Mathematicians and scientists have traditionally been very wary of the infinite (indeed, the appearance of infinities is considered an indication of the limitations of theories in modern physics), and Barrow presents any number of paradoxes which illustrate that, as he titles chapter four, “infinity is not a big number”: it is fundamentally different and requires a distinct kind of intuition if nonsensical results are to be avoided. One of the most delightful examples is Zhihong Xia's five-body configuration of point masses which, under Newtonian gravitation, expands to infinite size in finite time. (Don't worry: the finite speed of light, formation of an horizon if two bodies approach too closely, and the emission of gravitational radiation keep this from working in the relativistic universe we inhabit. As the author says [p. 236], “Black holes might seem bad but, like growing old, they are really not so bad when you consider the alternatives.”)
This is an enjoyable and enlightening read, but I found it didn't come up to the standard set by The Book of Nothing and The Constants of Nature (June 2003). Like the latter book, this one is set in a hideously inappropriate font for a work on mathematics: the digit “1” is almost indistinguishable from the letter “I”. If you look very closely at the top serif on the “1” you'll note that it rises toward the right while the “I” has a horizontal top serif. But why go to the trouble of distinguishing the two characters and then making the two glyphs so nearly identical you can't tell them apart without a magnifying glass? In addition, the horizontal bar of the plus sign doesn't line up with the minus sign, which makes equations look awful.
This isn't the author's only work on infinity; he's also written a stage play, Infinities, which was performed in Milan in 2002 and 2003.
So curious and counterintuitive are the notions associated with risk that understanding them took centuries. The ancients, who made such progress in geometry and other difficult fields of mathematics, were, while avid players of games of chance, inclined to attribute the outcome to the will of the Gods. It was not until the Enlightenment that thinkers such as Pascal, Cardano, the many Bernoullis, and others worked out the laws of probability, bringing the inherent randomness of games of chance into a framework which predicted the outcome, not of any given event—that was unknowable in principle, but the result of a large number of plays with arbitrary precision as the number of trials increased. Next was the understanding of the importance of uncertainty in decision making. It's one thing not to know whether a coin will come up heads or tails. It's entirely another to invest in a stock and realise that however accurate your estimation of the probabilistic unknowns affecting its future (for example, the cost of raw materials), it's the “unknown unknowns” (say, overnight bankruptcy due to a rogue trader in an office half way around the world) that can really sink your investment. Finally, classical economics always assumed that participants in the market behave rationally, but they don't. Anybody who thinks their fellow humans are rational need only visit a casino or watch them purchasing lottery tickets; they are sure in the long term to lose, and yet they still line up to make the sucker bet.
Somehow, I'd gotten it into my head that this was a “history of insurance”, and as a result this book sat on my shelf quite some time before I read it. It is much, much more than that. If you have any interest at all in investing, risk management in business ventures, or in the history of probability, statistics, game theory, and investigations of human behaviour in decision making, this is an essential book. Chapter 18 is one of the clearest expositions for its length that I've read of financial derivatives and both the benefits they have for prudent investors as well as the risks they pose to the global financial system. The writing is delightful, and sources are well documented in end notes and an extensive bibliography.
The hesitant reader is eased into the topic through a variety of easily-comprehended and yet startling results, expanding the concept of number from the natural numbers to the real number line (like calculus, complex numbers only poke their nose under the tent in a few circumstances where they absolutely can't be avoided), and then the author provides a survey of the most profound and intractable puzzles of number theory including the Goldbach conjecture and Riemann hypothesis, concluding with a sketch of Gödel's incompleteness theorems and what it all means.
Two chapters are devoted to the life and work of Ramanujan, using his notebooks to illustrate the beauty of an equation expressing a deep truth and the interconnections in mathematics this singular genius perceived, such as:
which relates the sequence of prime numbers (p_{i} is the ith prime number) to the ratio of the circumference to the diameter of a circle. Who could have imagined they had anything to do with one another? And how did 105 get into it?
This book is a pure joy, and a excellent introduction for those who “don't get it” of how mathematics can become a consuming passion for those who do. The only low spot in the book is chapter 9, which discusses the application of large prime numbers to cryptography. While this was much in the news during the crypto wars when the book was published in the mid-1990s, some of the information in this chapter is factually incorrect and misleading, and the attempt at a popular description of the RSA algorithm will probably leave many who actually understand its details scratching their heads. So skip this chapter.I bought this book shortly after it was published, and it sat on my shelf for a decade and a half until I picked it up and started reading it. I finished it in three days, enjoying it immensely, and I was already familiar with most of the material covered here. For those who are encountering it for the first time, this may be a door into a palace of intellectual pleasures they previously thought to be forbidding, dry, and inaccessible to them.
In this book, mathematician and philosopher William A. Dembski attempts to lay the mathematical and logical foundation for inferring the presence of intelligent design in biology. Note that “intelligent design” needn't imply divine or supernatural intervention—the “directed panspermia” theory of the origin of life proposed by co-discoverer of the structure of DNA and Nobel Prize winner Francis Crick is a theory of intelligent design which invokes no deity, and my perpetually unfinished work The Rube Goldberg Variations and the science fiction story upon which it is based involve searches for evidence of design in scientific data, not in scripture.
