Mircea Pitici, Roger Penrose - The Best Writing on Mathematics 2013-Princeton University Press (2014) - Filosofia (2025)

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<p>The BEST</p><p>WRITING on</p><p>MATHEMATICS</p><p>2013</p><p>The BEST</p><p>WRITING on</p><p>MATHEMATICS</p><p>2013</p><p>Mircea Pitici, Editor</p><p>FOREWORD BY</p><p>ROGER PENROSE</p><p>princeton univers ity press</p><p>princeton and oxford</p><p>Copyright © 2014 by Princeton University Press</p><p>Published by Princeton University Press, 41 William Street,</p><p>Princeton, New Jersey 08540</p><p>In the United Kingdom: Princeton University Press,</p><p>6 Oxford Street, Woodstock, Oxfordshire OX20 1TW</p><p>press.princeton.edu</p><p>All Rights Reserved</p><p>ISBN (pbk.) 978-0-691-16041-2</p><p>This book has been composed in Perpetua</p><p>Printed on acid-free paper. ∞</p><p>Printed in the United States of America</p><p>1 3 5 7 9 10 8 6 4 2</p><p>http://press.princeton.edu</p><p>For William P. Thurston</p><p>In memoriam</p><p>Contents</p><p>Foreword</p><p>Roger Penrose ix</p><p>Introduction</p><p>Mircea Pitici xv</p><p>The Prospects for Mathematics in a Multimedia Civilization</p><p>Philip J. Davis 1</p><p>Fearful Symmetry</p><p>Ian Stewart 23</p><p>E pluribus unum: From Complexity, Universality</p><p>Terence Tao 32</p><p>Degrees of Separation</p><p>Gregory Goth 47</p><p>Randomness</p><p>Charles Seife 52</p><p>Randomness in Music</p><p>Donald E. Knuth 56</p><p>Playing the Odds</p><p>Soren Johnson 62</p><p>Machines of the Infinite</p><p>John Pavlus 67</p><p>Bridges, String Art, and Bézier Curves</p><p>Renan Gross 77</p><p>Slicing a Cone for Art and Science</p><p>Daniel S. Silver 90</p><p>High Fashion Meets Higher Mathematics</p><p>Kelly Delp 109</p><p>viii Contents</p><p>The Jordan Curve Theorem Is Nontrivial</p><p>Fiona Ross and William T. Ross 120</p><p>Why Mathematics? What Mathematics?</p><p>Anna Sfard 130</p><p>Math Anxiety: Who Has It, Why It Develops,</p><p>and How to Guard against It</p><p>Erin A. Maloney and Sian L. Beilock 143</p><p>How Old Are the Platonic Solids?</p><p>David R. Lloyd 149</p><p>Early Modern Mathematical Instruments</p><p>Jim Bennett 163</p><p>A Revolution in Mathematics? What Really Happened</p><p>a Century Ago and Why It Matters Today</p><p>Frank Quinn 175</p><p>Errors of Probability in Historical Context</p><p>Prakash Gorroochurn 191</p><p>The End of Probability</p><p>Elie Ayache 213</p><p>An abc Proof Too Tough Even for Mathematicians</p><p>Kevin Hartnett 225</p><p>Contributors 231</p><p>Notable Texts 237</p><p>Acknowledgments 241</p><p>Credits 243</p><p>Foreword</p><p>Roger Penrose</p><p>Although I did not expect to become a mathematician when I was grow-</p><p>ing up—my first desire had been to be a train driver, and later it was</p><p>(secretly) to be a brain surgeon—mathematics had intrigued and ex-</p><p>cited me from a young age. My father was not a professional mathema-</p><p>tician, but he used mathematics in original ways in his statistical work</p><p>in human genetics. He clearly enjoyed mathematics for its own sake,</p><p>and he would often engage me and my two brothers with mathematical</p><p>puzzles and with aspects of the physical and biological world that had</p><p>a mathematical character. To me, it had become clear that mathemat-</p><p>ics was something to be enjoyed. It was evidently also something that</p><p>played an important part in the workings of the world, and one could</p><p>see this basic role not only in physics and astronomy, but also in many</p><p>aspects of biology.</p><p>I learned much of the beauties of geometry from him, and together</p><p>we constructed from cardboard not only the five Platonic solids but also</p><p>many of their Archimedean and rhombic cousins. This activity arose</p><p>from one occasion when, at some quite early age, I had been studying a</p><p>floor or table surface, tiled with a repeating pattern of ceramic regular</p><p>hexagons. I had wondered, somewhat doubtfully, whether they might,</p><p>if continued far enough into the distance, be able to cover an entire</p><p>spherical surface. My father assured me that they could not but told me</p><p>that regular pentagons, on the other hand, would do so. Perhaps there</p><p>was some seed of a thought of a possible converse to this, planted early</p><p>in my mind, about a possibility of using regular pentagons in a tiling of</p><p>the plane, that found itself realized about one third of a century later!</p><p>x Foreword</p><p>My earliest encounter with algebra came about also at an early age,</p><p>when, having long been intrigued by the identity 2 + 2 = 2  2, I had</p><p>hit upon 12</p><p>1 + 3 = 12</p><p>1  3. Wondering whether there might be other ex-</p><p>amples, and using some geometrical consideration concerning squares</p><p>and rectangles, or something—I had never done any algebra—I hit</p><p>upon some rather too-elaborate formula for what I had guessed might</p><p>be a general expression for the solution to this problem. Upon my</p><p>showing this to my older brother Oliver, he immediately showed me</p><p>how my formula could be reduced to b</p><p>1 1+a = 1, and he explained to me</p><p>how this formula indeed provided the general solution to a + b = a  b.</p><p>I was amazed by this power of simple algebra to transform and simplify</p><p>expressions, and this basic demonstration opened my eyes to the won-</p><p>ders of the world of algebra.</p><p>Much later, when I was about 15, I told my father that my mathemat-</p><p>ics teacher had informed us that we would be starting calculus on the</p><p>following day. Upon hearing this, a desperate expression came over</p><p>his face, and he immediately took me aside to explain calculus to me,</p><p>which he did very well. I could see that he had almost deprived himself</p><p>of the opportunity to be the first to introduce to me the joy and the</p><p>magic of calculus. I think that almost as great as my immediate fasci-</p><p>nation with this wonderful subject was my father’s passionate need to</p><p>relate to me in this mathematically important way. This method (and</p><p>through other intellectual pursuits such as biology, art, music, puzzles,</p><p>and games) seems to have been his only emotional route to his sons. To</p><p>try to communicate with him on personal matters was a virtual impos-</p><p>sibility. It was with this background that I had grown up to be comfort-</p><p>able with mathematics and to regard it as a friend and as a recreation,</p><p>and not something to be frightened or deterred by.</p><p>Yet there was an irony in store for me. Both my parents had been</p><p>medically trained and had decided that of their three sons, I was the</p><p>one to take over the family concerns with medicine (and, after all, I had</p><p>my secret ambition to be a brain surgeon). This possibility went by the</p><p>wayside, however, because of a decision that I had had to make at school</p><p>that entailed my giving up biology in favor of mathematics, much to the</p><p>displeasure of my parents. (It was my much younger sister Shirley who</p><p>eventually took up the banner of medicine, eventually becoming a pro-</p><p>fessor of cancer genetics.) My father was even less keen when I later ex-</p><p>pressed the desire to study mathematics, pure and simple, at university.</p><p>Foreword xi</p><p>He seemed to be of the opinion that to do just mathematics, without</p><p>necessarily applying it to some other scientific area of study, one had</p><p>to be a strange, introverted sort of person, with no other interests but</p><p>mathematics itself. In his desires and ambitions for his sons, my father</p><p>was, indeed, an emotionally complicated individual!</p><p>In fact, I think that initially mathematics alone was my true main</p><p>interest, with no necessity for it to relate to any other science or to</p><p>any aspect of the external physical world. Nor had I any great desire</p><p>to communicate my mathematical understandings to others. Yet, as</p><p>things developed, I began to feel a greater and greater need to relate</p><p>my mathematical interests to the workings of the outside world and</p><p>also, eventually, to communicate what understandings I had acquired</p><p>to the general public. I have, indeed, come to recognize the importance</p><p>of trying to convey to others an appreciation of not only the unique</p><p>value of mathematics but also its remarkable aesthetic qualities. Few</p><p>other people have had the kind of advantages that I have myself had,</p><p>with regard to mathematics, arising from my own curiously distinctive</p><p>mathematical background.</p><p>This volume serves such a purpose, providing accounts of many of</p><p>the achievements of mathematics. It is the fourth of a series of compen-</p><p>dia of previously published articles, aimed at introducing to the general</p><p>public various areas of mathematics and its multifarious applications.</p><p>It is, in addition, aimed also at other mathematicians, who may them-</p><p>selves work in areas other than</p><p>future? Did not mathematician and philosopher Alfred North</p><p>Whitehead write in one of his books that it was a mistake to believe that</p><p>one had constantly to think? Did not Descartes write that his specific</p><p>goal was to bring about a condition of automated reasoning? Do not the</p><p>rules, the paradigms, the recipes, the algorithms, the theorems, and</p><p>the generalizations of mathematics reduce once again the necessity for</p><p>thinking through a certain situation?</p><p>Experimental mathematics and visual theorems, all linked to com-</p><p>puter experiences, are increasing in frequency. There are now two</p><p>types of researchers: the first try to think before they compute; others</p><p>do the reverse. I cannot set relative values on these strategies, but it is</p><p>clear that the successful development of mathematics has in the past has</p><p>been enriched by the simultaneous use of both strategies.</p><p>A researcher in automatic intelligence (AI) has written me,</p><p>Your question “Is Thinking Obsolete” is very much to the point.</p><p>This has certainly been the trend in AI over to the past ten years</p><p>(just now beginning to reverse itself)—trying to accomplish</p><p>things through huge brute-force searches and statistical analyses</p><p>rather than through high-level reasoning.</p><p>We can also ask in this context, “Are certain parts of traditional</p><p>advanced mathematics obsolete?” For example, what portions of a the-</p><p>ory of differential equations retain value when numerical solutions are</p><p>Mathematics in a Multimedia Civilization 15</p><p>available on demand, and when, for many equations, computation is</p><p>far ahead of analytic or theorematic mathematics in explaining what is</p><p>going on? Yet, theory points directions in which we should look experi-</p><p>mentally; otherwise, we can wander at random fruitlessly.</p><p>Words or Mathematical Symbols vs. Icons</p><p>A semanticist, Mihai Nadin, has written a large book, The Civilization</p><p>of Illiteracy, on the contemporary decline of the printed word, how</p><p>the word is being displaced by the hieroglyphic or iconic modes of</p><p>communication. There is no doubt in my mind but that such a dis-</p><p>placement will have a profound effect on the inner texture of math-</p><p>ematics. Such a shift already happened 4,000 years ago. Numbers are</p><p>among the oldest achievements of civilization, predating, perhaps,</p><p>writing. In his famous book, Über Vorgreichischer Mathematik, Otto</p><p>Neugebauer “explains . . . how hieroglyphs and cuneiforms are writ-</p><p>ten and how this affects the forms of numbers and the operations</p><p>with numbers.” Another such shift occurred in the late Middle Ages,</p><p>when, with strong initial resistance, algebraic symbolisms began to</p><p>invade older texts.</p><p>Mathematics as Objective Description</p><p>vs. Mathematics by Fiat, or the Ideal</p><p>vs. the Constructed and the Virtual</p><p>Applied mathematics deals with descriptions, predictions, and pre-</p><p>scriptions. We are now in a sellers’ market for all three. Prescriptions</p><p>will boom. There may indeed be limits to what can be achieved by</p><p>mathematics and science (there are a number of books on this topic),</p><p>but I see no limits to the number of mathematizations that can be pre-</p><p>scribed and to which humans are asked to conform. In the current ad-</p><p>vanced state of the mathematization of society and human affairs, we</p><p>prescribe the systems we want to put in, from the supermarket to the</p><p>library, to the income tax, to stocks and bonds, to machines in the</p><p>medical examination rooms. All products, all human activities are now</p><p>wide open to prescriptive mathematizations. The potentialities and the</p><p>advantages envisaged and grasped by the corporate world will lead it to</p><p>pick up some of the developmental tab. And, as it does, the human foot</p><p>16 Philip J. Davis</p><p>will be asked, as with Cinderella’s sisters, to fit the mathematical shoe.</p><p>And if the shoe does not fit, tough for the foot.</p><p>What Is Proved vs. What Is Observed</p><p>This is the philosophical argument between Descartes and Giambat-</p><p>tista Vico. I venture that as regards the generality of makers and users</p><p>of mathematics, its proof aspect will diminish. Remember, mathemat-</p><p>ics does not and never did belong exclusively to those who happen to</p><p>call themselves mathematicians and who pursue a rather rigid notion of</p><p>mathematics. I would hope that the notion of proof will be expanded so</p><p>as to be acknowledged as simply one part of a larger notion of “math-</p><p>ematical evidence.”</p><p>The whole present corpus of mathematical experience and education</p><p>has come under attack from at least two different sociopolitical direc-</p><p>tions: European or Western mathematics vs. other national or ethnic</p><p>mathematics. We have today’s ethnomathematicians to thank for re-</p><p>minding us that different cultures, primitive and advanced, have had</p><p>different answers as to what mathematics is and how it should be pur-</p><p>sued and valued (e.g., ancient oriental mathematics was carried on in a</p><p>proof-free manner, and ancient Indian mathematics often expressed it-</p><p>self in verse.) More important than drawing on ancient, “non- Western”</p><p>material is the possibility that new “ethnic” splits, to be described mo-</p><p>mentarily, will emerge from within current practices. Will a civiliza-</p><p>tion of computer-induced illiteracy compel major paradigm shifts in</p><p>mathematics? Extrapolating from Nadin’s book, one might conclude</p><p>that this might arrive sooner than we think and perhaps more rapidly</p><p>than is good for us.</p><p>Male vs. Female Mathematics?</p><p>Mathematics has been perceived as an expression of male machismo.</p><p>Margaret Wertheim is a TV writer, a promoter of visual mathematics,</p><p>and a former student of math and physics. Let me quote from her book</p><p>Pythagoras’ Trousers:</p><p>One of the reasons more women do not go into physics is that they</p><p>find the present culture of this science and its almost antihuman</p><p>Mathematics in a Multimedia Civilization 17</p><p>focus, deeply alienating. . . . After six years of studying physics</p><p>and math at university, I realized that much as I loved the science</p><p>itself, I could not continue to operate within such an intellectual</p><p>environment. (p 15)</p><p>The bottom line of this Pythagoras’ Trousers is that if more women</p><p>were in mathematics and science (particularly in physics), then they</p><p>would create</p><p>an environment in which one could pursue the quest for math-</p><p>ematical relationships in the world around us, but within a more</p><p>human ethos. . . . The issue is not that physics is done by men,</p><p>but rather the kind of men who have tended to dominate it. . . .</p><p>Mathematical Man’s problem is neither his math nor his maleness</p><p>per se, but rather the pseudo religious ideals and self-image with</p><p>which he so easily becomes obsessed.</p><p>In point of fact, more women are entering mathematics and science,</p><p>and it will take at least several generations to observe whether or not</p><p>Wertheim’s vision will materialize.</p><p>The Apparent vs. the Occult</p><p>In a somewhat disturbing direction, some mathematicians and physi-</p><p>cists have been producing hermeticisms, apocalypses of various sorts,</p><p>final theories of everything, secret messages hidden in the Bible, every-</p><p>thing under the sun implied by Gödel’s theorem. I was shocked recently</p><p>to read that one of the mathematical societies in the United States had</p><p>published some of this kind of material—even though it was in a spirit</p><p>of “fun.” The old marriage of literacy and rationality, in place since the</p><p>Western Enlightenment, seems to be cracking a bit. Rationality can</p><p>shack up with fanaticisms. Is this part of a reaction of a mathematized</p><p>civilization with its claims to logical rigidity?</p><p>Soft Mathematics vs. Traditional Mathematics</p><p>I have picked up the term “soft mathematics” from Keith Devlin’s</p><p>popular book Goodbye, Descartes, which describes the difficulties of</p><p>the relationship among natural language, logic, and rationality. These</p><p>18 Philip J. Davis</p><p>difficulties, Devlin asserts, cannot be overcome by traditional math-</p><p>ematics of the Cartesian variety, and he hopes for the development of</p><p>a “soft mathematics”— not yet in existence—that “will involve a mix-</p><p>ture of mathematical reasoning,</p><p>and the less mathematically formal</p><p>kinds of reasoning used in the social sciences.” Devlin adds that, “per-</p><p>haps most of today’s mathematicians find it hard to accept the current</p><p>work on soft mathematics as ‘mathematics’ at all.” Nonetheless, some</p><p>see the development as inevitable, and Devlin uses as a credentialing</p><p>authority the mathematician-philosopher Gian-Carlo Rota. Rota comes</p><p>to a similar viewpoint through his phenomenological (e.g., Husserl and</p><p>Heidegger) orientation. After listing seven properties that phenome-</p><p>nologists believe are shared by mathematics (absolute truth; items, not</p><p>objects; nonexistence; identity; placelessness; novelty; and rigor), Rota</p><p>goes on to say,</p><p>Is it true that mathematics is at present the only existing discipline</p><p>that meets these requirements? Is it not conceivable that someday,</p><p>other new, altogether different theoretical sciences might come</p><p>into being that will share the same properties while being distinct</p><p>from mathematics?</p><p>Rota shares Husserl’s belief that a new Galilean revolution will come</p><p>about to create an alternative, soft mathematics, that will establish the-</p><p>oretical laws through idealizations that run counter to common sense.</p><p>And what is “common sense?” It may be closer than we think to</p><p>what George Bernard Shaw wrote in Androcles and the Lion, “People be-</p><p>lieve not necessarily because something is true but because in some</p><p>mysterious way it catches their imagination.” The vaunted, metaphysi-</p><p>cal and (I think) mythic unity of mathematics is further threatened by</p><p>self-contained, self-publishing chat groups. It was already threatened in</p><p>Poincaré’s day by the sheer size of the material available. The riches of</p><p>mathematics, without contemplative judgments, would, in the words</p><p>of Poincaré, “soon become an encumbrance and their increase produce</p><p>an accumulation as incomprehensible as all the unknown truths are to</p><p>those who are ignorant.”</p><p>The classic Euclidean mode of exposition and teaching, “definition,</p><p>theorem, proof,” has come under serious attack as not providing a re-</p><p>alistic description of how mathematics is created, grasped, or used.</p><p>Platonism and its various offspring, which have been the generally</p><p>Mathematics in a Multimedia Civilization 19</p><p>accepted philosophies of mathematicians, have come under serious at-</p><p>tack. Here are a few quotes that bear on this:</p><p>By giving mathematicians access to results they would never have</p><p>achieved on their own, computers call into question the idea of</p><p>a transcendental mathematical realm. They make it harder and</p><p>harder to insist as the Platonists do, that the heavenly content of</p><p>mathematics is somehow divorced from the earthbound methods</p><p>by which mathematicians investigate it. I would argue that the</p><p>earthbound realm of mathematics is the only one there is. And if</p><p>that is the case, mathematicians will have to change the way they</p><p>think about what they do. They will have to change the way they</p><p>justify it, formulate it and do it.</p><p>—Brian Rotman</p><p>I know that the great Hilbert said “We will not be driven out of</p><p>the paradise that Cantor has created for us.” And I reply: “I see no</p><p>need for walking in.”</p><p>—Richard Hamming</p><p>I think the Platonistic philosophy of mathematics that is cur-</p><p>rently claimed to justify set theory and mathematics more gener-</p><p>ally is thoroughly unsatisfactory, and that some other philosophy</p><p>grounded in inter-subjective human conceptions will have to be</p><p>sought to explain the apparent objectivity of mathematics.</p><p>—Solomon Feferman</p><p>In the end it wasn’t Gödel, it wasn’t Turing and it wasn’t my re-</p><p>sults that are making mathematics go in an experimental direc-</p><p>tion. The reason that mathematicians are changing their habits is</p><p>the computer.</p><p>—G. J. Chaitin</p><p>A number of philosophies, though buckets of ink are still spilled by</p><p>their adherents, have suffered from “dead-end-itis.” Is philosophy, in</p><p>general, irrelevant in today’s world? But philosophy will never disap-</p><p>pear for formulated or unformulated thoughts; it is what we believe</p><p>deep down to be the nature of things. And we shall have more of it</p><p>before we can say nunc dimittis.</p><p>20 Philip J. Davis</p><p>A Philosophy of Mathematics</p><p>Fostered by Multimedia</p><p>Philosopher George Berkeley (1685–1753) has resurfaced as the phi-</p><p>losopher of choice for virtual reality. “Esse est percipi” (to be is to be per-</p><p>ceived, and vice versa) is Berkeley’s tag line (often refuted). Cynthia M.</p><p>Grund is a mathematician, a philosopher, and a Scholar in Residence,</p><p>2007–2008, at Whitehall, Berkeley’s temporary residence in Middle-</p><p>town, Rhode Island. Using a virtual reality package known as “Second</p><p>Life,” she has immersed herself as avatar in a computer construction of</p><p>Whitehall:</p><p>By juxtaposing features borrowed directly from the original</p><p>Whitehall with novel features only available in a virtual envi-</p><p>ronment, but inspired by aspects of the original one, the project</p><p>explores and exemplifies the philosophical relationship between</p><p>real and virtual in an educational context, as well as philosophical</p><p>aspects of issues such as the relationship of user to avatar, relation-</p><p>ships among avatars, the relationship of avatars to “their” world,</p><p>to name only a few.</p><p>The experience of negotiating one’s virtual self though a 3-D, virtual</p><p>metaworld raises questions as to the relation among sight, touch, and the</p><p>other senses and of the relation between the perceiver and the perceived.</p><p>These questions have been discussed along neo-Berkeleyan lines.</p><p>III. A Personal Illumination</p><p>Here, then, are some of the “tensions of mathematical texture” that</p><p>I perceive. Today’s scientist or mathematician spends his or her days</p><p>in a way that is vastly different from 50 years ago, even 20 years ago.</p><p>Thinking now is accomplished differently. Science is now undergoing a</p><p>fundamental change; it may suffer in some respects, but it will certainly</p><p>create its own brave new world and proclaim new insights and ideal-</p><p>isms. I think there will be a widening to what has been traditionally</p><p>been considered to be valid mathematics. In the wake of this, the field</p><p>will again be split just as it was in the late 1700s, when it began to be</p><p>split into the pure and the applied. As a consequence, there will be the</p><p>“true believers,” pursuing the subject pretty much in the traditional</p><p>Mathematics in a Multimedia Civilization 21</p><p>manner, and the radical wave of “young Turks,” pursuing it in ways that</p><p>will raise the eyebrows and the hackles of those who will cry, “They are</p><p>traitors to the great traditions.”</p><p>In his autobiography, Elias Canetti, Nobelist in literature (1981),</p><p>speaks of an illumination he had as a young man. Walking along the</p><p>streets of Vienna, he saw in a flash that history could be explained by the</p><p>tension between the individual and the masses. Yes, we may consider</p><p>how mathematics in our multimedia age has affected separately both</p><p>the individual and society. But walking the streets of my hometown, I</p><p>had my own illumination: that the history of future mathematics will</p><p>be seen as the increased tension and increased interfusion, sometimes</p><p>productive, sometimes counterproductive, between the real and the</p><p>virtual. What meaning the future will give to these last two adjectives</p><p>and how these elements will play out are now most excellent questions</p><p>to put to the mathematical oracles.</p><p>Thanks</p><p>I thank Robert Barnhill, Fred Bisshopp, Bernhelm Booss-Bavnbek, Er-</p><p>nest Davis, John Ewing, Stuart Geman, David Gottlieb, Cynthia M.</p><p>Grund, John Guckenheimer, Arieh Iserles, David Mumford, Igor Naj-</p><p>feld, and Glen Pate.</p><p>Bibliography</p><p>V. Arnold, M. Atiyah, P. Lax, B. Mazur, eds., Mathematics: Frontiers and Perspectives. American</p><p>Mathematical Society, 2000.</p><p>Nicolaus Bernoulli, “De Usu Artis Conjectandi in Jure,” Dissertatio Inauguralis. Basel, 1709. Re-</p><p>printed in Jakob Bernoulli, Werke. vol. 3.</p><p>Bernhelm Booss-Bavnbek and Jens Høyrup, eds., Mathematics and War. Birkhauser, 2003.</p><p>G. J. Chaitin, The Limits of Mathematics. Springer Verlag, 1997.</p><p>Philip J. Davis, “Ein</p><p>Blick in die Zukunft: Mathematik in einer Multi-media-Zivilisation,” in:</p><p>Alles Mathematik, M. Aigner and E. Behrends, eds., Vieweg-Teubner, 2000.</p><p>Philip J. Davis, Mathematics and Common Sense. A. K. Peters, 2007.</p><p>Philip J. Davis, “Unity and Disunity in Mathematics,” European Math. Soc. Newsletter, March</p><p>2013.</p><p>Philip J. Davis and David Mumford, “Henri’s Crystal Ball,” Notices of the AMS, v. 55, No. 4,</p><p>April 2008, pp. 458–466.</p><p>Keith Devlin, Goodbye, Descartes: The End of Logic and the Search for a New Cosmology of the Mind.</p><p>John Wiley, New York, 1997.</p><p>Umberto Eco, How to Travel with a Salmon and Other Essays. Harcourt Brace, 1994.</p><p>Björn Engquist and Wilfried Schmidt, Mathematics Unlimited—2001 and Beyond. Springer, 2001.</p><p>22 Philip J. Davis</p><p>P. Etingof, V. Retach, I. M. Singer, eds., The Unity of Mathematics. Birkhauser, 2006.</p><p>William Everdell, The First Moderns. Univ. Chicago Press, 1996.</p><p>Hans Magnus Enzensberger, Critical Essays. New York, Continuum, 1982, esp. “The Industri-</p><p>alization of the Mind,” pp. 3–14.</p><p>Solomon Feferman,”Does mathematics need new axioms ?” 1999, available on the Internet.</p><p>Michael O. Finkelstein and Bruce Levin, Statistics for Lawyers. Springer Verlag, 1990.</p><p>Jeremy Gray, Henri Poincaré: A Scientific Biography. Princeton Univ. Press, 2013.</p><p>Cynthia M. Grund, Perception and Reality in – and out – of Second Life: Second Life as a Tool for</p><p>Reflection and Instruction at the University of Denmark, available at http://eunis.dk/papers/p10</p><p>.pdf (last visited Apr. 17, 2013).</p><p>Richard W. Hamming, Introduction to Applied Numerical Analysis. McGraw-Hill, 1971.</p><p>Richard W. Hamming, “Mathematics on a Distant Planet,” American Mathematical Monthly,</p><p>vol. 105, no. 7, Aug.–Sept. 1998, pp. 640–650.</p><p>Reuben Hersh, “Some Proposals for Reviving the Philosophy of Mathematics,” Advances in</p><p>Mathematics, v. 31, 1979. pp. 31–50. Reprinted in Tymoczko, Thomas, In the Philosophy of</p><p>Mathematics. Princeton Univ. Press, Princeton, N.J., pp. 9–28.</p><p>Reuben Hersh, What Is Mathematics, Really? Oxford U. Press, 1997.</p><p>David Mumford, Trends in the Profession of Mathematics. Presidential Address, International Math-</p><p>ematical Union, Jahresbuch der Deutsche Mathematischer Vereinigung (DMV), 1998.</p><p>Mihai Nadin, The Civilization of Illiteracy. Dresden Univ. Press, 1997.</p><p>Otto Neugebauer, Über Vorgreichischer Mathematik. Leipzig, 1929.</p><p>Henri Poincaré, The Future of Mathematics. (Address delivered at International Congress of</p><p>Mathematicians, 1908, Rome.) Translated and reprinted in Annual Report of the Smith-</p><p>sonian Institution, 1909, pp. 123–140. Available on the Internet.</p><p>Gian-Carlo Rota, Ten Remarks on Husserl and Phenomenology. Address delivered at the Provost’s</p><p>Seminar, MIT, 1998.</p><p>Brian Rotman, “The Truth about Counting,” The Sciences, Nov.–Dec. 1977.</p><p>Brian Rotman, Ad Infinitum: The Ghost in Turing’s Machine: Taking God out of Mathematics and</p><p>Putting the Body Back In. Stanford Univ. Press, 1993.</p><p>T. Tymoczko,ed., New Directions in the Philosophy of Mathematics. Birkhauser, 1986.</p><p>Wertheim, Margaret. (1995). Pythagoras’ Trousers. New York: Crown.</p><p>http://eunis.dk/papers/p10.pdf</p><p>http://eunis.dk/papers/p10.pdf</p><p>Fearful Symmetry</p><p>Ian Stewart</p><p>Tyger! Tyger! burning bright</p><p>In the forests of the night,</p><p>What immortal hand or eye</p><p>Could frame thy fearful symmetry?</p><p>In this opening verse of William Blake’s “The Tyger” from his Songs of</p><p>Experience of 1794, the poet is using “symmetry” as an artistic metaphor,</p><p>referring to the great cat’s awe-inspiring beauty and terrible form. But</p><p>the tiger’s form and markings are also governed by symmetry in a more</p><p>mathematical sense. In 1997, when delivering one of the Royal Institu-</p><p>tion’s televised Christmas Lectures, I took advantage of this connection</p><p>to bring a live tiger into the lecture theatre. I have never managed to</p><p>create quite the same focus from the audience in any lecture since then.</p><p>Taking the term literally, the only symmetry in a tiger is an approxi-</p><p>mate bilateral symmetry, something that it shares with innumerable</p><p>other living creatures, humans among them. A tiger viewed in a mirror</p><p>continues to look like a tiger. But the markings on the tiger are the vis-</p><p>ible evidence of a biological process of pattern formation that is closely</p><p>connected to mathematical symmetries. Nowhere is this more evident</p><p>than in one of the cat’s most prominent, and most geometric, features:</p><p>its elegant, cylindrical tail. A series of parallel circular stripes, running</p><p>round the tail, has continuous rotational symmetries, and (if extended</p><p>to an infinitely long tail) discrete translational symmetries as well.</p><p>The extent to which these symmetries are real is a standard model-</p><p>ing issue. A real tiger’s tail is furry, not a mathematical surface, and its</p><p>symmetries are not exact. Nevertheless, our understanding of the cat’s</p><p>markings needs to explain why they have these approximate symme-</p><p>tries, not just explain them away by observing that they are imperfect.</p><p>For the past few decades, biology has been so focused on the genetic</p><p>24 Ian Stewart</p><p>revolution, and the molecules whose activities determine much of the</p><p>form and behavior of living organisms, that it has to some extent lost</p><p>sight of the organisms themselves. But now that focus is starting to</p><p>change, and an old, somewhat discredited theory is being revived as a</p><p>consequence.</p><p>Sixty years ago, when Francis Crick and James Watson first worked</p><p>out the molecular structure of DNA (with vital input from Rosalind</p><p>Franklin and Maurice Wilkins), many biologists hoped that most of</p><p>the important features of a living organism could be deduced from its</p><p>DNA. But, as the molecular biology revolution matured, it became</p><p>clear that instead of being some fixed blueprint, DNA is more like a list</p><p>of ingredients for a recipe. An awful lot depends on precisely how those</p><p>ingredients are combined and cooked.