Is the Universe Actually Made of Math?

Cosmologist Max Tegmark says mathematical formulas create reality.

By Adam Frank|Monday, June 16, 2008
photography by Erika Larsen

Cosmologists are not your run-of-the-mill thinkers, and Max Tegmark is not your run-of-the-mill cosmologist. Throughout his career, Tegmark has made important contributions to problems such as measuring dark matter in the cosmos and understanding how light from the early universe informs models of the Big Bang. But unlike most other physicists, who stay within the confines of the latest theories and measurements, the Swedish-born Tegmark has a night job. In a series of papers that have caught the attention of physicists and philosophers around the world, he explores not what the laws of nature say but why there are any laws at all.

According to Tegmark, “there is only mathematics; that is all that exists.” In his theory, the mathematical universe hypothesis, he updates quantum physics and cosmology with the concept of many parallel universes inhabiting multiple levels of space and time. By posing his hypothesis at the crossroads of philosophy and physics, Tegmark is harking back to the ancient Greeks with the oldest of the old questions: What is real?

Tegmark has pursued this work despite some risk to his career. It took four tries before he could get an early version of the mathematical universe hypothesis published, and when the article finally appeared, an older colleague warned that his “crackpot ideas” could damage his reputation. But propelled by optimism and passion, he pushed on.

“I learned pretty early that if I focused exclusively on these big questions I’d end up working at McDonald’s,” Tegmark explains. “So I developed this Dr. Jekyll/Mr. Hyde strategy where officially, whenever I applied for jobs, I put forth my mainstream work. And then quietly, on the side, I pursued more philosophical interests.” The strategy worked. Today a professor at the Massachusetts Institute of Technology, Tegmark travels among the world’s top physicists. Backed by this well-earned credibility, his audacious ideas are sparking fascination and taking flight.

These days Tegmark is a busy man. With his wife, the Brazilian cosmologist Angelica de Oliveira-Costa, he balances science with the demands of raising two young boys. Our interviewer, theoretical astrophysicist Adam Frank of the University of Rochester in New York, finally caught up with Tegmark as he made his way home to Winchester, Massachusetts, from a conference at Stanford University. In a comic juxtaposition of the profound and the profane, they spoke about the nature of reality by cell phone for three hours as Tegmark jockeyed his way through an airport rental car return, security lines, and a long wait for a delayed flight. A riff on reality would brake to a halt so Tegmark could avoid being hit by a rental-agency van. Just as the conversation plunged into parallel universes, Tegmark would have to downshift the dialogue for the bewildered security guard checking his boarding pass. Tegmark’s infectious excitement over the big issues, from physics and philosophy to kids and cosmology, made for one hell of an afternoon’s ride.

Max, you have gained a reputation for thinking far outside the box even for a cosmologist. Have you always pondered deep questions of Life, the Universe, and Everything?
No. I was a very confused youth. I came to it all pretty late, and there was no one I talked about philosophy with as a teenager. I did have one friend in high school who did everything the opposite way from everyone else. If people were sending letters in rectangular envelopes, he would make triangular envelopes and send letters in those. I remember thinking, “That is cool. That is how I want to be.”

Is that why you decided to go into physics?
Actually, my dad is a mathematician, and he was always very encouraging about math, but physics was my single most boring subject in high school. So I began as an undergrad in economics.

That was an interesting choice....When did physics show up on your radar screen again?
A friend gave me a book, Surely You’re Joking, Mr. Feynman! by the physicist Richard Feynman. It was all about picking locks and picking up women. It had nothing to do with physics, but it struck me how between the lines it said loud and clear, “I love physics!” I couldn’t understand how this could be the same boring stuff from high school. It really piqued my curiosity.

How so?
If you see some mediocre guy walking down the street arm in arm with Cameron Diaz, you say to yourself, “I’m missing something here.” So I started reading Feynman’s Lectures on Physics and I was like, whoa! Why haven’t I realized this before?

So then you changed your major?
Umm, no. You don’t pay for college in Sweden, so I was able to do this kind of scam where I enrolled in a different university to do physics without telling them I was already in college for economics.

You were in two colleges at the same time?
Yeah. You can see I was confused. It got complicated at times. I would have exams in both places on the same day, and I’d have to bike really fast between them.

