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

by Max Tegmark

Like an ostrich with its head in the sand, we humans have repeatedly assumed that all we could see was all that existed, hubristically imagining ourselves at the center of everything.

Fortunately, nature has provided us with a particular type of stars this helpful, called Cepheid variables. Their luminosity oscillates over time as they pulsate in size, and Harvard astronomer Henrietta Swan Leavitt discovered in 1912 that their pulsation rate acts like a watt meter: the more days there are between successive pulses, the more watts of light are radiated.

To me, a key lesson from both Newton and Friedmann is this simple mantra: “Dare to extrapolate!” Specifically, take your current understanding of the laws of physics, apply them in a new uncharted situation, and ask whether they predict something interesting that we can observe.

The famous Soviet physicist Lev Landau once said that “cosmologists are often wrong, but never in doubt,” and we’ve seen many examples of this, from Aristarchos claiming the Sun was eighteen times too close, to Hubble claiming our Universe was expanding seven times too fast.

So a telescope is essentially a Fourier transformer.

Interestingly, inflation predicts that to a good approximation, our baby Universe was scale-invariant, too, in the sense that you couldn’t tell the difference between a random cubic centimeter of it and a greatly magnified piece of

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.

the simplest and most popular cosmological model today predicts that this person actually exists in a galaxy about meters from here. This proposition doesn’t even assume speculative modern physics, but merely that space is infinite and rather uniformly filled with matter. Your alter ego is simply a prediction of eternal inflation, which, as we’ve seen in the last chapter, agrees with all current observational

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

students in Level I parallel universes would learn the same thing in physics class but different things in history class.

And if you roll the dice enough times, even the most unlikely things are guaranteed to happen. 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.

In the same way, parallel universes aren’t optional in eternal inflation. They come as part of the package, and if you don’t like them, then you have to find a different mathematical theory that solves the bang problem, the horizon problem and the flatness problem, that generates the cosmic seed fluctuations—and doesn’t predict parallel universes. This, too, has proven difficult, which is why more and more of my colleagues are—often grudgingly—beginning to take parallel universes seriously.

Quantum mechanics limits the variety even at a fundamental level. As we’ll explore in the next two chapters, quantum mechanics adds a sort of intrinsic fuzziness to nature that makes it meaningless to talk about where things are beyond a certain level of precision. The result of this limitation is that the total number of ways in which our Universe can be arranged is finite. A conservative estimate, erring on the high side, is that there are at most possible ways in which a universe the size of ours can be arranged.1 An even more conservative bound, known as the holographic principle, says that a volume the size of our Universe can be arranged in, at most, ways.2 Otherwise, you’d have to pack so much stuff into it that it would form a black hole larger than itself.

As a matter of fact, there’s mounting evidence that this is exactly how things are. Not only does our “empty space” seem to be a sort of medium, but it appears to have way more than three phases—perhaps about 10500, and perhaps even infinitely many, which opens up the possibility that, in addition to curving, stretching and vibrating, our space may even be able to do something analogous to freezing and evaporating!

Why this new pessimism? Because history is repeating itself. The Level II multiverse does to the electron’s mass what other planets did to Earth’s mass, demoting it from being a fundamental property of nature to being merely part of our cosmic address. For any number that varies across the Level II multiverse, measuring its value simply narrows down the options for what particular universe we happen to be in.

If you increase the number of space dimensions beyond three, there can be neither stable solar systems nor stable atoms. For instance, going to a four-dimensional space changes Newton’s inverse-square law for the gravitational force to an inverse-cube law, for which there are no stable orbits whatsoever.

Garden-variety atoms such as carbon, nitrogen and oxygen (which together with hydrogen make up 96% of your body weight) are so cheap because garden-variety stars such as our Sun can produce them in their death throes, after which they can form new solar systems in a cosmic recycling event. Gold, on the other hand, is produced when a star dies in a supernova explosion so violent and rare that it, during a fraction of a second, releases about as much energy as all the other stars in our observable Universe combined. No wonder making gold eluded the alchemists.

This is the essence of the Heisenberg uncertainty principle: Werner Heisenberg showed that if you confine something to a small region of space, then it will have lots of random momentum, which tends to make it spread out and become less confined. In other words, an object can’t simultaneously have an exact position and an exact velocity!2 This means that if a hydrogen atom tries to collapse as in Figure 7.5 (left) by sucking the electron into the proton, then the increasingly confined electron will get enough momentum and speed to come flying back out to a higher orbit again.

After much debate and discussion, Bohr and Heisenberg came up with a remarkably radical remedy that became known as the 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.

So what was Everett’s radical idea? It’s amazingly simple to state: The wavefunction never collapses. Ever.

Where are these parallel universes? Whereas the Level I and Level II kinds are far away in our good old three-dimensional space, the Level III ones can be right here as far as these three dimensions are concerned, but separated from us in what mathematicians call Hilbert space, an abstract space with infinitely many dimensions where the wavefunction lives.

