Our Mathematical Universe

by Max Tegmark

  Past, Present and Future

  “Excuse me, but what’s the time?” I’m guessing that you, like me, have asked this question, as if there were such a thing as the time at a fundamental level. Yet you’ve probably never approached a stranger and asked, “Excuse me, but what’s the place?” If you were really hopelessly lost, you’d probably instead have said something like “Excuse me, but where am I?”—thereby acknowledging that you’re not asking about a property of space, but rather about a property of yourself: your location in space while you’re asking the question. Similarly, when you ask for the time, you’re not really asking about a property of time, but rather about your location in time. Spacetime contains all places and all times, so there’s no the time any more than there’s the place. It would therefore be more appropriate (scientifically if not socially) to ask, “When am I?” Spacetime is like a map of cosmic history without a “You are here” marker. If you need such a marker to orient yourself, I recommend a phone with both a clock and a GPS.

  When Einstein wrote that “The distinction between past, present, and future is only a stubbornly persistent illusion,” he was referring to the fact that these concepts have no objective meaning in spacetime. Figure 11.2 illustrates that when we think about the “present,” we mean the time slice through spacetime corresponding to the time when we’re having that thought. We refer to the “future” and “past” as the parts of spacetime above and below this slice. This is analogous to your use of the terms here, in front of me and behind me to refer to different parts of spacetime relative to your present position. The part that’s in front of you is clearly no less real than the part behind you—indeed, if you’re walking forward, some of what’s presently in front of you will be behind you in the future, and is presently behind various other people. Analogously, in spacetime, the future is just as real as the past—parts of spacetime that are presently in your future will in your future be in your past. Since spacetime is static and unchanging, no parts of it can change their reality status, and all parts must be equally real.1

  Figure 11.2: The distinction between past, present and future exists only in the frog perspective (right), not in the bird perspective of the mathematical structure (left)—in the latter, you can’t ask, “What time is it?”; merely, “When am I?”

  In summary, time is not an illusion, but the flow of time is. So is change. In spacetime, the future exists and the past doesn’t disappear. When we combine Einstein’s classical spacetime with quantum mechanics, we get quantum parallel universes as we saw in Chapter 8. This means that there are many pasts and futures that are all real—but this in no way diminishes the unchanging mathematical nature of the full physical reality.

  This is how I see it. However, although this idea of an unchanging reality is venerable and dates back to Einstein, it remains controversial and subject to vibrant scientific debate, with scientists I greatly respect expressing a spectrum of views. For example, in his book The Hidden Reality, Brian Greene expresses unease toward letting go of the notions that change and creation are fundamental, writing, “I’m partial to there being a process, however tentative … that we can imagine generating the multiverse.” Lee Smolin goes further in his book Time Reborn, arguing that not only is change real, but that indeed time may be the only thing that’s real. At the other end of the spectrum, Julian Barbour argues in his book The End of Time not only that change is illusory, but that one can even describe physical reality without introducing the time concept at all.

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  1In his book The Hidden Reality, Brian Greene further hammers home this conclusion by pointing out that, according to Einstein’s relativity theory, the horizontal slice delimiting past from future in Figure 11.2 gets tilted if you start moving; there clearly can’t be any fundamental distinction between past and future if you can reclassify a distant supernova explosion from already having happened to not yet having happened simply by walking faster.

  How Spacetime and “Stuff” Can Be Mathematical

  Earlier, we’ve seen how spacetime can be viewed as a mathematical structure. But what about all the stuff in spacetime, say, the book you’re reading right now? How can that be part of a mathematical structure?

  In recent years, we’ve seen that many things that seemed totally unrelated to mathematics, such as texts, sounds, images and movies, can be represented mathematically by computers and transmitted over the Internet as a bunch of numbers. Let’s take a closer look at how computers do this—because as we’ll soon see, nature is doing something rather analogous to represent all the stuff around us.

  I just typed the word word, and my laptop represented it in its memory as the four-number sequence “119 111 114 100.” It represents each lowercase letter by a number that’s 96 plus its order in the alphabet, so a = 97, w = 119. At the same time, my laptop is playing De Profundis by the Estonian composer Arvo Pärt, which it’s also representing as a sequence of numbers—these numbers are interpreted not as letters, but as the positions in which it puts the loudspeaker membranes at 44,100 different instances each second, which in turn causes the air vibrations that my ears and my brain interpret as sound. As soon as I hit the w key on my keyboard, my laptop displayed a visual image of a w on my screen, and this image is also represented by numbers. Although all my screen images look smooth and continuous, my screen is in fact made up of 1,920 × 1,200 pixels in a rectangular grid, as illustrated in Figure 11.3, and the color of each pixel is represented by three numbers, each between 0 and 255, specifying the intensities of red, green and blue light coming from the pixel; suitable combinations of these three colors can then produce all intensities of all the colors of the rainbow. Last night, when my kids and I watched a YouTube video, my laptop similarly divided not only the two spatial dimensions of my screen into pixels, but also the time dimension, slicing time into 30 frames per second.

