This review is an attempt to summarize and understand Tegmark's two grand theories, meant to solve the two great conundrums of reality:
1. First-Origin conundrum -- Why does any reality exist at all? and
2. Uniqueness conundrum -- Why does our particular type of reality exist (and not some other)?
Therefore, this review is mainly for those who have already read his book and are trying to decide whether or not his ideas are true (or even make sense).
After first treating the reader to a history of cosmology, inflation, and quantum physics (along with a wide variety of resulting multiverses), Tegmark arrives at his two grand theories.
The first is the Postal Code theory of the fundamental constants of our universe. According to Tegmark, there are 32 fundamental constants, with very precise values. If these values were slightly different, our universe would not have been stable enough to support life.
Tegmark informs us that, when the planets were first discovered, scientists tried to explain why they had the particular properties (sizes, distances from the sun) that they have. However, as more star systems were discovered, scientists realized that there is no deep explanation needed for our planets' size or orbits; other than the fact that we are located where we are; that is, a Postal Code that locates our particular planetary system within our particular galaxy.
Tegmark applies this Postal Code theory to explain the Uniqueness conundrum (actually, to explain away the need to explain it). Just as our planetary system is one of a great many, if multiverse theories are correct, then our universe is just one of a (possibly infinite) number of universes.
Recall that a level-I multiverse consists an infinite 'sea' of universes, all with the same laws of physics, but with a different set of fundamental constants. Most of these would be inhospitable for life. However, according to the Anthropic Principle (first coined by the theoretical astrophysicist Brandon Carter), we necessarily find ourselves in a universe with fundamental constants set to just the right values to support life (because, if not, we wouldn't now be around to wonder why these constants have the values that they have).
Recall that a level-II multiverse is also an infinite 'sea' of universes, but each with a different set of physical laws. A level-III multiverse is an infinitely branching set of alternative futures (and pasts) with each caused by the probabilistically distinct outcomes of the Schrödinger's wave equation. When a given quantum outcome is observed, the multiverse level-III answer to why that particular outcome (and not some other) is: "All the other outcomes actually did occur, but in different level-III alternative universes."
These 3 different levels of multiverses are compatible with each other since all could be true at once. There could exist a sea of different laws of physics, within which each would contain a sea of different physical constants, within which each would exhibit a tangle of branching futures and pasts.
If a level-I multiverse exists then there is no point in looking for an explanation for the particular values of these 32 fundamental constants. They are analogous to the sizes and orbits of our planets. In his Postal Code theory, when asked why Nature exhibits those specific, 32 fundamental values, the appropriate reply, according to Tegmark, should be to ask back: "Which universe are you referring to?" The answer to this question will be a Postal Code -- one giving the location of our particular Universe, within the hierarchy of the first 3 levels of multiverses.
The second grand theory in Tegmark's book is his "Realty=Mathematics" theory. Almost all scientists believe that mathematical models can be used as descriptions of reality. Tegmark, however, claims that reality actually is mathematics and nothing but mathematics. Tegmark claims that reality exists simply because mathematical structures exist and because reality consists only of mathematical structures. As a result, each different mathematical structure brings about a different reality. A mathematical structure is any configuration of mathematical entities and relationships. Tegmark's level-IV multiverse consists of an infinite set of different mathematical structures.
Tegmark believes that, at the fundamental level of reality, there is just mathematics. Suppose you see a photon ph1 in location loc1 at time t1. Suppose at time t2, you see the photon disappear at loc1 and re-appear at a nearby location loc2. It might seem that photon ph1 has traveled from loc1 to loc2 during time interval [t1, t2]. However, since you can't tag ph1 you can't be sure that it's the same photon. Nature could have simply made ph1 disappear at loc1 and then could have created a completely new photon ph2 in loc2 at time t2. There is no way to know. Since all fundamental particles are this way, they therefore behave only according to their mathematical properties. In this sense, they are not just described by mathematics, they are mathematical and only mathematical. That is, there is no property that they have that is not mathematical.
I am familiar with this point of view (that what appears to be an abstraction might actually be real) because, as a computer scientist, I concluded many years ago that, if our universe were a simulation (inside some supercomputer created by an advanced alien civilization), then we, as inhabitants within that simulated universe, would not be able to prove whether or not we are simulated beings.
