Our Mathematical Universe: My Quest for the Ultimate Nature of Reality

by: Max Tegmark, read in 2015

6 “[W]e’ll explore fascinating relations…and explore a crazy-sounding belief of mine…that…leads to a…collection of parallel universes…forcing us to relinquish many of our most deeply ingrained notions of reality.” Your beliefs force us? Really Max?
8 “I use [the word reality] to mean the ultimate nature of the outside physical world that we’re part of” This definition excludes some possibilities, e.g. Plato’s forms, the Mental World, and the Spiritual World.
11 “Feynman…: “I believe it is that […] all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart but repelling upon being squeezed into one another.””
14 “Reality means very different things to different people. I use the word to mean the ultimate nature of the outside physical world that we’re part of” “outside” of what? And who’s “we” white man?
43 In 1922 Alexander Friedmann predicted the Big Bang – was ignored.
45 In 1927 Georges Lemaitre predicted the Big Bang – was ignored.
50 “George Gamow[‘s] Ph.D. advisor in Leningrad was…Alexander Friedmann”
61 I think that “the exotic explanation” of a toroidal universe should not be ruled out so quickly. It might be right.
119 “[T]he simplest and most popular cosmological model today predicts that this person actually exists in a galaxy about 10^10^29 meters from here.” Did the two universes derive from a common precursor? If so, how did they get so far apart?
120 By defining ‘Physical Reality’ as everything that exists”, Max either tacitly, and unwarrantedly, assumes that nothing non-physical exists, or he has deliberately rendered the word ‘physical’ to be superfluous and meaningless.
121 “By our very definition of universe…” But Max, you have not defined ‘universe’!
121 “Yet the epistemological borderline between physics and metaphysics is defined by whether a theory is experimentally testable, not by whether it’s weird or involves unobservable entities.” Very sloppy, Max. Popper’s criterion defines ‘science’, not the “borderline” you mentioned.
121 “Technology-powered experimental breakthroughs have therefore expanded the frontiers of physics to incorporate ever more abstract (and at the time counterintuitive) concepts…” True. Now we need to allow consciousness-powered experiential breakthroughs to do the same.
121 “Suppose inflation really happened and made our space infinite.” What a HUGE presumptuous suggestion. First, you can’t define ‘infinite’ in a sensible way and if you take Cantor’s approach, inconsistency comes with it according to Gödel. Not a fruitful premise IMHO.
122 “[T]he seed fluctuations responsible for all cosmic structure were generated by quantum fluctuations which are for all practical purposes random” I don’t think you can usefully define either ‘practical’ or ‘random’ without a considerable conscious component.
123 “Although each of these outcomes occur in an infinite number of universes, some occur in a larger fraction than others” The notion of a “larger fraction” is just one example of a concept that can’t be defined in a consistent way given your acceptance of “infinity”.
126 “[T]he Level I multiverse, including your doppelgängers, is a logical consequence of eternal inflation.” Yes, but another logical consequence of eternality, i.e. infinity, is that it implies inconsistency.
126 “1. Infinite space and matter: Early on, there was an infinite space filled with hot expanding plasma.
2. Random seeds: Early on, a mechanism operated such that any region could receive any possible seed fluctuations, seemingly at random.” I doubt both of these assumptions. First, I doubt that space or anything else that is real can be infinite. Second, whence the “mechanism”? And how can ‘randomness’ be defined in the absence of consciousness? “Seemingly” to whom? To us now long after the fact? Or to some Primordial Consciousness?
128 “[O]ur telescopes don’t show infinite copies of the Milky Way lined up in tidy rows.” That pattern shouldn’t be expected. If the torus is large enough, there wouldn’t have been time for even one circuit. If the torus were smaller, then the paths of the circuits would not be regular since the paths around would be compound helices that would typically require many turns around the minor diameter before returning, if ever.
