Our Mathematical Universe
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So if the entities of this structure have no intrinsic properties, does the structure itself have any interesting properties (besides having eight elements)? In fact, it does: symmetries! In physics, we say that something has a symmetry if it remains unchanged when you transform it in some way. For example, we say that your face has mirror symmetry if it looks the same after being reflected left to right. In the same way, the mathematical structure in Figure 10.8 (middle) has mirror symmetry: if you swap elements 1 and 2, 3 and 4, 5 and 6, and 7 and 8, the drawing of the relations will still look the same. It also has some rotational symmetry, corresponding either to rotating the cube in the drawing by 90 degrees around one of its faces, by 120 degrees around one of its corners, or by 180 degrees around an edge center. Although we intuitively think of symmetry as having to do with geometry, you can in fact discover these same symmetries just by messing around with the table in the right panel of Figure 10.8: if you renumber the eight elements in certain ways and then re-sort the table by increasing row and column numbers, you end up with the exact same table that you started with.
A famous thorny issue in philosophy is the so-called infinite regress problem. For example, if we say that the properties of a diamond can be explained by the properties and arrangements of its carbon atoms, that the properties of a carbon atom can be explained by the properties and arrangements of its protons, neutrons and electrons, that the properties of a proton can be explained by the properties and arrangements of its quarks, and so on, then it seems that we’re doomed to go on forever trying to explain the properties of the constituent parts. The Mathematical Universe Hypothesis offers a radical solution to this problem: at the bottom level, reality is a mathematical structure, so its parts have no intrinsic properties at all! In other words, the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.1 The external physical reality is therefore more than the sum of its parts, in the sense that it can have many interesting properties while its parts have no intrinsic properties at all.
Mathematical Universe Cheat Sheet
Baggage Concepts and words that are invented by us humans for convenience, which aren’t necessary for describing the external physical reality
Mathematical structure Set of abstract entities with relations between them; can be described in a baggage-independent way
Equivalence Two descriptions of mathematical structures are equivalent if there’s a correspondence between them that preserves all relations; if two mathematical structures have equivalent descriptions, they are one and the same
Symmetry The property of remaining unchanged when transformed; for example, a perfect sphere is unchanged when rotated
External Reality Hypothesis The hypothesis that there exists an external physical reality completely independent of us humans
Mathematical Universe Hypothesis The hypothesis that our external physical reality is a mathematical structure; I argue that this follows from the External Reality Hypothesis
Computable Universe Hypothesis Our external physical reality is a mathematical structure defined by computable functions (Chapter 12)
Finite-Universe Hypothesis Our external physical reality is a finite mathematical structure (Chapter 12)
Table 10.2: Summary of key concepts linked to the mathematical-universe idea
The particular mathematical structures illustrated in Figures 10.7 and 10.8 belong to the family of mathematical structures known as graphs: abstract elements, some of which are connected pairwise. You can use other graphs to describe the mathematical structures corresponding to the dodecahedron and the other Platonic solids from Figure 7.2. Another example of a graph is the network of friends on Facebook: here the elements correspond to all the Facebook users, and two users are connected if they’re in a friend relation. Although mathematicians have studied graphs extensively, they constitute merely one of many different families of mathematical structures. We’ll discuss mathematical structures in greater detail in Chapter 12, but let’s first briefly look at a few more examples here, to get a sense for how diverse mathematical structures can be.
There are many mathematical structures corresponding to different types of numbers. For example, the so-called natural numbers 1, 2, 3,…together form a mathematical structure. Here the elements are the numbers and there are many different kinds of relations. Some relations (say, equals, is greater than and is divisible by) can hold between two numbers (“15 is divisible by 5,” say), some relations hold between three numbers (“17 is the sum of 12 and 5,” say) and some relations involve other numbers of numbers. Mathematicians have gradually discovered larger classes of numbers that form their own mathematical structures, such as integers (including negative numbers), rational numbers (including fractions), real numbers (including the square root of 2), complex numbers (including the square root of –1), and transfinite numbers (including infinite numbers). When I close my eyes and think of the number 5, it looks yellow to me. Yet in all these mathematical structures, the numbers themselves have no such intrinsic properties at all, and their only properties are given by their relations to other numbers—5 has the property that it’s the sum of 4 and 1, say, but it’s not yellow, and it’s not made of anything.
