The Hidden Reality Part 11

A possible way to obviate such challenges would be to use a different approach-call it the "ultra-reductionist strategy"-in which the simulation would proceed by a single set of fundamental equations, much as physicists imagine is the case for the real universe. Such simulations would take as input a mathematical theory of matter and the fundamental forces and a choice of "initial conditions" (how things were at the starting point of the simulation); the computer would then evolve everything forward in time, thereby avoiding the meshing issues of the emergent approach. But simulations of this kind would encounter their own computational problems, even beyond the staggering computational burden of simulating "everything," right down to the behavior of individual particles. If the equations our descendants have in their possession are similar to those we work with today-involving numbers that can vary continuously-then the simulations would necessarily invoke approximations. To exactly exactly follow a number as it varies continuously, we would need to track its value to an infinite number of decimal places (for instance, as such a quant.i.ty varies, say, from .9 to 1, it would pa.s.s through numbers like .9, .95, .958, .9583, .95831, .958317, and on and on, with an arbitrarily large number of digits required for full accuracy). That's something a computer with finite resources can't manage: it will run out of time and memory. So, even if follow a number as it varies continuously, we would need to track its value to an infinite number of decimal places (for instance, as such a quant.i.ty varies, say, from .9 to 1, it would pa.s.s through numbers like .9, .95, .958, .9583, .95831, .958317, and on and on, with an arbitrarily large number of digits required for full accuracy). That's something a computer with finite resources can't manage: it will run out of time and memory. So, even if the the deepest equations were used, it's still possible that computer-based calculations would inevitably be approximate, allowing errors to build up over time. deepest equations were used, it's still possible that computer-based calculations would inevitably be approximate, allowing errors to build up over time.*

Of course, by "error" I mean a deviation between what occurs in the simulation and the description inherent in the most refined physical theories the simulator has at his or her disposal. But to those like you who are within the simulation, the mathematical rules driving the computer would be would be your laws of nature. The issue, then, is not how closely the mathematical laws used by the computer model the external world; we're imagining that you don't observe the external world from within the simulation. Rather, the problem for a simulated universe is that when a computer's necessary approximations permeate otherwise exact mathematical equations, calculations easily lose their stability. Round-off errors, when acc.u.mulated over a great many computations, can yield inconsistencies. You and other simulated scientists might witness anomalous results from experiments; cherished laws might start yielding inaccurate predictions; measurements that had long since converged on a single widely confirmed result might start producing different answers. For long stretches, you and your simulated colleagues would think that you'd encountered evidence, much as your forebears had throughout the previous centuries and millennia, that your final theory wasn't so final after all. Collectively, you'd closely reexamine the theory, perhaps coming up with new ideas, equations, and principles that better described the data. But, a.s.suming the inaccuracies didn't result in contradictions that crashed the program, at some point you'd hit a wall. your laws of nature. The issue, then, is not how closely the mathematical laws used by the computer model the external world; we're imagining that you don't observe the external world from within the simulation. Rather, the problem for a simulated universe is that when a computer's necessary approximations permeate otherwise exact mathematical equations, calculations easily lose their stability. Round-off errors, when acc.u.mulated over a great many computations, can yield inconsistencies. You and other simulated scientists might witness anomalous results from experiments; cherished laws might start yielding inaccurate predictions; measurements that had long since converged on a single widely confirmed result might start producing different answers. For long stretches, you and your simulated colleagues would think that you'd encountered evidence, much as your forebears had throughout the previous centuries and millennia, that your final theory wasn't so final after all. Collectively, you'd closely reexamine the theory, perhaps coming up with new ideas, equations, and principles that better described the data. But, a.s.suming the inaccuracies didn't result in contradictions that crashed the program, at some point you'd hit a wall.