</p><p>The best tool for discovering what a process does if you know its</p><p>ingredients and how they interact is mathematics. So a few mavericks</p><p>tried to understand the growth and form of living creatures by using</p><p>mathematical techniques. Unfortunately, their ideas were overshad-</p><p>owed by the flood of results appearing in molecular biology, and the</p><p>new ideas looked old-fashioned in comparison, so they weren’t taken</p><p>seriously by mainstream biologists. Recent new results suggest that this</p><p>reaction was unwise.</p><p>The story starts in 1952, a year before Crick and Watson’s epic discov-</p><p>ery, when the mathematician and computing pioneer Alan Turing pub-</p><p>lished a theory of animal markings in a paper with the title “The Chemical</p><p>Basis of Morphogenesis.” Turing is famous for his wartime code-breaking</p><p>activities at Bletchley Park, the Turing test for artificial intelligence, and</p><p>the undecidability of the halting problem for Turing machines. But he</p><p>also worked on number theory and the markings on animals.</p><p>We are all familiar with the stripes on tigers and zebras, the spots</p><p>on leopards, and the dappled patches on some breeds of cow. These</p><p>patterns seldom display the exact regularity that people often expect</p><p>from mathematics, but nevertheless they have a distinctly mathematical</p><p>“feel.” Turing modeled the formation of animal markings as a process</p><p>that laid down a “pre-pattern” in the developing embryo. As the em-</p><p>bryo grew, this pre-pattern became expressed as a pattern of protein</p><p>pigments. He therefore concentrated on modeling the pre-pattern.</p><p>Turing’s model has two main ingredients: reaction and diffusion.</p><p>He imagined some system of chemicals, which he called morphogens.</p><p>Fearful Symmetry 25</p><p>At any given point on the part of the embryo that eventually becomes</p><p>the skin—in effect, the embryo’s surface—these morphogens react</p><p>together to create other chemical molecules. These reactions can be</p><p>modeled by ordinary differential equations. However, the skin also has</p><p>a spatial structure, and that is where diffusion comes into play. The</p><p>chemicals and their reaction products can also diffuse, moving across</p><p>the skin in any direction.</p><p>Turing wrote down partial differential equa-</p><p>tions for processes that combined these two features. We now call them</p><p>reaction–diffusion equations, or Turing equations.</p><p>The most important result to emerge from Turing’s model is that the</p><p>reaction–diffusion process can create striking and often complex pat-</p><p>terns. Moreover, many of these patterns are symmetric. A symmetry</p><p>of a mathematical object or system is a way to transform it so that the</p><p>end looks exactly the same as it started. The striped pattern on a tiger’s</p><p>tail, for example, has rotational and translational symmetries. It looks</p><p>the same if the cylindrical tail is rotated through any angle about its</p><p>axis, and it looks the same if the tail (here we assume that it is infinitely</p><p>long in both directions) is translated through integer multiples of the</p><p>distance between neighboring stripes at right angles to the stripes.</p><p>Turing’s theory fell out of favor because it did not specify enough</p><p>biological details—for example, what the morphogens actually are. Its</p><p>literal interpretation also failed to predict what would happen in vari-</p><p>ous experiments; for example, if the embryo developed at a different</p><p>temperature. It was also realized that many different equations can pro-</p><p>duce such patterns, not just the specific ones proposed by Turing. So</p><p>the occurrence of patterns like those seen in animals does not of itself</p><p>confirm Turing’s proposed mechanism for animal markings. Mathe-</p><p>matically, there is a large class of equations that give the same general</p><p>catalog of possible patterns. What distinguishes them is the details:</p><p>which patterns occur in which circumstances. Biologists tended to see</p><p>this result as an obstacle: How can you decide which equations are re-</p><p>alistic? Mathematicians saw it as an opportunity: Let’s try to find out.</p><p>Thinking along such lines, Jim Murray has developed more general</p><p>versions of Turing’s model and has applied them to the markings on</p><p>many animals, including big cats, giraffes, and zebras. Here the iconic</p><p>patterns are stripes (tiger, zebra) and spots (cheetah, leopard). Both pat-</p><p>terns are created by wavelike structures in the chemistry. Long, parallel</p><p>waves, like waves breaking on a seashore, produce stripes. A second</p><p>26 Ian Stewart</p><p>system of waves, at an angle to the first, can cause the stripes to break</p><p>up into series of spots. Mathematically, stripes turn into spots when the</p><p>pattern of parallel waves becomes unstable. Pursuing this pattern-mak-</p><p>ing led Murray to an interesting theorem: A spotted animal can have a</p><p>striped tail, but a striped animal cannot have a spotted tail.</p><p>Hans Meinhardt has studied many variants of Turing’s equations,</p><p>with particular emphasis on the markings on seashells. His elegant</p><p>book The Algorithmic Beauty of Seashells studies many different kinds of</p><p>chemical mechanism, showing that particular types of reactions lead</p><p>to particular kinds of patterns. For example, some of the reactants</p><p>inhibit the production of others, and some reactants activate the pro-</p><p>duction of others. Combinations of inhibitors and activators can cause</p><p>chemical oscillations, resulting in regular patterns of stripes or spots.</p><p>2</p><p>2</p><p>0</p><p>0</p><p>0</p><p>5</p><p>10</p><p>15</p><p>20</p><p>25</p><p>30</p><p>35</p><p>40</p><p>45</p><p>–2</p><p>–2</p><p>5</p><p>2</p><p>0</p><p>0</p><p>0</p><p>5</p><p>10</p><p>15</p><p>20</p><p>25</p><p>35</p><p>40</p><p>45</p><p>50</p><p>–5</p><p>–5</p><p>30</p><p>Figure 1. Stripes and spots in reaction–diffusion equations on a cylindrical</p><p>domain. Source: http://www-rohan.sdsu.edu/~rcarrete/teaching/M-596</p><p>_patt/lectures/lectures.html.</p><p>http://www-rohan.sdsu.edu/~rcarrete/teaching/M-596_patt/lectures/lectures.html</p><p>http://www-rohan.sdsu.edu/~rcarrete/teaching/M-596_patt/lectures/lectures.html</p><p>Fearful Symmetry 27</p><p>Meinhardt’s theoretical patterns compare well with those found on</p><p>real shells.</p><p>Stripes and spots can be obtained by solving Turing’s equations nu-</p><p>merically, and their existence can be inferred by analyzing the equa-</p><p>tions directly, using methods from bifurcation theory, which tells us</p><p>what happens if a uniform pattern becomes unstable. But why are</p><p>we seeing patterns at all? It is here that symmetry enters the picture.</p><p>Turing himself noticed that the tendency of Turing equations to cre-</p><p>ate patterns, rather than bland uniformity, can be explained using the</p><p>mathematics of symmetry-breaking. In suitably mathematical regions,</p><p>such as a cylinder or a plane, Turing equations are symmetric. If a so-</p><p>lution is transformed by a symmetry operation, it remains a solution</p><p>(though usually a different one). The uniform solution—corresponding</p><p>Figure 2. Cone shell pattern. Source: Courtesy of Shutterstock.com/Kasia</p><p>http://Shutterstock.com/Kasia</p><p>28 Ian Stewart</p><p>to something like a lion, the same color everywhere—possesses the</p><p>full set of symmetries. Stripes and spots do not, but they retain quite a</p><p>lot of them.</p><p>Symmetry-breaking is an important process in pattern formation,</p><p>which at first sight conflicts with a basic principle about symmetries in</p><p>physical systems stated by the physicist Pierre Curie: Symmetric causes</p><p>produce equally symmetric effects. This principle, if taken too literally,</p><p>suggests that Turing’s equations are incapable of generating patterns.</p><p>In his 1952 paper, Turing includes a discussion of this point, under the</p><p>heading “The Breakdown of Symmetry and Homogeneity.” He writes,</p><p>There appears superficially to be a difficulty [with the theory] . . .</p><p>An embryo in its spherical blastula stage has spherical symmetry</p><p>. . . A system which has spherical symmetry, and whose state is</p><p>changing because of chemical reactions and diffusion, will remain</p><p>spherical for ever . . . It certainly cannot result in an organism</p><p>such as a horse, which is not spherically symmetrical.</p><p>However, he goes on to point out that</p><p>There is a fallacy in this argument . . . The system may reach a</p><p>state of instability in which . . . irregularities . . . tend to grow. If</p><p>this happens a new and stable equilibrium is usually reached. . . .</p><p>For example, if a rod is hanging from a point a little above its cen-</p><p>tre of gravity it will be in stable equilibrium. If, however, a mouse</p><p>climbs up the rod, the equilibrium eventually becomes unstable</p><p>and the rod starts to swing.</p><p>To drive the point home, he then provides an analogous example in a</p><p>reaction–diffusion system, showing that instability of the uniform state</p><p>leads to stable patterns with some—though not all—of the symmetries</p><p>of the equations.</p><p>In fact, it is the reduced list of symmetries that makes the patterns</p><p>visible. The tiger’s stripes are visibly separated from each other be-</p><p>cause the translational symmetries are discrete. You can slide the pat-</p><p>tern through an integer multiple of the distance between successive</p><p>stripes—an integer number of wavelengths of the underlying chemical</p><p>pattern.</p><p>Turing put his finger on the fundamental reason for symmetry-</p><p>breaking: A fully symmetric state of the system may be unstable. Tiny</p><p>Fearful Symmetry 29</p><p>perturbations can destroy the symmetry. If so, this pattern does not</p><p>occur in practice. Curie’s principle is not violated because the pertur-</p><p>bations are not symmetric, but the principle is misleading because it is</p><p>easy to forget about the perturbations (because they are very small).</p><p>It turns out that in a symmetric system of equations, instability of the</p><p>fully symmetric state usually leads to stable patterns instead.</p><p>For example, if the domain of the equations is a circle, typical</p><p>broken-symmetry patterns are waves, like sine curves, with a wave-</p><p>length that is the circumference of the circle divided by an integer. On</p><p>a rectangular or cylindrical domain, symmetry-breaking can lead to</p><p>plane waves, corresponding to stripes, and superpositions of two plane</p><p>waves, corresponding to spots.</p><p>Most biologists found Turing’s ideas unsatisfactory. In particular,</p><p>his model did not specify what the supposed morphogens were. Biolo-</p><p>gists came to prefer a different approach to the growth and form of the</p><p>embryo, known as positional information. Here an animal’s body is</p><p>thought of as a kind of map, and its DNA acts as an</p><p>instruction book.</p><p>The cells of the developing organism look at the map to find out where</p><p>they are, and then at the book to find out what they should do when</p><p>they are in that location. Coordinates on the map are supplied by chem-</p><p>ical gradients: For example, a chemical might be highly concentrated</p><p>near the back of the animal and gradually fade away toward the front.</p><p>By “measuring” the concentration, a cell can work out where it is.</p><p>0.115</p><p>0.11</p><p>0.105</p><p>0.1</p><p>x</p><p>t</p><p>In</p><p>te</p><p>ns</p><p>ity</p><p>–2</p><p>–1</p><p>0</p><p>1</p><p>2</p><p>–2</p><p>–1</p><p>0</p><p>1</p><p>2</p><p>Figure 3. Patterns formed from one or more plane waves.</p><p>30 Ian Stewart</p><p>The main difference in viewpoint was that the mathematicians saw</p><p>biological development as a continuous process in which the animal</p><p>grew organically by following general rules controlled by specific in-</p><p>puts from the genes, whereas to the biologists it was more like making</p><p>a model out of chemical Lego bricks by following a plan laid out in the</p><p>DNA genetic instruction book.</p><p>Important evidence supporting the theory of positional information</p><p>came from transplant experiments, in which tissue in a growing em-</p><p>bryo is moved to a new location. For example, the wing bud of a chick</p><p>embryo starts to develop a kind of striped pattern that later becomes</p><p>the bones of the wing, and a mouse embryo starts to develop a similar</p><p>pattern that eventually becomes the digits that make up its paws. The</p><p>experimental results were consistent with the theory of positional in-</p><p>formation and were widely interpreted as confirming it.</p><p>Despite the apparent success of positional information, some mathe-</p><p>maticians, engineers, physicists, and computer scientists were not con-</p><p>vinced that a chemical gradient could provide accurate positions in a</p><p>robust manner. It is now starting to look as though they were right. The</p><p>experiments that seemed to confirm the positional information theory</p><p>turn out to have been a little too simple to reveal some effects that are</p><p>not consistent with it.</p><p>In December 2012, a team of researchers led by Rushikesh Sheth at</p><p>the University of Cantabria in Spain carried out more complex trans-</p><p>plant experiments involving a larger number of digits. They showed</p><p>that a particular set of genes affects the number of digits that a grow-</p><p>ing mouse embryo develops. Strikingly, as the effect of these genes de-</p><p>creases, the mouse grows more digits than usual—like a human with six</p><p>or seven fingers instead of five. This and other results are incompatible</p><p>with the theory of positional information and chemical gradients, but</p><p>they make complete sense in terms of Turing equations.</p><p>Other groups have discovered additional examples supporting Tur-</p><p>ing’s model and have identified the specific genes and morphogens in-</p><p>volved. In 2006, Stefanie Stick and co-workers reported experiments</p><p>showing that the spacing of hair follicles (from which hairs sprout) in</p><p>mice is controlled by a biochemical signaling system called WNT and</p><p>proteins in the DKK family that inhibit WNT.</p><p>In 2012, a group at King’s College London (Jeremy Green from the</p><p>Department of Craniofacial Development at King’s Dental Institute)</p><p>Fearful Symmetry 31</p><p>showed that ridge patterns inside a mouse’s mouth are controlled by a</p><p>Turing process. The team identified the pair of morphogens working</p><p>together to influence where each ridge is formed. They are FGF (fibro-</p><p>blast growth factor) and Shh (Sonic Hedgehog, so called because fruit</p><p>flies that lack the corresponding gene look spiky).</p><p>These results are likely to be followed by many others that provide</p><p>genetic underpinning for Turing’s model or more sophisticated vari-</p><p>ants. They show that genetic information alone cannot explain the</p><p>structural changes that occur in a developing embryo. More flexible</p><p>mathematical processes are also important. On the other hand, math-</p><p>ematical models become genuinely useful only when they are combined</p><p>with detailed and specific information about which genes are active</p><p>in a given process. Neither molecular biology nor mathematics alone</p><p>can explain the markings on animals, or how a developing organism</p><p>changes as it grows. It will take both, working together, to frame the</p><p>fearful symmetry of the tyger.</p><p>References</p><p>Andrew D. Economou, Atsushi Ohazama, Thantrira Porntaveetus, Paul T. Sharpe, Shigeru</p><p>Kondo, M. Albert Basson, Amel Gritli-Linde, Martyn T. Cobourne, and Jeremy B. A.</p><p>Green, “Periodic stripe formation by a Turing mechanism operating at growth zones in</p><p>the mammalian palate.” Nature Genetics 44 (2012) 348–351; doi: 10.1038/ng.1090.</p><p>Rushikesh Sheth, Luciano Marcon, M. Félix Bastida, Marisa Junco, Laura Quintana, Randall</p><p>Dahn, Marie Kmita, James Sharpe, and Maria A. Ros, “Hox genes regulate digit pattern-</p><p>ing by controlling the wavelength of a Turing-type mechanism.” Science 338 (2012) 6113,</p><p>1476–1480; doi: 10.1126/science.1226804; available at http://www.sciencemag.org</p><p>/content/338/6113/1476.full?sid=dc2a12e8-c1c7-4f2a-bc4e-97962850caa3.</p><p>Alan Turing, “The chemical basis of morphogenesis.” Phil. Trans. R. Soc. London B 237 (1952)</p><p>37–72.</p><p>http://www.sciencemag.org/content/338/6113/1476.full?sid=dc2a12e8-c1c7-4f2a-bc4e-97962850caa3</p><p>http://www.sciencemag.org/content/338/6113/1476.full?sid=dc2a12e8-c1c7-4f2a-bc4e-97962850caa3</p><p>E pluribus unum:</p><p>From Complexity, Universality</p><p>Terence Tao</p><p>Nature is a mutable cloud, which is always and never the same.</p><p>—Ralph Waldo Emerson, “History” (1841)</p><p>Modern mathematics is a powerful tool to model any number of</p><p>real-world situations, whether they be natural—the motion of celes-</p><p>tial bodies, for example, or the physical and chemical properties of a</p><p>material —or man-made: for example, the stock market or the voting</p><p>preferences of an electorate.1 In principle, mathematical models can be</p><p>used to study even extremely complicated systems with many inter-</p><p>acting components. However, in practice, only simple systems (ones</p><p>that involve only two or three interacting agents) can be solved pre-</p><p>cisely. For instance, the mathematical derivation of the spectral lines</p><p>of hydrogen, with its single electron orbiting the nucleus, can be given</p><p>in an undergraduate physics class; but even with the most powerful</p><p>computers, a mathematical derivation of the spectral lines of sodium,</p><p>with 11 electrons interacting with each other and with the nucleus, is</p><p>out of reach. (The three-body problem, which asks to predict the motion</p><p>of three masses with respect to Newton’s law of gravitation, is famously</p><p>known as the only problem to have ever given Newton headaches. Un-</p><p>like the two-body problem, which has a simple mathematical solution,</p><p>the three-body problem is believed not to have any simple mathemati-</p><p>cal expression for its solution and can only be solved approximately, via</p><p>numerical algorithms.) The inability to perform feasible computations</p><p>on a system with many interacting components is known as the curse of</p><p>dimensionality.</p><p>From Complexity, Universality 33</p><p>Despite this curse, a remarkable phenomenon often occurs once the</p><p>number of components becomes large enough: that is, the aggregate</p><p>properties of the complex system can mysteriously become predictable</p><p>again, governed by simple laws of nature. Even more surprisingly, these</p><p>macroscopic laws for the overall system are often largely independent of</p><p>their microscopic counterparts that govern the individual components</p><p>of that system. One could replace the microscopic components by com-</p><p>pletely different types of objects and obtain the same governing law at</p><p>the macroscopic level. When this phenomenon occurs, we say that the</p><p>macroscopic law is universal. The universality phenomenon has been ob-</p><p>served both empirically and mathematically in many different contexts,</p><p>several of which I discuss below. In some cases, the phenomenon is well</p><p>understood, but in many situations, the underlying source of universal-</p><p>ity is mysterious and remains an active area of mathematical research.</p><p>The U.S. presidential election of November 4, 2008, was a massively</p><p>complicated affair. More than 100</p><p>million voters from 50 states cast</p><p>their ballots; each voter’s decision was influenced in countless ways by</p><p>campaign rhetoric, media coverage, rumors, personal impressions of</p><p>the candidates, political discussions with friends and colleagues, or all</p><p>of these things. There were millions of “swing” voters who were not</p><p>firmly supporting either of the two major candidates; their final deci-</p><p>sions would be unpredictable and perhaps even random in some cases.</p><p>The same uncertainty existed at the state level: Although many states</p><p>were considered safe for one candidate or the other, at least a dozen</p><p>were considered “in play” and could have gone either way.</p><p>In such a situation, it would seem impossible to forecast accurately</p><p>the election outcome. Sure, there were electoral polls—hundreds of</p><p>them—but each poll surveyed only a few hundred or a few thousand</p><p>likely voters, which is only a tiny fraction of the entire population. And</p><p>the polls often fluctuated wildly and disagreed with each other; not all</p><p>polls were equally reliable or unbiased, and no two polling organiza-</p><p>tions used exactly the same methodology.</p><p>Nevertheless, well before election night was over, the polls had pre-</p><p>dicted the outcome of the presidential election (and most other elec-</p><p>tions taking place that night) quite accurately. Perhaps most spectacular</p><p>were the predictions of statistician Nate Silver, who used a weighted</p><p>analysis of all existing polls to predict correctly the outcome of the</p><p>presidential election in 49 out of 50 states, as well as in all of the 35</p><p>34 Terence Tao</p><p>U.S. Senate races. (The lone exception was the presidential election in</p><p>Indiana, which Silver called narrowly for McCain but which eventually</p><p>favored Obama by just 0.9 percent.)</p><p>The accuracy of polling can be explained by a mathematical law</p><p>known as the law of large numbers. Thanks to this law, we know that</p><p>once the size of a random poll is large enough, the probable outcomes</p><p>of that poll will converge to the actual percentage of voters who would</p><p>vote for a given candidate, up to a certain accuracy, known as the mar-</p><p>gin of error. For instance, in a random poll of 1,000 voters, the margin</p><p>of error is about 3 percent.</p><p>A remarkable feature of the law of large numbers is that it is univer-</p><p>sal. Does the election involve 100,000 voters or 100 million voters? It</p><p>doesn’t matter: The margin of error for the poll will remain 3 percent.</p><p>Is it a state that favors McCain to Obama 55 percent to 45 percent, or</p><p>Obama to McCain 60 percent to 40 percent? Is the state a homoge-</p><p>neous bloc of, say, affluent white urban voters, or is the state instead a</p><p>mix of voters of all incomes, races, and backgrounds? Again, it doesn’t</p><p>matter: The margin of error for the poll will still be 3 percent. The</p><p>only factor that makes a significant difference is the size of the poll; the</p><p>larger the poll, the smaller the margin of error. The immense complex-</p><p>ity of 100 million voters trying to decide between presidential candi-</p><p>dates collapses to just a handful of numbers.</p><p>The law of large numbers is one of the simplest and best understood</p><p>of the universal laws in mathematics and nature, but it is by no means the</p><p>only one. Over the decades, many such universal laws have been found</p><p>to govern the behavior of wide classes of complex systems, regardless of</p><p>the components of a system or how they interact with each other.</p><p>In the case of the law of large numbers, the mathematical under-</p><p>pinnings of the universality phenomenon are well understood and are</p><p>taught routinely in undergraduate courses on probability and statistics.</p><p>However, for many other universal laws, our mathematical understand-</p><p>ing is less complete. The question of why universal laws emerge so often</p><p>in complex systems is a highly active direction of research in mathemat-</p><p>ics. In most cases, we are far from a satisfactory answer to this ques-</p><p>tion, but as I discuss below, we have made some encouraging progress.</p><p>After the law of large numbers, perhaps the next most fundamental</p><p>example of a universal law is the central limit theorem. Roughly speaking,</p><p>this theorem asserts that if one takes a statistic that is a combination of</p><p>From Complexity, Universality 35</p><p>many independent and randomly fluctuating components, with no one</p><p>component having a decisive influence on the whole, then that statistic</p><p>will be approximately distributed according to a law called the normal</p><p>distribution (or Gaussian distribution) and more popularly known as the</p><p>bell curve. The law is universal because it holds regardless of exactly how</p><p>the individual components fluctuate or how many components there</p><p>are (although the accuracy of the law improves when the number of</p><p>components increases). It can be seen in a staggeringly diverse range</p><p>of statistics, from the incidence rate of accidents; to the variation of</p><p>height, weight, or other vital statistics among a species; to the financial</p><p>gains or losses caused by chance; to the velocities of the component</p><p>particles of a physical system. The size, width, location, and even the</p><p>units of measurement of the distribution vary from statistic to statistic,</p><p>but the bell curve shape can be discerned in all cases. This convergence</p><p>arises not because of any low-level, or microscopic, connection among</p><p>such diverse phenomena as car crashes, human height, trading prof-</p><p>its, or stellar velocities, but because in all these cases the high-level,</p><p>or macroscopic, structure is the same: namely, a compound statistic</p><p>formed from a combination of the small influences of many indepen-</p><p>dent factors. That the macroscopic behavior of a large, complex system</p><p>can be almost totally independent of its microscopic structure is the</p><p>essence of universality.</p><p>The universal nature of the central limit theorem is tremendously</p><p>useful in many industries, allowing them to manage what would other-</p><p>wise be an intractably complex and chaotic system. With this theorem,</p><p>insurers can manage the risk of, say, their car insurance policies with-</p><p>out having to know all the complicated details of how car crashes occur;</p><p>astronomers can measure the size and location of distant galaxies with-</p><p>out having to solve the complicated equations of celestial mechanics;</p><p>electrical engineers can predict the effect of noise and interference on</p><p>electronic communications without having to know exactly how this</p><p>noise is generated; and so forth. The central limit theorem, though, is</p><p>not completely universal; there are important cases when the theorem</p><p>does not apply, giving statistics with a distribution quite different from</p><p>the bell curve. (I will return to this point later.)</p><p>There are distant cousins of the central limit theorem that are uni-</p><p>versal laws for slightly different types of statistics. One example, Ben-</p><p>ford’s law, is a universal law for the first few digits of a statistic of large</p><p>36 Terence Tao</p><p>magnitude, such as the population of a country or the size of an account;</p><p>it gives a number of counterintuitive predictions: for instance, that any</p><p>given statistic occurring in nature is more than six times as likely to</p><p>start with the digit 1 than with the digit 9. Among other things, this</p><p>law (which can be explained by combining the central limit theorem</p><p>with the mathematical theory of logarithms) has been used to detect ac-</p><p>counting fraud because numbers that are made up, as opposed to those</p><p>that arise naturally, often do not obey Benford’s law (Figure 1).</p><p>In a similar vein, Zipf ’s law is a universal law that governs the largest</p><p>statistics in a given category, such as the largest country populations in</p><p>the world or the most frequent words in the English language. It asserts</p><p>that the size of a statistic is usually inversely proportional to its ranking;</p><p>thus, for instance, the tenth largest statistic should be about half the size</p><p>of the fifth largest statistic. (The law tends not to work so well for the</p><p>top two or three statistics, but becomes more accurate after that.) Un-</p><p>like the central limit</p><p>theorem and Benford’s law, which are fairly well</p><p>understood mathematically, Zipf’s law is primarily an empirical law; it</p><p>is observed in practice, but mathematicians do not have a fully satisfac-</p><p>tory and convincing explanation for how the law comes about and why</p><p>it is universal.</p><p>30</p><p>25</p><p>20</p><p>15</p><p>10</p><p>5</p><p>0</p><p>1 2 3 4 5</p><p>First Digit</p><p>Pe</p><p>rc</p><p>en</p><p>t</p><p>6 7 8 9</p><p>Figure 1. Histogram of the first digits of the populations of the 237 countries</p><p>of the world in 2010. The black dots indicate the Benford’s law prediction.</p><p>Source: Wikipedia, http://en.wikipedia.org/wiki/File:Benfords_law</p><p>_illustrated_by_world%27s_countries_population.png; used here under The</p><p>Creative Commons Attribution-ShareAlike license.</p><p>http://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world%27s_countries_population.png</p><p>http://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world%27s_countries_population.png</p><p>From Complexity, Universality 37</p><p>So far, I have discussed universal laws for individual statistics: com-</p><p>plex numerical quantities that arise as the combination of many smaller</p><p>and independent factors. But universal laws have also been found for</p><p>more complicated objects than mere numerical statistics. Take, for ex-</p><p>ample, the laws governing the complicated shapes and structures that</p><p>arise from phase transitions in physics and chemistry. As we learn in high</p><p>school science classes, matter comes in various states, including the</p><p>three classic states of solid, liquid, and gas, but also a number of exotic</p><p>states such as plasmas or superfluids. Ferromagnetic materials, such as</p><p>iron, also have magnetized and nonmagnetized states; other materials</p><p>become electrical conductors at some temperatures and insulators at</p><p>others. What state a given material is in depends on a number of factors,</p><p>most notably the temperature and, in some cases, the pressure. (For</p><p>some materials, the level of impurities is also relevant.) For a fixed value</p><p>of the pressure, most materials tend to be in one state for one range of</p><p>temperatures and in another state for another range. But when the mate-</p><p>rial is at or very close to the temperature dividing these two ranges, in-</p><p>teresting phase transitions occur. The material, which is not fully in one</p><p>state or the other, tends to split into beautifully fractal shapes known</p><p>as clusters, each of which embodies one or the other of the two states.</p><p>There are countless materials in existence, each with a different</p><p>set of key parameters (such as the boiling point at a given pressure).</p><p>There are also a large number of mathematical models·that physicists</p><p>and chemists use to model these materials and their phase transitions,</p><p>in which individual atoms or molecules are assumed to be connected to</p><p>some of their neighbors by a random number of bonds, assigned accord-</p><p>ing to some probabilistic rule. At the microscopic level, these models</p><p>can look quite different from each other. For instance, Figures 2 and 3</p><p>display the small-scale structure of two typical models. Figure 2 shows</p><p>a site percolation model on a hexagonal lattice in which each hexagon</p><p>(or site) is an abstraction of an atom or molecule randomly placed in</p><p>one of two states; the clusters are the connected regions of a single</p><p>color. Figure 3 shows a bond percolation model on a square lattice in</p><p>which the edges of the lattice are abstractions of molecular bonds that</p><p>each have some probability of being activated; the clusters are the con-</p><p>nected regions given by the active bonds.</p><p>If one zooms out to look at the large-scale structure of clusters</p><p>while at or near the critical value of parameters (such as temperature),</p><p>38 Terence Tao</p><p>the differences in microscopic structure fade away, and one begins</p><p>to see a number of universal laws emerging. Although the clusters</p><p>have a random size and shape, they almost always have a fractal struc-</p><p>ture; thus, if one zooms in on any portion of the cluster, the resulting</p><p>image more or less resembles the cluster as a whole. Basic statistics,</p><p>such as the number of clusters, the average size of the clusters, or</p><p>the frequency with which a cluster connects two given regions of</p><p>space, appear to obey some specific universal laws, known as power</p><p>laws (which are somewhat similar, though not quite the same, as Zipf’s</p><p>law). These laws arise in almost every mathematical model that has</p><p>been put forward to explain (continuous ) phase transitions and have</p><p>been observed many times in nature. As with other universal laws,</p><p>the precise microscopic structure of the model or the material may af-</p><p>fect some basic parameters, such as the phase transition temperature,</p><p>but the underlying structure of the law is the same across all models</p><p>and materials.