  At work, we physicists often simulate some event in 3-D, such as a hurricane, a supernova explosion or the formation of a solar system. To do this, we divide three-dimensional space into 3-D pixels (voxels). We also divide the 4-D spacetime into 4-D voxels. Each such 4-D voxel represents what’s happening at that place at that time by a group of numbers encoding everything that’s relevant, such as the temperature, the pressure, and the densities and velocities of different substances in the voxel. For example, in a simulation of our Solar System, a voxel corresponding to the center of the Sun will have an extremely large temperature number, and a voxel outside the Sun containing almost empty space will have a pressure number close to zero. The numbers in neighboring voxels satisfy certain relations that are captured by mathematical equations, and when a computer is performing a simulation, it’s using these relations to fill in missing numbers like a Sudoku player. If a computer is making a weather forecast, then the spacetime voxels corresponding to right now are filled in with measured numbers for air pressure, air temperature, etc. The computer then uses the relevant equations to calculate the numbers that go in the spacetime voxels corresponding to tomorrow and the rest of the week.

  Figure 11.3: Computers usually represent gray-scale images by a number at each point (pixel) of the image (rightmost panel). The larger the number, the more intense the light from the pixel, with 0 representing black (no light at all) and 255 representing white. Similarly, so-called fields in classical physics are represented by numbers at each point in spacetime, which, loosely speaking, specify the amount of “stuff” existing at each point.

  Although such simulations represent aspects of our external physical reality mathematically, they do so only approximately. Spacetime certainly isn’t made of the crude voxels we use to simulate tomorrow’s weather, which is one of the reasons why weather forecasts are often inaccurate. Yet this idea that there’s a bunch of numbers at each point in spacetime is quite deep, and I think it’s telling us something not merely about our description of reality, but about reality itself. One of the most fundamental concepts in modern physics is that of a field, whic
h is just this: something represented by numbers at each point in spacetime. For example, there’s a temperature field corresponding to the air around you: there’s a well-defined temperature at each point, totally independent of any human-invented voxels, and you can measure the temperature number by holding a thermometer there—or your finger, if you don’t need great accuracy. There’s also a pressure field: at each point, there’s a pressure number which you can measure with a barometer—or with your ear, which will hurt if the number is too extreme and which can detect sound if the pressure is fluctuating over time.

  We now know that neither of these two fields are truly fundamental: they’re merely different measures of how fast the air molecules are moving on average, so these numbers stop being well defined if you try to measure them on subatomic scales. However, there are other fields that seem to be quite fundamental, forming part of the very fabric of our external physical reality. As a first example, let’s look at the magnetic field. It’s represented by not one (like temperature) but three numbers at each point in spacetime, encoding both a strength and a direction. You’ve probably measured the magnetic field using a compass, watching its magnetic pointer align itself with Earth’s magnetic field, which points north. The pointer aligns itself faster if the magnetic field is stronger, such as near an MRI machine. A second example is the electric field, which is also represented by a triplet of numbers encoding strength and direction. An easy way to measure it is by the force it exerts on a charged object—like when your hair gets electrically attracted to a plastic comb. These electric and magnetic fields can be elegantly unified into what’s known as the electromagnetic field, represented by six numbers at each point in spacetime. As we discussed in Chapter 7, light is simply a wave rippling through the electromagnetic field, so if our physical world is a mathematical structure, then all the light in our Universe (which feels quite physical) corresponds to six numbers at each point in spacetime (which feels quite mathematical). These numbers obey the mathematical relations that we know as Maxwell’s equations, shown in Figure 10.4.

  There’s a caveat here: what I’ve just described was our understanding of electricity, magnetism and light in classical physics. Quantum mechanics complicates this picture, but without making it any less mathematical, replacing classical electromagnetism with quantum field theory, the bedrock of modern particle physics. In quantum field theory, the wavefunction specifies the degree to which each possible configuration of the electric and magnetic fields is real. This wavefunction is itself a mathematical object, an abstract point in Hilbert space.

  As we saw in Chapter 7, quantum field theory says that light is made of particles called photons, and, crudely speaking, the numbers constituting the electric and magnetic fields can be thought of as specifying how many photons there are at each time and place. Just as the strength of the electromagnetic field corresponds to the number of photons at each time and place, there are other fields corresponding to all the other elementary particles known. For example, the strengths of the electron field and the quark field relate to the numbers of electrons and quarks at each time and place. In this way, all motions of all particles in all of spacetime correspond, in classical physics, to a bunch of numbers at each point in a four-dimensional mathematical space—a mathematical structure. In quantum field theory, the wavefunction specifies the degree to which each possible configuration of each of these fields is real.

  As we discussed in Chapter 7, we physicists still haven’t found a mathematical structure that can describe all aspects of reality, including gravity, but so far, there’s no indication that string theory or any of the other most actively pursued candidates for such a description are any less mathematical than quantum field theory.

  Description Versus Equivalence

  Before moving on, there’s an important semantic issue that we need to sort out. Whereas most of my physics colleagues would say that our external physical reality is (at least approximately) described by mathematics, I’m arguing that it is mathematics (more specifically, a mathematical structure). In other words, I’m making a much stronger claim. Why?

