Our Mathematical Universe

by Max Tegmark

  * * *

  1Indeed, as pointed out by Ken Wharton in his paper “The Universe Is Not a Computer,” at http://arxiv.org/pdf/1211.7081.pdf, our laws of physics may be such that the past doesn’t uniquely determine the future, so the idea that our Universe can be simulated even in principle is a hypothesis that shouldn’t be taken for granted.

  Does a Simulation Really Need to Be Run?

  A deeper understanding of the relations between mathematical structures, formal systems and computations (the triangle in Figure 12.6) would shed light on many of the thorny issues we’ve encountered in this book. One such issue is the measure problem that plagued us in the last chapter, which is in essence the problem of how to deal with annoying infinities and predict probabilities for what we should observe. For example, since every universe simulation corresponds to a mathematical structure, and therefore already exists in the Level IV multiverse, does it in some meaningful sense exist “more” if it is also run on a computer? This question is further complicated by the fact that eternal inflation predicts an infinite space with infinitely many planets, civilizations and computers, some of which may be running universe simulations, and that the Level IV multiverse also includes an infinite number of mathematical structures that can be interpreted as computer simulations.

  The fact that our Universe (together with the entire Level III multiverse) may be simulatable by a quite short computer program calls into question whether it makes any ontological difference whether simulations are “run” or not. If, as I have argued, the computer need only describe and not compute the history, then the complete description would probably fit on a single memory stick, and no CPU power would be required. It would appear absurd that the existence of this memory stick would have any impact whatsoever on whether the multiverse it describes exists “for real.” Even if the existence of the memory stick mattered, some elements of this multiverse will contain an identical memory stick that would “recursively” support its own physical existence. This wouldn’t involve any Catch-22, chicken-or-the-egg problem regarding whether the stick or the multiverse was created first, since the multiverse elements are four-dimensional spacetimes, whereas “creation” is of course only a meaningful notion within a spacetime.

  So are we simulated? According to the MUH, our physical reality is a mathematical structure, and as such, it exists regardless of whether someone here or elsewhere in the Level IV multiverse writes a computer program to simulate/describe it. The only remaining question is then whether a computer simulation could make our mathematical structure in any meaningful sense exist even more than it already did. If we solve the measure problem, perhaps we’ll realize that simulating it would increase its measure slightly, by some fraction of the measure of the mathematical structure within which it’s simulated. My guess is that this would be a tiny effect at best, so if asked, “Are we simulated?,” I’d bet my money on “No!”

  Relation Between the MUH, the Level IV Multiverse and Other Hypotheses

  An interesting variety of ultimate-reality proposals have been put forth by various researchers at the interfaces between philosophy, information theory, computer science and physics, and for excellent recent overviews, I recommend Brian Greene’s book The Hidden Reality and Russell Standish’s book Theory of Nothing.

  On the philosophy side, the proposal closest to the Level IV multiverse is the theory of modal realism by the late philosopher David Lewis, which posits that “all possible worlds are as real as the actual world.” His late philosophy colleague Robert Nozick made a similar proposal termed the principle of fecundity. One common criticism of modal realism asserts that because it posits that all imaginable universes exist, it makes no testable predictions at all. The Level IV multiverse can be thought of as a smaller and more rigorously defined reality by virtue of replacing Lewis’s “all possible worlds” by “all mathematical structures.” The Level IV multiverse does not imply that all imaginable universes exist. We humans can imagine many things that are mathematically undefined and hence don’t correspond to mathematical structures. Mathematicians publish papers with existence proofs that demonstrate the mathematical consistency of various mathematical-structure descriptions precisely because to do this is difficult and not possible in all cases.

  On the computer-science side, the most closely related proposals are that our physical reality is some form of computer simulation or simulations, as we discussed earlier in this chapter. The relation is most clearly seen in Figure 12.6 where the two ideas correspond to two different vertices of the triangle: our reality is a computation according to the simulation hypothesis, as opposed to a mathematical structure according to the MUH. Under the simulation hypothesis, computations evolve our Universe, but under the MUH they merely describe it by evaluating its relations. According to the computational-multiverse theories explored by Jürgen Schmidhuber, Stephen Wolfram and others, the time evolution needs to be computable, while according to the Computable Universe Hypothesis (CUH), it’s the description (the relations) that must be computable. John Barrow and Roger Penrose have suggested that only structures complex enough for Gödel’s incompleteness theorem to apply can contain self-aware observers. Earlier, we saw that the CUH in a sense postulates the exact opposite.

