by Max Tegmark
THIS IS A BORZOI BOOK
PUBLISHED BY ALFRED A. KNOPF
Copyright © 2014 by Max Tegmark
All rights reserved. Published in the United States by Alfred A. Knopf, a division of Random House LLC, New York, and in Canada by Random House of Canada Limited, Toronto, Penguin Random House Companies.
www.aaknopf.com
Knopf, Borzoi Books, and the colophon are registered trademarks of Random House LLC.
Library of Congress Cataloging-in-Publication Data
Tegmark, Max, author.
Our mathematical universe : my quest for the ultimate nature of reality / Max Tegmark.—First edition.
pages cm
ISBN 978-0-307-59980-3 (hardback)
1. Cosmology—Mathematics. 2. Physics—Mathematics.
3. Plurality of worlds. I. Title.
QB981.T44 2013 523.10151—dc23 2013016020
eBook ISBN: 978-0-385-35049-5
Jacket image: sphere representing the edge of our observable Universe, seen 400,000 years after our Big Bang. Courtesy of the author and the Planck satellite team
Jacket design by Jason Booher
v3.1
TO MEIA,
WHO INSPIRED ME TO WRITE THIS BOOK
Contents
Cover
Title Page
Copyright
Dedication
Preface
1 What Is Reality?
Not What It Seems • What’s the Ultimate Question? • The Journey Begins
Part One: Zooming Out
2 Our Place in Space
Cosmic Questions • How Big Is Space? • The Size of Earth • Distance to the Moon • Distance to the Sun and the Planets • Distance to the Stars • Distance to the Galaxies • What Is Space?
3 Our Place in Time
Where Did Our Solar System Come From? • Where Did the Galaxies Come From? • Where Did the Mysterious Microwaves Come From? • Where Did the Atoms Come From?
4 Our Universe by Numbers
Wanted: Precision Cosmology • Precision Microwave-Background Fluctuations • Precision Galaxy Clustering • The Ultimate Map of Our Universe • Where Did Our Big Bang Come From?
5 Our Cosmic Origins
What’s Wrong with Our Big Bang? • How Inflation Works • The Gift That Keeps on Giving • Eternal Inflation
6 Welcome to the Multiverse
The Level I Multiverse • The Level II Multiverse • Multiverse Halftime Roundup
Part Two: Zooming In
7 Cosmic Legos
Atomic Legos • Nuclear Legos • Particle-Physics Legos • Mathematical Legos • Photon Legos • Above the Law? • Quanta and Rainbows • Making Waves • Quantum Weirdness • The Collapse of Consensus • The Weirdness Can’t Be Confined • Quantum Confusion
8 The Level III Multiverse
The Level III Multiverse • The Illusion of Randomness • Quantum Censorship • The Joys of Getting Scooped • Why Your Brain Isn’t a Quantum Computer • Subject, Object and Environment • Quantum Suicide • Quantum Immortality? • Multiverses Unified • Shifting Views: Many Worlds or Many Words?
Part Three: Stepping Back
9 Internal Reality, External Reality and Consensus Reality
External Reality and Internal Reality • The Truth, the Whole Truth and Nothing but the Truth • Consensus Reality • Physics: Linking External to Consensus Reality
10 Physical Reality and Mathematical Reality
Math, Math Everywhere! • The Mathematical Universe Hypothesis • What Is a Mathematical Structure?
11 Is Time an Illusion?
How Can Physical Reality Be Mathematical? • What Are You? • Where Are You? (And What Do You Perceive?) • When Are You?
12 The Level IV Multiverse
Why I Believe in the Level IV Multiverse • Exploring the Level IV Multiverse: What’s Out There? • Implications of the Level IV Multiverse • Are We Living in a Simulation? • Relation Between the MUH, the Level IV Multiverse and Other Hypotheses • Testing the Level IV Multiverse
13 Life, Our Universe and Everything
How Big Is Our Physical Reality? • The Future of Physics • The Future of Our Universe—How Will It End? • The Future of Life • The Future of You—Are You Insignificant?
Acknowledgments
Suggestions for Further Reading
Index
A Note About the Author
Preface
I’m truly grateful to everyone who has encouraged and helped me write this book, including:
my family, friends, teachers, colleagues and collaborators for support and inspiration over the years,
Mom for sharing her passion and curiosity for life’s big questions,
Dad for sharing his fascination and wisdom about mathematics and its meaning,
my sons, Philip and Alexander, for asking such great questions about the world and for unwittingly providing anecdotes for the book,
all the science enthusiasts around the world who’ve contacted me over the years with questions, comments and encouragement to pursue and publish my ideas,
my agents, John and Max Brockman, for convincing me to write this book and setting everything in motion,
those who gave me feedback on parts of the manuscript, including Mom, my brother Per, Josh Dillon, Marty Asher, David Deutsch, Louis Helm, Andrei Linde, Jonathan Lindström, Roy Link, David Raub, Shevaun Mizrahi, Mary New, Sandra Simpson, Carl Shulman and Jaan Tallinn.
the superheroes who commented on drafts of the entire book, namely Meia, Dad, Paul Almond, Julian Barbour, Phillip Helbig, Adrian Liu, Howard Messing, Dan Roberts, Edward Witten and my editor, Dan Frank, and most of all,
my beloved wife, Meia, my muse and fellow traveler, who’s given me more encouragement, support and inspiration than I ever dreamed of.
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 s
cience-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 reli
gions 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.
Click here to see a larger image.
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?
