by Brian Greene
The philosophical maneuver behind this trick is straightforward: defang the question. If you want to avoid explaining why one particular theory should be singled out over another, then don’t single it out. Nozick suggests that we imagine we’re part of a multiverse that comprises every possible universe.7 The multiverse would include not only the alternative evolutions emerging from the Quantum Multiverse, or the many bubble universes of the Inflationary Multiverse, or the possible stringy worlds of the Brane or Landscape Multiverse. These multiverses wouldn’t, on their own, fulfill Nozick’s proposal, because you’d still be left wondering: Why quantum mechanics? Or why inflation? Or why string theory? Instead, come up with any possible universe whatsoever—it could be made of the usual atomic species, but a universe made solely of melted mozzarella would serve just as well—and it has a place in Nozick’s scheme.
This is the last multiverse we will consider, since it’s the most expansive of all—the most expansive possible. Any multiverse that’s ever been or ever will be proposed is itself composed of possible universes, and will therefore be part of this mega-conglomerate, which I’ll call the Ultimate Multiverse. Within this framework, if you ask why our universe is governed by the laws our research reveals, the answer harks back to anthropics: there are other universes out there, all possible universes in fact, and we inhabit the one we do because it’s among those that support our form of life. In the other universes where we could live—of which there are many since, among other things, we can certainly survive sufficiently tiny changes to the various fundamental parameters of physics—there are people, much like us, asking the same question. And the same answer applies equally well to them. The point is that the attribute of existence affords a universe no special status, because in the Ultimate Multiverse all possible universes do exist. The question of why one set of laws describes a real universe—ours—while all others are sterile abstractions evaporates. There are no sterile laws. All sets of laws describe real universes.
Curiously, Nozick noted that within his multiverse there’d be a universe that consists of nothing. Absolutely nothing. Not empty space, but the nothing that Gottfried Leibniz referred to in his famous query “Why is there something rather than nothing?” Not that Nozick could have known, but for me this was an observation of particular resonance. When I was ten or eleven, I came upon Leibniz’s question and found it deeply troubling. I’d pace my room, trying to grasp what nothing would be, often with one hand hovering behind the back of my head, thinking that the struggle to do the impossible—see my hand—would help me grasp the meaning of total absence. Even now, to focus on absolute true nothingness makes my heart sink. Total nothingness, from our familiar vantage point of somethingness, entails the most profound loss. But because nothing also seems so vastly simpler than something—no laws at work, no matter at play, no space to inhabit, no time to unfurl—Leibniz’s question strikes many as right on the mark. Why isn’t there nothingness? Nothingness would have been decidedly elegant.
In the Ultimate Multiverse, a universe consisting of nothing does exist. As far as we can tell, nothingness is a perfectly logical possibility and so must be included in a multiverse that embraces all universes. Nozick’s answer to Leibniz, then, is that in the Ultimate Multiverse there is no imbalance between something and nothing that calls out for explanation. Universes of both types are part of this multiverse. A nothing universe is nothing to get exercised about. It’s only because we humans are something that the nothing universe eludes us.
A theoretician, trained to speak in mathematics, understands Nozick’s all-encompassing multiverse as one where all possible mathematical equations are realized physically. It’s a version of Jorge Luis Borges’ story “La Biblioteca de Babel,” in which the books of Babel are written in the language of mathematics, and so contain all possible sensible, non-self-contradictory strings of mathematical symbols.* Some of the books would spell out familiar formulae, such as the equations of general relativity and those of quantum mechanics, as applied to the known particles of nature. But such recognizable strings of mathematical characters would be extremely rare. Most books would contain equations no one has previously written down, equations that would normally be deemed pure abstractions. The idea of the Ultimate Multiverse is to shed this familiar perspective. No longer do most equations lie dormant, with only the lucky few mysteriously coaxed to life through physical instantiation. Instead, every book in the Library of Mathematical Babel is a real universe.
