Our Mathematical Universe

by Max Tegmark

  Things get even more intriguing when we start asking about the information content in small parts of these patterns. In the upper row of Figure 12.7, things are as we’d expect: smaller patterns are simpler and require less information to describe: we simply require one bit to describe each little black/white pixel. But in the bottom row, we see an example of the exact opposite! Here less is more, in the sense that the middle pattern is more complex than the left one, requiring more bits to describe. That’s because it’s no longer enough to simply say that it’s the binary digits of : we also need to specify at which of the digits our pattern starts—which in this case requires another 14 bits of information. In summary, we’ve seen that the whole can contain less information than the sum of its parts—and sometimes even less than one of its parts!

  Finally, the two rightmost patterns in Figure 12.7 each require 9 bits to describe. You and I know that the lower right pattern is hidden in the first 16,384 digits of , but for such a small pattern, this knowledge is no longer interesting or useful: there are only 29 = 512 possible patterns of length 9, so you’ll find our particular pattern hidden in most random-looking strings of thousands of zeros and ones.

  Figure 12.8 shows the beautiful mathematical structure known as the Mandelbrot fractal, which further illustrates these ideas. It has the remarkable property of having intricate patterns down to arbitrarily tiny scales, and although many of these patterns look similar, no two are identical. How complex are the two images shown? They each contain about a million pixels, which in turn are represented by three bytes of information (a byte is eight bits), suggesting that each image requires a few megabytes to describe. However, the left image can in fact be computed from a program only a few hundred bytes long, implementing repeated use of the simple computation z2 + c described in the figure caption.

  Figure 12.8: Despite its complex appearance with millions of elaborately colored pixels, the Mandelbrot fractal (left panel) has a very simple description: the points in the image correspond to what mathematicians call complex numbers c, and the colors encode how rapidly the complex number z blows up toward infinite size when you start with z = 0 and repeatedly keep squaring and adding c, i.e., repeatedly applying the simple equation z→z2 + c. Paradoxically, the right image requires more information to describe even though it’s simply a small part of the left one: if you cut the Mandelbrot fractal into about a hundred trillion trillion pieces, this is one of them, and the information contained in the right image basically tells you its address within the larger image, because the most economical way to specify it is as something like “Piece 31415926535897932384 of the Mandelbrot fractal.”

  The right image is also simple, being merely a tiny part of the left one. But it’s slightly more complex, requiring another 8 bytes to specify, with a 20-digit number, which of 1020 different parts it is. So once again, we see that less is more, in the sense that the apparent information content rises when we restrict our attention to a small part of the whole, thus losing the symmetry and simplicity that was inherent in the totality of all parts taken together. For an even simpler example of this, consider that the algorithmic information content of a typical trillion-digit number is substantial, since the shortest program that prints it can’t do much better than simply store all its trillion digits. Nonetheless, the list of all numbers 1, 2, 3,…can be generated by quite a trivial computer program, so the complexity of the whole set is smaller than that of a typical member.

  Let’s now return to our physical Universe and the near googol bits that appear required to specify it. Some scientists, such as Stephen Wolfram and Jürgen Schmidhuber, have wondered whether much of this complexity might also be a mere illusion, just as for the Mandelbrot fractal and the lower left pattern of Figure 12.7, resulting from a yet-to-be-discovered mathematical rule that’s very simple. Although I find this to be an elegant idea, I’d bet against it: I view it as unlikely that all the numbers that characterize our Universe, from the patterns in the WMAP cosmic microwave–background maps to the positions of grains of sand on a beach, can be reduced to almost nothing by a simple data-compression algorithm. Indeed, as we saw in Chapter 5, cosmological inflation explicitly predicts that the cosmic seed fluctuations, from which much of this information ultimately originates, are distributed like random numbers for which such dramatic data compression is impossible.

  These seed fluctuations specify all ways in which our early Universe differed from an easy-to-describe perfectly uniform plasma. Why do the patterns of cosmic seed fluctuations appear so random? We saw in Chapter 5 that, according to the cosmological standard model, inflation generates all possible patterns in different parts of space (in different universes throughout the Level I multiverse), and since we find ourselves in a rather typical part of this multiverse, we’ll see a random-looking pattern without any hidden regularities to help us compress the information. The situation is a lot like the bottom row of Figure 12.7, where our Universe (corresponding to the right panel) corresponds to just a small random-looking part of the Level I multiverse (corresponding to the left panel) with its simple description. In fact, if you flip back to Chapter 6, you’ll see that Figure 6.2 becomes equivalent to the bottom row of Figure 12.7 if we simply extend it to include more than a googolplex binary digits of , and expand the right panel to contain about a googol bits as our Universe does. It’s widely believed among mathematicians (albeit still unproven) that the digits of behave like random numbers, so that any possible pattern appears somewhere, just as universes with any possible initial conditions appear somewhere in the Level I multiverse. This means that a sequence of a googol digits from the digits of actually tells us nothing at all about , merely about where in the digit sequence we’re looking. Similarly, observing a googol bits of information about typical random-looking inflation-generated cosmic seed fluctuations only gives us information about where in the vast post-inflationary space we’re looking.