I find that when it comes to telling the truth, the whole truth, and nothing but the truth, it’s the second part that accounts for most of the differences in how they portray reality: what they omit. I think the same holds for our senses: although they can produce hallucinations and illusions, it’s their omissions that account for most of the discrepancy between the internal and external realities. My visual system omitted the information that distinguishes between black and teal suitcases, but even if you’re not color-blind, you’re missing out on the vast majority of the information that light carries. When I was taught in elementary school that all colors of light can be made up by mixing three primary colors red, green, and blue, I thought that this number three told us something fundamental about the external reality. But I was wrong: it teaches us only about the omissions of our visual system. Specifically, it tells us that our retina has three kinds of cone cells, which take the thousands of numbers that can be measured in a spectrum of light (see Figure 2.5 in Chapter 2) and keeps only three numbers, corresponding to the average light intensity across three broad ranges of wavelengths.

Today, the grand challenge of theoretical physics is unifying quantum mechanics with gravitation. Based on this historical progression of examples, I predict that the correct mathematical theory of quantum gravity will break all previous records in being difficult to interpret. Suppose that on the eve of the next quantum-gravity conference, our friend the genie broke into the lecture hall and scribbled the equations of the ultimate theory on a blackboard. Would any of the participants realize what was being erased the next morning? I doubt it!

find it amusing how strong the conformist herd mentality is among many physicists, given that we all pay lip service to thinking outside the box and challenging authority. I’d become acutely aware of this sociological situation already back in grad school: for example, Einstein’s revolutionary relativity theory never won the Nobel Prize,1 Einstein himself dismissed Friedmann’s expanding-universe discovery, and Hugh Everett never even got a job in physics. In other words, much more important discoveries than I could realistically hope to make were being dismissed.

I think of these composite objects as emergent, in the sense that they emerge as solutions of equations involving only more fundamental objects. This emergence is subtle and easy to miss because historically, the scientific process has mostly gone in the opposite direction: for example, 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.

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. How exactly would she do this? We don’t know, but I’m quite sure about what her first step would be: to calculate the symmetries of the mathematical structure.

To the bird, reality is the geometry of the pasta.

In summary, time is not an illusion, but the flow of time is. So is change. In spacetime, the future exists and the past doesn’t disappear.

Whereas most of my physics colleagues would say that our external physical reality is (at least approximately) described by mathematics, I’m arguing that it is mathematics (more specifically, a mathematical structure).

However, in broad brushstrokes, we might say this: You’re a pattern in spacetime. A mathematical pattern. Specifically, you’re a braid in spacetime—indeed one of the most elaborate braids known.

For example, about three-quarters of your body weight is water molecules, which get replaced every month or so, and your skin cells and red blood cells are replaced every few months. In spacetime, the trajectories of these particles joining and then leaving your body make a pattern reminiscent of the familiar silk strands attached to a corncob. At both ends of your spacetime braid, corresponding to your birth and death, all the threads gradually separate, corresponding to all your particles joining, interacting and finally going their own separate ways (Figure 11.4, right). This makes the spacetime structure of your entire life resemble a tree: at the bottom, corresponding to early times, is an elaborate system of roots corresponding to the spacetime trajectories of many particles, which gradually merge into thicker strands and culminate in a single tubelike trunk corresponding to your current body (with a remarkable braidlike pattern inside as we described above). At the top, corresponding to late times, the trunk splits into ever-finer branches, corresponding to your particles going their own separate ways once your life is over. In other words, the pattern of life has only a finite extent along the time dimension, with the braid coming apart into frizz at both ends.

You feel as if you’re observing this space and time from here and now, but all that space and time are just part of the reality model that you’re experiencing. This is why you subjectively feel that time flows even though it doesn’t.

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. Much more often, it will replicate merely your disembodied brain, surviving just long enough to replicate your current observer moment. And then it will do it again, infinitely many times over, so that for every copy of you that has evolved and lived a real life, there are infinitely many delusional disembodied Boltzmann brains who think that they’ve lived that same real life.

Before you get too worried about the ontological status of your body, here’s a simple test you can do to determine whether you’re a Boltzmann brain. Pause. Introspect. Examine your memories. In the Boltzmann-brain scenario, it’s indeed more likely that any particular memories that you have are false rather than real. However, for every set of false memories that could pass as having been real, very similar sets of memories with a few random crazy bits tossed in (say, you remembering Beethoven’s Fifth Symphony sounding like pure static) are vastly more likely, because there are vastly more disembodied brains with such memories. This is because there are vastly more ways of getting things almost right than getting them exactly right. Which means that if you really are a Boltzmann brain who at first thinks you’re not, then when you start jogging your memory, you should discover more and more utter absurdities. And after that, you’ll feel your reality dissolving, as your constituent particles drift back into the cold and almost empty space from which they came. In other words, if you’re still reading this, you’re not a Boltzmann brain. This means that something is fundamentally wrong with what we assumed about the future of our Universe, and that there’s a lesson to be learned.