Was it in college that you started to think about the bigger questions?
I was taking the one and only quantum physics class offered, and when I got to the chapter on measurement I felt sure that I was missing something.

You’re talking about the way the observer appears to affect the measurement of what’s being observed.
Right. There is this beautiful mathematical equation in quantum theory called the Schrödinger equation. It uses something called the wave function to describe the system you are studying—an atom, an electron, whatever—and all the possible ways that system can evolve. The usual perspective of quantum mechanics is that as soon as you measure something, the wave function literally collapses, going from a state that reflects all potential outcomes to a state that reflects only one: the outcome you see at the moment the measurement is done. It seemed crazy to me. I didn’t get why you were supposed to use the Schrödinger equation before you measured the atom, but then, while you’re measuring it, the equation doesn’t apply. So I got up my courage and knocked on the door of one of the most famous physicists in Sweden, a man on the Nobel committee, but he just blew me off. It wasn’t until years later that I had this revelation that it wasn’t me who didn’t get it; it was him!

It is a beautiful moment in the education of a scientist when you realize that these guys in higher positions of power still don’t have all of the answers. So you took your questions about the Schrödinger equation and the effect of measurement with you when you left for the United States and your Ph.D. at Berkeley?
That’s where it all started for me. I had this friend, Bill Poirier, and we spent hours talking about crazy ideas in physics. He was ribbing me because I argued that any fundamental description of the universe should be simple. To annoy him, I said there could be a whole universe that is nothing more than a dodecahedron, a 12-sided figure the Greeks described 2,500 years ago. Of course, I was just fooling around, but later, when I thought more about it, I got excited about the idea that the universe is really nothing more than a mathematical object. That got me thinking that every mathematical object is, in a sense, its own universe.

Right from the start you tried to get this radical idea of yours published. Were you worried about whether it would affect your career?
I anticipated problems and did not submit until I had accepted a postdoctoral appointment at Princeton University. My first paper got rejected by three journals. Finally I got a good referee report from Annals of Physics, but the editor there rejected the paper as being too speculative.

Wait—that is not supposed to happen. If the referee likes a paper, it usually gets accepted.
That’s what I thought. I was fortunate to be friends with John Wheeler, a Princeton theoretical physicist and one of my greatest physics heroes, who recently passed away. When I showed him the rejection letter, he said, “‘Extremely speculative’? Bah!” Then he reminded me that some of the original papers on quantum mechanics were also considered extremely speculative. So I wrote an appeal to Annals of Physics and included Wheeler’s comments. Finally the editors there published it.

Still, it wasn’t your bread and butter. You did your Ph.D. and postdoc in cosmology, a totally different subject.
It’s ironic that my cover for these more philosophical interests was cosmology, a field that has often been seen as flaky as well. But cosmology was gradually becoming more respectable because computer technology, space technology, and detector technology had combined to give us an avalanche of great information about the universe.

Let’s talk about your effort to understand the measurement problem by positing parallel universes—or, as you call them in aggregate, the multiverse. Can you explain parallel universes?
There are four different levels of multiverse. Three of them have been proposed by other people, and I’ve added a fourth—the mathematical universe.

What is the multiverse’s first level?
The level I multiverse is simply an infinite space. The space is infinite, but it is not infinitely old—it’s only 14 billion years old, dating to our Big Bang. That’s why we can’t see all of space but only part of it—the part from which light has had time to get here so far. Light hasn’t had time to get here from everywhere. But if space goes on forever, then there must be other regions like ours—in fact, an infinite number of them. No matter how unlikely it is to have another planet just like Earth, we know that in an infinite universe it is bound to happen again.

You’re saying that we must all have doppelgängers somewhere out there due to the mathematics of infinity.
That’s pretty crazy, right? But I’m not even asking you to believe in anything weird yet. I’m not even asking you to believe in any kind of crazy new physics. All you need for a level I multiverse is an infinite universe—go far enough out and you will find another Earth with another version of yourself.

So we are just at level I. What’s the next level of the multiverse?
Level II emerges if the fundamental equations of physics, the ones that govern the behavior of the universe after the Big Bang, have more than one solution. It’s like water, which can be a solid, a liquid, or a gas. In string theory, there may be 10500 kinds or even infinitely many kinds of universes possible. Of course string theory might be wrong, but it’s perfectly plausible that whatever you replace it with will also have many solutions.