If this simulation scenario happens to be the case, then the most fundamental unit of reality (for us, situated in our simulated universe) would be information. That is, the alien supercomputer would be manipulating bits of information which we, and our universe, would be composed of. Let us call this theory of reality: Reality = Information + Execution, or more concisely, Reality = Computation.
In this case, information would be more fundamental than electrons and photons. Some physicists actually do take this position -- that information is more fundamental in our universe than physics. For example, the physicist Viatko Vedral holds this point of view.
In traditional computer science, however, physical matter/energy is more fundamental than information because the computer makes use of patterns of matter/energy to create information. The computer also makes use of the laws of physics to manipulate this information over time. Executing the computer with different programs then creates different simulations.
A computer can simulate different realities. For example, a computer scientist can, by programming a computer, create a simulation of an virtual world (say, a 2-dimensional world) containing a population of 2-D artificial animals (which we will here term "animats"). Each animat's behavior (both how it senses and manipulates its environment) could be controlled by, say, a network of simulated neurons.
Students who take my graduate "Animats Modeling" class (in the CS dept. at UCLA) commonly build just such virtual universes. Animats can mate, produce offspring, and evolve, as the result of simulated mutations to their simulated genes.
Most computer scientists believe that, if we are virtual creatures, then the alien programmer must be currently executing the program that brings our universe into being. Without execution our universe would not come into existence or exhibit its dynamics.
For Tegmark's Reality=Mathematics theory to have a chance of being correct he needs to first eliminate the requirement that something must be executing; otherwise his theory would be identical to that of Reality=Computation.
The problem with the Reality=Computation theory is that it does not solve the First-Origin conundrum. The alien programmer will not be able to execute an infinite number of universes that are postulated in multiverse theories and, more importantly, we would still be left with having to explain how the matter, energy, and physical laws of the alien's universe came about. After all, the alien's computer needs its own universe (with its own laws of physics) in order to execute the program that creates our universe. Thus, we are still left with a First-Origin conundrum.
However, if Tegmark can eliminate the need for execution (i.e. dynamic changes in the memory of a computer over time) then he can replace Reality=Computation with Reality=Mathematics.
Tegmark achieves this by pointing out that Einstein's space-time theory views time as an illusion. Instead of time moving forward (like a river), time is statically laid out, just like another spatial dimension. In the space-time theory of reality, the past and future have equal status. They both co-exist within a single, static space-time geometry. That is, there is no special present "moment" that is moving along. Motion also does not exist (since motion is how time is measured). In his book Tegmark gives the example of the moon going around the Earth as the Earth goes around the Sun. A space-time diagram of this situation is displayed as a kind of bent slinky. All temporal dynamics have been eliminated.
Why do we still experience the illusion of motion (i.e., change over time) if there is no motion of any sort? This is not explained by Tegmark. Consider a movie. We know that the motion of the characters that we see in a movie is an illusion. The movie consists of a series of static frames that are displayed in sequence as time unfolds and so the illusion is not of motion; the illusion is of SMOOTH (as opposed to jerky) character motion. The explanation is that our brains create representations that transform the jerky motion into smooth, continuous character motion. If, however, the frames were never displayed over time (rather, all laid out statically, within a space-time coordinate system), then it would be difficult to explain why we experience the movie as unfolding (because neither we nor the projector would be moving). (It must be difficult to explain the illusion of motion -- given a space-time view of the universe -- because I have not yet come upon a reasonable explanation for this illusion in my readings on space-time theory.)
But let us leave this problem aside and accept Tegmark's premise: namely, that motion (and therefore time also) is an illusion. Thus, the execution of some computer (to create a virtual reality) is no longer required. All that is required now is the mathematical structure of space-time itself!
Where does this space-time structure come from? Well, it comes from mathematics, which supplies such structures. Tegmark now applies his Postal Code strategy: He states that all possible mathematical structures exist within a level-IV multiverse of mathematical structures. Since we exist, we must exist within one of these structures. Using the Anthropic Principle, we must exist within a mathematical reality that is structured coherently enough to support life.
Tegmark argues that mathematical structures exist independent of our awareness of them. For example, if we place 2 things next to 3 other things, we will get 5 things (whether or not we are there to notice). Since mathematical structures exist independent of our minds and since reality is fundamentally mathematical, voila!, reality comes into being simply because mathematical structures have their own independent existence!
