Four Levels of Universe

At Discover Magazine, a highly speculative piece about the nature of the Universe(s):

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.
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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.
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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.
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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.

Ah, the Platonic "Theory of Forms", that there exists somewhere an "ideal chair", and that all objects which we call "chairs" are more or less chairlike depending only on how much they resemble the ideal object, the ideal chair. Even such concepts as "sameness" and "difference" have reality in this ideal existence. They are real things.

Of course it gets difficult when the concept is "a concept that does not exist in the Ideal Universe". For the ideal Universe includes all concepts, including this one. So we have a concept that by definition simultaneously cannot exist as an Ideal, and one that has to.

I think the same critique may hold of the Type IV Universe as postulated, if I understand it correctly. Goedel Incompleteness states basically that any Mathematical system can either be complete, or consistent, but not both. And we can prove that - that there are either unprovable true statements, or some statements are both provably true and provably false. Either would be fatal to a mathematical universe. If I understand it correctly - which I may well not. I'm good at intuition, but this is one for a specialist in Pure Mathematics, not a beginner like me.

There is more of interest in the article though.

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.”

Science is a very Human activity. It's one where we never lose that childlike quality of wanting to know "why?".