Our Mathematical Universe

by Max Tegmark

  The Holy Grail of physics is to find what’s jocularly referred to as a “Theory of Everything,” or ToE, from which all else can be derived—this would replace the big question mark at the top of the theory tree. As we discussed in Chapter 7, we know that something is missing here because we lack a consistent theory unifying gravity with quantum mechanics. This ToE would be a complete description of the external physical reality that the External Reality Hypothesis assumes. At the beginning of this section, I argued that such a complete description must be devoid of any human baggage. This means that it must contain no concepts at all! In other words, it must be a purely mathematical theory, with no explanations or “postulates” as in quantum textbooks (mathematicians are perfectly capable of—and often pride themselves on—studying abstract mathematical structures that lack any intrinsic meaning or connection with physical concepts). Rather, an infinitely intelligent mathematician should be able to derive the entire theory tree of Figure 10.5 from these equations alone, by deriving the properties of the physical reality that they describe, the properties of its inhabitants, their perceptions of the world, and even the words they invent. This purely mathematical theory of everything could potentially turn out to be simple enough to describe with equations that fit on a T-shirt.

  Figure 10.5: Theories can be crudely organized into a family tree where each might, at least in principle, be derivable from more fundamental ones above it. For example, special relativity can be obtained from general relativity in the approximation that Newton’s gravitational constant G equals zero, classical mechanics can be derived from special relativity in the approximation that the speed of light c is infinite, and hydrodynamics with its concepts such as density and pressure can be derived from the classical physics of how particles bounce around. However, these cases where the arrows are well understood form a minority. Deriving biology from chemistry or psychology from biology appears unfeasible in practice. Only limited and approximate aspects of such subjects are mathematical, and it’s likely that all mathematical models found in physics so far are similarly approximations of limited aspects of reality.

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  All of this begs the question: is it actually possible to find such a description of the external reality that involves no baggage? If so, such a description of objects in this external reality and the relations between them would have to be completely abstract, forcing any words or symbols to be mere labels with no preconceived meanings whatsoever. Instead, the only properties of these entities would be those embodied by the relations between them.

  Mathematical Structures

  To answer this question, we need to take a closer look at mathematics. To a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them. Take the integers, for instance, or geometric objects such as the dodecahedron, a favorite of the Pythagoreans. This is in stark contrast to the way most of us first perceive mathematics—either as a sadistic form of punishment, or as a bag of tricks for manipulating numbers. Like physics, mathematics has evolved to ask broader questions.

  Modern mathematics is the formal study of structures that can be defined in a purely abstract way, without any human baggage. Think of mathematical symbols as mere labels without intrinsic meaning. It doesn’t matter whether you write, “Two plus two equals four,” “2 + 2 = 4,” or “Dos más dos es igual a cuatro.” The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don’t invent mathematical structures—we discover them, and invent only the notation for describing them. If an alien civilization gets interested in 3-D shapes with only flat identical faces, they might discover the five from Figure 7.2 that we Earthlings call Platonic solids. They might invent their own exotic names for them, but they can’t invent a sixth one—it simply doesn’t exist.

  In summary, there are two key points to take away from our discussion above:

  1. The External Reality Hypothesis implies that a “theory of everything” (a complete description of our external physical reality) has no baggage.

  2. Something that has a complete baggage-free description is precisely a mathematical structure.

  Taken together, this implies the Mathematical Universe Hypothesis, i.e., that the external physical reality described by the ToE is a mathematical structure.1 So the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object—one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical—including you.

  * * *

  1In the philosophy literature, John Worrall has coined the term structural realism as a compromise position between scientific realism and anti-realism; crudely speaking, stating that the fundamental nature of reality is correctly described only by the mathematical or structural content of scientific theories. This term has been interpreted and refined in different ways by different science philosophers, and Gordon McCabe has argued that the term universal structural realism should be used for my hypothesis that our physical Universe is isomorphic to a mathematical structure.

  What Is a Mathematical Structure?

  “Waaaaaaaaaaaaaaaaait a minute!” That’s what my friend Justin Bendich used to cry out whenever a physics claim raised urgent unanswered questions, and the Mathematical Universe Hypothesis raises three:

  • What exactly is a mathematical structure?

  • How exactly can our physical world be a mathematical structure?

  • Does this make any testable predictions?

  We’ll tackle the second question in Chapter 11 and the third one in Chapter 12. Let’s start by exploring the first question—we’ll return to it in more detail in Chapter 12.