After an exhaustive search through possible explanations, none of which was able to fully explain what was happening, an iconoclastic thinker might suggest a radically different idea. If the continuum laws that physicists had developed over many millennia were input to a powerful digital computer and used to generate a simulated universe, the errors built up from the inherent approximations would yield anomalies of the very kind being observed. "Are you suggesting that we're in a computer simulation?" you'd ask. "Yes," your colleague would answer. "Well, that's nutty," you'd say. "Really?" she'd reply. "Take a look." And she'd produce a monitor showing a simulated world, which she had programmed using those very same deep laws of physics, and-catching your breath after the shock of encountering a simulated world at all-you would see that the simulated scientists were indeed puzzling over the very same kind of strange data that troubled you.6 A Simulator who sought more a.s.siduously to conceal herself could, of course, use more aggressive tactics. As inconsistencies started to build, she might reset the program and erase the inhabitants' memory of the anomalies. So it would seem a stretch to claim that a simulated reality would reveal its true nature through glitches and irregularities. And certainly I'd be hard pressed to argue that inconsistencies, anomalies, unanswered questions, and stalled progress would reflect anything more than our own scientific failings. The sensible interpretation of such evidence would be that we scientists need to work harder and be more creative in seeking explanations. However, there is one serious conclusion that emerges from the fanciful scenario I've told. If and when we do generate simulated worlds, with apparently sentient inhabitants, an essential question will arise: Is it reasonable to believe that we occupy a rarefied place in the history of scientific-technological development-that we have become the very first creators of sentient simulations? We may may have-but if we're keen to go with the odds, we must consider alternative explanations that, in the grand scheme of things, don't require us to be so extraordinary. And there is a ready-made explanation that fits the bill. Once our own work convinces us that sentient simulations are possible, the guiding principle of "garden variety," discussed in have-but if we're keen to go with the odds, we must consider alternative explanations that, in the grand scheme of things, don't require us to be so extraordinary. And there is a ready-made explanation that fits the bill. Once our own work convinces us that sentient simulations are possible, the guiding principle of "garden variety," discussed in Chapter 7 Chapter 7, suggests that there's not just one such simulation out there but a swarming ocean of simulations, which const.i.tute a Simulated Multiverse. While the simulation we've created might be a landmark feat in the limited domain to which we have access, within the context of the entire Simulated Multiverse it's nothing special, having been achieved a gazillion times over. Once we accept that idea, we're led to consider that we too may be in a simulation, since that's the status of the vast majority of sentient beings in a Simulated Multiverse.

Evidence for artificial sentience and for simulated worlds is grounds for rethinking the nature of your own reality.

The Library of Babel.

During my first semester in college, I enrolled in an introductory philosophy course taught by the late Robert Nozick. From the very first lecture, it was a wild ride. Nozick was completing his voluminous Philosophical Explanations; Philosophical Explanations; he used the course as a dress rehearsal for many of the book's central arguments. Just about every cla.s.s shook my grasp on the world, sometimes vigorously. This was an unexpected experience-I'd thought that upending reality would be the purview solely of my physics courses. Yet, there was an essential difference between the two. The physics lectures challenged comfortable views by exposing strange phenomena that arise in wholly unfamiliar realms where things move fast, are extremely heavy, or are fantastically tiny. The philosophy lectures shook comfortable views by challenging the foundations of he used the course as a dress rehearsal for many of the book's central arguments. Just about every cla.s.s shook my grasp on the world, sometimes vigorously. This was an unexpected experience-I'd thought that upending reality would be the purview solely of my physics courses. Yet, there was an essential difference between the two. The physics lectures challenged comfortable views by exposing strange phenomena that arise in wholly unfamiliar realms where things move fast, are extremely heavy, or are fantastically tiny. The philosophy lectures shook comfortable views by challenging the foundations of everyday everyday experience. How do we know there's a real world out there? Should we trust our perceptions? What thread binds our molecules and atoms to preserve our personal ident.i.ty through time? experience. How do we know there's a real world out there? Should we trust our perceptions? What thread binds our molecules and atoms to preserve our personal ident.i.ty through time?

