BOOK REVIEW: The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner

The Unreasonable Effectiveness of Mathematics in the Natural SciencesThe Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Paul Wigner
My rating: 4 of 5 stars

Available online here

This brief essay asks why math proves so effective for describing / codifying physical laws, and whether our physical theories — built on (phenomenally successful) mathematics — offer the truth, the whole truth, and nothing but the truth.

There’s a popular story in which a drunk man is found on his hands and knees under a lamppost at night when a police officer comes along. The cops says, “What-cha doin’?” To which the drunk replies, “I dropped my keys, and I’m looking for them?” So, the cop says, “Well, they’re clearly not where you’re looking, why not look elsewhere?” And the drunk says, “Cuz this is where the light is.” I think this story can help us understand what Wigner is getting on about, if only we replace the drunk’s “light” with the scientist’s “elegant mathematics.” Wigner reflects upon why it should be that so many laws of nature seem to be independent from all but a few variables (which is the only way scientists could have discovered them –historically, mathematically, and realistically speaking.) On the other hand, could it be that Physics has led itself into epistemological cul-de-sacs by chasing elegant mathematics?

There’s no doubt that (for whatever the reason turns out to be) mathematics has been tremendously successful in facilitating the construction of theories that make predictions that can be tested with high levels of accuracy. However, that doesn’t mean that some of those theories won’t prove to be mirages.

A few of the examples used in this paper are somewhat esoteric and won’t be readily understood by the average (non-expert) reader. That said, Wigner puts his basic arguments and questions in reasonably clear (if academic) language. The essay is definitely worth reading for its thought-provoking insights.

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BOOK REVIEW: The Joy of X by Steven H. Strogatz

The Joy of X: A Guided Tour of Mathematics, from One to InfinityThe Joy of X: A Guided Tour of Mathematics, from One to Infinity by Steven H. Strogatz
My rating: 5 of 5 stars Page

This is a mile-wide and inch-deep overview of mathematics. That is to say, it shines a light on a wide variety of subdisciplines, running from counting through subjects like topology, using rudimentary examples to give the reader insight into the kind of problems that can be solved. The book employs graphics, intuitive examples, and step-by-step explanation to clarify mathematics for individuals who didn’t get on so well with the subject the first time around.

The book’s thirty short chapters are divided into six sections: numbers (arithmetic,) relationships (roots, powers, etc.,) shapes (geometry,) change (calculus,) data (statistics,) and frontiers (group theory, topology, analysis, etc.) Like most popular mathematics books, formulas and equations are avoided to the extent possible. Even the notes that elaborate for curious readers use mathematical notation sparingly.

If you’re looking to give math a second try, this wouldn’t be a bad overview to get started. I don’t think it would be of much benefit to anyone who’s stayed in touch with mathematics, but it makes a fine light overview of the subject.

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BOOK REVIEW: New Theories of Everything by John D. Barrow

New Theories of EverythingNew Theories of Everything by John D. Barrow
My rating: 5 of 5 stars

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This book reflects upon the various elements that any Theory of Everything (ToE) would have to reconcile. A ToE is the holy grail of physics, a theory that would unify the various forces to explain the nature of the universe as we experience it. There have been many attempts to achieve a ToE, but it remains elusive. There is the mathematically beautiful and elegant string theory that suffers that one drawback of having no experimental support. There are those who have given up on a ToE in the sense that the term is normally used, suggesting that the desired degree of unification isn’t possible and that the desire to think it must be is just wishful thinking.

Probably the most useful piece of information about this book for one considering reading it is its readability. As works of popular science go, it’s more challenging that most (but not as difficult as, for example, Hawking’s “A Brief History of Time.”) [I have little doubt that those who read physics textbooks will find it a walk in the park.] It has few equations, and the mathematics it does present is elementary. However, it does explore quite complicated ideas. The book uses graphics to assist, mostly diagrams, but many of these require thoughtful consideration in their own right.

The organization of the book is based on an eightfold way (no relation to the Buddhist eightfold path) – that is, eight ingredients with which a ToE must be consistent. The nine chapters of the book begin with a brief opening chapter that sets up the rest of the book by discussing what a ToE would really explain (“everything” isn’t necessarily the answer in a strict meaning of that word), what the eight components are, how pre-scientific ToE’s operated, as well as introducing the recurring concept of algorithmic compressibility. (The importance of compressibility lies in the idea that in order to make the equations describing the universe more concise it’s necessary that the data describing the universe be “compressible” – i.e. have some underlying order.)

