Our Mathematical Universe

by Max Tegmark

  Figure 11.11: In Figure 11.5, we saw how observer moment (c) feels like the continuation of observer moment (b) because it shares all its memories. However, (c) also feels like the continuation of (B), an observer moment belonging to a doppelgänger whose flight is identical except for a terrorist bomb that kills all passengers before they wake up. If there are no other doppelgängers, then the correct prediction for both (B) and (b) is that they’ll next perceive (c).

  Click here to see a larger image.

  As Figure 11.11 illustrates, subjective immortality doesn’t require quantum mechanics, merely parallel universes—it doesn’t matter whether the two airplanes in the figure are in different parts of our 3-D space (Level I multiverse) or different parts of our Hilbert space (Level III multiverse). So let’s quite generally consider any multiverse scenario where some mechanism kills half of all copies of you each second. After twenty seconds, only about one in a million (1 in 220) of your initial doppelgängers will still be alive. Up until that point, there have been 220 + 219 +…+ 4 + 2 + 1 ≈ 221 second-long observer moments, so only one in two million observer moments remembers surviving for twenty seconds. As Paul Almond has pointed out, this means that those surviving that long should rule out the entire premise (that they’re undergoing the immortality experiment) at 99.99995% confidence. In other words, we have a philosophically bizarre situation: you start with a correct theory for what’s going on, you make a prediction for what will happen (that you’ll survive), you observationally confirm that your prediction was correct, and you then nonetheless turn around and declare that the theory is ruled out! Moreover, as we discussed in Chapter 8, you’d start experiencing increasingly bizarre fluke coincidences the longer you waited, which kept saving your life in ever more unlikely-seeming ways—getting saved by power failures, asteroid impacts, etc., would probably suffice to make most people start questioning their assumptions about reality.…

  Infinite Problems

  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” and “infinitely small.” By infinitely big, I mean the idea that space can have infinite volume, that time can continue forever, and that there can be infinitely many physical objects. By infinitely small, I mean the continuum: the idea that even a liter of space contains an infinite number of points, that space can be stretched out indefinitely without anything bad happening, and that there are quantities in nature that can vary continuously. The two are closely related: we saw in Chapter 5 that inflation created an infinite volume by stretching continuous space indefinitely.

  We have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable Universe contains only about 1089 objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about sixteen decimal places.

  I remember distrusting infinity already as a teenager, and the more I’ve learned, the more suspicious I’ve become. Without infinity, there’d be no measure problem—we’d always calculate the same fractions regardless of what order we counted in. Without infinity, there’d be no quantum immortality.

  Among physicists, my skepticism toward infinity places me in a very small minority. Among mathematicians, infinity and the continuum used to be viewed with considerable suspicion. Carl Friedrich Gauss, sometimes referred to as “the greatest mathematician since antiquity,” had this to say two centuries ago: “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.” Criticizing the continuum and related ideas, his younger colleague Leopold Kronecker once said: “God made integers; all else is the work of man.” In the past century, however, infinity has become mathematically mainstream, with only a few vocal critics remaining—for example, the Canadian-Australian mathematician Norman Wildberger has posted an essay arguing that “real numbers are a joke.”

  So why are today’s physicists and mathematicians so enamored with infinity that it’s almost never questioned? Basically, because infinity is an extremely convenient approximation, and we haven’t discovered good alternatives. For example, consider the air in front of you. Keeping track of the positions and speeds of octillions of atoms would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum, a smooth substance that has a density, pressure and velocity at each point, you find that this idealized air obeys a beautifully simple equation that explains almost everything we care about from how sound waves propagate through air to how winds work. Yet despite all that convenience, air isn’t truly continuous. Could it be the same way for space, time and all the other building blocks of our physical word? We’ll explore that in the next chapter.

  THE BOTTOM LINE

  • Mathematical structures are eternal and unchanging: they don’t exist in space and time—rather, space and time exist in (some of) them. If cosmic history were a movie, then the mathematical structure would be the entire DVD.

  • The Mathematical Universe Hypothesis (MUH) implies that the flow of time is an illusion, as is change.

  • The MUH implies that creation and destruction are illusions, since they involve change.