  Initial Conditions Reinterpreted

  Above we worried about how to think about our initial conditions, and we now have a radical answer: this information isn’t fundamentally about our physical reality, but about our place in it. The vast complexity we observe is an illusion in the sense that the underlying reality is quite simple to describe, and what requires close to a googol bits to specify is just our particular address in the multiverse. We discussed in Chapter 6 how our Galaxy contains many solar systems with different numbers of planets, so that when we say ours has eight, we’re saying nothing fundamental about our Galaxy but something about our Galactic address. Because the Level I multiverse contains other Earths whose skies show all possible variations of cosmic microwave–background patterns or stellar constellations, the information contained in the WMAP map or a photo of the Big Dipper similarly tells us about our multiversal address. Analogously, the 32 physical constants from Chapter 10 tell us about our place in the Level II multiverse, if it exists. Although we thought all this information was about our physical reality, it was about us. The complexity is an illusion, existing only in the eye of the beholder.

  I first thought of these ideas while biking to work through Munich’s Englischer Garten back in 1995, and published them in a paper with the provocative title “Does our Universe in fact contain almost no information?” Now I realize that I should have dropped the word almost! Let me explain why. Our Level III multiverse reminds me more of the Mandelbrot fractal (Figure 12.8) than of our example (Figure 12.7), because its pieces exhibit lots of regularity. Whereas all possible patterns occur equally often in the digits of , many patterns (pictures of your friends, say) don’t occur anywhere in the Mandelbrot fractal. Just as most pieces of the Mandelbrot fractal seem to share a certain artistic style, dictated by that formula z2 + c, most of the post-inflationary universes in the Level III multiverse share regularities in their time development that follow from quantum mechanics. When I referred to “almost no information,” I meant the small amount of information needed to describe these regularities, specify
ing the mathematical structure that is the Level III multiverse. But in the light of the Mathematical Universe Hypothesis, not even this information tells us anything about the ultimate physical reality—rather, it simply tells us our address in the Level IV multiverse.

  Randomness Reinterpreted

  Okay, now that we’ve figured out how to interpret initial conditions, what about randomness? Here too the answer lies in the multiverse. We saw in Chapter 8 how the completely deterministic Schrödinger equation of quantum mechanics can give rise to apparent randomness from the subjective perspective of an observer in the Level III multiverse, and how the core process was a more general one having nothing to do with quantum mechanics: cloning. Specifically, randomness is simply how it feels when you’re cloned: you can’t predict what you’ll perceive next if there’ll be two copies of you perceiving different things. In Chapter 8 we saw that apparent randomness is caused by observer cloning in some cases. Now we see that it’s in fact caused by cloning in all cases, since the MUH banishes fundamental randomness, the other logically possible explanation.

  In other words, whereas apparently arbitrary initial conditions are caused by multiple universes, apparent randomness is caused by multiple yous. These two ideas merge into one and the same if we consider those parallel universes that contain a subjectively indistinguishable copy of you, so that there’s both multiple universes and multiple yous. Then when you measure the initial conditions of your Universe, this information will appear random to all your copies, and it doesn’t matter whether you interpret this information as coming from initial conditions or randomness—the information is the same. Observing which universe you’re in reveals which copy of you is doing the observing.

  How Complexity Suggests a Multiverse

  We’ve talked a lot about the complexity of our Universe, but what about the complexity of our mathematical structure?

  The MUH doesn’t specify whether the complexity of the mathematical structure in the bird perspective is low or high, so let’s consider both possibilities. If it’s extremely high, then our quest to figure out its specification is clearly doomed. In particular, if describing the structure requires more bits than describing our observable Universe, then we can’t even store the information about the structure in our Universe—it won’t fit. An example of such a high-complexity theory would be the standard model with its 32 parameters from Chapter 10 explicitly specified as real numbers, such as 1/α = 1/137.035999…, with infinitely many decimals lacking any simplifying pattern. Because even if one such parameter would require an infinite amount of information to store, the mathematical structure would be infinitely complex and impossible to specify in practice.

  Most physicists hope for a complete Theory of Everything that’s much simpler than this and can be specified by few enough bits to fit in a book, if not on a T-shirt—vastly fewer than the near googol bits needed to describe our Universe. Such a simple theory must predict a multiverse, regardless of whether the MUH is true or not. Why? Because this theory is by definition a complete description of reality: if it lacks enough bits to completely specify our Universe, then it must instead describe all possible combinations of stars, sand grains and such—so that the extra bits that describe our Universe simply encode which universe we’re in, like a multiversal postal code would. The address written on the envelope in Figure 12.5 would then have a relatively short bottom line, specifying the theory, but the address lines right above would contain nearly a googol characters.

  Are We Living in a Simulation?