After the intro chapter, the eight subsequent chapters are logically arranged into the aforementioned eightfold way. These are: 1.) laws, 2.) initial conditions, 3.) the nature of forces and particles, 4.) the constants of nature, 5.) symmetries and the breaking thereof, 6.) organizing principles, 7.) Bias and selection effects, and 8.) to what extent mathematics is integral to the universe. Some of these elements (e.g. the laws and constants) we are told couldn’t vary by much and allow us to still exist. So, the question addressed in the book isn’t only how can science get to a theory that explains the existence of a stable(-ish) universe, but further one that can support complex and intelligent life. The chapters on symmetry breaking and selection effects are particularly relevant to this discussion.

One of the most interesting discussions is the last. Chapter nine, entitled: “Is ‘pi’ really in the sky?” discusses the question of how fundamental mathematics is to the universe. It’s long been a topic of scientific intrigue that there seems to be no particular reason for mathematics to be as effective as it is at describing the way the universe works. The discussion has resulted in a wide range of replies from those who say the success of mathematics is more illusory and limited than it appears to be, to those who believe the universe not only is written in mathematics but is math (see: the work of Max Tegmark.) That is, some say that there are parts of a stable universe that must be orderly enough to be described mathematically and those are the only parts we truly understand as of yet. Others say mathematics is the bedrock of the universe.

I enjoyed this book, and found the organizational approach helped a great deal in thinking about the problem. I doubt I grasped everything the author was trying to convey, but it was a book piled high with food for thought for anyone interested in thinking about the nature of the universe. If you’re interested in the grand-scale questions, I’d recommend this book. That said, there are more readable takes on the subject out there if one is looking for light pop science fare.

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BOOK REVIEW: Our Mathematical Universe by Max Tegmark

Our Mathematical Universe: My Quest for the Ultimate Nature of RealityOur Mathematical Universe: My Quest for the Ultimate Nature of Reality by Max Tegmark
My rating: 4 of 5 stars

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In this book, physicist Max Tegmark makes an argument for the possibility of a reality in which the universe is a mathematical structure a theory that predicts a Level IV multiverse (i.e. one in which various universes all have different physical laws and aren’t spread out across one infinite space [i.e. not “side-by-side.”]) Nobel Laureate Eugene Wigner wrote a famous paper entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The article describes one of the great mysteries of science, namely, how come mathematics describes our universe so well and with such high precision. Tegmark’s answer is because the universe is fundamentally mathematical—or at least he suspects it could be.

The first chapter serves as an introduction, setting the stage by considering the core question with which the book is concerned, “What is reality?” The book then proceeds in three parts. The first, Chapters 2 through 6, discuss the universe at the scale of the cosmos. Chapters two and three consider space and time and answer such questions as how big is the universe and where did everything come from. Chapter 4 explores many examples of mathematics’ “unreasonable effectiveness” in explaining our universe with respect to expansion and background radiation and the like (a more extensive discussion is in Ch. 10.) The fifth chapter investigates the big bang and our universe’s inflation. The last chapter in part one introduces the idea of multiverses and how the idea of multiple universes acts as an alternative explanation to prevailing notions in quantum physics (e.g. collapsing wave functions)—and, specifically, Tegmark describes the details of the first two of four models of the multiverse (i.e. the ones in which parallel universes are out there spread out across and infinite space), leaving the other two for the latter parts of the book.

Part two takes readers from the cosmological scale to the quantum scale, reflecting upon the nature of reality at the smallest scales—i.e. where the world gets weird. Chapter 7 is entitled “Cosmic Legos” and, as such, it describes the building blocks of our world as well as the oddities, anomalies, and counter-intuitive characteristics of the quantum realm. Chapter 8 brings in the Level III approach to multiverses and explains how it negates the need for waveform collapse that mainstream physics requires we accept (i.e. instead of a random outcome upon observation, both [or multiple] outcomes transpire as universes split.)

