The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos - Brian Greene (2011)
Chapter 10. Universes, Computers, and Mathematical Reality
The Simulated and the Ultimate Multiverses
The parallel universe theories we considered in previous chapters emerged from mathematical laws developed by physicists in their pursuit of nature’s deepest workings. The credence accorded one set of laws or another varies widely—quantum mechanics is viewed as established fact; inflationary cosmology has observational support; string theory is thoroughly speculative—as does the type and logical necessity of the parallel worlds associated with each. But the pattern is clear. When we hand over the steering wheel to the mathematical underpinnings of the major proposed physical laws, we’re driven time and again to some version of parallel worlds.
Let’s now change tack. What happens if we seize the wheel? Can we humans manipulate the cosmic unfolding to volitionally create universes parallel to our own? If you believe, as I do, that the behavior of living beings is dictated by nature’s laws, then you may see this as no change of tack at all but simply as a narrowing of perspective, to the impact of physical law when funneled through human activity. This line of thought quickly engages thorny issues such as the age-old debate about determinism and free will, but that’s not a direction in which I want to head. Rather, my question is this: With the same sense of intent and control you feel when you choose a movie or a meal, might you create a universe?
The question sounds outlandish. And it is. I’ll tip you off now that in addressing it we will find ourselves in territory even more speculative than what we’ve already covered, and considering where we’ve been, that says a lot. But let’s have a little fun and see where it takes us. Let me lay out the perspective I’ll take. In contemplating universe creation, I’m less interested in practical constraints than in the possibilities made available by the laws of physics. So, when I speak of “you” creating a universe, what I really mean is you, or a distant descendant, or an army of such descendants possibly millennia down the road. These present or future humans will still be subject to the laws of physics, but I will imagine that they’re in possession of arbitrarily advanced technologies. I will also consider the creation of two distinct types of universes. The first type comprises the usual universes, ones that encompass an expanse of space and are filled with various forms of matter and energy. The second kind is less tangible: virtual computer-generated universes. The discussion will also naturally forge a link to a third multiverse proposal. This variety does not originate from thinking about universe creation, per se, but instead addresses the question of whether mathematics is “real” or is instead created by the mind.
To Create a Universe
Despite uncertainties in delineating the composition of the universe—What is the dark energy? What is the full list of fundamental particulate ingredients?—scientists are confident that were you to weigh everything that’s within our cosmic horizon, the tally would come in at about 10 billion billion billion billion billion billion grams. If the contents weighed significantly more or less than this, their gravitational influence on the cosmic microwave background radiation would cause the splotches in Figure 3.4 to be much larger or smaller, and that would conflict with refined measurements of their angular size. But the precise weight of the observable universe is secondary; my point is that it’s huge. So huge that the notion of us humans creating another such realm seems utterly fatuous.
Using big bang cosmology as our blueprint for universe formation, we find no guidance on how to clear this hurdle. In the standard big bang theory, the observable universe was ever-smaller at ever-earlier times, but the stupendous quantities of matter and energy we now measure were always present; they were just squeezed into an ever-smaller volume. If you want a universe like the one we see today, you have to start with raw material whose mass and energy are those we see today. The big bang theory takes such raw material as an unexplained given.1
In broad strokes, then, the big bang’s instructions for creating a universe like ours require that we gather a gargantuan amount of mass and compress it to a fantastically small size. But having achieved that, however improbable, we would face another challenge. How do we ignite the bang? It’s an obstacle that becomes only more daunting when we recall that the big bang is not an explosion that takes place within a static region of space; the big bang propels the expansion of space itself.
If the big bang theory were the pinnacle of cosmological thought, the scientific pursuit of universe creation would stop here. But it’s not. We’ve seen that the big bang theory has given way to the more robust inflationary cosmology, and inflation offers a strategy for going forward. With a powerful outward burst of spatial expansion being its trademark, the inflationary theory puts a bang in the big bang, and a big one at that; according to inflation, an anti-gravity blast is what set the outward expansion of space in motion. Of equal importance, as we’ll now see, inflation establishes that vast amounts of matter can be createdfrom the most modest of seeds.