  We’ve just seen how the Mathematical Universe Hypothesis changes our perspective on many fundamental questions. Let’s now turn to another such topic: that of simulated realities. Long a staple of science fiction, the idea that our external reality is some form of computer simulation has gained prominence with blockbuster movies such as The Matrix. Scientists such as Eric Drexler, Ray Kurzweil and Hans Moravec have argued that simulated minds are both possible and imminent, and some (for example Frank Tipler, Nick Bostrom and Jürgen Schmidhuber) have gone as far as discussing the probability that this has already happened—that we’re simulated.

  Why should you think you’re simulated? Well, many science-fiction authors have explored scenarios where future space colonization transforms much of the matter in our Universe into ultra-advanced computers that simulate huge numbers of observer moments subjectively indistinguishable from yours. Nick Bostrom and others have argued that in this case, it’s most likely that your current observer moment is in fact one of the simulated ones, since they’re more numerous. However, I think that this argument logically self-destructs: if the argument is valid, then your indistinguishable simulated copies can make it, too, implying that there are way more doubly simulated copies, and that you’re probably a simulation within a simulation. Making this argument repeatedly, you conclude that you’re probably a simulation within a simulation within a simulation, and so on, arbitrarily many levels down—a reductio ad absurdum. I think the logical mistake happens at the very first step: if you’re willing to assume that you’re simulated, then as emphasized by Phillip Helbig, the computational resources of your own (simulated) universe are irrelevant: what matters are the computational resources in the universe where the simulation is taking place, about which you know essentially nothing.

  Others have argued that it’s fundamentally impossible for our reality to be a simulation. Seth Lloyd has advanced the intermediate possibility that we live in an analog simulation performed by a quantum computer, albeit not a computer designed by anybody—rather, because the structure of quantum field theory is mathematically equivalent to that of a spatially distributed quantum computer. In a similar spirit, Konrad Zuse, John Barrow, Jürgen Schmidhuber, Stephen Wolfram and others have explored the idea that the laws of physics correspond to a classical computation. Let’s explore these ideas in the context of the Mathematical Universe Hypothesis.

  The Time Misconception

  Suppose that our Universe is indeed some form of computation. A common misconception in the universe-simulation literature is that our physical notion of a one-dimensional time must then necessarily be equated with the step-by-step one-dimensional flow of the computation. I’ll argue below that if the MUH is correct, then computations don’t need to evolve our Universe, but merely describe it (defining all its relations).

  The temptation to equate time steps with computational steps is understandable, given that both form a one-dimensional sequence where (at least for the non-quantum case) the next step is determined by the current state. However, this temptation stems from an outdated classical description of physics: there’s generically no natural and well-defined global time variable in Einstein’s general relativity, and even less so in quantum gravity where time is known to emerge only as an approximate property of certain “clock” subsystems. Indeed, linking frog-perspective time with computer time is unwarranted even within the context of classical physics. The rate of time flow perceived by an observer in the simulated universe is completely independent of the rate at which a computer runs the simulation, a point emphasized in Greg Egan’s science-fiction novel Permutation City. Moreover, as we discussed in the last chapter and as stressed by Einstein, it’s arguably more natural to view our Universe not from the frog perspective as a three-dimensional space where things happen, but from the bird perspective as a four-dimensional spacetime that merely is. There should therefore be no need for the computer to compute anything at all—it could simply store all the four-dimensional data, that is, encode all properties of the mathematical structure that is our Universe. Individual time slices could then be read out sequentially if desired, and the “simulated” world should still feel as real to its inhabitants as in the case where only three-dimensional data is stored and evolved. In conclusion: the role of the simulating computer isn’t to compute the history of our Universe, but to specify it.

  How specify it? The way in which the data are stored (the type of computer, the data for
mat, etc.) should be irrelevant, so the extent to which the inhabitants of the simulated universe perceive themselves as real should be independent of whatever method is used for data compression. The physical laws that we’ve discovered provide great means of data compression, since they make it sufficient to store the initial data at some time together with the equations and a program computing the future from these initial data. As emphasized on this page, the initial data might be extremely simple: popular initial states from quantum field theory with intimidating names such as the Hawking-Hartle wavefunction or the inflationary Bunch-Davies vacuum have very low algorithmic complexity, since they can be defined in brief physics papers, yet simulating their time evolution would simulate not merely one universe like ours, but a vast decohering collection of parallel ones. It’s therefore plausible that our Universe (and even the whole Level III multiverse) could be simulated by quite a short computer program.

  A Different Sort of Computation

  The previous example referred to our particular mathematical structure, with its quantum mechanics and so on. More generally, as we’ve discussed, a complete description of an arbitrary mathematical structure is by definition a specification of the relations between its elements. We saw earlier in this chapter that for these relations to be well defined, all these functions must be computable: there must exist a computer program that can compute the relations in a finite number of computational steps. Each relation of the mathematical structure is thus defined by a computation. In other words, if our world is a well-defined mathematical structure in this sense, then it’s indeed inexorably linked to computations, albeit computations of a different sort than those usually associated with the simulation hypothesis: these computations don’t evolve our Universe, but describe it by evaluating its relations.1