  • The MUH implies that it’s not only spacetime that is a mathematical structure, but also all the stuff therein, including the particles that we’re made of. Mathematically, this stuff seems to correspond to “fields”: numbers at each point in spacetime that encode what’s there.

  • The MUH implies that you’re a self-aware substructure that is part of the mathematical structure. In Einstein’s theory of gravity, you’re a remarkably complex braidlike structure in spacetime, whose intricate pattern corresponds to information processing and self-awareness. In quantum mechanics, your braid pattern branches like a tree.

  • The movielike subjective reality that you’re perceiving right now exists only in your head, as part of your brain’s reality model, and it includes not merely edited highlights of here and now, but also a selection of prerecorded distant and past events, giving the illusion that time flows.

  • You’re self-aware rather than just aware because your brain’s reality model includes a model of yourself and your relation to the outside world: your perceptions of a subjective vantage point you call “I” are qualia, just as your subjective perceptions of “red” and “sweet” are.

  • The theory that our external physical reality is perfectly described by a mathematical structure while still not being one is 100% unscientific in the sense of making no observable predictions whatsoever.

  • You should expect your current observer moment to be a typical one among all observer moments that feel like you. Such reasoning leads to controversial conclusions regarding the end of humanity, the stability of our Universe, the validity of cosmological inflation, and whether you’re a disembodied brain or simulation.

  • It also leads to the so-called measure problem, a serious scientific crisis that calls into question the ability of physics to predict anything at all.

  12

  The Level IV Multiverse

  What is it that breathes fire into the equations and makes a universe for them to describe?

  —Stephen Hawking

  Why I Believe
in the Level IV Multiverse

  Why These Equations, Not Others?

  Suppose that you’re a physicist, and that you discover how to unify all physical laws into a “Theory of Everything.” Using its mathematical equations, you’re able to answer the tough questions that keep today’s physicists awake at night, such as how quantum gravity works and how to solve the measure problem. A T-shirt with these equations becomes a best-seller, and you’re awarded the Nobel Prize. You’re elated, but the night before the award ceremony, you can’t sleep because you struggle with an embarrassing question from my hero John Wheeler that still remains unanswered: Why these particular equations, not others?

  In the last two chapters, I’ve argued for the Mathematical Universe Hypothesis (MUH), according to which our external physical reality is a mathematical structure, and this sharpens Wheeler’s question. Mathematicians have discovered a large number of mathematical structures, and Figure 12.1 illustrates some of the simplest ones as boxes. None of the ones in the figure match our physical reality, even though some of them may describe certain limited aspects of our world. In 1916, the box labeled “GENERAL RELATIVITY” was a serious candidate for being an exact match, containing within it not only space and time but also various forms of matter, but the discovery of quantum mechanics soon made clear that our own physical reality had features that this particular mathematical structure lacked. Fortunately, you can now extend the figure by adding the mathematical structure that you’ve discovered and will get your prize for, knowing that this new box in the figure is the box, the one that corresponds to our physical reality.

  Figure 12.1: Relationships between various basic mathematical structures. The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures—for instance, an algebra is a vector space that’s also a ring, and a Lie group is a group that’s also a manifold. The full family tree is probably infinite in extent—the figure shows merely a small sample near the bottom.

  Click here to see a larger image.

  At this point, I can hear John Wheeler’s friendly voice interject: But what about the other boxes? If your box corresponds to a physically existing reality, then why don’t they?

  All boxes are on an equal mathematical footing, corresponding to different mathematical structures, so why should some be more equal than others when it comes to physical existence? Could there really be a fundamental, unexplained existential asymmetry built into the very heart of reality, splitting mathematical structures into two classes—those with and without physical existence?

  Mathematical Democracy

  This question really bothered me that Berkeley evening back in 1990, when I first had the mathematical universe idea and told my friend Bill Poirier about it in the fifth-floor hallway outside our dorm rooms in International House. Until a lightbulb went off in my head and I realized that there’s a way out of this philosophical conundrum. I argued to Bill that complete mathematical democracy holds: that mathematical existence and physical existence are equivalent, so that all structures that exist mathematically exist physically as well. Then each other box in Figure 12.1 also describes a physically real universe—just a different one from the one we happen to inhabit. This can be viewed as a form of radical Platonism, asserting that all the mathematical structures in Plato’s “realm of ideas” exist “out there” in a physical sense.

