Our Mathematical Universe

by Max Tegmark

  This is deeply troubling. If our spacetime really contains these Boltzmann brains, then you’re basically 100% certain to be one of them! After all, the observer moment of the evolved you is in the same reference class as those of these brains, since they subjectively feel the same, so you should reason as if you’re a random one of these observer moments—and the disembodied ones outnumber the embodied one by infinity to one.…

  Before you get too worried about the ontological status of your body, here’s a simple test you can do to determine whether you’re a Boltzmann brain. Pause. Introspect. Examine your memories. In the Boltzmann-brain scenario, it’s indeed more likely that any particular memories that you have are false rather than real. However, for every set of false memories that could pass as having been real, very similar sets of memories with a few random crazy bits tossed in (say, you remembering Beethoven’s Fifth Symphony sounding like pure static) are vastly more likely, because there are vastly more disembodied brains with such memories. This is because there are vastly more ways of getting things almost right than getting them exactly right. Which means that if you really are a Boltzmann brain who at first thinks you’re not, then when you start jogging your memory, you should discover more and more utter absurdities. And after that, you’ll feel your reality dissolving, as your constituent particles drift back into the cold and almost empty space from which they came.

  In other words, if you’re still reading this, you’re not a Boltzmann brain. This means that something is fundamentally wrong with what we assumed about the future of our Universe, and that there’s a lesson to be learned. We’ll shortly explore that in the “measure problem” section.

  * * *

  1Note that many of the characters in The Matrix have simulated experiences in human brains; in contrast, the simulated people in the movie The Thirteenth Floor involve no human hardware whatsoever.

  The Doomsday Argument: Is the End Nigh?

  We’ve seen that the idea that you should be a typical observer is a powerful one, with surprising consequences. Another much debated consequence is the doomsday argument, which was first given by Brandon Carter in 1983.

  During World War II, the Allied forces successfully estimated the number of German tanks from their serial numbers. If the first captured tank had the serial number 50, then this ruled out the hypothesis that there were more than a thousand tanks with 95% confidence, since the probability of capturing one of the first fifty ones built was less than 5%. The key assumption is that the first tank captured can be thought of as a random one from the reference class of all tanks.

  Carter pointed out that if we assign each human a serial number at birth, then we can make exactly the same argument to estimate the total number of humans who will ever live. When I arrived on the scene in 1967, I was roughly the fifty-billionth person born, so if I’m a random human out of all people who’ll ever live, then I can rule out the hypothesis that more than a trillion humans will be born with 95% confidence. In other words, it’s highly unlikely that there’ll be more than a trillion humans born, because this would place me within the first 5% of humans to exist—something which we could explain only by invoking an unlikely fluke coincidence. Moreover, if the world population stabilizes at 10 billion with an eighty-year life expectancy, then humanity as we know it will with 95% certainty end before the year 10,000 AD.

  If I believe that our doomsday will be caused by nuclear weapons (or computer technology, biotech or any other technology that has existed only since after 1945), then my forecast gets gloomier: my birth rank since the dangers began is 1.6 billion, and I can rule out with 95% confidence that there will be another 32 billion births after me, around the year 2100. And that’s the 95% confidence limit—a more likely end date for humanity is right around now. To escape this pessimistic conclusion, I’d need to come up with some a priori reason for why I should be among the first 5% of all humans to be born under the shadow of these technologies. We’ll return to the existential risk posed by technology in Chapter 13.

  Some people take the doomsday argument very seriously. For example, when I had the pleasure of meeting Brandon Carter at a conference, he excitedly told me about the latest evidence that the population explosion was slowing, saying that he’d predicted that this would happen, and that this meant we should expect humanity to survive for longer. Others have criticized the argument on various grounds. For example, things get more subtle if there are other planets with people similar to us. Figure 11.10 illustrates such an example, where the total number of people ever born varies sharply from planet to planet. If you know this to be the case, then you should be more optimistic about the future than the standard doomsday argument suggests. Indeed, if I believed the more extreme theory that there are only two inhabited planets in spacetime, supporting a total of 10 billion and 10 quadrillion people from beginning to end, then the probability is 50% that I’m now on the planet that will eventually enjoy a quadrillion people.

  Unfortunately, this counterargument gives only false hope. I have no such information, and I have very good reason to believe that this two-planet theory is false: the observation that my birth rank is about 50 billion rules out the theory at more than 99.999999% confidence, since the probability of a random person being within the first 50 billion born is only 0.0000005%.

  Figure 11.10: If you know that your birth rank is 3 billion, you might think there’s only a 10% chance that more than 30 billion will ever live on your planet. But suppose you know that there are six planets similar to ours, where the total number of people born from the beginning to the end of their civilizations is 1, 2, 4, 8, 16 and 32 billion, respectively (each stick figure above represents a billion people). Then the probability that more than 30 billion will ever live on your planet is actually 25%; there are exactly four people who have your birth rank, you’re equally likely to be any of them, and 25% of them live on the highly successful bottom planet in the image above.

  Why Is Earth So Old?