</p><p>In contrast to more classical universal laws such as the central limit</p><p>theorem, our understanding of the universal laws of phase transition is</p><p>incomplete. Physicists have put forth some compelling heuristic argu-</p><p>ments that explain or support many of these laws (based on a powerful,</p><p>Figure 2. Site percolation model on a hexagonal lattice at the critical thresh-</p><p>old. Source: Michael Kozdron, http://stat.math.uregina.ca/~kozdron</p><p>/Simulations/Percolation/percolation.html; used here with permission from</p><p>Michael Kozdron.</p><p>http://stat.math.uregina.ca/~kozdron/Simulations/Percolation/percolation.html</p><p>http://stat.math.uregina.ca/~kozdron/Simulations/Percolation/percolation.html</p><p>From Complexity, Universality 39</p><p>but not fully rigorous, tool known as the renormalization group method),</p><p>but a completely rigorous proof of these laws has not yet been obtained</p><p>in all cases. This field is a very active area of research; for instance, in</p><p>August 2010, the Fields medal, one of the most prestigious prizes in</p><p>mathematics, was awarded to Stanislav Smirnov for his breakthroughs</p><p>in rigorously establishing the validity of these universal laws for some</p><p>key models, such as percolation models on a triangular lattice.</p><p>As we near the end of our tour of universal laws, I want to consider</p><p>an example of this phenomenon that is closer to my own area of re-</p><p>search. Here, the object of study is not a single numerical statistic (as in</p><p>the case of the central limit theorem) or a shape (as with phase transi-</p><p>tions), but a discrete spectrum: a sequence of points (or numbers, or</p><p>frequencies, or energy levels) spread along a line.</p><p>Figure 3. Bond percolation model on a square lattice at the critical threshold.</p><p>Note the presence of both very small clusters and extremely large clusters.</p><p>Source: Wikipedia, http://en.wikipedia.org/wiki/File:Bond_percolation</p><p>_p_51.png; used here under the Creative Commons Attribution-ShareAlike</p><p>license.</p><p>http://en.wikipedia.org/wiki/File:Bond_percolation_p_51.png</p><p>http://en.wikipedia.org/wiki/File:Bond_percolation_p_51.png</p><p>40 Terence Tao</p><p>Perhaps the most familiar example of a discrete spectrum is the</p><p>radio frequencies emitted by local radio stations; this set is a sequence</p><p>of frequencies in the radio portion of the electromagnetic spectrum,</p><p>which one can access by turning a radio dial. These frequencies are not</p><p>evenly spaced, but usually some effort is made to keep any two station</p><p>frequencies separated from each other to reduce interference.</p><p>Another familiar example of a discrete spectrum is the spectral lines</p><p>of an atomic element, which come from the frequencies that the elec-</p><p>trons in the atomic shells can absorb and emit, according to the laws of</p><p>quantum mechanics. When these frequencies lie in the visible portion</p><p>of the electromagnetic spectrum, they give individual elements their</p><p>distinctive colors, from the blue light of argon gas (which, confusingly,</p><p>is often used in neon lamps because pure neon emits orange-red light)</p><p>to the yellow light of sodium. For simple elements, such as hydrogen,</p><p>the equations of quantum mechanics can be solved relatively easily,</p><p>and the spectral lines follow a regular pattern; however, for heavier</p><p>elements, the spectral lines become quite complicated and not easy to</p><p>work out just from first principles.</p><p>An analogous, but less familiar, example of spectra comes from the</p><p>scattering of neutrons off of atomic nuclei, such as the uranium-238</p><p>nucleus. The electromagnetic and nuclear forces of a nucleus, when</p><p>combined with the laws of quantum mechanics, predict that a neutron</p><p>will pass through a nucleus virtually unimpeded for some energies but</p><p>will bounce off that nucleus at other energies, known as scattering res-</p><p>onances. The internal structures of such large nuclei are so complex</p><p>that it has not been possible to compute these resonances either theo-</p><p>retically or numerically, leaving experimental data as the only option.</p><p>These resonances have an interesting distribution; they are not in-</p><p>dependent of each other but instead seem to obey a precise repulsion</p><p>law that makes it unlikely that two adjacent resonances are too close to</p><p>each other—somewhat analogous to how radio station frequencies tend</p><p>to avoid being too close together, except that the former phenomenon</p><p>arises from the laws of nature rather than from government regulation</p><p>of the spectrum. In the 1950s, the renowned physicist and Nobel laure-</p><p>ate Eugene Wigner investigated these resonance statistics and proposed</p><p>a remarkable mathematical model to explain them, an example of what</p><p>we now call a random matrix model. The precise mathematical details of</p><p>these models are too technical to describe here, but in general, one can</p><p>From Complexity, Universality 41</p><p>view such models as a large collection of masses, all connected to each</p><p>other by springs of various randomly selected strengths. Such a mechan-</p><p>ical system will oscillate (or resonate) at a certain set of frequencies; and</p><p>the Wigner hypothesis asserts that the resonances of a large atomic nu-</p><p>cleus should resemble that of the resonances of a random matrix model.</p><p>In particular, they should experience the same repulsion phenomenon.</p><p>Because it is possible to rigorously prove repulsion of the frequencies of a</p><p>random matrix model, a heuristic explanation can be given for the same</p><p>phenomenon that is experimentally observed for nuclei.</p><p>Of course, an atomic nucleus does not actually resemble a large sys-</p><p>tem of masses and springs (among other things, it is governed by the</p><p>laws of quantum mechanics rather than of classical mechanics). Instead,</p><p>as we have since discovered, Wigner’s hypothesis is a manifestation of a</p><p>universal law that governs many types of spectral lines, including those</p><p>that ostensibly have little in common with atomic nuclei or random ma-</p><p>trix models. For instance, the same spacing distribution was famously</p><p>found in the waiting times between buses arriving at a bus stop in Cuer-</p><p>navaca, Mexico (without, as yet, a compelling explanation for why this</p><p>distribution emerges in this case).</p><p>Perhaps the most unexpected demonstration of the universality of</p><p>these laws came from the wholly unrelated area of number theory, and</p><p>in particular the distribution of the prime numbers 2, 3, 5, 7, 11, and</p><p>so on—the natural numbers greater than 1that cannot be factored into</p><p>smaller natural numbers. The prime numbers are distributed in an</p><p>irregular fashion through the integers; but if one performs a spectral</p><p>analysis of this distribution, one can discern certain long-term oscilla-</p><p>tions in the distribution (sometimes known as the music of the primes),</p><p>the frequencies of which are described by a sequence of complex num-</p><p>bers known as the (nontrivial) zeroes of the Riemann zeta function,</p><p>first studied by Bernhard Riemann in 1859. (For this discussion, it is</p><p>not important to know exactly what the Riemann zeta function is.) In</p><p>principle, these numbers tell us everything we would wish to know</p><p>about the primes. One of the most famous and important problems</p><p>in number theory is the Riemann hypothesis, which asserts that these</p><p>numbers all lie on a single line in the complex plane. It has many con-</p><p>sequences in number theory and, in particular, gives many important</p><p>consequences about the prime numbers. However, even the powerful</p><p>Riemann hypothesis does not settle everything on this subject, in part</p><p>42 Terence Tao</p><p>because it does not directly say much about how the zeroes are distrib-</p><p>uted on this line. But there is extremely strong numerical evidence that</p><p>these zeroes obey the same precise law that is observed in neutron scat-</p><p>tering and in other systems; in particular, the zeroes seem to “repel”</p><p>each other in a manner that matches the predictions of random matrix</p><p>theory with uncanny accuracy. The formal description of this law is</p><p>known as the Gaussian unitary ensemble (GUE) hypothesis. (The GUE</p><p>hypothesis is a fundamental example of a random matrix model.) Like</p><p>the Riemann hypothesis, it is currently unproven, but it has powerful</p><p>consequences for the distribution of the prime numbers.</p><p>The discovery of the GUE hypothesis, connecting the music of the</p><p>primes and the energy levels of nuclei, occurred at the Institute for</p><p>Advanced Study in 1972, and the story is legendary in mathematical</p><p>circles. It concerns a chance meeting between the mathematician Hugh</p><p>Montgomery, who had been working on the distribution of zeroes of</p><p>the zeta function (and more specifically, on a certain statistic relating</p><p>to that distribution known as the pair correlation function), and the</p><p>renowned physicist Freeman Dyson. In his book Stalking the Riemann</p><p>Hypothesis, mathematician and computer scientist Dan Rockmore de-</p><p>scribes that meeting:</p><p>As Dyson recalls it, he and Montgomery had crossed paths from</p><p>time to time at the Institute nursery when picking up and drop-</p><p>ping off their children. Nevertheless, they had not been formally</p><p>introduced. In spite of Dyson’s fame, Montgomery hadn’t seen</p><p>any purpose in meeting him. “What will we talk about?” is what</p><p>Montgomery purportedly said when brought to tea. Neverthe-</p><p>less, Montgomery relented and upon being introduced, the ami-</p><p>able physicist asked the young number theorist about his work.</p><p>Montgomery began to explain his recent results on the pair cor-</p><p>relation, and Dyson stopped him short—“Did you get this?” he</p><p>asked, writing down a particular mathematical formula. Mont-</p><p>gomery almost fell over in surprise: Dyson had written down</p><p>the sinc-infused pair correlation function. . . . Whereas Mont-</p><p>gomery had traveled a number theorist’s road to a “prime pic-</p><p>ture” of the pair correlation, Dyson had arrived at this formula</p><p>through the study of these energy levels in the mathematics of</p><p>matrices.2</p><p>From Complexity, Universality 43</p><p>The chance discovery by Montgomery and Dyson that the same uni-</p><p>versal law that governs random matrices and atomic spectra also ap-</p><p>plies to the zeta function was given substantial numerical support by</p><p>the computational work of Andrew Odlyzko beginning in the 1980s</p><p>(Figure 4). But this discovery does not mean that the primes are some-</p><p>how nuclear-powered or that atomic physics is somehow driven by the</p><p>prime numbers; instead, it is evidence that a single law for spectra is so</p><p>universal that it is the natural end product of any number of different</p><p>processes, whether from nuclear physics, random matrix models, or</p><p>number theory.</p><p>The precise mechanism underlying this law has not yet been fully</p><p>unearthed; in particular, we still do not have a compelling explana-</p><p>tion, let alone a rigorous proof, of why the zeroes of the zeta function</p><p>are subject to the GUE hypothesis. However, there is now a substantial</p><p>body of rigorous work (including some of my own work, and including</p><p>some substantial breakthroughs in just the past few years) that gives</p><p>1.0</p><p>0.6</p><p>0</p><p>0.50 1.0 1.5</p><p>Normalized Spacing</p><p>Nearest Neighbor Spacings</p><p>D</p><p>en</p><p>si</p><p>ty</p><p>2.0 2.5 3.0</p><p>0.8</p><p>0.4</p><p>0.2</p><p>Figure 4. Spacing distribution for a billion zeroes of the Riemann zeta</p><p>function, with the corresponding prediction from random matrix theory.</p><p>Source: Andrew M. Odlyzko, “The 1022nd Zero of the Riemann Zeta Function,”</p><p>in Dynamical, Spectral, and Arithmetic Zeta Functions, Contemporary Mathemat-</p><p>ics Series, no. 290, ed. Machiel van Frankenhuysen</p><p>and Michel L. Lapidus</p><p>( American Mathematical Society, 2001), 139–144, http://www.dtc.umn.edu</p><p>/~odlyzko/doc/zeta.10to22.pdf; used here with permission from Andrew</p><p>Odlyzko.</p><p>http://www.dtc.umn.edu/~odlyzko/doc/zeta.10to22.pdf</p><p>http://www.dtc.umn.edu/~odlyzko/doc/zeta.10to22.pdf</p><p>44 Terence Tao</p><p>support to the universality of this hypothesis by showing that a wide va-</p><p>riety of random matrix models (not just the most famous model of the</p><p>GUE) are all governed by essentially the same law for their spacings. At</p><p>present, these demonstrations of universality have not extended to the</p><p>number theoretic or physical settings, but they do give indirect support</p><p>to the law being applicable in those cases.</p><p>The arguments used in this recent work are too technical to give</p><p>here, but I do mention one of the key ideas, which my colleague Van</p><p>Vu and I borrowed from an old proof of the central limit theorem by</p><p>Jarl Lindeberg from 1922. In terms of the mechanical analogy of a</p><p>system of masses and springs (mentioned above), the key strategy was</p><p>to replace just one of the springs by another, randomly selected spring</p><p>and to show that the distribution of the frequencies of this system</p><p>did not change significantly when doing so. Applying this replacement</p><p>operation to each spring in turn, one can eventually replace a given</p><p>random matrix model with a completely different model while keep-</p><p>ing the distribution mostly unchanged—which can be used to show</p><p>that large classes of random matrix models have essentially the same</p><p>distribution.</p><p>This field is a very active area of research; for instance, simultane-</p><p>ously with Van Vu’s and my work from last year, László Erdös, Benjamin</p><p>Schlein, and Horng-Tzer Yau also gave a number of other demonstra-</p><p>tions of universality for random matrix models, based on ideas from</p><p>mathematical physics. The field is moving quickly, and in a few years</p><p>we may have many more insights into the nature of this mysterious</p><p>universal law.</p><p>There are many other universal laws of mathematics and nature; the</p><p>examples I have given are only a small fraction of those that have been dis-</p><p>covered over the years, from such diverse subjects as dynamical systems</p><p>and quantum field theory. For instance, many of the macroscopic laws of</p><p>physics, such as the laws of thermodynamics or the equations of fluid mo-</p><p>tion, are quite universal in nature, making the microscopic structure of</p><p>the material or fluid being studied almost irrelevant, other than via some</p><p>key parameters, such as viscosity, compressibility, or entropy.</p><p>However, the principle of universality does have definite limita-</p><p>tions. Take, for instance, the central limit theorem, which gives a bell</p><p>curve distribution to any quantity that arises from a combination of</p><p>many small and independent factors. This theorem can fail when the</p><p>From Complexity, Universality 45</p><p>required hypotheses are not met. The distribution of, say, the heights</p><p>of all human adults (male and female) does not obey a bell curve dis-</p><p>tribution because one single factor—gender—has so large an effect on</p><p>height that it is not averaged out by all the other environmental and</p><p>genetic factors that influence this statistic.</p><p>Another important way in which the central limit fails is when the</p><p>individual factors that make up a quantity do not fluctuate indepen-</p><p>dently of each other but are instead correlated, so that they tend to</p><p>rise or fall in unison. In such cases, “fat tails” (also known colloqui-</p><p>ally as “black swans”) can develop, in which the quantity moves much</p><p>further from its average value than the central limit theorem would</p><p>predict. This phenomenon is particularly important in financial model-</p><p>ing, especially when dealing with complex financial instruments such</p><p>as the collateralized debt obligations that were formed by aggregating</p><p>mortgages. As long as the mortgages behaved independently of each</p><p>other, the central limit theorem could be used to model the risk of</p><p>these instruments. However, in the recent financial crisis (a textbook</p><p>example of a black swan), this independence hypothesis broke down</p><p>spectacularly, leading to significant financial losses for many holders of</p><p>these obligations (and for their insurers). A mathematical model is only</p><p>as strong as the assumptions behind it.</p><p>A third way in which a universal law can break down is if the system</p><p>does not have enough degrees of freedom for the law to take effect. For</p><p>instance, cosmologists can use universal laws of fluid mechanics to de-</p><p>scribe the motion of entire galaxies, but the motion of a single satellite</p><p>under the influence of just three gravitational bodies can be far more</p><p>complicated (quite literally, rocket science).</p><p>Another instance where the universal laws of fluid mechanics break</p><p>down is at the mesoscopic scale: that is, larger than the microscopic scale</p><p>of individual molecules but smaller than the macroscopic scales for</p><p>which universality applies. An important example of a mesoscopic fluid</p><p>is the blood flowing through blood vessels; the blood cells that make</p><p>up this liquid are so large that they cannot be treated merely as an en-</p><p>semble of microscopic molecules but rather as mesoscopic agents with</p><p>complex behavior. Other examples of materials with interesting meso-</p><p>scopic behavior include colloidal fluids (such as mud), certain types of</p><p>nanomaterials, and quantum dots; it is a continuing challenge to math-</p><p>ematically model such materials properly.</p><p>46 Terence Tao</p><p>There are also many macroscopic situations in which no universal</p><p>law is known to exist, particularly in cases where the system contains</p><p>human agents. The stock market is a good example: Despite extremely</p><p>intensive efforts, no satisfactory universal laws to describe the move-</p><p>ment of stock prices have been discovered. (The central limit theorem,</p><p>for instance, does not seem to be a good model, as discussed earlier.)</p><p>One reason for this shortcoming is that any regularity discovered in the</p><p>market is likely to be exploited by arbitrageurs until it disappears. For</p><p>similar reasons, finding universal laws for macroeconomics appears to</p><p>be a moving target; according to Goodhart’ s law, if an observed statisti-</p><p>cal regularity in economic data is exploited for policy purposes, it tends</p><p>to collapse. (Ironically, Goodhart’s law itself is arguably an example of</p><p>a universal law.)</p><p>Even when universal laws do exist, it still may be practically impos-</p><p>sible to use them to make predictions. For instance, we have universal</p><p>laws for the motion of fluids, such as the Navier–Stokes equations, and</p><p>these are used all the time in such tasks as weather prediction. But these</p><p>equations are so complex and unstable that even with the most powerful</p><p>computers, we are still unable to accurately predict the weather more</p><p>than a week or two into the future. (By unstable, I mean that even small</p><p>errors in one’s measurement data, or in one’s numerical computations,</p><p>can lead to large fluctuations in the predicted solution of the equations.)</p><p>Hence, between the vast, macroscopic systems for which universal</p><p>laws hold sway and the simple systems that can be analyzed using the</p><p>fundamental laws of nature, there is a substantial middle ground of</p><p>systems that are too complex for fundamental analysis but too simple to</p><p>be universal—plenty of room, in short, for all the complexities of life</p><p>as we know it.</p><p>Notes</p><p>1. This essay benefited from the feedback of many readers of my blog. They commented</p><p>on a draft version that (together with additional figures and links) can be read at http://terry</p><p>tao.wordpress.com/2010/09/14/, A second draft of a non-technical article on universality.</p><p>2. Dan Rockmore, Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime</p><p>Numbers (New York: Pantheon Books, 2005).</p><p>http://terrytao.wordpress.com/2010/09/14/</p><p>http://terrytao.wordpress.com/2010/09/14/</p><p>Degrees of Separation</p><p>Gregory Goth</p><p>The idea of six degrees of separation—that is, that every person in</p><p>the world is no more than six people</p><p>away from every other person on</p><p>earth—has fascinated social scientists and laymen alike ever since Hun-</p><p>garian writer Frigyes Karinthy introduced the concept in 1929. (The</p><p>story has since been reprinted in 2006 in Princeton University Press’s</p><p>The Structure and Dynamics of Networks by Mark Newman, Albert-László</p><p>Barabási, and Duncan J. Watts, translated by Adam Makkai and edited</p><p>by Enikö Jankó.)</p><p>For the greater public, the cultural touchstone of the theory was</p><p>the 1990 play entitled Six Degrees of Separation by John Guare. Although</p><p>the drama was not an exploration of the phenomenon by any means, it</p><p>spawned countless versions of parlor games. For scientists, however,</p><p>the wellspring of the six degrees phenomenon, also called the small-</p><p>world problem, was a 1967 study undertaken by social psychologist</p><p>Stanley Milgram, in which a selected group of volunteers in the mid-</p><p>western United States was instructed to forward messages to a target</p><p>person in Boston. Milgram’s results, published in Psychology Today in</p><p>1967, were that the messages were delivered by “chains” that comprised</p><p>anywhere between 2 and 10 intermediaries; the mean was 5 (Figure 1).</p><p>In the ensuing years, the problem has become a perennial favorite</p><p>among researchers of many disciplines, from computer scientists ex-</p><p>ploring probabilistic algorithms for best use of network resources to</p><p>epidemiologists exploring the interplay of infectious diseases and net-</p><p>work theory.</p><p>Most recently, the vast architectural resources of Facebook and</p><p>Twitter have supplied researchers with something they never possessed</p><p>before—the capability to look at the small-world problem from both</p><p>the traditional algorithmic approach, which explores the probabilities</p><p>48 Gregory Goth</p><p>of how each person (or network node) in a chain seeks out the next</p><p>messenger using only the limited local knowledge they possess, and the</p><p>new topological approach, which can examine the entire structure of</p><p>a network as it also observes the progression of the algorithmic chains.</p><p>“It’s amazing how far we’ve come,” says Duncan Watts, a founding</p><p>partner at Microsoft Research New York City, who was until recently a</p><p>senior researcher at Yahoo! Watts is one of the world’s leading authori-</p><p>ties on the small-world problem, dating to the publication of “Collec-</p><p>tive Dynamics of ‘Small-World’ Networks,” coauthored with Steven</p><p>Strogatz, in Nature in 1998. At that time, Watts says, the largest avail-</p><p>able network, actors listed in the Internet Movie Database, contained</p><p>about 225,000 edge nodes (individual actors). A recent study by re-</p><p>searchers from Facebook and the University of Milan, however, looked</p><p>at 721 million Facebook users, who had 69 billion unique friendships</p><p>among them, and revealed an average of 3.74 intermediaries between</p><p>a source and target user, suggesting an even smaller world than Mil-</p><p>gram’s original study showed.</p><p>“In fact, the whole motivation of the thing I did with Strogatz was</p><p>precisely that you couldn’t do the exercise Facebook just did,” Watts</p><p>says. “Now the empirical exercise is possible. That’s a remarkable</p><p>change.”</p><p>A Similarity of Results</p><p>One must consider the large variety of online communities and com-</p><p>pare the small-world experiments performed on them to Milgram’s</p><p>method—sending a message via terrestrial delivery routes—in order</p><p>to fully appreciate the similarity of results across the board. Whereas</p><p>the Facebook experiment yielded approximately four degrees of sepa-</p><p>ration, work by distinguished scientist Eric Horvitz of Microsoft Re-</p><p>search and Stanford University assistant professor Jure Leskovec, on</p><p>connections between users of the Microsoft Instant Messaging network,</p><p>yielded an average 6.6 degrees of separation between any two users. In</p><p>their 2009 paper “Social Search in ‘Small-World’ Experiments” exam-</p><p>ining the algorithmic approach, Watts, Sharad Goel, and Roby Muha-</p><p>mad discovered that roughly half of all chains can be completed in six or</p><p>seven steps, “thus supporting the ‘six degrees of separation’ assertion,”</p><p>they wrote, “but on the other hand, estimates of the mean are much</p><p>Degrees of Separation 49</p><p>longer, suggesting that for at least some of the population, the world is</p><p>not ‘small’ in the algorithmic sense.”</p><p>Discovering the reason why “the world is not ‘small’ in the algorith-</p><p>mic sense” presents a wide swath of fertile ground for those research-</p><p>ers, including Watts and Leskovec, who are still plumbing the many</p><p>vectors of network navigation.</p><p>One ironic, or counterintuitive, factor in examining the small-world</p><p>problem as online communities grow ever larger is that the experi-</p><p>ments’ attrition rates are also vastly greater than in the past. For in-</p><p>stance, Watts says that only 12% of those who signed up for a joint</p><p>small-world experiment at Yahoo! and Facebook completed their</p><p>chains, compared with 75% of those who participated in Milgram’s</p><p>experiment and the 35% who completed chains in a 2001–2002 ex-</p><p>periment run by Watts.</p><p>However, Watts says the data they have should allow them still to</p><p>answer the questions they care about most, which are about exploring</p><p>the efficiency of intermediary connections selected.</p><p>“We know how far you are from the target, Facebook knows how far</p><p>your friends are from the target, and we know who you picked, so we</p><p>can establish whether you made the right choice,” Watts says. “So we</p><p>can get most of the science out of it, it’s just a little bummer that the</p><p>attrition was so bad.”</p><p>The logic behind finding the most efficient paths may produce pay-</p><p>offs unforeseen for both theoretical modeling and production networks</p><p>such as search engine optimization. Finding the best ways to determine</p><p>those paths, though, will necessitate a leap from the known models of</p><p>small-world networks to a better understanding of the intermediary</p><p>steps between any two endpoints of a chain.</p><p>Leskovec says that, given constants from graph theory, the diameter</p><p>of any given network will grow logarithmically with its size; that is, the</p><p>difference between five and six degrees of separation mandates a graph</p><p>an order of magnitude larger or denser. Jon Kleinberg, Tisch University</p><p>professor in the department of computer science at Cornell University,</p><p>whose “The Small-World Phenomenon: An Algorithmic Perspective”</p><p>is regarded as one of the problem’s seminal modeling documents, says</p><p>this basic property is precisely what makes the small-world theory so</p><p>appealing while also presenting the research community the greatest</p><p>challenge inherent in it.</p><p>50 Gregory Goth</p><p>“It’s something that still feels counterintuitive when you first en-</p><p>counter it,” Kleinberg says. “It makes sense in the end: I know 1,000</p><p>people, and my friend knows 1,000 people—and you don’t have to</p><p>multiply 1,000 by itself too many times for it to make sense.”</p><p>However, this logarithmic progression also precludes the ability to</p><p>examine or design intermediate levels of scale, Kleinberg says. “We</p><p>thought the right definition of distance was going to be ‘Here I am, and</p><p>how many steps do I have to go to get to you?’ but that turns out not to</p><p>be. We need some other measure and I think that remains an interest-</p><p>ing open question that people are actively looking at: Is there some kind</p><p>of smoother scale here? Who are the 10,000 people closest to me? The</p><p>100,000?</p><p>“We need a much more subtle way to do that, and it is going to</p><p>require some sophisticated mathematical ideas and sophisticated com-</p><p>binational ideas—what is the right definition of distance when you’re</p><p>looking at social networks? It’s not just how many steps I have to go.</p><p>That’s an important question in everyday life and when you’re design-</p><p>ing some online system.”</p><p>Mozart Meets the Terminator</p><p>Recent research is beginning to use the short-path principles of social</p><p>search in the online systems discussed by Kleinberg. In “Degrees of</p><p>Separation in Social Networks,” presented at the Fourth International</p><p>Symposium on Combinatorial Search 2011, researchers from Shiraz</p><p>University,</p><p>the ones being described here. With</p><p>regard to this latter purpose, I can vouch for its success. For in my own</p><p>case, I write as a mathematician whose professional interests lie in areas</p><p>almost entirely outside those described here, and upon reading these</p><p>articles I have certainly had my perspectives broadened in several ways</p><p>that I had not expected.</p><p>The breadth of the ideas that we find here is considerable, ranging</p><p>over many areas, such as the philosophy of mathematics, the issue of</p><p>why mathematics is so important in education and society, and whether</p><p>its public perception has changed in recent years; perhaps it should now</p><p>be taught fundamentally differently, and there is the issue of the extent</p><p>to which the modern technological world might even have thoroughly</p><p>changed the very nature of our subject. We also find fascinating his-</p><p>torical accounts, from achievements made a thousand years or so be-</p><p>fore the ancient Greeks, to the deep insights and occasional surprising</p><p>xii Foreword</p><p>errors made in more modern historical times, and of the wonderful</p><p>mathematical instruments that played important roles in their societ-</p><p>ies. We find unexpected connections with geometry, both simple and</p><p>highly sophisticated, in artistic creations of imposing magnitude and</p><p>to the fashionable adornment of individual human beings. We learn of</p><p>the roles of symmetry in animals and in art, and of the use of art in il-</p><p>lustrating the value of mathematical rigor. There is much here on the</p><p>role of randomness and how it is treated by statistics, which is a subject</p><p>of ubiquitous importance throughout science and of importance also in</p><p>everyday life.</p><p>Yet I was somewhat surprised that, throughout this great breadth of</p><p>mathematical application, I find no mention of that particular area of</p><p>the roles of mathematics that I myself find so extraordinarily remark-</p><p>able, and to which I have devoted so much of my own mathematical</p><p>energies. This area is the application of sophisticated mathematics to</p><p>the inner workings of the physical world wherein we find ourselves.</p><p>It is true that with many situations in physics, very large numbers of</p><p>elementary constituents (e.g., particles) are involved. This truth applies</p><p>to the thermodynamic concepts of heat and entropy and to the detailed</p><p>behavior of gases and other fluids. Many of the relevant concepts are</p><p>then governed by the laws of large numbers—that is, by the principles</p><p>of statistics—and this issue is indeed addressed here from various dif-</p><p>ferent perspectives in several of these articles.