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Our Mathematical Universe

Our Mathematical Universe
My Quest for the Ultimate Nature of Reality

by Max Tegmark

Alfred A. Knopf, 2014

432 pages, hardback

Review by Jim Walker

Max Tegmark works as a professor at MIT specializing in cosmology, especially information theory as applied to Cosmic Microwave Background experiments such as COBE, QMAP, and WMAP, and to galaxy redshift surveys such as the Las Campanas Redshift Survey, the 2dF Survey and the Sloan Digital Sky Survey. Tegmark has certainly earned his place in mathematics so he certainly has the qualifications to speak about the philosophical aspects of mathematics and the universe. However, it should be noted that this book does not represent the views of most physicists and mathematicians and that Tegmark's views are controversial. Some might even call them crackpot ideas. But the difference between crackpot ideas and just ideas is that Tegmark does not claim that his ideas are factual, but rather, hypotheses (even though, unfortunately, he does claim to believe in them). Philosophers might call this metaphysics or perhaps a better term: metamathematics.

Mathematics to most people represents a language. As Galileo once wrote, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written." However, to Tegmark, mathematics also represents the subject for which the language of mathematics points to, and this subject amounts to nothing but relationships between structures. For example, we know that reality consists of subatomic particles existing in spacetime, yet the only way to describe these subatomic structures are in the form of mathematical language. Consider that the whole of quantum mechanics consists of energy, momentum, charge and other conserved quantities and every one of these properties consists of nothing but numbers. Tegmark questions, "So what are quantum numbers like energy and charge made of? Nothing--they're just numbers!" Likewise, if you look at the other end of reality, the universe as a whole and the possibility of multiverses, again, the only way to describe these large structures are in the language of mathematics.

At first the idea that the smallest and the largest structures in the universe can only be resolved through mathematics seems like a revelation, but on second thought, not so much. Human perception is limited by our evolutionary upbringing. Just because we can no longer connect on a personal level to things out of range of our senses does not imply that they are made of math. For example, if an aircraft flies out of range of our eyesight, we can still detect it as a blip on a radar screen, but even though this blip represents a mathematical point says nothing about what it is made of (or not made of). We can only detect subatomic particles and the Cosmic Microwave Background through scientific instruments but just because they are out of range of human perception says nothing at all about the ultimate nature of reality. Moreover, like human perception, our scientific instruments, themselves, have limitations from which they, too, are out of range from the smallest to the largest.

Perhaps the most controversial part of this book involves multiverses. Tegmark goes into length about the various theories of multiverses which Tegmark calls Level I, Level II, Level III and Level IV universes. Tegmark basically believes in all of them because mathematical theories predict them. In fact, according to Tegmark, the evidence for the Big Bang inflation (and the latest data from the Cosmic Microwave Background provides good evidence for inflation), is the best evidence for Level I multiverses because the theory of inflation predicts an infinite number of multiverses. In fact physicists have to go to great length to talk themselves out of the predictions of multiverses, not because they have evidence against multiverses, or that the math predicts it, but because, well, it's just so outlandish and there's really no hard evidence for them (although this is quickly changing because multiverses explain more than a single universe does). But note that there isn't a scrap of evidence that our universe is the only one either, so that leaves us with a question of which comes closer to the truth. It seems to me the best approach would be to leave out belief and just accept both hypotheses as possible and just look for evidence to support one or the other and, so far, the only evidence comes from mathematical evidence.

This reminds me of the ancient and still ongoing debate about zero and infinity. It took thousands of years to accept zero as a numerical number, and in many ways, zero resembles infinity (and vice versa). In fact these two ideas destroyed the Aristotelian view of physics and it wasn't until the fall of the European Dark Ages that thinking about zero and infinity helped open the way to the scientific revolution. I suspect that investigations into infinite universes might do the same for the future of physics.

