Our Mathematical Universe

by Max Tegmark

  Reality model Your brain’s model of the external reality; this is the internal reality that you perceive

  Bird perspective Your perspective on the external reality when studying the abstract mathematical equations that describe it

  Frog perspective Your subjective perspective of the physical world (your internal reality)

  Table 9.1: Key terms introduced in this chapter that we’ll use later on

  Physics: Linking External Reality to Consensus Reality

  We’ve seen above that the consensus reality is quite different from the internal reality, and that the challenge of linking the two is as hard as understanding consciousness. However, as we’ve seen in the earlier parts of this book, the consensus reality is also quite different from the external reality, making it crucial not to conflate the two. Indeed, in my opinion, the history of modern physics shows that in several of the greatest breakthroughs, the hardest part wasn’t doing the math, but understanding how these two realities are related.

  When Einstein discovered special relativity in 1905, many of the key equations had already been written down by Hendrik Lorentz and others. However, what required Einstein’s genius was figuring out the relation between the mathematics and the measurements. He realized that the lengths and durations appearing in the mathematical description of the external reality differ from those measured in the consensus reality, and that the difference depends on motion: if an airplane flies over a group of people, then in their consensus reality, it will be shorter than before it took off, and its onboard clocks will run slower.1

  When Einstein discovered general relativity a decade later, Bernhard Riemann and others had already developed key parts of the mathematical formalism. However, the crowning achievement was again so difficult that it required Einstein’s insight: understanding that curved space in the mathematical description of the external reality corresponded to gravitation in the consensus reality. To appreciate how hard this was, imagine that on the eve of his death, Isaac Newton was approached by a genie who granted him one last wish. After some contemplation, Newton made up his mind:

  “Please tell me what the state-of-the-art equations of gravity will be in three hundred years.”

  The genie scribbled down the complete equations of general relativity on a sheet of paper, and being a kind genie, it also explained how to translate them into the old-fashioned mathematical notation of the time. Would it be obvious to Newton how to interpret this as a generalization of his own theory?

  The difficulty of linking external reality to consensus reality reached a new record high with the discovery of quantum mechanics, manifested in the fact that we physicists still argue about how to interpret the theory today, about a century after its inception. As we saw in Chapter 8, the external reality is described by a Hilbert space where a wavefunction changes deterministically over time, whereas the consensus reality is one where things happen seemingly at random, with probability distributions that can be computed to great accuracy from the wavefunction. It took over thirty years from the birth of quantum mechanics before Everett showed how these two realities could be reconciled, and the world had to wait another decade for the discovery of decoherence, which was crucial for reconciling the presence of macrosuperpositions in the external reality with their absence in the consensus reality.

  Today, the grand challenge of theoretical physics is unifying quantum mechanics with gravitation. Based on this historical progression of examples, I predict that the correct mathematical theory of quantum gravity will break all previous records in being difficult to interpret. Suppose that on the eve of the next quantum-gravity conference, our friend the genie broke into the lecture hall and scribbled the equations of the ultimate theory on a blackboard. Would any of the participants realize what was being erased the next morning? I doubt it!

  In summary, our quest to understand reality splits into two parts that can be tackled separately: the grand challenge for cognitive science is to link our consensus reality with our internal reality, and the grand challenge for physics is to link our consensus reality with our external reality. We’ve seen that, although the former challenge is daunting, so is the latter. Our consensus reality appears to have impenetrably solid and stationary objects, but all except a quadrillionth of the volume of a rock is empty space between particles in restless schizophrenic vibration. Our consensus reality feels like a three-dimensional stage where events unfold over time, but as we’ll explore in Chapter 11, Einstein’s work suggests that change is an illusion, time being merely the fourth dimension of an unchanging spacetime that just is, never created and never destroyed, containing our cosmic history as a DVD contains a movie. The quantum world feels random, but as we saw in the last chapter, Everett’s work suggests that randomness, too, is an illusion, being simply the way our minds feel when cloned into diverging parallel universes. The quantum-gravity world feels—well, here we physicists still have a looooooong way to go.

  In the remainder of this book, we’re going to focus on the physics quest, and push it to its logical extreme: given what we know about our consensus reality, what’s the external reality like? What’s its ultimate nature?

  THE BOTTOM LINE

  • I’ve argued that, although there’s only one true reality, there are several complementary perspectives on it.

  • In the internal reality of your mind, the only information you have about the external reality is the small sample transmitted through your senses.

  • This information is distorted in many ways, and arguably tells you as much about how your senses and your brain work as it tells you about the external reality.

  • The mathematical description of the external reality that theoretical physics has uncovered appears very different from the way we perceive this external reality.

  • Midway between the internal and external realities lies the “consensus reality,” the shared description of the physical world that all self-aware observers agree on.