Recall from Chapter 3 that in the inflationary approach, a universe like ours—a hole in the cosmic Swiss cheese—formed when the inflaton’s value rolled down its potential energy curve, bringing to a close the phenomenal outward surge in our vicinity. As the inflaton’s value dropped, the energy it contained was transformed into a bath of particles uniformly filling our bubble. That’s where the matter we see originated. Progress, for sure, but the insight raises the next question: What’s the source of the inflaton’s energy?
It comes from gravity. Remember that inflationary expansion is much like viral replication: a high-valued inflaton field drives the region it inhabits to rapidly grow, and in doing so creates an increasingly large spatial volume that is itself infused with a high-valued inflaton field. And because a uniform inflaton field contributes a constant energy per unit volume, the larger the volume it fills, the more energy it embodies. The driving force behind the expansion is gravity—in its repulsive guise—and so gravity is the source of the ever-larger energy the region contains.
Inflationary cosmology can thus be thought of as creating a sustained energy flow from the gravitational field to the inflaton field. This might seem like one more passing of the energy buck—where does gravity get its energy?—but the situation is a good deal better than that. Gravity is different from the other forces because where there’s gravity, there’s a virtually unlimited reservoir of energy. It’s a familiar idea expressed in unfamiliar language. When you jump off a cliff, your kinetic energy—the energy of your motion—gets ever larger. Gravity, the force driving your motion, is the energy’s source. In any realistic situation, you will hit the ground, but in principle you could fall arbitrarily far, tumbling down an increasingly long rabbit hole, while your kinetic energy grows ever larger. The reason gravity can supply such unlimited quantities of energy is that, much like the U.S. Treasury, it has no fear of debt. As you fall and your energy gets ever more positive, gravity compensates by its energy becoming ever more negative. You know intuitively that the gravitational energy is negative because to climb out of the rabbit hole, you need to exert positive energy—pushing with your legs, pulling with your arms; that’s how you repay the energy debt gravity incurred on your behalf.2
The essential conclusion is that as an inflaton-filled region rapidly grows, the inflaton extracts energy from the gravitational field’s inexhaustible resources, resulting in the region’s energy rapidly growing too. And because the inflaton field supplies the energy that’s converted into ordinary matter, inflationary cosmology—unlike the big bang model—does not need to posit the raw material for generating planets, stars, and galaxies. Gravity is matter’s sugar daddy.
The only independent energy budget required by inflationary cosmology is what’s needed to create an initial inflationary seed, a small spherical nugget of space filled with a high-valued inflaton field that gets the inflationary expansion rolling in the first place. When you put in numbers, the equations show that the nugget need be only about 10–26 centimeters across and filled with an inflaton field whose energy, when converted to mass, would weigh less than ten grams.3 Such a tiny seed would, faster than a flash, undergo spectacular expansion, growing far larger than the observable universe while harboring ever-increasing energy. The inflaton’s total energy would quickly soar beyond what’s necessary to generate all the stars in all the galaxies we observe. And so, with inflation in the cosmological driver’s seat, the impossible starting point of the big bang’s recipe—gather more than 1055 grams and squeeze the whole lot into an infinitesimally small speck—is radically transformed. Gather ten grams of inflaton field and squeeze it into a lump that’s about 10–26 centimeters across. That’s a lump you could put in your wallet.
This approach, nevertheless, presents daunting challenges. For one thing, the inflaton remains a purely hypothetical field. Cosmologists freely incorporate the inflaton field into their equations, but unlike with electron and quark fields, there is as yet no evidence that the inflaton field exists. For another, even if the inflaton proves real, and even if we one day develop the means to manipulate it much as we do the electromagnetic field, still the density of the requisite inflaton seed would be enormous: about 1067 times that of an atomic nucleus. Although the seed would weigh less than a handful of popcorn, the compressive force we would need to apply is trillions and trillions of times beyond what we can now muster.
But this is just the kind of technological hurdle that we’re imagining an arbitrarily advanced civilization might one day overcome. So, if our distant descendants one day harness the inflaton field and develop extraordinary compressors capable of producing such dense nuggets, will we have attained the status of universe creators? And, as we contemplate such a step toward Olympus, should we worry that if we artificially set off new inflationary realms, our own corner of space may be swallowed by the ballooning expanse? Alan Guth and a number of collaborators investigated these questions in a series of papers, and found both good news and bad. Start with the last question, as that’s where we’ll find the good news.