You certainly won't find any theology here. What you will find is logical and mathematical arguments which sometimes ascend (or descend, if you wish) into prose like (p. 153), “Thus, if P characterizes the probability of E_{0} occurring and f characterizes the physical process that led from E_{0} to E_{1}, then P∘f^{ −1} characterizes the probability of E_{1} occurring and P(E_{0}) ≤ P∘f^{ −1}(E_{1}) since f(E_{0}) = E_{1} and thus E_{0} ⊂ f^{ −1}(E_{1}).” OK, I did cherry-pick that sentence from a particularly technical section which the author advises readers to skip if they're willing to accept the less formal argument already presented. Technical arguments are well-supplemented by analogies and examples throughout the text.
Dembski argues that what he terms “complex specified information” is conclusive evidence for the presence of design. Complexity (the Shannon information measure) is insufficient—all possible outcomes of flipping a coin 100 times in a row are equally probable—but presented with a sequence of all heads, all tails, alternating heads and tails, or a pattern in which heads occurred only for prime numbered flips, the evidence for design (in this case, cheating or an unfair coin) would be considered overwhelming. Complex information is considered specified if it is compressible in the sense of Chaitin-Kolmogorov-Solomonoff algorithmic information theory, which measures the randomness of a bit string by the length of the shortest computer program which could produce it. The overwhelming majority of 100 bit strings cannot be expressed more compactly than simply by listing the bits; the examples given above, however, are all highly compressible. This is the kind of measure, albeit not rigorously computed, which SETI researchers would use to identify a signal as of intelligent origin, which courts apply in intellectual property cases to decide whether similarity is accidental or deliberate copying, and archaeologists use to determine whether an artefact is of natural or human origin. Only when one starts asking these kinds of questions about biology and the origin of life does controversy erupt!
Chapter 3 proposes a “Law of Conservation of Information” which, if you accept it, would appear to rule out the generation of additional complex specified information by the process of Darwinian evolution. This would mean that while evolution can and does account for the development of resistance to antibiotics in bacteria and pesticides in insects, modification of colouration and pattern due to changes in environment, and all the other well-confirmed cases of the Darwinian mechanism, that innovation of entirely novel and irreducibly complex (see chapter 5) mechanisms such as the bacterial flagellum require some external input of the complex specified information they embody. Well, maybe…but one should remember that conservation laws in science, unlike invariants in mathematics, are empirical observations which can be falsified by a single counter-example. Niels Bohr, for example, prior to its explanation due to the neutrino, theorised that the energy spectrum of nuclear beta decay could be due to a violation of conservation of energy, and his theory was taken seriously until ruled out by experiment.
Let's suppose, for the sake of argument, that Darwinian evolution does explain the emergence of all the complexity of the Earth's biosphere, starting with a single primordial replicating lifeform. Then one still must explain how that replicator came to be in the first place (since Darwinian evolution cannot work on non-replicating organisms), and where the information embodied in its molecular structure came from. The smallest present-day bacterial genomes belong to symbiotic or parasitic species, and are in the neighbourhood of 500,000 base pairs, or roughly 1 megabit of information. Even granting that the ancestral organism might have been much smaller and simpler, it is difficult to imagine a replicator capable of Darwinian evolution with an information content 1000 times smaller than these bacteria, Yet randomly assembling even 500 bits of precisely specified information seems to be beyond the capacity of the universe we inhabit. If you imagine every one of the approximately 10^{80} elementary particles in the universe trying combinations every Planck interval, 10^{45} times every second, it would still take about a billion times the present age of the universe to randomly discover a 500 bit pattern. Of course, there are doubtless many patterns which would work, but when you consider how conservative all the assumptions are which go into this estimate, and reflect upon the evidence that life seemed to appear on Earth just about as early as environmental conditions permitted it to exist, it's pretty clear that glib claims that evolution explains everything and there are just a few details to be sorted out are arm-waving at best and propaganda at worst, and that it's far too early to exclude any plausible theory which could explain the mystery of the origin of life. Although there are many points in this book with which you may take issue, and it does not claim in any way to provide answers, it is valuable in understanding just how difficult the problem is and how many holes exist in other, more accepted, explanations. A clear challenge posed to purely naturalistic explanations of the origin of terrestrial life is to suggest a prebiotic mechanism which can assemble adequate specified information (say, 500 bits as the absolute minimum) to serve as a primordial replicator from the materials available on the early Earth in the time between the final catastrophic bombardment and the first evidence for early life.
With all of these connections, there's a strong temptation for an author to wander off into fields not generally considered part of algebra (for example, analysis or set theory); Derbyshire is admirable in his ability to stay on topic, while not shortchanging the reader where important cross-overs occur. In a book of this kind, especially one covering such a long span of history and a topic so broad, it is difficult to strike the right balance between explaining the mathematics and sketching the lives of the people who did it, and between a historical narrative and one which follows the evolution of specific ideas over time. In the opinion of this reader, Derbyshire's judgement on these matters is impeccable. As implausible as it may seem to some that a book about algebra could aspire to such a distinction, I found this one of the more compelling page-turners I've read in recent months.
