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

by Max Tegmark

Darwin’s theory thus makes the testable prediction that whenever we use technology to glimpse reality beyond the human scale, our evolved intuition should break down. We’ve repeatedly tested this prediction, and the results overwhelmingly support Darwin.

reality can have many different connotations. I use it to mean the ultimate nature of the outside physical world that we’re part of,

To me, the question “What is reality?” represents the ultimate detective story, and I consider myself incredibly fortunate to be able to spend so much of my time pursuing it.

One could indeed argue that space is a mathematical object, in the sense that its only intrinsic properties are mathematical properties—properties such as dimensionality, curvature and topology.

The highest form of ignorance is when you reject something you don’t know anything about. —Wayne Dyer

If we think of space as expanding, then we can rephrase this by saying that nothing is allowed to move faster than light through space, but space itself is free to stretch however fast it wants to.

If we take inflation seriously, then we need to start correcting people claiming that inflation happened shortly after our Big Bang, because it happened before it, creating it. It is inappropriate to define our Hot Big Bang as the beginning of time, because we don’t know whether time actually had a beginning, and because the early stages of inflation were neither strikingly hot nor big nor much of a bang. I think that the early stages of inflation are better thought of a Cold Little Swoosh, because at that time our universe was not that hot (getting a thousand times hotter once inflation ended), not that big (less massive than an apple and less than a billionth of the size of a proton) and not much of a bang (with expansion velocities a trillion trillion times slower than after inflation).

How could an infinite space get created in a finite time? It sounds impossible. But as I mentioned, inflation is like a magic show where seemingly impossible tricks happen through creative use of the laws of physics. Indeed, inflation can do something even better, which I think is its most amazing trick of all: it can create an infinite volume inside a finite volume! Specifically, it can start with something smaller than an atom and create an infinite space inside of it, containing infinitely many galaxies, without affecting the exterior space.

our Universe is a spherical region with Earth at the center. The stuff near the edges of our Universe, from which light has only now reached us after a 14-billion-year space journey, is currently about 5 × 1026 meters away from us.1 As far as we currently know, our Universe contains about 1011 galaxies, 1023 stars, 1080 protons and 1089 photons (particles of light).

With infinitely many Level I parallel universes created by inflation, quantum fluctuations effectively rolled the dice infinitely many times, guaranteeing with 100% certainty that your life would occur in one of them. Indeed, in infinitely many of them, since even a tiny fraction of an infinite number is still an infinite number.

This raises an interesting philosophical point that will come back and haunt us in Chapter 11: if there are indeed many copies of “you,” with identical past lives and memories, this kills the traditional notion of determinism: you can’t predict your own future—even if you have complete knowledge of the entire past and future history of the cosmos! The reason you can’t is that there’s no way for you to determine which of these copies is “you” (they all feel that they are). Yet their lives will typically begin to differ eventually, so the best you can do is predict probabilities for what you’ll experience from now on.

Parallel universes are not a theory, but a prediction of certain theories.

So far, our entire discussion in this chapter has been in the context of inflation. But does the Level I multiverse stand and fall with inflation? No! For there to be no Level I parallel universes at all, there must be no space whatsoever beyond the region we can see. I don’t have a single science colleague who’s argued for such a small space, and someone arguing for it could be likened to an ostrich with its head in the sand, claiming that only what it can see can exist.

we’ve seen how inflation predicts that space is flat and the spots in the cosmic microwave background should have an average size around a degree, and that the experiments described in Chapter 4 confirmed this.

If we’re living in a random habitable universe, the numbers should still look random, but with a probability distribution that favors habitability. By combining predictions about how the numbers vary across the multiverse with the relevant physics of galaxy formation and so on, we can make statistical predictions for what we should actually observe, and such predictions have so far agreed fairly well with data for dark energy, dark matter and neutrinos (Figure 6.9). Indeed, Steven Weinberg’s first prediction of a non-zero dark-energy density was made this way.