Guth, together with Steven Blau and Eduardo Guendelman, showed that there’s no need to be concerned about an artificial phase of inflationary expansion ripping through our existing environment. The reason has to do with pressure. If an inflationary seed were created in the laboratory, it would harbor the inflaton field’s characteristic positive energy and negative pressure, but it would be surrounded by ordinary space in which the inflaton field’s value, and its pressure, would be zero (or nearly so).
We usually don’t ascribe much power to zero, but in this case zero makes all the difference. Zero pressure is larger than negative pressure, and so the pressure outside the seed would be larger than the pressure inside. This would subject the seed to a net external force pressing upon it, much like what your eardrums experience when deep-sea diving. It is the pressure differential is powerful enough to prevent the seed from expanding into the surrounding environment.
But this does not prevent the inflaton’s drive to expand. If you blow air into a balloon while tightly clasping its surface, the balloon will bubble out from between your hands. The inflaton seed can behave similarly. The seed can generate a new expanding spatial realm that sprouts from the original spatial environment, as illustrated by the little growing sphere in Figure 10.1. The calculations show that once the new expanding realm reaches a critical size, its umbilical cord to the parent space severs, as in the final image of Figure 10.1, and an independent inflating universe is born.
As enticing as the process might be—the artificial creation of a new universe—the view from the laboratory wouldn’t live up to the advance billing. It’s a relief that the inflationary bubble would not gobble up the surrounding environment, but the flip side is that there would be little evidence of the creation itself. A universe that expands by generating new space, which then detaches from ours, is a universe we can’t see. Indeed, as the new universe pinches off, its sole residue would be a deep gravitational well—you can see this in the last image of Figure 10.1—which would appear to us as a black hole. And since we have no capacity to see beyond a black hole’s edge, we wouldn’t even be assured that our experiment had been a success; without access to the new universe, we would have no means of establishing observationally that the universe had been created at all.
Figure 10.1 Because of the greater pressure in the ambient environment, an inflationary seed is forced to expand into newly formed space. As the bubble universe grows, it detaches from the parent environment, yielding a separate, expanding spatial domain. To someone in the ambient environment, the process looks like the formation of a black hole.
Physics protects us, but the price for safety is total separation from our handiwork. And that’s the good news.
The bad news for aspiring universe creators is a more sobering result derived by Guth and his MIT colleague Edward Farhi. Their careful mathematical treatment showed that the sequence depicted in Figure 10.1 requires an additional ingredient. Much as some balloons require that you give a strong initial burst of air, after which they more easily inflate, Guth and Farhi found that the nascent universe in Figure 10.1needs a strong kick-start to get the inflationary expansion off and running. So strong that there’s only one entity that can provide it: a white hole. A white hole, the opposite of a black hole, is a hypothetical object that spews matter out rather than drawing it in. This requires conditions so extreme that known mathematical methods break down (much as is the case at the center of a black hole); suffice it to say, no one anticipates generating white holes in the laboratory. Ever. Guth and Farhi found a fundamental wrench in the universe-creation works.
A number of research groups have since suggested possible ways of skirting the problem. Guth and Farhi, joined by Jemal Guven, found that by creating the inflationary seed through a quantum tunneling process (similar to what we discussed in the context of the Landscape Multiverse) the white hole singularity can be avoided; but the probability for the quantum tunneling process to happen is so fantastically small that there’s essentially no chance of its happening over timescales that anyone would consider worth contemplating. A group of Japanese physicists, Nobuyuki Sakai, Ken-ichi Nakao, Hideki Ishihara, and Makoto Kobayashi, showed that a magnetic monopole—a hypothetical particle that has either the north pole or the south pole of a standard bar magnet—might catalyze inflationary expansion, also avoiding singularities; but after nearly forty years of intense searching, no one has yet found a single one of these particles.*
As of today, then, the summary is that the door to creating new universes remains open, but only barely. Given the proposals’ heavy reliance on hypothetical elements, future developments may well shut this door permanently. But if they don’t—or, perhaps, if subsequent work makes a stronger case for the possibility of universe creation—would there be motivation to proceed? Why create a universe if there’s no way to see it, or interact with it, or even know for sure that it was created? Andrei Linde, famous not just for his deep cosmological insights but also for his flair for mock drama, has noted that the allure of playing god would simply prove irresistible.