  • This cleanly splits what Douglas Adams jocularly called “the ultimate question of life, the universe and everything” into two parts that can be tackled separately: the challenge for the physical sciences is deriving the consensus reality from the external reality, and the challenge for the cognitive sciences is to derive the internal reality from the consensus reality.

  • The rest of this book is focused on the first of these two challenges.

  * * *

  1Einstein realized that whereas observers sharing the same location and motion will share a common consensus reality, two groups moving relative to one another will have different consensus realities. In other words, there can be many different consensus realities, but their differences are explained by physical effects that have nothing to do with consciousness or the internal structure of the observers.

  10

  Physical Reality and Mathematical Reality

  Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth.

  —Galileo Galilei, The Assayer, 1623

  The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and … there is no rational explanation for it.

  —Eugene Wigner, 1960

  Whoa! It’s Friday morning in Princeton, and I’ve just finished reading emails about a book project, a broken oven and a quantum-suicide debate, when I find this gem in my inbox from a senior professor I know:

  Date: December 4, 1998, 7:17:42 EST

  Subject: Not an easy e-mail to write…

  Dear Max,

  .… your crackpot papers are not helping you. First, by submitting them to good journ
als and being unlucky so that they get published, you remove the “funny” side of them.… I am the Editor of the leading journal … and your paper would have never passed. This might not be that important except that colleagues perceive this side of your personality as a bad omen on future development.… You must realize that, if you do not fully separate these activities from your serious research, perhaps eliminating them altogether, and relegate them to the pub or similar places, you may find your future in jeopardy.

  I’d had cold water poured on me before, but this was one of those great moments when I realized I’d set a new personal record, the new high score to try to top. When I forwarded this email to my dad, who’s greatly inspired my scientific pursuits, he responded with a Dante quote: “Segui il tuo corso et lascia dir le genti!” Italian for “Follow your own path and let people talk!”

  I find it amusing how strong the conformist herd mentality is among many physicists, given that we all pay lip service to thinking outside the box and challenging authority. I’d become acutely aware of this sociological situation already back in grad school: for example, Einstein’s revolutionary relativity theory never won the Nobel Prize,1 Einstein himself dismissed Friedmann’s expanding-universe discovery, and Hugh Everett never even got a job in physics. In other words, much more important discoveries than I could realistically hope to make were being dismissed. So I faced a dilemma back in grad school: I’d fallen in love with physics precisely because I was fascinated with the biggest questions, yet it seemed clear that if I just followed my heart, then my next job would be at McDonald’s.

  I didn’t want to choose between my passion and my career, so I developed a secret strategy that ended up working surprisingly well, letting me have my cake and eat it, too. I called it my “Dr. Jekyll/Mr. Hyde Strategy,” and it exploited a sociological loophole. Whereas Giordano Bruno was burned at the stake in 1600 for his views (which included heresies such as space being infinite) and Galileo was condemned to lifelong house arrest for arguing that the Earth orbits the Sun, today’s sanctions are milder. If you’re interested in big philosophical-sounding questions, most physicists will treat you in much the same way as if you’re captivated by computer games: what you do after work is your own business and won’t be held against you as long as it doesn’t distract you from your day job, and as long as you don’t talk too much about it at work. So whenever authority figures asked what I worked on, I transformed into the respectable Dr. Jekyll and told them that I worked on mainstream topics in cosmology, such as those of Chapter 4, involving lots of measurements and numbers and blah blah blah. But secretly, when nobody was watching, I’d transform into the evil Mr. Hyde and do what I really wanted to do: pursue the ultimate nature of reality as in Chapters 6, 8 and most of the rest of this book. To allay fears, I put a blurb on my website about having some “side interests,” joking that every time I’d written ten mainstream papers, I’d allow myself to indulge in writing one wacky one. This was very convenient, since I was the only one who kept count.… When I graduated from Berkeley, I’d published eight papers, but half of them were written by Mr. Hyde, so I omitted these from my Ph.D. thesis. I really liked my Berkeley thesis advisor, Joe Silk, but to be on the safe side, I made sure he was far from the laser printer before printing the Hyde papers, and I showed them to him only after he’d officially signed my thesis.…2 And I stuck with this strategy: whenever I applied for jobs and research grants, I only mentioned Dr. Jekyll’s work, while on the side, I kept doing research on these big questions that set me on fire—in a good, non-Bruno kind of way.