  Baggage and Equivalent Descriptions

  Earlier, we described how we humans add baggage to our descriptions. Now let’s look at the opposite: how mathematical abstraction can remove baggage and strip things down to their bare essence. Consider the particular sequence of chess moves that have become known as the “Immortal Game,” where white spectacularly sacrifices both rooks, a bishop and the queen to checkmate with the three remaining minor pieces as shown in Figure 10.6. Here on Earth, this game was first played in 1851 by Adolf Anderssen and Lionel Kieseritzky. However, the same game is replayed annually in the town of Marostica, Italy, with live players dressed as chess pieces, and it’s regularly repeated by countless chess enthusiasts around the world. Some players (including my brother Per, his son Simon and my son Alexander in Figure 10.6) use pieces made of wood, while others use pieces of marble or plastic with different shapes and sizes. Some boards are brown and beige, some are black and white, and some are virtual, being mere 3-D or 2-D computer graphics as in Figure 10.6. Yet there’s a sense in which none of these details matter: when chess aficionados call the Immortal Game beautiful, they’re not referring to the attractiveness of the players, the board, or the pieces, but to a more abstract entity, which we might call the abstract game, or the sequence of moves.

  Figure 10.6: An abstract game of chess is independent of the colors and shapes of the pieces, and of whether its moves are described on a physically existing board, by stylized computer-rendered images, or by so-called algebraic chess notation—it’s still the same chess game. Analogously, a mathematical structure is independent of the symbols used to describe it.

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  Let’s look in detail at how we humans go about describing such abstract entities. First of all, a description needs to be specific, so we invent objects, words or other symbols to correspond to the abstract idea
s: for example, in the United States, we name the chess piece that can move diagonally a “bishop.” Second, it’s obvious that this name is arbitrary, and that other names would have worked just as well—indeed, this piece is call a fou (fool) in French, strelec (shooter) in Slovak, löpare (runner) in Swedish and fil (elephant) in Persian. However, we can reconcile the uniqueness of the Immortal Game with the multiplicity of possible descriptions of it by introducing the powerful idea of equivalence:

  1. We define what we mean by two descriptions being equivalent.

  2. We say that if two descriptions are equivalent, then they’re describing one and the same thing.

  For example, we agree that any two descriptions of a chess position are equivalent if the only difference between them lies in the sizes of the pieces, or in the names that the players give to the pieces in their native language.

  Any word, concept or symbol that appears in some but not all of the equivalent descriptions is clearly optional and therefore baggage. So if we want to get down to the bare essence of the Immortal Game, then how much baggage can we strip away? Clearly a lot, since computers are able to play chess without having any notion of human language or human concepts such as the colors, textures, sizes and names of chess pieces. To fully understand how far we can go, we need to make a more rigorous definition of equivalence:

  Equivalence: Two descriptions are equivalent if there’s a correspondence between them that preserves all relations.

  Chess involves abstract entities (different chess pieces and different squares on the board) and relations between them. For example, one relation that a piece may have to a square is that the former is standing on the latter. Another relation that a piece may have to a square is that it’s allowed to move there. For example, the two center panels in Figure 10.6 are equivalent by our definition: there’s a correspondence between the three-dimensional and two-dimensional pieces and boards such that whenever a 3-D piece stands on a particular square, the corresponding 2-D piece stands on the corresponding square. Similarly, a description of a chess position given purely verbally in English is equivalent to a description given purely verbally in Spanish if you can provide a dictionary specifying the correspondence between the English and Spanish words, and if using it to translate the Spanish description produces the English description.

  When newspapers and websites print chess games, they customarily use yet another equivalent description: so-called algebraic chess notation (Figure 10.6, right). Here pieces are represented not by objects or words, but by single letters; bishop is equivalent to B, for example, and squares are represented by a letter specifying the column and a number specifying the row. Since the abstract game description in Figure 10.6 (right) is equivalent to a description in the form of a movie of the game being played on a physical board, everything in the latter description that isn’t in the former description is mere baggage—from the physical existence of a board to the shapes, colors, and names of the pieces. Even the specifics of algebraic chess notation are baggage: when computers play chess, they typically use other abstract chess-position descriptions, involving certain patterns of zeros and ones in their memory. So what is it that’s left when you strip away all this baggage? What is it that’s described by all these equivalent descriptions? The Immortal Game itself, 100% pure, with no additives.