Tegmark's solution to the First-Origin conundrum can now be summarized as:
a. Mathematical structures exist independently of anyone's mind.
b. Reality=Mathematics postulates that every mathematical structure gives rise to a reality that conforms to the mathematics of that structure.
c. Given that Reality=Mathematics, there must exist a multiverse of every possible mathematical structure (i.e., Tegmark's level IV multiverse).
d. As a result we will find ourselves existing within one of these mathematically structured universes and the explanation for "why this particular mathematical structure?" is that our Postal Code specifies also which mathematical structure we inhabit. In addition, it must be a mathematical structure stable enough to support life (due to the Anthropic Principle).
If Tegmark's Reality=Mathematics theory is correct, then he would have explained the first (and most difficult) conundrum; namely, the First-Origin of reality (which includes all level I, II and III multiverses).
I have spent much of my review summarizing my understanding of Tegmark's theories of reality. I would now like to offer a critique of these theories. This critique I term the "Ugly Math" critique.
Tegmark seems to only consider BEAUTIFUL mathematical structures when discussing his level-IV multiverse. I am now going to examine the nature of mathematics more closely. Since Tegmark does not mention "UGLY Mathematics" in his level-IV multiverse, I am going to create a level-V multiverse; namely, a multiverse which contains universes brought into existence by the existence of UGLY mathematical structures.
I argue here that mathematicians tend to concentrate on just the beautiful mathematical structures and avoid the ugly ones. By "beautiful" I mean mathematical structures that are: concise, self-consistent, symmetrical, have broad scope or generality, are useful, appear to be true, and so on. In contrast, ugly mathematical structures lack one or more of these elements of beauty.
Consider the function of addition (+). This function consists of an infinite mapping of pairs of numbers into a corresponding single number. For example, the function + includes the following 3 mappings:
(2, 3) --> 5
(-4, 33) --> 29
(3.2, 1.1) --> 4.3
+ is a very beautiful and useful mathematical structure and appears in every elementary math text book. However, let us consider some ugly variants of +. Consider the function +ugly1,1. I define this function to be:
+ugly1,1: same as + for every pair of numbers except, if you attempt to add 7 and 19, you get -3.
So +ugly1 is just like + except that it behaves differently on one particular number pair. This single exception makes it a different function.
I am sure that most mathematicians never consider such a function. They don't consider +ugly1,1 because it lacks the conciseness of +; it lacks the consistency of +; it lacks the generality of +, and so on. However, I must emphasize:
+ugly1,1 is just as much a mathematical structure as is +!
How many ugly variants of the function + are there? Well there exists only one + function but there are an infinite number of ugly variants! For example, I can define +ugly1,2 as follows:
+ugly1,2: same as +ugly1,1 except if you try to add (12300.3 + 21.5) you get 12021
So this function has two pairs (among an infinite number of pairs) that deviate from +.
Clearly, there exist an enormous number of +ugly functions. They can deviate from + in terms of the value that a given pair will map to and they can also deviate from + in terms of the number of pairs that happen to be exceptional. For every beautiful mathematical structure there will exist a countless number of ugly variants and all of these ugly variants are just as mathematical as the beautiful ones!
Concentration on beauty occurs in many areas outside of mathematics. For example, visual artists tend to focus on images that are beautiful. However, the vast majority of images are ugly -- they look like noise on an analog TV late at night. The beautiful images (e.g., the Mona Lisa, a cartoon sketch, a photograph of a child, a minimalist, impressionist or surrealist painting, etc.) constitute only a tiny, tiny fraction of the abstract image space, which consists overwhelmingly of noisy, blurry, dirty, fuzzy, mushy, unrecognizable images.
Likewise, mathematicians tend to concentrate so much on beautiful mathematical structures that they forget that the vast majority of abstract mathematical structures are exceedingly ugly (incoherent, non-generalizable, untrue, useless, non-concise, etc.). For every beautiful structure (appearing in some mathematical textbook or theoretical physics book) I can generate an INFINITE number of ugly mathematical structures. These ugly structures far, far, far outnumber the beautiful ones.
If the theory of Reality=Mathematics is true, then the number of ugly universes that exist (created within my level-V multiverse) far outnumber all the other universes existing within Tegmark's level-IV multiverse. If every beautiful mathematical structure gives rise to some universe, then so also must every ugly mathematical structure.
