by Max Tegmark
Because of our education system, many people equate mathematics with arithmetic. Yet like physics, mathematics has evolved to ask broader questions. For example, when I quoted Galileo above, he spoke of geometric figures such as circles and triangles being mathematical, so when you look around you, do you see any geometric patterns or shapes? Here again, human-made designs like the rectangular shape of this book don’t count. But try throwing a pebble and watch the beautiful shape that nature makes for its trajectory! Galileo made a remarkable discovery illustrated in Figure 10.2: the trajectories of anything you throw have the same shape, called an upside-down parabola. Moreover, the shape of a parabola can be described by a very simple mathematical equation: y = x2, where x is the horizontal position and y is the vertical position (the height). Depending on the initial speed and direction, the shape can be stretched both vertically and horizontally, but it always remains a parabola.
When we observe how things move around in orbits in space, we discover another recurring shape, as illustrated in Figure 10.3: the ellipse. The equation x2 + y2 = 1 describes the points on a circle, and an ellipse is simply a stretched circle. Depending on the initial speed and direction of the orbiting object and the mass of the thing it’s orbiting around, the shape of the orbit can be both stretched and tilted, but it always remains an ellipse. Moreover, these two shapes are related: the tip of a very elongated ellipse is shaped almost exactly like a parabola, so in fact, all of these trajectories are simply parts of ellipses.1
Figure 10.2: When you throw something up into the air, its trajectory always has the same shape, called an upside-down parabola, if it doesn’t collide with anything and air resistance is unimportant.
Figure 10.3: When something is orbiting something else due to gravity, its orbit always has the same shape, called an ellipse, which is simply a circle that’s stretched in one direction (that’s if there’s no source of friction, and you ignore Einstein’s corrections to Newton’s gravity theory, which are usually tiny unless you’re near a black hole). The orbit is an ellipse even for dramatically different objects, say, for a comet orbiting the Sun (left), a white dwarf stellar corpse orbiting Sirius A, the brightest star in our night sky (center), and a star orbiting the monster black hole at the center of our Galaxy (right), which is four million times more massive than our Sun. (Right panel courtesy of Reinhard Genzel and Rainer Schödel)
Click here to see a larger image.
We humans have gradually discovered many additional recurring shapes and patterns in nature, involving not only motion and gravity, but also areas as disparate as electricity, magnetism, light, heat, chemistry, radioactivity and subatomic particles. These patterns are summarized by what we call our laws of physics. Just as the shape of an ellipse, all these laws can be described using mathematical equations, as illustrated in Figure 10.4. Why is that?
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1Indeed, if you prevented the basketball in Figure 10.2 from hitting the ground by compressing our entire planet into a black hole at the center, then the parabolic part of the ball’s trajectory would remain the same, and would extend into a full ellipse around the black hole.
Numbers
Equations aren’t the only hints of mathematics that are built into nature: there are also numbers. As opposed to human creations such as the page numbers in this book, I’m now talking about numbers that are basic properties of our physical reality. For example, how many pencils can you arrange so that they’re all perpendicular (at 90 degrees) to each other? The answer is 3—for instance, by placing them along the 3 edges emanating from a corner of your room. Where did that number 3 come sailing in from? We call this number the dimensionality of our space, but why are there 3 dimensions rather than 4 or 2 or 42? And why are there, as far as we can tell, exactly 6 kinds of quarks in our Universe? As we saw in Chapter 7, there are many additional whole numbers (so-called integers) built into nature that describe what types of elementary particles exist.
Figure 10.4: Just as art and poetry can capture a lot in just a few symbols, so can the equations of physics. From left to right, top to bottom, these masterpieces describe electromagnetism, near–light speed motion, gravity, quantum mechanics and our expanding Universe. We still haven’t found equations for a unified theory of everything.
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As if that weren’t enough mathematical goodies, there are also quantities encoded in nature that aren’t whole numbers, but require decimals to write out. Nature encodes 32 such fundamental numbers according to my latest count. Does the number shown when you stand on your bathroom scale count as such a number? No, that number doesn’t count, because it’s measuring something (your mass) that changes from day to day and therefore isn’t a basic property of our Universe. What about the mass of a proton, 1.672622 × 10−27kg, or the mass of an electron, 9.109382 × 10−31kg, which seem to stay perfectly constant over time? They don’t count either, because they’re measuring the number of kilograms, and that’s just a rather arbitrary unit of mass that we humans have made up. But if you divide one of these last two numbers by the other, then you get something truly fundamental: the proton is about 1836.15267 more massive than the electron.1 1836.15267 is a pure number, just as π or , in the sense that it’s a quantity that doesn’t involve any human units of measurement such as grams, meters, seconds or volts. Why is it close to 1836? Why not 2013? Why not 42? The short answer is that we don’t know, but that we think we can in principle calculate this number and every other fundamental constant of nature ever measured from just the 32 numbers listed in Table 10.1.