The final part is where Tegmark dives into his own theory. The first two parts having outlined what we know about the universe, and some of the major remaining mysteries left unexplained or unsubstantiated by current theories, Tegmark now makes his argument for why the Mathematical Universe Hypothesis (MUH) is at least as effective at explaining reality as any out there, and how it might eliminate some daunting mysteries.

Chapter 9 goes back to the topic of the first chapter, namely the nature of reality and the differences between our subjective internal reality, objective external reality, and a middling consensus reality. Chapter 10 also elaborates on the nature of reality, but this time by exploring mathematical and physical reality. Here he elaborates on how the universe behaves mathematically and explains the nature of mathematical structures—which is important as he is arguing the universe and everything in it may be one. Chapter 11 is entitled, “Is Time and Illusion?” and it proposes there is a block of space-time and our experience of time is an artifact of how we ride our world lines through it—in this view we are braids in space-time of the most complex kind observed. A lot of this chapter is about what we are and are not. Chapter 12 explains the Level IV multiverse (different laws for each universe) and what it does for us that the others do not. Chapter 13 is a bit different. It describes how we might destroy ourselves or die out, but that, it seems, is mostly a set up for a pep talk. You see, Tegmark has hypothesized a universe in which one might feel random and inconsequential, and so he wants to ensure the reader that that isn’t the case so that we don’t decide to plop down and watch the world burn.

While this book is about 4/5ths pop science physics book, the other 1/5th is a memoir of Tegmark’s trials and tribulations in coloring outside the lines with his science. All and all, I think this serves the book. The author avoids coming off as whiny in the way that scientists often do when writing about their challenges in obtaining funding and / or navigating a path to tenure that is sufficiently novel but not so heterodox as to be scandalous. There’s just enough to give you the feeling that he’s suffered for his science without making him seem ungrateful or like he has a martyr complex.

Graphics are presented throughout (photos, computer renderings, graphs, diagrams, etc.), and are essential because the book deals in complex concepts that aren’t easily translated from mathematics through text description and into a layman’s visualization. The book has endnotes to expand and clarify on points, some of which are mathematical—though not all. It also has recommended reading section to help the reader expand their understanding of the subject.

I enjoyed this book and found it to be loaded with food-for-thought. If Tegmark’s vision of the universe does prove to be meritorious, it will change our approach to the world. And, if not, it will make good fodder for sci-fi.

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BOOK REVIEW: Fooling Houdini by Alex Stone

Fooling Houdini: Magicians, Mentalists, Math Geeks, and the Hidden Powers of the MindFooling Houdini: Magicians, Mentalists, Math Geeks, and the Hidden Powers of the Mind by Alex Stone
My rating: 4 of 5 stars

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This book operates on two levels. The first is the autobiography of a magician, telling a tale from being gonged off stage at the “Magic Olympics” through a rising obsession with the craft before rolling into his redemption. On a second level, it’s a history of magic in the modern age (although there are occasional forays into more ancient history.) The author tells of the magicians that inspired him, some of whom he learned from personally and some were from the preceding generation, such as Dai Vernon—the magician who actually fooled Houdini. However, the book’s title doesn’t come from Vernon’s feat with the Ambitious Card Trick, but is instead a more general statement about the challenge of tricking magicians—an accomplishment a great deal more prestigious than fooling a pod of eight year olds at little Timmy’s birthday party.

Stone was a science writer turned Physics graduate student, and so the science of magic and mentalism comes out frequently. However, this book is distinct from one such as “Sleights of Mind” by Macknik & Martinez-Conde, which is focused entirely upon conveying the science of how magic tricks work (primarily neuroscience with a focus on how the sense organs and brain interact to a magician’s advantage.) In truth, I expected this book to more along the lines of “Sleights of Mind.” However, in a way, it’s a good thing that it wasn’t. Stone reviews the science that Macknik and Martinez-Conde drill down into enough so that it’s a good review if one has read that book (I had) or an introduction if one hasn’t. What Stone does a great deal more of is describing the perfection one’s craft. Along the way he shows us a blind card handler with a preternatural capacity for tactile control of the deck, he takes us to clown college to improve showmanship, and he meets up with some street hustlers of the 3-card monte variety.

