p.17
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.
p.56
Although Newton first applied this idea only to motion and gravity, the concept spread like wildfire and was gradually applied to other topics such as light, gases, liquids, solids, electricity and magnetism. People boldly extrapolated not only to the macrocosmos of space, but also to the microcosmos, finding that many properties of gases and other substances could be explained by applying Newton’s laws of motion to the atoms that they were made of. The scientific revolution had begun.
p.61
I find it ironic that even Einstein, one of the most creative scientists ever, whose trademark was questioning unquestioned assumptions and authorities, failed to question the most important authority of all: himself, and his prejudice that we live in an eternal unchanging universe. Instead, in what he later described as his greatest blunder, he changed his equations by adding an extra term that allowed our Universe to be static and eternal. In a double irony, it now seems as if this extra term is really there in the form of the cosmic dark energy we’ll discuss later, but with a different value that doesn’t make our Universe static.
p.107
Not only is our planet a part of a solar system and our Solar System part of a galaxy, but our Galaxy is part of a cosmic web of galaxy groups, clusters, superclusters and gigantic filamentary structures.
These patterns in the galaxy distribution are really the same patterns that we saw manifest themselves in cosmic microwave–background maps—only billions of years later, amplified by gravity. In a region of space where gas was once 0.001% denser than its surroundings, causing a spot in the WMAP map (see Figure 3.4), there might today be a cluster of one hundred galaxies. In this sense, we can think of the cosmic microwave–background fluctuations as the cosmic DNA, the blueprints for what our Universe will grow to become. By comparing the slight past clustering seen in the cosmic microwave background to the strong current clustering seen in a 3-D galaxy map, we can measure the detailed nature of the stuff whose gravity caused the clustering to grow between then and now.
p.121
The missing mass is ghostly, being both invisible and able to pass through us undetected. Its gravitational effects suggest that it consists of two separate substances of opposite character, dubbed dark matter and dark energy: Dark matter clusters, dark energy doesn’t. Dark matter dilutes as it expands, dark energy doesn’t. Dark matter attracts, dark energy repels. Dark matter helps galaxies form, dark energy sabotages.
p.134
Another puzzling feature of inflation is that it causes accelerated expansion. In high school, I was taught that gravity is an attractive force, so if I have a bunch of expanding stuff, then shouldn’t gravity instead decelerate the expansion, trying to ultimately reverse the motion and pull things back together? Again Einstein comes to the rescue with a loophole, this time from his general relativity theory, which says that gravity is caused not only by mass, but also by pressure. Since mass can’t be negative, the gravity from mass is always attractive. But positive pressure also causes attractive gravity, which means that negative pressure causes repulsive gravity! We just saw that an inflating substance has huge negative pressure. Alan Guth calculated that the repulsive gravitational force caused by its negative pressure is three times stronger than the attractive gravitational force caused by its mass, so the gravity of an inflating substance will blow it apart!
p.137
Well, one of the beauties of inflation is that it connects the smallest and largest scales: during the early stages of inflation, the region of space that now contains our Milky Way Galaxy was much smaller than an atom, so quantum effects could have been important. And indeed they were: as we’ll see in Chapter 7, the so-called 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. As we saw in the last chapter, gravitational instability took care of the rest, amplifying these fluctuations from the tiny 0.002%-level amplitudes with which quantum mechanics had endowed them into the spectacular galaxies, galaxy clusters and superclusters that now adorn our night sky.
p.143
We see that even our Big Bang is just a small part of something much grander, a treelike structure that’s still growing. In other words, what we’ve called our Big Bang wasn’t the ultimate beginning, but rather the end—of inflation in our part of space...Eternal inflation gives a clear answer: space isn’t just huge—it’s infinite. With infinitely many galaxies, stars and planets.
p.144
1. In almost all parts of space, inflation will eventually end in a Big Bang like ours.
2. There will nonetheless be some points in space where inflation never ends.
3. The total inflating volume increases forever, doubling at regular intervals.
4. The total post-inflationary volume containing galaxies also increases forever, doubling at regular intervals.
p.149
Inflation theory says that our Universe grew much like a human baby: an accelerating growth phase, in which the size doubled at regular intervals, was followed by a more leisurely decelerating growth phase.