  In March 2005, I had the pleasure of meeting Nick Bostrom at a conference in California, and we soon discovered that we share not only Swedish childhoods but also a fascination with big questions. After some good wine, our conversation turned to doomsday scenarios. Could the Large Hadron Collider create a miniature black hole that would end up gobbling up Earth? Could it create a “strangelet” that could catalyze the conversion of Earth into strange quark matter? MIT colleagues of mine whose calculations I trust have concluded that there’s negligible risk, but what if we’ve overlooked something? What used to reassure me the most was the fact that nature is vastly more violent than any of our human-made machines: for example, cosmic-ray particles created near monster black holes routinely slam into Earth with over a million times more energy than our accelerators can deliver, and 4.5 billion years after its formation, Earth is still alive and well. So Earth is clearly very robust, and I needn’t worry. For the same reason, I shouldn’t worry about other cosmic doomsday scenarios, such as space “freezing” into another lower-energy phase as per Chapter 5, with a cosmic death bubble containing this uninhabitable new kind of space expanding with the speed of light, destroying all people in its path at exactly the same instant they saw it come: if we’re still here after all this time, such events must be nonexistent or very rare.

  Then a terrible thought hit me: my reassuring argument was flawed! Suppose each planet has a 50% chance of getting destroyed each day. Then the vast majority will be gone within weeks, but in an infinite space with infinitely many planets, there’ll always be an infinite number remaining, whose inhabitants can be blissfully unaware of the grim fate that awaits them. And if I’m simply a random observer in spacetime, then I expect myself to be one of these naive people who don’t realize that they’re like lambs about to be slaughtered. In other words, the fact that my region of space hasn’t yet been destroyed tells me nothing, because all living observers are in regions of space that haven’t been destroyed. I got really nervous. I felt as if I were i
n a zoo in front of a pack of hungry lions, and had just realized that the fence I thought protected me was an optical illusion—and one that the lions couldn’t see.

  Nick and I agonized about this for a while, until we managed to come up with a different anti-doomsday argument that wasn’t flawed. Earth formed about 9 billion years after our Big Bang, and it’s now fairly clear that our Galaxy (and other similar galaxies elsewhere) harbors a large number of Earth-like planets that formed several billion years earlier. This means that when we consider all observers similar to us in all of spacetime, a significant fraction of them exist long before us. Now, in a scenario where planets get randomly destroyed with some short half-life (say, a day, a year or a millennium), then almost all observer moments will happen very early on, and it’s extremely unlikely for us to find ourselves on a planet that formed at such a leisurely pace relatively late in the game. We decided to write a paper about it, and worked on it late into the night in a hotel lounge. When I finally drifted off to sleep, I did so knowing that with 99.9% confidence, neither death bubbles, black holes, nor strangelets would get us for another billion years.

  Unless, of course, we humans do something stupid of a kind that nature hasn’t already tried.…

  Why Aren’t You Younger?

  We just saw that if there were some terrible instability built into physics that made most planets short-lived, then we should expect to find ourselves on one of the first habitable planets to form, not on this slowpoke planet of ours. So that depressing theory is ruled out. Unfortunately for inflation, Alan Guth realized that under some reasonable-sounding assumptions, it predicts the same thing! Bothered by his brainchild predicting a much younger Earth, he called this the youngness paradox. Around when I became his colleague at MIT back in 2004, I spent a lot of time worrying about how to make predictions in a multiverse. I wrote a paper on this topic that painfully broke all my past length records, and was surprised to discover that the youngness paradox was even more extreme than we’d thought.

  As we saw in Chapter 5, inflation typically goes on forever doubling the volume of space every 10−38 seconds or so, creating a messy spacetime with countless Big Bangs occurring at different times and countless planets forming at different times. We saw that an observer on any given planet will consider her Big Bang to be the moment when inflation ended in her part of space; for me personally, the delay between my Big Bang and my current observer moment is about 14 billion years. Now let’s consider all simultaneous observer moments: for some, the time since their Big Bang is 13 billion years, for some it’s 15, etc. Because of the frenetic volume doubling, there will be 21038 times as many Big Bangs happening one second later, because the volume doubled 1038 times during that extra second. Similarly, there are 21038 times more observers in the galaxies they form. This means that if I’m a random observer moment out of all currently occurring ones, then I’m 21038 times more probable to find myself in a one-second-younger universe, whose Big Bang happened one second more recently! That’s about one with a hundred trillion trillion trillion zeros times more likely. My planet should be younger, my body should be younger, and everything should appear to have formed and evolved in haste.

  A part of space that experienced its Big Bang more recently will be hotter, because it’s had less time to cool off, so finding ourselves in a relatively cool universe is highly unlikely and we have a coolness problem: when I worked out the probability of measuring the cosmic microwave–background temperature to be less than three degrees above absolute zero, I got 10−1056, so when the COBE-satellite measured this temperature to be 2.725 Kelvin, this measurement ruled out our whole inflation-based story with 99.999 … 999% confidence, where there are a hundred million trillion trillion trillion trillion nines after the decimal point. Not good … In the hall of shame for disagreements between theory and experiment, this crushes even the hydrogen-atom stability problem from Chapter 7 (28 nines) and the dark-energy problem from Chapter 4 (123 nines). Welcome to the measure problem!

