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.