  Everything I’ve said so far in this chapter suggests that our external physical reality can be described by a mathematical structure. If a future physics textbook contains the coveted Theory of Everything (ToE), then its equations are a complete description of the mathematical structure that is the external physical reality. I’m writing is rather than corresponds to here, because if two structures are equivalent, then there’s no meaningful sense in which they’re not one and the same, as emphasized by the Israeli philosopher Marius Cohen.1 Recall the powerful mathematical notion of equivalence that we described in Chapter 10, which embodies the very essence of mathematical structures: if two complete descriptions are equivalent, then they’re describing one and the same thing.2 This means that if some mathematical equations completely describe both our external physical reality and a mathematical structure, then our external physical reality and the mathematical structure are one and the same, and then the Mathematical Universe Hypothesis is true: our external physical reality is a mathematical structure.

  Remember that two mathematical structures are equivalent if you can pair up their entities in a way that preserves all relations. If you can thus pair up every entity in our external physical reality with a corresponding one in a mathematical structure (“This electric-field strength here in physical space corresponds to this number in the mathematical structure,” for example), then our external physical reality meets the definition of being a mathematical structure—indeed, that same mathematical structure.

  We saw in Chapter 10 that if someone wishes to avoid accepting the Mathematical Universe Hypothesis, they can do so by rejecting the External Reality Hypothesis that there’s an external physical reality completely independent of us humans. They could then argue that our Universe is somehow made of stuff perfectly described by a mathematical structure, but which also has other properties that aren’t described by it, and can’t be described in an abstract, human-independent, baggage-free way. However, I think this viewpoint would make the famous science philosopher Karl Popper from Chapter 6 turn in his grave, since he emphasized that scientific theories must have observable effects. In contrast, since the mathematical description is supposedly perfect, accounting for everything that can be observed, those additional bells and whistles that would make our Universe nonmathematical would by definition have no observable effects whatsoever, rendering them 100% unscientific.

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  1Marius Cohen, “On the Possibility of Reducing Actuality to a Pure Mathematical Structure” (master’s thesis, Ben Gurion University of the Negev, Israel, 2003).

  2If you have a mathematics background and are familiar with the notion of isomorphism, you can restate this argument as follows. From the definition of a mathematical structure, it follows that if there’s an isomorphism between a mathematical structure and another structure (a one-to-one correspondence between the two that respects the relations), then they’re one and the same. If our external physical reality is isomorphic to a mathematical structure, it therefore fits the definition of being a mathematical structure.

  What Are You?

  We’ve now seen how both spacetime and the stuff in it can be viewed as being part of a mathematical structure. But what about us? Our thoughts, our emotions, our self-awareness, and that deep existential feeling I am—none of this feels the least bit mathematical to me. Yet we too are made of the same kinds of elementary particles that make up everything else in our physical world, which we’ve argued is purely mathematical. How can we reconcile this?

  In my opinion, we don’t yet fully understand what we are. Moreover, as we discussed in Chapter 9, we don’t really need to fully understand the mysteries of consciousness to understand our external physical reality. Nonetheless, I feel that modern physics has provided some tantalizing hints about fruitful ways of viewing ourselves, so let’s explore this topic further.

&
nbsp; The Braid of Life

  George Gamow, the cosmology pioneer whom we encountered in Chapter 3, titled his autobiography My World Line, a phrase also used by Einstein to refer to paths through spacetime. However, your own world line strictly speaking isn’t a line: it has a non-zero thickness and it’s not straight. Let’s first consider the roughly 1029 elementary particles (quarks and electrons) that your body is made of. Together, they form a tubelike shape through spacetime, analogous to the spiral shape of the Moon’s orbit (Figure 11.1) but more complicated, reflecting the fact that your motion from birth to death is more complicated than the Moon’s. For example, if you’re swimming laps in a pool, that part of your spacetime tube has a zigzag shape, and if you’re using a playground swing, that part of your spacetime tube has a serpentine shape.

  However, the most interesting property of your spacetime tube isn’t its bulk shape, but its internal structure, which is remarkably complex. Whereas the particles that constitute the Moon are stuck together in a rather static arrangement, many of your particles are in constant motion relative to one another.

  Consider, for example, the particles that make up your red blood cells. As your blood circulates through your body to deliver the oxygen you need, each red blood cell traces out its own unique tube shape through spacetime, corresponding to a complex itinerary through your arteries, capillaries and veins with regular returns to your heart and lungs. These spacetime tubes of different red blood cells are intertwined to form a braid pattern (Figure 11.4, middle panel) which is more elaborate than anything you’ll ever see in a hair salon: whereas a classic braid consists of three strands with perhaps thirty thousand hairs each, intertwined in a simple repeating pattern, this spacetime braid consists of trillions of strands (one for each red blood cell), each composed of trillions of hairlike elementary-particle trajectories, intertwined in a complex pattern that never repeats. In other words, if you imagine spending a year giving a friend a truly crazy hairdo, braiding his hair by separately intertwining not strands but all the individual hairs, the pattern you’d get would still be very simple in comparison.