If you’re still struggling to make inner peace with parallel universes, here’s another way of thinking about them that might help. Alan Guth mentioned it in a recent MIT talk, but it has nothing to do with inflation. When we discover an object in nature, the scientific thing to do is look for a mechanism that created it. Cars are created by car factories, rabbits are created by rabbit parents and solar systems are created from gravitational collapse in giant molecular clouds. So it’s quite reasonable to assume that our Universe was created by some sort of universe-creation mechanism (perhaps inflation, perhaps something totally different). Now here’s the thing: all the other mechanisms we mentioned naturally produce many copies of whatever they create; a cosmos containing only one car, one rabbit, and one solar system would seem quite contrived. In the same vein, it’s arguably more natural for the correct universe-creation mechanism, whatever it is, to create many universes rather than just the one we inhabit.

Inflation aside, there might be other mechanisms that create universes. An idea proposed by Richard Tolman and John Wheeler and recently elaborated on by Paul Steinhardt and Neil Turok is that our cosmic history is cyclic, going through an infinite series of Big Bangs. If it exists, the ensemble of such incarnations would also form a multiverse, perhaps with a diversity similar to that of Level II. However, the cyclic models are ruled out by the gravitational wave observations of BICEP2.

So what’s everything ultimately made of? Based on the current state-of-the-art experimental evidence, the answer is clear: we simply don’t know yet, but there’s good reason to suspect that everything we know of so far—including the very fabric of spacetime itself—is ultimately made up of some more fundamental building blocks.

particle physics seems to prefer the former: every reaction that isn’t forbidden (for violating some conservation law) appears to actually occur in nature. This means that we can think of the fundamental Legos of particle physics as being not the particles themselves, but the conserved quantities! So particle physics is simply rearranging energy, momentum, charge and other conserved quantities in new ways.

A cat has energy and charge, too, but it also has many other properties besides these numbers such as its name, smell and personality—so it would sound crazy to say that the cat is a purely mathematical object completely described by those two numbers. Our elementary-particle friends, on the other hand, are completely described by their quantum numbers, and appear to have no intrinsic properties at all besides these numbers! In this sense, we’ve now come full circle back to Plato’s idea: the fundamental Legos out of which everything is made appear to be purely mathematical in nature, having no properties except mathematical properties.

De Broglie’s thesis made waves, and in November 1925, Erwin Schrödinger gave a seminar about it in Zurich. When he was finished, Peter Debye said in effect: “You speak about waves, but where is the wave equation?” Schrödinger went on to produce and publish his famous wave equation (Figure 7.4), the master key for so much of modern physics. An equivalent formulation involving tables of numbers called matrices was provided by Max Born, Pasqual Jordan and Werner Heisenberg around the same time. With this new powerful mathematical underpinning, quantum theory made explosive progress.

When we physicists describe something mathematically, we usually need to describe two separate things: 1. Its state at a given time. 2. The equation describing how this state will change over time.

In summary, Schrödinger altered the classical description of the world in two ways: 1. The state is described not by positions and velocities of the particles, but by a wavefunction. 2. The change of this state over time is described not by Newton’s or Einstein’s laws, but by the Schrödinger equation.

Copenhagen interpretation, which to this day is taught and advocated in most quantum-mechanics textbooks. A key part of it is to add a loophole to the second item mentioned above, postulating that change is only governed by the Schrödinger equation part of the time, depending on whether an observation is taking place. Specifically, if something is not being observed, then its wavefunction changes according to the Schrödinger equation, but if it is being observed, then its wavefunction collapses so that you find the object only in one place. This collapse process is both abrupt and fundamentally random, and the probability that you find the particle in any particular place is given by the square of the wavefunction. The wavefunction collapse thus conveniently gets rid of schizophrenic superpositions and explains our familiar classical world where we see things in only one place at a time.

So what was Everett’s radical idea? It’s amazingly simple to state: The wavefunction never collapses. Ever. In other words, the wavefunction that fully describes our Universe just changes deterministically at all times, always governed by the Schrödinger equation, regardless of whether there are observations taking place or not. So the Schrödinger equation rules supreme, without ifs, ands or buts.

There’s nothing random at all about the Schrödinger equation: if you know the wavefunction of our Universe right now, it will in principle let you predict what the wavefunction will be at any time in the future.