Six “math primers” interspersed in the text provide the fundamentals the reader needs to understand the chapters which follow. While excellent refreshers, readers who have never encountered these concepts before may find the primers difficult to comprehend (but then, they probably won't be reading a history of algebra in the first place). Thirty pages of end notes not only cite sources but expand, sometimes at substantial length, upon the main text; readers should not deprive themselves this valuable lagniappe.
For a book devoted to one of the most finicky topics in pure mathematics, there are a dismaying number of typographical errors, and not just in the descriptive text. Even some of the LaTeX macros used to typeset the book are bungled, with “@”-form \index entries appearing explicitly in the text. Many of the errors would have been caught by a spelling checker, and there are a number of rather obvious typesetting errors in equations. As the book contains an abundance of “magic numbers” related to the various problems which may figure in computer searches, I would make a point to independently confirm their accuracy before launching any extensive computing project.
Mathematicians are human, and mathematical research is a human activity like any other, so regardless of the austere crystalline perfection of the final product, the process of getting there can be as messy, contentious, and consequently entertaining as any other enterprise undertaken by talking apes. This book chronicles ten of the most significant and savage disputes in the history of mathematics. The bones of contention vary from the tried-and-true question of priority (Tartaglia vs. Cardano on the solution to cubic polynomials, Newton vs. Leibniz on the origin of the differential and integral calculus), the relation of mathematics to the physical sciences (Sylvester vs. Huxley), the legitimacy of the infinite in mathematics (Kronecker vs. Cantor, Borel vs. Zermelo), the proper foundation for mathematics (Poincaré vs. Russell, Hilbert vs. Brouwer), and even sibling rivalry (Jakob vs. Johann Bernoulli). A final chapter recounts the incessantly disputed question of whether mathematicians discover structures that are “out there” (as John D. Barrow puts it, “Pi in the Sky”) or invent what is ultimately as much a human construct as music or literature.
The focus is primarily on people and events, less so on the mathematical questions behind the conflict; if you're unfamiliar with the issues involved, you may want to look them up in other references. The stories presented here are an excellent antidote to the retrospective view of many accounts which present mathematical history as a steady march forward, with each generation building upon the work of the previous. The reality is much more messy, with the directions of inquiry chosen for reasons of ego and national pride as often as inherent merit, and the paths not taken often as interesting as those which were. Even if you believe (as I do) that mathematics is “out there”, the human struggle to discover and figure out how it all fits together is interesting and ultimately inspiring, and this book provides a glimpse into that ongoing quest.
Thus began one of the most flabbergasting examples of “mission creep” in human intellectual history, which set the style for much of mathematics publication and education in subsequent decades. Working collectively and publishing under the pseudonym “Nicolas Bourbaki” (after the French general in the Franco-Prussian War Charles Denis Bourbaki), the “analysis textbook” to be assembled by a small group over a few years grew into a project spanning more than six decades and ten books, most of multiple volumes, totalling more than seven thousand pages, systematising the core of mathematics in a relentlessly abstract and austere axiomatic form. Although Bourbaki introduced new terminology, some of which has become commonplace, there is no new mathematics in the work: it is a presentation of pre-existing mathematical work as a pedagogical tool and toolbox for research mathematicians. (This is not to say that the participants in the Bourbaki project did not do original work—in fact, they were among the leaders in mathematical research in their respective generations. But their work on the Bourbaki opus was a codification and grand unification of the disparate branches of mathematics into a coherent whole. In fact, so important was the idea that mathematics was a unified tree rooted in set theory that the Bourbaki group always used the word mathématique, not mathématiques.)
Criticisms of the Bourbaki approach were many: it was too abstract, emphasised structure over the content which motivated it, neglected foundational topics such as mathematical logic, excluded anything tainted with the possibility of application (including probability, automata theory, and combinatorics), and took an eccentric approach to integration, disdaining the Lebesgue integral. These criticisms are described in detail, with both sides fairly presented. While Bourbaki participants had no ambitions to reform secondary school mathematics education, it is certainly true that academics steeped in the Bourbaki approach played a part in the disastrous “New Math” episode, which is described in chapter 10.
The book is extravagantly illustrated, and has numerous boxes and marginal notes which describe details, concepts, and the dramatis personæ in this intricate story. An appendix provides English translations of documents which appear in French in the main text. There is no index.
La version française reste disponible.
From prehistoric times humans have had the need to count things, for example, the number of sheep in a field. This could be done by establishing a one-to-one correspondence between the sheep and something else more portable such as one's fingers (for a small flock), or pebbles kept in a sack. To determine whether a sheep was missing, just remove a pebble for each sheep and if any remained in the sack, that indicates how many are absent. At a slightly more abstract level, one could make tally marks on a piece of bark or clay tablet, one for each sheep. But all of this does not imply number as an abstraction independent of individual items of some kind or another. Ancestral humans don't seem to have required more than the simplest notion of numbers: until the middle of the 20th century several tribes of Australian aborigines had no words for numbers in their languages at all, but counted things by making marks in the sand. Anthropologists discovered tribes in remote areas of the Americas, Pacific Islands, and Australia whose languages had no words for numbers greater than four.