Nozick’s suggestion, in this mathematical framing, provides a concrete answer to a long-debated question. For centuries, mathematicians and philosophers have wondered whether mathematics is discovered or invented. Are mathematical concepts and truths “out there,” waiting for an intrepid explorer to stumble upon them? Or, since that explorer is more than likely sitting at a desk, pencil in hand, scribbling arcane symbols furiously across a page, are the resulting mathematical concepts and truths invented as part of the mind’s search for order and pattern?
At first sight, the uncanny way that a great many mathematical insights find application to physical phenomena provides compelling evidence that math is real. Examples abound. From general relativity to quantum mechanics, physicists have found that various mathematical discoveries are tailor-made for physical applications. Paul Dirac’s prediction of the positron (the anti-particle of the electron) provides a simple but impressive case in point. In 1931, upon solving his quantum equations for the motion of electrons, Dirac found that the math provided an “extraneous” solution—apparently describing the motion of a particle just like the electron except that it carried a positive electric charge (whereas the electron’s charge is negative). In 1932, that very particle was discovered by Carl Anderson through a close study of cosmic rays bombarding earth from space. What began as Dirac’s manipulation of mathematical symbols in his notebooks concluded in the laboratory with the experimental discovery of the first species of antimatter.
The skeptic can counter, however, that mathematics still emanates from us. We were shaped by evolution to find patterns in the environment; the better we could do that, the better we could predict how to find the next meal. Mathematics, the language of pattern, emerged from our biological fitness. And with that language, we’ve been able to systematize the search for new patterns, going well beyond those relevant for mere survival. But mathematics, like any of the tools we developed and utilized through the ages, is a human invention.
My view on mathematics periodically changes. When I’m in the throes of a mathematical investigation that’s going well, I often feel that the process is one of discovery, not invention. I know of no more exciting experience than watching the disparate pieces of a mathematical puzzle suddenly coalesce into a single coherent picture. When it happens, there’s a feeling that the picture was there all along, like a grand vista hidden by the morning fog. On the other hand, when I more objectively survey mathematics, I’m less convinced. Mathematical knowledge is the literary output of humans conversant in the unusually precise language of mathematics. And as is surely the case with literature produced in one of the world’s natural languages, mathematical literature is the product of human ingenuity and creativity. That’s not to say that other intelligent life-forms wouldn’t come upon the same mathematical results we’ve found; they very well might. But that could easily reflect similarities in our experiences (such as the need to count, the need to trade, the need to survey, and so on) and so would provide minimal evidence that math has a transcendent existence.
A number of years ago, in a public debate on the subject, I said that I could imagine an alien encounter during which, in response to learning of our scientific theories, the aliens remark, “Oh, math. Yeah, we tried that for a while. At first it seemed promising, but ultimately it was a dead end. Here, let us show you how it really works.” But, to continue with my own vacillation, I don’t know how the aliens would actually finish the sentence, and with a broad enough definition of mathematics
(e.g., logical deductions following from a set of assumptions), I’m not even sure what kind of answers wouldn’t amount to math.
The Ultimate Multiverse is unequivocal on the issue. All math is real in the sense that all math describes some real universe. Across the multiverse, all math gets its due. A universe governed by Newton’s equations and populated solely by solid billiard balls (with no additional internal structure) is a real universe; an empty universe with 666 spatial dimensions governed by a higher-dimensional version of Einstein’s equations is a universe too. If the aliens happened to be right, there would also be universes whose description would stand outside mathematics. But let’s hold that possibility off to the side. A multiverse realizing all mathematical equations will be enough to keep us occupied; that’s what the Ultimate Multiverse gives us.
