The first book I ever read was "Godel, Escher, Bach: An Eternal Golden Braid" (GEB) by Douglas Hofstadter. Of course it's not really the first book I ever read, but I consider it to be, since it's the first book I actually paid attention to. It set me off on a years-long reading frenzy to which I owe much of my current knowledge and happiness. I read GEB in early 2013, and despite reading many books since, it has remained my favorite. Until today.

Today, I finished Max Tegmark's "Our Mathematical Universe," which replaces GEB as my favorite book. But only slightly: GEB is a very close second.

Our Mathematical Universe begins with a breathtaking popular science overview of inflation theory and the evidence for it, and then delves into Tegmark's Mathematical Universe Hypothesis. Even if you aren't interested in the Mathematical Universe Hypothesis, the book is worth reading just for the extremely well-done explanation of established science.

Tegmark builds up a heirarchy of multiverses, some of which are implied by already-established scientific theories, others are more controversial:

  • The Level I multiverse: Our universe is really big and if you go far enough, you'll find an area of space that looks so similar to our own area of space that we can call it a parallel universe.

  • The Level II multiverse: Like the Level I multiverse, but really, really far away from us there might be regions of space that obey different effective laws of physics. In the same way that water can be in a state of solid, liquid, and gas, space might analogously have states resulting in different laws of physics.

  • The Level III multiverse: The many-worlds interpretation of quantum mechanics.

  • The Level IV multiverse: All mathematical structures.

Tegmark's Mathematical Universe Hypothesis (MUH) is essentially this:

The universe we inhabit not only is described by a mathematical structure, but is a mathematical structure. Moreover, all mathematical structures physically exist.

I find it more intuitive to phrase it like this:

Mathematical existence is equivalent to physical existence.

Or, in the way that makes the most sense to me:

Some mathematical structures describe worlds that contain self-aware beings. Just by the existence of the mathematical structure, those self-aware beings are indeed concious and self-aware. You don't have to add any special sauce to the mathematical structure to make it "real."

I won't summarize Tegmark's arguments for the MUH, or its implications. If you don't have time to read the book and would like a quick technical read, see Tegmark's "Is the "theory of everything" merely the ultimate ensemble theory?" I highly recommend reading that paper even if you're going to read the book.

I have a more personal connection to this book than I do to most books. After I learned about computability and the Church-Turing thesis in university, I couldn't help but think of everything in terms of computation. In some of my courses, especially philosophy ones, I couldn't help but think, "You're trying to understand this in human terms. No wonder it's not making sense. Just think about it in terms of computation and then it will!"

Understanding computability theory is liberating because it broadens your view of what's possible in principle. Before I understood computability, I thought in terms of what's practically possible, so lots of crazy ideas that I should have taken seriously seemed out of the question. Once you have a good understanding of computation, it's easy to imagine a Turing machine running for a doubly-exponential number of steps, using a doubly-exponential amount of memory, in order to find the answer to some problem. Once you can imagine mathematical objects with that sort of scale, all of the crazy ideas suddenly seem mundane.

This kind of thinking eventually lead me to an idea similar to, and probably equivalent to, the MUH. Rather than being mathematical structures as Tegmark puts it, to me universes are merely computations. Not a simulation, carried out by some alien life form, rather just the computation, expressed, for example, by writing down a Turing machine whose execution describes how the universe changes over time. You don't even have to do the computation, the mathematical existence of the Turing machine is enough.

After keeping the idea in the back of my mind for years, I came across the above-linked paper of Tegmark's. It was so refreshing to see the crazy ideas I'd been having written down, and by an MIT professor no less! I immediately bought "Our Mathematical Universe", and I was not disappointed. Nothing is missing. Every argument and counter-argument that I had ever thought of regarding my pet Turing machine universe theory is considered in the book, and there are lots of new arguments and counterarguments that I hadn't considered.

One of Tegmark's really great insights is that what appears randomness to an observer is really the observer being cloned. Here's a thought experiment to illustrate the point: After you fall asleep, a precise clone of you, including all of your mental state, is created. When you wake up, the experimenter gives you a number. One clone is given 0, the other is given 1. Before you go to sleep, because the cloning process is so precise, you have no way to predict which number you'll see when you wake up, so what you experience is randomness, even though from the experimenter's point of view, the process is deterministic.

Another way subjective randomness can arise, which I don't think Tegmark mentions in the book, is given by computational complexity. Complexity theory can put a lower bound on the amount of time a computer needs to compute a function. If nature computes the solutions to laws of physics faster than you can in your brain, then you'll have no prediction in advance of what is going to happen, and you'll experience randomness. You could, in principle, keep computing your prediction even after what you're trying to predict happened, to prove to yourself that it wasn't really random. That could keep you busy for a long time, though, since if P != NP then the same cryptography that's protecting your connection to this website could make it longer than the age of the universe!

There are a lot of great ideas in "Our Mathematical Universe," and not just ones related to the MUH. If you like the intersection of mathematics, physics, and computer science, I highly recommend it. If the conclusion is hard for you to stomach, you can think of it as hard science fiction, and you'll enjoy it just as much.