Throughout the course of the book are ups and downs that maintain the tension. In fact, one chapter is actually entitled “It’s Annoying and I Asked You to Stop,” about the inevitable point at which a magician’s obsession with improving his/her skills stops being cute to loved ones. There is also a chapter about Stone’s [almost] being blackballed from the magic community for revealing secrets in a general readership magazine (I guess that’s a muggle-mag?) An important part of the story is Stone’s search for a Yoda, a wizened member of the magic community who can give him the deeper insight needed to fool a room of experts. He eventually finds said individual, but is not quickly adopted. (It has a hero’s journey feel through this part.)

I thought that the author did a good job of building an interesting story arc within a work of nonfiction. This increases the book’s readability, particular if one has no particular interest in magic. One need not be knowledgeable about the discipline to find the story interesting and to learn some fascinating tidbits. If nothing else, one will learn how con men cull marks, so one can avoid falling prey to their potent psychology (though I expect the subset of readers of books and those tricked into gambling 3-card monte is probably not huge.)

One area in which a reader might be dissatisfied is in the coverage given to mentalism and math-based tricks. The alliterative subtitle makes reference to “magicians, mentalists, and math geeks…” but the bulk of the book is about close-up magic; mentalism and mathematical methods don’t come in until the last few chapters. If you’re expecting that the coverage of those topics will be on par with that of close-up magic, this may not be the book for you. Still, while this was different from I expected, it didn’t hurt my impression of the book.

I enjoyed this book, and received some intriguing insights from it. I’d recommend it for those interested in magic and in particular the craft and science of it. Even if you aren’t that interested in magic, you might find the story of one man’s development of his discipline to be worth reading.

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BOOK REVIEW: Antifragile by Nassim Nicholas Taleb

Antifragile: Things That Gain from DisorderAntifragile: Things That Gain from Disorder by Nassim Nicholas Taleb

My rating: 4 of 5 stars

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Nassim Nicholas Taleb has a gift for uncovering simple and fascinating topics that have remained buried–not because they are unfathomable, but–because of the institutional blinders and group-think present in academia (at least within the social sciences.) I don’t mean to diminish what Dr. Taleb does by saying these are simple ideas, it takes a great intellect to not only recognize the ideas others have missed but to clarify them for a broad audience and to unravel the challenging ideas that must be made clear as one moves beyond the crux of the idea. Furthermore, it takes a bold writer to push these ideas out into the open against brute institutional antagonism. (If Taleb hadn’t written books that were highly readable and that presented the ideas in a manner readily digested by a broad audience, he’d likely still be being completely ignored by academicians.)

By “simple” I mean ideas that can be captured in a single sentence—often a pithy one at that. In his second book (his first work for popular audiences), Fooled by Randomness, the idea was that randomness is more pervasive than most people imagine and that false explanations are often built for chance occurrences. Black Swan told us that statistical forecasting fails catastrophically when one has “800 pound gorillas” in the data set (e.g. if one is comparing countries—a situation in which one will virtually always be in, as Taleb calls it, “Extremistan.”) The book in question, Antifragile, is built around the notion that some entities get stronger when subjected to stressors and disorder.

One can see many “antifragile” elements in one’s own body. A muscle subjected to exercise often gets tiny tears in fibers, but when the body does its repair work those fibers will be stronger than ever before. Wolff’s Law tells us that bones subjected to an increased load will increase their density. In fact, our bodies are testaments to the concept of antifragility on many levels. For this reason, Taleb uses many examples from the field of medicine—in addition to those from disciplines more closely related to his own, e.g. finance, economics, and risk. A lot of the medical discussion deals with the proclivity of Western medicine towards interventionism (in contrast to the “first, do no harm” motto often heard.) An example with which many people are familiar is that of the over prescription of antibiotics. While there are obviously cases for which antibiotics are necessary and beneficial, prescribing them willy-nilly robs the body of antifragility (i.e. if the body defeats the infection itself, it has inborn resistance.)

As with other of Dr. Taleb’s writings, I found Antifragile to be interesting as well as informative. The author does a good job of providing examples to elucidate and bolster his arguments and puts it all together in a readable package. He also does a great job of pulling examples and discussions from a number of different fields. This book doesn’t read like it’s about an Economics or Business subfield as much as it’s a book that can teach you something applicable to whatever your field might be. The book also covers a number of other critical but related ideas, such as the value of heuristics in decision-making, how antifragility can be increased (and fragility reduced), and the ethical issues involved.