I don’t know that it would. Admittedly, it would be thrilling to have so thoroughly grasped nature’s laws that we could reenact the most pivotal of all events. I suspect, however, that by the time we can seriously consider universe creation—if that time ever comes—our scientific and technical advancements would have made available so many other spectacular undertakings, whose results we could not just imagine but truly experience, that the intangible nature of universe creation would make it much less interesting.
The appeal would surely be stronger were we to learn how to manufacture universes that we could see or even interact with. For “real” universes, in the usual sense of a universe constituted from the standard ingredients of space, time, matter and energy, we don’t yet have any strategy for doing so that’s compatible with the laws of physics as we currently understand them.
But what if we set aside real universes and consider virtual ones?
The Stuff of Thought
A couple of years ago, I had a bout of feverish flu that came with hallucinations far more vivid than any ordinary dream or nightmare. In one that has stayed with me, I’d find myself with a group of people sitting in a sparse hotel room, locked in a hallucination within the hallucination. I was absolutely certain that days and weeks went by—until I was thrust back into the primary hallucination, where I’d learn, shockingly, that hardly any time had passed at all. Each time I felt myself drifting back to the room, I resisted strenuously, since I knew from previous iterations that once there I’d be swallowed whole, unable to recognize the realm as false until I found myself back in the primary hallucination, where I’d again be distraught to learn that what I’d thought real was illusory. Periodically, when the fever subsided, I’d pull out one level further, back to ordinary life, and realize that all those translocations had been taking place within my own swirling mind.
I don’t usually learn much from having a fever. But this experience added immediacy to something which, to that point, I’d largely understood only in the abstract. Our grip on reality is more tenuous than day-today life can lead us to believe. Modify normal brain function just a bit, and the bedrock of reality may suddenly shift; though the outside world remains stable, our perception of it does not. This raises a classic philosophical question. Since all of our experiences are filtered and analyzed by our respective brains, how sure are we that our experiences reflect what’s real? In the framing philosophers like to use: How do you know you’re reading this sentence, and not floating in a vat on a distant planet, with alien scientists stimulating your brain to produce the thoughts and experiences you deem real?
These issues are central to epistemology, a philosophical subfield that asks what constitutes knowledge, how we acquire it, and how sure we are that we have it. Popular culture has brought these scholarly pursuits to a wide audience in films such as The Matrix, The Thirteenth Floor, and Vanilla Sky, tussling with them in entertaining and thought-provoking ways. So, in looser language, the question we’re asking is: How do you know you’re not hooked into the Matrix?
The bottom line is that you can’t know for sure. You engage the world through your senses, which stimulate your brain in ways your neural circuitry has evolved to interpret. If someone artificially stimulates your brain so as to elicit electrical crackles exactly like those produced by eating pizza, reading this sentence, or skydiving, the experience will be indistinguishable from the real thing. Experience is dictated by brain processes, not by what activates those processes.
Going a step further, we can consider dispensing with the sloppiness of biological material altogether. Might all your thoughts and experiences be nothing more than a simulation that leverages software and circuitry sufficiently elaborate to mimic ordinary brain function? Are you convinced of the reality of flesh, blood, and the physical world, when actually your experience is only a crowd of electrical impulses firing through a hyper-advanced supercomputer?
An immediate challenge in considering such scenarios is that they easily set off a spiraling skeptical collapse; we wind up trusting nothing, not even our powers of deductive reasoning. My first response to questions like the ones just posed is to work out how much computer power you’d need to stand a chance of simulating a human brain. But if I am indeed part of such a simulation, why should I believe anything I read in neurobiology texts? The books would be simulations too, written by simulated biologists, whose findings would be dictated by the software running the simulation and thus could easily be irrelevant to the workings of “real” brains. The very notion of a “real” brain might itself be computer-generated artifice. Once you can’t trust your knowledge base, reality quickly sails to sea.
We’ll return to these concerns, but I don’t want them to sink us—at least, not yet. So, for the time being, let’s drop anchor. Imagine that you are real flesh and blood, and so am I, and that everything you and I take to be real, in the everyday sense of the term, is real. With all that assumed, let’s take up the question of computers and brainpower. What, roughly, is the processing speed of the human brain, and how does it compare with the capacity of computers?
