Our Mathematical Universe

by Max Tegmark

  • We probably don’t live in a simulation.

  • The MUH is in principle testable and falsifiable.

  13

  Life, Our Universe and Everything

  This is the way the world ends

  Not with a bang but a whimper.

  —T. S. Eliot, “The Hollow Men”

  The future ain’t what it used to be.

  —Yogi Berra

  Figure 13.1: When we ask what everything is made of and zoom in to ever smaller scales, we find that the ultimate building blocks of matter are mathematical structures, objects whose properties are mathematical properties. When we ask how big everything is and zoom out to ever-larger scales, we end up at the same place: in the realm of mathematical structures, indeed a Level IV multiverse of all mathematical structures.

  How Big Is Our Physical Reality?

  I feel honored that you, my dear reader, have joined my reality-exploration journey all the way to this last chapter. We’ve traveled far, from the extra-galactic macrocosm to the subatomic microcosm, encountering a grander reality than I ever dreamed of in my wildest childhood dreams, with four different levels of parallel universes.

  How does this all fit together? Figure 13.1 shows how I think about it. In the first part of the book, we pursued the question “How big is everything?” and explored ever-larger scales: we’re on a planet in a galaxy in a universe that I think is in a doppelgänger-laden Level I multiverse in a more diverse Level II multiverse in a quantum-mechanical Level III multiverse in a Level IV multiverse of all mathematical structures. In the second part of the book, we pursued the question “What’s everything made of?” and explored ever smaller scales: we’re made of cells made of molecules made of atoms made of elementary particles, which are purely mathematical structures in the sense that their only properties are mathematical properties. Although we don’t yet know what if anything these particles are made of, string theory and its leading competitors all suggest that any more fundamental building blocks are purely mathematical as well. In this sense, even though our two intellectual expeditions set off in opposite directions, toward the large and the small, respectively, they ended up in the same place: in the realm of mathematical structures. Whereas all roads were said to lead to Rome, our two roads to reality both lead to mathematics. This elegant confluence reflects the fact that one mathematical structure can contain others within it, explaining all the mathematical regularities that physics has uncovered as aspects or approximations of the grand mathematical structure that is our full external reality. On the largest and smallest scales, the mathematical fabric of reality becomes evident, while it remains easy to miss on the intermediate scales that we humans are usually aware of.1

  The Case for a Smaller Reality

  I’ve painted a picture for you of our ultimate physical reality as I see it. Personally, I find this reality breathtakingly beautiful and inspiringly grand. But is it real? Or could it be that the picture is misleading, with much of the grandeur being mere mirages? Do you really live in a multiverse? Or is the whole question a silly one, lying beyond the pale of science? Let me give you my two cents.

  Multiverse ideas have traditionally received short shrift from the establishment: we’ve seen that Giordano Bruno with his infinite-space multiverse got burned at the stake in 1600 and Hugh Everett with his quantum multiverse got burned on the physics job market in 1957. As I mentioned, I’ve even felt some of the heat firsthand, with senior colleagues suggesting that my multiverse-related publications were nuts and would ruin my career. There’s been a sea change in recent years, however. Parallel universes are now all the rage, cropping up in books, movies and even jokes: “You passed your exam in many parallel universes—but not in this one.”

  This airing of ideas certainly hasn’t led to a consensus among scientists, but it’s made the multiverse debate much more nuanced and, in my opinion, more interesting, with scientists moving beyond shouting sound bites past each other and genuinely trying to understand opposing points of view. A nice example of this is a recent anti-multiverse article in Scientific American by the relativity pioneer George Ellis, which I highly recommend reading (see http://tinyurl.com/antiverse).

  As we discussed in Chapter 6, we use the term our Universe to mean the spherical region of space from which light has had time to reach us during the 14 billion years since our Big Bang. When talking about parallel universes, we distinguished between four different levels: Level I (other such regions far away in space where the apparent laws of physics are the same, but where history played out differently because things started out differently), Level II (regions of space where even the apparent laws of physics are different), Level III (parallel worlds elsewhere in Hilbert space where quantum reality plays out) and Level IV (totally disconnected realities governed by different mathematical equations). In his critique, George Ellis classifies many of the arguments in favor of these multiverse levels and argues that they all have problems. Here’s my summary of his main anti-multiverse arguments:

  1. Inflation may be wrong (or not eternal).

  2. Quantum mechanics may be wrong (or not unitary).

  3. String theory may be wrong (or lack multiple solutions).

  4. Multiverses may be unfalsifiable.

  5. Some claimed multiverse evidence is dubious.

  6. Fine-tuning arguments may assume too much.

  7. It’s a slippery slope to even bigger multiverses.

  (George didn’t actually mention argument 2 in his article, but I’m adding it here because I think he would have if the editor had allowed him more than six pages.)

  What’s my take on this critique? Interestingly, I agree with all of these seven statements—and nonetheless, I’ll still happily bet my life savings on the existence of a multiverse!

