by Brian Greene
There are other types of mathematical functions, however, for which a computer simulation can be absolutely precise. They’re part of a class called 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 to the laws of physics that you learned in high school. (Typically, continuous space 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 appetite. 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 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 assured, 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 constituting 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 spawn 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 Jürgen 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 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 assured 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 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 encompassing 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 prickly issues raised in the middle of the last century by penetrating thinkers like Kurt Gödel, Alan Turing, and Alonzo Church. Gödel’s famous 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 Gödel’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 Gödel’s theorem; this wing of the Library of Mathematical Babel, this version of the Ultimate Multiverse, would be free of Gödel’s ghost. Maybe that’s what singles out computable functions.
Would 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. Space 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 space is fundamentally discrete; for now, the question is open. A similar limited understanding applies to intervals of time. The discoveries recounted in Chapter 9, which yield information capacity of one bit per Planck area in any region of space, constitute 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.
The Roots of Reality
In the Simulated Multiverse, there’s no ambiguity regardi
ng 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 anticipate 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 space 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. 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 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 quantities 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 assures us that in any finite region of space 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 assume 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 space. 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 seashore, diverting myself in now and then finding a smoother pebble or prettier shell 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 circumstances. 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 assumption of science is that regularities and patterns exist on all scales, but as Newton himself anticipated, there’s no reason to expect the patterns we directly encounter to be recapitulated on all scales.
Th
e 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 passing 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.