Although Tegmark has not convinced me that our universe is purely mathematical, his ideas inspire and make you want to think. Some of Tegmark's thoughts have helped clarify my own guesses about what our Universe is made of. Instead of everything consisting of pure math, my hypothesis suggests that there are only two things in the universe. Putting it in Chinese philosophical terms, you could call it yin and yang. In mathematical terms, zero and infinity. In physics terms (my preference): space and energy. The mixing of space and energy, creates spacetime (Hilbert space). The particulate nature of energy can transform into subatomic particles that forms atoms, molecules, gases, stars, planets, galaxies, and living things, etc. This is not something that I believe in but only a hypothesis that makes the most sense to me (at this time).

Tegmark also covers a lot of other subjects including life in the universe and speculations on whether or not we are alone in the universe. Here is one area where I disagree with him. According to Tegmark, we are alone in the universe. His calculations are based on the age of the universe, the distance between possible civilizations, and the fact that no one has ever observed evidence for life anywhere else in the universe. However, his flaw (as I see it) comes from his assumptions that intelligent life would colonize other habitable planets and that the fraction of civilizations that can colonize would do so. I think this is an unwarranted assumption. Why in the world would an intelligent civilization that learned to live in space want to colonize other planets? A species that evolves to live in a different environment rarely returns to its previous environment. It takes greater energy to colonize a planet and it would have to fight gravity as well as protect it from bombardment of asteroids, comets, and other flying objects. If a civilization can live in space, why not take advantage of it and live in an area free from danger? However, to do so would require knowledge of extreme energy conversation and they would live in a different time frame (sending out signals per very long duty cycles) which would give reason why we have not detected them (see Death and Time Traveling for a detailed explanation on this).

This is not just a book about science but mostly metaphysics and speculation and that's what's fun about it. Tegmark gives the reader a lot to think about and allows us to play with these ideas while providing explanations to lay people about the cutting edge of scientific discoveries. If Tegmark is correct about infinite universes the implications are staggering, so staggering that it's difficult to even comprehend. It would also mean an infinite "you"s, along with your friends and family members. It would mean that the multiveres are deterministic (even though to inhabitants it would seem indetermined). It would also imply that gods only exist in imagination. I highly recommend this book for people looking for possible explanations about life and the universe and who like to think for themselves.


A few quotes from the book:

No matter how emphatically we scientists claim to be rational seekers of truth, we're as prone as anyone to human foibles such as prejudice, peer pressure and herd mentality. p. 50

[B]ack when our entire Universe was a fusion reactor, it fused about 25% of its mass into helium. When you measure the helium fraction of distant intergalactic gas by studying its spectrum with a telescope, you find. . . 25%! To me, this finding is just as impressive as discovering a fossilized Tyrannosaurus rex femur. p. 63

[A]fter spending its first 7 billion years slowing down, the cosmic expansion started speeding up again and has accelerated ever since! p. 77

Albert Einstein allegedly said: "In theory, theory and practice are the same. In practice, they are not." p.92

[T]he whole inflation process, from beginning to end, could have been almost instantaneous by human standards, requiring less than about 10-35 seconds, less time than light takes to travel a trillionth of the size of a proton. p. 103

Heisenberg uncertainty principle of quantum mechanics prevents any substance, including the inflating material, from being completely uniform. If you try to make it uniform, quantum effects force it to start wiggling around, spoiling the uniformity. When inflation stretched a subatomic region into what became our entire observable Universe, the density fluctuations that quantum mechanics had imprinted were stretched as well, to sizes of galaxies and beyond. p. 107

The fact that space expands inside doesn't necessarily increase the amount of room it all takes as seen from outside: remember that Einstein allows space to stretch and produce more volume from nothing, without taking it from someplace else. p. 116

[I]f there are indeed many copies of "you," with identical past lives and memories, this kills the traditional notion of determinism: you can't predict your own future--even if you have complete knowledge of the entire past and future history of the cosmos! The reason you can't is that there's no way for you to determine which of these copies is "you" (they all feel that they are). Yet their lives will typically begin to differ eventually, so the best you can do is predict probabilities for what you'll experience from now on. p. 123

[T]he best evidence for the Level I multiverse is the evidence we have for inflation. p. 126

