Our Mathematical Universe

by Max Tegmark

  Figure 11.6 illustrates how we can reformulate this as a sensible question even without the notions of change or the flow of time. The eight observer moments illustrated belong to two different people, one diving and one skiing, each corresponding to a long braidlike pattern in spacetime. Comparing the eight observer moments reveals some interesting relations between them, where the current visual impressions (right frame on each film strip) of some observer moments match fresh memories (middle frame) in others, and fresh memories in some match older memories (left frame) in others. This uniquely defines two separate time sequences of observer moments, corresponding to the left and right columns of strips, with later times corresponding to higher up in the figure.

  Consider all observer moments in all of spacetime. The ones that are natural to call your future perceptions are the ones that can be similarly matched with your current observer moment, fitting together like pieces of a puzzle. Specifically, they should share your current memories in the correct order (with some reasonable allowance for forgetting and misremembering), and with additional memories added to the sequence. Suppose, for example, that you’re the diver who’s just seen the giant turtle swim up toward the right (Figure 11.6, left column, second observer moment from the top) and want to predict your future. As a thought experiment, suppose also that you’re infinitely intelligent and have figured out which mathematical structure our Universe is, and have calculated what all of its observer moments are and how they subjectively feel. You realize that the only one that matches with your current observer moment and has an extra second’s worth of perceptions at the end is the top-left observer moment in the figure. You therefore make the prediction that this is what you’ll perceive in one second: in one second, you’re going to see the giant turtle turn and start swimming toward you. In this way, you recover the traditional scientific notion of causality: that you can predict the future from the present.

  Where Are You? (And What Do You Perceive?)

  We’ve now seen how our physical reality can be a mathematical structure, including space, time, stuff and even you yourself. We’ve also seen how you, at least in principle, might be able to make predictions about your future by analyzing observer moments, matching them up like pieces in a puzzle. To predict things in practice, this observer-moment approach often reduces to “business as usual” physics. Suppose, for example, that you do the experiment illustrated in Figure 10.2, throwing a basketball up into the air and studying its motion. If you assume (1) that Einstein’s equations of gravity describe this motion, and (2) that there’s no other person who subjectively feels exactly like you, with the exact same life memories, then you know that the only future observer moments that smoothly match with your present one are ones where you see the ball fly in a parabolic trajectory as in the figure, so that’s what you predict that you’ll perceive. How did you know that it was going to be a parabola and not another shape, say, a spiral? By solving Einstein’s equations and getting a parabola as the solution.

  Predicting Your Future, Revisited

  We’ve seen, however, that the second assumption is likely to be false: if either the Level I or the Level III multiverse exists, then there are other people who subjectively feel exactly like you, and the problem of predicting your future gets much more interesting! I was sneaky when I chose the heading “Where Are You (And What Do You Perceive)?” because I want to ask the question also when the word you is interpreted in the plural sense. As we’ll see, it gets particularly tricky when the number of yous increases or decreases.

  Let’s continue our thought experiment where you know every detail about the mathematical structure that we inhabit. Then predicting your future boils down to three steps:

  1. Find all self-aware entities in it.

  2. Figure out what they subjectively perceive so that you know which ones might be you, and what they perceive in the future.

  3. Predict what you’ll subjectively perceive in the future (probabilities for different options).

  Amusingly, as we’ll see below, all three of these steps involve daunting unsolved problems!

  Finding Self-Awareness

  Let’s start with the first step. Given some mathematical structure that is our external physical reality, perhaps including a multiverse, how do we find all self-aware entities in it? We discussed how we humans correspond to certain complex braidlike patterns in spacetime. However, we don’t want to limit our exploration of self-awareness to our own human form of life, so let’s use the more general term self-aware substructure (or “SAS” for short) to refer to any part of a mathematical structure that has subjective perceptions. We’ll also use observer as a synonym from time to time, but will stick with SAS whenever we need to remind ourselves to avoid anthropocentrism.

  So how do we find SASs in a mathematical structure? The short answer is clearly that we don’t yet know—science simply hasn’t advanced to that point. We can’t even answer the question in the particular case we’re most familiar with: our own spacetime. First of all, we don’t know what mathematical structure we inhabit, since a self-consistent model of quantum gravity remains conspicuous with its absence. Second, even if we knew our mathematical structure, we wouldn’t know what to do with it to find its SASs.

  Imagine that a friendly visiting alien gives you an “SAS-buster,” a convenient handheld device that looks a bit like a metal detector but which makes a loud beeping noise whenever it detects an SAS. You play around with it and find that it beeps quietly when you point it at a goldfish, more loudly when you point it at a cat and with ear-piercing volume when you point it at yourself, but that it remains dead silent when you point it at a cucumber, a car or a corpse. How might this SAS-buster work?