Why should there be more than one kind of universe coming out of the Big Bang?
Inflationary cosmology, which is our best theory for what happened right after the Big Bang, says that a tiny chunk of space underwent a period of rapid expansion to become our universe. That became our level I multiverse. But other chunks could have inflated too, from other Big Bangs. These would be parallel universes with different kinds of physical laws, different solutions to those equations. This kind of parallel universe is very different from what happens in level I.

Well, in level I, students in different parallel universes might learn a different history from our own, but their physics would still be the same. Students in level II parallel universes learn different history and different physics. They might learn that there are 67 stable elements in the periodic table, not the 80 we have. Or they might learn there are four kinds of quarks rather than the six kinds we have in our world.

Do these level II universes inhabit different dimensions?
No, they share the same space, but we could never communicate with them because we are all being swept away from each other as space expands faster than light can travel.

OK, on to level III.
Level III comes from a radical solution to the measurement problem proposed by a physicist named Hugh Everett back in the 1950s. [Everett left physics after completing his Ph.D. at Prince­ton because of a lackluster response to his theories.] Everett said that every time a measurement is made, the universe splits off into parallel versions of itself. In one universe you see result A on the measuring device, but in another universe, a parallel version of you reads off result B. After the measurement, there are going to be two of you.

So there are parallel me’s in level III as well.
Sure. You are made up of quantum particles, so if they can be in two places at once, so can you. It’s a controversial idea, of course, and people love to argue about it, but this “many worlds” interpretation, as it is called, keeps the integrity of the mathematics. In Everett’s view, the wave function doesn’t collapse, and the Schrödinger equation always holds.

The level I and level II multiverses all exist in the same spatial dimensions as our own. Is this true of level III?
No. The parallel universes of level III exist in an abstract mathematical structure called Hilbert space, which can have infinite spatial dimensions. Each universe is real, but each one exists in different dimensions of this Hilbert space. The parallel universes are like different pages in a book, existing independently, simultaneously, and right next to each other. In a way all these infinite level III universes exist right here, right now.

That brings us to the last level: the level IV multiverse intimately tied up with your mathematical universe, the “crackpot idea” you were once warned against. Perhaps we should start there.
I begin with something more basic. You can call it the external reality hypothesis, which is the assumption that there is a reality out there that is independent of us. I think most physicists would agree with this idea.

The question then becomes, what is the nature of this external reality?
If a reality exists independently of us, it must be free from the language that we use to describe it. There should be no human baggage.

I see where you’re heading. Without these descriptors, we’re left with only math.
The physicist Eugene Wigner wrote a famous essay in the 1960s called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In that essay he asked why nature is so accurately described by mathematics. The question did not start with him. As far back as Pythagoras in the ancient Greek era, there was the idea that the universe was built on mathematics. In the 17th century Galileo eloquently wrote that nature is a “grand book” that is “written in the language of mathematics.” Then, of course, there was the great Greek philosopher Plato, who said the objects of mathematics really exist.

How does your mathematical universe hypothesis fit in?
Well, Galileo and Wigner and lots of other scientists would argue that abstract mathematics “describes” reality. Plato would say that mathematics exists somewhere out there as an ideal reality. I am working in between. I have this sort of crazy-sounding idea that the reason why mathematics is so effective at describing reality is that it is reality. That is the mathematical universe hypothesis: Mathematical things actually exist, and they are actually physical reality.

OK, but what do you mean when you say the universe is mathematics? I don’t feel like a bunch of equations. My breakfast seemed pretty solid. Most people will have a hard time accepting that their fundamental existence turns out to be the subject they hated in high school.
For most people, mathematics seems either like a sadistic form of punishment or a bag of tricks for manipulating numbers. But like physics, mathematics has evolved to ask broad questions.These days mathematicians think of their field as the study of “mathematical structures,” sets of abstract entities and the relations between them. What has happened in physics is that over the years more complicated and sophisticated mathematical structures have proved to be invaluable.

Can you give a simple example of a mathematical structure?
The integers 1, 2, 3 are a mathematical structure if you include operations like addition, subtraction, and the like. Of course, the integers are pretty simple. The mathematical structure that must be our universe would be complex enough for creatures like us to exist. Some people think string theory is the ultimate theory of the universe, the so-called theory of everything. If that turns out to be true, then string theory will be a mathematical structure complex enough so that self-awareness can exist within it.