Many of the regularities that we used to view as fundamental laws, which by definition hold anywhere and anytime, have turned out to be merely effective laws, local bylaws that can vary from place to place, corresponding to different knob settings defining space in different phases. p. 138

[E]vidence for eternal inflation (of which there's plenty) is evidence for the Level II multiverse, because the former predicts the latter. p. 138

So it's quite reasonable to assume that our Universe was created by some sort of universe-creation mechanism (perhaps inflation, perhaps something totally different). Now here's the thing: all the other mechanisms we mentioned naturally produce many copies of whatever they create; a cosmos containing only one car, one rabbit, and one solar system would seem quite contrived. In the same vein, it's arguably more natural for the correct universe-mechanism, whatever it is, to create many universes rather than just the one we inhabit. p.151

[W]e can think of the fundamental Legos of particle physics as being not the particles themselves, but the conserved quantities! So particle physics is simply rearranging energy, momentum, charge and other conserved quantities in new ways. p. 165

So what are quantum numbers like energy and charge made of? Nothing--they're just numbers! p. 165

Sometimes it's hard to reconcile what I believe with what I feel. p. 191

I concluded that quantum mechanics requires secrecy: an object can only be found in two places at once in a quantum superposition as long as its position is kept secret from the rest of the world. p. 199

But even if a single photon bounces off the object, the information about its whereabouts is out: it gets encoded in the subsequent position of the photon. p. 199

I was wondering whether you needed a human observer to collapse the wavefunction, or if a robot would suffice. Now I was convinced that consciousness had nothing to do with it, since even a single particle could do the trick: a single photon bouncing off of an object had the same effect as if a person observed it. I realized that quantum observation isn't about consciousness, but simply about the transfer of information. p. 199

[S]o a bowling ball that's in two places at once will have its quantum superposition ruined even before I have a chance to become consciously aware of it. p. 199

I also did the math for another model that Penrose had proposed, where the quantum computation was done not by neurons but by microtubules, parts of the scaffolding in cells, and found that they suffered decoherence after about 10-13 (100 quadrillionths) of a second. For my thoughts to correspond to a quantum computation, they'd need to finish before decoherence kicked in, so I'd need to be able to think fast enough to have 10,000,000,000,000 thoughts each second. Perhaps Roger Penrose can think that fast, but I sure can't. . . . p.207

[D]ecoherence does one more important thing for us: not only does it explain why large objects never seem to be in two places at once, but it also explains why conventional states (such as being in only one place) are so special: out of all the states that quantum mechanics allows for large objects, these conventional states are the ones that are most robust to decoherence, and therefore the ones that survive. p. 211

If the object isn't interacting with anything, its entropy stays constant. p. 212

If the object interacts with you, then you typically get more information about it, and its entropy decreases... p. 212

If the object interacts with the environment, however you typically lose information about it, so its entropy increases... p. 212

In summary here's how I informally think about this: the entropy of an object decreases while you look at it and increases while you don't.

According to a recent estimate, more than a quarter of the U.S. gross national product is now based on inventions made possible by quantum mechanics, from lasers to computer chips. p. 228

All physics theories have two parts: mathematical equations and words that tell us what they mean. p. 229

To me, the most interesting question is what the math part is, and specifically whether the simplest math of all (just Schrödinger equation with no exceptions is enough. So far, there isn't a shred of experimental evidence to the contrary, yet many of the quantum interpretations add lengthy "words" part to talk away the parallel universes. So when you pick your own favorite interpretation, it really comes down to what bothers you the most: a profusion of worlds or a profusion of words. p 229

In Chapter 7, we talked about how everything is made of particles, and how particles are in a sense purely mathematical objects. In this chapter, we've seen that in quantum mechanics, there's something that is arguably even more fundamental: the wavefunction and the infinite-dimensional place called Hilbert space where it lives. p. 229

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. p. 241

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. p 241

[T]his idea that there's a bunch of numbers at each point in spacetime is quite deep, and I think it's telling us something not merely about our description of reality, but about reality itself. One of the most fundamental concepts in modern physics is that of a field, which is just this: something represented by numbers at each point in spacetime.