With the emergence of settled human populations and the increasingly complex interactions of trade between villages and eventually cities, a more sophisticated notion of numbers was required. A merchant might need to compute how many kinds of one good to exchange for another and to keep records of his inventory of various items. The earliest known written records of numerical writing are Sumerian cuneiform clay tablets dating from around 3400 B.C. These tablets show number symbols formed from two distinct kinds of marks pressed into wet clay with a stylus. While the smaller numbers seem clearly evolved from tally marks, larger numbers are formed by complicated combinations of the two symbols representing numbers from 1 to 59. Larger numbers were written as groups of powers of 60 separated by spaces. This was the first known instance of a positional number system, but there is no evidence it was used for complicated calculations—just as a means of recording quantities.
Ancient civilisations: Egypt, Hebrew, Greece, China, Rome, and the Aztecs and Mayas in the Western Hemisphere all invented ways of writing numbers, some sophisticated and capable of representing large quantities. Many of these systems were additive: they used symbols, sometimes derived from letters in their alphabets, and composed numbers by writing symbols which summed to the total. To write the number 563, a Greek would write “φξγ”, where φ=500, ξ=60, and γ=3. By convention, numbers were written with letters in descending order of the value they represented, but the system was not positional. This made the system clumsy for representing large numbers, reusing letters with accent marks to represent thousands and an entirely different convention for ten thousands.
How did such advanced civilisations get along using number systems in which it is almost impossible to compute? Just imagine a Roman faced with multiplying MDXLIX by XLVII (1549 × 47)—where do you start? You don't: all of these civilisations used some form of mechanical computational aid: an abacus, counting rods, stones in grooves, and so on to actually manipulate numbers. The Sun Zi Suan Jing, dating from fifth century China, provides instructions (algorithms) for multiplication, division, and square and cube root extraction using bamboo counting sticks (or written symbols representing them). The result of the computation was then written using the numerals of the language. The written language was thus a way to represent numbers, but not compute with them.
Many of the various forms of numbers and especially computational tools such as the abacus came ever-so-close to stumbling on the place value system, but it was in India, probably before the third century B.C. that a positional decimal number system including zero as a place holder, with digit forms recognisably ancestral to those we use today emerged. This was a breakthrough in two regards. Now, by memorising tables of addition, subtraction, multiplication, and division and simple algorithms once learned by schoolchildren before calculators supplanted that part of their brains, it was possible to directly compute from written numbers. (Despite this, the abacus remained in common use.) But, more profoundly, this was a universal representation of whole numbers. Earlier number systems (with the possible exception of that invented by Archimedes in The Sand Reckoner [but never used practically]) either had a limit on the largest number they could represent or required cumbersome and/or lengthy conventions for large numbers. The Indian number system needed only ten symbols to represent any non-negative number, and only the single convention that each digit in a number represented how many of that power of ten depending on its position.
Knowledge diffused slowly in antiquity, and despite India being on active trade routes, it was not until the 13th century A.D. that Fibonacci introduced the new number system, which had been transmitted via Islamic scholars writing in Arabic, to Europe in his Liber Abaci. This book not only introduced the new number system, it provided instructions for a variety of practical computations and applications to higher mathematics. As revolutionary as this book was, in an era of hand-copied manuscripts, its influence spread very slowly, and it was not until the 16th century that the new numbers became almost universally used. The author describes this protracted process, about which a great deal of controversy remains to the present day.
Just as the decimal positional number system was becoming established in Europe, another revolution in notation began which would transform mathematics, how it was done, and our understanding of the meaning of numbers. Algebra, as we now understand it, was known in antiquity, but it was expressed in a rhetorical way—in words. For example, proposition 7 of book 2 of Euclid's Elements states:
If a straight line be cut at random, the square of the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
Now, given such a problem, Euclid or any of those following in his tradition would draw a diagram and proceed to prove from the axioms of plane geometry the correctness of the statement. But it isn't obvious how to apply this identity to other problems, or how it illustrates the behaviour of general numbers. Today, we'd express the problem and proceed as follows:
Once again, faced with the word problem, it's difficult to know where to begin, but once expressed in symbolic form, it can be solved by applying rules of algebra which many master before reaching high school. Indeed, the process of simplifying such an equation is so mechanical that computer tools are readily available to do so.
Or consider the following brain-twister posed in the 7th century A.D. about the Greek mathematician and father of algebra Diophantus: how many years did he live?
“Here lies Diophantus,” the wonder behold.
Through art algebraic, the stone tells how old;
“God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then one-seventh ere marriage begun;
In five years there came a bounding new son.
Alas, the dear child of master and sage
After attaining half the measure of his father's life chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.”
Oh, go ahead, give it a try before reading on!
Today, we'd read through the problem and write a system of two simultaneous equations, where x is the age of Diophantus at his death and y the number of years his son lived. Then:
Plug the second equation into the first, do a little algebraic symbol twiddling, and the answer, 84, pops right out. Note that not only are the rules for solving this equation the same as for any other, with a little practice it is easy to read the word problem and write down the equations ready to solve. Go back and re-read the original problem and the equations and you'll see how straightforwardly they follow.