  This devious strategy worked beyond my wildest expectations, and I’m extremely grateful that I get to work at a university with brilliant colleagues and students without having to stop thinking about my greatest interests. But now I feel that I have a debt to pay to the science community, and that the time has come to pay my dues! If we imagine all research topics arranged in front of us, in a metaphorical space, then there’s a border delimiting what’s mainstream physics from what’s not. The amazing thing about this border is that, as illustrated in Figure 10.1, it’s continually shifting! In some places, it has contracted, with theories from alchemy to astrology leaving the mainstream. In other places, it has expanded to reclassify ideas such as relativity theory and the germ theory of disease from speculative minority views to mainstream science. I’ve long believed that there are additional topics that physicists can usefully contribute to even though they at first sound rather philosophical, and I’ve now had tenure long enough that I can’t make excuses: I feel that I now have a moral obligation to more junior scientists to bring Mr. Hyde out of the academic closet and do my part to push the boundary a little. That’s why Anthony Aguirre and I started the Foundational Questions Institute I mentioned in Chapter 8, http://fqxi.org. And that’s why I’m writing this book.

  Figure 10.1: The boundary of what’s considered mainstream keeps shifting.

  Click here to see a larger image.

  So what paper of mine was it that triggered that “stop or you’ll ruin your career”? What topic was it about that was so far outside the current mainstream boundary of Figure 10.1 that this professor felt the need to bring me back into the fold? It was about the core idea of this book: that our physical world is a giant mathematical object. And this is the chapter where we’re going to start exploring it.

  * * *

  1At nobelprize.org, you can read that the Nobel Prize in Physics was awarded to Albert Einstein in 1922 “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.” However, a Swedish colleague of mine on the Nobel Committee once showed me the less-publicized full version of the award text. In my translation of it below, I’ve boldfaced a hilarious caveat that some curmudgeons presumably inserted to reflect their misgivings about relativity theory, today universally hailed as one of the greatest triumphs of the human mind:

  THE ROYAL SWEDISH ACADEMY OF SCIENCE has at its meeting on November 9, 1922, in accordance with the regulations in the November 27, 1895, will of ALFRED NOBEL, decided to, independently of the value that, after possible confirmation, may be attributed to the relativity and gravitation theory, award the prize that for 1921 is given to the person who within the domain of physics has made the most important discovery or invention, to ALBERT EINSTEIN for his contributions to Theoretical Physics, especially his discovery of the photoelectric effect.

  2I also timed Mr. Hyde’s papers strategically. Just as politicians like to discreetly reveal unpopular news on a Friday afternoon to give people time to forget before next week’s news cycle, I wrote that alleged crackpot paper during the summer of 1996, right after I’d been offered my Princeton postdoc job, because I knew that this would give people the maximum time to forget before I had to apply for jobs again.

  Math, Math Everywhere!

  What’s the answer to the ultimate question of life, the universe, and everything? In Douglas Adams’s science-fiction spoof The Hitchhiker’s Guide to the Galaxy, the answer was found to be 42; the hardest part turned out to be finding the real question. Indeed, although our inquisitive ancestors undoubtedly asked such big questions, their search for a “theory of everything” evolved as their knowledge grew. As the ancient Greeks replaced myth-based explanations with mechanistic models of the Solar System, their emphasis shifted from asking why to asking how.

  Since then, the scope of our questioning has dwindled in some areas and mushroomed in others, as illustrated in Figure 10.1. Some questions were abandoned as naive or misguided, such as explaining the sizes of planetary orbits from first principles, which was popular during the Renaissance. The same may happen to currently trendy pursuits such as predicting the amount of dark energy in the cosmos, if it turns out that the amount in our neighborhood is a historical accident as we discussed in Chapter 6. Yet our ability to answer other questions has surpassed earlier generations’ wildest expectations: Newton would have been amazed to know that we’d one day me
asure the age of our Universe to an accuracy of 1%, and comprehend the microworld well enough to make an iPhone.

  I find it very appropriate that Douglas Adams joked about 42, because mathematics has played a striking role in these successes.1 The idea that our Universe is in some sense mathematical goes back at least to the Pythagoreans of ancient Greece, and has spawned centuries of discussion among physicists and philosophers. In the seventeenth century, Galileo famously stated that our Universe is a “grand book” written in the language of mathematics. More recently, the physics Nobel laureate Eugene Wigner argued in the 1960s that “the unreasonable effectiveness of mathematics in the natural sciences” demanded an explanation.

  * * *

  1I’ve switched from collecting stamps to collecting cool questions whose answer is 42. Here are my favorites so far:

  1. At what latitude was this book written?

  2. What’s the radius of the rainbow, in degrees?

  3. At most how many percent of the gas around it can a black hole gobble up?

  Feeding a black hole turns out to be a lot like feeding a baby: most of the material comes flying back at great speeds.… Black holes can eat at most 1–1/ ≈ 42% of the gas around them.

  Shapes, Patterns and Equations

  Below we’re going to explore a really extreme explanation. However, first we need to clear up what exactly it is that we’re trying to explain. Please stop your reading for a few moments and look around you. Where’s all this math that we’re going on about? Isn’t math all about numbers? You can probably spot a few numbers here and there, for example the page numbers of this book, but these are just symbols invented and printed by people, so they can hardly be said to reflect our Universe being mathematical in any deep way.