  Although the minimalistic user manual that came with the SAS-buster merely refers to “a proprietary algorithm,” my guess is that part of what it does is measure both the complexity and the information content of the object you point it toward. The complexity of something is usually defined as the smallest number of bits required to fully describe it (a bit is a zero or a one). For example, a diamond describable as 1024 carbon atoms arranged in a perfectly regular lattice pattern has very low complexity compared to a hard drive with a terabyte of random numbers, since the latter can’t be described with less than a terabyte (about 8 × 1012 bits) of information. Yet that hard drive is much less complex than your brain, where more than a hundred quadrillion (1017) bits of information are needed just to describe the state of its synapses alone.

  However, a hard drive wouldn’t be self-aware no matter how big it was, so complexity alone clearly isn’t enough to make an SAS. I suspect that another quantity that the SAS-buster measures is the information content of the object you point it at. There are rigorous mathematical definitions of information content in mathematics and physics, tracing back to the work of Claude Shannon and John von Neumann over half a century ago. Whereas the complexity of an object measures how complicated it is to describe, its information content1 measures the extent to which it describes the rest of the world. In other words, information is a measure of how much meaning complexity has. If you fill your hard drive with random numbers, then it contains no information about the outside world, but if you fill it with history books or with movie clips of your family, then it does. Your brain contains a vast amount of information about the outside world, both in the form of memories of distant times and places and in the form of its continually updated model of what’s happening around you right now. When a person dies, the information content of the electrical firing patterns of their neurons vanishes as this entire electrical system shuts down, and before long, the information content stored chemically and biologically in their synapses begins to disappear as well.

  Yet complexity and information content still aren’t sufficient to guarantee self-awareness—for example, a video camera has both without being self-aware in any meaningful sense. This means that the SAS-buster needs to look for additional ingredients of self-awareness that ar
e harder to understand. For example, Figure 11.7 suggests that an SAS needs to be able not only to store information, but also to process it in some form of computation, and that a high degree of interconnectedness may be required in the information processing. The neuroscientist Giulio Tononi has made an intriguing proposal for how to quantify the required interconnectedness, described in the publications by Koch and Tononi in the “Suggestions for Further Reading” section. The core idea is that for an information processing system to be conscious, it needs to be integrated into a unified whole that can’t be decomposed into nearly independent parts.2 This means that all parts need to compute jointly with lots of information about each other—otherwise there would be more than one independent consciousness, such as in a room full of people or, perhaps, in the two brain halves of a patient whose connecting corpus callosum has been cut. If there are fairly independent parts that are too simple, then these won’t be conscious at all, like the independent pixels of a video camera.

  Generations of physicists and chemists have studied what happens when you group together vast numbers of atoms, finding that their collective behavior depends on the pattern in which they’re arranged: the key difference between a solid, a liquid and a gas lies not in the types of atoms, but in their arrangement. My guess is that we’ll one day understand consciousness as yet another phase of matter. I’d expect there to be many types of consciousness just as there are many types of liquids, but in both cases, they share certain characteristic traits that we can aim to understand.

  As a baby step toward consciousness, let’s first consider memory—what traits does it have? For a substance to be useful for storing information, it clearly needs to have a large repertoire of possible long-lived states. Solids do, whereas liquids and gases don’t: if you engrave someone’s name on a gold ring, the information will still be there years later, but if you engrave it on the surface of a pond, it will be lost within a second as the water surface changes its shape. Another desirable trait of a memory substance is that it’s not only easy to read from (as a gold ring), but also easy to write to: altering the state of your hard drive or your synapses requires less energy than engraving gold.

  What traits should we ascribe to “computronium,” the most general substance that can process information as a computer? Rather than just remain immobile as a gold ring, it must exhibit complex dynamics so that it’s future state depends in some complicated (and hopefully controllable/programmable) way on the present state. Its atom arrangement must be less ordered than a rigid solid where nothing interesting changes, but more ordered than a liquid or a gas. At the microscopic level, computronium doesn’t need to be very complicated, because computer scientists have shown that as long as a device can perform certain basic logic operations, it’s universal: it can be programmed to perform the same computation as any other computer with enough time and memory.

  What about “perceptronium,” the most general substance that feels subjectively self-aware? If Tononi is right, then it should not merely have the traits of computronium, but also the property that its information is indivisible, forming a unified whole. So when our SAS-buster analyzes a room full of atoms, it will first discover which ones are strongly connected to others and classify the connected atom groups as objects, say, a bench with two people on it. It will then identify parts of these objects that meet the criteria for computronium: say two brains and two cell phone CPUs. Finally, it will determine that there’s only perceptronium in the two brains, and that these are two separate pieces that are rather disconnected from one another, one corresponding to the consciousness of each person.