Don’t worry about the intimidating-sounding technical names of the numbers in this table, which are irrelevant for what we’re getting at here. The point is that there’s something very mathematical about our Universe, and that the more carefully we look, the more math we seem to find. Apropos constants of nature, there are hundreds of thousands of pure numbers that have been measured across all areas of physics, ranging from ratios of masses of elementary particles to ratios of characteristic wavelengths of light emitted by different molecules, and using sufficiently powerful computers to solve the equations describing the laws of nature, it seems that every single one of these numbers can be computed from the 32 in Table 10.1. Some of the computations and some of the measurements are really difficult and haven’t been done yet, and perhaps when that happens, some of the decimals won’t match between theory and experiment. That sort of discrepancy has happened repeatedly in the past, and has typically been resolved in one of three ways:
1. Someone discovered a mistake in the experiment.
2. Someone discovered a mistake in the calculation.
3. Someone discovered a mistake in our laws of physics.
Table 10.1: Every fundamental property of nature ever measured can be computed from the 32 numbers in this table—at least in principle. Some of these numbers have been measured very accurately, while others haven’t yet been experimentally determined. The detailed meaning of these numbers doesn’t matter for our discussion, but if you’re interested, you’ll find it explained in my paper at http://arxiv.org/abs/astro-ph/0511774. But what determines these numbers?
Click here to see a larger image.
In the third case, a more fundamental law of physics was usually found, as when Newton’s equations for gravity were replaced by Einstein’s, explaining why the motion of Mercury around the Sun isn’t quite a perfect ellipse. In all cases, the feeling that there’s something mathematical about nature was further strengthened.
If you discover an even more accurate law of physics in the future, it might decrease the number of parameters from the 32 in Table 10.1 by allowing you to compute some of the numbers from others in the table, or it might increase them by adding new ones, say, involving the masses of new kinds of particles that might be discovered by the Large Hadron Collider outside Geneva.
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1If you wonder why the ratio can be measured more accurately than the two masses se
parately, the reason is that the two measurement errors are very strongly related (correlated).
More Clues
So what do we make of all these hints of mathematics in our physical world? Most of my physics colleagues take them to mean that nature is for some reason described by mathematics, at least approximately, and leave it at that. In his book Is God a Mathematician?, the astrophysicist Mario Livio concludes that “scientists have selected what problems to work on based on those problems being amenable to a mathematical treatment.” But I’m convinced that there’s more to it than that.
First of all, why does math describe nature so well? I agree with Wigner that it demands an explanation. Second, throughout this book, we’ve come across clues suggesting that nature isn’t just described by mathematics, but that some aspects of it are mathematical:
1. In Chapters 2–4, we saw that the very fabric of our physical world, space itself, is a purely mathematical object in the sense that its only intrinsic properties are mathematical properties—numbers such as dimensionality, curvature and topology.
2. In Chapter 7, we saw that all the “stuff” in our physical world is made of elementary particles, which in turn are purely mathematical objects in the sense that their only intrinsic properties are mathematical properties—numbers listed in Table 7.1 such as charge, spin and lepton number.
3. In Chapter 8, we saw that there’s something that’s arguably even more fundamental than our three-dimensional space and the particles within it: the wavefunction and the infinite-dimensional place called Hilbert space where it lives. Whereas particles can be created and destroyed, and can be in several places at once, there is, was and always will be only one wavefunction, moving through Hilbert space as determined by the Schrödinger equation—and the wavefunction and Hilbert space are purely mathematical objects.
What does this all mean? Now let me share with you what I think it means, and let’s see if it makes more sense to you than to that professor who said it would ruin my career.
The Mathematical Universe Hypothesis
I was quite fascinated by all these mathematical clues back in grad school. One Berkeley evening in 1990, while my friend Bill Poirier and I were sitting around speculating about the ultimate nature of reality, I suddenly had an idea for what it all meant: that our reality isn’t just described by mathematics—it is mathematics, in a very specific sense that I’ll describe below. Not just aspects of it, but all of it, including you.1 This idea sounds rather crazy and far-fetched, so after telling Bill about it, I mulled it over for many years before writing that first paper about it.
Before delving into the details, here’s my logical framework for thinking about this business. First there are two hypotheses, one seemingly innocuous and one seemingly radical:
Second, I have an argument that, with a sufficiently broad definition of mathematical structure, the former implies the latter.
My starting assumption, the External Reality Hypothesis, isn’t too controversial: I’d guess that the majority of physicists favor this long-standing idea, though it’s still debated. Metaphysical solipsists reject it flat out, and supporters of the Copenhagen interpretation of quantum mechanics may reject it on the grounds that there’s no reality without observation. Assuming that an external reality exists, physics theories aim to describe how it works. Our most successful theories, such as general relativity and quantum mechanics, describe only parts of this reality: gravity, for instance, or the behavior of subatomic particles. In contrast, the Holy Grail of theoretical physics is a theory of everything—a complete description of reality.
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1Roger Penrose expresses similar sentiments in his book The Road to Reality.