Even if we are not stuck in a skeptical morass, this is a difficult question. Brain function is largely an uncharted territory. But just to get a glimpse of the terrain, however foggy, consider some numbers. The human retina, a thin slab of 100 million neurons that’s smaller than a dime and about as thick as a few sheets of paper, is one of the best-studied neuronal clusters. The robotics researcher Hans Moravec has estimated that for a computer-based retinal system to be on a par with that of humans, it would need to execute about a billion operations each second. To scale up from the retina’s volume to that of the entire brain requires a factor of roughly 100,000; Moravec suggests that effectively simulating a brain would require a comparable increase in processing power, for a total of about 100 million million (1014) operations per second.4 Independent estimates based on the number of synapses in the brain and their typical firing rates yield processing speeds within a few orders of magnitude of this result, about 1017 operations per second. Although it’s difficult to be more precise, this gives a sense of the numbers that come into play. The computer I’m now using has a speed that’s about a billion operations per second; today’s fastest supercomputers have a peak speed of about 1015 operations per second (a statistic that no doubt will quickly date this book). If we use the faster estimate for brain speed, we find that a hundred million laptops, or a hundred supercomputers, approach the processing power of a human brain.
Such comparisons are likely naïve: the mysteries of the brain are manifold, and speed is only one gross measure of function. But most everyone agrees that one day we will have raw computing capacity equal to, and likely far in excess of, what biology has provided. Futurists contend that such technological leaps will yield a world so far beyond familiar experience that we lack the capacity to imagine what it will be like. Invoking an analogy with phenomena that lie outside the bounds of our most refined physical theories, they call this visionary roadblock a singularity. One broad-brush prognosis holds that the surpassing of brainpower by computers will completely blur the boundary between humans and technology. Some anticipate a world run rampant with thinking and feeling machines, while those of us still based in old-fashioned biology routinely upload our brain content, safely storing knowledge and personalities in silico, complete with backup drives, for unlimited durations.
This vision may well be hyperbolic. There’s little dispute regarding projections of computer power, but the obvious unknown is whether we will ever leverage such power into a radical fusion of mind and machine. It’s a modern-day question with ancient roots; we’ve been thinking about thinking for thousands of years. How is it that the external world generates our internal responses? Is your sensation of color the same as mine? How about your sensations of sound and touch? What exactly is that voice we hear in our heads, the stream of internal chatter we call our conscious selves? Does it derive from purely physical processes? Or does consciousness arise from a layer of reality that transcends the physical? Penetrating thinkers through the ages, Plato and Aristotle, Hobbes and Descartes, Hume and Kant, Kierkegaard and Nietzsche, James and Freud, Wittgenstein and Turing, among countless others, have tried to illuminate (or debunk) processes that animate the mind and create the singular inner life available through introspection.
A great many theories of mind have emerged, differing in ways significant and subtle. We won’t need the finer points, but just to get a feel for where the trails have led, here are a few: dualist theories, of which there are many varieties, maintain that there’s an essential nonphysical component vital to mind. Physicalist theories of mind, of which there are also many varieties, deny this, emphasizing instead that underlying each unique subjective experience is a unique brain state. Functionalist theories go further in this direction, suggesting that what really matters to making a mind are the processes and functions—the circuits, their interconnections, their relationships—and not the particulars of the physical medium within which these processes take place.
Physicalists would largely agree that were you to faithfully replicate my brain by whatever means—molecule by molecule, atom by atom—the end product would indeed think and feel as I do. Functionalists would largely agree that were you to focus on higher-level structures—replicating all my brain connections, preserving all brain processes while changing only the physical substrate through which they occur—the same conclusion would hold. Dualists would largely disagree on both counts.
The possibility of artificial sentience clearly relies on a functionalist viewpoint. A central assumption of this perspective is that conscious thought is not overlaid on a brain but rather is the very sensation generated by a particular kind of information processing. Whether that processing happens within a three-pound biological mass or within the circuits of a computer is irrelevant. The assumption could be wrong. Maybe a bundle of connections needs a substrate of wrinkled wet matter if it’s to gain self-awareness. Maybe you need the actual physical molecules that constitute a brain, not just the processes and connections those molecules facilitate, if conscious thought is to animate the inanimate. Maybe the kinds of information processing that computers carry out will always differ in some essential way from brain functioning, preventing the leap to sentience. Maybe conscious thought is fundamentally nonphysical, as claimed by various traditions, and so lies permanently beyond the reach of technological innovation.