  * * *

  1What I’m casually calling the information content of an object is in technical terms called the mutual information between the object and the rest of the world.

  2This is closely linked to so-called redundancy and error-correcting codes used in bar codes, hard drives, mobile telephony and other modern information technology: you use more bits than the minimum needed, which encode your information in a clever collective way such that none of your information is lost even if you lose any modest fraction of your bits. Our brain appears to use a similarly redundant architecture, since it doesn’t seem to depend critically on any single neuron, and keeps functioning well even if a modest number of neurons die. Perhaps part of the reason that consciousness evolved is that such redundancy is evolutionarily useful.

  Computing the Internal Reality: What Has History Taught Us?

  Once you’ve found a self-aware entity with your SAS-buster, the next step is to calculate what it subjectively perceives. In the language of Chapter 9, we wish to compute its internal reality from the external reality. This is a tough challenge with which we have limited experience, since physics has historically tended to focus on the opposite problem: given our subjective perceptions, we’ve looked for mathematical equations that could describe them. For example, Newton observed the motion of the Moon and came up with a law of gravity that explained it. Nonetheless, I feel that the history of physics has taught us many valuable lessons about how the internal and external realities are related: below are seven examples.

  Don’t panic

  Although the problem is unsolved and very hard, we saw in Chapter 9 that we can conveniently split it into two parts: we physicists can limit ourselves to starting with the external reality and predicting the consensus reality that all reasonable observers agree on, leaving the quest for the internal reality to neuroscientists and psychologists. For most of the tricky predict-the-future questions that we’ll encounter below, we’ll see that the distinction between the consensus reality and the internal reality doesn’t matter. Moreover, the history of physics has provided useful case studies such as classical mechanics, general relativity and quantum mechanics, where we know both the key equations and how it feels to be governed by them.

  We perceive that which is stable

  We humans replace the bulk of both our “hardware” (e.g., our cells) and our “software” (e.g., our memories) many times in our life span. Nonetheless, we perceive ourselves as stable and permanent. Likewise, we perceive objects other than ourselves as permanent. Or rather, what we perceive as objects are those aspects of the world that display a certain permanence. For instance, when observing the ocean, we perceive the moving waves as objects because they display a certain permanence, even though the water itself is only bobbing up and down. Similarly, as we saw in Chapter 8, we perceive only those aspects of the world that are fairly stable against quantum decoherence.

  We perceive ourselves as local

  Both relativity and quantum mechanics illustrate that you perceive yourself as being “local” even if you aren’t. Although in the external reality of general relativity, you’re an extended braidlike pattern in a static four-dimensional spacetime, you nonetheless perceive yourself as localized at a particular place and time in a three-dimensional world where things happen. As we discussed previously, your basic perceptions are observer moments, each of which corresponds to a particular localized part of your braid pattern rather than to the whole thing, your whole life.

  Quantum mechanics teaches us the same lesson: if you enter a quantum superposition of being in two separate places at once in the external reality (the mathematical Hilbert space where the Schrödinger equation rules), then as we saw in Chapter 8, both of these copies of you will perceive an inner reality where they’re in a well-defined location.

  We perceive ourselves as unique

  In Chapter 8, we also saw that we perceive ourselves as unique and isolated systems even if we aren’t. We saw that even if quantum mechanics effectively clones us so that we end up in several macroscopically different places at once, intricately entangled with other systems, we perceive ourselves as remaining unique and isolated, retaining an independent and distinct identity. What appears as “observer branching” in the external reality is perceived as merely a slight randomness in the internal reality.

  The same thing happens
with classical cloning as in Figure 8.3: cloning with determinism is perceived as uniqueness with randomness. In other words, our well-defined local and unique identity exists only in our internal reality; at a fundamental level, it’s an illusion.

  We perceive ourselves as immortal(?)

  In Chapter 8, we also discussed the possibility that the Level I and/or Level III multiverses make us feel immortal. In summary, the relation between the external and internal realities is quite subtle when the number of copies of you increases or decreases:

  • When the number of yous increases, you perceive subjective randomness.

  • When the number of yous decreases, you perceive subjective immortality.

  The latter is particularly controversial, and whether it’s a correct inference or not may hinge on the resolution of the so-called measure problem that we’ll describe further on.

  We perceive that which is useful

  Why do we perceive the world as stable and ourselves as local and unique? Here’s my guess: because it’s useful. It appears that we humans have evolved self-awareness in the first place because certain aspects of our world are somewhat predictable, so that being good at modeling the world, predicting the future and making smart decisions increases our reproductive success. Self-awareness would then be a side effect of this advanced information processing. More generally, any SAS that’s either evolved or engineered with a purpose might have self-awareness as a by-product of having an internal model of the world and itself.