128 “[W]e can look for a recognizable object…and then look for the same object…in the opposite direction. Such searches have also come up empty-handed.” We shouldn’t expect the image to show up in the opposite direction. A little reflection while looking at a bagel makes this clear. The simplistic video-game-screen analogy does not completely or correctly reflect the geometry or the topology of a torus.
130 “According to classical physics, a universe can be arranged in infinitely many different ways…” Only if it contains an infinite number of constituents.
130 “[T]he total number of ways in which our Universe can be arranged is finite.” As I was saying.
131 “Although 10^10^118 is huge beyond astronomical, it’s still nothing compared with infinity.” It’s reassuring that Max acknowledges this fact. He seems to have ignored it in some previous comments.
146 “[E]ight planets whose orbits were 1.84, 2.51, 4.33, 12.7, 24.7, 51.1, and 76.5 times larger than that of the innermost planet… “
148 “If you increase the number of space dimensions beyond three, there can be neither stable solar systems nor stable atoms. For instance, going to a four-dimensional space changes Newton’s inverse square law for the gravitational force to an inverse-cube law, for which there are no stable orbits whatsoever.” Not true. If physical constituents are confined to a 4D spacetime manifold, then they behave just as they seem to in our world.
148 “Changing the number of time dimensions isn’t as absurd as you might think, and Einstein’s theory of general relativity can handle this just fine.” I never thought the idea was absurd.
148 “The entire Level II multiverse exists in a single space, so how can the dimensionality vary within it?” Here’s how: the “single space” could consist of, say, eleven dimensions with embedded manifolds of many different dimensionalities
149 “Well, according to the most popular string-theory models, it’s only the apparent dimensionality that varies: the true space always has nine dimensions, but we don’t notice six of them because they’re microscopically curled up…” Yes, popular, but it’s too bad the purveyors of this idea don’t understand, or at least consider, embedded manifolds.
152 “In so-called braneworld scenarios, another three-dimensional world could be quite literally parallel to ours, a short distance away in a higher dimension. However, I don’t think that such a world (brane) deserves to be called a parallel universe separate from our own, since it can interact with it gravitationally much as we do with dark matter.” You may rule out the designation semantically, but the possibility still exists that gravity doesn’t interact between them, and even that the other forces do also.
152 “In my opinion, our job as scientists isn’t to tell our Universe how to work in order to conform to our human prejudice, but to look at it with open minds and try to figure out how it actually works.” Good. Now I suggest that Tegmark take his own advice.
153 I’m surprised and disappointed that Max did not discuss Everett’s Many Worlds. Are they equivalent to, or related to, the Level II Multiverse? If so, are they causally related by entanglement? If not, how do you resolve the mystery of quantum state reduction?
165 “[S]ome particle physicists like to glibly answer the question “What’s a particle?” by saying, “It’s an element of an irreducible representation of the symmetry group of the Lagrangian.”“
167 “[T]he strength of the oscillation (the maximum number of volts you measure) determines the intensity of the light.“
167 “…the intensity (the number of photons).” Which is it?
174 “Mathematically, the uncertainty for each quantity is defined as the standard deviation of its probability distribution.“
176 “[A] wavefunction, written ?,…describes the extent to which the particle is in different places.“
178 “This wavefunction [collapse] process is both abrupt and fundamentally random…” How do we know that??
178 “If wavefunction collapse really happened, then this would mean that a fundamental randomness was built into the laws of nature. Einstein was deeply unhappy about this interpretation…” Logically and obviously there is a 3rd possibility: Rather than being “random” the outcome of the “collapse” could be the result of a “designer’s” deliberate and conscious choice without “whom” the notion of “observation” is meaningless. In the spirit (so to speak) of the rest of Max’s book, let’s pursue the logical implications as far as they lead whatever they are and however absurd they seem.
181 “[S]uch weirdness amplification happens regularly even within your brain. Whether a given neuron fires at a given time depends on whether the sum of all its input signals exceeds a certain threshold, and this can make neural networks highly unstable…” Why not use this as a starting point for a theory of consciousness? We sorely need such a theory.