p.153
our Universe is a spherical region with Earth at the center. The stuff near the edges of our Universe, from which light has only now reached us after a 14-billion-year space journey, is currently about 5 × 10^26 meters away from us.1 As far as we currently know, our Universe contains about 10^11 galaxies, 10^23 stars, 10^80 protons and 10^89 photons (particles of light).
p.158
...we consider it [general relativity] a successful scientific theory and take seriously also its predictions for things we can’t observe—for example, that space continues inside black-hole event horizons and that (contrary to early misconceptions) nothing funny happens right at the horizon. [1 Although you can, in principle, enter a black hole and observe what happens inside if its tidal forces don’t “spaghettify” you first]
General relativity is a rigid mathematical theory with no adjustments possible; you have to either accept all its predictions, or you have to start over from scratch and invent a different mathematical theory that agrees with all of general relativity’s successful predictions while simultaneously predicting that black holes can’t exist. This turns out to be extremely difficult, and so far, all such attempts have failed. In the same way, parallel universes aren’t optional in eternal inflation.
p.169
But could it be that what we humans think of as empty space is also some form of medium? Then the last laugh would be on us! As a matter of fact, there’s mounting evidence that this is exactly how things are. Not only does our “empty space” seem to be a sort of medium, but it appears to have way more than three phases—perhaps about 10^500, and perhaps even infinitely many, which opens up the possibility that, in addition to curving, stretching and vibrating, our space may even be able to do something analogous to freezing and evaporating!
p.172
In the same way, detailed study of the smallest building blocks of nature suggests to us that, with enough energy, they could be rearranged in a way such that our Universe would operate differently—we’ll explore the workings of these building blocks in the next chapter. Eternal inflation would have provided enough energy for the quantum fluctuations to actually make all such possible rearrangements in different Level I multiverses...The Level II multiverse takes this concept to a new level by downgrading many more fundamental
laws to effective laws, as we’ll explore next.
p.174
PHYSICAL REALITY - Everything that exists, Level IV Multiverse
SPACE - The part of physical reality that’s continuously connected to what we can observe; with eternal inflation, Level II Multiverse
OUR UNIVERSE - The part of physical reality we can in principle observe; quantum complications aside, this is the spherical region of space from which light has had time to reach us during the 14 billion years since our Big Bang
PARALLEL UNIVERSE - A part of physical reality that can in principle be observed from somewhere else but not from here—parallel universes are not a theory, but a prediction of certain theories
MULTIVERSE - A collection of universes
LEVEL I MULTIVERSE - Distant regions of space that are currently but not forever unobservable; they have the same effective laws of physics but may have different histories
LEVEL II MULTIVERSE - Distant regions of space that are forever unobservable because space between here and there keeps inflating; they obey the same fundamental laws of physics, but their effective laws of physics may differ
LEVEL III MULTIVERSE - Different parts of quantum Hilbert space (Chapter 8); same diversity as Level II
LEVEL IV MULTIVERSE - All mathematical structures (Chapter 12), corresponding to different fundamental laws of physics
FUNDAMENTAL LAWS - The mathematical equations that govern physics
EFFEECTIVE LAWS - Particular solution to the mathematical equations that describe physics; can be mistaken for fundamental laws if the same solution is implemented throughout universe
FINE-TUNING - Physical constants in the effective laws having values in a very narrow range allowing life; observed fine-tuning is arguable evidence for the Level II multiverse
p.176
If rotating the dark-energy knob in Figure 6.6 by a full turn would vary the density across the full range, then the actual knob setting for our Universe is about 10−123 of a turn away from the halfway point. That means that if you want to tune the knob to allow galaxies to form, you have to get the angle by which you rotate it right to over 120 decimal places! Although this sounds like an impossible fine-tuning task, some mechanism appears to have done precisely this for our Universe.