With the rise of ever more sophisticated technologies, the questions have become sharper and the pathway toward answers more tangible. A number of research groups have already taken the initial steps toward simulating a biological brain on a computer. For example, the Blue Brain Project, a joint venture between IBM and the École Polytechnique Fédérale in Lausanne, Switzerland, is dedicated to modeling brain function on IBM’s fastest supercomputer. Blue Gene, as the supercomputer is called, is a more powerful version of Deep Blue, the computer that triumphed in 1997 over the world chess champion Garry Kasparov. Blue Brain’s approach is not all that different from the scenarios I just described. Through painstaking anatomical studies of real brains, researchers are gathering ever more precise insight into the cellular, genetic, and molecular structure of neurons and their interconnections. The project aims to encode such understanding, for now mostly at the cellular level, in digital models simulated by the Blue Gene computer. To date, researchers have drawn on results from tens of thousands of experiments focused on a pinhead-sized section of a rat brain, the neocortical column, to develop a three-dimensional computer simulation of roughly 10,000 neurons communicating through some 10 million interconnections. Comparisons between the response of a real rat’s neocortical column and the computer simulation to the same stimuli show an encouraging fidelity of the synthetic model. This is far from the 100 billion neurons firing away in a typical human head, but the project’s leader, the neuroscientist Henry Markram, anticipates that before 2020 the Blue Brain Project, leveraging processing speeds that are projected to increase by a factor of more than a million, will achieve a full simulated model of the human brain. Blue Brain’s goal is not to produce artificial sentience, but rather to have a new investigative tool for developing treatments for various forms of mental illness; still, Markram has gone out on a limb to speculate that, when completed, Blue Brain may very well have the capacity to speak and to feel.
Regardless of the outcome, such hands-on explorations are pivotal to our theories of mind; I’m quite certain that the issue of which, if any, of the competing perspectives are on target cannot be settled through purely hypothetical speculation. In practice, too, challenges are immediately evident. Suppose a computer one day professes to be sentient—how would we know whether it really is? I can’t even verify such claims of sentience when made by my wife. Nor she with me. That’s a burden arising from consciousness being a private affair. But because our human interactions yield abundant circumstantial evidence supporting the sentience of others, solipsism quickly becomes absurd. Computer interactions may one day reach a similar point. Conversing with computers, consoling and cajoling them, may one day convince us that the simplest explanation for their apparent conscious self-awareness is that they are indeed conscious and self-aware.
Let’s take a functionalist viewpoint, and see where it leads.
If we ever create computer-based sentience, some would likely implant the thinking machines in artificial human bodies, creating a mechanical species—robots—that would be integrated into conventional reality. But my interest here is in those who would be drawn by the purity of electrical impulses to program simulated environments populated by simulated beings that would exist within a computer’s hardware; instead of C-3PO or Data, think Sims or Second Life, but with inhabitants who have self-aware and responsive minds. The history of technological innovation suggests that iteration by iteration, the simulations would gain verisimilitude, allowing the physical and experiential characteristics of the artificial worlds to reach convincing levels of nuance and realism. Whoever was running a given simulation would decide whether the simulated beings knew that they existed within a computer; simulated humans who surmised that their world was an elaborate computer program might find themselves taken away by simulated technicians in white coats and confined to simulated locked wards. But probably the vast majority of simulated beings would consider the possibility that they’re in a computer simulation too silly to warrant attention.
You may well be having that very reaction right now. Even if you accept the possibility of artificial sentience, you may be persuaded that the overwhelming complexity of simulating an entire civilization, or just a smaller community, renders such feats beyond computational reach. On this point, it’s worth looking at some more numbers. Our distant descendants will likely fashion ever-larger quantities of matter into vast computing networks. So allow imagination free rein. Think big. Scientists have estimated that a present-day high-speed computer the size of the earth could perform anywhere from 1033 to 1042 operations per second. By comparison, if we assume that our earlier estimate of 1017 operations per second for a human brain is on target, then an average brain performs about 1024 total operations in a single hundred-year life span. Multiply that by the roughly 100 billion people who have ever walked the planet, and the total number of operations performed by every human brain since Lucy (my archaeology friends tell me I should say “Ardi”) is about 1035. Using the conservative estimate of 1033 operations per second, we see that the collective computational capacity of the human species could be achieved with a run of less than two minutes on an earth-sized computer.