But self-awareness includes the feeling of being alive. That seems pretty hard to capture in mathematics.
To understand the concept, you have to distinguish two ways of viewing reality. The first is from the outside, like the overview of a physicist studying its mathematical structure. The second way is the inside view of an observer living in the structure. You can think of a frog living in the landscape as the inside view and a high-flying bird surveying the landscape as the outside view. These two perspectives are connected to each other through time.

In what way does time provide a bridge between the two perspectives?
Well, all mathematical structures are abstract, immutable entities. The integers and their relations to each other, all these things exist outside of time.

Do you mean that there is no such thing as time for these structures?
Yes, from the outside. But you can have time inside some of them. The integers are not a mathematical structure that includes time, but Einstein’s beautiful theory of relativity certainly does have parts that correspond to time. Einstein’s theory has a four-dimensional mathematical structure called space-time, in which there are three dimensions of space and one dimension of time.

So the mathematical structure that is the theory of relativity has a piece that explicitly describes time or, better yet, is time. But the integers don’t have anything similar.
Yes, and the important thing to remember is that Einstein’s theory taken as a whole represents the bird’s perspective. In relativity all of time already exists. All events, including your entire life, already exist as the mathematical structure called space-time. In space-time, nothing happens or changes because it contains all time at once. From the frog’s perspective it appears that time is flowing, but that is just an illusion. The frog looks out and sees the moon in space, orbiting around Earth. But from the bird’s perspective, the moon’s orbit is a static spiral in space-time.

The frog feels time pass, but from the bird’s perspective it’s all just one eternal, unalterable mathematical structure.
That is it. If the history of our universe were a movie, the mathematical structure would correspond not to a single frame but to the entire DVD. That explains how change can be an illusion.

Of course, quantum mechanics with its notorious uncertainty principle and its Schrödinger equation will have to be part of the theory of everything.
Right. Things are more complicated than just relativity. If Einstein’s theory described all of physics, then all events would be predetermined. But thanks to quantum mechanics, it’s more interesting.

But why do some equations describe our universe so perfectly and others not so much?
Stephen Hawking once asked it this way: “What is it that breathes fire into the equations and makes a universe for them to describe?” If I am right and the cosmos is just mathematics, then no fire-breathing is required. A mathematical structure doesn’t describe a universe, it is a universe. The existence of the level IV multiverse also answers another question that has bothered people for a long time. John Wheeler put it this way: Even if we found equations that describe our universe perfectly, then why these particular equations and not others? The answer is that the other equations govern other, parallel universes, and that our universe has these particular equations because they are just statistically likely, given the distribution of mathematical structures that can support observers like us.

These are pretty broad and sweeping ideas. Are they just philosophical musings, or is there something that can actually be tested?
Well, the hypothesis predicts a lot more to reality than we thought, since every mathematical structure is another universe. Just as our sun is not the center of the galaxy but just another star, so too our universe is just another mathematical structure in a cosmos full of mathematical structures. From that we can make all kinds of predictions.

So instead of exploring just our universe, you look to all possible mathematical structures in this much bigger cosmos.

If the mathematical universe hypothesis is true, then we aren’t asking which particular mathematical equations describe all of reality anymore. Instead we have to figure out how to separate the frog’s view of the universe—our observations—from the bird’s view. Once we distinguish them we can determine whether we have uncovered the true structure of our universe and figure out which corner of the mathematical cosmos is our home.
Max, this is pretty rarefied territory. On a personal level, how do you reconcile this pursuit of ultimate truth with your everyday life?
Sometimes it’s quite comical. I will be thinking about the ultimate nature of reality and then my wife says, “Hey, you forgot to take out the trash.” The big picture and the little picture just collide.

Your wife is a respected cosmologist herself. Do you ever talk about this over breakfast cereal with your kids?
She makes fun of me for my philosophical “bananas stuff,” but we try not to talk about it too much. We have our kids to raise.

Do your theories help with raising your kids, or does that also seem like two different worlds?
The overlap with the kids is great because they ask the same questions I do. I did a presentation about space for my son Alexander’s preschool when he was 4. I showed them videos of the moon landing and brought in a rocket. Then one little kid put up his hand and said: “I have a question. Does space end or go on forever?” I was like, “Yeah, that is exactly what I am thinking about now.”

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