</p><p>However, it is often the case that such statistical considerations leave</p><p>us very far from what is needed, and a proper understanding of the</p><p>underlying laws themselves is fundamentally needed. Indeed, in appro-</p><p>priate circumstances (i.e., when the physical behavior is in sufficiently</p><p>“clean” systems), a precision is found that is extraordinary between the</p><p>observed physical behavior and the calculated behavior that is expected</p><p>from the known physical laws. This precision already exists, for ex-</p><p>ample, in modern treatments that use powerful computer techniques</p><p>for the ancient 17th century laws of Isaac Newton. But in appropriate</p><p>circumstances, the agreement can be far more impressive when the</p><p>appropriate mathematically sophisticated laws of general relativity or</p><p>quantum field theory are brought into play.</p><p>These matters are often hard to explain in the general terms that</p><p>could meet the criteria of this collection, and one can understand the</p><p>omission here of such extraordinary achievements of mathematics. It</p><p>Foreword xiii</p><p>must be admitted, also, that there is much current activity of consider-</p><p>able mathematical sophistication that, though it is ostensibly concerned</p><p>with the workings of the actual physical world, has little, if any, direct</p><p>observational connection with it. Accordingly, despite the considerable</p><p>mathematical work that is currently being done in these areas—much</p><p>of it of admittedly great interest with regard to the mathematics it-</p><p>self—this work may be considered from the physical point of view to</p><p>be somewhat dubious, or tenuous at best because it has no observa-</p><p>tional support as things stand now. Nevertheless, it is within physics,</p><p>and its related areas, such as chemistry, metallurgy, and astronomy,</p><p>that we are beginning to witness the deep and overreaching command</p><p>of mathematics, when it is aided by the computational power of modern</p><p>electronic devices.</p><p>Introduction</p><p>Mircea Pitici</p><p>In the fourth annual volume of The Best Writing on Mathematics series, we</p><p>present once again a collection of recent articles on various aspects re-</p><p>lated to mathematics. With few exceptions, these pieces were published</p><p>in 2012. The relevant literature I surveyed to compile the selection is vast</p><p>and spread over many publishing venues; strict limitation to the time</p><p>frame offered by the calendar is not only unrealistic but also undesirable.</p><p>I thought up this series for the first time about nine years ago. Quite</p><p>by chance, in a fancy, and convinced that such a series existed, I asked</p><p>in a bookstore for the latest volume of The Best Writing on Mathematics.</p><p>To my puzzlement, I learned that the book I wanted did not exist—and</p><p>the remote idea that I might do it one day was born timidly in my mind,</p><p>only to be overwhelmed, over the ensuing few years, by unusual ad-</p><p>versity, hardship, and misadventures. But when the right opportunity</p><p>appeared, I was prepared for it; the result is in your hands.</p><p>Mathematicians are mavericks—inventors and explorers of sorts;</p><p>they create new things and discover novel ways of looking at old things;</p><p>they believe things hard to believe, and question what seems to be obvi-</p><p>ous. Mathematicians also disrupt patterns of entrenched thinking; their</p><p>work concerns vast streams of physical and mental phenomena from</p><p>which they pick the proportions that make up a customized blend of ab-</p><p>stractions, glued by tight reasoning and augmented with clues glanced</p><p>from the natural universe. This amalgam differs from one mathemati-</p><p>cian to another; it is “purer” or “less pure,” depending on how little or</p><p>how much “application” it contains; it is also changeable, flexible, and</p><p>adaptable, reflecting (or reacting to) the social intercourse of ideas that</p><p>influences each of us.</p><p>xvi Introduction</p><p>When we talk about mathematics, or when we teach it, or when</p><p>we write about it, many of us feign detachment. It is almost a cultural</p><p>universal to pretend that mathematics is “out there,” independent of</p><p>our whims and oddities. But doing mathematics and talking or writing</p><p>about it are activities neither neutral nor innocent; we can only do them</p><p>if we are engaged, and the engagement marks not only us (as thinkers</p><p>and experimenters) but also those who watch us, listen to us, and think</p><p>with us. Thus mathematics always requires full participation; without</p><p>genuine involvement, there is no mathematics.</p><p>Mathematicians are also tinkerers—as all innovators are. They try,</p><p>and often succeed, in creating effective tools that guide our minds in</p><p>the search for certainties. Or they err; they make judgment mishaps or</p><p>are seduced by the illusion of infallibility. Sometimes mathematicians</p><p>detect the errors themselves; occasionally, others point out the prob-</p><p>lems. In any case, the edifice mathematicians build should be up for</p><p>scrutiny, either by peers or by outsiders.</p><p>And here comes a peculiar aspect that distinguishes mathemat-</p><p>ics among other intellectual domains: Mathematicians seek validation</p><p>inside their discipline and community but feel little need (if any) for</p><p>validation coming from outside. This professional chasm surrounding</p><p>much of the mathematics profession is inevitable up to a point because</p><p>of the nature of the discipline. It is a Janus-faced curse of the ivory</p><p>tower, and it is unfortunate if we ignore it. On the contrary, I believe</p><p>that we should address it. Seeking the meaning and the palpable reason-</p><p>ing underlying every piece of mathematics,</p><p>Carnegie Mellon University, and the University of Alberta</p><p>designed a search algorithm, tested on Twitter, intended for uses be-</p><p>yond social search.</p><p>For example, they reported in Voice over Internet Protocol (VoIP)</p><p>networks, when a user calls another user in the network, he or she is</p><p>first connected to a VoIP carrier, a main node in the network. The VoIP</p><p>carrier connects the call to the destination either directly or, more</p><p>commonly, through another VoIP carrier.</p><p>“The length of the path from the caller to the receiver is important</p><p>since it affects both the quality and price of the call,” the researchers</p><p>noted. “The algorithms that are developed in this paper can be used to</p><p>find a short path (fewest carriers) between the initial (sender) and the</p><p>goal (receiver) nodes in the network.”</p><p>Degrees of Separation 51</p><p>These algorithms, such as greedy algorithms enhanced by geographic</p><p>heuristics, or probabalistic bidirectional methods, have the potential to</p><p>cut some of the overhead, and cost, of network search sessions, such as</p><p>the sample VoIP session, the authors believe.</p><p>Leskovec’s most recent work based on small-world algorithms ex-</p><p>plores the paths that humans take in connecting concepts that, on the</p><p>surface, seem rather disparate, such as Wolfgang Amadeus Mozart and</p><p>the Terminator character from the science-fiction films starring Arnold</p><p>Schwarzenegger.</p><p>“As a human, I sort of know how the knowledge fits together,” Les-</p><p>kovec says. “If I want to go from Mozart to Terminator and I know Mo-</p><p>zart was from Austria and Schwarzenegger was from Austria, maybe</p><p>I can go through the Austrian connection. A computer that is truly</p><p>decentralized has no clue; it has no conception that getting to Schwar-</p><p>zenegger is good enough.”</p><p>Interestingly enough, Leskovec says, computers fared better than</p><p>humans on average on solving such search chains, but humans also were</p><p>less likely to get totally lost and were capable of forming backup plans,</p><p>which the Web-crawling agents could not do. Effectively, he says, the</p><p>payoff of such research is “understanding how humans do this, what</p><p>kind of cues are we using, and how to make the cues more efficient or</p><p>help us recognize them, to help us understand where we are, right now,</p><p>in this global network.”</p><p>Further Reading</p><p>Backstrom, L., Boldi, P., Rosa, M., Ugander, J., and Vigna, S. “Four degrees of separation,”</p><p>http://arxiv.org/abs/1111.4570, Jan. 6, 2012.</p><p>Bakhshandeh, R., Samadi, M., Azimifar, Z., and Schaeffer, J. “Degrees of separation in social</p><p>networks,” Proceedings of the Fourth International Symposium on Combinatorial Search, Barce-</p><p>lona, Spain, July 15–16, 2011.</p><p>Goel, S., Muhamad, R., and Watts, D. “Social search in ‘small-world’ experiments,” 18th</p><p>International World Wide Web Conference, Madrid, Spain, April 20–24, 2009.</p><p>Kleinberg, J. “The small-world phenomenon: An algorithmic perspective,” 32nd ACM Sym-</p><p>posium on Theory of Computing, Portland, OR, May 21–23, 2000.</p><p>West, R., and Leskovec, J. “Human wayfinding in information networks,” 22nd Interna-</p><p>tional World Wide Web Conference, Lyon, France, April 16–20, 2012.</p><p>Randomness</p><p>Charles Seife</p><p>Our very brains revolt at the idea of randomness. We have evolved as a</p><p>species to become exquisite pattern-finders; long before the advent of</p><p>science, we figured out that a salmon-colored sky heralds a dangerous</p><p>storm or that a baby’s flushed face likely means a difficult night ahead.</p><p>Our minds automatically try to place data in a framework that allows</p><p>us to make sense of our observations and use them to understand and</p><p>predict events.</p><p>Randomness is so difficult to grasp because it works against our</p><p>pattern- finding instincts. It tells us that sometimes there is no pattern</p><p>to be found. As a result, randomness is a fundamental limit to our in-</p><p>tuition; it says that there are processes we can’t predict fully. It’s a con-</p><p>cept that we have a hard time accepting, even though it’s an essential</p><p>part of the way the cosmos works. Without an understanding of ran-</p><p>domness, we are stuck in a perfectly predictable universe that simply</p><p>doesn’t exist outside our heads.</p><p>I would argue that only once we understand three dicta—three laws</p><p>of randomness—can we break out of our primitive insistence on pre-</p><p>dictability and appreciate the universe for what it is, rather than what</p><p>we want it to be.</p><p>The First Law of Randomness:</p><p>There Is Such a Thing as Randomness</p><p>We use all kinds of mechanisms to avoid confronting randomness. We</p><p>talk about karma, in a cosmic equalization that ties seemingly uncon-</p><p>nected events together. We believe in runs of luck, both good and ill,</p><p>and that bad things happen in threes. We argue that we are influenced</p><p>Randomness 53</p><p>by the stars, by the phases of the moon, by the motion of the planets in</p><p>the heavens. When we get cancer, we automatically assume that some-</p><p>thing—or someone—is to blame.</p><p>But many events are not fully predictable or explicable. Disasters</p><p>happen randomly, to good people as well as to bad ones, to star-crossed</p><p>individuals as well as those who have a favorable planetary alignment.</p><p>Sometimes you can make a good guess about the future, but random-</p><p>ness can confound even the most solid predictions. Don’t be surprised</p><p>when you’re outlived by the overweight, cigar-smoking, speed-fiend</p><p>motorcyclist down the block.</p><p>What’s more, random events can mimic nonrandom ones. Even</p><p>the most sophisticated scientists can have difficulty telling the dif-</p><p>ference between a real effect and a random fluke. Randomness can</p><p>make placebos seem like miracle cures, or harmless compounds ap-</p><p>pear to be deadly poisons, and can even create subatomic particles out</p><p>of nothing.</p><p>The Second Law of Randomness:</p><p>Some Events Are Impossible to Predict</p><p>If you walk into a Las Vegas casino and observe the crowd gathered</p><p>around the craps table, you’ll probably see someone who thinks he’s on</p><p>a lucky streak. Because he’s won several rolls in a row, his brain tells</p><p>him he’s going to keep winning, so he keeps gambling. You’ll probably</p><p>also see someone who’s been losing. The loser’s brain, like the win-</p><p>ner’s, tells him to keep gambling. Since he’s been losing for so long, he</p><p>thinks he’s due for a stroke of luck; he won’t walk away from the table,</p><p>for fear of missing out.</p><p>Contrary to what our brains are telling us, there’s no mystical force</p><p>that imbues a winner with a streak of luck, nor is there a cosmic sense</p><p>of justice that ensures that a loser’s luck will turn around. The universe</p><p>doesn’t care one whit whether you’ve been winning or losing; each roll</p><p>of the dice is just like every other.</p><p>No matter how much effort you put into observing how the dice have</p><p>been behaving or how meticulously you have been watching for people</p><p>who seem to have luck on their side, you get absolutely no information</p><p>about what the next roll of a fair die will be. The outcome of a die roll is</p><p>54 Charles Seife</p><p>entirely independent of its history. And as a result, any scheme to gain</p><p>some sort of advantage by observing the table is doomed to fail. Events</p><p>like these—independent, purely random events—defy any attempts to</p><p>find a pattern because there is none to be found.</p><p>Randomness provides an absolute block against human ingenuity; it</p><p>means that our logic, our science, our capacity for reason can penetrate</p><p>only so far in predicting the behavior of the cosmos. Whatever methods</p><p>you try, whatever theory you create, whatever logic you use to predict</p><p>the next roll of a fair die, there’s always a 5/6 chance you are wrong.</p><p>Always.</p><p>The Third Law of Randomness:</p><p>Random Events Behave Predictably in Aggregate</p><p>Even If They’re Not Predictable Individually</p><p>Randomness is daunting; it sets limits where even the most sophisti-</p><p>cated theories cannot go, shielding elements of nature from even our</p><p>most determined inquiries. Nevertheless, to say that something is ran-</p><p>dom is not equivalent to saying that we can’t understand it. Far from it.</p><p>Randomness follows its own set of rules—rules that make the be-</p><p>havior</p><p>of a random process understandable and predictable.</p><p>These rules state that even though a single random event might be</p><p>completely unpredictable, a collection of independent random events is</p><p>extremely predictable—and the larger the number of events, the more</p><p>predictable they become. The law of large numbers is a mathematical</p><p>theorem that dictates that repeated, independent random events con-</p><p>verge with pinpoint accuracy upon a predictable average behavior. An-</p><p>other powerful mathematical tool, the central limit theorem, tells you</p><p>exactly how far off that average a given collection of events is likely to</p><p>be. With these tools, no matter how chaotic, how strange, a random</p><p>behavior might be in the short run, we can turn that behavior into</p><p>stable, accurate predictions in the long run.</p><p>The rules of randomness are so powerful that they have given phys-</p><p>ics some of its most sacrosanct and immutable laws. Though the atoms</p><p>in a box full of gas are moving at random, their collective behavior</p><p>is described by a simple set of deterministic equations. Even the laws</p><p>Randomness 55</p><p>of thermodynamics derive their power from the predictability of large</p><p>numbers of random events; they are indisputable only because the rules</p><p>of randomness are so absolute.</p><p>Paradoxically, the unpredictable behavior of random events has given</p><p>us the predictions in which we are most confident.</p><p>Randomness in Music*</p><p>Donald E. Knuth</p><p>Patterns that are perfectly pure and mathematically exact have a strong</p><p>aesthetic appeal, as advocated by Pythagoras and Plato and their in-</p><p>numerable intellectual descendants. Yet a bit of irregularity and un-</p><p>predictability can make a pattern even more beautiful. I was reminded</p><p>of this fact as I passed by two decorative walls while walking yesterday</p><p>from my home to my office: One wall, newly built, tries to emulate</p><p>the regular rectangular pattern of a grid, but it looks sterile and un-</p><p>attractive to my eyes; the other wall consists of natural stones that fit</p><p>together only approximately yet form a harmonious unity that I find</p><p>quite pleasing.</p><p>I noticed similar effects when I was experimenting years ago with</p><p>the design of computer-generated typefaces for the printing of beautiful</p><p>books. A design somehow “came to life” when it was not constrained to</p><p>be rigidly consistent.1</p><p>Similar examples abound in the musical realm, as well as in the</p><p>world of visual images. For example, I’m told that people who synthe-</p><p>size music electronically discovered long ago that rhythms are more</p><p>exciting if they don’t go exactly “l, 2, 3, 4” but rather miss the beat</p><p>very slightly and become what a mathematician might call “l + d1, 2 +</p><p>d2, 3 + d3, 4 +d4.” Although the discrepancies d mount to only a few</p><p>milliseconds, positive or negative, they enliven the music significantly</p><p>by comparison with the deadly and monotonous pulsing that you hear</p><p>when the ds are entirely zero.</p><p>Singers and saxophone players know better than to hit the notes of a</p><p>melody with perfect pitch.</p><p>* This article is based on an informal talk given to the Stanford Music Affiliates on May 9,</p><p>1990.</p><p>Randomness in Music 57</p><p>Furthermore, we can take liberties with the “ideal” notes them-</p><p>selves. In an essay called “Chance in artistic creation,” published in</p><p>1894,2 August Strindberg recounted the following experience:</p><p>A musician whom I knew amused himself by tuning his piano</p><p>haphazardly, without any rhyme or reason. Afterwards he played</p><p>Beethoven’s Sonate Pathétique by heart. It was an unbelievable de-</p><p>light to hear an old piece come back to life. How often had I previ-</p><p>ously heard this sonata, always the same way, never dreaming that</p><p>it was capable of being developed further!</p><p>And the notion of planned imperfection is not limited to variations in</p><p>the performance of a given composition; it extends also to the choices</p><p>of notes that a composer writes down. The main purpose of my talk</p><p>today is to describe a way by which you could build a simple machine</p><p>that will produce random harmonizations of any given melody.</p><p>More precisely, I’ll show you how to produce 2n + 2n – l different har-</p><p>monizations of any n-note melody, all of which are pretty good. A ma-</p><p>chine can easily generate any one of them, chosen at random, when its</p><p>user plays the melody on a keyboard with one hand.</p><p>The method I shall describe was taught to me in 1969 by David</p><p>Kraehenbuehl (1923–1997), when I audited his class on keyboard</p><p>harmony at Westminster Choir College. It is extremely simple, al-</p><p>though you do need to understand the most elementary aspects of</p><p>music theory. I shall assume that you are familiar with ordinary music</p><p>notation.</p><p>Kraehenbuehl’s algorithm produces four-part harmony from a given</p><p>melody, where the top three parts form “triadic” chords and the bot-</p><p>tom part supplies the corresponding bass notes.</p><p>A triad is a chord that consists of three notes separated by one-note</p><p>gaps in the scale. Thus the triads are</p><p>and others that differ only by being one or more octaves higher or</p><p>lower. The bottom note of a triad is called its “root,” and the other two</p><p>notes are called its “third” and its “fifth.”</p><p>These notions apply to any clef and to any key signature. For ex-</p><p>ample, with the treble clef and in the key of G major, the seven triads</p><p>58 Donald E. Knuth</p><p>are known by more precise names such as a “D major triad,’’ etc.; but</p><p>we don’t need to concern ourselves with such technical details.</p><p>The important thing for our purposes is to consider what happens</p><p>when individual notes of a triad move up or down by an octave. If we</p><p>view these chords modulo octave jumps, we see that they make a differ-</p><p>ent shape on the staff when the root tone is moved up an octave so that</p><p>the third tone becomes lowest; this change gives us the first inversion of</p><p>the triad. And if the root and third are both moved up an octave, leav-</p><p>ing the fifth tone lowest, we obtain the second inversion:</p><p>Root position First inversion Second inversion</p><p>Even though two-note gaps appear between adjacent notes of the first</p><p>and second inversions, these chords are still regarded as triads, because</p><p>octave jumps don’t change the name of a note on its scale: An A is still</p><p>an A, etc., and no two notes of an inverted triad have adjacent names.</p><p>Music theorists have traditionally studied three-note chords by fo-</p><p>cusing their attention first on the root of each triad, and next on the</p><p>bottom note, which identifies the inversion. Kraehenbuehl’s innovation</p><p>was to concentrate rather on the top note because it’s the melody note.</p><p>He observed that each melody note in the scale comes at the top of</p><p>three triadic chords—one in root position (0), one in first inversion</p><p>(1), and one in second inversion (2):</p><p>0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2</p><p>Furthermore, said Kraehenbuehl, there’s a natural way to add a</p><p>fourth part to this three-part harmony by simply repeating the root</p><p>note an octave or two lower. For example, in the key of C, we get</p><p>0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2</p><p>Randomness in Music 59</p><p>as the four-part harmonizations of melody notes A, A, A, B, B,</p><p>B, . . . , G, G, G. This rule almost always works well; but like all</p><p>good rules it has an exception: When the bass note turns out to be</p><p>the leading tone of the scale (which is one below the tonic), we should</p><p>change it to the so-called dominant tone (which is two notes lower).</p><p>Thus Kraehenbuehl’s correction to the natural rule yields the 21</p><p>four-part chords</p><p>0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2</p><p>in the key of C, when B is the leading tone; and it yields</p><p>0 1 2 0 1 2 0 1 2 1 2 00 1 2 0 1 2 0 1 2</p><p>in the key of G, because F# is the leading tone in that case. Notice that</p><p>when the bass note is corrected by shifting it down from a leading</p><p>tone in root position, it produces a chord with four separate pitches</p><p>(the so-called “dominant seventh chord” of its key), so it’s no longer</p><p>a triad.</p><p>Okay, now we know three good ways to harmonize any given</p><p>melody note in any given key. Kraehenbuehl completed his method</p><p>by</p><p>pointing out that the same principles apply to melodies with any</p><p>number of notes, provided only that we never use the same inversion twice</p><p>in a row. If one note has been harmonized with, say, the first inver-</p><p>sion, the next note should be harmonized with either the root posi-</p><p>tion or the second inversion; and so on. With this procedure there</p><p>are three choices for the first chord, and two choices for every chord</p><p>thereafter.</p><p>Let’s test his algorithm by trying it out on a familiar melody:</p><p>60 Donald E. Knuth</p><p>“London Bridge is falling down, my fair lady” has eleven notes, so Krae-</p><p>henbuehl has given us 3·2·2·2·2·2·2·2·2·2·2 = 3,072 ways to harmonize</p><p>it. The binary representations of three fundamental constants,</p><p>r = 3 + (0.00100100001111110110101010001 . . .)2,</p><p>e = 2 + (0.10110111111000010101000101100 . . .)2,</p><p>z = 1 + (0.10011110001101110111100110111 . . .)2,</p><p>serve to define three more-or-less random sequences of suitable inver-</p><p>sions, if we prefer mathematical guidance to coin-flipping. Namely, we</p><p>can use the integer part of the constant to specify the first chord, then</p><p>we can change the number of inversions by +1 or -1 (modulo 3) for</p><p>each successive binary digit that is 0 or 1, respectively. This procedure</p><p>gives us three new harmonizations of that classic British theme:</p><p>0 1 2 1 2 0 2 0 1 2 0</p><p>�</p><p>Variation 1. harmonized by p</p><p>2 1 2 1 0 1 0 2 1 0 2</p><p>Variation 2. harmonized by e</p><p>1 0 1 2 1 0 2 1 2 0 1</p><p>Variation 3. harmonized by</p><p>Amazing. Kraehenbuehl’s algorithm seems far too simple to be correct,</p><p>yet it really works!</p><p>Randomness in Music 61</p><p>Of course, there’s a little glitch at the end, because we have only one</p><p>chance in three of ending on a chord that’s stable and “resolved.” No</p><p>problem: In such a case we can just repeat the last melody note. With</p><p>this extension, variations 1 and 2 will end nicely, with</p><p>2 0 1 0 2 1</p><p>and</p><p>I can hardly wait for somebody to build me a keyboard that will per-</p><p>form such harmonizations automatically. After all, it’s basically just a</p><p>small matter of programming.</p><p>Notes</p><p>1. See, for example, my book Digital Typography, pages 57–59, 286–287, 324–325, 386,</p><p>391–396; also The METAFONTbook, pages 183–185.</p><p>2. August Strindberg, “Du Hasard dans la production artistique,” La Revue des revues 11</p><p>(Nov. 15, 1894), 265–270.</p><p>Playing the Odds</p><p>Soren Johnson</p><p>One of the most powerful tools a designer can use when developing</p><p>games is probability, using random chance to determine the outcome</p><p>of player actions or to build the environment in which play occurs. The</p><p>use of luck, however, is not without its pitfalls, and designers should be</p><p>aware of the tradeoffs involved—what chance can add to the experi-</p><p>ence and when it can be counterproductive.</p><p>Failing at Probability</p><p>One challenge with using randomness is that humans are notoriously</p><p>poor at evaluating probability accurately. A common example is the</p><p>gambler’s fallacy, which is the belief that odds even out over time. If the</p><p>roulette wheel comes up black five times in a row, players often believe</p><p>that the odds of it coming up black again are quite small, even though</p><p>clearly the streak makes no difference whatsoever. Conversely, people</p><p>also see streaks where none actually exist—the shooter with a “hot</p><p>hand” in basketball, for example, is a myth. Studies show that, if any-</p><p>thing, a successful shot actually predicts a subsequent miss.</p><p>Also, as designers of slot machines and massively multiplayer online</p><p>games are quite aware, setting odds unevenly between each progressive</p><p>reward level makes players think that the game is more generous than</p><p>it really is. One commercial slot machine had its payout odds published</p><p>by wizardofodds.com in 2008:</p><p>1:1 per 8 plays</p><p>2:1 per 600 plays</p><p>5:1 per 33 plays</p><p>20:1 per 2,320 plays</p><p>http://wizardofodds.com</p><p>Playing the Odds 63</p><p>80:1 per 219 plays</p><p>150:1 per 6,241 plays</p><p>The 80:1 payoff is common enough to give players the thrill of beating</p><p>the odds for a “big win” but still rare enough that the casino is at no</p><p>risk of losing money. Furthermore, humans have a hard time estimat-</p><p>ing extreme odds—a 1 percent chance is anticipated too often, and 99</p><p>percent odds are considered to be as safe as 100 percent.</p><p>Leveling the Field</p><p>These difficulties in estimating odds accurately actually work in the</p><p>favor of the game designer. Simple game-design systems, such as the</p><p>dice-based resource-generation system in Settlers of Catan, can be tanta-</p><p>lizingly difficult to master with a dash of probability.</p><p>In fact, luck makes a game more accessible because it shrinks the</p><p>gap—whether in perception or in reality—between experts and nov-</p><p>ices. In a game with a strong luck element, beginners believe that no</p><p>matter what, they have a chance to win. Few people would be willing</p><p>to play a chess grand master, but playing a backgammon expert is much</p><p>more appealing—a few lucky throws can give anyone a chance.</p><p>In the words of designer Dani Bunten, ‘‘Although most players hate</p><p>the idea of random events that will destroy their nice safe predictable</p><p>strategies, nothing keeps a game alive like a wrench in the works. Do not</p><p>allow players to decide this issue. They don’t know it but we’re offering</p><p>them an excuse for when they lose (‘It was that damn random event that</p><p>did me in!’) and an opportunity to ‘beat the odds’ when they win.”</p><p>Thus luck serves as a social lubricant—the alcohol of gaming, so</p><p>to speak—that increases the appeal of multiplayer gaming to audi-</p><p>ences that would not normally be suited for cutthroat head-to-head</p><p>competition.</p><p>Where Luck Fails</p><p>Nonetheless, randomness is not appropriate for all situations or even all</p><p>games. The “nasty surprise” mechanic is never a good idea. If a crate</p><p>provides ammo and other bonuses when opened but explodes 1 percent</p><p>of the time, the player has no chance to learn the probabilities in a</p><p>64 Soren Johnson</p><p>safe manner. If the explosion occurs early enough, the player may stop</p><p>opening crates immediately. If it happens much later, the player may</p><p>feel unprepared and cheated.</p><p>Also, when randomness becomes just noise, the luck simply detracts</p><p>from the player’s understanding of the game. If a die roll is made every</p><p>time a StarCraft Marine shoots at a target, the rate of fire simply appears</p><p>uneven. Over time, the effect of luck on the game’s outcome is negli-</p><p>gible, but the player has a harder time grasping how strong a Marine’s</p><p>attack actually is with all the extra random noise.</p><p>Furthermore, luck can slow down a game unnecessarily. The board</p><p>games History of the World and Small World have a similar conquest me-</p><p>chanic, except that the former uses dice and the latter does not (until</p><p>the final attack). Making a die roll with each attack causes a History of</p><p>the World turn to last at least three or four times as long as a turn in</p><p>Small World. The reason is not just the logistical issues of rolling so many</p><p>dice—knowing that the results of one’s decisions are completely pre-</p><p>dictable allows one to plan out all the steps at once without worrying</p><p>about contingencies. Often, handling contingencies is a core part of the</p><p>game design, but game speed is an important factor too, so designers</p><p>should be sure that the tradeoff is worthwhile.</p><p>Finally, luck is inappropriate for calculations to determine victory.</p><p>Unlucky rolls feel the fairest the longer players are given to react to them</p><p>before the game’s end. Thus the earlier luck plays a role, the better for</p><p>the perception of game balance. Many classic card games—pinochle,</p><p>bridge, and hearts—follow a standard model of an initial random dis-</p><p>tribution of cards that establishes the game’s “terrain,” followed by a</p><p>luck-free series of tricks that determines the winners and losers.</p><p>Probability Is Content</p><p>Indeed, the idea that randomness can provide an initial challenge to be</p><p>overcome plays an important role in many classic games, from simple</p><p>games such as Minesweeper to deeper ones such as NetHack and Age of Em-</p><p>pires. At their core, solitaire and Diablo are not so different—both</p><p>pres-</p><p>ent a randomly generated environment that the player needs to navigate</p><p>intelligently for success.</p><p>An interesting recent use of randomness is Spelunky, which is indie</p><p>developer Derek Yu’s combination of the random level generation of</p><p>Playing the Odds 65</p><p>NetHack with the game mechanics of 2-D platformers such as Lode Run-</p><p>ner. The addictiveness of the game comes from the unlimited number</p><p>of new caverns to explore, but frustration can emerge from the wild</p><p>difficulty of certain unplanned combinations of monsters and tunnels.</p><p>In fact, pure randomness can be an untamed beast, creating game</p><p>dynamics that throw an otherwise solid design out of balance. For ex-</p><p>ample, Civilization III introduced the concept of strategic resources that</p><p>were required to construct certain units—chariots need horses, tanks</p><p>need oil, and so on. These resources were sprinkled randomly across</p><p>the world, which inevitably led to large continents with only one clus-</p><p>ter of iron controlled by a single artificial intelligence (AI) opponent.</p><p>Complaints of being unable to field armies for lack of resources were</p><p>common among the community.