Our Mathematical Universe

1 What Is Reality?

… trees are made of air, primarily. When they are burned, they go back to air, and in the flaming heat is released the flaming heat of the Sun which was bound in to convert the air into tree. And in the ash is the small remnant of the part which did not come from air, that came from the solid earth, instead.

—Richard Feynman

There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.

—William Shakespeare, Hamlet, act 1, scene 5

Not What It Seems

A second later, I died. I stopped pedaling and hit the brakes, but it was too late. Headlights. Grille. Forty tons of steel, honking furiously, like a modern-day dragon. I saw the panic in the truck driver’s eyes. I felt time slow down and my life pass before me, and my very last thought in life was “I hope this is just a nightmare.” Alas, I felt in my gut that this was for real.

But how could I be completely sure that I wasn’t dreaming? What if, just before impact, I’d perceived something that didn’t happen except in dreamland, say my dead teacher Ingrid alive and well, sitting behind me on my bike rack? Or what if, five seconds earlier, a pop-up window had appeared in the upper left corner of my visual field, with the text line “Are you sure it’s a good idea to zoom out of this underpass without looking to the right?” materializing above clickable Continue and Cancel buttons? Had I watched enough movies like The Matrix and The Thirteenth Floor, I might have started wondering whether my whole life had been a computer simulation, questioning some of my most basic assumptions about the nature of reality. But I experienced no such things, and died certain that my problem was real. After all, how much more solid and real than a forty-ton truck can something get?

However, not everything is the way it first seems, and this goes even for trucks and reality itself. Such suggestions come not only from philosophers and science-fiction authors, but also from physics experiments. Physicists have known for a century that solid steel is really mostly empty space, because the atomic nuclei that make up 99.95% of the mass are tiny balls that fill up merely 0.0000000000001% of the volume, and that this near-vacuum only feels solid because the electrical forces that hold these nuclei in place are very strong. Moreover, careful measurements of subatomic particles have revealed that they appear able to be in different places at the same time, a well-known puzzle at the heart of quantum physics (we’ll explore this in Chapter 7). But I’m made of such particles, so if they can be in two places at once, then can’t I as well? Indeed, about three seconds before the accident, I was subconsciously deciding whether to simply look to the left where I’d always turned on my way to Blackebergs Gymnasium, my Swedish high school, since there was never any traffic on this cross street, or whether to also look to the right just in case. My ill-fated snap decision that morning in 1985 ended up being a very close call. It all came down to whether a single calcium atom would enter a particular synaptic junction in my prefrontal cortex, causing a particular neuron to fire an electrical signal that would trigger a whole cascade of activity by other neurons in my brain, which collectively encoded “Don’t bother.” So if that calcium atom started in two slightly different places at once, then half a second later, my pupils would have been pointing in two opposite directions at once, two seconds later my bike would have been in two different places at once, and before long, I’d have been dead and alive at once. The world’s leading quantum physicists argue passionately about whether this really happens, effectively splitting our world into parallel universes with different histories, or whether the so-called Schrödinger equation, the supreme law of quantum motion, needs to be somehow amended. So did I really die? I just barely made it in this particular universe, but did I die in another equally real universe where this book never got written? If I’m both dead and alive, then can we somehow amend our notion of what reality is so that it all makes sense?

If you feel that what I’ve just put forth sounds absurd and that physics has muddied the waters, it gets even worse if we consider how I personally perceived this. If I’m in these two different places in two parallel universes, then one version of me will survive. If you apply the same argument to all other ways I can die in the future, it seems there will always be at least one parallel universe where I never die. Since my consciousness exists only where I’m alive, does that mean that I’ll subjectively feel immortal? If so, will you, too, find yourself subjectively immortal, eventually the oldest person on Earth? We’ll answer these questions in Chapter 8.