p.186
In summary, this means that although the fundamental equations of physics (those of string theory, perhaps) remain valid throughout the Level II multiverse, the apparent laws of physics that observers will uncover can change from one Level I multiverse to another. In other words, these apparent laws are universal not in the dictionary sense of “always applicable,” but only in the literal sense of “applicable in our Universe.” They’re multiversal only at Level I, not at Level II. The fundamental equations, however, are multiversal even at Level II
p.187
My colleague Eddie Farhi likes to call Alan Guth “The Enabler,” because eternal inflation enables everything that can happen to actually happen: inflation produces space for it to take place and creates initial conditions allowing the story to play itself out. In other words, inflation is a process converting potentiality into reality.
p.188
My friend Andrew Hamilton from Chapter 4 may have uncovered such a universe-creation mechanism: he’s investigated an instability that occurs inside black holes shortly after they form, and it may be violent enough to trigger inflation that would create a Level I multiverse—which would be entirely contained inside the original black hole, but its inhabitants would probably neither know nor care about this fact.
p.197
1 There are 80 kinds of stable atoms, containing all numbers of protons from 1 (hydrogen) through 82 (lead), except for 43 (technetium) and 61 (promethium), which are radioactive and unstable. Many of these atoms have more than one stable version corresponding to different numbers of neutrons (so-called isotopes); the total number of stable atomic isotopes is 257. There are about 338 isotopes found naturally here on Earth, if we also count about 30 isotopes with half lives longer than 80 million years and about 50 more short-lived ones.
p.201
if we could slam them [particles] together with vastly (perhaps ten trillion times) more energy than today, we’d discover that everything is made of tiny vibrating strings, and that different types of vibrations of the same basic type of strings would correspond to different types of particles a bit like different types of vibrations of a guitar string correspond to different musical notes. The rival theory known as loop quantum gravity suggests that everything is made not of strings, but of a so-called spin network of quantized loops of excited gravitational fields
p.205
Particle-Physics Cheat Sheet
MOMENTUM - The punch something packs if it crashes into somethingor, more rigorously, the amount of time it would take you to stop it times the average force with which you’d need to push it
ANGULAR MOMENTUM - How much something spins or, more rigorously, the amount of time it would take you to make it stop spinning times the average torque (twisting force) you’d need to use
SPIN - The angular momentum of a single particle spinning around its center
CONSERVED QUANTITY - Quantity that remains constant over time and can neither be created nor destroyed. Examples: energy, momentum, angular momentum, electric charge
ATOM - Electrons orbiting around a nucleus of protons and neutrons; the number of protons in an atom determines its name (1 = hydrogen, 2 = helium, etc.)
ELECTRON - Negatively charged particle that electric currents are made of
PROTON - Positively charged particle found in atomic nuclei, made of two up quarks and a down quark
NEUTRON - Particle without electric charge that’s found in atomic nuclei, made of two down quarks and an up quark
PHOTON - Particle of light
GLUON - Particle that help glue quark triplets together into protons and neutrons
NEUTRINO - Particle that’s so stealthy that it can usually pass right through Earth without interacting with anything
FERMION - Particle that can’t be in the same place and state as an identical particle. Examples: electrons, quarks, neutrinos
BOSON - Particle that likes to be in the same place and state as an identical particle. Examples: photons, gluons, Higgs particle
p.218
Specifically, if you experimentally go looking for the electron, you find that the square of the wavefunction gives the probability that you’ll find it in different places, so some physicists like to think of the wavefunction as describing a probability cloud or probability wave. In particular, you’ll never find a particle in places where its wavefunction equals zero.
p.220
After much debate and discussion, Bohr and Heisenberg came up with a remarkably radical remedy that became known as the Copenhagen interpretation, which to this day is taught and advocated in most quantum-mechanics textbooks. A key part of it is to add a loophole to the second item mentioned above, postulating that change is only governed by the Schrödinger equation part of the time, depending on whether an observation is taking place. Specifically, if something is not being observed, then its wavefunction changes according to the Schrödinger equation, but if it is being observed, then its wavefunction collapses so that you find the object only in one place. This collapse process is both abrupt and fundamentally random, and the probability that you find the particle in any particular place is given by the square of the wavefunction.
p.221
Quantum-Mechanics Cheat Sheet
WAVEFUNCTION - Mathematical entity describing the quantum state of an object. The wavefunction of a particle describes the extent to which it’s in different places
SUPERPOSITION - Quantum-mechanical situation where something is in more than one state at once, for example in two different places
SCHRODINGER EQUATION - Equation that lets us predict how the wavefunction will change in the future
HILBERT SPACE - Abstract mathematical space where the wavefunction lives
WAVEFUNCTION COLLAPSE - Hypothesized random process whereby the wavefunction changes abruptly in violation of the Schrödinger equation, giving a measurement a definite outcome. Lack of wavefunction collapse implies Hugh
Everett’s Level III multiverse
MEASUREMENT PROBLEM - The controversial question of what happens to the wavefunction during a quantum measurement: does it collapse or not?