If you have a mathematics background and are familiar with the notion of isomorphism, you can restate this argument as follows. From the definition of a mathematical structure, it follows that if there's an isomorphism between a mathematical structure and another structure (a one-to-one correspondence between the two that respects the relations), then they're one and the same. If our external physical reality is isomorphic to a mathematical structure, it therefore fits the definition of being a mathematical structure. p. 280 (footnote)

I think that consciousness is the way information feels when being processed in certain complex ways, and that the particular kind of consciousness that we humans subjectively perceive arises when your brain's model of you is interacting with your brain's model of the world. p. 289 (Figure 11.7)

The key difference lies not in the neurons that carry this information, but in the patterns whereby they're connected. p. 290

I'm arguing that your perceptions of having a self, that subjective vantage point that you call "I," are qualia just as your subjective perceptions of "red" or "green" are. In short, redness and self-awareness are both qualia. p. 290

[I]nformation is a measure of how much meaning complexity has. p. 294

My guess is that we'll one day understand consciousness as yet another phase of matter. p. 295

[T]here's a beautiful mathematical toolkit known as Bayesian decision theory which generalizes the true/false dichotomy to allow shades of gray: each possible assumption gets assigned a number between zero and one, the probability which which you think it's correct, and there's a simple formula for how to update these probabilities whenever you make new observations. p. 301

I view the measure problem as the greatest crisis in physics today. p. 314

In the context of the MUH, there's thus no fire-breathing required, since the point isn't that a mathematical structure describes a universe, but that it is a universe. Moreover, there's no making required either. You can't make a mathematical structure--it simply exists.

It's striking that many of the continuum models of classical mathematical physics (for example, the equations describing waves, diffusion or liquid flow) are known to be mere approximations of an underlying discrete collection of atoms. Quantum-gravity research suggests that even classical spacetime breaks down on very small scales. We therefore can't be sure that quantities and quantum wavefunction amplitudes) aren't mere approximations of something discrete. p. 334

Symmetry properties are among the very few types of properties that every mathematical structure possesses, and they can manifest themselves as physical symmetries to the structure's inhabitants. p. 337

The MUH provides the answer that our physical reality has symmetry properties because it's a mathematical structure, and mathematical structures have symmetry properties. p. 338

Regardless of whether anything seems random to an observer, it must ultimately be an illusion, not existing at the fundamental level, because there's nothing random about mathematical structure. p. 340

In other words, our successful theories aren't mathematics approximating physics, but mathematics approximating mathematics. p. 355

Paul Dirac in 1931: "The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical feature in terms of physical entities." p. 355

[I]f we're impressed by the successful predictions of inflation or quantum mechanics so far, then we also need to take seriously their other predictions, including the Level I and Level III multiverses. p. 361

As a theoretical physicist, I judge the elegance and simplicity of a theory not by its ontology, but by the elegance and simplicity of its mathematical equations--and it's quite striking to me that the mathematically simplest theories tend to give us multiverses. It's proven remarkably hard to write down a theory that produces exactly the universe we see and nothing more. p. 361

In other words, we basically get stuck with all these parallel universes as soon as we accept that there;s an external reality independent of us. p. 364

If the Mathematical Universe Hypothesis is correct, then there isn't much to say about the future of our physical reality as a whole: since it exists outside of space and time, it can't end or disappear any more than it can get created or change. p. 366

Now here's what bothers me. The Schrödinger equation of quantum mechanics that we encountered in Chapter 7 implies that information can't be created or destroyed.

As we've seen, the fundamental mathematical equations that appear to govern our physical reality make no reference to meaning, so a universe devoid of life would arguably have no meaning at all. . . . So in this sense, our Universe doesn't give life meaning, but life gives our Universe meaning. p. 391




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 the Earth · Distance to the Moon · Distance to the Sun and 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 Realty 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 Multiverse · What's Out There? · Implications of the Level IV Multiverse · Are We Living in a Simulation? · Relation Between Testing the Level IV Multiuniverse

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?

Suggestions for Further Reading


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