Another large class of mathematical structures corresponds to different types of spaces. For example, the three-dimensional Euclidean space that we learned about in school is a mathematical structure. Here the elements are points in the 3-D space and real numbers that are interpreted as distances and angles. There are many different kinds of relations. For example, three points can satisfy the relation that they lie on a line. There’s a different mathematical structure corresponding to Euclidean space with four dimensions and with any other number of dimensions. Mathematicians have also discovered many other types of more general spaces that form their own mathematical structures, like so-called Minkowski space, Riemann spaces, Hilbert spaces, Banach spaces and Hausdorff spaces. Many people used to think that our three-dimensional physical space was a Euclidean space. However, we saw in Chapter 2 that Einstein put an end to that. First his special relativity theory said that we live in a Minkowski space (including time as a fourth dimension), then his general relativity said that we instead live in a Riemann space, which meant that it could be curved. Then, as we saw in Chapter 7, quantum mechanics came along and said that we’re really living in a Hilbert space. Again, the points in these spaces aren’t made of anything, and have no color, texture or other intrinsic properties whatsoever.
Although the collection of known mathematical structures is large and exotic, and even more remain to be discovered, every single mathematical structure can be analyzed to determine its symmetry properties, and many have interesting symmetry. Intriguingly, one of the most important discoveries in physics has been that our physical reality also has symmetries built into it: for example, the laws of physics have rotational symmetry, which means that there’s no special direction in our Universe that you can call “up.” They also appear to have translation (sideways shifting) symmetry, meaning that there’s no special place that we can call the center of space. Many of the spaces just mentioned have beautiful symmetries, some of which match the observed symmetries of our physical world. For example, Euclidean space has both rotational symmetry (meaning that you can’t tell the difference if the space gets rotated) and translational symmetry (meaning that you can’t tell the difference if the space gets shifted sideways). The four-dimensional Minkowski space has even more symmetry: you can’t even tell the difference if you do a type of generalized rotation between the space and time dimensions—and Einstein showed that this explains why time appears to slow down if you travel near the speed of light, as mentioned in the last chapter. Many more subtle symmetries of nature have been discovered in the last century, and these symmetries form the foundations of Einstein’s rela
tivity theories, quantum mechanics, and the standard model of particle physics.
Note that these symmetry properties that are so important in physics come precisely from the lack of intrinsic properties of the building blocks of reality, that is, from the very essence of what it means for it to be a mathematical structure. If you take a colorless sphere and paint part of it yellow, then its rotational symmetry gets destroyed. Similarly, if the points in a three-dimensional space had any properties that made some points intrinsically different from others, then the space would lose its rotational and translational symmetry. “Less is more,” in the sense that the less properties the points have, the more symmetry the space has.
If the Mathematical Universe Hypothesis is correct, then our Universe is a mathematical structure, and from its description, an infinitely intelligent mathematician should be able to derive all these physics theories. How exactly would she do this? We don’t know, but I’m quite sure about what her first step would be: to calculate the symmetries of the mathematical structure.
At the beginning of this chapter, you saw the grim prognostication that my publications on the relation between mathematics and physics were too crazy and would ruin my career. I’ve now told you about the first part of these ideas, arguing that our external physical reality is a mathematical structure, which sounds quite crazy indeed. However, that was just the warm-up—it’s going to get much crazier later, when we examine the implications and testable predictions of the Mathematical Universe Hypothesis! Among other things, we’ll be led inexorably to a new multiverse so large that it makes even the Level III Multiverse of quantum mechanics pale in comparison. But before that, we need to answer a burning question. Our physical world is changing over time, whereas mathematical structures don’t change—they just exist. So how can our world possibly be a mathematical structure? We’ll tackle that in the next chapter.
THE BOTTOM LINE
• Since antiquity, people have puzzled over why our physical world can be so accurately described by mathematics.
• Ever since, physicists have kept discovering more shapes, patterns and regularities in nature that are describable by mathematical equations.
• The fabric of our physical reality contains dozens of pure numbers, from which all measured constants can in principle be calculated.
• Some key physical entities such as empty space, elementary particles and the wavefunction appear to be purely mathematical in the sense that their only intrinsic properties are mathematical properties.
• The External Reality Hypothesis (ERH)—that there exists an external physical reality completely independent of us humans—is accepted by most but not all physicists.
• With a sufficiently broad definition of mathematics, the ERH implies the Mathematical Universe Hypothesis (MUH) that our physical world is a mathematical structure.
• This means that our physical world not only is described by mathematics, but that it is mathematical (a mathematical structure), making us self-aware parts of a giant mathematical object.
• A mathematical structure is an abstract set of entities with relations between them. The entities have no “baggage”: they have no properties whatsoever except these relations.
• A mathematical structure can have many interesting properties—for example, symmetries—even though neither its entities nor its relations have any intrinsic properties whatsoever.
• The MUH solves the infamous infinite regress problem where the properties of nature can only be explained from the properties of its parts, which require further explanation, ad infinitum: the properties of nature stem not from properties of its ultimate building blocks (which have no properties at all), but from the relations between these building blocks.
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1Our brain may provide another example of where properties stem mainly from relations. According to the so-called concept cell hypothesis in neuroscience, particular firing patterns in different groups of neurons correspond to different concepts. The main difference between the concept cells for “red,” “fly” and “Angelina Jolie” clearly don’t lie in the types of neurons involved, but in their relations (connections) to other neurons.