183 Good summary of quantum theory.
187 Hugh Everett’s thesis URL
190 What is the justification for the “infinite” number of dimensions for Hilbert space? Only a finite number of interactions and their combinations could have happened since the Big Bang
192 “If you repeat this experiment with four cards…” Very confusing. I think he really means repeat the experiment 4 times with 1 card.
195 “[E]very time something fundamentally random appears to happen to you, which couldn’t have been predicted even in principle, it’s a sign that you’ve been cloned.” How about positing this as a fundamental premise: waveform “collapse” is equivalent to this appearance, or observation?
195 “[Hugh Everett] didn’t get a job in physics, became rather bitter and withdrawn, smoked and drank too much, and died of an early heart attack in 1982.“
196 “Everett’s thesis left one important question unanswered: if a large object can really be in two places at once, why don’t we ever observe that?” An understanding of the properties of manifolds explains this completely and, to me, convincingly. See bottom drawing in Figure 8.1
196 “Sure, if you measure its position, the two copies of you in the two resulting parallel universes will each find it in a definite place.” Yes, but you can’t measure the position of the other guy even though he might be mere inches away. He is outside your manifold.
197 “Large objects…never display wavelike properties that make so-called quantum interference patterns.” There is no interference pattern because there is no interference. Being in separate and distinct manifolds they can’t interact.
197 “It wasn’t just Everett’s thesis that lacked an answer to this puzzle—there were no answers in my textbooks either.” Then I suggest you look elsewhere or else think it through for yourself. Think about the implications and consequences of manifolds.
210 “[T]he 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.” That’s a fair assumption. Just remember this assumption holds for a radio too.
211 “…if the complex neuron-firing patterns in your brain have anything to do with consciousness,…” Kudos for acknowledging that you are making this assumption. It may be false.
211 “The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations—then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation—well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.“
212 “[H]ere’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.” In other words, consciousness is Maxwell’s Demon: It breaks the 2nd Law
213 Read this article.
228 “The Austrian animal behaviorist Konrad Lorenz mused that important scientific discoveries go through three phases: first they’re completely ignored, then they’re violently attacked, and finally they’re brushed aside as well known.“
238 Max’s three views of reality correspond pretty well with Penrose’s three worlds: Mental, Physical, Platonic. Max should close the loop by acknowledging that the mentality of the “Internal Reality” invents, discovers, derives, and “understands” the Mathematical descriptions of his “External Reality”.
239 I don’t see any difference between his “Frog perspective”, his “Reality Model”, or even his ”Internal Reality”.
239 “[T]he challenge for cognitive science is to derive the internal reality from the consensus reality.” With respect, I believe this is a hopelessly counter-productive and inadequate approach.
241 “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.” That’s fine as far as it goes. What’s missing is an explanatory link between his “internal” and “external reality”: The basic questions of philosophy: ontology and epistemology. What exists? How did existence come to be? Who or what knows? Or can know? How can that knower know? Completing this link closes Penrose’s loop.
243 “[T]he universe…is written in the language of mathematics and its characters are triangles, circles, and other geometric figures…--Galileo Galilei” It is noteworthy that Galileo made this observation before Descartes showed that geometrical objects, thus the “universe”, could be described by equations. Do we need another such jump?
243 Crackpot letter to Max.
244 “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.” It has ceased to amuse me.
246 Interesting diagram showing science/pseudoscience boundary. Needs work.
246 “Anthony Aguirre and I started the Foundational Questions Institute I mentioned in Chapter 8, http://fqxi.org“
255 “……such a description must be expressible as a form that’s devoid of any human baggage like “particle,” observation or other English words.
255 “Since the ball is made of elementary particles (quarks and electrons), you could in principle describe its motion without making any reference to basketballs: Particle 1 moves in a parabola…” But you just said that words like “particle” are “human baggage” and can’t be allowed in our description. You are ignoring the involvement of consciousness to the detriment of your theory.