Are you surprised that physics has uncovered our reality to be much stranger than we’d imagined? Well, it’s actually not surprising if we take Darwinian evolution seriously! Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic orbits of flying rocks (explaining our penchant for baseball). A cavewoman thinking too hard about what matter is ultimately made of might fail to notice the tiger sneaking up behind her and get cleaned right out of the gene pool. Darwin’s theory thus makes the testable prediction that whenever we use technology to glimpse reality beyond the human scale, our evolved intuition should break down. We’ve repeatedly tested this prediction, and the results overwhelmingly support Darwin. At high speeds, Einstein realized that time slows down, and curmudgeons on the Swedish Nobel committee found this so weird that they refused to give him the Nobel Prize for his relativity theory. At low temperatures, liquid helium can flow upward. At high temperatures, colliding particles change identity; to me, an electron colliding with a positron and turning into a Z-boson feels about as intuitive as two colliding cars turning into a cruise ship. On microscopic scales, particles schizophrenically appear in two places at once, leading to the quantum conundrums mentioned above. On astronomically large scales—surprise!—weirdness strikes again: if you intuitively understand all aspects of black holes, I think you’re in a minority of one, and should immediately put down this book and publish your findings before someone scoops you on the Nobel Prize for quantum gravity. Zoom out to still larger scales, and more weirdness awaits, with a reality vastly grander than everything we can see with our best telescopes. As we’ll explore in Chapter 5, the leading theory for what happened early on is called cosmological inflation, and it suggests that space isn’t merely really, really big, but actually infinite, containing infinitely many exact copies of you, and even more near-copies living out every possible variant of your life in two different types of parallel universes. If this theory proves to be true, it means that even if there’s something wrong with the quantum physics argument I gave above for a copy of me never making it to school, there will be infinitely many other Maxes on solar systems far out there in space who lived identical lives up until that fateful moment, and then decided not to look to the right.

In other words, discoveries in physics challenge some of our most basic ideas about reality both when we zoom into the microcosm and when we zoom out to the macrocosm. As we’ll explore in Chapter 11, many ideas about reality get challenged even on the intermediate scale of us humans if we use neuroscience to delve into how our brains work.

Last but not least, we know that mathematical equations offer a window into the workings of nature, as metaphorically illustrated in Figure 1.1. But why has our physical world revealed such extreme mathematical regularity that astronomy superhero Galileo Galilei proclaimed nature to be “a book written in the language of mathematics,” and Nobel Laureate Eugene Wigner stressed the “unreasonable effectiveness of mathematics in the physical sciences” as a mystery demanding an explanation? Answering this question is the main goal of this book, as its title suggests. In Chapters 10–12, we’ll explore the fascinating relations between computation, mathematics, physics and mind, and explore a crazy-sounding belief of mine that our physical world not only is described by mathematics, but that it is mathematics, making us self-aware parts of a giant mathematical object. We’ll see that this leads to a new and ultimate collection of parallel universes so vast and exotic that all the above-mentioned bizarreness pales in comparison, forcing us to relinquish many of our most deeply ingrained notions of reality.

Figure 1.1: When we look at reality through the equations of physics, we find that they describe patterns and regularities. But to me, mathematics is more than a window on the outside world: in this book, I’m going to argue that our physical world not only is described by mathematics, but that it is mathematics: a mathematical structure, to be precise.

What’s the Ultimate Question?

For as long as our human ancestors have walked the Earth, they’ve undoubtedly wondered what reality is all about, pondering deep existential questions. Where did everything come from? How will it all end? How big is it all? These questions are so captivating that virtually all human cultures across the globe have grappled with them and passed on answers from generation to generation, in the form of elaborate creation myths, legends and religious doctrines. As Figure 1.2 illustrates, these questions are so difficult that no global consensus has emerged on the answers. Instead of all cultures converging on a unique worldview that could potentially be the ultimate truth, their answers have differed greatly, and at least some of these differences appear to reflect their differences in lifestyle. For example, creation myths from ancient Egypt, where the Nile River kept the land fertile, all have our world emerge from water. In my native Sweden, on the other hand, where fire and ice used to strongly affect survival, Norse mythology held that life originated from (surprise!) fire and ice.