COPENHAGEN INTERPRETATION - A set of assumptions including that the wavefunction collapses during measurements
EVERETT INTERPRETATION - The assumption that the wavefunction never collapses—implies the Level III multiverse (Chapter 8)
DECOHERENCE - A censorship effect derivable from the Schrödinger equation, whereby superpositions become unobservable unless they’re kept secret from the rest of the world—makes the wavefunction appear to collapse during measurements even if it actually doesn’t (Chapter 8)
QUANTUM IMMORTALITY - The idea that we’re subjectively eternal if the Level III multiverse exists. I suspect that there’s no quantum immortality because the continuum is an illusion
p.222
Another aspect of collapse that caused consternation was that observation was upgraded to such a central concept. When Bohr exclaimed, “No reality without observation!” it seemed to put humans back on center stage. After Copernicus, Darwin and others had gradually deflated our human hubris and warned against our egocentric tendencies to assume that everything revolved around us, the Copenhagen interpretation made it seem as if we humans in some sense created reality by just looking at it.
p.237
3 The wavefunction corresponds to a single point in this infinite-dimensional space, and the Schrödinger equation says that this point will orbit around the center of the space at a fixed distance.
p.245
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. If the secret gets out, all quantum superposition effects become unobservable, and it’s for all practical purposes as if it’s either here or there and you simply don’t know which. If a lab technician measures the position and writes it
down, the information is obviously out. 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.246
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. Finally I understood why we never see macroscopic objects in two places at once even if they’re in two places at once: it’s not because they’re big, but because they’re hard to isolate!...And if you can’t make the information secret again, then the quantum superposition can’t be restored. Now I finally understood why Level III parallel universes stay parallel!
p.258
It’s convenient to decompose the world into three parts: the part corresponding to your subjective perceptions (the subject), the part being studied (the object), and everything else (the environment). As indicated, the interactions between these three parts cause qualitatively very different effects, providing a unified picture including both decoherence and apparent wavefunction collapse.
p.259
After all, talking about observations without mentioning the mind feels a bit like discussing nearsightedness
without mentioning the eye. I think the explanation is that, since we don’t understand how consciousness works, most physicists feel uncomfortable even talking about it, fearing that they’ll get regarded as too philosophical. Personally, I feel that just because we don’t understand something doesn’t mean that we can ignore it and still expect to get correct answers...However, to understand Figure 8.8, the details of how your mind works don’t matter at all: the only assumption I’m making here is that your subjective consciousness results in some way from the remarkably complicated motions of the particles that make up your brain, and that these particles obey the Schrödinger equation just as all other particles do.
p.261
The way I see it, these two great controversies of quantum mechanics and thermodynamics are linked, in the sense that they can both be resolved in one fell swoop if we use the standard quantum-mechanics definition of entropy (given by John von Neumann), reject wavefunction collapse, and consider all parts of reality: subject, object and environment.
p.262
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. Decoherence is simply a measurement that you don’t know the outcome of. More rigorously, we can reformulate the second law of thermodynamics in a more nuanced way:
1. The object’s entropy can’t decrease unless it interacts with the subject.
2. The object’s entropy can’t increase unless it interacts with the environment.
p.265
It was a great pleasure and encouragement to talk to you in Copenhagen as I believe you share my belief that under and behind quantum mechanics lies some deep and wonderful principle yet to be discovered, as Einstein’s great geometric idea threw unexpected light on the power and scope of Newton’s supposedly all-embracing theory. The likelihood of such a discovery is surely proportional to our belief that there is something there to be discovered. -- John Wheeler
p.285
The wavefunction and Hilbert space, which constitute arguably the most fundamental physical reality, are purely mathematical objects.
p.292
et’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.
p.293
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.