257 “As mentioned, all these theories have two components: mathematical equations and words that explain how they’re connected to what we observe. For example, we saw in Chapter 8 how quantum mechanics, as usually presented in textbooks, has both components: math such as the Schrödinger equation as well as fundamental postulates written out in plain English, such as the wavefunction-collapse postulate.” Again, you are leaving out a third, and most important, component: consciousness. It is consciousness, and consciousness alone, that can see Chapter 8; present QM in textbooks; invent, write, read, and understand equations and postulates; compose, read, and understand “plain English”. Without this involvement, there is no theory and no mathematics. If reality is mathematical, then without consciousness there is no reality. Therefore, consciousness must be a fundamental component of reality. Max’s hypotheses imply it.
257 “[I]n principle, everything could have been derived from the fundamental theory at the top of the tree…” Not without consciousness. “Derivation” is a conscious act.
258 “I argued that such a complete description must be devoid of any human baggage. This means that it must contain no concepts at all!” But max, mathematics is nothing but concepts!
258 “[I]t 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).” Max is conflating the notion of “concepts” with the “meaning” in a physical sense. The former is a necessary part of mathematics; the latter is a necessary part of science. If you eliminate the latter (baggage) you are left with Hilbert’s 6th problem.
259 “All of this begs the question: is it actually possible to find such a description of the external reality that involves no baggage?” This is Hilbert’s 6th problem. Dr. Dick may have solved it. But if the definition of “baggage” includes any and all conscious involvement, then it is impossible. “Description” is meaningless in the absence of consciousness.
259 “Modern mathematics is the formal study of structures that can be defined in a purely abstract way, without any human baggage.” With respect, I think the use of the term “baggage” ruins this definition. I like mine better: Mathematics is the combined efforts of a group of people who have agreed to be very careful in their use of language.
259 Interesting perspective on the distinction between mathematical discovery and invention.
260 “[I]f you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure.” Fair enough. And I believe this is equivalent to Plato’s view: External reality is the Platonic world of forms.
280 “[I]f some mathematical equations completely describe both our external physical reality and a mathematical structure, then our external physical reality and the mathematical structure are one and the same, and then the Mathematical Universe Hypothesis is true: our external physical reality is a mathematical structure.” It seems to me that the qualifier “completely” is daunting. E.g. it would seem that the phenomenon of experience might be missing from one.
281 “[I]f someone wishes to avoid accepting the Mathematical Universe Hypothesis, they can do so by rejecting the External Reality Hypothesis that there’s an external physical reality completely independent of us humans. They could then argue that our Universe is somehow made of stuff perfectly described by a mathematical structure, but which also has other properties that aren’t described by it and can’t be described in an abstract, human—independent, baggage-free way. However, I think this viewpoint would make the famous science philosopher Karl Popper from Chapter 6 turn in his grave, since he emphasized that scientific theories must have observable effects. In contrast, since the mathematical description is supposedly perfect, accounting for everything that can be observed, those additional bells and whistles that would make our Universe nonmathematical would by definition have no observable effects whatsoever, rendering them 100% unscientific.” This logic cries out for a definition of “us humans”. Exactly where does conscious experience exist? Popper insists on observable effects, but what is his justification for ruling out the possibility of additional unobservable effects?
281 “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 physical reality.” Um! Yes. As I was saying, if you close Penrose’s loop, which I think you must, then you do need to understand consciousness.
283 “There’s broad consensus in the scientific community that we still don’t understand how this [self-awareness] works, so it’s fair to say that we humans don’t yet fully understand what we are.” The “consensus” view, in this case, is no doubt right.
283 “However, in broad brushstrokes, we might say this: You’re a pattern in spacetime.” You could say that, but I disagree. Your description is of a remote controlled vehicle i.e. controlled by a remote operator who exists beyond Popper’s view.
285 “All your subjective perceptions exist in spacetime, just as every scene of a movie exists on its DVD.” Be careful, Max. You have conflated “scenes” and “perceptions”. The scenes exist in the DVD track but they are perceived only when the DVD is played and observed by a perceiver.