My primary criticism is that the book overdoes the jabs at scholars and economists. I can understand where Taleb might have some pent-up rage against many academics. He has certainly had to weather a lot of equally petty assaults from the academics who loath him. The work of many a social scientist and economist looks pretty silly to those who grasp the concepts Taleb is presenting. Still, we got it. Halfway through the book, one wonders why Taleb is still so vigorously and maniacally whipping such a skeletal horse. While it’s hard to imagine anyone less strong-willed than Dr. Taleb could get these messages out in the face of the institutionalized opposition he faced, the flip side is that he will probably strike you as a pretentious jackass on occasion.

The book is organized into seven sections (each of multiple chapters.) It begins by describing antifragility and then proceeds through relevant concepts like optionality, nonlinearity, via negativa, and ethics. The book has handy appendices for those who prefer graphic or mathematical representations. (Like all popular science / social science works, there’s an attempt to keep the overly technical and visually intimidating material out of the body of the work.) There is also a works cited section.

I’d recommend this book for those interested in wonky type books.

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BOOK REVIEW: The Simpsons and Their Mathematical Secrets by Simon Singh

The Simpsons and Their Mathematical SecretsThe Simpsons and Their Mathematical Secrets by Simon Singh

My rating: 4 of 5 stars

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It will come as no surprise that television comedy writers are disproportionately Ivy League educated individuals. What may come as a surprise is that a number of comedies—particularly animated series—have a large number of technically and mathematically educated individuals on their writing staffs. Mathematicians, computer scientists, engineers, and physicists regularly work in hidden humor that only a math geek could love—or get—into episodes of The Simpsons and Futurama. Singh’s book explores the subtle mathematical references and humor that swoosh over the heads of most viewers.

While the title doesn’t mention Futurama, it should be noted that there are four chapters devoted to that series. (This in contrast to the 14 chapters dedicated to the much older show, The Simpsons.)

Let’s assume that nerds can be categorized into three sets: nerds, super-nerds, and mega-nerds. This book takes as its core demographic the largest of these groups, run-of-the-mill nerds. How does one define these three apparently arbitrary designations? A mega-nerd would see the humor in the equation scrawled on a blackboard in the background as he (or she) watched an episode of The Simpsons. (All Hail, King of the Nerds!) A super-nerd wouldn’t get many of these jokes as he (or she) watched, but he would freeze-frame the scene, and would have enough mathematical skill to decipher the cryptic jokes. A regular nerd misses the joke altogether, but is interested enough to take the time to read an explanation of these obscure references. (These categories are contrasted with the typical TV viewer, who doesn’t get the joke, but is blissful in his ignorance.)

While much of the book is devoted to these series’ mathematical gags—which range from the elementary to the arcane—Singh offers interesting insight into the writing process on shows with a team that mixes traditional writers (English and Literature majors) with mathematical types. One of the most interesting behind-the-scenes questions is why mathematical writers work so well for the The Simpsons? Futurama, being a science fiction series–and thus aimed at the geek/nerd nexus, isn’t so much a surprise, but Homer and his family don’t have any motive to be particularly mathematical—with the possible exception of the occasional reference by brainy Lisa. The chapters are arranged by various mathematical themes, such as prime numbers, pi, statistics, topology, etc.

There are some ancillary sections that deserve mention. First, there are a series of “quizzes” that consist of jokes with the set ups written as the question and the punchline serving as the answer. These jokes get progressively more complicated—starting with crude elementary school jokes (e.g. “Why did 5 eat 6?”) and ranging to the truly obscure (e.g. “What’s big, grey, and proves the uncountability of the decimal numbers?” The answer, if you’re wondering, is “Cantor’s Diagonal Elephant.”) Second, there are five appendices that are used to go into more mathematical depth on some of the topics under discussion. This is written as a book for the masses, and so attempts are made to minimize and simplify equations. There are equations and graphic representations, but they’re kept at a relatively elementary level of mathematics.

I enjoyed reading this book and would recommend it for anyone who—like me–kind of likes mathematics, but finds it more palatable with a spoonful of sugar. In this case, the sugar is the discussion of the humorous scenes of these two comedies.

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