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.
p.294
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.
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.
p.299
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 selfaware observers agree on
INTERNAL REALITY - The way you subjectively perceive the external reality
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)
p.314
As if that weren’t enough mathematical goodies, there are also quantities encoded in nature that aren’t whole numbers, but require decimals to write out. Nature encodes 32 such fundamental numbers according to my latest count.
p.320
When we derive the consequences of a theory, we introduce new concepts and words for them, such as protons, atoms, molecules, cells and stars, because they’re convenient. It’s important to remember, however, that it’s we humans who create these concepts; in principle, everything could be calculated without this baggage.
p.321
Another striking fact is that you can often predict the existence of such name-worthy entities mathematically, from the equations governing their parts. In this way, the whole Lego-like hierarchy of structures that we discussed in Chapter 7 can be predicted, from elementary particles to atoms to molecules, and what we humans add are merely catchy names for the objects at each level. For example, if you solve the Schrödinger equation for five or fewer quarks, it turns out that there are only two fairly stable ways for them to be arranged: either as a clump of two up quarks and a down quark or as a clump of two down quarks and an up quark, and we humans have added the baggage of calling these two types of clumps “protons” and “neutrons” for convenience. Similarly, if you apply the Schrödinger equation to such clumps, it turns out that there are only 257 stable ways for them to be assembled together. We humans have added the baggage of calling these proton/neutron assemblies “atomic nuclei,” and have also invented specific names for each kind: hydrogen, helium, etc. The Schrödinger equation also lets you calculate all the ways of putting atoms together into larger objects, but this time, there turn out to be so many different stable objects that it’s inconvenient to name them all—instead, we’ve just named important classes of objects (such as “molecules” and “crystals”), and the most common or interesting objects in each class (e.g., “water,” “graphite,” “diamond”). I think of these composite objects as emergent, in the sense that they emerge as solutions of equations involving only more fundamental objects. This emergence is subtle and easy to miss because historically, the scientific process has mostly gone in the opposite direction:
p.323
At the beginning of this section, I argued that such a complete description must be devoid of any human baggage. This means that it must contain no concepts at all! In other words, it must be a purely mathematical theory, with no explanations or “postulates” as in quantum textbooks (mathematicians are perfectly capable of—and often pride themselves on —studying abstract mathematical structures that lack any intrinsic meaning or connection with physical concepts). Rather, an infinitely intelligent mathematician should be able to derive the entire theory tree of Figure 10.5 from these equations alone, by deriving the properties of the physical reality that they describe, the properties of its inhabitants, their perceptions of the world, and even the words they invent. This purely mathematical theory of everything could potentially turn out to be simple enough to describe with equations that fit on a T-shirt.
p.325
Modern mathematics is the formal study of structures that can be defined in a purely abstract way, without any human baggage. Think of mathematical symbols as mere labels without intrinsic meaning. It doesn’t matter whether you write, “Two plus two equals four,” “2 + 2 = 4,” or “Dos más dos es igual a cuatro.” The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don’t invent mathematical structures—we discover them, and invent only the notation for describing them.
In summary, there are two key points to take away from our discussion above:
1. The External Reality Hypothesis implies that a “theory of everything” (a complete description of our external physical reality) has no baggage.
2. Something that has a complete baggage-free description is precisely a mathematical structure.
p.326
Taken together, this implies the Mathematical Universe Hypothesis, i.e., that the external physical reality described by the ToE is a mathematical structure.1 So the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure.
p.333
Mathematical Universe Hypothesis offers a radical solution to this problem: at the bottom level, reality is a mathematical structure, so its parts have no intrinsic properties at all! In other words, the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.1 The external physical reality is therefore more than the sum of its parts, in the sense that it can have many interesting properties while its parts have no intrinsic properties at all.