  Let’s start with the first four. As we saw in Chapter 6, inflation naturally produces the Level I multiverse, and if you add in string theory with a landscape of possible solutions, you get Level II as well. As we saw in Chapter 8, quantum mechanics in its mathematically simplest collapse-free (“unitary”) form gives you Level III. So if these theories are ruled out, then key evidence for these multiverses collapses. Remember: Parallel universes are not a theory—they’re predictions of certain theories.

  To me, the key point is that if theories are scientific, then it’s legitimate science to work out and discuss all their consequences even if they involve unobservable entities. For a theory to be falsifiable, we need not be able to observe and test all its predictions, merely at least one of them. My answer to argument 4 is therefore that what’s scientifically testable are our mathematical theories, not necessarily their implications, and that this is quite okay. As we discussed in Chapter 6, because Einstein’s theory of general relativity has successfully predicted many things that we can observe, we also take seriously its predictions for things we can’t observe—for example, what happens inside black holes. Likewise, if we’re impressed by the successful predictions of inflation or quantum mechanics so far, then we also need to take seriously their other predictions, including the Level I and Level III multiverses. George even mentions the possibility that eternal inflation may one day be ruled out: to me, this is simply an argument that eternal inflation is a scientific theory.

  String theory certainly hasn’t come as far as inflation and quantum mechanics in terms of establishing itself as a testable scientific theory. However, I suspect that we’ll be stuck with a Level II multiverse even if string theory turns out to be a red herring. It’s quite common for mathematical equations to have multiple solutions, and as long as the fundamental equations describing our reality do, then eternal inflation generically creates huge regions of space that physically realize each of these solutions, as we saw in Chapter 6. For example, the equations governing water molecules, which have nothing to do with string theory, permit the three solutions corresponding to steam, liquid water and ice, and if space itself can similarly exist in different phases, inflation will tend
to realize them all.

  George lists a number of observations purportedly supporting multiverse theories that are dubious at best, such as evidence that certain constants of nature aren’t really constant, and evidence in the cosmic microwave–background radiation of collisions with other universes or strangely connected space. I totally share his skepticism of these claims. In all these cases, however, the controversies have been about the analysis of the data, much as it was in the 1989 cold-fusion debacle. To me, the very fact that scientists are making these measurements and arguing about data details is further evidence that this is within the pale of science: this is precisely what separates a scientific controversy from a nonscientific one!

  We saw in Chapter 6 that our Universe appears surprisingly fine-tuned for life in the sense that if you tweaked many of our constants of nature by just a tiny amount, life as we know it would be impossible. Why? If there’s a Level II multiverse where these “constants” take all possible values, it’s not surprising that we find ourselves in one of the rare universes that are inhabitable, just as it’s not surprising that we find ourselves living on Earth rather than Mercury or Neptune. George objects to the fact that you need to assume a multiverse theory to draw this conclusion, but that’s how we test any scientific theory: we assume that it’s true, work out the consequences, and discard the theory if the predictions fail to match the observations. Some of the fine-tuning appears extreme enough to be quite embarrassing—for example, we saw that we need to tune the dark energy to about 123 decimal places to make habitable galaxies. To me, an unexplained coincidence can be a telltale sign of a gap in our scientific understanding. Dismissing it by saying, “We just got lucky—now stop looking for an explanation!” is not only unsatisfactory, but also tantamount to ignoring a potentially crucial clue.

  George argues that if we take seriously that anything that could happen does happen, we’re led down a slippery slope toward even larger multiverses, such as the Level IV one. Since this is my favorite multiverse level, and I’m one of the very few proponents of it, this is a slope that I’m happy to slide down!

  George also mentions that multiverses may fall foul of Occam’s razor by introducing unnecessary complications. As a theoretical physicist, I judge the elegance and simplicity of a theory not by its ontology, but by the elegance and simplicity of its mathematical equations—and it’s quite striking to me that the mathematically simplest theories tend to give us multiverses. It’s proven remarkably hard to write down a theory that produces exactly the universe we see and nothing more.

  Finally, there’s an anti-multiverse argument that I commend George for avoiding, but which is in my opinion the most persuasive one of all for most people: parallel universes just seem too weird to be real. But as we discussed in Chapter 1, this is exactly what we should expect: evolution endowed us with intuition only for those everyday aspects of physics that had survival value for our distant ancestors, leading to the prediction that whenever we use technology to glimpse reality beyond the human scale, our evolved intuition should break down. We’ve seen this happen again and again with counterintuitive features of relativity theory, quantum mechanics, etc., and should expect the ultimate theory of physics, whatever it turns out to be, to feel weirder still.

  * * *

  1This expansion of our ontology in physics is reminiscent of the expansion of our ontology in mathematics over the past centuries. Mathematicians call this generalization: the insight that what we’re studying is part of a larger structure.