Figure 1.2: Many cosmological questions that we’ll tackle in this book have fascinated thinkers throughout the ages, but no global consensus has emerged. The classification above is based on a 2011 presentation by MIT grad student David Hernandez for my cosmology class. Because such simplistic taxonomies are strictly impossible, they should be taken with a large grain of salt: many religions have multiple branches and interpretations, and some fall into multiple categories. For example, Hinduism contains aspects of all three creation options shown: according to one legend, both the creation god Brahma (depicted) and our Universe emerged from an egg, which in turn may have originated from water.

Other big questions tackled by ancient cultures are at least as radical. What is real? Is there more to reality than meets the eye? Yes! was Plato’s answer over two millennia ago. In his famous cave analogy, he likened us to people who’d lived their entire lives shackled in a cave, facing a blank wall, watching the shadows cast by things passing behind them, and eventually coming to mistakenly believe that these shadows were the full reality. Plato argued that what we humans call our everyday reality is similarly just a limited and distorted representation of the true reality, and that we must free ourselves from our mental shackles to begin comprehending it.

If my life as a physicist has taught me anything at all, it’s that Plato was right: modern physics has made abundantly clear that the ultimate nature of reality isn’t what it seems. But if reality isn’t what we thought it was, then what is it? What’s the relation between the internal reality of our mind and the external reality? What’s everything ultimately made of? How does it all work? Why? Is there a meaning to it all, and if so, what? As Douglas Adams put it in his sci-fi spoof The Hitchhiker’s Guide to the Galaxy: “What’s the answer to the ultimate question of life, the universe, and everything?”

Thinkers throughout the ages have offered a fascinating spectrum of responses to the question “What is reality?”—either attempting to answer it or attempting to dismiss it. Here are some examples (this list makes no claims to be complete, and not all alternatives are mutually exclusive).

This book (and indeed my scientific career) is my personal attempt to tackle this question. Part of the reason that thinkers have offered such a broad spectrum of answers is clearly that they’ve chosen to interpret the question in different ways, so I owe you an explanation of how I interpret it and how I approach it. The word reality can have many different connotations. I use it to mean the ultimate nature of the outside physical world that we’re part of, and I’m fascinated by the quest to understand it better. So what’s my approach?

One evening back in high school, I started reading Agatha Christie’s detective novel Death on the Nile. Although I was painfully aware that my alarm clock would go off at seven a.m., I couldn’t for the life of me put it down until the mystery had been solved, around four a.m. I’ve been inexorably drawn to detective stories ever since I was a kid, and when I was around twelve, I started a detective club with my classmates Andreas Bette, Matthias Bothner and Ola Hansson. We never captured any criminals, but the idea of solving mysteries captured my imagination. To me, the question “What is reality?” represents the ultimate detective story, and I consider myself incredibly fortunate to be able to spend so much of my time pursuing it. In the chapters ahead, I’ll tell you about other occasions when my curiosity has kept me up in the wee hours of the morning, totally unable to stop reading until the mystery was resolved. Except that I wasn’t reading a book, but what my hand was writing, and what I was writing was a trail of mathematical equations that I knew would ultimately lead me to an answer.

Some Responses to “What Is Reality?”
The question has a meaningful answer.	Elementary particles in motion
	Earth, wind, fire, air and quintessence
	Atoms in motion
	Elementary particles in motion
	Strings in motion
	Quantum fields in curved spacetime
	M-theory (substitute your favorite capital letter …)
	A divine creation
	A social construct
	A neurophysiological construct
	A dream
	Information
	A simulation (à la The Matrix)
	A mathematical structure
	The Level IV Multiverse
The question lacks a meaningful answer.	There is a reality, but we humans can’t fully know it: we have no access to what Immanuel Kant called “das Ding an sich.”
	Reality is fundamentally unknowable.
	Not only don’t we know it, but we couldn’t express it if we did. Science is nothing but a story (postmodern answer by Jacques Derrida and others).
	Reality is all in our head (constructivist answer).
	Reality doesn’t exist (solipsism).