  Testing the Level IV Multiverse

  We have argued that the External Reality Hypothesis (ERH), which says that there is an external physical reality completely independent of us humans, implies the Mathematical Universe Hypothesis (MUH), which says that our external physical reality is a mathematical structure, which in turn implies the existence of the Level IV multiverse. Therefore, the most straightforward way to strengthen or weaken our evidence for the Level IV multiverse is to further study and test the ERH. While the jury is still out on the ERH, I think it’s fair to say that most of my physics colleagues subscribe to it, and that the recent successes of the standard models of particle physics and cosmology do little to suggest that our ultimate physical reality, whatever it is, fundamentally revolves around us humans and can’t exist without us. That said, let’s nonetheless explore two ways of potentially testing the MUH and the Level IV multiverse more directly.

  The Typicality Prediction

  As we saw in Chapter 6, the discovery that a physical parameter seems fine-tuned to allow life can be interpreted as evidence of a multiverse where the parameter takes a broad range of values, because this interpretation makes it unsurprising that a habitable universe like ours exists, and predicts that this is where we’ll find ourselves. In particular, we saw that some of the strongest evidence for a Level II multiverse comes from the observed fine-tuning of the dark-energy density. Could there be fine-tuning evidence even for Level IV, at least in principle?

  At a 2005 physics conference in Cambridge, while my friend Anthony Aguirre and I were taking a late-evening walk through the quaint courtyards of Trinity College, I suddenly realized that the answer was yes. Here’s why.

  Suppose you’re getting out of your friend’s car after she’s driven you to a town you know nothing about, and you notice a confusing zoo of signs (see Figure 12.9) banning parking everywhere on the street except for the one place where she parked. She explains that, as part of an antipollution campaign, the new mayor has ordered ten signs to be randomly placed on each street, each one banning parking on the whole street either to the left or to the right side of the sign. After doing some math, you realize that this crazy random process will typically ban parking everywhere on a street, with only about a 1% chance that there’s an allowed space;1 this happens only if all signs with left arrows get placed to the left of all signs with right arrows.

  Figure 12.9: If a street has lots of randomly placed signs, each banning parking on the whole street on either the left or right side of the sign, it’s quite unlikely that parking will be allowed anywhere on it: this happens only if all left arrows end up to the left of all right arrows, as in the top panel example. Similarly, if a universe has a physical parameter that must satisfy
lots of constraints for life to be allowed (bottom panel), it’s a priori unlikely that there’s any habitable range of parameter values. Situations such as those illustrated here can therefore be interpreted as evidence for the existence of many streets or many mathematical structures in a Level IV multiverse, respectively.

  Click here to see a larger image.

  What are you to make of this? Is it just a lucky coincidence? If you abhor unexplained coincidences as a typical scientist does, then you’ll lean toward the one interpretation that doesn’t require a wild stroke of luck: that there are many streets in this strange town, probably in the ballpark of a hundred or more. This makes it likely that there’s legal parking on some street, and since your friend knows the town, it’s totally unsurprising that this is where your friend has chosen to park. This fine-tuning example differs from those of Chapter 6 because what appears to be fine-tuned isn’t something continuous, such as the dark-energy density, but rather something discrete: all the directions of the left- or right-pointing arrows conspire in a surprising way.

  My parking example was admittedly silly, but as the lower panel of Figure 12.9 illustrates, we observe something rather similar in our Universe. The horizontal axis shows a parameter related to the recently discovered Higgs particle, and recent work by John Donoghue, Craig Hogan, Heinz Oberhummer and their collaborators has shown that, much like the dark-energy density, it appears highly fine-tuned: it’s about sixteen orders of magnitude smaller than one might naturally expect, yet changing it by even a percent up or down dramatically reduces the amount of either carbon or oxygen produced by stars. Increasing it by 18% radically reduces fusion of hydrogen into any other atoms by stars, while reducing it by 34% makes hydrogen atoms decay into neutrons as their proton gobbles up their electron. Reducing it fivefold makes even lone protons decay to neutrons, guaranteeing a universe with no atoms at all.

  How should we interpret this? Well, first of all, this looks like further evidence for a Level II multiverse across which some physical parameters vary. Just as this can explain why we find a dark-energy density that’s just right to allow galaxies to form, this can clearly also explain why we find Higgs properties that are just right to allow more complex atoms than hydrogen to form—and it’s not surprising that we’re in one of the relatively few universes with both interesting atoms and interesting galaxies if life requires at least a minimal level of complexity.

  But Figure 12.9 raises a second question as well: why do the five arrows in the bottom panel conspire to allow any habitable range of Higgs properties? This could well be a fluke: five random arrows would allow some range with 19% probability, so we need only invoke a small amount of luck. Moreover, because of how nuclear physics works, these five arrows aren’t independent, so I don’t view this particular five-arrow example as strong evidence of anything. However, it’s perfectly plausible that further physics research could uncover more striking fine-tuning of this discrete type with, say, ten or more arrows conspiring to allow a habitable range for some physical parameter or parameters.2 And if this happens, then we can argue just as for the top panel: that this is evidence for the existence not of other streets, but of other universes where the laws of physics are different, giving quite different requirements for life! In some cases, these universes might exist in the Level II multiverse, in a region where the same fundamental laws of physics give rise to a different phase of space with other effective laws. In other instances, however, this might be proven to be impossible, in which case these other universes would have to obey different fundamental laws, corresponding to different mathematical structures in the Level IV multiverse. In other words, while we currently lack direct observational support for the Level IV multiverse, it’s possible that we may get some in the future.