What’s the measure problem telling us? Here’s what I think: that 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: ∞. In fact, I have two suspects: “infinitely big”

Criticizing the continuum and related ideas, his younger colleague Leopold Kronecker once said: “God made integers; all else is the work of man.”

Why these particular equations, not others?

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. The first three levels correspond to noncommunicating parallel universes within the same mathematical structure: 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, and Level III, Everett’s “Many Worlds,” involves noncommunicating parts of the Hilbert space of quantum mechanics. Whereas all the parallel universes at Levels I, II and III obey the same fundamental mathematical equations (describing quantum mechanics, inflation, etc.), Level IV parallel universes dance to the tunes of different equations, corresponding to different mathematical structures.

Many mathematical structures—the dodecahedron, for example—lack the complexity to support any kind of self-aware substructures, so it’s likely that the Level IV multiverse resembles a vast and mostly uninhabitable desert, with life confined to rare oases, bio-friendly mathematical structures such as the one we inhabit.

It’s been widely speculated that even our own Universe may exhibit some form of spacetime discreteness that’s hidden away on such small scales that we haven’t yet noticed.

Are infinities undecidability, potential insistency, and the measure problem really inherent in the ultimate physical reality, or are they merely mirages, artifacts of our playing with fire and using powerful mathematical tools that are more convenient to work with than those that actually describe our Universe?

It’s striking that many of the continuum models of classical mathematical physics (for example, the equations describing waves, diffusion or liquid flow) are known to be mere approximations of an underlying discrete collection of atoms. Quantum-gravity research suggests that even classical spacetime breaks down on very small scales. We therefore can’t be sure that quantities that we still treat as continuous (such as spacetime, field strengths and quantum wavefunction amplitudes) aren’t mere approximations of something discrete.

Regardless of whether anything seems random to an observer, it must ultimately be an illusion, not existing at the fundamental level, because there’s nothing random about a mathematical structure. Yet the physics textbooks on my office bookshelves are full of that word: quantum measurements are said to produce random outcomes, and the heat in a cup of coffee is alleged to be caused by random motion of its molecules. Again traditional physics embraces something that the MUH rejects: what are we to make of this?

Suppose that our Universe is indeed some form of computation. A common misconception in the universe-simulation literature is that our physical notion of a one-dimensional time must then necessarily be equated with the step-by-step one-dimensional flow of the computation. I’ll argue below that if the MUH is correct, then computations don’t need to evolve our Universe, but merely describe it (defining all its relations).

Looking toward the future, there are two possibilities. 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. For example, a convincing demonstration that there’s such a thing as fundamental randomness in the laws of nature (as opposed to deterministic observer cloning that merely feels random subjectively) would therefore refute the MUH. If I’m right, on the other hand, then there’ll be no roadblock in our quest to understand reality, and we’re limited only by our imagination!

“We just got lucky—now stop looking for an explanation!” is not only unsatisfactory, but also tantamount to ignoring a potentially crucial clue.

Forecasts suggest that about a billion years from now, this solar brightening will start having a catastrophic effect on Earth’s biosphere, and that a runaway greenhouse effect will eventually boil off our oceans, much like what has already happened on Venus.

From my cosmological perspective, however, I find our performance pathetic, and can’t give more than a D: the long-term potential for life is literally astronomical, yet we humans have no convincing plans for dealing with even the most-urgent existential risks, and we devote a minuscule fraction of our attention and resources to developing such plans.

Changes in our human society result from a complex set of forces pushing in different directions, often working against each other. From a physics perspective, the easiest way to change a complex system is to find an instability, where the effect of pushing with a small force gets amplified into a major change. For example, we saw that a gentle nudge to an asteroid can prevent it from hitting Earth a decade later. Analogously, the easiest way for a single person to affect society is by exploiting an instability, as captured by numerous physics-based metaphors: an idea can be a “spark in a powder keg,” “spread like wildfire,” have a “domino effect,” or “snowball out of control.”1 For example, if you want to tackle the existential risk from killer asteroids, the hard way is to build an asteroid deflector–rocket system. The easier way is to spend much less money building an early-warning system, knowing that once you have information about an incoming asteroid, raising money for the rocket system will be easy.

All too often, schools resemble museums, reflecting the past rather than shaping the future. The curriculum should shift from one watered down by consensus and lobbying to skills our century needs for relationships, health, contraception, time management, critical thinking and recognizing propaganda. For youngsters, learning a global language and typing should trump long division and writing cursive. In the Internet age, my own role as a classroom teacher has changed. I’m no longer needed as a conduit of information, which my students can simply download on their own. Rather, my key role is inspiring a scientific lifestyle, curiosity and desire to learn more.