</p><p>For Civilization IV, the problem was solved by adding a minimum</p><p>amount of space between certain important resources so that two</p><p>sources of iron never could be within seven tiles of each other. The re-</p><p>sult was a still unpredictable arrangement of resources around the globe</p><p>but without the clustering that could doom an unfortunate player. On</p><p>the other hand, the game actively encouraged clustering for less impor-</p><p>tant luxury resources—incense, gems, and spices—to promote inter-</p><p>esting trade dynamics.</p><p>Showing the Odds</p><p>Ultimately, when considering the role of probability, designers need to</p><p>ask themselves, “How is luck helping or hurting the game?” Is random-</p><p>ness keeping the players pleasantly off balance so that they can’t solve</p><p>the game trivially? Or is it making the experience frustratingly unpre-</p><p>dictable so that players are not invested in their decisions?</p><p>One factor that helps to ensure the former is making the probabil-</p><p>ity as explicit as possible. The strategy game Armageddon Empires based</p><p>combat on a few simple die rolls and then showed the dice directly on-</p><p>screen. Allowing the players to peer into the game’s calculations in-</p><p>creases their comfort level with the mechanics, which makes chance a</p><p>tool for the player instead of a mystery.</p><p>Similarly, with Civilization IV, we introduced a help mode that showed</p><p>the exact probability of success in combat, which drastically increased</p><p>player satisfaction with the underlying mechanics. Because humans</p><p>66 Soren Johnson</p><p>have such a hard time estimating probability accurately, helping them</p><p>make a smart decision can improve the experience immensely.</p><p>Some deck-building card games, such as Magic: The Gathering or Do-</p><p>minion, put probability in the foreground by centering the game ex-</p><p>perience on the likelihood of drawing cards in the player’s carefully</p><p>constructed deck. These games are won by players who understand</p><p>the proper ratio of rares to commons, knowing that each card is drawn</p><p>exactly once each time through the deck. This concept can be extended</p><p>to other games of chance by providing, for example, a virtual “deck of</p><p>dice” that ensures that the distribution of die rolls is exactly even.</p><p>Another interesting—and perhaps underused—idea from the dis-</p><p>tant past of gaming history is the element-of-chance game option from</p><p>the turn-based strategy game Lords of Conquest. The three options avail-</p><p>able—low, medium, and high—determined whether luck was used</p><p>only to break ties or to play a larger role in resolving combat. The ap-</p><p>propriate role of chance in a game ultimately is a subjective question,</p><p>and giving players the ability to adjust the knobs themselves can open</p><p>up the game to a larger audience with a greater variety of tastes.</p><p>Machines of the Infinite</p><p>John Pavlus</p><p>On a snowy day in Princeton, New Jersey, in March 1956, a short,</p><p>owlish-looking man named Kurt Gödel wrote his last letter to a dying</p><p>friend. Gödel addressed John von Neumann formally even though the</p><p>two had known each other for decades as colleagues at the Institute for</p><p>Advanced Study in Princeton. Both men were mathematical geniuses,</p><p>instrumental in establishing the U.S. scientific and military supremacy</p><p>in the years after World War II. Now, however, von Neumann had can-</p><p>cer, and there was little that even a genius like Gödel could do except</p><p>express a few overoptimistic pleasantries and then change the subject:</p><p>Dear Mr. von Neumann:</p><p>With the greatest sorrow I have learned of your illness. . . . As</p><p>I hear, in the last months you have undergone a radical treatment</p><p>and I am happy that this treatment was successful as desired, and</p><p>that you are now doing better. . . .</p><p>Since you now, as I hear, are feeling stronger, I would like to</p><p>allow myself to write you about a mathematical problem, of which</p><p>your opinion would very much interest me. . . .</p><p>Gödel’s description of this problem is utterly unintelligible to non-</p><p>mathematicians. (Indeed, he may simply have been trying to take von</p><p>Neumann’s mind off of his illness by engaging in an acutely specialized</p><p>version of small talk.) He wondered how long it would take for a hypo-</p><p>thetical machine to spit out answers to a problem. What he concluded</p><p>sounds like something out of science fiction:</p><p>If there really were [such] a machine . . . this would have conse-</p><p>quences of the greatest importance. Namely, it would obviously</p><p>mean that . . . the mental work of a mathematician concerning</p><p>Yes-or-No questions could be completely replaced by a machine.</p><p>68 John Pavlus</p><p>By “mental work,” Gödel didn’t mean trivial calculations like add-</p><p>ing 2 and 2. He was talking about the intuitive leaps that mathemati-</p><p>cians take to illuminate entirely new areas of knowledge. Twenty-five</p><p>years earlier, Gödel’s now famous incompleteness theorems had forever</p><p>transformed mathematics. Could a machine be made to churn out simi-</p><p>lar world-changing insights on demand?</p><p>A few weeks after Gödel sent his letter, von Neumann checked into</p><p>Walter Reed Army Medical Center in Washington, D.C., where he</p><p>died less than a year later, never having answered his friend. But the</p><p>problem would outlive both of them. Now known as P versus NP,</p><p>Gödel’s question went on to become an organizing principle of mod-</p><p>ern computer science. It has spawned an entirely new area of research</p><p>called computational complexity theory—a fusion of mathematics, sci-</p><p>ence, and engineering that seeks to prove, with total certainty, what</p><p>computers can and cannot do under realistic conditions.</p><p>But P versus NP is about much more than just the plastic-and-</p><p>silicon contraptions we call computers. The problem has practical im-</p><p>plications for physics and molecular biology, cryptography, national</p><p>security, evolution, the limits of mathematics, and perhaps even the</p><p>nature of reality. This one question sets the boundaries for what, in</p><p>theory, we will ever be able to compute. And in the 21st century, the</p><p>limits of computation look more and more like the limits of human</p><p>knowledge itself.</p><p>The Bet</p><p>Michael Sipser was only a graduate student, but he knew someone</p><p>would solve the P versus NP problem soon. He even thought he might</p><p>be the one to do it. It was the fall of 1975, and he was discussing the</p><p>problem with Leonard Adleman, a fellow graduate student in the com-</p><p>puter science department at the University of California, Berkeley. “I</p><p>had a fascination with P versus NP, had this feeling that I was somehow</p><p>able to understand it in a way that went beyond the way everyone else</p><p>seemed to be approaching it,” says Sipser, who is now head of the math-</p><p>ematics department at the Massachusetts Institute of Technology. He</p><p>was so sure of himself that he made a wager that day with Adleman:</p><p>P versus NP would be solved by the end of the 20th century, if not</p><p>sooner. The terms: one ounce of pure gold.</p><p>Machines of the Infinite 69</p><p>Sipser’s bet made a kind of poetic</p><p>sense because P versus NP is itself</p><p>a problem about how quickly other problems can be solved. Sometimes</p><p>simply following a checklist of steps gets you to the end result in rela-</p><p>tively short order. Think of grocery shopping: You tick off the items</p><p>one by one until you reach the end of the list. Complexity theorists</p><p>label these problems P, for “polynomial time,” which is a mathemati-</p><p>cally precise way of saying that no matter how long the grocery list</p><p>becomes, the amount of time that it will take to tick off all the items</p><p>will never grow at an unmanageable rate.</p><p>In contrast, many more problems may or may not be practical to</p><p>solve by simply ticking off items on a list, but checking the solution is</p><p>easy. A jigsaw puzzle is a good example: Even though it may take effort</p><p>to put together, you can recognize the right solution just by looking at</p><p>it. Complexity theorists call these quickly checkable, “jigsaw puzzle-</p><p>like” problems NP.</p><p>Four years before Sipser made his bet, a mathematician named Ste-</p><p>phen Cook had proved that these two kinds of problems are related:</p><p>Every quickly solvable P problem is also a quickly checkable NP prob-</p><p>lem. The P versus NP question that emerged from Cook’s insight—</p><p>and that has hung over the field ever since—asks if the reverse is also</p><p>true: Are all quickly checkable problems quickly solvable as well? In-</p><p>tuitively speaking, the answer seems to be no. Recognizing a solved</p><p>jigsaw puzzle (“Hey, you got it!”) is hardly the same thing as doing</p><p>all the work to find the solution. In other words, P does not seem to</p><p>equal NP.</p><p>What fascinated Sipser was that nobody had been able to mathemati-</p><p>cally prove this seemingly obvious observation. And without a proof, a</p><p>chance remained, however unlikely or strange, that all NP problems</p><p>might actually be P problems in disguise. P and NP might be equal—</p><p>and because computers can make short work of any problem in P, P</p><p>equals NP would imply that computers’ problem-solving powers are</p><p>vastly greater than we ever imagined. They would be exactly what</p><p>Gödel described in his letter to von Neumann: mechanical oracles that</p><p>could efficiently answer just about any question put to them, so long as</p><p>they could be programmed to verify the solution.</p><p>Sipser knew that this outcome was vanishingly improbable. Yet</p><p>proving the opposite, much likelier, case—that P is not equal to NP—</p><p>would be just as groundbreaking.</p><p>70 John Pavlus</p><p>Like Gödel’s incompleteness theorems, which revealed that mathe-</p><p>matics must contain true but unprovable propositions, a proof showing</p><p>that P does not equal NP would expose an objective truth concern-</p><p>ing the limitations of knowledge. Solving a jigsaw puzzle and recog-</p><p>nizing that one is solved are two fundamentally different things, and</p><p>there are no shortcuts to knowledge, no matter how powerful our</p><p>computers get.</p><p>Proving a negative is always difficult, but Gödel had done it. So</p><p>to Sipser, making his bet with Adleman, 25 years seemed like more</p><p>than enough time to get the job done. If he couldn’t prove that P did</p><p>not equal NP himself, someone else would. And he would still be one</p><p>ounce of gold richer.</p><p>Complicated Fast</p><p>Adleman shared Sipser’s fascination, if not his confidence, because</p><p>of one cryptic mathematical clue. Cook’s paper establishing that P</p><p>problems are all NP had also proved the existence of a special kind of</p><p>quickly checkable type of problem called NP-complete. These prob-</p><p>lems act like a set of magic keys: If you find a fast algorithm for solving</p><p>one of them, that algorithm can also unlock the solution to every other</p><p>NP problem and prove that P equals NP.</p><p>There was just one catch: NP-complete problems are among the</p><p>hardest anyone in computer science had ever seen. And once discov-</p><p>ered, they began turning up everywhere. Soon after Cook’s paper</p><p>appeared, one of Adleman’s mentors at Berkeley, Richard M. Karp,</p><p>published a landmark study showing that 21 classic computational prob-</p><p>lems were all NP-complete. Dozens, then hundreds, soon followed. “It</p><p>was like pulling a finger out of a dike,” Adleman says. Scheduling air</p><p>travel, packing moving boxes into a truck, solving a Sudoku puzzle, de-</p><p>signing a computer chip, seating guests at a wedding reception, playing</p><p>Tetris, and thousands of other practical, real-world problems have been</p><p>proved to be NP-complete.</p><p>How could this tantalizing key to solving P versus NP seem so com-</p><p>monplace and so uncrackable at the same time? “That’s why I was inter-</p><p>ested in studying the P versus NP problem,” says Adleman, who is now</p><p>a professor at the University of Southern California. “The power and</p><p>breadth of these computational questions just seemed deeply awesome.</p><p>Machines of the Infinite 71</p><p>The Basics of Complexity</p><p>How long will it take to solve tHat problem? That’s the question</p><p>that researchers ask as they classify problems into computational</p><p>classes. As an example, consider a simple sorting task: Put a list of</p><p>random numbers in order from smallest to largest. As the list gets</p><p>bigger, the time it takes to sort the list increases at a manageable</p><p>rate—as the square of the size of the list, perhaps. This puts it in</p><p>class “P” because it can be solved in polynomial time. Harder ques-</p><p>tions, such as the “traveling salesman” problem [see next box in this</p><p>article], require exponentially more time to solve as they grow more</p><p>complex. These “NP-complete” problems quickly get so unwieldy</p><p>that not even billions of processors working for billions of years can</p><p>crack them.</p><p>NP</p><p>These problems</p><p>have one</p><p>essential quality:</p><p>if you are given an</p><p>answer to an NP</p><p>problem, you can quickly</p><p>verify whether the answer</p><p>is true or false.</p><p>NP-complete</p><p>This subset of di�cult-to-solve</p><p>NP problems acts as a master key:</p><p>every NP problem can be translated</p><p>into any NP-complete problem.</p><p>Thus, if someone were to find a</p><p>quick solution to an NP-complete</p><p>problem, he or she would be able</p><p>to quickly solve all NP problems.</p><p>P would equal NP.</p><p>P</p><p>These problems can be</p><p>quickly solved. Note that all</p><p>quickly solvable problems</p><p>are also quickly verifi able, so</p><p>all P problems are also NP.</p><p>The reverse is almost</p><p>certainly not true.</p><p>M</p><p>or</p><p>e</p><p>D</p><p>i�</p><p>c</p><p>ul</p><p>t t</p><p>o</p><p>So</p><p>lv</p><p>e</p><p>What Kind of Problem Is It?</p><p>Venn diagram. Reproduced with permission. Copyright © 2012 Scien-</p><p>tific American, a division of Nature America, Inc. All Rights Reserved.</p><p>72 John Pavlus</p><p>But we certainly didn’t understand them. And it didn’t seem like we</p><p>would be understanding them anytime soon.” (Adleman’s pessimism</p><p>about P versus NP led to a world-changing invention: A few years</p><p>after making his bet, Adleman and his colleagues Ronald Rivest and</p><p>Adi Shamir exploited the seeming incommensurability of P and NP to</p><p>create their eponymous RSA encryption algorithm, which remains in</p><p>wide use for online banking, communications, and national security</p><p>applications.)</p><p>NP-complete problems are hard because they get complicated fast.</p><p>Imagine that you are a backpacker planning a trip through a number</p><p>of cities in Europe, and you want a route that takes you through each</p><p>city while minimizing the total distance you will need to travel. How</p><p>do you find the best route? The simplest method is just to try out each</p><p>possibility. With five cities to visit, you need to check only 12 possible</p><p>routes. With 10 cities, the number of possible routes mushrooms to</p><p>more than 180,000. At 60 cities, the number of paths exceeds the num-</p><p>ber of atoms in the known universe. This computational nightmare is</p><p>known as the traveling salesman problem, and in more than 80 years of</p><p>intense study, no one has ever found a general way to solve it that works</p><p>better than trying every possibility one at a time.</p><p>That is the perverse essence of NP-completeness—and of P versus</p><p>NP: Not only are all NP-complete problems equally impossible to solve</p><p>except in the simplest cases—even if your computer has more memory</p><p>than God and the entire lifetime of the universe to work with—they</p><p>seem to pop up everywhere. In</p><p>fact, these NP-complete problems don’t</p><p>just frustrate computer scientists. They seem to put limits on the capa-</p><p>bilities of nature itself.</p><p>Nature’s Code</p><p>The pioneering Dutch programmer Edsger Dijkstra understood that</p><p>computational questions have implications beyond mathematics. He</p><p>once remarked that “computer science is no more about computers</p><p>than astronomy is about telescopes.” In other words, computation is a</p><p>behavior exhibited by many systems besides those made by Google and</p><p>Intel. Indeed, any system that transforms inputs into outputs by a set of</p><p>discrete rules—including those studied by biologists and physicists—</p><p>can be said to be computing.</p><p>Machines of the Infinite 73</p><p>The Swedish Salesman</p><p>If you’re feeling ambitious on your next trip to Sweden, consider</p><p>seeing it all. Researchers have proved that the route pictured here</p><p>is the shortest possible path that crosses through every one of the</p><p>country’s 24,978 cities, towns, and villages. Researchers don’t</p><p>expect anyone to make the actual trip, but the search techniques</p><p>they developed to solve it will help in other situations where in-</p><p>vestigators need to find the optimal path through a complicated</p><p>landscape—in microchip design or genome sequencing, for in-</p><p>stance. Image courtesy of William J. Cook.</p><p>Population centers</p><p>(dots)</p><p>Optimum path</p><p>(line)</p><p>74 John Pavlus</p><p>In 1994 mathematician Peter Shor proved that cleverly arranged</p><p>subatomic particles could break modern encryption schemes. In 2002</p><p>Adleman used strands of DNA to find an optimal solution to an in-</p><p>stance of the traveling salesman problem. And in 2005 Scott Aaronson,</p><p>an expert in quantum computing who is now at MIT’s Computer Sci-</p><p>ence and Artificial Intelligence Laboratory, used soap bubbles, of all</p><p>things, to efficiently compute optimal solutions to a problem known</p><p>as the Steiner tree. These are all exactly the kinds of NP problems on</p><p>which computers should choke their circuit boards. Do these natural</p><p>systems know something about P versus NP that computers don’t?</p><p>“Of course not,” Aaronson says. His soap bubble experiment was ac-</p><p>tually a reductio ad absurdum of the claim that simple physical systems</p><p>can somehow transcend the differences between P and NP problems.</p><p>Although the soap bubbles did “compute” perfect solutions to the mini-</p><p>mum Steiner tree in a few instances, they quickly failed as the size of</p><p>the problem increased, just as a computer would. Adleman’s DNA-strand</p><p>experiment hit the same wall. Shor’s quantum algorithm does work in</p><p>all instances, but the factoring problem that it cracks is almost certainly</p><p>not NP-complete. Therefore, the algorithm doesn’t provide the key that</p><p>would unlock every other NP problem. Biology, classical physics, and</p><p>quantum systems all seem to support the idea that NP-complete prob-</p><p>lems have no shortcuts. And that would only be true if P did not equal NP.</p><p>“Of course, we still can’t prove it with airtight certainty,” Aaron-</p><p>son says. “But if we were physicists instead of complexity theorists,</p><p>‘P does not equal NP’ would have been declared a law of nature long</p><p>ago—just like the fact that nothing can go faster than the speed of</p><p>light.” Indeed, some physical theories about the fundamental nature of</p><p>the universe—such as the holographic principle, suggested by Stephen</p><p>Hawking’s work on black holes—imply that the fabric of reality itself</p><p>is not continuous but made of discrete bits, just like a computer (see “Is</p><p>Space Digital?” by Michael Moyer, Scientific American, February 2012).</p><p>Therefore, the apparent intractability of NP problems—and the limita-</p><p>tions on knowledge that this implies—may be baked into the universe</p><p>at the most fundamental level.</p><p>Brain Machine</p><p>So if the very universe itself is beholden to the computational limits</p><p>imposed by P versus NP, how can it be that NP-complete problems</p><p>Machines of the Infinite 75</p><p>seem to get solved all the time—even in instances where finding these</p><p>solutions should take trillions of years or more?</p><p>For example, as a human fetus gestates in the womb, its brain wires</p><p>itself up out of billions of individual neurons. Finding the best ar-</p><p>rangement of these cells is an NP-complete problem—one that evo-</p><p>lution appears to have solved. “When a neuron reaches out from one</p><p>point to get to a whole bunch of other synapse points, it’s basically</p><p>a graph- optimization problem, which is NP-hard,” says evolutionary</p><p>neurobiologist Mark Changizi. Yet the brain doesn’t actually solve the</p><p>problem—it makes a close approximation. (In practice, the neurons</p><p>consistently get within 3 percent of the optimal arrangement.) The</p><p>Caenor habditis elegans worm, which has only 302 neurons, still doesn’t</p><p>have a perfectly optimal neural-wiring diagram, despite billions on bil-</p><p>lions of generations of natural selection acting on the problem. “Evolu-</p><p>tion is constrained by P versus NP,” Changizi says, “but it works anyway</p><p>because life doesn’t always require perfection to function well.”</p><p>And neither, it turns out, do computers. That modern computers</p><p>can do anything useful at all—much less achieve the wondrous feats we</p><p>all take for granted on our video-game consoles and smartphones—is</p><p>proof that the problems in P encompass a great many of our computing</p><p>needs. For the rest, often an imperfect approximating algorithm is good</p><p>enough. In fact, these “good enough” algorithms can solve immensely</p><p>complex search and pattern-matching problems, many of which are</p><p>technically NP-complete. These solutions are not always mathemati-</p><p>cally optimal in every case, but that doesn’t mean they aren’t useful.</p><p>Take Google, for instance. Many complexity researchers consider</p><p>NP problems to be, in essence, search problems. But according to</p><p>Google’s director of research Peter Norvig, the company takes pains</p><p>to avoid dealing with NP problems altogether. “Our users care about</p><p>speed more than perfection,” he says. Instead Google researchers op-</p><p>timize their algorithms for an even faster computational complexity</p><p>category than P (referred to as linear time) so that search results ap-</p><p>pear nearly instantaneously. And if a problem comes up that cannot</p><p>be solved in this way? “We either reframe it to be easier, or we don’t</p><p>bother,” Norvig says.</p><p>That is the legacy and the irony of P versus NP. Writing to von Neu-</p><p>mann in 1956, Gödel thought the problem held the promise of a future</p><p>filled with infallible reasoning machines capable of replacing “the men-</p><p>tal work of a mathematician” and churning out bold new truths at the</p><p>76 John Pavlus</p><p>push of a button. Instead decades of studying P versus NP have helped</p><p>build a world in which we extend our machines’ problem-solving pow-</p><p>ers by embracing their limitations. Lifelike approximation, not mechan-</p><p>ical perfection, is how Google’s autonomous cars can drive themselves</p><p>on crowded Las Vegas freeways and IBM’s Watson can guess its way to</p><p>victory on Jeopardy.</p><p>Gold Rush</p><p>The year 2000 came and went, and Sipser mailed Adleman his ounce</p><p>of gold. “I think he wanted it to be embedded in a cube of Lucite, so he</p><p>could put it on his desk or something,” Sipser says. “I didn’t do that.”</p><p>That same year the Clay Mathematics Institute in Cambridge, Mass.,</p><p>offered a new bounty for solving P versus NP: $1 million. The prize</p><p>helped to raise the problem’s profile, but it also attracted the attention</p><p>of amateurs and cranks; nowadays, like many prominent complexity</p><p>theorists, Sipser says, he regularly receives unsolicited e-mails asking</p><p>him to review some new attempt to prove that P does not equal NP—</p><p>or worse, the opposite.</p><p>Although P versus NP remains unsolved, many complexity research-</p><p>ers still think it will yield someday. “I never really gave up on it,” Sipser</p><p>says. He claims to still pull out pencil and paper from time to time and</p><p>work on it—almost for recreation, like a dog chewing on a favorite</p><p>bone. P versus NP is, after all, an NP problem itself: The only way to</p><p>find the answer is to keep searching. And while that answer may never</p><p>come, if it does, we will know</p><p>it when we see it.</p><p>More to Explore</p><p>The Efficiency of Algorithms. Harry R. Lewis and Christos H. Papadimitriou in Scientific Ameri-</p><p>can, Vol. 238, No. 1, pp 96–109, January 1978.</p><p>“The History and Status of the P versus NP Question.” Michael Sipser in Proceedings of the</p><p>Twenty-Fourth Annual ACM Symposium on Theory of Computing, pp 603–618, 1992.</p><p>“The Limits of Reason.” Gregory Chaitin in Scientific American, Vol. 294, No. 3, pp 74–81,</p><p>March 2006.</p><p>“The Limits of Quantum Computers.” Scott Aaronson in Scientific American, Vol. 298, No. 3,</p><p>pp 62–69, March 2008.</p><p>Bridges, String Art, and Bézier Curves</p><p>Renan Gross</p><p>The Jerusalem Chords Bridge</p><p>The Jerusalem Chords Bridge, in Israel, was built to make way for the</p><p>city’s light rail train system. However, its design took into consideration</p><p>more than just utility—it is a work of art, designed as a monument. Its</p><p>beauty rests not only in the visual appearance of its criss-cross cables,</p><p>but also in the mathematics that lies behind it. Let us take a deeper look</p><p>into these chords.</p><p>The Jerusalem Chords Bridge is a suspension bridge, which means</p><p>that its entire weight is held from above. In this case, the deck is con-</p><p>nected to a single tower by powerful steel cables. The cables are con-</p><p>nected in the following way: The ones at the top of the tower support</p><p>the center of the bridge, and the ones at the bottom support the further</p><p>away sections, so that the cables cross each other.</p><p>Despite the fact that they draw out discrete, straight lines, we notice</p><p>a remarkable feature: The outline of the cables’ edges seems strikingly</p><p>smooth. Does it obey any known mathematical formula?</p><p>To find out the shape that the edges make, we have to devise a</p><p>mathematical model for the bridge. As the bridge itself is quite com-</p><p>plex, featuring a curved deck and a two-part leaning tower, we have</p><p>to simplify things. Although we lose a little accuracy and precision,</p><p>we gain in mathematical simplicity, and we still capture the beautiful</p><p>essence of the bridge’s form. Afterward, we will be able to generalize</p><p>our simple description and apply it to the real bridge structure.</p><p>This is the core of modeling—taking only the important features</p><p>from the real world and translating them into mathematics.</p><p>78 Renan Gross</p><p>Figure 2. The Jerusalem Chords Bridge as seen from below.</p><p>Figure 1. The Jerusalem Chords Bridge at night. Image: Petdad [7].</p><p>Bridges, String Art, and Bézier Curves 79</p><p>Chord Analysis</p><p>Let’s look at a coordinate system, (x,y). The x axis corresponds to the</p><p>base of the bridge, and the y axis to the tower from which it hangs.</p><p>Taking the tower to span from 0 to 1 on our y axis, and the deck</p><p>from 0 to 1 on the x axis, we make n marks uniformly spaced on each</p><p>axis. From each mark on the x axis, we draw a straight line to the y axis,</p><p>so that the first mark on the x axis is connected to the nth on the y axis,</p><p>the second on the x axis to the (n - 1)st on the y axis and so on. These</p><p>lines represent our chords. Let’s also assume that the x and y axes meet</p><p>at a right angle. This is not a perfect picture of reality, for the cables are</p><p>not evenly spaced and the tower and deck are not perpendicular, but it</p><p>simplifies things.</p><p>The outline formed by the chords is essentially made out of the inter-</p><p>sections of one cable and the one adjacent to it: You connect each inter-</p><p>section point with the one after it by a straight line. The more chords</p><p>there are, the smoother the outline becomes. So the smooth curve that</p><p>is hinted at by the chords, the intersection envelope, is the outline you</p><p>would get from infinitely many chords.</p><p>Forgoing the detailed calculations (which you can find at Plus Inter-</p><p>net magazine [8]), we find that all the points on this curve have coor-</p><p>dinates of the type</p><p>, ( ) )t1−( , ) (x y t 2 2=</p><p>(0,1)</p><p>(0,0)(0,0) (1,0)</p><p>Figure 3. Coordinate axes superimposed on the bridge.</p><p>80 Renan Gross</p><p>for all t between 0 and 1. Hence,</p><p>( ) ( )y x x1 2= −</p><p>“Well, is this it?” we ask. Is the shape going to remain an unnamed</p><p>mathematical relation? In fact, no! Though it is not easy to see at first,</p><p>this is actually the equation for a parabola! To this you might reply that</p><p>the equation for a parabola is this:</p><p>( )y x ax bx c2= + +</p><p>which differs vastly from our result, and you would be correct. How-</p><p>ever, with a little work it can be shown that if we define R = x + y and</p><p>S = x - y, we can rewrite our unfamiliar equation as</p><p>( )R S S</p><p>2 2</p><p>12</p><p>= +</p><p>(See [9] for the details.)</p><p>And this indeed conforms to our well-known parabola equation. By</p><p>replacing our variables x and y with S and R, we have actually rotated</p><p>our coordinate system by 45 degrees. But this new coordinate system</p><p>needn’t frighten us. As we can see, the parabola equation is all the same.</p><p>The result we got—that the outline of the cables is essentially par-</p><p>abolic—is certainly satisfying, for the parabola is such a simple and</p><p>elegant shape. But it also leaves us a bit puzzled. Is there a reason for</p><p>0.9</p><p>0.7</p><p>0.6</p><p>0.5</p><p>0.3</p><p>0.1</p><p>0</p><p>0.10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1</p><p>0.8</p><p>0.4</p><p>0.2</p><p>Figure 4. Our axes with evenly spaced chords.</p><p>Bridges, String Art, and Bézier Curves 81</p><p>this simplicity? Why should it be parabolic, and not some other curve?</p><p>If we changed the shape of the bridge—say, to make the tower more</p><p>leaning—how would it affect our curve? Is there any way in which we</p><p>can make amends for our simplifications and assumptions that we had</p><p>to perform earlier?</p><p>An Unlikely Answer</p><p>The answers to our questions originate from a surprising field: that of</p><p>automobile design. Back in the 1960s, engineer Pierre Bézier [10] used</p><p>special curves to specify how he wanted car parts to look. These curves</p><p>are called Bézier curves. We shall now take a look at what they have to</p><p>offer us.</p><p>35</p><p>–10 –5 0 5</p><p>30</p><p>25</p><p>20</p><p>15</p><p>10</p><p>5</p><p>10</p><p>1</p><p>0 0.2 0.80.4 0.6</p><p>0.8</p><p>0.6</p><p>0.4</p><p>0.2</p><p>0</p><p>1</p><p>0 2 84 6 10 12</p><p>14</p><p>12</p><p>10</p><p>8</p><p>6</p><p>4</p><p>2</p><p>0</p><p>14</p><p>Figure 5. Top left: The parabola R(S) = S2/2 + 1/2. Top right: The same</p><p>para bola tilted by 45 degrees. Bottom: Zoom on the square defined by x and</p><p>y ranging from 0 to 1 in the tilted parabola, which is the region that repre-</p><p>sents the bridge.</p><p>82 Renan Gross</p><p>We all know that between any two points there can be only one</p><p>straight line; hence, we can define a specific line using only two points.</p><p>In a similar fashion, a Bézier curve is defined by any number of points,</p><p>called control points. Unlike a straight line, it does not pass through all</p><p>of the points. Rather, it starts at the first point and ends at the last, but</p><p>it does not necessarily go through all the others. Instead, the points act</p><p>as “weights,” which direct the flow of the curve from the initial point</p><p>to the last.</p><p>The number of points is used to define what is called the degree of</p><p>the curve. A two-point linear Bézier curve has degree 1 and is just an</p><p>ordinary straight line; a three-point quadratic Bézier curve has degree 2</p><p>and is a parabola; and in general, a curve of degree n has n + 1 control</p><p>points.</p><p>A nice way of visualizing the construction of a Bézier curve is to</p><p>imagine a pencil that starts drawing from the first control point to the</p><p>last. On the way, it is attracted to the various control points, but the</p><p>level of attraction changes as the pencil goes along. It is initially most</p><p>attracted to the first control points, so as the pencil starts drawing,</p><p>it heads off in their direction. As it progresses, it becomes more and</p><p>more attracted to the later control points, until it finally reaches the</p><p>last point. At any given time while we draw, we can ask, “What per-</p><p>centage of the curve has the pencil drawn already?” This percentage is</p><p>called the curve parameter and is marked by t.