  * * *

  1If there are n random signs, the probability that any parking is allowed is (n + 1)/2n: once the signs have been placed, there are 2n ways of orienting the left/right arrows, and only n + 1 of these ways corresponds to all the left arrows being to the left of all the right arrows.

  2It’s easy to generalize this discrete fine-tuning definition to the case where more than one parameter can vary.

  The Mathematical-Regularity Prediction

  We’ve mentioned Wigner’s famous 1960 essay where he argued that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious,” and that “there is no rational explanation for it.” The Mathematical Universe Hypothesis provides this missing explanation. It explains the utility of mathematics for describing the physical world as a natural consequence of the fact that the latter is a mathematical structure, and we’re simply uncovering this bit by bit. The various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of certain aspects of more-complex mathematical structures. In other words, our successful theories aren’t mathematics approximating physics, but mathematics approximating mathematics.

  One of the key testable predictions of the Mathematical Universe Hypothesis is that physics research will uncover further mathematical regularities in nature. This predictive power of the mathematical-universe idea was expressed by Paul Dirac in 1931: “The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities.”

  How successful has this prediction been so far? Two millennia after the Pythagoreans promulgated the basic idea of a mathematical universe, further discoveries made Galileo describe nature as being “a book written in the language of mathematics.” Then much more far-reaching mathematical regularities were uncovered, ranging from the motions of planets to the properties of atoms, prompting those awestruck endorsements by Dirac and Wigner. After this, the standard models of particle physics and cosmology revealed new “unreasonable” mathematical order to a spectacular extent, from the microcosm of elementary particles to the macrocosm of the early Universe, arguably enabling all physics measurements ever made to be successfully calculated from the 32 numbers listed in Table 10.1. I know of no other compelling explanation for this trend than that the physical world really is completely mathematical.

  Looking toward the future, there are two possibilities. If I’m wrong and the MUH is false, then physics will eventually hit an insurmountable roadblock beyond which no further progress is possible: there would be no further mathematical regularities left to discover even though we still lacked a complete description of our physical reality. For example, a convincing demonstration that there’s such a thing as fundamental randomness in the laws of nature (as opposed to deterministic observer cloning that merely feels random subjectively) would therefore refute the MUH. If I’m right, on the other hand, then there’ll be no roadblock in our quest to understand reality, and we’re limited only by our imagination!

  THE BOTTOM LINE

  • The Mathematical Universe Hypothesis implies that mathematical existence equals physical existence.

  • This means that all structures that exist mathematically exist physically as well, forming the Level IV multiverse.

  • The parallel universes we’ve explored form a nested four-level hierarchy of increasing diversity: Level I (unobservably distant regions of space), Level II (other post-inflationary regions), Level III (elsewhere in quantum Hilbert space) and Level IV (other mathematical structures).

  • Intelligent life appears to be rare, with most of Levels I, II and IV being uninhabitable.

  • Exploring the Level IV multiverse doesn’t require rockets or telescopes, merely computers and ideas.

  • The simplest mathematical structures can be listed by a computer in telephone-book fashion, with each one having its own unique number.

  • Mathematical
structures, formal systems and computations are closely related, suggesting that they’re all aspects of the same transcendent structure whose nature we still haven’t fully understood.

  • The Computable Universe Hypothesis (CUH) that the mathematical structure that is our external physical reality is defined by computable functions may be needed for the MUH to make sense, as Gödel incompleteness and Church-Turing uncomputability will otherwise correspond to unsatisfactorily defined relations in the mathematical structure.

  • The Finite Universe Hypothesis (FUH) that our external physical reality is a finite mathematical structure implies the CUH and eliminates all concerns about reality being undefined.

  • The CUH/FUH may help solve the measure problem and explain why our Universe is so simple.

  • The MUH implies that there are no undefined initial conditions: initial conditions tell us nothing about physical reality, merely about our address in the multiverse.

  • The MUH implies that there’s no fundamental randomness: randomness is simply the way cloning feels subjectively.

  • The MUH implies that most of the complexity we observe is an illusion, existing only in the eye of the beholder, being merely information about our address in the multiverse.

  • A collection of things can be simpler to describe than one of its parts.

  • Our multiverse is simpler than our Universe, in the sense that it can be described with less information, and the Level IV multiverse is simplest of all, requiring essentially no information to describe.