Our Mathematical Universe

by Max Tegmark

  THE BOTTOM LINE

  • In the mathematically simplest quantum theory, there’s something 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.

  • In this theory, particles can be created and destroyed, and can be in several places at once, but there is, was, and always will be only one wavefunction, moving through Hilbert space as determined by the Schrödinger equation.

  • This mathematically simplest quantum theory, where the Schrödinger equation always rules, predicts the existence of parallel universes where you live out countless variations of your life.

  • It also implies that quantum randomness is an illusion, caused by quantum cloning of you.

  • There’s nothing quantum about apparent randomness, which happens even if you’re classically cloned.

  • This mathematically simplest quantum theory also predicts a censorship effect called decoherence, which hides most such weirdness from us, mimicking wavefunction collapse.

  • Decoherence happens constantly in your brain, debunking popular suggestions about “quantum consciousness.”

  • This quantum multiverse is unified with the spatial multiverse from Chapter 6, so that a wavefunction for a system describes its infinite copies throughout space, and quantum uncertainty reflects your ignorance about which particular copy you’re observing.

  • If we live in an infinite uniform space as in the cosmological standard model, then it doesn’t matter whether the wavefunction ultimately collapses: all of Everett’s many worlds are indistinguishable, and collapse doesn’t prevent all quantum outcomes from actually happening.

  • The quantum multiverse arguably makes you subjectively immortal, in which case you’ll eventually find yourself the oldest person on the planet, and this may not even require quantum mechanics, merely the Level I multiverse in an infinite space. But I don’t think so, as I’ll explain in Chapter 11.

  • The wavefunction and Hilbert space, which constitute arguably the most fundamental physical reality, are purely mathematical objects.

  * * *

  1You’ll find references for all these interpretations in http://arxiv.org/abs/1008.1066.

  Part Three

  STEPPING BACK

  9

  Internal Reality, External Reality and Consensus Reality

  Sweet exists by convention, bitter by convention, color by convention; atoms and void [alone] exist in reality.

  —Democritus, ca. 400 B.C.

  “Nooooo! My suitcase!”

  They were already boarding my flight from Boston to Philadelphia, where I was supposed to help with a BBC documentary about Hugh Everett, when I realized that my hand wasn’t holding a suitcase. I ran back to the security checkpoint.

  “Did someone just forget a black roll-on bag here?”

  “No,” said the guard.

  “But there it is—that’s my suitcase, right there!”

  “That’s not a black suitcase,” said the guard. “That’s a teal suitcase.”

  Until then, I’d never realized how color-blind I was, and it was quite humbling to realize that many assumptions I’d previously made about reality—and my wardrobe—were dead wrong. How could I ever trust what my senses told me about the outside world? And if I couldn’t, then how could I hope to ever know anything with certainty about the external reality? After all, everything I know about the outside world and my untrustworthy senses, I’ve learned from my senses. This puts me on the same shaky epistemological footing as a prisoner who’s spent his whole life in solitary confinement, whose only information about the outside world and his untrustworthy prison guard is what his prison guard has told him. More generally, how can I trust what my conscious perceptions tell me about the world if I don’t understand how my mind works?

  This basic dilemma has been eloquently explored by philosophers throughout the ages, including titans such as Plato, René Descartes, David Hume and Immanuel Kant. Socrates said: “The only true wisdom is in knowing you know nothing.” So how can we make further progress in our quest to understand reality?

  So far in this book, we’ve taken a physics approach to exploring our external physical reality, zooming out to the transgalactic macrocosm and zooming in to the subatomic microcosm, attempting to understand things in terms of their basic building blocks such as elementary particles. However, all we have direct knowledge of are instead qualia, the basic building blocks of our conscious perception,1 exemplified by the redness of a rose, the sound of a cymbal, the smell of a steak, the taste of a tangerine or the pain of a pinprick. So don’t we also need to understand consciousness before we can fully understand physics? I used to answer “yes,” thinking that we could never figure out the elusive “theory of everything” for our external physical reality without first understanding the distorting mental lens through which we perceive it. But I’ve changed my mind, and in this brief interlude chapter, I want to tell you why.

  * * *

  1For introductions to the vast literature on consciousness by psychologists, neuroscientists, philosophers and others, I recommend the books about the mind in the “Suggestions for Further Reading” section.

  External Reality and Internal Reality

  Perhaps you’re thinking, Okay, Max, but I’m not color-blind. And I’m looking at the external reality right now with my own eyes, and I’d have to be paranoid to think it’s not the way it looks. But please try these simple experiments:

  Experiment 1: Turn your head from left to right a few times.

  Experiment 2: Move your eyes from left to right a few times, without moving your head.