I’m a physicist, and I’m taking a physics approach to the mysteries of reality. To me, this means starting with great questions such as “How big is our Universe?” and “What’s everything made of ?” and treating them exactly like detective mysteries: combining clever observations and reasoning and persistently following each clue wherever it leads.

The Journey Begins

A physics approach? Isn’t that a great way to turn something exciting into something boring? When the person sitting next to me on a plane asks what I do, I have two options. If I feel like talking, I’ll say, “Astronomy,” which infallibly triggers an interesting conversation.* If I don’t, I’ll say, “Physics,” at which point they typically say something like “Oh, that was my worst subject in high school,” and leave me alone for the rest of the flight.

Indeed, physics was also my least-favorite subject in high school. I still remember my very first physics class. With a monotonous and sedative voice, our teacher announced that we were going to learn about density. That density was mass divided by volume. So if the mass was blah and the volume was blah, then we could calculate that the density was blah blah. After that point, all I remember is a big blur. And that whenever his experiments failed, he’d blame humidity and say, “It worked this morning.” And that some friends of mine couldn’t figure out why their experiment wasn’t working until they discovered that I’d mischievously attached a magnet underneath their oscilloscope. …

When the time came to apply for college, I decided against physics and other technical fields, and ended up at the Stockholm School of Economics, focusing on environmental issues. I wanted to do my small part to make our planet a better place, and felt that the main problem wasn’t that we lacked technical solutions, but that we didn’t properly use the technology we had. I figured that the best way to affect people’s behavior was through their wallets, and was intrigued by the idea of creating economic incentives that aligned individual egoism with the common good. Alas, I soon grew disillusioned, concluding that economics was largely a form of intellectual prostitution where you got rewarded for saying what the powers that be wanted to hear. Whatever a politician wanted to do, he or she could find an economist as advisor who had argued for doing precisely that. Franklin D. Roosevelt wanted to increase government spending, so he listened to John Maynard Keynes, whereas Ronald Reagan wanted to decrease government spending, so he listened to Milton Friedman.

Then my classmate Johan Oldhoff gave me the book that changed everything: Surely You’re Joking, Mr. Feynman! I never got to meet Richard Feynman, but he’s the reason I switched to physics. Although the book wasn’t really about physics, dwelling more on topics such as how to pick locks and how to pick up women, I could read between the lines that this guy just loved physics. Which really intrigued me. If you see a mediocre-looking guy walking arm in arm with a gorgeous woman, you probably wonder if you’re missing something. Presumably, she’s seen some hidden quality in him. Suddenly I felt the same way about physics: what did Feynman see that I’d missed in high school?

I just had to solve this mystery, so I sat down with volume 1 of The Feynman Lectures on Physics, which I found in Dad’s bookcase, and started reading: “If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words?”

Whoa—this guy was nothing like my high-school physics teacher! Feynman continued: “I believe it is that [ … ] all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart but repelling upon being squeezed into one another.”

A lightbulb went off in my head. I read on and on and on, spellbound. I felt like I was having a religious experience. I finally got it! I had the epiphany that explained what I’d been missing all along, and what Feynman had realized: physics is the ultimate intellectual adventure, the quest to understand the deepest mysteries of our Universe. Physics doesn’t take something fascinating and make it boring. Rather, it helps us see more clearly, adding to the beauty and wonder of the world around us. When I bike to work in the fall, I see beauty in the trees tinged with red, orange and gold. But seeing these trees through the lens of physics reveals even more beauty, captured by the Feynman quote that opens this chapter. And the deeper I look, the more elegance I glimpse: we’ll see in Chapter 3 how the trees ultimately come from stars, and we’ll see in Chapter 8 how studying their building blocks suggests their existence in parallel universes.

At this time, I had a girlfriend studying physics at the Royal Institute of Technology, and her textbooks seemed so much more interesting than mine. Our relationship didn’t last, but my love for physics did. Since college was free in Sweden, I enrolled in her university without telling the Stockholm School of Economics administrators about my secret double life. My detective investigation had officially begun, and this book is my report a quarter of a century later.