While I was hanging around after cla.s.s one day, Nozick asked me what I was interested in, and I brazenly told him that I wanted to work on quantum gravity and unified theories. This was generally a conversation stopper, but for Nozick it presented a chance to educate a young mind by revealing a new perspective. "What drives your interest?" he asked. I told him that I wanted to find eternal truths, to help understand why things are the way they are. Naive and bl.u.s.tery, for sure. But Nozick listened graciously and then took the idea further. "Let's say you find the unified theory," he said. "Would that really provide the answers you're looking for? Wouldn't you still be left asking why that particular theory, and not another, was the correct theory of the universe?" He was right, of course, but I replied that in the search for explanations there might come a point when we would just have to accept certain things as given. That was just where Nozick wanted me to go; in writing Philosophical Explanations Philosophical Explanations he had developed an alternative to this view. It's based on what he called the principle of fecundity and is an attempt to frame explanations without "accepting certain things as given"; without, as Nozick explains it, accepting anything as brute-force truth. he had developed an alternative to this view. It's based on what he called the principle of fecundity and is an attempt to frame explanations without "accepting certain things as given"; without, as Nozick explains it, accepting anything as brute-force truth.

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. 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 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 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. 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 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 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. 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 s.p.a.ce, 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 s.p.a.ce to inhabit, no time to unfurl-Leibniz's question strikes many as right on the mark. Why isn't there nothingness? Why isn't there nothingness? Nothingness would have been decidedly elegant. Nothingness would have been decidedly elegant.

In the Ultimate Multiverse, a universe consisting of nothing does 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. 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-encompa.s.sing 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 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 is a real universe. 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 s.p.a.ce. 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 a.s.sumptions), I'm not even sure what kind of answers wouldn't wouldn't amount to math. 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 b.a.l.l.s (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 a.s.sumption 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 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 quant.i.tative 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 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 quant.i.tative 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 is reality. 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 ch.o.r.eographed 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. 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 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. 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 ent.i.ties 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 Ma.s.sachusetts Inst.i.tute 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 Schrodinger 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 multiverse 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 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 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 cla.s.sical physics (a.s.suming, as we do in cla.s.sical physics, that s.p.a.cetime is continuous) a batted ball pa.s.ses 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. 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.*

There are other types of mathematical functions, however, for which a computer simulation can be absolutely precise. They're part of a cla.s.s called computable functions computable functions, which are functions that can be evaluated by a computer running through a finite set of discrete instructions. The computer may need to cycle through the collection of steps repeatedly but sooner or later it will produce the exact answer. No originality or novelty is needed at any step; it's just a matter of grinding out the result. In practice, then, to simulate the motion of a batted ball, computers are programmed with equations that are computable approximations computable approximations to the laws of physics that you learned in high school. (Typically, continuous s.p.a.ce and time are approximated on a computer by a fine grid.) to the laws of physics that you learned in high school. (Typically, continuous s.p.a.ce and time are approximated on a computer by a fine grid.) By contrast, a computer trying to calculate a noncomputable function will churn away indefinitely without coming to an answer, regardless of its speed or memory capacity. Such would be the case for a computer seeking the exact continuous trajectory of that batted ball. For a more qualitative example, imagine a simulated universe in which a computer is programmed to provide a wonderfully efficient simulated chef who provides meals for all those simulated inhabitants-and only those simulated inhabitants-who don't cook for themselves. As the chef furiously bakes, fries, and broils, he works up quite an appet.i.te. The question is: Whom does the computer charge with feeding the chef?10 Think about it, and it makes your head hurt. The chef can't cook for himself as he Think about it, and it makes your head hurt. The chef can't cook for himself as he only only cooks for those who don't cook for themselves, but if the chef doesn't cook for himself, he is among those for whom he is meant to cook. Rest a.s.sured, the computer's head would hardly fare better than yours. Noncomputable functions are much like this example: they stymie a computer's ability to complete its calculations, and so the simulation being run by the computer would hang. The successful universes const.i.tuting the Simulated Multiverse would therefore be based on computable functions. cooks for those who don't cook for themselves, but if the chef doesn't cook for himself, he is among those for whom he is meant to cook. Rest a.s.sured, the computer's head would hardly fare better than yours. Noncomputable functions are much like this example: they stymie a computer's ability to complete its calculations, and so the simulation being run by the computer would hang. The successful universes const.i.tuting the Simulated Multiverse would therefore be based on computable functions.