285 “Let’s refer to each such perception as an “observer moment”” You have not shown that “observer moments” exist in spacetime.
285 “How does the connecting work? Specifically, is there some sort of rule for which observer moments feel connected, and why does this connected sequence of observer moments subjectively feel like time flowing?” Beon Theory explains it.
289 “I think that consciousness is the way information feels when being processed in certain complex ways…” This idea is just as incomplete and ambiguous as Nagel’ request for a metaphor explaining “What it is like to…” How does the “information feel” to whom?
289 “The arrows above indicate information flow. For example, information input from your senses continually helps your world model track key aspects of what’s actually going on in the external reality, and information output via your motor cortex controls your muscles to affect the external reality, say, by turning a page in this book.” You could easily program an identical information flow into a computer and it would not make the computer or the program conscious.
290 “Which leads to the ultimate consciousness question: who’s looking at your brain’s reality model, to give rise to subjective consciousness?” Good question.
290 “Here’s my guess: nobody!” Unsatisfactory answer.
290 “I think that consciousness is the way information feels when being processed in certain complex ways.” Again, “feels” to whom? Machines can’t feel.
290 “In short, redness and self-awareness are both qualia.” And both are equally unexplained.
293 Good thought experiment trying to identify SASes (Self-Aware Substructures). But consider a parallel, and IMHO equivalent, thought experiment of going back in time, presenting a lot of modern devices to Isaac Newton and asking him to figure out which ones are radios.
293 “[M]y guess is that part of what it does is measure both the complexity and the information content of the object you point it toward.” That would be a bad guess in my thought experiment. A computer would pass and a crystal radio would fail.
293 “There are rigorous mathematical definitions of information content in mathematics and physics, tracing back to the work of Claude Shannon and John von Neumann over half a century ago.” Rigorous though they might be, they are nonetheless inadequate IMHO because they leave out the very conscious component you are trying to explore.
294 “Koch and Tononi[:]…for an information processing system to be conscious, it needs to be integrated into a unified whole that can’t be decomposed into nearly independent parts…this is closely linked to so-called redundancy and error-correcting codes used in bar codes, hard drives, mobile telephony and other modern information technology.” Then bar code readers and hard drives should be self-aware. They aren’t.
295 “My guess is that we’ll one day understand consciousness as yet another phase of matter.“
295 “What about “perceptronium,” the most general substance that feels subjectively self-aware? If Tononi is right, then it should not merely have the traits of computronium, but also the property that its information is indivisible, forming a unified whole.” This is an incomplete and unsatisfactory description of “subjective self-awareness”.
297 “We perceive ourselves as local” This suggests John’s “Stylus Guy”.
298 “[O]ur well-defined local and unique identity exists only in our internal reality; at a fundamental level, it’s an illusion.” Illusion?? Exactly who is deceived? Stylus Guy explains the uniqueness and isolation much better.
298 “Self-awareness would then be a side effect of this advanced information processing. More generally, any SAS that’s either evolved or engineered with a purpose might have self-awareness as a by-product of having an internal model of the world and itself.” I doubt that that is possible.
299 Max’s argument for why the immune system is not self-aware is not convincing.
300 “To me, science is all about understanding reality and our place in it.” Yes, unfortunately that’s what it’s all about. Too bad scientists are oblivious to the incompleteness evident in the cavalier use of the term “our”. They overlook Stylus Guy.
300 “[M]y guess is that it’s to help accomplish this task that we’ve been fortunate enough to evolve consciousness.” Bad guess IMHO
300 Bayesian Decision Theory. Sort of like my quantitative view of belief: numbers from 0 to 1.
301 Max’s “awesome remote control for reality” would be possible only for someone like Stylus Guy.
301 Good graphic illustration of Max’s view of causality, immortality, and randomness.
302 “While my senses are hard at work recording new information to be consciously perceived in future observer moments, the conscious part of my mind is busy using my scientific-reasoning algorithm to update my assumptions about more subtle and abstract aspects of reality.” It is an unwarranted assumption that the senses and consciousness operate in the same temporal dimension.
