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

by Max Tegmark

In other words, nature contains many types of entities that are almost begging to be named. Sure enough, virtually every human population on Earth has a word for star in its own language, often invented independently to reflect its own cultural and linguistic tradition.

we humans knew of stars before realizing that they were made of atoms, we knew of atoms before realizing that they were made of electrons, protons and neutrons, and we knew of neutrons before we discovered quarks. For every emergent object that’s important to us humans, we create baggage in the form of new concepts.

Crudely speaking, as we move down the tree, the number of words goes up while the number of equations goes down, dropping to near zero for highly applied fields such as medicine and sociology. In contrast, theories near the top are highly mathematical, and physicists are still struggling to understand the concepts, if any, in terms of which we can understand them.

Wow

The Holy Grail of physics is to find what’s jocularly referred to as a “Theory of Everything,” or ToE, from which all else can be derived—this would replace the big question mark at the top of the theory tree.

It doesn’t matter whether you write, “Two plus two equals four,” “2 + 2 = 4,” or “Dos más dos es igual a cuatro.” 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.

Wow

we can reconcile the uniqueness of the Immortal Game with the multiplicity of possible descriptions of it by introducing the powerful idea of equivalence:

Concept of equivalence

We define what we mean by two descriptions being equivalent. We say that if two descriptions are equivalent, then they’re describing one and the same thing.

Concept of equivalence

Any word, concept or symbol that appears in some but not all of the equivalent descriptions is clearly optional and therefore baggage. So if we want to get down to the bare essence of the Immortal Game, then how much baggage can we strip away?

Equivalence: Two descriptions are equivalent if there’s a correspondence between them that preserves all relations.

when computers play chess, they typically use other abstract chess-position descriptions, involving certain patterns of zeros and ones in their memory. So what is it that’s left when you strip away all this baggage? What is it that’s described by all these equivalent descriptions? The Immortal Game itself, 100% pure, with no additives.

Mathematical structure: Set of abstract entities with relations between them

It is important not to confuse the description with that which is described: even the most abstract-looking description of a mathematical structure is still not the structure itself. Rather, the structure corresponds to the class of all equivalent descriptions of it.

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.

Main

The external physical reality is therefore more than the sum of its parts, in the sense that it can have many interesting properties while its parts have no intrinsic properties at all.

Main

when you ask for the time, you’re not really asking about a property of time, but rather about your location in time.

One of the most fundamental concepts in modern physics is that of a field, which is just this: something represented by numbers at each point in spacetime. For example, there’s a temperature field corresponding to the air around you:

The key point is simply that you can be an unchanging pattern in spacetime—the specific details of this pattern are less important for the points we’re making. This pattern is part of the mathematical structure that is our Universe, and the relations between different parts of the pattern are encoded in mathematical equations.

This means that how an observer moment subjectively feels depends only on what’s right there in that localized region of spacetime—not on what’s elsewhere in space (such as the external reality you see around you), and not on what’s elsewhere in time (such as what you experienced a few seconds ago).

It contains lots of information about the actual external physical reality—as long as you aren’t dreaming or hallucinating—but still constitutes only a very heavily edited version of reality,

Reality

when you watch news on TV, you’re not watching distant parts of space directly: you’re watching merely an edited movie about these parts of space. Similarly, you’re not watching the past, but an edited movie about the

words, I’m arguing that your perceptions of having a self, that subjective vantage point that you call “I,” are qualia just as your subjective perceptions of “red” or “green” are. In short, redness and self-awareness are both qualia.

qualia

Koch and Tononi in the “Suggestions for Further Reading” section. The core idea is that for an information processing system to be conscious, it needs to be integrated into a unified whole that can’t be decomposed into nearly independent parts.

we physicists can limit ourselves to starting with the external reality and predicting the consensus reality that all reasonable observers agree on, leaving the quest for the internal reality to neuroscientists and psychologists.

Main

I find it interesting that our bodily defense against microscopic enemies (our highly complex immune system) doesn’t appear to be self-aware even though our defense against macroscopic enemies (our brain controlling various muscles) does.