Multiverse Rationalization
Where the Ultimate Multiverse differs from the other parallel universe proposals we’ve encountered is in the reasoning that leads to its consideration. The multiverse theories in previous chapters were not dreamed up to solve a problem or answer a question. Some of them do, or at least claim to, but they weren’t developed for that purpose. We’ve seen that some theorists believe the Quantum Multiverse resolves the quantum measurement problem; some believe the Cyclic Multiverse addresses the question of time’s beginning; some believe the Brane Multiverse clarifies why gravity is so much weaker than the other forces; some believe the Landscape Multiverse gives insight into the observed value of dark energy; some believe the Holographic Multiverse explains data emerging from the collision of heavy atomic nuclei. But such applications are secondary. Quantum mechanics was developed to describe the microrealm; inflationary cosmology was developed to make sense of observed properties of the cosmos; string theory was developed to mediate between quantum mechanics and general relativity. The possibility that these theories generate various multiverses is a by-product.
The Ultimate Multiverse, by contrast, carries no explanatory weight apart from its assumption of a multiverse. It achieves precisely one goal: cleaving from our to-do list the project of finding an explanation for why our universe adheres to one set of mathematical laws and not another, and it accomplishes this singular feat precisely by introducing a multiverse. Cooked up specifically to address one issue, the Ultimate Multiverse lacks the independent rationale characterizing the multiverses discussed in earlier chapters.
That’s my view, but not everyone agrees. There’s a philosophical perspective (coming from the structural realist school of thought) that suggests physicists may have fallen prey to a false dichotomy between mathematics and physics. It’s common for theoretical physicists to speak of mathematics providing a quantitative language for describing physical reality; I’ve done so on most every page of this book. But maybe, this perspective suggests, math is more than just a description of reality. Maybe math is reality.
It’s a peculiar idea. We are not used to thinking of solid reality as being constructed from intangible mathematics. The simulated universes of the previous section provide a concrete and enlightening way to think about it. Consider that most celebrated of knee-jerk reactions, in which Samuel Johnson responded to Bishop Berkeley’s claim that matter is a figment of the mind’s conjuring by kicking a large stone. Imagine, however, that unbeknownst to Dr. Johnson, his kick happened within a hypothetical, high-fidelity computer simulation. In that simulated world, Dr. Johnson’s experience of the stone would be just as convincing as in the historical version. Yet, the computer simulation is nothing but a chain of mathematical manipulations that take the computer’s state at one moment—a complex arrangement of bits—and, according to specified mathematical rules, evolve those bits through subsequent arrangements.
Which means that were you to intently study the mathematical transformations the computer carried out during Dr. Johnson’s demonstration, you’d see, right there in the math, the kick and the rebound of his foot, as well as the thought and the famous articulation “I refute it thus.” Hook the computer to a monitor (or some futuristic interface), and you would see that the mathematically choreographed dancing bits yield Dr. Johnson and his kick. But don’t let the simulation’s bells and whistles—the computer’s hardware, the fancy interface, and so on—obscure the essential fact: underneath the hood, there’d be nothing but math. Change the mathematical rules, and the dancing bits would tap out a different reality.
Now, why stop there? I put Dr. Johnson in a simulation only because that context provides an instructive bridge between mathematics and Dr. Johnson’s reality. But the deeper point of this perspective is that the computer simulation is an inessential intermediate step, a mere mental stepping-stone between the experience of a tangible world and the abstraction of mathematical equations. The mathematics itself—through the relationships it creates, the connections it establishes, and the transformations it embodies—contains Dr. Johnson, both his actions and his thoughts. You don’t need the computer. You don’t need the dancing bits. Dr. Johnson is in the mathematics.8
And once you take on board the idea that mathematics itself can, through its inherent structure, embody any and all aspects of reality—sentient minds, heavy rocks, vigorous kicks, stubbed toes—you’re led to envision that our reality is nothing but math. In this way of thinking, everything you’re aware of—the sensation of holding this book, the thoughts you’re now having, the plans you’re making for dinner—is the experience of mathematics. Reality is how math feels.
To be sure, this perspective requires a conceptual leap not everyone will be persuaded to take; personally, it’s a leap I’ve not taken. But for those who do, the worldview sees math as not just “out there,” but as the only thing that’s “out there.” A body of mathematics, be it Newton’s equations, those of Einstein, or any others, doesn’t become real when physical entities arise that instantiate it. Mathematics—all mathematics—already is real; it doesn’t require instantiation. Different collections of mathematical equations are different universes. The Ultimate Multiverse is thus the by-product of this perspective on mathematics.