And that’s with today’s technology. Quantum computing—harnessing all the distinct possibilities represented in a quantum probability wave so as to do many different calculations simultaneously—has the capacity to increase processing speeds by spectacular factors. Although we are still very far from mastering this application of quantum mechanics, researchers have estimated that a quantum computer no bigger than a laptop has the potential to perform the equivalent of all human thought since the dawn of our species in a tiny fraction of a second.
To simulate not just individual minds but also their interactions among themselves and with an evolving environment, the computational load would grow orders of magnitude larger. But a sophisticated simulation could cut computational corners with minimal impact on quality. Simulated humans on a simulated earth won’t be bothered if the computer simulates only things lying within the cosmic horizon. We can’t see beyond that range, so the computer can safely ignore it. More boldly, the simulation might simulate stars beyond the sun only during simulated nights, and then only when the simulated local weather resulted in clear skies. When no one’s looking, the computer’s celestial simulator routines could take a break from working out the appropriate stimulus to provide each and every person who could look skyward. A sufficiently well-structured program would keep track of the mental states and intentions of its simulated inhabitants, and so would anticipate, and appropriately respond to, any impending stargazing. The same goes for simulating cells, molecules, and atoms. For the most part, they’d be necessary only for simulated specialists of one scientific persuasion or another, and then only when such specialists were in the act of studying these exotic realms. A computationally cheaper replica of familiar reality that adjusts the simulation’s degree of detail on an as-needed basis would be adequate.
Such simulated worlds would forcefully realize Wheeler’s vision of information’s primacy. Generate circuits that carry the right information and you’ve generated parallel realities that are as real to their inhabitants as this one is to us. These simulations constitute our eighth variety of multiverse, which I’ll call the Simulated Multiverse.
Are You Living in a Simulation?
The idea that universes might be simulated on computers has a long history, dating as far back as suggestions made in the 1960s by the computer pioneer Konrad Zuse and the digital guru Edward Fredkin. I worked at IBM during five summers spanning college and graduate school; my boss, the late John Cocke, himself a revered computer specialist, spoke frequently of Fredkin’s view that the universe was nothing but a giant computer chugging along, executing something akin to cosmic Fortran. The idea struck me as taking the digital paradigm to a ridiculous extreme. Through the years, I hardly gave it a thought—until I encountered, much more recently, a simple but curious conclusion by the Oxford philosopher Nick Bostrom.
To appreciate Bostrom’s point (one that Moravec had also hinted at), begin with a straightforward comparison: the difficulty of creating a real universe versus the difficulty of creating a simulated universe. To create a real one, as we’ve discussed, presents enormous obstacles. And if we succeeded, the resulting universe would be beyond our ability to see, which invites the question of what motivated us to create it in the first place.
The creation of a simulated universe is a wholly different enterprise. The march toward increasingly powerful computers, running ever more sophisticated programs, is inexorable. Even with today’s rudimentary technology, the fascination of creating simulated environments is strong; with more capability it’s hard to imagine anything but more intense interest. The question is not whether our descendants will create simulated computer worlds. We’re already doing it. The unknown is how realistic the worlds will become. Should there be an inherent obstacle to generating artificial sentience, all bets are off. But Bostrom, assuming that realistic simulations prove possible, makes a simple observation.
Our descendants are bound to create an immense number of simulated universes, filled with a great many self-aware, conscious inhabitants. If someone can come home at night, kick back, and fire up the create-a-universe software, it’s easy to envision that they’ll not only do so, but do so often. Think about what this scenario might entail. One future day, a cosmic census that takes account of all sentient beings might find that the number of flesh-and-blood humans pales in comparison with those made of chips and bytes, or their future equivalents. And, Bostrom reasons, if the ratio of simulated humans to real humans were colossal, then brute statistics suggests that we are not in a real universe. The odds would overwhelmingly favor the conclusion that you and I and everyone else are living within a simulation, perhaps one created by future historians with a fascination for what life was like back on twenty-first-century earth.
You may object that we have now run headlong into the skeptical quicksand we planned at the outset to avoid. Once we conclude that there’s a high likelihood that we’re living in a computer simulation, how do we trust anything, including the very reasoning that led to the conclusion? Well, our confidence in a great many things might diminish. Will the sun rise tomorrow? Maybe, as long as whoever is running the simulation doesn’t pull the plug. Are all our memories trustworthy? They seem so, but whoever is at the keyboard may have a penchant for adjusting them from time to time.