Hugh Everett’s work is still controversial, but I think that he was right and that the wavefunction never collapses. I also think that he’ll one day be recognized as a genius on par with Newton and Einstein—at least in most parallel universes. Unfortunately, in this particular universe, his work was almost completely dismissed and ignored for over a decade. He didn’t get a job in physics, became rather bitter and withdrawn, smoked and drank too much, and died of an early heart attack in 1982.

Since I believe that Hugh Everett’s parallel universes are real, I can’t help thinking about what they’re like.

As brilliant as it was, Everett’s thesis left one important question unanswered: if a large object can really be in two places at once, why don’t we ever observe that? Sure, if you measure its position, the two copies of you in the two resulting parallel universes will each find it in a definite place. But that answer turns out not to be good enough, because careful experiments show that large objects never act like they’re in two places at once, even if you don’t look at them. In particular, they never display wavelike properties that make so-called quantum interference patterns. It wasn’t just Everett’s thesis that lacked an answer to this puzzle—there were no answers in my textbooks either.

an object can only be found in two places at once in a quantum superposition as long as its position is kept secret from the rest of the world. If the secret gets out, all quantum superposition effects become unobservable, and it’s for all practical purposes as if it’s either here or there and you simply don’t know which. If a lab technician measures the position and writes it down, the information is obviously out. But even if a single photon bounces off the object, the information about its whereabouts is out: it gets encoded in the subsequent position of the photon.

Sweet exists by convention, bitter by convention, color by convention; atoms and void [alone] exist in reality. —Democritus, ca. 400 B.C.

The difficulty of linking external reality to consensus reality reached a new record high with the discovery of quantum mechanics, manifested in the fact that we physicists still argue about how to interpret the theory today, about a century after its inception. As we saw in Chapter 8, the external reality is described by a Hilbert space where a wavefunction changes deterministically over time, whereas the consensus reality is one where things happen seemingly at random, with probability distributions that can be computed to great accuracy from the wavefunction. It took over thirty years from the birth of quantum mechanics before Everett showed how these two realities could be reconciled, and the world had to wait another decade for the discovery of decoherence, which was crucial for reconciling the presence of macrosuperpositions in the external reality with their absence in the consensus reality.

there are also quantities encoded in nature that aren’t whole numbers, but require decimals to write out. Nature encodes 32 such fundamental numbers according to my latest count. Does the number shown when you stand on your bathroom scale count as such a number? No, that number doesn’t count, because it’s measuring something (your mass) that changes from day to day and therefore isn’t a basic property of our Universe. What about the mass of a proton, 1.672622 × 10−27kg, or the mass of an electron, 9.109382 × 10−31kg, which seem to stay perfectly constant over time? They don’t count either, because they’re measuring the number of kilograms, and that’s just a rather arbitrary unit of mass that we humans have made up. But if you divide one of these last two numbers by the other, then you get something truly fundamental: the proton is about 1836.15267 more massive than the electron.1 1836.15267 is a pure number, just as π or , in the sense that it’s a quantity that doesn’t involve any human units of measurement such as grams, meters, seconds or volts. Why is it close to 1836? Why not 2013? Why not 42? The short answer is that we don’t know, but that we think we can in principle calculate this number and every other fundamental constant of nature ever measured from just the 32 numbers listed in Table 10.1.

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. It is crucial not to conflate the language of mathematics (which we invent) with the structures of mathematics (which we discover).

The External Reality Hypothesis implies that a “theory of everything” (a complete description of our external physical reality) has no baggage. 2. Something that has a complete baggage-free description is precisely a mathematical structure. 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.

Everything in our world is purely mathematical—including you.

Figure 10.6: An abstract game of chess is independent of the colors and shapes of the pieces, and of whether its moves are described on a physically existing board, by stylized computer-rendered images, or by so-called algebraic chess notation—it’s still the same chess game. Analogously, a mathematical structure is independent of the symbols used to describe it.

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. Table 10.2 summarizes the relations between these and other key concepts linked to the mathematical-universe idea. Figure 10.7: Three equivalent descriptions of the same mathematical structure, which mathematicians would call an “ordered graph with four elements.” Each description contains some arbitrary baggage, but the structure that they all describe is 100% baggage-free: its four entities have no properties whatsoever except the relations that hold between them, and the relation has no properties whatsoever except the information about which elements it relates.

