Is the Universe Made of Math?

The following is excerpted from a Discover Magazine interview, by Adam Frank, with cosmologist Max Tegmark. The full article can be found here.

Is the Universe Actually Made of Math?

Unconventional cosmologist Max Tegmark says mathematical formulas create reality.

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.

Go far enough out and you will find another Earth with another version of yourself.

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.

Why?

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.

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