Nevertheless, Bostrom notes, the conclusion that we’re in a simulation does not fully sever our grasp on the true underlying reality. Even if we believe that we’re in a simulation, we can still identify one feature that the underlying reality definitely possesses: it allows for realistic computer simulations. After all, according to our belief, we’re in one. The unbridled skepticism generated by the suspicion that we’re simulated aligns with that very knowledge and so fails to undermine it. While it was useful when we began to weigh anchor and declare the reality of all that seems real, it wasn’t necessary. Logic alone can’t ensure that we’re not in a computer simulation.
The only way to dodge the conclusion that we’re likely living in a simulation is to leverage intrinsic weaknesses in the reasoning. Maybe sentience can’t be simulated, full stop. Or maybe, as Bostrom also suggests, civilizations en route to the technological mastery necessary to create sentient simulations will inevitably turn that technology inward and destroy themselves. Or maybe when our distant descendants gain the capacity to create simulated universes they choose not to do so, perhaps for moral reasons or simply because other currently inconceivable pursuits prove so much more interesting that, much as we noted with universe creation, universe simulation falls by the wayside.
These are among numerous loopholes, but whether they’re large enough for the proverbial truck to drive through, who knows?* If not, you might want to spice up your life a bit, make your mark. Whoever is running the simulation is bound to get tired of wallflowers. Being a cynosure would seem a likely path toward longevity.5
Seeing Beyond a Simulation
If you were living in a simulation, could you figure that out? The answer depends in no small part on who is running your simulation—call him or her the Simulator—and the manner in which your simulation was programmed. The Simulator, for instance, might choose to let you in on the secret. One day while taking a shower you might hear a gentle “dingding,” and when you’d cleared the shampoo from your eyes you’d see a floating window in which your smiling Simulator would appear and introduce herself. Or maybe this revelation would happen on a worldwide scale, with giant windows and a booming voice surrounding the planet, announcing that there is in fact an All Powerful Programmer up in the heavens. But even if your Simulator shied away from exhibitionism, less obvious clues might turn up.
Simulations allowing for sentient beings would certainly have reached a minimum fidelity threshold, but as they do with designer clothes and cheap knockoffs, quality and consistency would likely vary. For example, one approach to programming simulations—call it the “emergent strategy”—would draw on the accumulated mass of human knowledge, judiciously invoking relevant perspectives as dictated by context. Collisions between protons in particle accelerators would be simulated using quantum field theory. The trajectory of a batted ball would be simulated using Newton’s laws. The reactions of a mother watching her child’s first steps would be simulated by melding insights from biochemistry, physiology, and psychology. The actions of governmental leaders would fold in political theory, history, and economics. Being a patchwork of approaches focused on different aspects of simulated reality, the emergent strategy would need to maintain internal consistency as processes nominally construed to lie in one realm spilled over into another. A psychiatrist needn’t fully grasp the cellular, chemical, molecular, atomic, and subatomic processes underlying brain function—which is a good thing for psychiatry. But in simulating a person, the challenge for the emergent strategy would be to consistently meld coarse and fine levels of information, ensuring for example that emotional and cognitive functions interface sensibly with physiochemical data. This kind of cross-border meshing takes place in all phenomena and has always compelled science to seek deeper, more unified explanations.
Simulators employing emergent strategies would have to iron out mismatches arising from the disparate methods, and they’d need to ensure that the meshing was smooth. This would require fiddles and tweaks which, to an inhabitant, might appear as sudden, baffling changes to the environment with no apparent cause or explanation. And the meshing might fail to be fully effective; the resulting inconsistencies could build over time, perhaps becoming so severe that the world became incoherent, and the simulation crashed.
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 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 quantity varies, say, from .9 to 1, it would pass 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 thedeepest 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 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 accumulated 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, assuming 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 assiduously 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 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, suggests that there’s not just one such simulation out there but a swarming ocean of simulations, which constitute 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; he used the course as a dress rehearsal for many of the book’s central arguments. Just about every class 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 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 identity through time?
While I was hanging around after class 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. Naïve and blustery, 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 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.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.
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 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 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.
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.*
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 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 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.