Once you have transformed a mass of words into symbols, they invite you to discover new ways in which they apply. What is the solution of the equation x+4=0? In antiquity many would have said the equation is meaningless: there is no number you can add to four to get zero. But that's because their conception of number was too limited: negative numbers such as −4 are completely valid and obey all the laws of algebra. By admitting them, we discovered we'd overlooked half of the real numbers. What about the solution to the equation x² + 4 = 0? This was again considered ill-formed, or imaginary, since the square of any real number, positive or negative, is positive. Another leap of imagination, admitting the square root of minus one to the family of numbers, expanded the number line into the complex plane, yielding the answer 2i as we'd now express it, and extending our concept of number into one which is now fundamental not only in abstract mathematics but also science and engineering. And in recognising negative and complex numbers, we'd come closer to unifying algebra and geometry by bringing rotation into the family of numbers.
This book explores the groping over centuries toward a symbolic representation of mathematics which hid the specifics while revealing the commonality underlying them. As one who learned mathematics during the height of the “new math” craze, I can't recall a time when I didn't think of mathematics as a game of symbolic transformation of expressions which may or may not have any connection with the real world. But what one discovers in reading this book is that while this is a concept very easy to brainwash into a 7th grader, it was extraordinarily difficult for even some of the most brilliant humans ever to have lived to grasp in the first place. When Newton invented calculus, for example, he always expressed his “fluxions” as derivatives of time, and did not write of the general derivative of a function of arbitrary variables.
Also, notation is important. Writing something in a more expressive and easily manipulated way can reveal new insights about it. We benefit not just from the discoveries of those in the past, but from those who created the symbolic language in which we now express them.
This book is a treasure chest of information about how the language of science came to be. We encounter a host of characters along the way, not just great mathematicians and scientists, but scoundrels, master forgers, chauvinists, those who preserved precious manuscripts and those who burned them, all leading to the symbolic language in which we so effortlessly write and do mathematics today.
History has not been kind to this work of Archimedes. Only two centuries after the copy of his work was made, the parchment on which it was written was scrubbed of its original content and re-written with the text of a Christian prayer book, which to the unaided eye appears to completely obscure the Archimedes text in much of the work. To compound the insult, sometime in the 20th century four full-page religious images in Byzantine style were forged over pages of the book, apparently in an attempt to increase its market value. This, then, was a bogus illustration painted on top of the prayer book text, which was written on top of the precious words of Archimedes. In addition to these depredations of mankind, many pages had been attacked by mold, and an ill-advised attempt to conserve the text, apparently in the 1960s, had gummed up the binding, including the gutter of the page where Archimedes's text was less obscured, with an intractable rubbery glue.
But from what could be read, even in fragments, it was clear that the text, if it could be extracted, would be of great significance. Two works, “The Method” and “Stomachion”, have their only known copies in this text, and the only known Greek text of “On Floating Bodies” appears here as well. Fortunately, the attempt to extract the Archimedes text was made in the age of hyperspectral imaging, X-ray fluorescence, and other nondestructive technologies, not with the crude and often disastrous chemical potions applied to attempt to recover such texts a century before.
This book, with alternating chapters written by the curator of manuscripts at the Walters and a Stanford professor of Classics and Archimedes scholar, tells the story of the origin of the manuscript, how it came to be what it is and where it resides today, and the painstaking efforts at conservation and technological wizardry (including time on the synchrotron light source beamline at SLAC) which allowed teasing the work of Archimedes from the obscuration of centuries.
What has been found so far has elevated the reputation of Archimedes even above the exalted position he already occupied in the pantheon of science. Analysis of “The Method” shows that Archimedes anticipated the use of infinitesimals and hence the calculus in his proof of the volume of curved solids. The “Stomachion”, originally thought to be a puzzle devoid of serious mathematical interest, turns out to be the first and only known venture of Greek mathematics into the realm of combinatorics.
If you're interested in rare books, the origins of mathematical thought, applications of imaging technology to historical documents, and the perilous path the words of the ancients traverse to reach us across the ages, there is much to fascinate in this account. Special thanks to frequent recommender of books Joe Marasco, who not only brought this book to my attention but mailed me a copy! Joe played a role in the discovery of the importance of the “Stomachion”, which is chronicled in the chapter “Archimedes at Play”.
Coxeter was one of the last generation to be trained in classical geometry, and he continued to do original work and make striking discoveries in that field for decades after most other mathematicians had abandoned it as mined out or insufficiently rigorous, and it had disappeared from the curriculum not only at the university level but, to a great extent, in secondary schools as well. Coxeter worked in an intuitive, visual style, frequently making models, kaleidoscopes, and enriching his publications with numerous diagrams. Over the many decades his career spanned, mathematical research (at least in the West) seemed to be climbing an endless stairway toward ever greater abstraction and formalism, epitomised in the work of the Bourbaki group. (When the unthinkable happened and a diagram was included in a Bourbaki book, fittingly it was a Coxeter diagram.) Coxeter inspired an increasingly fervent group of followers who preferred to discover new structures and symmetry using the mind's powers of visualisation. Some, including Douglas Hofstadter (who contributed the foreword to this work) and John Horton Conway (who figures prominently in the text) were inspired by Coxeter to carry on his legacy. Coxeter's interactions with M. C. Escher and Buckminster Fuller are explored in two chapters, and illustrate how the purest of mathematics can both inspire and be enriched by art and architecture (or whatever it was that Fuller did, which Coxeter himself wasn't too sure about—on one occasion he walked out of a new-agey Fuller lecture, noting in his diary “Out, disgusted, after ¾ hour” [p. 178]).