This is presumably because the aspects of our world that are relevant in the former case are so different (e.g., smaller length scales, longer time scales) from that of the latter that sophisticated, logical thinking and the accompanying self-awareness aren’t needed.

However, like any powerful tool, the SSSA must be used with caution. For example, why aren’t you an ant? If we take carbon-based life-forms on Earth as our reference class, our over ten quadrillion (1016) six-legged friends outnumber us bipeds by more than a million to one, so doesn’t that imply that your current observer moment is a million times more likely to be that of an ant than that of a human?

Exactly

We’re living proof that atoms can be put together in an elaborate pattern that subjectively feels self-aware.

you should live life to its fullest and do novel and interesting things. That way, in case you’re a simulation, whoever created you will be less likely to get bored and switch you off.

Ludwig Boltzmann realized that if you leave a warm object alone for enough time, even most unlikely arrangements of atoms will occur by chance. The time it will take for the particles to spontaneously rearrange themselves into a self-aware brain is extremely long, but if you wait long enough, it will happen.

???

In the standard cosmological model, this random rearranging goes on forever, so it will randomly produce an exact replica of you who subjectively feels exactly like you do, complete with false memories of having lived your entire life.

Boltzmann brain

If our spacetime really contains these Boltzmann brains, then you’re basically 100% certain to be one of them!

inflation typically goes on forever doubling the volume of space every 10−38 seconds or so, creating a messy spacetime with countless Big Bangs occurring at different times and countless planets forming at different times.

we now have a number of different measures that appear to give different but reasonable predictions, with no obvious way to choose between them. If the probabilities we predict depend on the measure we assume, and we can assume a measure giving almost any answer we want, then we really haven’t predicted anything at all.

Measure problem

there’s a fundamentally flawed assumption at the very foundation of modern physics. The failures of classical mechanics required switching to quantum mechanics, and I think that today’s best theories similarly need a major shakeup. Nobody knows for sure where the root of the problem lies, but I have my suspicions. Here’s my prime suspect: ∞.

We have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable Universe contains only about 1089 objects (mostly photons).

If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about sixteen decimal places.

In other words, the idea is that there’s a fourth level of parallel universes that’s vastly larger than the three we’ve encountered so far, corresponding to different mathematical structures.

Level I simply means distant regions from which light hasn’t yet had time to reach us,

Level II covers regions that are forever unreachable because of the cosmological inflation of intervening space,

Level III, Everett’s “Many Worlds,” involves noncommunicating parts of the Hilbert sp...

Level IV parallel universes dance to the tunes of different equations, corresponding to different mathematical structures.

a mathematical structure is simply a set of abstract elements with relations between them.

The initial-condition puzzle and the randomness puzzle are linked, and raise a pressing question. By a crude estimate, it takes almost a googol (10100) bits of information to specify the actual state of every particle in our Universe right now.

If not randomness

when you measure the initial conditions of your Universe, this information will appear random to all your copies, and it doesn’t matter whether you interpret this information as coming from initial conditions or randomness—the information is the same.

The information is the same

The way in which the data are stored (the type of computer, the data format, etc.) should be irrelevant, so the extent to which the inhabitants of the simulated universe perceive themselves as real should be independent of whatever method is used for data compression.

the External Reality Hypothesis (ERH), which says that there is an external physical reality completely independent of us humans, implies the Mathematical Universe Hypothesis (MUH), which says that our external physical reality is a mathematical structure, which in turn implies the existence of the Level IV multiverse.

If I’m wrong and the MUH is false, then physics will eventually hit an insurmountable roadblock beyond which no further progress is possible: there would be no further mathematical regularities left to discover even though we still lacked a complete description of our physical reality.

It’s quite common for mathematical equations to have multiple solutions, and as long as the fundamental equations describing our reality do, then eternal inflation generically creates huge regions of space that physically realize each of these solutions,

the equations governing water molecules, which have nothing to do with string theory, permit the three solutions corresponding to steam, liquid water and ice, and if space itself can similarly exist in different phases, inflation will tend to realize them all.

Our best theory for what put the bang into our Big Bang, the inflation theory from Chapter 5, says that there was an awful lot of rapid space-stretching going on in our early Universe, with some regions getting much more stretched than others.