</p><p>P1</p><p>P1</p><p>P1</p><p>P0</p><p>P0</p><p>P0</p><p>P2</p><p>P2 P3</p><p>Degree 1</p><p>Degree 2</p><p>Degree 3</p><p>Figure 6. Bézier curves with degrees 1, 2, and 3.</p><p>Bridges, String Art, and Bézier Curves 83</p><p>How does all of this relate to our parabolic bridge? The connection</p><p>is revealed when we take a look at how to actually draw a Bézier curve.</p><p>One way of drawing</p><p>it is to follow a mathematical formula that gives</p><p>the coordinates of the curve. We will skip over that (you can have a</p><p>look at the formula at [11]) and instead move on to the second method:</p><p>constructing a Bézier curve recursively. In this method, to construct</p><p>an nth degree curve, we use two (n – 1)st degree curves. It is best to</p><p>illustrate this method with an example.</p><p>Suppose we have a 3rd degree curve. It is defined by 4 points: P0, P1,</p><p>P2, and P3. From these we create two new groups: all points except the</p><p>last, and all points except the first. We now have</p><p>Group 1: P0, P1, and P2</p><p>Group 2: P1, P2, and P3</p><p>Each of these groups defines a 2nd degree Bézier curve. Remember</p><p>how we talked about using a pencil that moves from the first point to</p><p>P1</p><p>P0</p><p>P2</p><p>P3</p><p>P1</p><p>P0</p><p>P2</p><p>P3</p><p>P1</p><p>P0</p><p>P2</p><p>P3</p><p>P1</p><p>P0</p><p>P2</p><p>P3</p><p>Figure 7. Drawing a Bézier curve.</p><p>84 Renan Gross</p><p>the last? Now, suppose that you have two pencils, and you draw both of</p><p>these 2nd degree curves at the same time. The first curve starts from</p><p>P0 and finishes at P2, and the second starts at P1 and ends at P3. At any</p><p>given time during the pencils’ journeys, you can connect their positions</p><p>with a straight line.</p><p>So, while these two are drawing, think of a third pencil. This pen-</p><p>cil is always somewhere on the line connecting the two current posi-</p><p>tions of the other pencils, and it moves along at the same rate as the</p><p>other two. At the start, it is on the line connecting P0 and P1, and since</p><p>both pencils have moved along 0% of their curves, the third pencil has</p><p>moved along 0% of this line, which puts it at P0. After the other two</p><p>pencils have moved along, say, 36% of their curves and are at points Q 0</p><p>and Q 1, the third pencil is on the line from Q 0 to Q 1, at the point that</p><p>marks 36% of that line. When the other two pencils have finished their</p><p>journey, so they are at points P2 and P3 and have traveled 100% of the</p><p>way, the third pencil is on the line from P2 to P3 at 100% along the way,</p><p>which puts it at P3.</p><p>There is a very good question to be asked here. We have just de-</p><p>scribed how to create an nth degree curve, but doing so requires</p><p>P1</p><p>P0</p><p>Start</p><p>P2</p><p>P3</p><p>P1</p><p>Q0 Q1</p><p>P0</p><p>36% of the way</p><p>P2</p><p>P3</p><p>P1</p><p>Q0</p><p>Q1</p><p>P0</p><p>62% of the way</p><p>P2</p><p>P3</p><p>P1</p><p>Q0</p><p>Q1</p><p>P0</p><p>Finished</p><p>P2</p><p>P3</p><p>Figure 8. Building a cubic Bézier curve using quadratic curves. The P0 to P2</p><p>and P1 to P3 curves are the second-degree quadratic curves, whereas the P0 to</p><p>Q 1 curve is the third-degree cubic. This is the curve we want to construct.</p><p>The points Q 0 and Q 1 go along the two second-degree curves. Our drawing</p><p>pencil always goes along the line connecting Q 0 and Q 1.</p><p>Bridges, String Art, and Bézier Curves 85</p><p>drawing (n - 1)st degree curves. How do we know how to do that?</p><p>Luckily, we can apply exactly the same process to these assistant curves</p><p>as well. We can build them out of two lesser degree curves. By repeat-</p><p>ing this process, we eventually reach a curve that we do know how</p><p>to draw. This is the linear, 1st degree curve—it is just a straight line,</p><p>which we have no problem drawing at all. Thus, all complex Bézier</p><p>curves can be drawn using a composition of many straight lines.</p><p>Applying Bézier Curves</p><p>Now that we are a bit familiar with Bézier curves, we can return to our</p><p>original questions: What made the bridge’s shape come out parabolic?</p><p>How can we extend our model to fix the assumptions we made?</p><p>It turns out that our beautiful Jerusalem Chords Bridge is nothing</p><p>but a quadratic Bézier curve! To see this, let’s go back to our coordinate</p><p>system representing the bridge and draw a quadratic Bézier curve with</p><p>the control points P0 = (0,1), P1 = (0,0), and P2 = (1,0). Using our re-</p><p>cursive process, the quadratic curve is formed from two straight lines:</p><p>the line from P0 to P1 (the y axis from 1 down to 0) and from P1 to P2</p><p>(the x axis from 0 up to 1).</p><p>Now suppose that the first pencil has traveled down the y axis by a</p><p>distance t to the point (0,1 - t). In the same time, the second pencil has</p><p>traveled along the x axis to the point (0,t). The third pencil is therefore</p><p>on the line Lt from (0,1 - t) to (t,0), 100  t% along the way. Thus, the</p><p>Bézier curve meets all of the lines Lt for t between 0 and 1. These lines</p><p>(or at least n of them) correspond to our bridge chords.</p><p>Now setting t = 0.5, we see that the halfway point P = (0.25,0.25)</p><p>of the line L0.5, lies on our Bézier curve. Figure 10 shows that P also lies</p><p>on the outline formed by the lines Lt. (See [12] if you’re not convinced</p><p>by the picture). This method is enough to show that the Bézier curve</p><p>and the outline are one and the same curve. As Figure 10 shows, any</p><p>other parabolic curve that meets all the Lt misses the point P and crosses</p><p>the line L0.5 twice.</p><p>Appreciating this fact allows us to deal with some of the previous</p><p>model’s inaccuracies. First, we assumed that the axes were perpen-</p><p>dicular to each other, even though the tower and the deck of the bridge</p><p>are actually at an angle. Now we see that this angle does not matter.</p><p>The argument we just used continues to hold if we increase the angle</p><p>86 Renan Gross</p><p>between the axes by rotating the y axis (and any other line radiating out</p><p>from the point (0,0) in between the x and y axes) counterclockwise by</p><p>the necessary amount. We know that any quadratic Bézier curve is a</p><p>parabola, so the bridge outline is still a parabola. Second, we see that</p><p>it does not matter whether the chords in our model are evenly spaced</p><p>P</p><p>1</p><p>0 0.2</p><p>Pencil positions at t = 0.5</p><p>First pencil</p><p>Second pencil</p><p>Third pencil at P</p><p>0.80.4 0.6</p><p>0.8</p><p>0.6</p><p>0.4</p><p>0.2</p><p>0</p><p>1</p><p>P0</p><p>P2P1</p><p>Figure 9. The arrows inside the x and y axes represent the distance t traveled</p><p>along the axes for t = 0.5. The diagonal line is the line L0.5.</p><p>P</p><p>1</p><p>0 0.2</p><p>y = x</p><p>0.80.4 0.6</p><p>0.8</p><p>0.6</p><p>0.4</p><p>0.2</p><p>0</p><p>1</p><p>Figure 10. The lines that resemble netting represent some of the lines Lt.</p><p>The diagonal line is L0.5. The parabola inside the netting illustrates that any</p><p>parabolic shape apart from the outline curve misses the point P.</p><p>Bridges, String Art, and Bézier Curves 87</p><p>or not: They just represent some of the straight lines Lt whose outlines</p><p>define our Bézier curve. Third, the chords coming out of the tower do</p><p>not span its entire length, but stop about halfway. This just means that</p><p>we have to look at a partial Bézier curve. If we extend the deck onward,</p><p>we still have a parabola, with P2 as (2,0). Then the outline of the chords</p><p>is just the portion of the parabola that spans until (0,1).</p><p>We can now rest peacefully, knowing the underlying reason for the</p><p>parabolic shape of the Jerusalem Chords Bridge outline. Somehow, a</p><p>curve that was used in the 1960s for designing car parts managed to</p><p>sneak its way into 21st century bridges!</p><p>Béziers Everywhere!</p><p>The abundance of uses for Bézier curves is far greater than just for cars</p><p>and bridges. It finds its way into many more fields and applications.</p><p>One such field is that of string art, in which strings are spread across</p><p>a board filled with nails. Although the strings can only make straight</p><p>lines, a great many of them at different angles can generate Bézier curve</p><p>outlines, just like the chords in the bridge do.</p><p>Another interesting appearance of Bézier curves is in computer</p><p>graphics. Whenever you use the pen tool, common in many image</p><p>manipulation programs, you are drawing Bézier curves. More impor-</p><p>tantly, many computer fonts use Bézier curves to define how to draw</p><p>their letters. Each letter is defined by up to several dozen control points</p><p>P0 = (0,1)</p><p>P1 = (0,0)P1 = (0,0)</p><p>P2 = (2,0)</p><p>Figure 11. The correct Bézier curve applied to the Jerusalem Chords</p><p>Bridge.</p><p>88 Renan Gross</p><p>and is drawn using a series of 3rd to 5th degree Bézier curves. This</p><p>makes the letters scalable: They are clearly drawn and presented, no</p><p>matter how much you zoom in on them.</p><p>This is the beauty of mathematics. It appears in places we would</p><p>never expect and connects fields that appear entirely unrelated. Con-</p><p>sidering the fact that the curves were initially</p><p>used for designing au-</p><p>tomobile parts, this is truly a display of the interdisciplinary nature of</p><p>mathematics.</p><p>Figure 12. A pattern made from strings. [13].</p><p>Figure 13. Some of the Bézier control points used to make “a” and “g” in the</p><p>FreeSerif font (simplified).</p><p>Bridges, String Art, and Bézier Curves 89</p><p>Sources</p><p>[1] http://plus.maths.org/content/taxonomy/term/800</p><p>[2] http://plus.maths.org/content/category/tags/bezier-curve</p><p>[3] http://plus.maths.org/content/taxonomy/term/902</p><p>[4] http://plus.maths.org/content/taxonomy/term/674</p><p>[5] http://plus.maths.org/content/taxonomy/term/338</p><p>[6] http://plus.maths.org/content/taxonomy/term/21</p><p>[7] http://en.wikipedia.org/wiki/File:Jerusalem_Chords_Bridge.JPG</p><p>[8] http://plus.maths.org/content/finding-intersection-envelope</p><p>[9] http://plus.maths.org/content/changing-variables</p><p>[10] http://en.wikipedia.org/wiki/Pierre_Bézier</p><p>[11] http://plus.maths.org/content/formula-bezier-curve</p><p>[12] http://plus.maths.org/content/point-025025-lies-outline-curve</p><p>[13] http://www.stringartfun.com/product.php/7/free-boat-pattern</p><p>[14] http://sarcasticresonance.wordpress.com</p><p>Slicing a Cone for Art and Science</p><p>Daniel S. Silver</p><p>Albrecht Dürer (1471–1528), master painter and printmaker of the</p><p>German Renaissance, never thought of himself as a mathematician. Yet</p><p>he used geometry to uncover nature’s hidden formulas for beauty. His</p><p>efforts influenced renowned mathematicians, including Gerolamo Car-</p><p>dano and Niccolo Tartaglia, as well as famous scientists such as Galileo</p><p>Galilei and Johannes Kepler.</p><p>We praise Leonardo da Vinci and other Renaissance figures for em-</p><p>bracing art and science as a unity. But for artists such as Leonardo and</p><p>Dürer, there was little science to embrace. Efforts to draw or paint</p><p>directly from nature required an understanding of physiology and op-</p><p>tics that was not found in the ancient writings of Galen or Aristotle. It</p><p>was not just curiosity but also need that motivated Dürer and his fellow</p><p>Renaissance artists to initiate scientific investigations.</p><p>Dürer’s nature can seem contradictory. Although steadfastly reli-</p><p>gious, he sought answers in mathematics. He was outwardly modest</p><p>but inwardly vain. He fretted about money and forgeries of his work,</p><p>yet to others he appeared to be a simple man, ready to help fellow</p><p>artists.</p><p>Concern for young artists motivated Dürer to write an ambitious</p><p>handbook for all varieties of artists. It has the honor of being the</p><p>first serious mathematics book written in the German language. Its</p><p>title, Underweysung der Messung, might be translated as A Manual of</p><p>Measurement. Walter Strauss, who translated Dürer’s work into En-</p><p>glish, gave the volume a pithy and convenient moniker: the Painter’s</p><p>Manual.</p><p>Dürer begins his extraordinary manual with apologetic words, an</p><p>inversion of the famous warning of Plato’s Academy: Let no one un-</p><p>trained in geometry enter here:</p><p>Slicing a Cone for Art and Science 91</p><p>The most sagacious of men, Euclid, has assembled the founda-</p><p>tion of geometry. Those who understand him well can dispense</p><p>with what follows here, because it is written for the young and for</p><p>those who lack a devoted instructor.</p><p>The manual was organized into four books and printed in Nürem-</p><p>berg in 1525, just three years before the artist’s death. It opens with</p><p>the definition of a line, and it closes with a discussion of elaborate me-</p><p>chanical devices for accurate drawing in perspective. In between can</p><p>be found descriptions of spirals, conchoids, and other exotic curves.</p><p>Constructions of regular polygons are given. Cut-out models (“nets”)</p><p>of polyhedra are found. There is also an important section on typog-</p><p>raphy, containing a modular construction of the Gothic alphabet. An</p><p>artist who wishes to draw a bishop’s crozier can learn how to do it</p><p>with a compass and ruler. An architect who wants to erect a monument</p><p>might find some sort of inspiration in Dürer’s memorial to a drunkard,</p><p>a humorous design complete with coffin, beer barrel, and oversized</p><p>drinking mug.</p><p>Scholarly books of the day were generally written in Latin. Dürer</p><p>wrote Underweysung der Messung in his native language because he wanted</p><p>it to be accessible to all German readers, especially those with limited</p><p>formal education. But there was another reason: Dürer’s knowledge of</p><p>Latin was rudimentary. Others later translated Underweysung der Mes-</p><p>sung into several different languages, including Latin.</p><p>There was no reason to expect that Dürer should have been fluent</p><p>in Latin. As the son of a goldsmith, he was lucky to have gone to school</p><p>at all. Fortunately for the world, Dürer displayed his unusual intel-</p><p>ligence at an early age. “My father had especial pleasure in me, because</p><p>Figure 1. A detail from Albrecht Dürer’s Melencolia I from 1514 shows a</p><p>magic square, in which each row, column, and main diagonal sum to the</p><p>same total, in this case 34.</p><p>92 Daniel S. Silver</p><p>he said that I was diligent in trying to learn,” he recalled. He was sent</p><p>to school, possibly the nearby St. Sebald parochial school, where he</p><p>learned to read and write. He and his fellow students carried slates or</p><p>wax writing tablets to class. (Johannes Gutenberg had invented a print-</p><p>ing press only 40 years before, and books were still a luxury.) Learning</p><p>was a slow, oral process.</p><p>When Dürer turned 13, he was plucked from school so that he could</p><p>begin learning his father’s trade. At that age, he produced a self-portrait</p><p>that gives a hint of his emerging artistic skill. Self-portraits at the time</p><p>were rare. Dürer produced at least 11 more during his lifetime.</p><p>What might have inspired a tradesman’s son to study the newly re-</p><p>discovered works of ancient Greek mathematicians such as Euclid and</p><p>Apollonius? Part of the answer can be found in the intellectual atmo-</p><p>sphere of Nüremberg at the time. In 1470, Anton Koberger founded</p><p>the city’s first printing house. One year later, he became Dürer’s god-</p><p>father. Science and technology were so appreciated in Nüremberg that</p><p>the esteemed astronomer Johannes Müller von Königsberg, also known</p><p>as Regiomontanus (1436–1476), settled there and built an observatory.</p><p>The rest of the answer can be found in the dedication of the Paint-</p><p>er’s Manual: “To my especially dear master and friend, Herr Wilbolden</p><p>Pirckheymer, I, Albert Dürer wish health and happiness.” This master,</p><p>whose name is more commonly spelled Willibald Pirckheimer (1470–</p><p>1530), was a scion of one of Nüremberg’s most wealthy and powerful</p><p>families. He was enormous in many ways, both physically and in per-</p><p>sonality, as well as boastful and argumentative. He was also a deeply</p><p>knowledgeable humanist with a priceless library.</p><p>Pirckheimer’s house was a gathering place for Nüremberg’s bril-</p><p>liant minds. Despite the wide difference between their social rankings,</p><p>Dürer and Pirckheimer became lifelong friends. Pirckheimer depended</p><p>on Dürer to act as a purchasing agent during his travels, scouting for</p><p>gems and other valuable items. Dürer depended on Pirckheimer for ac-</p><p>cess to rare books and translation from Greek and Latin.</p><p>The word “Messung” meant more to Dürer than simple measure-</p><p>ment. “Harmony” might have been closer to the mark. In his youth, pos-</p><p>sibly in 1494, Dürer had marveled over a geometrically based drawing of</p><p>male and female figures by the Venetian artist Jacopo de’ Barbari (about</p><p>1440–1516). Despite the fact that de’ Barbari was unwilling to share his</p><p>methods—or maybe because of it—Dürer became convinced that the</p><p>Slicing a Cone for Art and Science 93</p><p>secrets of beauty might be found by means of mathematics. Dürer was</p><p>only 23 years old at the time. He devoted the remaining three decades of</p><p>his life to the search, for as he reflected some years later, “I would rather</p><p>have known what [de’ Barbari’s] opinions were than to have seen a new</p><p>kingdom.” Geometry, recovered from ancient works, lit his way.</p><p>The gravity of Dürer’s quest can be sensed in his enigmatic engraving</p><p>Melencolia I, shown in Figure 2. Now approaching the 500th anniversary</p><p>of its creation,</p><p>Melencolia I has been the subject of more academic debate</p><p>than any other print in history. Is the winged figure dejected because</p><p>she has tried but failed to discover beauty’s secret? She holds in her hand</p><p>an open compass. Above her head is a magic square, the first to be seen in</p><p>Western art. (In a magic square, the numbers in each row and column,</p><p>as well as the two main diagonals, add to the same total, in this case 34.</p><p>In this one, the date of the engraving, 1514, appears in the lowest row.)</p><p>Clearly Dürer’s mathematical interests were not limited to geometry.</p><p>When he wrote the Painter’s Manual, Dürer was approaching the end</p><p>of a successful career. As a young man eager to learn more about the</p><p>new science of perspective and to escape outbreaks of plague at home,</p><p>he had made two trips to Italy. After the first journey, his productivity</p><p>soared. Dürer’s self-portrait of 1498 radiates an expanding confidence.</p><p>(It was not the first time that plague encouraged scientific discovery,</p><p>nor was it the last. In 1666 Isaac Newton escaped an outbreak of plague</p><p>at Cambridge University, returning to his mother’s farm, where he had</p><p>the most profitable year that science has ever known.)</p><p>During his second visit to Italy, Dürer met with fellow artists, in-</p><p>cluding the great master Giovanni Bellini, who praised his work. Dürer</p><p>came to the conclusion that German artists could rise to the heights</p><p>of the Italians, but only if they learned the foundations of their art.</p><p>Such a foundation would prevent mistakes—and such a foundation</p><p>required geometry. He returned with an edition of Euclid that bears</p><p>his inscription: “I bought this book at Venice for one ducat in the year</p><p>1507— Albrecht Dürer.”</p><p>Dürer purchased a house in Nüremberg and began to study math-</p><p>ematics. The Painter’s Manual was not the book that he had originally</p><p>planned to write. He had started work on Vier Bücher von Menschlicher</p><p>Proportion (“Four Books on Human Proportion”), but soon realized that</p><p>the mathematical demands that it placed on young readers were too</p><p>great. The Painter’s Manual was intended as a primer.</p><p>94 Daniel S. Silver</p><p>Figure 2. Melencolia I has been heavily debated among art historians. Is the</p><p>angel’s dejection caused by her inability to discover beauty’s secret? The</p><p>engraving reflects Dürer’s mathematical interests. Dürer’s mistaken belief</p><p>that ellipses were egg-shaped is reflected in the shape of the bell open-</p><p>ing. His quest to extend the mathematics behind beauty to artists led him</p><p>to publish a primer that ended up influencing scientists as well as artists.</p><p>Source: Wikimedia Commons.</p><p>Slicing a Cone for Art and Science 95</p><p>Figure 3. Dürer produced at least</p><p>a dozen self-portraits during his</p><p>lifetime. The first, at age 13 (top</p><p>left), hinted at his emerging artistic</p><p>gifts. A second, produced in 1498</p><p>at age 27 (top right), showed him</p><p>at the peak of a successful career,</p><p>when his confidence was expand-</p><p>ing and his productivity was</p><p>soaring. A final self-portrait in</p><p>1522, at age 51 (bottom), shows the</p><p>artist after his body was ravaged</p><p>by a disease that killed him a few</p><p>years later. Source: Wikimedia</p><p>Commons.</p><p>96 Daniel S. Silver</p><p>Work on the Painter’s Manual, too, was temporarily halted when, in</p><p>1523, Dürer acquired 10 books from the library of Nüremberg math-</p><p>ematician Bernhard Walther (1430–1504). Walther had been a student</p><p>of Regiomontanus and had acquired important books and papers from</p><p>him. But Walther was a moody man who denied others access to this</p><p>valuable cache. Walther died, but his library remained with his execu-</p><p>tors for two decades. Finally its contents had been released for sale.</p><p>Dürer’s precious purchases were chosen and appraised by Pirckheimer.</p><p>It took Dürer two more years to absorb the ideas these books con-</p><p>tained. The completion of the Painter’s Manual would just have to wait.</p><p>It would be a book for artists, or so Dürer thought. Nevertheless he</p><p>allowed himself to be carried aloft by mathematics. “How is it that two</p><p>lines which meet at an acute angle which is made increasingly smaller will</p><p>nevertheless never join together, even at infinity?” he asks (and proceeds</p><p>to give a strange explanation). Later he writes: “If you wish to construct</p><p>a square of the same area as a triangle with unequal sides, proceed as fol-</p><p>lows.” It is difficult to imagine any artist of the 16th century making use</p><p>of such ideas. These are the thoughts of a compulsive theoretician.</p><p>Time for Dürer to complete his Painter’s Manual was running out.</p><p>In December 1520, he had foolishly trekked to the swamps of Zeeland</p><p>in the southwestern Netherlands, hoping to inspect a whale that had</p><p>washed ashore. Alas, the whale had already washed away by the time he</p><p>arrived. It was not a healthy place to visit, and the chronic illness that</p><p>he contracted there eventually killed him after eight painful years.</p><p>Dürer’s self-portrait of 1522 contrasts disturbingly with his earlier</p><p>one. In the words of Strauss: “It represents Dürer himself in the nude,</p><p>with thinned, disheveled hair and drooping shoulders, his body ravaged</p><p>by his lingering disease.” He fashioned himself as the Man of Sorrows.</p><p>No Matter How You Slice It</p><p>The subject [Conic Sections] is one of those which</p><p>seem worthy of study for their own sake.</p><p>—Apollonius of Perga</p><p>Although there is much in the Painter’s Manual that rewards close ex-</p><p>amination, one specific area worthy of concentration is Dürer’s treat-</p><p>ment of conic sections. The techniques that Dürer found to draw them</p><p>Slicing a Cone for Art and Science 97</p><p>anticipate the field of descriptive geometry that Gaspard Monge (1746–</p><p>1818) developed later. The curves themselves would accompany a revo-</p><p>lution in astronomy.</p><p>“The ancients have shown that one can cut a cone in three ways and</p><p>arrive at three differently shaped sections,” Dürer writes toward the</p><p>end of Book I. “I want to teach you how to draw them.”</p><p>Menaechmus (circa 350 b.c.), who knew Plato and tutored Alex-</p><p>ander the Great, is thought to have discovered conic sections (often</p><p>called simply “conics”). He found them while trying to solve the famous</p><p>Delian problem of “doubling the cube.” According to legend, terrified</p><p>citizens of the Greek island of Delos were told by an oracle that plague</p><p>would depart only after they had doubled the size of Apollo’s cubical</p><p>altar. Assuming that the altar had unit volume, the task of doubling it</p><p>amounted to constructing a new edge of length precisely equal to the</p><p>cube root of 2. Although the legend is doubtful, the Delian problem</p><p>was certainly studied in Plato’s Academy. Plato insisted on an exact</p><p>solution accomplished using only ruler and compass.</p><p>a b c</p><p>Figure 4. A plane slicing through a cone can produce several different shapes,</p><p>called conic sections. Top and side views of the cone show how altering the</p><p>angle of the plane results in a parabola (a), ellipse, (b) or hyperbola (c). Dürer’s</p><p>Painter’s Manual aimed to show artists how to draw these shapes correctly.</p><p>98 Daniel S. Silver</p><p>Ingenious ruler-and-compass constructions abound in the Painter’s</p><p>Manual. Dürer’s construction of a regular pentagon is particularly note-</p><p>worthy. The construction method came not from Euclid but rather</p><p>from one that had been taught by Ptolemy and is found in his Almagest.</p><p>In 1837, the French mathematician Pierre Wantzel (1814–1848) proved</p><p>that doubling the cube with ruler and compass is impossible. However,</p><p>Menaechmus changed the rules of the game and managed to win. By</p><p>inter secting a right-angled cone with a plane perpendicular to its side,</p><p>he produced a curve that was later called a parabola. Then by inter-</p><p>secting two parabolas, chosen carefully, Menaechmus produced a line</p><p>segment of length equal to the cube root of 2. (The parabola can be</p><p>described by a simple equation y2 = 4px. The positive number p is called</p><p>the latus rectum.)</p><p>Menaechmus looked at other sorts of cones. When the cone’s angle</p><p>was either less than or greater than 90 degrees, two new types of curves</p><p>resulted</p><p>as well as conveying them</p><p>in natural language, are means that bridge at least part of the gap that</p><p>separates theoretical mathematics from the general public; such means</p><p>demythologize the widespread belief that higher mathematics is, by its</p><p>epistemic status and technical difficulty, inaccessible to the layperson.</p><p>Mathematics and its applications are scrutable only as far as math-</p><p>ematicians are explicit with their own assumptions, claims, results, and</p><p>interpretations. When these elements of openness are missing math-</p><p>ematicians not only fail to disrupt patterns of entrenched thinking but</p><p>also run the risk of digging themselves new trenches. Writing about</p><p>mathematics offers freedoms of explanation that complement the dense</p><p>texture of meaning captured by mathematical symbols.</p><p>Talking plainly about mathematics also has inestimable educational</p><p>and social value. A sign of a mature mind is the ability to hold at the</p><p>Introduction xvii</p><p>same time opposite ideas and to juggle with them, analyze them, re-</p><p>fine them, corroborate them, compromise with them, and choose</p><p>among them. Such intellectual dexterity has moral and practical con-</p><p>sequences, for the lives of the individuals as well as for the life of soci-</p><p>ety. Mathematical thinking is eminently endowed to prepare the mind</p><p>for these habits, but we almost never pay attention to this aspect, we</p><p>rarely notice it, and we seldom talk about it. We use contradiction as a</p><p>trick of the (mathematical) trade and as a routine method of proof. Yet</p><p>opposing ideas, contrasts, and complementary qualities are intimately</p><p>interwoven into the texture of mathematics, from definitions and el-</p><p>ementary notions to highly specialized mathematical practice; we use</p><p>them implicitly, tacitly, all the time.</p><p>As a reader of this book, you will have a rewarding task in identi-</p><p>fying in the contributions some of the virtues I attribute to writing</p><p>about mathematics—and perhaps many others. With each volume in</p><p>this series, I put together a book I like to read, the book I did not find</p><p>in the bookstore years ago. If I include a contribution here, it does not</p><p>mean that I necessarily concur in the opinions expressed in it. Whether</p><p>we agree or disagree with other people’s views, our polemics gain in</p><p>substance if we aim to comprehend and address the highest quality of</p><p>the opposing arguments.</p><p>Overview of the Volume</p><p>In a sweeping panoramic view of the likely future trajectory of math-</p><p>ematics, Philip Davis asks pertinent questions that illuminate some</p><p>of the myriad links connecting mathematics to its applications and to</p><p>other practical domains and offers informed speculations on the multi-</p><p>faceted mathematical imprint on our ever more digitized world.</p><p>Ian Stewart explains recent attempts made by mathematicians to re-</p><p>fine and reaffirm a theory first formulated by Alan Turing, stating, in</p><p>its most general formulation, that pattern formation is a consequence</p><p>of symmetry breaking.</p><p>Terence Tao observes that many complex systems seem to be gov-</p><p>erned by universal principles independent of the laws that govern the</p><p>interaction between their components; he then surveys various aspects</p><p>of several well-studied mathematical laws that characterize phenomena</p><p>as diverse as spectral lines of chemical elements, political elections,</p><p>xviii Introduction</p><p>celestial mechanics, economic changes, phase transitions of physical</p><p>materials, the distribution of prime numbers, and others.</p><p>A diversity of contexts, with primary focus on social networking,</p><p>is also Gregory Goth’s object of attention; in his article he examines</p><p>the “small-world” problem—the quest to determine the likelihood that</p><p>two individuals randomly chosen from a large group know each other.</p><p>Charles Seife argues that humans’ evolutionary heritage, cultural</p><p>mores, and acquired preconceptions equip us poorly for expecting and</p><p>experiencing randomness; yet, in an echo of Tao’s contribution, Seife</p><p>observes that in aggregate, randomness of many independent events</p><p>does obey immutable mathematical rules.</p><p>Writing from experience, Donald Knuth shows that the deliberate</p><p>and methodical coopting of randomness into creative acts enhances the</p><p>beauty and the originality of the result.</p><p>Soren Johnson discusses the advantages and the pitfalls of using</p><p>chance in designing games and gives examples illustrative of this</p><p>perspective.</p><p>John Pavlus details the history, the meaning, and the implications of</p><p>the P versus NP problem that underpins the foundations of computa-</p><p>tional complexity theory and mentions the many interdisciplinary areas</p><p>that are connected through it.</p><p>Renan Gross analyzes the geometry of the Jerusalem Chords Bridge</p><p>and relates it to the mathematics used half a century ago by the French</p><p>mathematical engineer Pierre Bézier in car designs—an elegantly sim-</p><p>ple subject that has many other applications.</p><p>Daniel Silver presents Albrecht Dürer’s Painter’s Manual as a pre-</p><p>cursor work to projective geometry and astronomy; he details some</p><p>of the mathematics in the treaty and in Dürer’s artworks, as well as</p><p>the biographical elements that contributed to Dürer’s interaction with</p><p>mathematics.</p><p>Kelly Delp writes about the late William Thurston’s little-known</p><p>but intensely absorbing collaboration with the fashion designer Dai Fu-</p><p>jiwara and his Issey Mayake team; she describes the topological notions</p><p>that, surprisingly, turned out to be at the confluence of intellectual</p><p>passions harbored by two people so different in background and living</p><p>on opposite sides of the world.