  Did you notice how the first time, the external reality appeared to rotate, and the second time, it appeared to stay still, even though your eyeballs rotated both times? This proves that what your mind’s eye is looking at isn’t the external reality, but a reality model stored in your brain! If you looked at the image recorded by a rotating video camera, you’d clearly see it move as it did in Experiment 1. But your eyes are a form of biological video camera, so Experiment 2 shows that your consciousness isn’t directly perceiving the images formed on their retinas. Rather, as neuroscientists have now studied in great detail, the information recorded by your retinas gets processed in highly complex ways and is used to continually update an elaborate model of the outside world that’s stored in your brain. Take another look in front of you, and you’ll see that, thanks to this advanced information processing, your reality model is three-dimensional even though the raw images from your retinas are two-dimensional.

  I don’t have a light switch near my bed, so I’ll often take a good look at my bedroom and all the obstacles littering the floor, then turn off the light and walk to my bed. Try it yourself: put down this book, stand up, look around, and then walk a few steps with your eyes closed. Can you “see”/“feel” the objects in the room moving relative to you? That’s your reality model being updated, this time using information from your leg movements rather than from your eyes. Your brain continuously updates its reality model using any useful information it can get hold of, including sound, touch, smell and taste.

  Let’s call this reality model your internal reality, because it’s the way you subjectively perceive the external reality from the internal vantage point of your mind. This reality is internal also in the sense that it exists only internally to you: your mind feels as if it’s looking at the outside world, while it’s actually looking only at a reality model inside your head—which in turn is continually tracking what’s outside your brain via elaborate but automatic processes that you’re not consciously aware of.

  It’s absolutely crucial that we don’t conflate this internal reality with the external reality that it’s tracking, because the two are very different. My brain’s internal reality is like the dashboard of my car: a convenient summary of the most useful information. Just as my car’s dashboard tells me my speed, fuel level, mot
or temperature, and other things useful for a driver to be aware of, my brain’s dashboard/reality model tells me my speed and position, my hunger level, the air temperature, highlights of my surroundings and other things useful for the operator of a human body to be aware of.

  The Truth, the Whole Truth and Nothing but the Truth

  Once my car’s dashboard malfunctioned and sent me to the garage with its “CHECK ENGINE” indicator illuminated even though nothing was wrong. Similarly, there are many ways in which a person’s reality model can malfunction and differ from the true external reality, giving rise to illusions (incorrect perceptions of things that do exist in the external reality), omissions (nonperception of things that do exist in the external reality) and hallucinations (perceptions of things that don’t exist in the external reality). If we swear under oath to tell the truth, the whole truth and nothing but the truth, we should be aware that our perceptions might violate all three with illusions, omissions and hallucinations, respectively.

  So metaphorically speaking, the “CHECK ENGINE” incident was my car hallucinating—or experiencing phantom pain. I recently discovered that my car also suffers from an illusion: based on its speedometer reading, it thinks it’s always driving two miles per hour faster than it really is. That’s not bad compared to the vast list of human illusions that cognitive scientists have discovered, which afflict all our senses and distort our internal reality. If your version of this figure is in color rather than black and white, you’ll probably see the lower dot in the left panel as orange and the upper dot as somewhat brown. Figure 9.1 shows two examples of optical illusions, where our visual system creates an internal reality different from the external reality. In the external reality, the light from both of them has identical properties, with a wavelength around 600 nanometers. If a spotlight beamed out such light, it would be orange light. What about brown? Have you ever seen a spotlight or a laser pointer produce a brown beam? Well, you never will, because there’s no such thing as brown light! The color brown doesn’t exist in the external reality, but only in your internal reality: it’s simply what you perceive when seeing dim orange light against a darker background.

  For fun, I sometimes compare how the same news story is reported online by MSNBC, FOX News, the BBC, Al Jazeera, Pravda and elsewhere. I find that when it comes to telling the truth, the whole truth, and nothing but the truth, it’s the second part that accounts for most of the differences in how they portray reality: what they omit. I think the same holds for our senses: although they can produce hallucinations and illusions, it’s their omissions that account for most of the discrepancy between the internal and external realities. My visual system omitted the information that distinguishes between black and teal suitcases, but even if you’re not color-blind, you’re missing out on the vast majority of the information that light carries. When I was taught in elementary school that all colors of light can be made up by mixing three primary colors red, green, and blue, I thought that this number three told us something fundamental about the external reality. But I was wrong: it teaches us only about the omissions of our visual system. Specifically, it tells us that our retina has three kinds of cone cells, which take the thousands of numbers that can be measured in a spectrum of light (see Figure 2.5 in Chapter 2) and keeps only three numbers, corresponding to the average light intensity across three broad ranges of wavelengths.