When the “new math” craze took hold in the 1960s, Coxeter immediately saw it for the disaster it was to be become and involved himself in efforts to preserve the intuitive and visual in mathematics education. Unfortunately, the power of a fad promoted by purists is difficult to counter, and a generation and more paid the price of which Coxeter warned. There is an excellent discussion at the end of chapter 9 of the interplay between the intuitive and formalist approaches to mathematics. Many modern mathematicians seem to have forgotten that one proves theorems in order to demonstrate that the insights obtained by intuition are correct. Intuition without rigour can lead to error, but rigour without intuition can blind one to beautiful discoveries in the mathematical objects which stand behind the austere symbols on paper.
The main text of this 400 page book is only 257 pages. Eight appendices expand upon technical topics ranging from phyllotaxis to the quilting of toilet paper and include a complete bibliography of Coxeter's publications. (If you're intrigued by “Morley's Miracle”, a novel discovery in the plane geometry of triangles made as late as 1899, check out this page and Java applet which lets you play with it interactively. Curiously, a diagram of Morley's theorem appears on the cover of Coxeter's and Greitzer's Geometry Revisited, but is misdrawn—the trisectors are inexact and the inner triangle is therefore not equilateral.) Almost 90 pages of endnotes provide both source citations (including Web links to MathWorld for technical terms and the University of St. Andrews biographical archive for mathematicians named in the text) and detailed amplification of numerous details. There are a few typos and factual errors (for example, on p. 101 the planets Uranus and Pluto are said to have been discovered in the nineteenth century when, in fact, neither was: Herschel discovered Uranus in 1781 and Tombaugh Pluto in 1930), but none are central to the topic nor detract from this rewarding biography of an admirable and important mathematician.
Finite groups, which govern symmetries among a finite number of discrete items (as opposed to, say, the rotations of a sphere, which are continuously valued), can be arbitrarily complicated, but, as shown by Galois, can be decomposed into one or more simple groups whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire group itself: these are the fundamental kinds of symmetries or, as this book refers to them, the “atoms of symmetry”, and there are only five categories (four of the five categories are themselves infinite). The fifth category are the sporadic groups, which do not fit into any of the other categories. The first was discovered by Émile Mathieu in 1861, and between then and 1873 he found four more. As group theory continued to develop, mathematicians kept finding more and more of these sporadic groups, and nobody knew whether there were only a finite number or infinitely many of them…until recently.
Most research papers in mathematics are short and concise. Some group theory papers are the exception, with two hundred pagers packed with dense notation not uncommon. The classification theorem of finite groups is the ultimate outlier; it has been likened to the Manhattan Project of pure mathematics. Consisting of hundreds of papers published over decades by a large collection of authors, it is estimated, if every component involved in the proof were collected together, to be on the order of fifteen thousand pages, many of which are so technical those not involved in the work itself have extreme difficulty understanding them. (In fact, a “Revision project” is currently underway with the goal of restating the proof in a form which future generations of mathematicians will be able to comprehend.) The last part of the classification theorem, itself more than a thousand pages in length, was not put into place until November 2004, so only then could one say with complete confidence that there were only 26 sporadic groups, all of which are known.
While these groups are “simple” in the sense of not being able to be decomposed, the symmetries most of them represent are of mind-boggling complexity. The order of a finite group is the number of elements it contains; for example, the group of permutations on five items has an order of 5! = 120. The simplest sporadic group has an order of 7920 and the biggest, well, it's a monster. In fact, that's what it's called, the “monster group”, and its order is (deep breath):
In one of those “take your breath away” connections between distant and apparently unrelated fields of mathematics, the divisors of the order of the monster are precisely the 15 supersingular primes, which are intimately related to the j-function of number theory. Other striking coincidences, or maybe deep connections, link the monster group to the Lorentzian geometry of general relativity, the multidimensional space of string theory, and the enigmatic properties of the number 163 in number theory. In 1983, Freeman Dyson mused, “I have a sneaking hope, a hope unsupported by any facts or any evidence, that sometime in the twenty-first century physicists will stumble upon the Monster group, built in some unsuspected way into the structure of the universe.” Hey, stranger things have happened.
This book, by a professional mathematician who is also a talented populariser of the subject, tells the story of this quest. During his career, he personally knew almost all of the people involved in the classification project, and leavens the technical details with biographical accounts and anecdotes of the protagonists. To avoid potentially confusing mathematical jargon, he uses his own nomenclature: “atom of symmetry” instead of finite simple group, “deconstruction” instead of decomposition, and so on. This sometimes creates its own confusion, since the extended quotes from mathematicians use the standard terminology; the reader should refer to the glossary at the end of the book to resolve any such puzzlement.
The essential message of the book, explained by example in a wide variety of contexts is (and I'll be rather more mathematical here in the interest of concision) is that while many (but certainly not all) natural phenomena can be well modelled by a Gaussian (“bell curve”) distribution, phenomena in human society (for example, the distribution of wealth, population of cities, book sales by authors, casualties in wars, performance of stocks, profitability of companies, frequency of words in language, etc.) are best described by scale-invariant power law distributions. While Gaussian processes converge rapidly upon a mean and standard deviation and rare outliers have little impact upon these measures, in a power law distribution the outliers dominate.