</p><p>Fiona and William Ross tell the brief history of Jordan’s curve theo-</p><p>rem, hint at some tricky cases that defy the simplistic intuition behind</p><p>Introduction xix</p><p>it, and, most remarkably, illustrate the nonobvious character of the</p><p>theorem with arresting drawings penned by Fiona Ross.</p><p>To answer pressing questions about the need for widespread math-</p><p>ematics education, Anna Sfard sees mathematics as a narrative means</p><p>to comprehend the world, which we humans developed for our con-</p><p>venience; she follows up on this perspective by arguing that the story</p><p>we tell (and teach) with mathematics needs to change, in sync with the</p><p>unprecedented changes of our world.</p><p>Erin A. Maloney and Sian L. Beilock examine the practical and psy-</p><p>chological consequences of states of anxiety toward mathematical ac-</p><p>tivities. They contend that such feelings appear early in schooling and</p><p>tend to recur in subsequent years, have a dual cognitive and social basis,</p><p>and negatively affect cognitive performance. The authors affirm that</p><p>the negative effect of mathematical anxiety can be alleviated by certain</p><p>deliberate practices, for instance by writing about the emotions that</p><p>cause anxiety.</p><p>David R. Lloyd reviews arguments put forward by proponents of the</p><p>idea that the five regular polyhedral shapes were well known, as math-</p><p>ematical objects, perhaps one millennium before Plato—in Scotland,</p><p>where objects of similar configurations and markings have been discov-</p><p>ered. He concludes that the objects, genuine and valuable aesthetically</p><p>and anthropologically, do not substantiate the revisionist claims at least</p><p>as far as in the mathematical knowledge they reveal.</p><p>In the interaction between the material culture of mathematical in-</p><p>struments and the mathematics that underlined it from the 16th to the</p><p>18th centuries in Western Europe, Jim Bennett decodes subtle recipro-</p><p>cal influences that put mathematics in a nodal position of a network of</p><p>applied sciences, scientific practices, institutional academics, entrepre-</p><p>neurship, and commerce.</p><p>Frank Quinn contrasts the main features of mathematics before and</p><p>after the profound transformations that took place at the core of the</p><p>discipline roughly around the turn of the 19th century; he contends</p><p>that the unrecognized magnitude of those changes led to some current</p><p>fault lines (for instance between teaching and research needs in univer-</p><p>sities, between school mathematics and higher level mathematics, and</p><p>others) and in</p><p>from their intersection with a plane. A century later, Apollo-</p><p>nius of Perga (262–190 b.c.) called the three curves parabola, ellipse, and</p><p>hyperbola, choosing Greek words meaning, respectively, comparison,</p><p>fall short, and excess. Echoes are heard today in the English words such</p><p>as parable, ellipsis, and hyperbole.</p><p>Today there is debate as to whether the terms originated with</p><p>Apollonius. In any event, they were likely adapted from earlier termi-</p><p>nology of Pythagoras (570–circa 495 b.c.) concerning a construction</p><p>known as “application of areas.” The interpretation in terms of angle</p><p>is historically inaccurate but mathematically equivalent and simpler</p><p>to state.</p><p>Apollonius’ accomplishments went beyond nomenclature. He made</p><p>a discovery that afforded a lovely simplification. Instead of using three</p><p>different types of cones, as Menaechmus did, Apollonius used a single</p><p>cone. Then by allowing the plane to slice the cone at different angles,</p><p>he produced all three conics.</p><p>Associated with an ellipse or hyperbola are a pair of special points</p><p>called foci. (For a circle, a special case of an ellipse, the distance be-</p><p>tween the two foci is zero.) Distances to the foci determine the curves</p><p>in a simple way: The ellipse consists of those points such that the sum</p><p>of distances to the foci is constant. Likewise, the hyperbola consists of</p><p>points such that the difference is a constant.</p><p>Slicing a Cone for Art and Science 99</p><p>Tilt a glass of water toward you and observe the shape of the water’s</p><p>edge. It is an ellipse. So is the retinal image of a circle viewed from a</p><p>generic vantage point.</p><p>Johannes Kepler (1571–1630) made the profound discovery that the</p><p>orbit of Mars is an ellipse with the Sun at one focus. Kepler introduced</p><p>the word “focus” into the mathematics lexicon in 1604. It is a Latin</p><p>word meaning hearth or fireplace. What word could be more appropri-</p><p>ate for the location of the Sun?</p><p>Kepler’s letter to fellow astronomer David Fabricius (1564–1617),</p><p>dated October 11, 1605, reveals that Kepler had read Dürer’s descrip-</p><p>tion of conics:</p><p>Figure 5. Dürer gave detailed instructions in his manual for how to transcribe</p><p>the cutting of a cone with a plane (seen in side and top view, left) into an ellipse.</p><p>He called ellipses “egg lines” because he believed, mistakenly, that they were</p><p>wider on the bottom than on the top. (Unless otherwise indicated, all photo-</p><p>graphs are courtesy of the author.)</p><p>100 Daniel S. Silver</p><p>So, Fabricius, I already have this: That the most true path of the</p><p>planet [Mars] is an ellipse, which Dürer also calls an oval, or cer-</p><p>tainly so close to an ellipse that the difference is insensible.</p><p>In fact, Dürer used a more flavorful term for an ellipse, as we will see.</p><p>Nature’s parabolas and hyperbolas are less apparent than the ellipse.</p><p>A waterspout and the path of a cannonball have parabolic trajectories.</p><p>Figure 6. Johannes Kepler’s Astronomia Nova from 1609 features a sketch of</p><p>the retrograde motion of the planet Mars when viewed from Earth. Kepler</p><p>made the discovery that the orbit of Mars is an ellipse with the Sun at one</p><p>focus. His correspondence makes it clear that he had read Dürer’s description</p><p>of conics. Image courtesy of Linda Hall Library of Science, Engineering &</p><p>Technology.</p><p>Slicing a Cone for Art and Science 101</p><p>The wake generated by a boat can assume the form of a hyperbola, but</p><p>establishing that fact requires more mathematics—or a boat.</p><p>Egg Lines</p><p>The significance of Dürer’s treatment of conics is the technique that he</p><p>used for drawing them, a fertile method of parallel projection. Art histo-</p><p>rian Erwin Panofsky observed that the technique was “familiar to every</p><p>architect and carpenter but never before applied to the solution of a</p><p>purely mathematical problem.” In brief, Dürer viewed a cut cone from</p><p>above as well as from the side, then projected downward. His trick was</p><p>to superimpose the two views and then transfer appropriate measure-</p><p>ments using dividers. In this way he relocated the curve from the cone</p><p>to a two-dimensional sheet of paper.</p><p>Dürer’s method was correct, but the master draftsman blundered</p><p>while transferring measurements. He mistakenly believed that the el-</p><p>lipse was wider at the bottom of the cone than at the top, an under-</p><p>standable error considering the shape of the cone. As he transferred the</p><p>distances with his divider, his erroneous intuition took hold of his hand.</p><p>Dürer writes, “The ellipse I call eyer linie [egg line] because it looks</p><p>like an egg.” Egg lines for ellipses can indeed be spotted in Dürer’s</p><p>work, such as in the bell in Melencolia I. Dürer knew no German equiva-</p><p>lent of the Greek word “ellipse.” The appellation he concocted drew</p><p>attention to his error, and the egg line persisted in German works for</p><p>nearly a century.</p><p>It is easy to understand why Kepler had an interest in Dürer’s flawed</p><p>analysis of the ellipse. For 10 years beginning in 1601, Kepler struggled</p><p>to understand the orbit of Mars, a problem that had defeated Regio-</p><p>montanus. Until he understood that the orbit was an ellipse, Kepler</p><p>believed that it was some sort of oval. In fact, he specifically used the</p><p>word “oval,” a descendant of the Latin word “ovum” meaning egg.</p><p>Kepler was not the first to believe that a planet’s orbit might be</p><p>egg-shaped. Georg von Peuerbach (1423–1461), a teacher of Regio-</p><p>montanus, had said as much in Theoricae novae planetarum. Published</p><p>in Nüremberg in 1473 and reprinted 56 times, Peuerbach’s treatise</p><p>influenced both Copernicus and Kepler. The 1553 edition, published</p><p>by Erasmus Reinhold (1511–1553), a pupil of Copernicus, included a</p><p>102 Daniel S. Silver</p><p>comment about Mercury’s orbit that might have caused Kepler to go</p><p>back to the Painter’s Manual:</p><p>Mercury’s [orbit] is egg-shaped, the big end lying toward his apo-</p><p>gee, and the little end towards its perigee.</p><p>Later Kepler had this to say about the orbit of Mars:</p><p>The planet’s orbit is not a circle but [beginning at the aphelion] it</p><p>curves inward little by little and then [returns] to the amplitude of</p><p>a circle at [perihelion]. An orbit like this is called an oval.</p><p>Burning Mirrors</p><p>It is a safe bet that few artists in 16th century Germany felt the need</p><p>to draw a parabola. In what seems like a marketing effort, Dürer tells</p><p>his readers how it can be fashioned into a weapon of mass destruction.</p><p>The story that Archimedes set fire to an invading Roman fleet dur-</p><p>ing the siege of Syracuse was well known in Dürer’s time. Dioclese, a</p><p>contemporary of Archimedes, explained the principle in his book On</p><p>Burning Mirrors, preserved by Muslims in the 9th century.</p><p>Dioclese had observed something special about the parabola that</p><p>had escaped the notice of Apollonius: On its axis of symmetry there</p><p>is a point—a single focus—with the property that if a line parallel to</p><p>the axis reflects from the parabola with the same angle with which it</p><p>strikes, the reflected line passes through the focus. In physical terms,</p><p>a mirror in the shape of a paraboloid, a parabola revolved about its axis,</p><p>gathers all incoming light at the focus. Collect enough light, and what-</p><p>ever is at the focus will become hot.</p><p>Making an effective burning mirror is not a simple matter. Un-</p><p>less the parabola used is sufficiently wide, the mirror does not collect</p><p>enough light. Dürer writes,</p><p>If you plan to construct a burning mirror of paraboloid shape, the</p><p>height of the cone you have to use should not exceed the diameter</p><p>of the base—or this cone should be of the shape of an equilateral</p><p>triangle.</p><p>Dürer goes on to explain why the angle of incidence of a light beam</p><p>striking a mirror is equal to the angle of reflection. An elegant drawing</p><p>Slicing a Cone for Art and Science 103</p><p>of an artisan (possibly the artist) holding a pair of dividers does little to</p><p>help matters (Figure 7). Dürer probably sensed that he was getting into</p><p>a rough technical patch. He concludes the section desperately:</p><p>The cause of this has been explained by mathematicians. Whoever</p><p>wants to know it can look it</p><p>up in their writings. But I have drawn</p><p>my explanation . . . in the figure below.</p><p>Burning mirrors might have sounded useful to readers of the Painter’s</p><p>Manual. The first scientific evidence that Archimedes’ mirrors might</p><p>not have been such a hot idea had to wait for 12 years until René Des-</p><p>cartes expressed doubts in his treatise Dioptrique. Nevertheless, since</p><p>the time of Archimedes, burning mirrors, whatever their effectiveness,</p><p>were constructed in a more practical, approximate fashion with sec-</p><p>tions of a sphere. (In 1668, Isaac Newton designed the first reflecting</p><p>telescope on the principle of the burning mirror, with an eyepiece near</p><p>the focus. He substituted a spherical mirror to simplify its construc-</p><p>tion.) It seems that Dürer liked parabolas and was determined to write</p><p>about them.</p><p>Dürer invented German names for the parabola and the hyperbola</p><p>as well as the ellipse. The parabola he called a Brenn Linie (“burn line”).</p><p>“And the hyperbola I shall call gabellinie [fork line],” he writes, but he</p><p>offers no explanation for his choice. Nor, it seems, has a reason been</p><p>suggested by anyone else. Dürer might have been paying tribute to</p><p>the many gabled houses of which Nüremberg was proud, the artist’s</p><p>own home near the Thiergärtnerthor included. In the Painter’s Manual,</p><p>Dürer constructs the hyperbola but has little to say about it.</p><p>Much of what Dürer knew about conic sections came from Johannes</p><p>Werner (1468–1522). A former student of Regiomontanus, Werner</p><p>was an accomplished instrument maker. He made contributions to ge-</p><p>ography, meteorology, and mathematics. A lunar impact crater named</p><p>in his honor is not far from a crater named Regiomontanus.</p><p>Werner’s Libellus super viginti duobus elementis conicis was published in</p><p>1522, at the time when Dürer was studying conics. The volume’s 22</p><p>theorems were intended to introduce the author’s work on the Delian</p><p>problem. From handwritten notes, it appears that Werner died during</p><p>his book’s printing. (Werner’s book soon became rare. It is reported</p><p>that the Danish astronomer Tycho Brahe could not find a copy for sale</p><p>anywhere in Germany.)</p><p>104 Daniel S. Silver</p><p>Figure 7. Dürer used a method called parallel projection when transcribing</p><p>figures, such as this parabola, from three to two dimensions. He viewed a cut</p><p>cone from above as well as from the side, then projected downward, super-</p><p>imposing the two views and then transferring appropriate measurements using</p><p>dividers (top). Dürer argued that parabolas correctly angled could be used as</p><p>burning mirrors, heating what is at the focus (bottom left). He tried to explain</p><p>light angles with a drawing of an artisan (top right). However, he miscalculated</p><p>the placement of the focal point, so a correction had to be manually pasted into</p><p>each copy of his 1525 publication. The original mistake is revealed behind the</p><p>correction when the page is backlit (bottom right).</p><p>focus</p><p>Slicing a Cone for Art and Science 105</p><p>Since 1508, Werner had been serving as priest at the Church of</p><p>St. John, not far from Dürer’s house. Like Pirckheimer, Werner ac-</p><p>quired some of the rare books and papers that had been in Walther’s</p><p>possession. However, Werner knew no Greek and probably relied on</p><p>Pirckheimer for translation. (His commentary on Ptolemy, published</p><p>in 1514, is dedicated to Pirckheimer.) Like Dürer, Werner would have</p><p>been a frequent visitor to Pirckheimer’s house.</p><p>I believe that Dürer was inspired by Werner’s novel construction</p><p>of the parabola (Figure 8). The cone that he used was an oblique cone</p><p>with vertex directly above a point on the base circle. A cut by a vertical</p><p>plane produced the parabola. Regularly spaced, circular cross sections</p><p>of the cone are in the lower diagram, each tangent to a point that lies</p><p>directly below the vertex of the cone. The cutting plane is seen in pro-</p><p>file as a line through points labeled b and f. By transferring the segments</p><p>Figure 8. Johannes Werner, a contemporary of Dürer, made contributions to</p><p>geography, meteorology, and mathematics. His parabola (left), was published</p><p>in 1522 at a time when Dürer was studying conics. It is likely that Dürer’s</p><p>parabola, published in 1525 (right), was influenced by Werner’s. The method</p><p>of parallel projections that is often credited to Dürer might well have derived</p><p>from Werner’s construction. However, Werner used an oblique cone, whereas</p><p>Dürer’s was a right cone, so Werner’s formula for the location of the focus no</p><p>longer applied.</p><p>106 Daniel S. Silver</p><p>cut by the circles along the line, Werner produced the semi-arcs trans-</p><p>verse to the line through k and n in the upper figure.</p><p>Had Dürer seen the picture, which is likely, Germany’s master of</p><p>perspective would have had no trouble imagining the tangent circles</p><p>stacked in three dimensions, the smallest coming closest to his eye. The</p><p>method of parallel projections that is often credited to Dürer might</p><p>well have derived from Werner’s construction.</p><p>In Appendix Duodecima of his book, Werner explains the reflective</p><p>properties of the parabola to his audience. He also tells the reader how</p><p>to locate the focus: Its distance from the vertex is one quarter of the</p><p>length of the segment ab. (The length of ab, which is equal to the length</p><p>of kn, is the latus rectum of the parabola—the distance between the slic-</p><p>ing plane and the vertex of the cone.)</p><p>But Dürer used a right cone with vertex directly above the center</p><p>of the circular base, so his cross-sectional circles became concentric</p><p>rather than tangent to a single point, as in Werner’s diagram.</p><p>Unfortunately for Dürer, Werner’s formula for the location of the</p><p>focus no longer applied. Whether Dürer computed the distance incor-</p><p>rectly or merely guessed, we do not know. However, in every copy of</p><p>the 1525 publication a small piece of paper with the correct drawing</p><p>had to be pasted by hand over the erroneous one. By holding the final</p><p>product up to the light, Dürer’s mistake is revealed.</p><p>Dürer and Creativity</p><p>For Albrecht Dürer, questions of technique eventually gave way to</p><p>those of philosophy. In 1523, he wondered at the way “one man may</p><p>sketch something with his pen on half a sheet of paper in one day . . .</p><p>and it turns out to be better and more artistic than another’s big work</p><p>at which its author labors with the utmost diligence for a whole year.”</p><p>The belief that divine genius borrows the body of a fortunate art-</p><p>ist was common in Dürer’s time. According to Panofsky, Leonardo da</p><p>Vinci would have been perplexed had someone called him a genius. But</p><p>Dürer had begun to see the creative process differently. For him, it</p><p>became one of synthesis governed by trained intuition.</p><p>Dürer’s last name was likely derived from the German word tür,</p><p>meaning door. (His father was born in the Hungarian town of Ajtas,</p><p>which is related to the Hungarian word for door, ajitó.)</p><p>Slicing a Cone for Art and Science 107</p><p>It is a fitting name for someone who opened a two-way passage be-</p><p>tween mathematics and art. As Panofsky observed, “While [the Painter’s</p><p>Manual] familiarized the coopers and cabinet-makers with Euclid and</p><p>Ptolemy, it also familiarized the professional mathematicians with what</p><p>may be called ‘workshop geometry.’ ”</p><p>Figure 9. Dürer engraving from 1514, titled St. Jerome in His Study, is noted</p><p>for its use of unusual mathematical perspective, which invites the viewer into</p><p>the snug chamber. It is rich with symbolism related to the theological and</p><p>contemplative aspects of life in Dürer’s time.</p><p>108 Daniel S. Silver</p><p>Dürer used geometry to search for beauty, but he never regarded</p><p>mathematics as a substitute for aesthetic vision. It was a tool to help</p><p>the artist avoid errors. However, the Painter’s Manual demonstrates</p><p>that mathematics and, in particular, geometry, meant much more to</p><p>him. Four centuries after its publication, poet Edna St. Vincent Mil-</p><p>lay wrote, “Euclid alone has looked on Beauty bare.” Dürer might have</p><p>agreed.</p><p>Bibliography</p><p>Coolidge, Julian L. 1968. A History of the Conic Sections and Quadric</p><p>Surfaces. New York: Dover</p><p>Publications.</p><p>Dürer, Albrecht. 1525. A Manual of Measurement [Underweysung der Messung]. Translated by</p><p>Walter L. Strauss, 1977. Norwalk, Conn.: Abaris Books.</p><p>Eves, Howard. 1969. An Introduction to the History of Mathematics, Third Edition. Toronto: Holt,</p><p>Rinehart and Winston.</p><p>Guppy, Henry, ed. 1902. The Library Association Record, Volume IV. London: The Library Association.</p><p>Heaton, Mrs. Charles. 1870. The History of the Life of Albrecht Dürer of Nürnberg. London: Mac-</p><p>millan and Co.</p><p>Herz-Fischler, Roger. 1990. “Durer’s paradox or why an ellipse is not egg-shaped.” Mathemat-</p><p>ics Magazine 63(2): 75–85.</p><p>Kepler, Johannes. 1937. Johannes Kepler, Gesammelte Werke. Vol. 15, letter 358, l. 390–392,</p><p>p. 249. Walter von Dyck and Max Caspar, eds. Munich: C. H. Beck.</p><p>Knowles Middleton, William Edgar. 1961. “Archimedes, Kircher, Buffon, and the burning-</p><p>mirrors.” Isis 52(4): 533–543.</p><p>Koyré, Alexander. 2008. The Astronomical Revolution. Translated by R.E.W. Maddison. Lon-</p><p>don: Routledge.</p><p>Pack, Stephen F. 1966. Revelatory Geometry: The Treatises of Albrecht Dürer. Master’s thesis,</p><p>School of Architecture, McGill University.</p><p>Panofsky, Erwin. 1955. The Life and Art of Albrecht Dürer, Fourth Edition. Princeton, N.J.:</p><p>Princeton University Press.</p><p>Rupprich, Hans. 1972. “Wilibald Pirckheimer.” In Pre-Reformation Germany, Gerald Strauss,</p><p>ed. London: Harper and Row.</p><p>Russell, Francis. 1967. The World of Dürer. New York: Time Incorporated.</p><p>Strauss, Gerald, ed. 1972. Pre-Reformation Germany. London: Harper and Row.</p><p>Thausing, Moriz. 1882. Albert Dürer: His Life and Works. Translated by Fred A. Eaton, 2003.</p><p>London: John Murray Publishers.</p><p>Toomer, Gerald J. 1976. “Diocles on burning mirrors.” In Sources in the History of Mathematics</p><p>and the Physical Sciences 1. New York: Springer.</p><p>Werner, Johannes. 1522. Libellus super viginti duobus elementis conicis. Vienna, Austria: Alantsee.</p><p>Westfall, Richard S. 1995. “The Galileo Project: Albrecht Dürer.” http://galileo.rice.edu</p><p>/Catalog/NewFiles/duerer.html.</p><p>Wörz, Adèle Lorraine. 2006. The Visualization of Perspective Systems and Iconology in Dürer’s</p><p>Works. Ph.D. dissertation, Department of Geography, Oregon State University.</p><p>http://galileo.rice.edu/Catalog/NewFiles/duerer.html</p><p>http://galileo.rice.edu/Catalog/NewFiles/duerer.html</p><p>High Fashion Meets Higher Mathematics</p><p>Kelly Delp</p><p>Try the following experiment. Get a tangerine and attempt to take the</p><p>peel off in one piece. Lay the peel flat and see what you notice about the</p><p>shape. Repeat several times. This can be done with many types of citrus</p><p>fruit. Clementines work especially well.</p><p>Cornell mathematics professor William P. Thurston used this exper-</p><p>iment to help students understand the geometry of surfaces. Thurston,</p><p>who won the Fields Medal in 1982, was well known for his geometric</p><p>insight. In the early 1980s, he made a conjecture, called the geometri-</p><p>zation conjecture, about the possible geometries for three-dimensional</p><p>manifolds. Informally, an n-dimensional manifold is a space that locally</p><p>looks like ℝn.</p><p>Although Thurston proved the conjecture for large classes of three-</p><p>manifolds, the general case remained one of the most important out-</p><p>standing problems in geometry and topology for 20 years. In 2003</p><p>Grigori Perelman proved the conjecture. The geometrization conjec-</p><p>ture implies the Poincaré conjecture, so with his solution Perelman</p><p>became the first to solve one of the famed Clay Millennium Problems.</p><p>(The November 2009 issue of Math Horizons ran a feature on Perelman.)</p><p>The story of Thurston’s geometrization conjecture and the reso-</p><p>lution of the Poincaré conjecture drew attention from reporters and</p><p>other writers outside of the mathematical community. One person who</p><p>happened upon an account of Thurston and his work was the creative</p><p>director of House of Issey Miyake, fashion designer Dai Fujiwara. In a</p><p>letter to Thurston, Fujiwara described how he felt a connection with</p><p>the geometer, as he had used the same technique of peeling fruit to</p><p>explain clothing design to students new to the subject. Designers also</p><p>practice the art of shaping surfaces from two-dimensional pieces.</p><p>110 Kelly Delp</p><p>Fujiwara felt that Thurston’s three-dimensional geometries could</p><p>provide a theme for Issey Miyake’s ready-to-wear fashion line. Thur-</p><p>ston, who in 1991 had organized (along with his mother, Margaret</p><p>Thurston) what was perhaps the first mathematical sewing class as part</p><p>of the Geometry and Imagination Workshop, agreed that there was</p><p>potential for connection. Thus the collaboration was born. The Issey</p><p>Miyake collection inspired by Thurston’s eight geometries debuted on</p><p>the runway at Paris Fashion Week in spring 2010.</p><p>Geometry of Surfaces</p><p>Before discussing the fashion show and the geometry of three- manifolds,</p><p>we discuss geometries of two-dimensional objects, or surfaces. Ex-</p><p>amples of surfaces include the sphere, the torus, and the Möbius</p><p>band. Any (orientable) surface can be embedded in ℝ3, and this fact</p><p>Figure 1. Dai Fujiwara and William P. Thurston, Paris Fashion Week, spring</p><p>2010. Photo by Nick Wilson.</p><p>Fashion Meets Mathematics 111</p><p>Beauty Is Truth, Truth Beauty—That Is All Ye</p><p>Know on Earth, and All Ye Need to Know</p><p>This famous and provocative quotation of John Keats is echoed</p><p>on the emblem of the Institute for Advanced Study, where I took</p><p>my first job after graduate school. After reading an account</p><p>of my mathematical discoveries concerning eight geometries</p><p>that shape all three-dimensional topology, Dai Fujiwara made</p><p>the leap to write to me, saying that he felt in his bones that my</p><p>insights could give inspiration to his design team at Issey Mi-</p><p>yake. He observed that we are both trying to understand the</p><p>best three-dimensional forms of two-dimensional surfaces, and</p><p>he noted that we each, independently, had come around to ask-</p><p>ing our students to peel oranges to explore these relationships.</p><p>This notion resonated strongly with me, for I have long been</p><p>fascinated (from a distance) by the art of clothing design and its</p><p>connections to mathematics.</p><p>Many people think of mathematics as austere and self-con-</p><p>tained. To the contrary, mathematics is a very rich and very</p><p>human subject, an art that enables us to see and understand deep</p><p>interconnections in the world. The best mathematics uses the</p><p>whole mind, embraces human sensibility, and is not at all limited</p><p>to the small portion of our brains that calculates and manipulates</p><p>symbols. Through pursuing beauty we find truth, and where we</p><p>find truth we discover incredible beauty.</p><p>The roots of creativity tap deep within to a place we all share,</p><p>and I was thrilled that Dai Fujiwara recognized the deep com-</p><p>monality underlying his efforts and mine. Despite literally and</p><p>figuratively training and working on opposite ends of the Earth,</p><p>we had a wonderful exchange of ideas when he visited me at Cor-</p><p>nell. I feel both humbled and honored that he has taken up the</p><p>challenge to create beautiful clothing inspired by the beautiful</p><p>theory that is dear to my heart.</p><p>—William P. Thurston</p><p>112 Kelly Delp</p><p>allows us to measure distances on the surface. Let’s think about the</p><p>sphere with radius 1 centered at the origin. This sphere is described by</p><p>the familiar equation x2 + y2 + z2 = 1. Choose any two points on the</p><p>sphere, say the north pole (0, 0, 1) and the south pole (0, 0, –1).</p><p>There are two natural ways to define the distance between these</p><p>points. The first way is to assign the distance between points to be the</p><p>usual Euclidean distance in ℝ3. In this metric, the distance between the</p><p>poles is 2, the length of the diameter.</p><p>For another metric, assign the distance to be the minimum length</p><p>of any path on the sphere that starts at one pole and ends at the other;</p><p>this distance would be the length of the shortest arc of a great circle be-</p><p>tween them. Now the distance between the poles would be r. This lat-</p><p>ter metric is more appropriate; after all, when traveling from Buffalo to</p><p>Sydney, the best way is not</p><p>by drilling through the center of the Earth.</p><p>Two manifolds are topologically equivalent if there is a continuous</p><p>bijection, with a continuous inverse, between them. The bijection is</p><p>called a homeomorphism, and we say that the manifolds are homeomor-</p><p>phic. Under this equivalence relation, all of the surfaces in Figure 2</p><p>are spheres.</p><p>Each of these spheres can be equipped with a metric from ℝ3, as pre-</p><p>viously described, by measuring the shortest path in the surface. Even</p><p>though they are topologically equivalent, as metric spaces they are very</p><p>different. Two surfaces are metrically equivalent if there is a distance-</p><p>preserving map, called an isometry, between them. One quantity that</p><p>is preserved under isometries is Gaussian curvature. Recall that the</p><p>Gaussian curvature is a function k from a surface S to the real num-</p><p>bers, where k(p) is the product of the principal curvatures at p; roughly</p><p>speaking, k(p) gives a measure of the amount and type of bending of the</p><p>surface at a point p. At a point of positive curvature, all of the (locally</p><p>length minimizing) curves through p bend in the same direction; in</p><p>Figure 2. Spheres.</p><p>Fashion Meets Mathematics 113</p><p>negative curvature, the surface has curves that bend in opposite direc-</p><p>tions. A surface that contains a straight line through a point p gives an</p><p>example of zero curvature (Figure 3).</p><p>You should be able to identify points of negative curvature in the</p><p>right two spheres in Figure 2. The second sphere has only positive cur-</p><p>vature, though the curvature is greater at the north pole than at the</p><p>equator. The first sphere is the most symmetric and has constant cur-</p><p>vature for k(p) = 1 for all p.</p><p>The round sphere is one of three model two-dimensional geome-</p><p>tries. The other two model geometries are the Euclidean plane, which</p><p>has constant zero curvature, and the hyperbolic plane, which has con-</p><p>stant curvature of –1. Every (compact and smooth) surface supports a</p><p>metric of exactly one type of constant curvature: positive, negative, or</p><p>zero. Although the sphere has many different metrics, it cannot have</p><p>a Euclidean or hyperbolic metric. We give examples of surfaces of the</p><p>latter two types.</p><p>Euclidean surfaces, such as a torus, can be constructed from pieces</p><p>of the Euclidean plane. We can start with a rectangle in the Euclid-</p><p>ean plane, a sheet of paper works nicely as a model, and tape together</p><p>opposite sides of the piece of paper (mathematically, this is done by</p><p>p</p><p>p</p><p>p</p><p>κ(p) > 0 κ(p) = 0 κ(p)</p><p>orbifolds with metrics modeled on a specific geometry. For our</p><p>purposes, it should be sufficient to understand two related examples.</p><p>Let’s start with the two-dimensional case. A cone, with cone angle of</p><p>2r/3, is an example of a Euclidean orbifold with one singular point of</p><p>order, 3. This cone can be constructed in two ways. In a method simi-</p><p>lar to the construction of the torus, we can cut a wedge from the circle</p><p>with angle 2r/3 and tape up the sides. We could also cut just one slit</p><p>from the edge of the circle to the center, and then roll up the disk so</p><p>that it wraps around itself three times. Mathematically, this process can</p><p>be described as taking the quotient of the disk by a rotation. Away from</p><p>the cone point, every point has a small neighborhood so that the metric</p><p>looks just like a small disk in ℝ2.</p><p>The higher dimensional analog of the cone can be constructed from a</p><p>solid cylinder. Again, we can think about the construction in two ways:</p><p>either as cutting a wedge and gluing opposite sides, or by this process of</p><p>rolling up the cylinder so that it wraps around itself three times. We see</p><p>that in three-dimensional spaces our singular sets can be one- dimensional.</p><p>We have a whole line segment of singularities labeled with a 3.</p><p>The important point is that a particular three-dimensional orbifold</p><p>can belong to at most one of the eight geometric classes and that sin-</p><p>gular sets can be one-dimensional. Sometimes these singular sets have</p><p>3</p><p>rotate</p><p>by 2π/3</p><p>φ</p><p>Figure 6. Euclidian cone.