The discussion suggests an overlap between the Simulated and Ultimate Multiverses. Consider a scaled-down version of the Ultimate Multiverse that includes only universes arising from computable functions. Then, rather than merely being posited as a resolution to one particular question-Why is this universe real, while other possible universes are not?-the scaled-down version of the Ultimate Multiverse can emerge from a process. An army of future computer users, perhaps not much different in temperament from today's Second Life enthusiasts, could sp.a.w.n this multiverse through their insatiable fascination with running simulations based on ever-different equations. These users wouldn't generate all universes contained in the Mathematical Library of Babel, because the ones based on noncomputable functions wouldn't get off the ground. But the users would continually work their way through the library's computable wing.

The computer scientist Jurgen Schmidhuber, extending earlier ideas of Zuse, has come to a similar conclusion from a different angle. Schmidhuber realized that it's actually easier to program a computer to generate all possible computable universes than it is to program individual computers to generate them one by one. To see why, imagine programming a computer to simulate baseball games. For each game, the amount of information you'd need to supply is vast: every detail about every player, physical and mental, every detail about the stadium, the umpires, the weather, and so on. And each new game you simulate requires you to specify yet another mountain of data. However, if you decide to simulate not one or a few games, but every every game imaginable, your programming job would be far easier. You'd just need to set up one master program that systematically makes its way through every possible variable-those that affect players, the environment, and all other relevant features-and let the program run. Finding any one particular game in the resulting voluminous output would be a challenge, but you'd be a.s.sured that sooner or later every possible game would appear. game imaginable, your programming job would be far easier. You'd just need to set up one master program that systematically makes its way through every possible variable-those that affect players, the environment, and all other relevant features-and let the program run. Finding any one particular game in the resulting voluminous output would be a challenge, but you'd be a.s.sured that sooner or later every possible game would appear.

The point is that whereas specifying one member of a large collection requires a great deal of information, specifying the entire collection can often be much easier. Schmidhuber found that this conclusion applies to simulated universes. A programmer hired to simulate a collection of universes based on specific sets of mathematical equations could take the easy way out: much like the baseball enthusiast, he could opt to write a single, relatively short program that would generate all all computable universes, and turn the computer loose. Somewhere among the resulting gargantuan collection of simulated universes, the programmer would find those he'd been hired to simulate. I wouldn't want to be paying for computer usage by the hour as the turnaround time for generating these simulations would similarly be gargantuan. But I'd happily pay the programmer by the hour since the instruction set to generate all computable universes would be much less intensive than that required to yield any one universe in particular. computable universes, and turn the computer loose. Somewhere among the resulting gargantuan collection of simulated universes, the programmer would find those he'd been hired to simulate. I wouldn't want to be paying for computer usage by the hour as the turnaround time for generating these simulations would similarly be gargantuan. But I'd happily pay the programmer by the hour since the instruction set to generate all computable universes would be much less intensive than that required to yield any one universe in particular.11 Either of these scenarios-a great many users simulating a great many universes, or a master program that simulates them all-is how the Simulated Multiverse might be generated. And because the resulting universes would be based on a wide variety of different mathematical laws, we can equivalently think of these scenarios as generating part of the Ultimate Multiverse: the part encompa.s.sing universes based on computable mathematical functions.*

The drawback of generating only part of the Ultimate Multiverse is that this downsized version less effectively addresses the issue that inspired Nozick's principle of fecundity in the first place. If all possible universes don't exist, if the entire Ultimate Multiverse is not generated, the question resurfaces of why some equations come to life and others don't. Specifically, we're left wondering why universes based on computable equations hog the spotlight.