p.334
MATHEMATICAL UNIVERSE CHEAT SHEET
----------------------------------
BAGGAGE - Concepts and words that are invented by us humans for convenience, which aren’t necessary for describing the external physical reality
MATHEMATICAL STRUCTURE - Set of abstract entities with relations between them; can be described in a baggage-independent way
EQUIVALENCE - Two descriptions of mathematical structures are equivalent if there’s a correspondence between them that preserves all relations; if two mathematical structures have equivalent descriptions, they are one and the same
SYMMETRY - The property of remaining unchanged when transformed; for example, a perfect sphere is unchanged when rotated
EXTERNAL REALITY HYPOTHESIS - The hypothesis that there exists an external physical reality completely independent of us humans
MATHEMATICAL UNIVERSE HYPOSTHESIS - The hypothesis that our external physical reality is a mathematical structure; I argue that this follows from the External Reality Hypothesis
COMPUTABLE UNIVERSE HYPOTHESIS - Our external physical reality is a mathematical structure defined by computable functions (Chapter 12)
FINITE UNIVERSE HYPOTHESIS - Our external physical reality is a finite mathematical structure (Chapter 12)
p.336
Intriguingly, one of the most important discoveries in physics has been that our physical reality also has symmetries built into it: for example, the laws of physics have rotational symmetry, which means that there’s no special direction in our Universe that you can call “up.” They also appear to have translation (sideways shifting) symmetry, meaning that there’s no special place that we can call the center of space.
p.338
This means that our physical world not only is described by mathematics, but that it is mathematical (a mathematical structure), making us self-aware parts of a giant mathematical object.
The MUH solves the infamous infinite regress problem where the properties of nature can only be explained from the properties of its parts, which require further explanation, ad infinitum: the properties of nature stem not from properties of its ultimate building blocks (which have no properties at all), but from the relations between these building blocks.
p.352
In my opinion, we don’t yet fully understand what we are. Moreover, as we discussed in Chapter 9, we don’t really need to fully understand the mysteries of consciousness to understand our external physicalreality. Nonetheless, I feel that modern physics has provided some tantalizing hints about fruitful ways of viewing ourselves, so let’s explore this topic further.
p.353
However, in broad brushstrokes, we might say this: You’re a pattern in spacetime. A mathematical pattern. Specifically, you’re a braid in spacetime—indeed one of the most elaborate braids known.
p.388
In fairness to inflation, I don’t feel that there’s any competing cosmological theory on the market that does any better, so I don’t view this as an argument against inflation per se. I simply feel strongly that we need to solve the measure problem, and my guess is that once we solve it, some form of inflation will still remain. Moreover, the measure problem isn’t limited to inflation, but crops up in any theory with infinitely many observers.
p.390
In fact, I have two suspects: “infinitely big” and “infinitely small.” By infinitely big, I mean the idea that space can have infinite volume, that time can continue forever, and that there can be infinitely many physical objects. By infinitely small, I mean the continuum: the idea that even a liter of space contains an infinite number of points, that space can be stretched out indefinitely without anything bad happening, and that there are quantities in nature that can vary continuously. The two are closely related: we saw in Chapter 5 that inflation created an infinite volume by stretching continuous space indefinitely.
p.391
Carl Friedrich Gauss, sometimes referred to as “the greatest mathematician since antiquity,” had this to say two centuries ago: “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.”
p.392
-Mathematical structures are eternal and unchanging: they don’t exist in space and time—rather, space and time exist in (some of) them. If cosmic history were a movie, then the mathematical structure would be the entire DVD.
- The Mathematical Universe Hypothesis (MUH) implies that the flow of time is an illusion, as is change.
- The MUH implies that creation and destruction are illusions, since they involve change.
- The MUH implies that it’s not only spacetime that is a mathematical structure, but also all the stuff therein, including the particles that we’re made of. Mathematically, this stuff seems to correspond to “fields”: numbers at each point in spacetime that encode what’s there.
- The MUH implies that you’re a self-aware substructure that is part of the mathematical structure. In Einstein’s theory of gravity, you’re a remarkably complex braidlike structure in spacetime, whose intricate pattern corresponds to information processing and self-awareness. In quantum mechanics, your braid pattern branches like a tree.
- The movielike subjective reality that you’re perceiving right now exists only in your head, as part of your brain’s reality model, and it includes not merely edited highlights of here and now, but also a selection of prerecorded distant and past events, giving the illusion that time flows.