  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. Figure 12.2 illustrates this four-level multiverse hierarchy, one of the core ideas of this book.

  How the Mathematical Universe Hypothesis Implies the Level IV Multiverse

  If the theory that the Level IV multiverse exists is correct, then since it has no free parameters whatsoever, all properties of all parallel universes (including the subjective perceptions of self-aware substructures in them) could in principle be derived by an infinitely intelligent mathematician. But is this theory correct? Does the Level IV multiverse really exist?

  Figure 12.2: The parallel universes described in this book form a four-level hierarchy, where each multiverse is a single member among many at the level above it.

  Click here to see a larger image.

  Interestingly, in the context of the Mathematical Universe Hypothesis (MUH), the existence of the Level IV multiverse isn’t optional. As we discussed in detail in the previous chapter, the MUH says that a mathematical structure is our external physical reality, rather than being merely a description thereof. This equivalence between physical and mathematical existence means that if a mathematical structure contains a self-aware substructure, it will perceive itself as existing in a physically real universe, just as you and I do (albeit generically a universe with different properties from ours). Stephen Hawking famously asked, “What is it that breathes fire into the equations and makes a universe for them to describe?” In the context of the MUH, there’s thus no fire-breathing required, since the point isn’t that a mathematical structure describes a universe, but that it is a universe. Moreover, there’s no making required either. You can’t make a mathematical structure—it simply exists. It doesn’t exist in space and time—space and time may exist in it. In other words, all structures that exist mathematically have the same ontological status, and the most interesting question isn’t which ones exist physically (they all do), but which ones contain life—and perhaps us. 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. Analogously, we saw evidence in Chapter 6 that the Level II multiverse is mostly barren wasteland, with self-awareness confined to the tiny “Goldilocks” fraction of space where the value of the dark-energy density and other physical parameters are just right for life. In the Level I multiverse, the story appears to repeat itself, with life flourishing mainly in the tiny fraction of space that lies near planetary surfaces. So we humans are in a very privileged place indeed!

  Exploring the Level IV Multiverse: What’s Out There?

  Our Local Neighborhood

  Let’s spend some time exploring the Level IV multiverse and the diverse zoo of mathematical structures that it contains, beginning in our local neighborhood. Although we still don’t know exactly which mathematical structure we inhabit, it’s not hard to imagine many small modifications that might give other valid mathematical structures. For example, the standard model of particle physics involves certain symmetries that mathematicians denote SU(3) × SU(2) × U(1), and if we replace them by different symmetries, we’ll end up with a mathematical structure with different kinds of particles and forces, where quarks, electrons and photons are replaced by other entities with novel properties. In some mathematical structures, there’s no light. In others, there’s no gravity. In Einstein’s mathematical description of spacetime, the numbers 1 and 3 that respectively specify the number of time and space dimensions can be replaced by different values of your choice.

  Although we discussed in Chapter 6 how inflation in a single mathematical structure with its single set of fundamental laws of physics can give rise
to different effective laws of physics in different parts of space, forming a Level II multiverse, we’re now talking about something more radical, where even the fundamental laws are different—where there’s no quantum mechanics, say. If string theory can be rigorously defined mathematically, then there’s a mathematical structure where string theory is the “correct Theory of Everything” in that structure, but everywhere else in the Level IV multiverse, it’s not.

  When contemplating the Level IV multiverse, we need to let our imagination fly, unencumbered by our preconceptions of what laws of physics are supposed to be like. Consider space and time: rather than being continuous as our world suggests, they can be discrete, as in the computer games PAC-MAN and Tetris, or in John Conway’s Game of Life, where motion can occur only in jerky jumps. As long as all user input is turned off so that the time evolution can be deterministically computed, these games all correspond to valid mathematical structures. For example, Figure 12.3 shows the 3-D Tetris clone called FRAC mentioned in Chapter 3, which I wrote with my friend Per Bergland in 1990, and if you play it without touching the keyboard (which isn’t the best high-score strategy…), then the entire game from start to finish is determined by simple mathematical rules in the program, which makes it a mathematical structure that’s part of the Level IV multiverse. 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.