Consider this example. Suppose you wish to determine the mean height of adult males in the United States. If you go out and pick 1000 men at random and measure their height, then compute the average, absent sampling bias (for example, picking them from among college basketball players), you'll obtain a figure which is very close to that you'd get if you included the entire male population of the country. If you replaced one of your sample of 1000 with the tallest man in the country, or with the shortest, his inclusion would have a negligible effect upon the average, as the difference from the mean of the other 999 would be divided by 1000 when computing the average. Now repeat the experiment, but try instead to compute mean net worth. Once again, pick 1000 men at random, compute the net worth of each, and average the numbers. Then, replace one of the 1000 by Bill Gates. Suddenly Bill Gates's net worth dwarfs that of the other 999 (unless one of them randomly happened to be Warren Buffett, say)—the one single outlier dominates the result of the entire sample.
Power laws are everywhere in the human experience (heck, I even found one in AOL search queries), and yet so-called “social scientists” (Thomas Sowell once observed that almost any word is devalued by preceding it with “social”) blithely assume that the Gaussian distribution can be used to model the variability of the things they measure, and that extrapolations from past experience are predictive of the future. The entry of many people trained in physics and mathematics into the field of financial analysis has swelled the ranks of those who naïvely assume human action behaves like inanimate physical systems.
The problem with a power law is that as long as you haven't yet seen the very rare yet stupendously significant outlier, it looks pretty much like a Gaussian, and so your model based upon that (false) assumption works pretty well—until it doesn't. The author calls these unimagined and unmodelled rare events “Black Swans”—you can see a hundred, a thousand, a million white swans and consider each as confirmation of your model that “all swans are white”, but it only takes a single black swan to falsify your model, regardless of how much data you've amassed and how long it has correctly predicted things before it utterly failed.
Moving from ornithology to finance, one of the most common causes of financial calamities in the last few decades has been the appearance of Black Swans, wrecking finely crafted systems built on the assumption of Gaussian behaviour and extrapolation from the past. Much of the current calamity in hedge funds and financial derivatives comes directly from strategies for “making pennies by risking dollars” which never took into account the possibility of the outlier which would wipe out the capital at risk (not to mention that of the lenders to these highly leveraged players who thought they'd quantified and thus tamed the dire risks they were taking).
The Black Swan need not be a destructive bird: for those who truly understand it, it can point the way to investment success. The original business concept of Autodesk was a bet on a Black Swan: I didn't have any confidence in our ability to predict which product would be a success in the early PC market, but I was pretty sure that if we fielded five products or so, one of them would be a hit on which we could concentrate after the market told us which was the winner. A venture capital fund does the same thing: because the upside of a success can be vastly larger than what you lose on a dud, you can win, and win big, while writing off 90% of all of the ventures you back. Investors can fashion a similar strategy using options and option-equivalent investments (for example, resource stocks with a high cost of production), diversifying a small part of their portfolio across a number of extremely high risk investments with unbounded upside while keeping the bulk in instruments (for example sovereign debt) as immune as possible to Black Swans.
There is much more to this book than the matters upon which I have chosen to expound here. What you need to do is lay your hands on this book, read it cover to cover, think it over for a while, then read it again—it is so well written and entertaining that this will be a joy, not a chore. I find it beyond charming that this book was published by Random House.
The author's central thesis, illustrated from real-world examples, tests you perform on yourself, and scholarship in fields ranging from philosophy to neurobiology, is that the human brain evolved in an environment in which assessment of probabilities (and especially conditional probabilities) and nonlinear outcomes was unimportant to reproductive success, and consequently our brains adapted to make decisions according to a set of modular rules called “heuristics”, which researchers have begun to tease out by experimentation. While our brains are capable of abstract thinking and, with the investment of time required to master it, mathematical reasoning about probabilities, the parts of the brain we use to make many of the important decisions in our lives are the much older and more instinctual parts from which our emotions spring. This means that otherwise apparently rational people may do things which, if looked at dispassionately, appear completely insane and against their rational self-interest. This is particularly apparent in the world of finance, in which the author has spent much of his career, and which offers abundant examples of individual and collective delusional behaviour both before and after the publication of this work.
But let's step back from the arcane world of financial derivatives and consider a much simpler and easier to comprehend investment proposition: Russian roulette. A diabolical billionaire makes the following proposition: play a round of Russian roulette (put one cartridge in a six shot revolver, spin the cylinder to randomise its position, put the gun to your temple and pull the trigger). If the gun goes off, you don't receive any payoff and besides, you're dead. If there's just the click of the hammer falling on an empty chamber, you receive one million dollars. Further, as a winner, you're invited to play again on the same date next year, when the payout if you win will be increased by 25%, and so on in subsequent years as long as you wish to keep on playing. You can quit at any time and keep your winnings.
