Our Mathematical Universe: My Quest for the Ultimate Nature of Reality

by Max Tegmark

…trees are made of air, primarily. When they are burned, they go back to air, and in the flaming heat is released the flaming heat of the Sun which was bound in to convert the air into tree. And in the ash is the small remnant of the part which did not come from air, that came from the solid earth, instead.

had the epiphany that explained what I’d been missing all along, and what Feynman had realized: physics is the ultimate intellectual adventure, the quest to understand the deepest mysteries of our Universe. Physics doesn’t take something fascinating and make it boring. Rather, it helps us see more clearly, adding to the beauty and wonder of the world around us.

My dad once gave me the following advice: “If you have a tough question that you can’t answer, first tackle a simpler question that you can’t answer.”

In summary, an inflating substance produces an antigravity force that blows it apart, and the energy that this antigravity force expends to expand the substance creates enough new mass for the substance to retain constant density. This process is self-sustaining, and the inflating substance keeps doubling its size over and over again. In this way, inflation creates everything we can observe with our telescopes from almost nothing. This prompted Alan Guth to refer to our Universe as “the ultimate free lunch”: inflation predicts that its total energy is very close to zero!

In fact, Alan Guth and collaborators even explored the speculative possibility of doing this trick yourself for real: creating in your laboratory something that looks like a small black hole from the outside and that looks like an infinite universe from the inside—as to whether this is really possible, the jury is still out. If you’re harboring demiurgic urges, I highly recommend Brian Greene’s instructions for “aspiring universe creators” in his book The Hidden Reality.

Although is huge beyond astronomical, it’s still nothing compared with infinity. This means that if eternal inflation made a space containing infinitely many Level I parallel universes, then you’ll find it containing all possibilities. Specifically, you’ll have to check on average about universes until you find a copy of any particular kind of universe, as illustrated in Figure 6.2. So if you could travel in a straight line until you reached the closest identical copy of our own Universe, you’d need to journey about universe diameters. If you’re willing to

The entire Level II multiverse exists in a single space, so how can the dimensionality vary within it? Well, according to the most popular string-theory models, it’s only the apparent dimensionality that varies: the true space always has nine dimensions, but we don’t notice six of them because they’re microscopically curled up in the spirit of the cylinder from Figure 2.7: if you travel a tiny distance along one of these six hidden dimensions, you find yourself back where you started. Supposedly, all nine dimensions started out curled up, and then in our patch of space, inflation stretched three of them out to astronomical size while leaving six of them tiny and invisible. Elsewhere in the Level II multiverse, inflation stretched out different numbers of dimensions, creating worlds that seem anywhere from zero-dimensional to nine-dimensional.

The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don’t invent mathematical structures—we discover them, and invent only the notation for describing them. It is crucial not to conflate the language of mathematics (which we invent) with the structures of mathematics (which we discover).

Taken together, this implies the Mathematical Universe Hypothesis, i.e., that the external physical reality described by the ToE is a mathematical structure.1 So the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object—one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical—including you.

The Mathematical Universe Hypothesis offers a radical solution to this problem: at the bottom level, reality is a mathematical structure, so its parts have no intrinsic properties at all! In other words, the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.

Generations of physicists and chemists have studied what happens when you group together vast numbers of atoms, finding that their collective behavior depends on the pattern in which they’re arranged: the key difference between a solid, a liquid and a gas lies not in the types of atoms, but in their arrangement. My guess is that we’ll one day understand consciousness as yet another phase of matter. I’d expect there to be many types of consciousness just as there are many types of liquids, but in both cases, they share certain characteristic traits that we can aim to understand.

Because the Level I multiverse contains other Earths whose skies show all possible variations of cosmic microwave–background patterns or stellar constellations, the information contained in the WMAP map or a photo of the Big Dipper similarly tells us about our multiversal address. Analogously, the 32 physical constants from Chapter 10 tell us about our place in the Level II multiverse, if it exists. Although we thought all this information was about our physical reality, it was about us. The complexity is an illusion, existing only in the eye of the beholder.