So what is reality, then? My goal with this audaciously titled chapter isn’t to arrogantly try to sell you on an ultimate answer (although we’ll explore intriguing possibilities in the last part of the book), but rather to invite you along on my personal journey of exploration, and to share with you my excitement and reflections about these mind-expanding mysteries. Like me, I think you’ll conclude that whatever reality is, it’s wildly different from what we once thought, and a fascinating enigma at the very heart of our everyday lives. I hope you will, like me, find that this places everyday problems such as parking tickets and heartaches in a refreshing perspective, making it easier to take them in stride and focus on enjoying life and its mysteries to the fullest.

When I first discussed my ideas for this book with John Brockman, now my book agent, he gave me clear marching orders: “I don’t want a textbook—I want your book.” So this book is a scientific autobiography of sorts: although it’s more about physics than it’s about me, it’s certainly not your standard popular science book that attempts to survey physics in an objective way, reflecting the community consensus and giving equal space to all opposing viewpoints. Rather, it’s my personal quest for the ultimate nature of reality, which I hope you’ll enjoy seeing through my eyes. Together, we’ll explore the clues that I personally find the most fascinating, and try to figure out what it all means.

We’ll begin our journey by surveying how the whole context of the question “What is reality?” has been transformed by recent scientific breakthroughs, with physics shedding new light on our external reality from the largest (Chapters 2–6) to the smallest (Chapters 7–8) scales. In Part I of the book, we’ll pursue the question “How big is our Universe?” and seek its ultimate conclusion by traveling out to ever-larger cosmic scales, exploring both our cosmic origins and two types of parallel universes, finding hints that space is in a sense mathematical. In Part II of the book, we’ll relentlessly pursue the question “What’s everything made of ?” by journeying into the subatomic microcosm, examining a third kind of parallel universe and finding hints that the ultimate building blocks of matter are also in a sense mathematical. In Part III of the book, we’ll take a step back and consider what all this might mean for the ultimate nature of reality. We’ll begin by arguing that our failure to understand consciousness doesn’t stand in the way of a complete understanding of the external physical reality. We’ll then delve into my most radical and controversial idea: that the ultimate reality is purely mathematical, demoting familiar notions such as randomness, complexity, and even change to the status of illusions, and implying that there’s a fourth and ultimate level of parallel universes. We’ll wrap up our journey in Chapter 13 by returning home, exploring what this all means for the future prospects of life in our Universe, for us humans, and for you personally. You’ll find our travel planner in Figure 1.3 with my reading tips. A fascinating journey awaits us. Let’s begin!

Figure 1.3: How to read this book. If you’ve read lots of modern popular-science books and feel that you already understand curved space, our Big Bang, the cosmic microwave background, dark energy, quantum mechanics etc., then you may consider skipping Chapters 2, 3, 4 and 7 after reviewing the “Bottom Line” boxes that follow them, and if you’re a professional physicist, you might consider skipping Chapter 5 as well. But many concepts that may sound familiar are startlingly subtle, and if you can’t answer all of questions 1–16 in Chapter 2, I hope you’ll learn from the early material as well and see how the later chapters logically build on it.

THE BOTTOM LINE

I feel that the most important lesson physics has taught us about the ultimate nature of reality is that, whatever it is, it’s very different from how it seems.
In Part I of this book, we’ll zoom out and explore physical reality on the largest scales, from planets to stars, galaxies, superclusters, our Universe and two possible levels of parallel universes.
In Part II of the book, we’ll zoom in and explore physical reality on the smallest scales, from atoms to their even more fundamental building blocks, encountering a third level of parallel universes.
In Part III, we’ll take a step back and examine the ultimate nature of this strange physical reality, investigating the possibility that it’s ultimately purely mathematical, specifically a mathematical structure that’s part of a fourth and ultimate level of parallel universes.
Reality means very different things to different people. I use the word to mean the ultimate nature of the outside physical world that we’re part of, and ever since I was a kid, I’ve been inspired and fascinated by the quest to understand it better.
This book is about my personal journey to explore the nature of reality—please join me!

Max Tegmark