11
Is Time an Illusion?
The distinction between past, present, and future is only a stubbornly persistent illusion.
—Albert Einstein, 1955
Time is an illusion, lunchtime doubly so.
—Douglas Adams, The Hitchhiker’s Guide to the Galaxy
If you’re like me, then you’re disturbed by unanswered questions. The last chapter raised many, so it’s valid for you to question what I’ve said. For example, I argued that our external physical reality is a mathematical structure, but what does this really mean? This physical reality is constantly changing—leaves move in the wind and planets orbit the Sun—while mathematical structures are static: an abstract dodecahedron always has had and always will have exactly twelve pentagonal faces. How can something changing possibly be something unchanging? Another urgent question concerns how you personally fit into this supposed mathematical structure—how can your self-awareness, thoughts and feelings be part of a mathematical structure?
How Can Physical Reality Be Mathematical?
Timeless Reality
Einstein can help us answer these questions. He taught us that there are two equivalent ways of thinking about our physical reality: either as a three-dimensional place called space, where things change over time, or as a four-dimensional place called spacetime that simply exists, unchanging, never created and never destroyed.1 These two perspectives correspond to the frog and bird perspectives on reality that we discussed in Chapter 9: the latter is the outside overview of a physicist studying its mathematical structure, like a bird surveying a landscape from high above; the former is the inside view of an observer living in this structure, like a frog living in the landscape surveyed by the bird.
Figure 11.1: The Moon’s orbit around the Earth. We can equivalently think of this either as a position in space that changes over time (right), or as an unchanging spiral shape in spacetime (left), corresponding to a mathematical structure. The snapshots of space (right) are simply horizontal slices of spacetime (left).
Mathematically, spacetime is a space with four dimensions, the first three being our familiar dimensions of space, and the fourth dimension being time. Figure 11.1 illustrates this idea. Here, I’ve drawn it so that the time dimension is in the vertical direction and the space dimensions are in the horizontal directions. To avoid confusion, I’ve plotted only two of the three space dimensions, labeled x and y, because smoke starts pouring out of my ears if I try to visualize four-dimensional objects.… The figure shows the Moon moving around Earth in a circular orbit—to keep things legible, I’ve drawn the orbit much smaller than to scale and made several simplifications.2 The right panel shows the frog perspective: five snapshots of space with the Moon in different positions, while Earth remains in the same place. The left panel shows the bird perspective: here the motion of the frog perspective is replaced by unchanging shapes in spacetime. Since Earth isn’t moving, it’s at the sample place in space for all time, and therefore makes a vertical cylinder in spacetime. The Moon is more interesting, manifesting itself as a spiral in spacetime that encodes where it is at different times. Please look at the left and right panels until you’ve figured out how they’re related, since this is crucial for the rest of our discussion. To get snapshots of space (right) from spacetime (left), you simply make horizontal slices through spacetime at the times you’re interested in.
Note that spacetime doesn’t exist within space and time—rather, space and time exist within it. I’m arguing that our external physical reality is a mathematical structure, which is by definition an abstract, immutable entity existing outside of space and time. This mathematical structure corresponds to the bird perspective of our reality, not the frog perspective, so it should contain spacetime, not just space. The mathematical structure co
ntains additional elements as well, as we’ll get to below, corresponding to the stuff contained in our spacetime. However, this doesn’t alter its timeless nature: if the history of our Universe were a chess game, the mathematical structure would correspond not to a single position, but to the entire game (Figure 10.6). If the history of our Universe were a movie, the mathematical structure would correspond not to a single frame but to the entire DVD. So from the bird’s perspective, trajectories of objects moving in four-dimensional spacetime resemble a tangle of spaghetti. Where the frog sees something moving with constant velocity, the bird sees a straight strand of uncooked spaghetti. Where the frog sees the Moon orbit Earth, the bird sees the rotini-like spiral of Figure 11.1. Where the frog sees hundreds of billions of stars moving around in our Galaxy, the bird sees hundreds of billions of intertwined spaghetti strands. To the frog, reality is described by Newton’s laws of motion and gravitation. To the bird, reality is the geometry of the pasta.
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1This idea of time as the fourth dimension as an unchanging reality has been promoted and explored by many, including H. G. Wells in his 1895 novel The Time Machine. Julian Barbour gives an interesting account of the idea and its history in his book The End of Time.
2To keep things simple, Figure 11.1 ignores the fact that both the Earth and the Moon are rotating, that the Moon’s orbit is slightly oblong (it’s an ellipse rather than a perfect circle), and that the Moon’s gravitational pull causes Earth to undergo circular motion, too, with a radius that’s about 74% of Earth’s radius.