd memories. But suppose I remember having done that test in the past few minutes. The Boltzmann brain hypothesis suggests it’s much more likely for me to have randomly acquired the memory of having passed the test than for me to actually be have done the test. Maybe there’s a way to turn Tegmark’s argument into something rigorous, but it isn’t obvious.

He gives a surprising argument that the differences between the Everett and Copenhagen interpretations of quantum mechanics don’t matter much, because unrelated reasons involving multiverses lead us to expect results comparable to the Everett interpretation even if the Copenhagen interpretation is correct.

It’s a bit hard to figure out what the book’s target audience is – he hides the few equations he uses in footnotes to make it look easy for laymen to follow, but he also discusses hard concepts such as universes with more than one time dimension with little attempt to prepare laymen for them.

The first few chapters are intended for readers with little knowledge of physics. One theme is a historical trend which he mostly describes as expanding our estimate of how big reality is. But the evidence he provides only tells us that the lower bounds that people give keep increasing. Looking at the upper bound (typically infinity) makes that trend look less interesting.

The book has many interesting digressions such as a description of how to build Douglas Adams’ infinite improbability drive.