A famous thorny issue in philosophy is the so-called infinite regress problem. For example, if we say that the properties of a diamond can be explained by the properties and arrangements of its carbon atoms, that the properties of a carbon atom can be explained by the properties and arrangements of its protons, neutrons and electrons, that the properties of a proton can be explained by the properties and arrangements of its quarks, and so on, then it seems that we’re doomed to go on forever trying to explain the properties of the constituent parts. 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.1 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.

Mathematical Universe Cheat Sheet Baggage Concepts and words that are invented by us humans for convenience, which aren’t necessary for describing the external physical reality Mathematical structure Set of abstract entities with relations between them; can be described in a baggage-independent way Equivalence Two descriptions of mathematical structures are equivalent if there’s a correspondence between them that preserves all relations; if two mathematical structures have equivalent descriptions, they are one and the same Symmetry The property of remaining unchanged when transformed; for example, a perfect sphere is unchanged when rotated External Reality Hypothesis The hypothesis that there exists an external physical reality completely independent of us humans Mathematical Universe Hypothesis The hypothesis that our external physical reality is a mathematical structure; I argue that this follows from the External Reality Hypothesis Computable Universe Hypothesis Our external physical reality is a mathematical structure defined by computable functions (Chapter 12) Finite-Universe Hypothesis Our external physical reality is a finite mathematical structure (Chapter 12) Table 10.2: Summary of key concepts linked to the mathematical-universe idea

When I close my eyes and think of the number 5, it looks yellow to me. Yet in all these mathematical structures, the numbers themselves have no such intrinsic properties at all, and their only properties are given by their relations to other numbers—5 has the property that it’s the sum of 4 and 1, say, but it’s not yellow, and it’s not made of anything.

Another large class of mathematical structures corresponds to different types of spaces.

For example, three points can satisfy the relation that they lie on a line. There’s a different mathematical structure corresponding to Euclidean space with four dimensions and with any other number of dimensions. Mathematicians have also discovered many other types of more general spaces that form their own mathematical structures, like so-called Minkowski space, Riemann spaces, Hilbert spaces, Banach spaces and Hausdorff spaces. Many people used to think that our three-dimensional physical space was a Euclidean space. However, we saw in Chapter 2 that Einstein put an end to that. First his special relativity theory said that we live in a Minkowski space (including time as a fourth dimension), then his general relativity said that we instead live in a Riemann space, which meant that it could be curved. Then, as we saw in Chapter 7, quantum mechanics came along and said that we’re really living in a Hilbert space. Again, the points in these spaces aren’t made of anything, and have no color, texture or other intrinsic properties whatsoever.

If the Mathematical Universe Hypothesis is correct, then our Universe is a mathematical structure, and from its description, an infinitely intelligent mathematician should be able to derive all these physics theories.

The fabric of our physical reality contains dozens of pure numbers, from which all measured constants can in principle be calculated. • Some key physical entities such as empty space, elementary particles and the wavefunction appear to be purely mathematical in the sense that their only intrinsic properties are mathematical properties. • The External Reality Hypothesis (ERH)—that there exists an external physical reality completely independent of us humans—is accepted by most but not all physicists. • With a sufficiently broad definition of mathematics, the ERH implies the Mathematical Universe Hypothesis (MUH) that our physical world is a mathematical structure. • This means that our physical world not only is described by mathematics, but that it is mathematical (a mathematical structure), making us self-aware parts of a giant mathematical object.

The MUH solves the infamous infinite regress problem where the properties of nature can only be explained from the properties of its parts, which require further explanation, ad infinitum: the properties of nature stem not from properties of its ultimate building blocks (which have no properties at all), but from the relations between these building blocks.

The distinction between past, present, and future is only a stubbornly persistent illusion. —Albert Einstein, 1955

Time is an illusion, lunchtime doubly so. —Douglas Adams, The Hitchhiker’s Guide to the Galaxy

Figure 11.1: The Moon’s orbit around the Earth. We can equivalently think of this either as a position in space that changes over time (right), or as an unchanging spiral shape in spacetime (left), corresponding to a mathematical structure. The snapshots of space (right) are simply horizontal slices of spacetime (left).

smoke starts pouring out of my ears if I try to visualize four-dimensional objects.…