  Figure 9.1: Optical illusions. In the left panel, squares A and B have the same shade of gray, and the two dots have identical color. In the right panel, look at the black dot while moving your head forward and backward, and see the circles move.

  Moreover, wavelengths of light outside of the narrow range 400–700 nanometers go completely undetected by our visual system, and it came as quite a shock when human-built detectors revealed that our external reality was vastly richer than we’d realized, teeming with radio waves, microwaves, x-rays, and gamma rays. And vision isn’t the only one of our senses that’s guilty of omissions: we can’t hear the ultrasound chirping of mice, bats and dolphins; we’re oblivious to most faint scents that dominate the olfactory inner reality of dogs, and so on. Although some animal species capture more visual, auditory, olfactory, gustatory or other sensory information than we humans do, they’re all unaware of the subatomic realm, the galaxy-spangled cosmos, and the dark energy and dark matter that, as we saw in Chapter 4, makes up 96% of our external reality.

  Consensus Reality

  In the first two parts of this book, we’ve seen how our physical world can be remarkably well described by mathematical equations, fueling the hope that one day equations can be found for a “theory of everything,” perfectly describing our external reality on all scales. The ultimate triumph of physics would be to start with the external reality from the “bird perspective” of a mathematician studying these equations (which are ideally simple enough to fit on her T-shirt) and to derive from them her internal reality, the way she subjectively perceives it from her “frog perspective” inside the external reality. To accomplish this would clearly require a detailed understanding of how consciousness works, including illusions, omissions, hallucinations and other complications.

  However, between the external reality and the internal reality, there’s also a third and intermediate consensus reality, as illustrated in Figure 9.2. This is the version of reality that we life-forms here on Earth all agree on: the 3-D positions and motions of macroscopic objects, and other everyday attributes of the world for which we have a shared description in terms of familiar concepts from classical physics. Table 9.1 summarizes these reality descriptions and perspectives and how they’re interrelated.

  Each of us has our own personal inner reality, perceived from the subjective perspective of our own position, orientation and state of mind, and distorted by our personal cognitive biases: in your inner reality, dreams are real and the world turns upside down when you stand on your head. In contrast, the consensus reality is shared. When you give your friend driving directions to your place, you do your best to transform your description from one involving subjective concepts from your inner reality (such as “here” and “in the direction I’m facing”) to shared concepts from the consensus reality (such as “on 70 Vassar Street” and “north”). Since we scientists need to be precise and quantitative when we refer to our shared consensus reality, we try extra-hard to be objective: we say that light has a “600-nanometer wavelength” instead of “orange color” and that something has “CH3COOC5H11 molecules” instead of “banana flavor.” The consensus reality isn’t free from some shared illusions relative to the external reality, as we’ll elaborate on below: for example, cats, bats and robots also experience the same quantum randomness and relativistic time dilation. However, it’s by definition free from illusions that are unique to biological minds, and therefore decouples from the issue of how our human consciousness works. The internal reality may feel teal deficient to me, black and white to a seal, iridescent to a bird seeing four primary colors, and still more different to a bee seeing polarized light, a bat using sonar, a blind person with keener touch and hearing, or the latest robotic vacuum cleaner, but we all agree on whether the door is open.

  Figure 9.2: We can view reality in three interrelated ways: from the bird perspective of a mathematician studying the equations describing it, from the subjective frog perspective of a self-aware observer in it, and from the intermediate consensus perspective in which we usually describe it to one another (as classical objects moving in 3-D, say). The ultimate quest for understanding splits conveniently into two parts, which can be tackled separately: for physics to reveal how the external reality relates to the consensus reality (including complications such as observer cloning appearing as randomness and rapid motion appearing as time slowdown) and for cognitive science to reveal how the consensus reality relates to the internal reality (including qualia and complications such as illusions, omissions and hallucinations).

  Click here to see a larger image.

&nb
sp; This is why I’ve changed my mind: although understanding the detailed nature of human consciousness is a fascinating challenge in its own right, it’s not necessary for a fundamental theory of physics, which need “only” derive the consensus reality from its equations. In other words, what Douglas Adams called “the ultimate question of life, the universe and everything” splits cleanly into two parts that can be tackled separately: the challenge for physics is deriving the consensus reality from the external reality, and the challenge for cognitive science is to derive the internal reality from the consensus reality. These are two great challenges for the third millennium. They’re each daunting in their own right, and I’m relieved that we need not solve them simultaneously.

  Reality Cheat Sheet

  External reality The physical world, which I believe would exist even if we humans didn’t

  Consensus reality The shared description of the physical world that self-aware observers agree on

  Internal reality The way you subjectively perceive the external reality