- You’re self-aware rather than just aware because your brain’s reality model includes a model of yourself and your relation to the outside world: your perceptions of a subjective vantage point you call “I” are qualia, just as your subjective perceptions of “red” and “sweet” are.
- The theory that our external physical reality is perfectly described by a mathematical structure while still not being one is 100% unscientific in the sense of making no observable predictions whatsoever.
- You should expect your current observer moment to be a typical one among all observer moments that feel like you. Such reasoning leads to controversial conclusions regarding the end of humanity, the stability of our Universe, the validity of cosmological inflation, and whether you’re a disembodied brain or simulation.
- It also leads to the so-called measure problem, a serious scientific crisis that calls into question the ability of physics to predict anything at all.
p.400
Interestingly, in the context of the Mathematical Universe Hypothesis (MUH), the existence of the Level IV multiverse isn’t optional. As we discussed in detail in the previous chapter, the MUH says that a mathematical structure is our external physical reality, rather than being merely a description thereof. This equivalence between physical and mathematical existence means that if a mathematical structure contains a self-aware substructure, it will perceive itself as existing in a physically real universe, just as you and I do (albeit generically a universe with different properties from ours). Stephen Hawking famously asked, “What is it that breathes fire into the equations and makes a universe for them to describe?” In the context of the MUH, there’s thus no firebreathing 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 doesn’t exist in space and time—space and time may exist in it. In other words, all structures that exist mathematically have the same ontological status, and the most interesting question isn’t which ones exist physically (they all do), but which ones contain life—and perhaps us.
p.409
All such uncertainties about undecidability and inconsistency apply only to mathematical structures with infinitely many elements. Are infinities, undecidability and potential inconsistency really inherent in the ultimate physical reality, or are they merely mirages, artifacts of our playing with fire and using powerful mathematical tools that are more convenient to work with than those that actually describe our Universe? More specifically, how well defined do mathematical structures need to be to be real, i.e., to be members of the Level IV multiverse?
p.415
I’ve drawn a question mark at the center of the triangle to suggest that the three vertices (mathematical structures, formal systems and computations) are simply different aspects of one underlying transcendent structure whose nature we still don’t fully understand. This structure (perhaps restricted to the defined/decidable/halting part as per the CUH) exists “out there” in a baggage-free way, and is both the totality of what has mathematical existence and the totality of what has physical existence.
p.416
If we turn our attention to some particular mathematical structure on the master list that serves as our atlas of the Level IV multiverse, how can we derive the physical properties that a self-aware observer in it would perceive it to have? In other words, how would an infinitely intelligent mathematician start with its mathematical definition and derive the physics description that we called the “consensus reality” in Chapter 9?1 We argued in Chapter 10 that her first step would be to calculate what symmetries the mathematical structure has. 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.The question of what she should calculate next when exploring an arbitrary structure is largely uncharted territory, but I find it striking that in the particular mathematical structure that we inhabit, further study of its symmetries has led to a gold mine of further insights. The German mathematician Emmy Noether proved in 1915 that each continuous symmetry of our mathematical structure leads to a so-called conservation law of physics, whereby some quantity is guaranteed to stay constant— and thereby has the sort of permanence that might make self-aware observers take note of it and give it a “baggage” name. All the conserved quantities that we discussed in Chapter 7 correspond to such symmetries: for example, energy corresponds to time-translation symmetry (that our laws of physics stay the same for all time), momentum corresponds to space-translation symmetry (that the laws are the same everywhere), angular momentum corresponds to rotation symmetry (that empty space has no special “up” direction) and electric charge corresponds to a certain symmetry of quantum mechanics. The Hungarian physicist Eugene Wigner went on to show that these symmetries also dictated all the quantum properties that particles can have, including mass and spin. In other words, between the two of them, Noether and Wigner showed that, at least in our own mathematical structure, studying the symmetries reveals what sort of “stuff” can exist in it.
p.418
...the Level IV multiverse provides a very different starting point for the subject, and this causes most traditional physics concepts to be reinterpreted. As we just saw, some concepts such as symmetries retain their central status. In contrast, other concepts, such as initial conditions, complexity and randomness, get reinterpreted as mere illusions, existing only in the mind of the beholder and not in the external physical reality.