To continue along this chapter's highly speculative path, maybe the computable/noncomputable division is telling us something. Computable mathematical equations avoid the p.r.i.c.kly issues raised in the middle of the last century by penetrating thinkers like Kurt G.o.del, Alan Turing, and Alonzo Church. G.o.del's famous incompleteness theorem incompleteness theorem shows that certain mathematical systems necessarily admit true statements that can't be proved within the mathematical system itself. Physicists have long wondered about the possible implications of G.o.del's insights for their own work. Might physics, too, necessarily be incomplete, in the sense that some features of the natural world would forever elude our mathematical descriptions? In the context of the downsized Ultimate Multiverse, the answer is no. Computable mathematical functions, by definition, lie squarely within the bounds of calculation. They are the very functions that admit a procedure by which a computer can successfully evaluate them. And so, if all the universes in a multiverse were based on computable functions, they all would also do an end run around G.o.del's theorem; this wing of the Library of Mathematical Babel, this version of the Ultimate Multiverse, would be free of G.o.del's ghost. Maybe that's what singles out computable functions. shows that certain mathematical systems necessarily admit true statements that can't be proved within the mathematical system itself. Physicists have long wondered about the possible implications of G.o.del's insights for their own work. Might physics, too, necessarily be incomplete, in the sense that some features of the natural world would forever elude our mathematical descriptions? In the context of the downsized Ultimate Multiverse, the answer is no. Computable mathematical functions, by definition, lie squarely within the bounds of calculation. They are the very functions that admit a procedure by which a computer can successfully evaluate them. And so, if all the universes in a multiverse were based on computable functions, they all would also do an end run around G.o.del's theorem; this wing of the Library of Mathematical Babel, this version of the Ultimate Multiverse, would be free of G.o.del's ghost. Maybe that's what singles out computable functions.

Would our our universe find a place in this multiverse? That is, if and when we put our hands on the final laws of physics, will those laws describe the cosmos using mathematical functions that are computable? Not just approximately computable functions, as is the case with the physical laws we work with today. But exactly computable? No one knows. If so, developments in physics should drive us toward theories in which the continuum plays no role. Discreteness, the core of the computational paradigm, should prevail. s.p.a.ce surely seems continuous, but we've only probed it down to a billionth of a billionth of a meter. It's possible that with more refined probes we will one day establish that s.p.a.ce is fundamentally discrete; for now, the question is open. A similar limited understanding applies to intervals of time. The discoveries recounted in universe find a place in this multiverse? That is, if and when we put our hands on the final laws of physics, will those laws describe the cosmos using mathematical functions that are computable? Not just approximately computable functions, as is the case with the physical laws we work with today. But exactly computable? No one knows. If so, developments in physics should drive us toward theories in which the continuum plays no role. Discreteness, the core of the computational paradigm, should prevail. s.p.a.ce surely seems continuous, but we've only probed it down to a billionth of a billionth of a meter. It's possible that with more refined probes we will one day establish that s.p.a.ce is fundamentally discrete; for now, the question is open. A similar limited understanding applies to intervals of time. The discoveries recounted in Chapter 9 Chapter 9, which yield information capacity of one bit per Planck area in any region of s.p.a.ce, const.i.tute a major step in the direction of discreteness. But the issue of how far the digital paradigm can be taken remains far from settled.12 My guess is that whether or not sentient simulations ever come to be, we will indeed find that the world is fundamentally discrete. My guess is that whether or not sentient simulations ever come to be, we will indeed find that the world is fundamentally discrete.

The Roots of Reality.

In the Simulated Multiverse, there's no ambiguity regarding which universe is "real"-that is, which universe lies at the root of the branching tree of simulated worlds. It's the one that houses those computers which, should they crash, would bring down the entire multiverse. A simulated inhabitant might simulate his or her own set of universes on simulated computers, as might the inhabitants of those simulations, but there are still real computers on which all these layered simulations appear as an avalanche of electrical impulses. There's no uncertainty about what facts, patterns, and laws are, in the traditional sense, real: they're the ones at work in the root universe.