Now suppose a hundred people sign up for this proposition, begin to play the game year after year, and none chooses to take their winnings and walk away from the table. (For connoisseurs of Russian roulette, this is the variety of the game in which the cylinder is spun before each shot, not where the live round continues to advance each time the hammer drops on an empty chamber: in that case there would be no survivors beyond the sixth round.) For each round, on average, 1/6 of the players are killed and out of the game, reducing the number who play next year. Out of the original 100 players in the first round, one would expect, on average, around 83 survivors to participate in the second round, where the payoff will be 1.25 million.
What do we have, then, after ten years of this game? Again, on average, we expect around 16 survivors, each of whom will be paid more than seven million dollars for the tenth round alone, and who will have collected a total of more than 33 million dollars over the ten year period. If the game were to go on for twenty years, we would expect around 3 survivors from the original hundred, each of whom would have “earned” more than a third of a billion dollars each.
Would you expect these people to be regular guests on cable business channels, sought out by reporters from financial publications for their “hot hand insights on Russian roulette”, or lionised for their consistent and rapidly rising financial results? No—they would be immediately recognised as precisely what they were: lucky (and consequently very wealthy) fools who, each year they continue to play the game, run the same 1 in 6 risk of blowing their brains out.
Keep this Russian roulette analogy in mind the next time you see an interview with the “sizzling hot” hedge fund manager who has managed to obtain 25% annual return for his investors over the last five years, or when your broker pitches a mutual fund with a “great track record”, or you read the biography of a businessman or investor who always seems to have made the “right call” at the right time. All of these are circumstances in which randomness, and hence luck, plays an important part. Just as with Russian roulette, there will inevitably be big winners with a great “track record”, and they're the only ones you'll see because the losers have dropped out of the game (and even if they haven't yet they aren't newsworthy). So the question you have to ask yourself is not how great the track record of a given individual is, but rather the size of the original cohort from which the individual was selected at the start of the period of the track record. The rate hedge fund managers “blow up” and lose all of their investors' money in one disastrous market excursion is less than that of the players blown away in Russian roulette, but not all that much. There are a lot of trading strategies which will yield high and consistent returns until they don't, at which time they suffer sudden and disastrous losses which are always reported as “unexpected”. Unexpected by the geniuses who devised the strategy, the fools who put up the money to back it, and the clueless journalists who report the debacle, but entirely predictable to anybody who modelled the risks being run in the light of actual behaviour of markets, not some egghead's ideas of how they “should” behave.
Shall we try another? You go to your doctor for a routine physical, and as part of the laboratory work on your blood, she orders a screening test for a rare but serious disease which afflicts only one person in a thousand but which can be treated if detected early. The screening test has a 5% false positive rate (in 5% of the people tested who do not actually have the disease, it erroneously says that they do) and a 0% false negative rate (if you have the disease, the test will always report that you do). You return to the doctor's office for the follow-up visit and she tells you that you tested positive for the disease. What is the probability you actually have it?
Even when we make decisions with our higher cognitive facilities rather than animal instincts, it's still easy to get it wrong. While the mathematics of probability and statistics have been put into a completely rigorous form, there are assumptions in how they are applied to real world situations which can lead to the kinds of calamities one reads about regularly in the financial press. One of the reasons physical scientists transmogrify so easily into Wall Street “quants” is that they are trained and entirely comfortable with statistical tools and probabilistic analysis. The reason they so frequently run off the cliff, taking their clients' fortunes in the trailer behind them, is that nature doesn't change the rules, nor does she cheat. Most physical processes will exhibit well behaved Gaussian or Poisson distributions, with outliers making a vanishingly small contribution to mean and median values. In financial markets and other human systems none of these conditions obtain: the rules change all the time, and often change profoundly before more than a few participants even perceive they have; any action in the market will provoke a reaction by other actors, often nonlinear and with unpredictable delays; and in human systems the Pareto and other wildly non-Gaussian power law distributions are often the norm.
We live in a world in which randomness reigns in many domains, and where we are bombarded with “news and information” which is probably in excess of 99% noise to 1% signal, with no obvious way to extract the signal except with the benefit of hindsight, which doesn't help in making decisions on what to do today. This book will dramatically deepen your appreciation of this dilemma in our everyday lives, and provide a philosophical foundation for accepting the rôle randomness and luck plays in the world, and how, looked at with the right kind of eyes (and investment strategy) randomness can be your friend.
There are so many wonders here I shall not attempt to list them but simply commend this book to your own exploration and enjoyment. But one example…it's obvious that a non-convex room with black walls cannot be illuminated by a single light placed within it. But what if all the walls are mirrors? It is possible to design a mirrored room such that a light within it will still leave some part dark (p. 263)? The illustration of the Voderberg tile on p. 268 is unfortunate; the width of the lines makes it appear not to be a proper tile, but rather two tiles joined at a point. This page shows a detailed construction which makes it clear that the tile is indeed well formed and rigid.
I will confess, as a number nerd more than a geometry geek, that this book comes in second in my estimation behind the author's Penguin Book of Curious and Interesting Numbers, one single entry of which motivated me to consume three years of computer time in 1987–1990. But there are any number of wonders here, and the individual items are so short you can open the book at random and find something worth reading you can finish in a minute or so. Almost all items are given without proof, but there are citations to publications for many and you'll be able to find most of the rest on MathWorld.