Max Tegmark of the Massachusetts Institute of Technology, who has been a strong promoter of the Ultimate Multiverse (which he has called the Mathematical Universe Hypothesis), justifies this view through a related consideration. The deepest description of the universe should not require concepts whose meaning relies on human experience or interpretation. Reality transcends our existence and so shouldn’t, in any fundamental way, depend on ideas of our making. Tegmark’s view is that mathematics—thought of as collections of operations (like addition) that act on abstract sets of objects (like the integers), yielding various relations between them (like 1 + 2 = 3)—is precisely the language for expressing statements that shed human contagion. But what, then, could possibly distinguish a body of mathematics from the universe it depicts? Tegmark argues that the answer is nothing. Were there some feature that did distinguish math from the universe, it would have to be non-mathematical; otherwise it could be absorbed into the mathematical depiction, erasing the purported distinction. But, according to this line of thought, if the feature were non-mathematical, it must bear a human imprint, and so can’t be fundamental. Thus, there’s no distinguishing what we conventionally call the mathematical description of reality from its physical embodiment. They are the same. There’s no switch that turns math “on.” Mathematical existence is synonymous with physical existence. And since this would be true for any and all math, this provides another road leading us to the Ultimate Multiverse.
While all these arguments are curious to contemplate, I remain skeptical. In evaluating a given multiverse proposal, I’m partial to there being a process, however tentative—a fluctuating inflaton field, collisions between braneworlds, quantum tunneling through the string theory landscape, a wave evolving via the Schrödinger equation—that we can imagine generating the multiverse. I prefer to ground my thinking in a sequence of events that can, at least in principle, result in the given multiv
erse unfolding. For the Ultimate Multiverse, it’s hard to imagine what such a process could be; the process would need to yield different mathematical laws in different domains. In the Inflationary and Landscape Multiverses, we’ve seen that the details of how the laws of physics manifest themselves can vary from universe to universe, but this is because of environmental differences, such as the values of certain Higgs fields or the shape of the extra dimensions. The underlying mathematical equations, operating across all the universes, are the same. So what process, operating within a given set of mathematical laws, can change those mathematical laws? Like the number five desperately trying to be six, it seems plainly impossible.
However, before settling on that conclusion, consider this: there can be domains that appear as though they are governed by different mathematical rules. Think again about simulated worlds. In discussing Dr. Johnson above, I invoked a computer simulation as a pedagogical device to explain how mathematics may embody the essence of experience. But if we consider such simulations in their own right, as we do in the Simulated Multiverse, we see that they offer just the process we need: although the computer hardware on which a simulation is run is subject to the usual laws of physics, the simulated world itself will be founded on the mathematical equations the user happens to choose. From simulation to simulation, the mathematical laws can and generally will vary.
As we will now see, this provides a mechanism for generating a particular privileged part of the Ultimate Multiverse.
Simulating Babel
Earlier, I noted that for the kinds of equations we typically study in physics, computer simulations yield only approximations to the mathematics. Such is generally the case when continuous numbers confront a digital computer. For example, in classical physics (assuming, as we do in classical physics, that spacetime is continuous) a batted ball passes through an infinite number of different points as it travels from home plate to left field.9 Keeping track of a ball through an infinity of locations, and of an infinity of possible speeds at those locations, will always remain beyond reach. At best, computers can perform highly refined but still approximate calculations, tracking a ball every millionth or billionth or trillionth of a centimeter, for instance. That’s fine for many purposes, but it’s still an approximation. Quantum mechanics and quantum field theory, by introducing various forms of discreteness, help in some ways. But both make extensive use of continuously varying numbers (values of probability waves, values of fields, and so on). The same reasoning holds for all the other standard equations of physics. A computer can approximate the math, but it can’t simulate the equations exactly.*