Could a physical reality actually exist this is governed by an ugly mathematical structure? Consider an ugly variant of the + function. We could imagine a universe in which, every time creatures within that universe add two very large numbers A and B, they do not get (A+B); instead they get, say, ((A+B)-1). They are left with one less than what they started with. There are two possibilities in this case: (a) Maybe a missing element has been transformed into something that they cannot currently measure, or (b) maybe their reality is such that, when they have enough of something, they simply get one less when they try to combine them.
I claim that my Reality=UglyMath theory (with its level-V multiverse) is scientific because (like Tegmark's level-IV multiverse) it is also potentially testable and falsifiable.
If a level IV multiverse exists (i.e. Reality=Mathematics) then a level-V multiverse must also exist (i.e. an infinite sea of ugly-math realities) because UglyMath is just as mathematical as is beautiful math. It is overwhelmingly more likely that we live in an ugly-math reality than in a beautiful-math reality. However, the Anthropic Principle states that our reality must be beautiful (consistent, stable) enough so that our universe enabled human life to develop. However, since the ugly-math realities are so much more common, we must conclude also that our universe, wherever possible, will consist of ugly-mathematical structures (but not so ugly as to preclude what we already observe in our universe).
This is a testable (and therefore falsifiable) prediction. Thus, my theory of a level-V multiverse is a scientific (as opposed to religious or metaphysical) theory.
I am not a physicist, but where I would look first (in an attempt to falsify my Reality=UglyMath theory) is in the area of dark energy and the accelerating expansion of the universe. I would look there because that is an area in which our universe could exhibit ugliness without that ugliness having caused our universe to be too unstable to support the development of life. Since ugly mathematical structures will be much more common that beautiful ones, they should dominate any reality. I predict that the pattern of dark energy acceleration should have fluctuated a lot over time (speeding up, then slowing down, then speeding up). This "ugliness" should be more likely if Tegmark's Reality=Mathematics is true and if ALL mathematics structures are considered (not just the beautiful ones). If, however, the acceleration of the expansion of the universe is non-fluctuating, then it is much more likely that my hypothesis, and thus Tegmark's hypothesis also, are both false.
Notice that Tegmark uses the systematicity and stability of our universe as evidence FOR his hypothesis, because he is just considering just mathematically beautiful structures. I view this same systematicity and stability to be evidence AGAINST Tegmark's hypothesis because ugly mathematical structures should always dominate over beautiful ones. If it is the case that mathematics brings about reality, then reality should overwhelmingly tend toward ugliness in its laws (except for where it would violate the Anthropic principle).
I would like to conclude with some additional comments: concerning (a) the effect that Tegmark's Postal Code might have on scientific methodology and (b) the relationship of mathematics to thought and thought's relation to reality.
Regarding methodology: Currently, when a scientist encounters some fundamental feature of reality, the scientist attempts to explain it by postulating a theory in which the observed feature is a necessary result of the theory. In the Postal Code approach, however, fundamental features of reality could, instead, always be explained away (by claiming that there exists, somewhere else, an alternative reality with that feature). This approach could result in a failure to create new theories that might actually explain (as opposed to explain away) some fundamental feature of our reality.
Regarding thought and reality: I do not believe (as Tegmark seems to) that mathematical structures exist separate from our minds. I view the space of all mathematical structures as a subspace in a larger, space of thoughts. Thoughts are representational structures that intelligent minds encode and manipulate in order to survive within their environments. Some of those thoughts are mathematical; however, others (e.g. involving human actions, relationships, political plans, emotions, desires, etc.) are not mathematical.
For example, the symbolic structure:
Believes(Agent(John), Gave(Agent (Fred), Recipient(Mary), Object(Ring),Loc(LincolnMemorial), Time(2/21/2014)),Time(2/22/2014))
is not itself about mathematical objects. It represents the fact that, on Feb. 22, 2014, John believes that Fred gave Mary a ring the day before at the Lincoln Memorial.
The fact that this structure exists in someone's mind (encoded, say, as neuronal firing patterns) or exists inside some artificially intelligent robot (encoded, say, as a symbolic structure) does not force us to conclude that this Believes structure is actually true. It should be clear to everyone that, while there is an enormous space of possible thoughts (including paranoid, delusional and nonsensical thoughts), only a tiny portion of these thoughts will accurately describe some aspect of reality.