305 “We’re living proof that atoms can be put together in an elaborate pattern that subjectively feels self-aware.” This is evidence but not proof. What exactly “feels self-aware”?
307 The idea of Boltzmann Brains is an example of the nonsense that comes with the acceptance of the notion of infinity.
313 “We cosmologists use the term measure to refer to an observer-moment ordering scheme, or, more generally, to a method for calculating probabilities from annoying infinities.” I suggest that you admit the obvious fact that the notion of infinity introduces inconsistency into the body of mathematics. This should not be allowed. We should expunge infinity from mathematics.
313 “[I]t appears flawed to talk about the reference class of all observer moments at a fixed time.” As Russell’s Paradox demonstrates, it is “flawed” to talk about the notion of “all” at all!
314 “It’s proven remarkably difficult to find one [alternative measure] that works with eternal inflation” IMHO that is because eternal implies infinite.
314 “If the probabilities we predict depend on the measure we assume, and we can assume a measure giving almost any answer we want, then we really haven’t predicted anything at all…In fact, I view the measure problem as the greatest crisis in physics today.“
314 “All predictions are revoked, including those predictions that made us take inflation seriously in the first place! Self-destruction complete.” Good reason to dump infinity.
314 “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 suggest you consider astronomically large >4D C-Y spaces
314 “Moreover, the measure problem isn’t limited to inflation, but crops up in any theory with infinitely many observers.” As I was saying. In fact it crops up any time infinity is mentioned.
316 “Nobody knows for sure where the root of the problem lies, but I have my suspicions. Here’s my prime suspect: 8“ WONDERFUL! You really had me going, Max.
316 “Without infinity, there’d be no measure problem” Whew!
317 “Norman Wildberger has posted an essay arguing that “real numbers are a joke.”“
319 “…an embarrassing question from my hero John Wheeler that still resonates unanswered, Why these particular equations, not others?” I think Dr. Dick has answered this question.
319 “In `1916 the box labeled “GENERAL RELATIVITY” was a serious candidate for being an exact match, containing within it not only space and time but also various forms of matter, but the discovery of quantum mechanics soon made clear that our own physical reality had features that this particular mathematical structure lacked. Fortunately, you can now extend the figure by adding the mathematical structure that you’ve discovered and will get your prize for, knowing that this new box in the figure is the box, the one that corresponds to our physical reality.” I suggest we develop Practical Number math and develop GR within that. Quantum graininess will then be a natural feature.
320 “All boxes are on an equal mathematical footing, corresponding to different mathematical structures, so why should some be more equal than others when it comes to physical existence? Could there really be a fundamental, unexplained existential asymmetry built into the very heart of reality, splitting mathematical structures into two classes—those with and without physical existence?” The answer is a resounding “YES”. The asymmetry is revealed in Penrose’s three worlds and in Descartes’ Dualism. The crucial world you are overlooking is the Mental world which alone can populate the Platonic world (your Mathematical Structures) as well as “Breathe[] fire into the equations” to create a physical world. This is Greylorn’s Beon, John’s Stylus Guy, and my PC.
321 “In 1990,…I first had the mathematical universe idea…“
321 “This can be viewed as a form of radical Platonism, asserting that all the mathematical structures in Plato’s “realm of ideas” exist “out there” in a physical sense.” Good!
321 “…all properties of all parallel universes (including the subjective perceptions of self-aware substructures in them) could in principle be derived by an infinitely intelligent mathematician.” Why contaminate the idea of this “mathematician” with the admittedly flawed notion of infinity?
323 “…all structures that exist mathematically have the same ontological status…” Huge unwarranted assumption. This implies an ontological starting point vastly more complex (and unlikely) than any religion’s creator or Big Bang initial conditions.