However, typical simulated scientists across the Simulated Multiverse may have a different perspective. If these scientists are allowed sufficient autonomy-if the simulants rarely if ever tinker with inhabitants' memories or disrupt the natural flow of events-then, to judge by our own experiences, we can antic.i.p.ate that they will make great progress in uncovering the mathematical code that propels their world. And they will treat that code as their laws of nature. Nevertheless, their laws won't necessarily be identical to the laws governing the real universe. Their laws merely need to be good enough, in the sense that when they're simulated on a computer they yield a universe with sentient inhabitants. If there are many distinct sets of mathematical laws that qualify as good enough, there could well be an ever-growing population of simulated scientists convinced of mathematical laws that, far from being fundamental, were simply chosen by whoever has programmed their simulation. If we are typical inhabitants in such a multiverse, this reasoning suggests that what we normally think of as science, a discipline charged with revealing fundamental truths about reality-the root reality operating at the base of the tree-would be undermined.

It's an uncomfortable possibility, but not one that keeps me up at night. Until I get my breath taken away by seeing a sentient simulation, I won't consider seriously the proposition that I am now in one. And, taking the long view, even if sentient simulations are achieved one day-itself a big if-I can well imagine that when a civilization's technical capabilities first enable such simulations, their appeal would be tremendous. But would that appeal be long-lived? I suspect the novelty of creating artificial worlds whose inhabitants are kept unaware of their simulated status would wear thin; there's just so much reality TV you can watch.

Instead, if I allow my imagination to run free within this speculative territory, my sense is that staying power would reside with applications that developed interactions between the simulated and the real worlds. Perhaps simulated inhabitants would be able to migrate into the real world or be joined in the simulated world by their real biological counterparts. In time, the distinction between real and simulated beings might become anachronistic. Such seamless unions strike me as a more probable outcome. In that case, the Simulated Multiverse would contribute to the expanse of reality-our expanse of reality, our real reality-in the most tangible way. It would become an intrinsic part of what we mean by "reality."

*Ironically, an explanation for why magnetic monopoles have not been found (even though they are predicted by many approaches to unified theories) is that their population was diluted by the rapid expansion of s.p.a.ce that takes place in inflationary cosmology. The suggestion now being made is that magnetic monopoles may themselves play a role in initiating future inflationary episodes.

*Another loophole arises from an incarnation of the measure problem from Chapter 7 Chapter 7. If the number of real (nonvirtual) universes is infinite (if we're part of, say, the Quilted Multiverse), then there will be an infinite collection of worlds like ours in which descendants run simulations, yielding an infinite number of simulated worlds. Even though it would still seem that the number of simulated worlds would vastly outnumber the real ones, we saw in Chapter 7 Chapter 7 that comparing infinities is a treacherous business. that comparing infinities is a treacherous business.

*A theory that allows for only a finite number of distinct states within a finite spatial volume (in accord, for example, with the entropy bounds discussed in the previous chapter) can still involve continuous quant.i.ties as part of its mathematical formalism. This is the case, for instance, with quantum mechanics: the probability wave's value can vary continuously even when only finitely many different outcomes are possible.

*Borges allows for books with all possible character strings, without regard to meaning.

*When we discussed the Quilted Multiverse (Chapter 2), I stressed that quantum physics a.s.sures us that in any finite region of s.p.a.ce there are only finitely many different ways in which matter can arrange itself. Nevertheless, the mathematical formalism of quantum mechanics involves features that are continuous and that hence can a.s.sume infinitely many values. These features are things we can't directly observe (such as the height of a probability wave at a given point); it's with respect to the distinct results that measurements can acquire that there are only finitely many possibilities.

*Max Tegmark has noted that the entirety of a simulation, run from start to finish, is itself a collection of mathematical relations. Thus, if one believes that all mathematics is real, so is this collection. In turn, from this perspective there's no need to actually run any computer simulations since the mathematical relations each would produce are already real. Also, note that the focus on evolving a simulation forward in time, however intuitive, is overly restrictive. The computability of a universe should be evaluated by examining the computability of the mathematical relations that define its entire history, whether or not these relations describe the unfolding of the simulation through time.

CHAPTER 11.

The Limits of Inquiry.

Multiverses and the Future.