Reducing the Baggage Allowance
My personal quest for this theory begins with an extreme argument about what it’s allowed to look like: If we assume that reality exists independently of humans, then for a description to be complete, it must also be well defined according to nonhuman entities—aliens or supercomputers, say—that lack any understanding of human concepts. Put differently, such a description must be expressible in a form that’s devoid of any human baggage like “particle,” observation or other English words.
In contrast, all physics theories that I’ve been taught have two components: mathematical equations and “baggage” —words that explain how the equations are connected to what we observe and intuitively understand. When we derive the consequences of a theory, we introduce new concepts and words for them, such as protons, atoms, molecules, cells and stars, because they’re convenient. It’s important to remember, however, that it’s we humans who create these concepts; in principle, everything could be calculated without this baggage. A hypothetical ideal supercomputer could calculate how the state of our Universe changes over time without interpreting what’s happening in human terms, simply figuring out how all the particles would move or how the wavefunction would change.
For example, suppose the basketball trajectory in Figure 10.2 is that of a beautiful buzzer beater that wins you the game, and that you later want to describe what it looked like to a friend. Since the ball is made of elementary particles (quarks and electrons), you could in principle describe its motion without making any reference to basketballs:
• Particle 1 moves in a parabola.
• Particle 2 moves in a parabola.
• …
• Particle 138,314,159,265,358,979,323,846,264 moves in a parabola.
That would be slightly inconvenient, however, because it would take you longer than the age of our Universe to say it. It would also be redundant, since all the particles are stuck together and move as a single unit. That’s why we humans have invented a word ball to refer to the entire unit, enabling us to save time by simply describing the motion of the whole unit once and for all.
The ball was designed by humans, but it’s quite analogous for composite objects that aren’t man-made, such as molecules, rocks and stars: inventing words for them is convenient both for saving time, and for providing concepts or so-called shorthand abstractions in terms to understand the world more intuitively. Although useful, such words are all optional baggage: for example, I’ve used the word star repeatedly in this book, but you could in principle replace every occurrence of it by a definition in terms of its building blocks, say, gravitationally bound clump of about 1057 atoms, some of which are undergoing nuclear fusion. In other words, nature contains many types of entities that are almost begging to be named. Sure enough, virtually every human population on Earth has a word for star in its own language, often invented independently to reflect its own cultural and linguistic tradition. I suspect that most alien civilizations in distant solar systems have also invented a name or symbol for star even if they don’t communicate using sounds.
Another striking fact is that you can often predict the existence of such name-worthy entities mathematically, from the equations governing their parts. In this way, the whole Lego-like hierarchy of structures that we discussed in Chapter 7 can be predicted, from elementary particles to atoms to molecules, and what we humans add are merely catchy names for the objects at each level. For example, if you solve the Schrödinger equation for five or fewer quarks, it turns out that there are only two fairly stable ways for them to be arranged: either as a clump of two up quarks and a down quark or as a clump of two down quarks and an up quark, and we humans have added the baggage of calling these two types of clumps “protons” and “neutrons” for convenience. Similarly, if you apply the Schrödinger equation to such clumps, it turns out that there are only 257 stable ways for them to be assembled together. We humans have added the baggage of calling these proton/neutron assemblies “atomic nuclei,” and have also invented specific names for each kind: hydrogen, helium, etc. The Schrödinger equation also lets you calculate all the ways of putting atoms together into larger objects, but this time, there turn out to be so many different stable objects that it’s inconvenient to name them all—instead, we’v
e just named important classes of objects (such as “molecules” and “crystals”), and the most common or interesting objects in each class (e.g., “water,” “graphite,” “diamond”).
I think of these composite objects as emergent, in the sense that they emerge as solutions of equations involving only more fundamental objects. This emergence is subtle and easy to miss because historically, the scientific process has mostly gone in the opposite direction: for example, we humans knew of stars before realizing that they were made of atoms, we knew of atoms before realizing that they were made of electrons, protons and neutrons, and we knew of neutrons before we discovered quarks. For every emergent object that’s important to us humans, we create baggage in the form of new concepts.
The same pattern of emergence and human baggage creation can be seen in Figure 10.5. Here I’ve crudely organized scientific theories into a family tree where each might, at least in principle, be derivable from more fundamental ones above it. As mentioned, all these theories have two components: mathematical equations and words that explain how they’re connected to what we observe. For example, we saw in Chapter 8 how quantum mechanics, as usually presented in textbooks, has both components: math such as the Schrödinger equation as well as fundamental postulates written out in plain English, such as the wavefunction-collapse postulate. At each level in the hierarchy of theories, new concepts (e.g., protons, atoms, cells, organisms, cultures) are introduced because they’re convenient, capturing the essence of what is going on without recourse to the more fundamental theory above it. It’s we humans who introduce these concepts and the words for them: in principle, everything could have been derived from the fundamental theory at the top of the tree, although such an extreme reductionist approach is often useless in practice. Crudely speaking, as we move down the tree, the number of words goes up while the number of equations goes down, dropping to near zero for highly applied fields such as medicine and sociology. In contrast, theories near the top are highly mathematical, and physicists are still struggling to understand the concepts, if any, in terms of which we can understand them.