  The Measure Problem: Physics in Crisis

  Something just went terribly wrong, but what exactly? Did this really rule out eternal inflation? Let’s take a closer look. We asked a reasonable question about what a typical observer should expect to measure—we picked the particular example of the cosmic microwave temperature. Because we considered eternal inflation, we analyzed a spacetime containing many observer moments measuring many different temperatures, so we couldn’t predict just one unique answer, merely probabilities for different temperature ranges. This, per se, isn’t the end of the world: we saw in Chapter 7 how quantum mechanics predicts only probabilities, not definite answers, and is nonetheless a perfectly testable and successful scientific theory. Rather, the problem was that the probabilities we computed told us that what we in fact observe is ridiculously unlikely, so that the underlying theory is ruled out.

  Could there be a mistake in our probability calculation? The math is straightforward in principle: the probabilities are simply the fractions of all observer moments in our reference class that measure various temperatures. If there are only five such observer moments and they observe 1, 2, 5, 10 and 12 degrees above absolute zero, then the fraction measuring less than three is two out of five, 2/5 = 40%—easy! But what if, as eternal inflation predicts, there are infinitely many such observer moments, and the fraction measuring less than three degrees is infinity divided by infinity? How do we make sense of that?

  Mathematicians have developed an elegant scheme called taking limits, which in many cases can make sense of ∞/∞. For example, what fraction of all the counting numbers 1, 2, 3,…are even? There are infinitely many numbers and infinitely many of them are even, so the fraction is ∞/∞. But if we count only the first n numbers, we get a sensible answer that depends slightly on our counting cutoff n. If we keep increasing n, we find that the fraction jiggles around less and less as n grows. If we now take the limit where n approaches infinity, we get a well-defined answer that doesn’t depend on n at all: exactly half of the numbers are even.

  This seems like a sensible answer, but infinities are treacherous: the fraction of the numbers that’s even depends on the order in which we count them! If we instead order the numbers 1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16 and so on, then the same limit scheme gives the answer that 2/3 of the numbers are even! Because as we proceed down this list of numbers, we encounter two even numbers for every odd number. We didn’t cheat, since all even and odd numbers eventually show up in our list; we merely reordered them. In the same way, by reordering the numbers appropriately, I can prove to you that the even fraction is one divided by your phone number.…

  Analogously, the fraction of all the infinitely many observers in spacetime who make a particular observation depends on the order in which you count them! We cosmologists use the term measure to refer to an observer-moment ordering scheme, or, more generally, to a method for calculating probabilities from annoying infinities. The crazy probabilities I computed for the coolness problem corresponded to a particular measure, and most of my colleagues guess that the problem isn’t with inflation but with the measure: somehow, it appears flawed to talk about the reference class of all observer moments at a fixed time.

  The last few years have seen an avalanche of interesting papers proposing alternative measures. It’s proven remarkably difficult to find one that works with eternal inflation: some measures flunk the coolness problem; others fail by predicting that you’re a Boltzmann brain; yet others predict that we should see our sky warped by giant black holes. Alex Vilenkin recently told me that he was getting disheartened: a few years ago, he’d hoped that only one measure would avoid all these pitfalls, and that it would be so simple and elegant that it convinced us all. Instead, we now have a number of different measures that appear to give different but reasonable predictions, with no obvious way to choose between them. If the probabilities we predict depend on the measure we assume, and we can assume a measure giving almost any answer we want, the
n we really haven’t predicted anything at all.

  I share Alex’s concern. In fact, I view the measure problem as the greatest crisis in physics today. The way I see it, inflation has logically self-destructed. As we saw in Chapter 5, we started taking inflation seriously because it made correct predictions: it predicted that typical observers should measure space around them to be flat rather than curved (the flatness problem); they should measure their cosmic microwave–background temperature to be similar in all directions (the horizon problem); they should measure a power spectrum similar to what the WMAP satellite saw, etc. But then it predicted infinitely many observers measuring different things with probabilities depending on a measure that we don’t know. Which in turn means that inflation, strictly speaking, isn’t predicting anything at all about what typical observers should see. All predictions are revoked, including those predictions that made us take inflation seriously in the first place! Self-destruction complete. Our inflationary baby Universe has grown into an unpredictable teenager.

  In fairness to inflation, I don’t feel that there’s any competing cosmological theory on the market that does any better, so I don’t view this as an argument against inflation per se. I simply feel strongly that we need to solve the measure problem, and my guess is that once we solve it, some form of inflation will still remain. Moreover, the measure problem isn’t limited to inflation, but crops up in any theory with infinitely many observers. As an example, let’s revisit collapse-free quantum mechanics. The quantum-immortality argument from Chapter 8 hinges crucially on there being infinitely many observers, so that some always survive, which means that we can’t trust any of the conclusions until the measure problem has been solved.