Isaac Newton cracked the scientific enterprise wide open. He discovered that a few mathematical equations could describe the way things move, both here on earth and up in s.p.a.ce. Considering the power and simplicity of his results, one could easily have imagined that Newton's equations reflected eternal truths etched into the bedrock of the cosmos. But Newton himself didn't think so. He believed that the universe was far more rich and mysterious than his laws implied; later in life he famously reflected, "I do not know what I may appear to the world, but to myself I seem to have only been a boy playing on the seash.o.r.e, diverting myself in now and then finding a smoother pebble or prettier sh.e.l.l than ordinary, whilst the great ocean of truth lay before me all undiscovered." The centuries since have abundantly affirmed this.

I'm glad. Had Newton's equations enjoyed unlimited reach, accurately describing phenomena in any context however big or small, heavy or light, fast or slow, the subsequent scientific odyssey would have taken on a distinctly different character. Newton's equations teach us much about the world, but their unlimited validity would have meant that the cosmic flavor was vanilla through and through. Once you understood physics on everyday scales, you'd be done. The same story would have held all the way up and all the way down.

In continuing Newton's explorations, scientists have ventured into realms far beyond the reach of his equations. What we've learned has required sweeping changes in our understanding of the nature of reality. Such changes are not made lightly. They are closely examined by the community of scientists, and they are often sharply resisted; only when the evidence reaches a critical abundance is the new view embraced. Which is just as it should be. There's no need to rush to judgment. Reality will wait.

The central fact, most forcefully emphasized by the last hundred years of theoretical and experimental progress, is that common experience fails to be a trustworthy guide for excursions that wander beyond everyday circ.u.mstances. But for all the radically new physics encountered in extreme conditions-described by general relativity, quantum mechanics, and, should it prove correct, string theory-the fact that radically new ideas would be required is not surprising. The basic a.s.sumption of science is that regularities and patterns exist on all scales, but as Newton himself antic.i.p.ated, there's no reason to expect the patterns we directly encounter to be recapitulated on all scales.

The surprise would have been to find no surprises.

The same is undoubtedly true regarding what physics will reveal in the future. A given generation of scientists can never know whether the long view of history will judge their work as a diversion, as pa.s.sing fascination, as a stepping-stone, or as having revealed insights that will stand the test of time. Such local uncertainty is balanced by one of physics' most gratifying features-global stability-that is, new theories generally do not erase those they supplant. As we've discussed, while new theories may require acclimation to new perspectives on the nature of reality, they almost never render past discoveries irrelevant. Instead, they incorporate and extend them. Because of this, the story of physics has maintained an impressive coherence.

In this book we've explored a candidate for the next major development in this story: the possibility that our universe is part of a multiverse. The journey has taken us through nine variations on the multiverse theme, which are summarized in Table 11.1 Table 11.1. Although the various proposals differ widely in detail, they all suggest that our commonsense picture of reality is only part of a grander whole. And they all bear the indelible mark of human ingenuity and creativity. But determining whether any of these ideas goes beyond mathematical musings of the human mind will require more insight, knowledge, calculation, experiment, and observation than we've so far achieved. A final reckoning on whether parallel universes will be written into the next chapter of physics' story must therefore also await the perspective that only the future can bring.

Table 11.1 Summary of Various Versions of Parallel Universes Summary of Various Versions of Parallel Universes PARALLELUNIVERSEPROPOSAL: Quilted Multiverse Quilted Multiverse DESCRIPTION: Conditions in an infinite universe necessarily repeat across s.p.a.ce, yielding parallel worlds. Conditions in an infinite universe necessarily repeat across s.p.a.ce, yielding parallel worlds.

PARALLELUNIVERSEPROPOSAL: Inflationary Multiverse Inflationary Multiverse EXPLANATION: Eternal cosmological inflation yields an enormous network of bubble universes, of which our universe would be one. Eternal cosmological inflation yields an enormous network of bubble universes, of which our universe would be one.

PARALLELUNIVERSEPROPOSAL: Brane Multiverse Brane Multiverse EXPLANATION: In string/M-theory's braneworld scenario, our universe exists on one three-dimensional brane, which floats in a higher-dimensional expanse potentially populated by other branes-other parallel universes. In string/M-theory's braneworld scenario, our universe exists on one three-dimensional brane, which floats in a higher-dimensional expanse potentially populated by other branes-other parallel universes.