  Baggage and Mathematical Structures

  Our case study involving abstract chess pieces, board squares and relations between them was an example of a much more general concept: a mathematical structure. This is a standard concept in modern mathematical logic. I’ll give a more technical description in Chapter 12, but this nontechnical definition is all we need for now:

  Mathematical structure: Set of abstract entities with relations between them

  To understand what this means, let’s consider a couple of examples. Figure 10.7 (left) is a description of a mathematical structure with four entities, some of which are related by the relation likes to. In the figure, the entity Philip is represented by an image with many intrinsic properties, such as being brown-haired. In contrast, the entities of a mathematical structure are purely abstract, which means that they have no intrinsic properties whatsoever. This means that whatever symbols we use to represent them are mere labels whose properties are irrelevant: to avoid the mistake of attributing properties of the symbols to the abstract entities that they symbolize, let’s consider the more spartan description in the middle panel. This description is equivalent to the first one, because if you apply the correspondence given by the dictionary Philip = 1, Alexander = 2, ski = 3, skate = 4 and likes to = R, all relations are preserved. For example, “Alexander likes to skate” translates to “2 R 4,” which is indeed a relation that holds in the middle panel.

  Just as chess games can be described using symbols alone, without any graphics, so can mathematical structures. For example, the right panel of Figure 10.7 gives a third equivalent description of our mathematical structure in terms of a four-by-four table of numbers. In this table, an entry of 1 means that the relation (likes to) holds between the element corresponding to that row and the element corresponding to that column, so the fact that there’s a 1 in the third column of the first row means that “Philip likes to ski.” There are clearly many more equivalent ways of describing this mathematical structure, but there’s only one unique mathematical structure that’s described by all these equivalent descriptions. In summary, any particular description of a mathematical structure contains baggage, but the structure itself doesn’t. It is important not to confuse the description with that which is described: even the most abstract-looking description of a mathematical structure is still not the structure itself. Rather, the structure corresponds to the class of all equivalent descriptions of it. Table 10.2 summarizes the relations between these and other key concepts linked to the mathematical-universe idea.

  Figure 10.7: Three equivalent descriptions of the same mathematical structure, which mathematicians would call an “ordered graph with four elements.” Each description contains some arbitrary baggage, but the structure that they all describe is 100% baggage-free: its four entities have no properties whatsoever except the relations that hold between them, and the relation has no properties whatsoever except the information about which elements it relates.

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  Symmetry and Other Mathematical Properties

  Some mathematicians enjoy debating what mathematics really is, and there’s certainly no consensus. However, a quite popular definition of mathematics is “the formal study of mathematical structures.” In this vein, mathematicians have identified large numbers of interesting mathematical structures, ranging from familiar ones such as the cube, the icosahedron (Figure 7.2), and the integers (the whole numbers) to ones with exotic names like Banach spaces, orbifolds and pseudo-Riemannian manifolds.

  One of the most important things that mathematicians do when studying mathematical structures is prove theorems about their properties. But what properties can a mathematical structure have if its entities and relations aren’t allowed to have any intrinsic properties at all?

  Figure 10.8: The center panel describes a mathematical structure with eight elements (symbolized by dots) and relations between them (symbolized by lines). You can interpret these elements as the corners of a cube and the relation as specifying which corners are connected by an edge, but this interpretation is completely optional baggage—the right panel gives an equivalent description of this same mathematical structure without any graphics or geometry—for example, the fact that there’s a 1 in the fifth column of the sixth row means that a relation holds between elements 5 and 6. This mathematical structure has many interesting properties, including both mirror symmetry and certain rotation symmetries. In contrast, the mathematical structure described in the left panel has no relations and no interesting properties at all except its cardinality of 8, the number of elements it contains.

  Consider the mathematical structure
described by the left panel of Figure 10.8. It has no relations at all between its entities, so there’s nothing to distinguish any one entity from any other. This means that this mathematical structure has no properties whatsoever except its cardinality, the number of entities that it has. Mathematicians call this mathematical structure “the set of 8 elements,” and its only property is having eight elements—a pretty boring structure!

  The middle panel of Figure 10.8 describes a different and more interesting mathematical structure with eight elements, which includes a relation. One description of this structure is that the elements are the corners of a cube and the relation specifies which corners are connected by an edge. However, remember not to confuse the description with that which is described: the mathematical structure itself has no intrinsic properties such as size, color, texture or composition—it has only these eight related entities which you can optionally interpret as cube corners. Indeed, the right panel of Figure 10.8 gives an equivalent definition of this mathematical structure without making any reference to geometric notions such as cubes, corners or edges.