326 URL for Max’s Mathematical Universe paper.
327 “A nice feature of this list is that it can also handle mathematical structures with infinitely many elements.” I disagree. Mathematica has never handled an infinite number of anything. At best it handles large finite numbers which are putatively part of an infinite set, which in fact, doesn’t exist.
327 “Mathematical structures involving infinitely many points on a continuum,…can often be well approximated by finite structures…” Nay, “can only be well approximated…”
327 “…that is how my colleagues and I perform most of our theoretical-physics calculations in practice.” Is there any other way???
330 “However, embracing infinity opens a Pandora’s box of ontological problems.” Yes! In particular, mathematical inconsistency. (Gödel)
331 “Alternatively, Kurt Gödel’s work might make us worry that the MUH makes no sense with infinite mathematical structures because our Universe would be somehow inconsistent or undefined.” This is not just a worry, it’s an indisputable fact.
331 “If one accepts the mathematician David Hilbert’s dictum that “mathematical existence is merely freedom from contradiction,” then an inconsistent structure would not exist mathematically,” let alone physically as in the MUH.” Yes, I accept the dictum, but I am baffled as to how Hilbert could have accepted Cantor’s nonsense.
332 “All such uncertainties about undecidability and inconsistency apply only to mathematical structures with infinitely many elements.” Yes! Take the hint!
332 I think all computable structures are finite structures, and vice-versa.
333 “I have no doubt that nature knows what it’s doing.” Hmmm. What exactly is this “nature” that can “know”? How about beon, Stylus Guy, or PC?
333 “By banishing the continuum altogether, perhaps the CUH may also help downsize the inflationary landscape and resolve the cosmological measure problem” Yes! Exactly what I have been suggesting.
334 “However, my guess is that if the CUH turns out to be correct, it will instead be because the rest of the mathematical landscape was a mere illusion, fundamentally undefined and simply not existing in any meaningful sense.” Good guess!
334 “Even approaches attempting to banish the classical spacetime continuum by discretizing or quantizing it tend to maintain continuous variables in other aspects of the theory, such as the strength of the electromagnetic field or the amplitude of the quantum wavefunction.” I think Practical Numbers can solve this problem.
335 “…we may have multiple layers of effective continuous and discrete descriptions on top of what’s ultimately a discrete computable structure.” We certainly may.
337 “…how can we derive the physical properties [of some particular mathematical structure] 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?” Dr. Dick claims to have answered this question.
338 Symmetries and conservation laws. Noether and Wigner.
347 “Making this argument repeatedly, you conclude that you’re probably a simulation within a simulation within a simulation, and so on, arbitrarily many levels down—a reductio ad absurdum” A hasty conclusion. Consider Matryoshka Dolls
355 “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 features in terms of physical entities.“
370 “Our space isn’t expanding uniformly: indeed, some regions, such as our Galaxy, aren’t expanding at all.“
382 “I believe that consciousness is the way information feels when being processed.” Feels to whom?
390 “…education and adherence to a scientific lifestyle is arguably deteriorating further in many countries, including the United States. Why? Clearly because there are powerful forces pushing in the opposite direction, and they’re pushing more effectively.” I agree.
390 “Corporations concerned that a better understanding of certain scientific issues would harm their profits have an incentive to muddy the waters, as do fringe religious groups concerned that questioning their pseudo-scientific claims would erode their power.” These pale in comparison to the influence of the 60s counter culture.
390 “…the anti-scientific coalition…” What is this phantom? AGW??
392 “I find it utterly remarkable that it’s possible for a bunch of particles to be self-aware…” That may be because it isn’t true. I think Max Tegmark is something more than a bunch of particles.
395 “There’s probably no highly intelligent life within…our own Galaxy or its immediate vicinity.“
398 “Evidence suggests that there’s no other life-form as advanced as us humans in our entire Universe.“
398 “Although it’s easy to feel insignificant in our vast cosmos, the entire future of life in our Universe will arguably be decided on our planet in our lifetime—by you, me and our fellow passengers on Spaceship Earth. Let’s make a difference!“

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