PARALLELUNIVERSEPROPOSAL: Cyclic Multiverse Cyclic Multiverse EXPLANATION: Collisions between braneworlds can manifest as big bang-like beginnings, yielding universes that are parallel in time. Collisions between braneworlds can manifest as big bang-like beginnings, yielding universes that are parallel in time.

PARALLELUNIVERSEPROPOSAL: Landscape Multiverse Landscape Multiverse EXPLANATION: By combing inflationary cosmology and string theory, the many different shapes for string theory's extra dimensions give rise to many different bubble universes. By combing inflationary cosmology and string theory, the many different shapes for string theory's extra dimensions give rise to many different bubble universes.

PARALLELUNIVERSEPROPOSAL: Quantum Multiverse Quantum Multiverse EXPLANATION: Quantum mechanics suggests that every possibility embodied in its probability waves is realized in one of a vast ensemble of parallel universes. Quantum mechanics suggests that every possibility embodied in its probability waves is realized in one of a vast ensemble of parallel universes.

PARALLELUNIVERSEPROPOSAL: Holographic Multiverse Holographic Multiverse EXPLANATION: The holographic principle a.s.serts that our universe is exactly mirrored by phenomena taking place on a distant bounding surface, a physically equivalent parallel universe. The holographic principle a.s.serts that our universe is exactly mirrored by phenomena taking place on a distant bounding surface, a physically equivalent parallel universe.

PARALLELUNIVERSEPROPOSAL: Simulated Multiverse Simulated Multiverse EXPLANATION: Technological leaps suggest that simulated universes may one day be possible. Technological leaps suggest that simulated universes may one day be possible.

PARALLELUNIVERSEPROPOSAL: Ultimate Multiverse Ultimate Multiverse EXPLANATION: The principle of fecundity a.s.serts that every possible universe is a real universe, thereby obviating the question of why one possibility-ours-is special. These universes instantiate all possible mathematical equations. The principle of fecundity a.s.serts that every possible universe is a real universe, thereby obviating the question of why one possibility-ours-is special. These universes instantiate all possible mathematical equations.

As with the metaphorical book of nature, so with the book you're reading. In this last chapter, I'd be delighted to pull all the pieces together and answer the subject's most essential question: Universe or multiverse? But I can't. That's the nature of explorations that brush the edge of knowledge. Instead, to catch a glimpse of where the multiverse concept might be headed, as well as to emphasize the essential highlights of where it now stands, here are five central questions with which physicists will continue to grapple in the years ahead.

Is the Copernican Pattern Fundamental?

Regularities and patterns, evident in observations and in mathematics, are essential to formulating physical laws. Patterns of a different sort, in the nature of the physical laws accepted by each successive generation, are also revealing. Such patterns reflect how scientific discovery has shifted humankind's perspective on its place in the cosmic order. Over the course of nearly five centuries, the Copernican progression has been a dominant theme. From the rising and setting of the sun to the motion of constellations across the night sky to the leading role we each play in our mind's inner world, experience abounds with clues suggesting that we're a central hub around which the cosmos revolves. But the objective methods of scientific discovery have steadily corrected this perspective. At nearly every turn, we've found that were we not here, the cosmic order would hardly differ. We've had to give up our belief in earth's centrality among our cosmic neighbors, the sun's centrality in the galaxy, the Milky Way's centrality among the galaxies, and even the centrality of protons, neutrons, and electrons-the stuff of which we're made-in the cosmic recipe. There was a time when evidence contrary to long-held collective delusions of grandeur was viewed as a frontal a.s.sault on human worth. With practice, we've gotten better at valuing enlightenment.

The trek in this book has been toward what may be the capstone Copernican correction. Our universe itself may not be central to any cosmic order. Much as with our planet, star, and galaxy, our universe may merely be one among a great many. The idea that reality based on a multiverse extends the Copernican pattern and perhaps completes it is cause for curiosity. But what elevates the multiverse concept above idle speculation is a key fact that we've now repeatedly encountered. Scientists have not been on a hunt for ways to extend the Copernican revolution. They've not been plotting in