  The Case for a Greater Reality

  Having looked at anti-multiverse arguments, let’s now analyze the pro-multiverse case a bit more closely. I’m going to argue that all the controversial issues melt away if we accept the External Reality Hypothesis from Chapter 10: There exists an external physical reality completely independent of us humans. Suppose that this hypothesis is correct. Then most multiverse critique rests on some combination of the following three dubious assumptions:

  Assumptions 1 and 2 appear to be motivated by little more than human hubris. The Omnivision Assumption effectively redefines the word exists to be synonymous with what’s observable by us humans, which is akin to being an ostrich with its head in the sand. Those who insist on the Pedagogical-Reality Assumption will typically have rejected comfortingly familiar childhood notions such as Santa Claus, Euclidean space, the Tooth Fairy and creationism—but have they really worked hard enough to free themselves from comfortingly familiar notions that are more deeply rooted? In my personal opinion, our job as scientists is to try to figure out how the world works, not to tell it how to work based on our philosophical preconceptions.

  If the Omnivision Assumption is false, then there are by definition things that exist despite being unobservable even in principle. Because our Universe definition includes everything that’s in principle observable, this means that our Universe isn’t all that exists, so we live in a multiverse. If the Pedagogical-Reality Assumption is false, then the objection that multiverses are too weird makes no logical sense. If the No-Copy Assumption is false, then there’s no fundamental reason why there can’t be copies of you elsewhere in the external reality—indeed, we’ve seen in Chapters 6 and 8 how both eternal inflation and collapse-free quantum mechanics provide mechanisms for creating them.

  Moreover, we argued in Chapter 10 that the External Reality Hypothesis implies the Mathematical Universe Hypothesis: that our external physical reality is a mathematical structure. In Chapter 12 we saw how this in turn implies the Level IV multiverse, which contains all other multiverse levels within it. In other words, we basically get stuck with all these parallel universes as soon as we accept that there’s an external reality independent of us.

  In summary, we’ve seen throughout this book how humanity’s self-image has evolved. We humans have long had a tendency toward hubris, arrogantly imagining ourselves at center stage, with everything revolving around us, but we’ve repeatedly been proven wrong: it is instead we who are revolving around the Sun, which is itself revolving around the center of one galaxy among countless others in a universe that may in turn be but one in a four-level multiverse hierarchy. I hope this makes us humbler. However, whereas we humans had overestimated our physical powers in the grand scheme of things, we had underestimated our mental powers! Our ancestors thought they were forever grounded, and could never truly understand the nature of the stars and what lay beyond. Then they realized how far they could get without flying into space to examine celestial objects—by letting their human minds fly. Thanks to breakthroughs in physics, we’re gaining ever-deeper insights into the very nature of reality. We’ve found ourselves inhabiting a reality far grander than our ancestors ever dreamed of, and this means that our future potential for life is much grander than we thought. With physical resources nearly limitless, it’s our future ingenuity that will make the key difference; so our destiny is in our own hands.

  The Future of Physics

  If I’m wrong and the Mathematical Universe Hypothesis is false, it means that fundamental physics is doomed to eventually hit a roadblock beyond which we can’t understand our physical reality any better, because it lacks a mathematical description. If I’m right, then there’s no roadblock, and everything is in principle understandable to us. I think that would be wonderful, because then we’ll be limited only by our own imagination.

  By our imagination and our willingness to do hard work, to be more specific. As we mentioned in Chapter 10, the answer Douglas Adams gave to his ultimate question of life, the Universe, and everything was hardly an answer that laid all questions to rest. Similarly, the answer I’m proposing to the question about the ultimate nature of reality (“It’s all math,” or more specifically, “It’s the Level IV multiverse”) leaves most of our traditional big questions unanswered. Instead of getting answered, most questions get rephrased. For example, “What are the equations of quantum gravity?” turns into “Where in the Level IV multiverse are we?”—a question that appears
as difficult to answer as the original. So the ultimate question about physical reality would change. We’d abandon as misguided the question of which particular mathematical equations describe all of reality, and instead ask how to compute the frog’s view of our Universe—our observations—from the bird’s view. That would determine whether we’ve uncovered the true structure of our particular Universe, and help us figure out which corner of the mathematical cosmos is our home.

  This situation where fundamental questions can be easier to answer than less-fundamental ones is actually typical in physics: if we find the correct equations describing quantum gravity, they’ll provide a deeper understanding about what space, time and matter are, but they won’t help us model global climate change more accurately, even though they in principle explain all the relevant physics of weather. The devil is in the details, and figuring out these details often requires hard work that’s rather independent of the ultimate underlying theory.

  In this spirit, we’re going to devote the rest of this book to exploring some specific big questions that bring us farther and farther from fundamental physics and closer and closer to home. Since the earlier parts of the book focused heavily on our past, it’s fitting to end our journey by focusing on our future.

  The Future of Our Universe—How Will It End?

  If the Mathematical Universe Hypothesis is correct, then there isn’t much to say about the future of our physical reality as a whole: since it exists outside of space and time, it can’t end or disappear any more than it can get created or change. However, if we head closer to home and zoom in to the particular mathematical structure that we inhabit, which contains space and time within it, then things get much more interesting. Here in our neck of the woods, things are such that they appear to change from the vantage point of observers such as us, and it’s natural to ask what will ultimately happen.