I maintain that all mathematical structures are a subset of all possible thoughts. Thus, the existence of mathematical structures (in human minds or in intelligence computers or in intelligent alien minds) does not imply that they have an objective existence apart from those minds.
If minds have conceptual structures that correspond accurately with reality, then those structures will enable those minds to better survive within that reality. For example, if a vehicle is coming at you at high speed and you fail to manipulate internal conceptual representations (about the fact that the vehicle exists and its predicted trajectory) then you will fail to decide to jump out of the vehicle's way and you won't be around to continue having thoughts.
Mathematical structures are also thoughts and thus exist ONLY within minds. It is a mistake to conclude that mathematical thoughts are independent of minds just because some of them happen to maintain a very good correspondence with a wide range of different aspects of reality.
When I teach natural language processing (in a graduate-level course at UCLA titled "Language & Thought") I tell my students that, although I might loosely state:
"A written word W1 has meaning M1."
What I ACTUALLY mean is:
"No written word 'has' a meaning. A written word is just scratches on a piece of paper. Those scratches, when viewed by a human eye, trigger some concept in the mind of that viewer. Thus, meanings exist only in minds, not on pages."
When we see the word "eats" that word triggers, in our minds, thoughts concerning the act of eating; the consequences of eating, etc. The meaning of the word "eat" is not in the word itself, but rather in our minds. In cases where our thoughts do not correspond to reality, then we say that our thoughts are false. Communication among humans is possible only because there is enough overlap in the conceptual structures that get triggered in different minds when different humans encounter the same sequences of words.
People do not normally conclude that, since a given concept can be conceivably thought, there must exist some reality in which that concept is true. Likewise, mathematical thoughts, no matter how beautiful, only exist within minds. When someone places mathematical symbols on a piece of paper, they are just scratches. There is no meaning in them; rather they trigger meanings in appropriately prepared minds. When the eye of a mathematician (with appropriate background knowledge, etc.) sees those scratches, they trigger mathematical thoughts in that mathematician's mind.
I could have postulated a level-VI multiverse, consisting of an infinite sea of all possible thoughts. I could have then claimed that all such thoughts create their own alternative realities and thus claimed that Reality=Thought.
This level-VI multiverse would contain thoughts both about mathematical structures and also about non-mathematical relations, such as, Eats(Agent(John), Object(steak)). But notice that if I were to postulate this level-VI multiverse, I would have to also include in it all weird and crazy thoughts, such as Eats(Agent(steak), Object(John)).
I did not do this (perhaps missing out on a super grand idea). I did not do this because it seems to me that such a theory would be the ultimate form of solipsism and I prefer to believe that reality exists independent of our thoughts about reality.
Let us return to my level-V ugly-math multiverse. A reason for not accepting a level-V multiverse is that it could (like Tegmark's level-IV multiverse) also have a negative effect on scientific methodology. Currently, when deviations from a given theory are found in scientific measurements, scientists first check to make sure that their instruments are properly calibrated. In contrast, if they were to accept the premise that Reality=UglyMath, then they would tend to look for ugly theories to explain those deviations. For example, if a planet did not follow a perfect, elliptical orbit then, instead of first looking for an unseen body influencing that orbit, scientists might simply replace the elliptical formula with an ugly elliptical formula (one containing exceptions in the formula that occur exactly where the deviations in measurements were observed).
Tegmark's book is very insightful, thought-provoking and enjoyable to read; so I do highly recommend it. As he himself has said, he may be wrong (and I have attempted to shown him wrong, by extending his theory, along with making a falsifiable prediction concerning that extension).
I do not have a solution to the two conundrums that Tegmark attempts to solve: (a) First-Origin -- why anything at all exists and (b) Uniqueness -- why our reality is the particular way it is. As to the uniqueness problem, I think that level I, II and III multiverses are a possibility but, given Ockham's Razor, physicists should first consider theories with fewer (or perhaps no) infinities in them. Scientists should accept multiverses only when they are absolutely forced to; that is, when no finite alternatives exist. Perhaps that time has actually arrived in physics and the Postal Code approach is the only viable approach.
In any case, let us not confuse thoughts (embedded within minds and referring to aspects of reality) with reality itself.
Tegmark's book has failed to convince me that mathematical structures cause our reality to come into existence (let alone bringing into existence an infinite number of alternative realities).
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