</p><p>Fashion Meets Mathematics 117</p><p>several components, which are linked together. The three-sphere S 3,</p><p>which is the set of points distance 1 from the origin in ℝ4, is a three-</p><p>dimensional manifold and is the model space for one of the eight ge-</p><p>ometries. However, for the orbifold S3, with a one-dimensional singular</p><p>set, the metric class depends on how the singular set is sitting inside</p><p>S 3. A table of links is shown in Figure 8. Each link corresponds to the</p><p>orbifold S 3 with the given link as a singular set of order 2. Each of the</p><p>orbifolds carries a different one of the eight geometries.</p><p>Thurston drew the links in Figure 8. They were one of the many</p><p>ideas that he shared with Fujiwara. The links intrigued Fujiwara, and</p><p>3</p><p>rotate</p><p>by 2π/3</p><p>φ</p><p>Figure 7. Three-dimensional Euclidian orbifold.</p><p>Euclidean:</p><p>Borromean Rings</p><p>Hyperbolic:</p><p>5×3 Turkshead</p><p>H2×RSpherical:</p><p>3 Hopf Circles</p><p>Nil:</p><p>4 Hopf Circles</p><p>SL(2,R):</p><p>5 Hopf Circles</p><p>S2×RSolv</p><p>Figure 8. Illustrations of orbifold representatives of the eight geometries.</p><p>118 Kelly Delp</p><p>Figure 9. Issey Miyake’s spring 2010</p><p>collection on the runway at Paris</p><p>Fashion Week. Courtesy of Frede-</p><p>rique Dumoulin/Issey Miyake.</p><p>Fashion Meets Mathematics 119</p><p>they appeared as an integral part of several of the pieces in the fashion</p><p>line, as seen on the models on the runway.</p><p>In an article written for the fashion magazine Idoménée, Thurston</p><p>gave the following comment about the collection:</p><p>The design team took these drawings as their starting theme and</p><p>developed from there with their own vision and imagination. Of</p><p>course it would have been foolish to attempt to literally illustrate</p><p>the mathematical theory—in this setting, it’s neither possible nor</p><p>desirable. What they attempted was to capture the underlying</p><p>spirit and beauty. All I can say is that it resonated with me.</p><p>Acknowledgment</p><p>I am but one of many who were influenced by Bill Thurston and sad-</p><p>dened by his death on August 21, 2012. I am grateful for the time we</p><p>spent playing with mathematics.</p><p>The Jordan Curve Theorem Is Nontrivial</p><p>Fiona Ross and William T. Ross</p><p>1. Introduction</p><p>The classical Jordan curve theorem (JCT) says,</p><p>Every Jordan curve (a non-self-intersecting continuous loop in the</p><p>plane) separates the plane into exactly two components.</p><p>It is often mentioned just in passing in courses ranging from lib-</p><p>eral arts mathematics courses, where it is an illuminating example of</p><p>an “obvious” statement that is difficult to prove, to undergraduate and</p><p>graduate topology and complex analysis, where it tends to break the</p><p>flow. In complex analysis, it is especially given short shrift. There are</p><p>several reasons for this short shrift. For one, a professor has bigger fish</p><p>to fry. There are the theorems of Cauchy, Hadamard, Morera, and the</p><p>like, which comprise the nuts and bolts of complex analysis, and so the</p><p>Jordan curve theorem appears to be a mere curiosity. Second, there is</p><p>insufficient time to even outline a proof that an uninitiated student can</p><p>really appreciate. Third, the result is clearly “obvious,” and professors</p><p>do not want to put themselves, or their students, through a complicated</p><p>proof of a theorem which seems to need no proof at all.</p><p>A complex analysis teacher who recognizes the need to at the least</p><p>mention the JCT might hastily draw a circle or an ellipse on the board</p><p>to point out the interior and exterior regions—knowing full well that</p><p>the students are not impressed. The slightly more ambitious teacher</p><p>might make some weak attempt (without practicing this before class) at</p><p>drawing something more complicated and invite the students to iden-</p><p>tify the interior and exterior regions. The students always do so quickly</p><p>The Jordan Curve Theorem 121</p><p>and are still not impressed and don’t see the real difficulties. Maybe the</p><p>teacher gives a slightly more formal “proof” of the JCT:</p><p>Start from a point not on the curve and draw a straight line from</p><p>that point to the outside of the whole drawing. If the line meets</p><p>the curve an odd number of times, you are on the interior. If the</p><p>line meets an even number of times, you are on the exterior.</p><p>For most curves a (nonartist) teacher might draw in class, this</p><p>“counting crossings” method is relatively easy, and students still don’t</p><p>see the real difficulties or why this is not really a valid proof of the JCT.</p><p>The Jordan curve theorem lesson usually ends with a few mumbled</p><p>words like, “Well, trust me on this one. Things can get pretty compli-</p><p>cated out there, and to make this all exact takes a lot of work. So, let’s</p><p>move on to . . .”</p><p>We certainly understand the issues mentioned above in teaching the</p><p>Jordan curve theorem and the need to move on to more relevant topics.</p><p>We write this note for those who wish to appreciate both the artistic</p><p>exploration and the history behind the JCT. Thus we unite two lines</p><p>of discourse on the nature of nontriviality. In our opinion, the JCT is a</p><p>wonderful result because it exposes us to amazing, pathological, coun-</p><p>terintuitive examples, such as nowhere differentiable curves or curves</p><p>with positive area (see below). So, in a way, not appreciating the JCT</p><p>is driven by a lack of imagination, in thinking that Jordan curves are</p><p>nothing more than circles or a couple of wavy lines a teacher hastily</p><p>draws on a blackboard.</p><p>The JCT not only inspires mathematicians to dream up these fantas-</p><p>tical examples, which almost mock the technical definition of a curve,</p><p>but it also inspires artists. The open curve artwork of Mø Morales (see</p><p>http://virtualmo.com) and the maze art of Berg [l] are well known and</p><p>much admired. Kaplan [10] and Pedersen and Singh [12] have developed</p><p>algorithms for computer-generated mazes and labyrinths. Perhaps most</p><p>relevant are the computer-generated Jordan curve artwork of Bosch</p><p>[2] and the TSP art of Bosch and Kaplan [3]. Before becoming familiar</p><p>with these artists’ work, the first author (Fiona) made the drawings de-</p><p>scribed here to show that Jordan curves are not the cold, abstract, bor-</p><p>ing objects we might think they are. Instead, they can tell a story. They</p><p>can help mathematicians make a better case to their students, and to</p><p>122 Ross and Ross</p><p>themselves, that there is something nontrivial going on here— indeed,</p><p>something beautiful. We see through some hand-drawn artwork that</p><p>the interior and exterior regions guaranteed by the JCT are not so easy</p><p>to identify, and when they are identified, they can be just wonderful to</p><p>look at.</p><p>Just as curves can be used</p><p>to tell a story, the proof of the JCT is a</p><p>great story by itself—from the Bohemian monk who had to convince</p><p>us that the JCT needed a proof, to the Frenchman who gave us an un-</p><p>convincing one, to the mathematicians who opened up the Pandora’s</p><p>box of terrifying pathological examples of curves, to the final proof, to</p><p>Jordan’s redemption. The artwork inspired by the JCT, and how it can</p><p>inspire mathematicians and teachers, as well as the history of the result,</p><p>are illuminated here.</p><p>2. The Artwork</p><p>The black-and-white hand drawings that are Jordan curves drawn by</p><p>the first author (Figures 1 and 2), were created to give the reader an op-</p><p>portunity to explore the idea of a curve as leading the viewer through</p><p>a story and also to invite the reader to apply the “proof” of the JCT</p><p>mentioned earlier (counting the number of crossings needed to exit</p><p>the drawing) to determine which parts of the figures lie in the interior</p><p>or exterior. As a visual artist, the first author has always found inspira-</p><p>tion in the way curves appear in the structure of the natural world.</p><p>The curves that appear in geology and biology, such as anafractuous</p><p>(twisting and turning), concentra, labyrinthine, phyllotaxy (dynamic</p><p>spirals), vermiculate (wormlike arrangements), and ripple and dune</p><p>formations have provided a starting point for many of her artworks.</p><p>In exploring the physical and metaphorical structure of unicursal laby-</p><p>rinths, the first author created a series of self-portraits in landscapes</p><p>rendered from a single line. The most famous of unicursal, or single</p><p>path, labyrinths features prominently in the mythological tale of The-</p><p>seus and the Minotaur. Labyrinth forms are also used as guides to medi-</p><p>tation and prayer, giving the person who travels the labyrinth a sense of</p><p>their very selves as the most complex of labyrinths.</p><p>The authors began a discussion of the JCT after the second author</p><p>(William) saw one of the first author’s unicursal labyrinths (Blue Intro-</p><p>verted Unicursal Labyrinth, Figures 3 and 4). In this drawing, the first</p><p>The Jordan Curve Theorem 123</p><p>author made a point of not uniting the curve that formed the laby-</p><p>rinth. Just as there is no metaphorical separation between the journey</p><p>and the traveler, there was, deliberately, no separation of the interior</p><p>or exterior of the labyrinth on the paper. Intrigued by the conver-</p><p>sations with the first author, who is a mathematician, and by Felix</p><p>Klein’s statement,</p><p>Everyone knows what a curve is, until he has studied enough</p><p>mathematics to become confused through the countless number</p><p>of possible exceptions,</p><p>the first author combined the idea of a curve leading the viewer through</p><p>a meaningful story together with the mathematical notion of the</p><p>Figure 1. Fiona Ross, When We Could Be Diving for Pearls, 9¾  6, 2011,</p><p>Micron ink on Denril paper.</p><p>124 Ross and Ross</p><p>complexities encountered when trying to prove the JCT. She began</p><p>a new series of drawings (When we could be diving for pearls (Figure 1)</p><p>and A thread in the labyrinth (Figure 2) rendered from a single (nonin-</p><p>tersecting) curve without beginning or end, i.e., a Jordan curve. In</p><p>these new drawings, the unicursal labyrinth is no longer open, as in the</p><p>earlier Blue Introverted Unicursal Labyrinth (Figures 3 and 4), but closed.</p><p>The interior and exterior spaces in the new drawings are difficult to</p><p>identify. Making it difficult to identify inside and outside spaces gives</p><p>the “closed” drawings in Figures 1 and 2 a different meaning than the</p><p>“open” drawings of Figures 3 and 4. The Jordan curve drawings have a</p><p>more secretive content. They seem to hold their figurative breath as the</p><p>inked fingers of the external space unsuccessfully reach in and touch the</p><p>interior, whereas the non-Jordan curve drawings invite entry into the</p><p>most remote areas of the work. The first author finds both the forms</p><p>of content (open and closed) interesting to work with and is grateful</p><p>to be given the opportunity to point out where art and mathematics</p><p>Figure 2. Fiona Ross, A Thread in the Labyrinth, 6  6, 2011, Micron ink on</p><p>Denril paper.</p><p>The Jordan Curve Theorem 125</p><p>intersect. The Jordan curve drawings specifically created for this ar-</p><p>ticle (Figures 1 and 2) were hand drawn with Micron ink markers or</p><p>graphite on Denril drafting film.</p><p>3. A Historical Perspective of the Jordan Curve Theorem</p><p>Just as the JCT artwork in the previous section shows that curves can</p><p>tell a tale, the JCT itself is an interesting story. The JCT says that the</p><p>plane is separated into two components by a Jordan curve.</p><p>We traditionally call the bounded component the “interior” and the</p><p>unbounded component the “exterior.” Bernard Bolzano [8,13] realized</p><p>that the problem was nontrivial and officially posed it as a theorem</p><p>needing a proof. Here is Bolzano’s “prophetic version of the celebrated</p><p>Jordan curve theorem” (8, p. 285):</p><p>Figure 3. Fiona Ross, Blue Introverted Unicursal Labyrinth, 14  8, 2010, Ink</p><p>on Yupo paper.</p><p>126 Ross and Ross</p><p>If a closed line lies in a plane and if by means of a connected line</p><p>one joins a point of the plane which is enclosed within the closed</p><p>line with a point which is not enclosed within it, then the con-</p><p>nected line must cut the closed line.</p><p>Bolzano also realized that the current notions of curve at the time were</p><p>in desperate need of proper definitions! The first proof, and hence the</p><p>name Jordan curve theorem, was given in 1887 by Camille Jordan in</p><p>his book Cours d’analyse de l’École Polytechnique [9] but was regarded by</p><p>many to be incorrect. As an interesting side note, Thomas Hales [7]</p><p>believes that the charges of incorrectness against Jordan’s original proof</p><p>Figure 4. Blue Introverted Unicursal Labyrinth, detail.</p><p>The Jordan Curve Theorem 127</p><p>were trumped up and that the errors are merely aesthetic. With a few</p><p>relatively minor changes, he claims to make Jordan’s proof rigorous.</p><p>If the Jordan curve is a polygon, one can prove the JCT using the</p><p>“counting crossings” method mentioned earlier. This proof, once</p><p>cleaned up a bit to make it completely rigorous, is the standard way to</p><p>prove the JCT for polygons.</p><p>When the curve is not a polygon, all sorts of pathologies can occur.</p><p>For the student, who may be reminded of defining the interior of the</p><p>curve by using the rule learned from calculus: “When you travel around</p><p>the curve, the interior is on your left,” it is important to notice that one</p><p>is tacitly assuming that there is a continuously turning normal to the</p><p>curve (your left hand). There are examples of functions f: [0, 1]  ℝ</p><p>that are continuous everywhere but differentiable nowhere, i.e., Weier-</p><p>strass’ famous example [5]</p><p>( ) ( ),cosxf a b xn</p><p>n</p><p>n</p><p>0</p><p>r=</p><p>3</p><p>=</p><p>/</p><p>where 0 1 is an odd integer, and ab > 1 + (3r/2) (Fig-</p><p>ure 5). The graph of f is jagged everywhere and can be used to define</p><p>a Jordan curve that is also jagged everywhere. Such a curve does not</p><p>have a well-defined tangent (and is hence normal). These types of jag-</p><p>ged curves also serve to point out how our informal proof of the JCT</p><p>Figure 5. A Weierstrass nowhere differentiable function.</p><p>2</p><p>1</p><p>–2</p><p>0.20 0.4 0.6 0.8 1.0</p><p>0</p><p>–1</p><p>128 Ross and Ross</p><p>for polygonal type regions (count the number of times you cross the</p><p>curve) breaks down since you might cross the curve an infinite number</p><p>of times when exiting the drawing, making the crossing parity (i.e.,</p><p>odd or even) undefined.</p><p>Even more pathological is an example of Osgood [11] of a Jordan</p><p>curve with positive area. Jordan showed that if the curve is rectifiable</p><p>(of finite length), then it has zero area. These positive-area Osgood</p><p>curves are quite wild and fascinating, and the reader is directed to a</p><p>wonderful Mathematica Demonstration by Robert Dickau [4], where</p><p>one constructs a Jordan curve with any desired (finite) area.</p><p>Otto Veblen [15] finally gave what many regard as the first correct</p><p>proof of the Jordan curve theorem, and others have followed with dif-</p><p>ferent proofs, including ones using formal logic [6].</p><p>4.</p><p>Final Thought</p><p>In this article, we have seen that a Jordan curve is much more than a</p><p>circle or ellipse. It can be jagged at every point or perhaps impossible</p><p>to visually determine the interior from the exterior or perhaps even</p><p>of positive area, and, as seen through the artwork in this article, can</p><p>lead the viewer through a story. In fact, the Jordan Curve Theorem</p><p>is a wonderful story in itself—a seemingly obvious result, which Bol-</p><p>zano had to convince people was not and whose original proof was not</p><p>convincing and which inspired mathematicians to create pathological,</p><p>counterintuitive, and fascinating examples. It is then fitting that Jordan</p><p>curves inspire art and, more importantly, help mathematicians make a</p><p>better case to the students that the JCT is non-trivial both as a math-</p><p>ematical result and as a work of art.</p><p>References</p><p>[1] C. Berg, Amazing Art: Wonders of the Ancient World. Harper Collins, New York, 2001.</p><p>[2] R. Bosch, Simple-closed-curve sculptures of knots and links, J. Math. Arts 4 (2010), pp. 57–</p><p>71. http://www.tandfonline.com/doi/abs/10.1080/17513470903459575#.UYQMp0</p><p>pTn0U.</p><p>[3] R. Bosch and C. Kaplan, TSP Art, Bridges Conference Proceedings 2005, 303–310,</p><p>http://www.cgl.uwaterloo.ca/~csk/projects/tsp/.</p><p>[4] R. Dickau, Knopp’s Osgood Curve Construction, Wolfram Demonstrations Project,</p><p>http://demonstrations.wolfram.com.KnoppsOsgoodCurveConstruction (user needs the</p><p>Mathematica player installed to see this demonstration.</p><p>http://www.tandfonline.com/doi/abs/10.1080/17513470903459575#.UYQMp0pTn0U</p><p>http://www.tandfonline.com/doi/abs/10.1080/17513470903459575#.UYQMp0pTn0U</p><p>The Jordan Curve Theorem 129</p><p>[5] B. R. Gelbaum and M. H. J. Olmsted, Counterexamples in analysis, Corrected reprint of the</p><p>second (1965), Dover Publications Inc., Mineola, N.Y., 2003.</p><p>[6] T. C. Hales, The Jordan curve theorem, formally and informally, Amer. Math. Monthly 114</p><p>(2007), pp. 882–894.</p><p>[7] T. C. Hales, Jordan’s proof of the Jordan curve theorem, Studies in Logic, Grammar and Rheto-</p><p>ric 10 (2007), pp.45–60.</p><p>[8] D. M. Johnson, Prelude to dimension theory: the geometrical investigations of Bernard Bolzano,</p><p>Arch. History Exact Sci. 17 (1977), pp. 262–295.</p><p>[9] C. Jordan, Cours d’analyse de l’École Polytechnique, Gauthier-Villars, Paris, 1887.</p><p>[10] C. Kaplan, Vortex maze construction, J. Math. Arts 1 (2007), pp. 7–20.</p><p>[11] W. F. Osgood, A Jordan curve of positive area. Trans. Amer. Math. Soc. 4(1) (1903), pp.</p><p>107–112.</p><p>[12] H. Pedersen and K. Singh, Organic labyrinths and mazes, NPAR Conference Proceedings,</p><p>(2006), 79–86, http://dx.doi.org/10.1145/1124728.1124742.</p><p>[13] B. B. Van Rootselaar, Dictionary of Scientific Biography; Vol. II, Charles Scribners Sons,</p><p>New York, 1970, pp.273–279.</p><p>[14] H. Sagan, A geometrization of Lebesgue’s space-filling curve, Math. Intelligencer 15 (1993),</p><p>pp. 37–43.</p><p>[15] O. Veblen, Theory on plane curves in non-metrical analysis situs, Trans. Amer. Math. Soc. 6</p><p>(1905), pp. 83–98.</p><p>Why Mathematics? What Mathematics?</p><p>Anna Sfard</p><p>“Why do I have to learn mathematics? What do I need it for?” When I</p><p>was a school student, it never occurred to me to ask these questions,</p><p>nor do I remember hearing it from any of my classmates. “Why do I</p><p>need history?”—yes. “Why Latin?” (yes, as a high school student I was</p><p>supposed to study this ancient language)—certainly. But not, “Why</p><p>mathematics?” The need to deal with numbers, geometric figures, and</p><p>functions was beyond doubt, and mathematics was unassailable.</p><p>Things changed. Today, every other student seems to ask why we</p><p>need mathematics. Over the years, the quiet certainty of the math-</p><p>ematics learner has disappeared: No longer do young people take it</p><p>for granted that everybody has to learn math, or at least the particular</p><p>mathematics curriculum that is practiced with only marginal variations</p><p>all over the world. The questions, “Why mathematics? Why so much of</p><p>it? Why ‘for all’?” are now being asked by almost anybody invested,</p><p>or just interested, in the business of education. Almost, but not all.</p><p>Whereas the question seems to be bothering students, parents, and,</p><p>more generally, all the “ordinary people” concerned about the cur-</p><p>rent standards of good education, the doubt does not seem to cross the</p><p>minds of those who should probably be the first to wonder: mathemat-</p><p>ics educators, policy makers, and researchers. Not only are mathemat-</p><p>ics educators and researchers convinced about the importance of school</p><p>mathematics, they also know how to make the case for it. If asked, they</p><p>all come up with a number of reasons, and their arguments look more</p><p>or less the same, whatever the cultural background of their presenters.</p><p>Yet these common arguments are almost as old as school mathemat-</p><p>ics itself, and those who use them do not seem to have considered the</p><p>possibility that, as times change, these arguments might have become</p><p>unconvincing.</p><p>Why Mathematics? What Mathematics? 131</p><p>Psychologically, this attitude is fully understandable. After all, at</p><p>stake is the twig on which the mathematics education community has</p><p>woven its nest. And yet, as the wonderings about the status of school</p><p>mathematics are becoming louder and louder, the need for a revision of</p><p>our reasons can no longer be ignored. In what follows, I respond to this</p><p>need by taking a critical look at some of the most popular arguments</p><p>for the currently popular slogan, “Mathematics for all.” This analysis</p><p>is preceded by a proposal of how to think about mathematics so as to</p><p>loosen the grip of clichés and to shed off hidden prejudice. It is fol-</p><p>lowed by my own take on the question of what mathematics to teach,</p><p>to whom, and how.</p><p>What Is Mathematics?</p><p>To justify the conviction that competence in mathematics is a condi-</p><p>tion for good citizenship, one must first address the question of what</p><p>mathematics is and what role it has been playing in the life of Western</p><p>society.1 Here is a proposal: I believe that it might be useful to think</p><p>about any type of human knowing, mathematics included, as an activity</p><p>of, or a potential for, telling certain kinds of stories about the world.</p><p>These stories may sometimes appear far removed from anything we can</p><p>see or touch, but they nevertheless are believed to remain in close rela-</p><p>tionship to the tangible reality and, in the final account, are expected</p><p>to mediate all our actions and improve the ways in which we are going</p><p>about our human affairs. Since mathematical stories are about objects</p><p>that cannot be seen, smelled, or touched, it may be a bit difficult to see</p><p>that the claim of practical usefulness applies to mathematics as much</p><p>as to physics or biology. But then it suffices to recall the role of, say,</p><p>measurements and calculations in almost any task a person or a society</p><p>may wish to undertake to realize that mathematical stories are, indeed,</p><p>a centerpiece of our universal world-managing toolkit. And I have used</p><p>just the simplest, most obvious example.</p><p>So, as the activity of storytelling, mathematics is not much different</p><p>from any other subject taught in school. Still, its narratives are quite</p><p>unlike those told in history, physics, or geography. The nature of the</p><p>objects these stories are about is but one aspect of the apparent dis-</p><p>similarity. The way the narratives are constructed and deemed as en-</p><p>dorsable (“valid” or “true”) makes a less obvious, but certainly not any</p><p>132 Anna Sfard</p><p>less important, difference. It is thus justified to say that mathematics is a</p><p>discourse—a special way of communicating, made unique by its vocabu-</p><p>lary, visual means, routine ways of doing things, and the resulting set</p><p>of endorsed narratives—of stories believed to faithfully reflect the real</p><p>state of affairs. By presenting mathematics in this way (see also Sfard</p><p>2008), I am moving away from the traditional vision of mathematics</p><p>as given to people by the world itself. Although definitely constrained</p><p>by external reality, mathematics is to a great extent a matter of human</p><p>decisions and choices, and of contingency</p><p>rather than of necessity. This</p><p>notion means that mathematical communication can and should be</p><p>constantly monitored for its effects. In particular, nothing that regards</p><p>the uses of mathematics is written in stone, and there is no other au-</p><p>thority than us to say what needs to be preserved and what must be</p><p>changed. This conceptualization, therefore, asks for a critical analysis</p><p>of our common mathematics-related educational practices.</p><p>Why Mathematics? Deconstructing</p><p>Some Common Answers</p><p>Three arguments for the status of mathematics as a sine qua non of</p><p>school curricula can usually be heard these days in response to the</p><p>question of why mathematics: the utilitarian, the political, and the</p><p>cultural. I will call these three motives “official,” to distinguish them</p><p>from yet another one, which, although not any less powerful than the</p><p>rest, is never explicitly stated by the proponents of the slogan “math-</p><p>ematics for all.”</p><p>The Utilitarian Argument: Mathematics</p><p>Helps in Dealing with Real-Life Problems</p><p>Let me say it again: Mathematics, just as any other domain of human</p><p>knowledge, is the activity of describing—thus understanding—the</p><p>world in ways that can mediate and improve our actions. It is often</p><p>useful to tell ourselves some mathematical stories before we act and to</p><p>repeat them as we act, while also forging some new ones. With their</p><p>exceptionally high level of abstraction and the unparalleled capacity for</p><p>generalization, mathematical narratives are believed to be a universal</p><p>tool, applicable in all domains of our lives. And indeed, mathematics</p><p>Why Mathematics? What Mathematics? 133</p><p>has a long and glorious history of contributions to the well-being of hu-</p><p>mankind. Ever since its inception, it has been providing us with stories</p><p>that, in spite of their being concerned with the universe of intangible</p><p>objects, make us able to deal with the reality around us in particularly</p><p>effective and useful ways. No wonder, then, that mathematics is consid-</p><p>ered indispensable for our existence. And yet, whereas this utilitarian</p><p>argument holds when the term “our existence” is understood as refer-</p><p>ring to the life of human society as a whole, it falls apart when it comes</p><p>to individual lives.</p><p>I can point to at least two reasons because of which the utility claim</p><p>does not work at the individual level. First, it is enough to take a critical</p><p>look at our own lives to realize that we do not, in fact, need much math-</p><p>ematics in our everyday lives. A university professor recently said in a</p><p>TV interview that in spite of his sound scientific-mathematical back-</p><p>ground he could not remember the last time he had used trigonometry,</p><p>derivatives, or mathematical induction for any purpose. His need for</p><p>mathematical techniques never goes beyond simple calculation, he said.</p><p>As it turns out, even those whose profession requires more advanced</p><p>mathematical competency are likely to say that whatever mathematical</p><p>tools they are using, the tools have been learned at the job rather than</p><p>in school.</p><p>The second issue I want to point to may be at least a partial ex-</p><p>planation for the first: People do not necessarily recognize the ap-</p><p>plicability of even those mathematical concepts and techniques with</p><p>which they are fairly familiar. Indeed, research of the past few decades</p><p>(Brown et al. 1989; Lave 1988; and Lave and Wenger 1991) brought</p><p>ample evidence that having mathematical tools does not mean know-</p><p>ing when and how to use them. If we ever have recourse to math-</p><p>ematical discourse, it is usually in contexts that closely resemble those</p><p>in which we encountered this discourse for the first time. The major-</p><p>ity of the school-learned mathematics remains in school for the rest</p><p>of our lives. These days, this phenomenon is known as situatedness of</p><p>learning, that is the dependence of the things we know on the context</p><p>in which they have been learned. To sum up, not only is our everyday</p><p>need for school mathematics rather limited, but also the mathematics</p><p>that we could use does not make it easily into our lives. All this pulls</p><p>the rug from under the feet of those who defend the idea of teaching</p><p>mathematics to all because of its utility.</p><p>134 Anna Sfard</p><p>The Political Argument:</p><p>Mathematics Empowers</p><p>Because of the universality of mathematics and its special usefulness,2</p><p>the slogan “knowledge is power,” which can now be translated into</p><p>“discourses are power,” applies to this special form of talk with a par-</p><p>ticular force. Ever since the advent of modernity, with its high respect</p><p>for, and utmost confidence in, human reason, mathematics has been</p><p>one of the hegemonic discourses of Western society. In this positivisti-</p><p>cally minded world, whatever is stated in mathematical terms tends</p><p>to override any other type of argument (just recall, for instance, what</p><p>counts as decisive “scientific evidence” in the eyes of the politician), and</p><p>the ability to talk mathematics is thus considered an important social</p><p>asset, indeed, a key to success. But the effectiveness of mathematics</p><p>as a problem-solving tool is only a partial answer to the question of</p><p>where this omnipotence of mathematical talk comes from. Another</p><p>relevant feature of mathematics is its ability to impose linear order on</p><p>anything quantifiable. Number-imbued discourses are perfect settings</p><p>for decision- making and, as such, they are favored by many, and es-</p><p>pecially by politicians (and it really does not matter that all too often,</p><p>politicians can only speak pidgin mathematics; the lack of competency</p><p>is not an obstacle for those who know their audience and are well aware</p><p>of the fact that numbers do not have to be used correctly to impress).</p><p>The second pro-math argument, one that I called political, can</p><p>now be stated in just two words: Mathematics empowers. Indeed, if</p><p>mathematics is the discourse of power, mathematical competency is</p><p>our armor and mathematical techniques are our social survival skills.</p><p>When we wonder whether mathematics is worth our effort, at stake is</p><p>our agency as individuals and our independence as members of society:</p><p>If we do not want to be pushed around by professional number-jugglers,</p><p>we must be able to juggle numbers with them and do it equally well, if</p><p>not better. Add to this idea the fact that in our society mathematics is</p><p>a gatekeeper to many coveted jobs and is thus a key to social mobility,</p><p>and you cannot doubt the universal need for mathematics any longer.</p><p>Now it is time for my counterarguments. The claim that “mathe-</p><p>matics empowers” is grounded in the assumption that mathematics is</p><p>a privileged discourse, a discourse likely to supersede any other. But</p><p>should the hegemony of mathematics go unquestioned? On a closer</p><p>Why Mathematics? What Mathematics? 135</p><p>look, not each of its uses may be for the good of those whose well-being</p><p>and empowerment we have in mind when we require “mathematics for</p><p>all.” For example, when mathematics, so effective in creating useful</p><p>stories about the physical reality around us, is also applied in crafting</p><p>stories about children (as in “This is a below average student”) and plays</p><p>a decisive role in determining the paths their lives are going to take,</p><p>the results may be less than helpful. More often than not, the numeri-</p><p>cal tags with which these stories label their young protagonists, rather</p><p>than empowering the student, may be raising barriers that some of the</p><p>children will never be able to cross. The same happens when the ability</p><p>to participate in mathematical discourse is seen as a norm and the lack</p><p>thereof as pathology and a symptom of a general insufficiency of the</p><p>child’s “potential.” I will return to all this when presenting the “unoffi-</p><p>cial” argument for obligatory school mathematics. For now, the bottom</p><p>line of what was written so far is simple: We need to remember that</p><p>by embracing the slogan “mathematics empowers” as is, without any</p><p>amendments, we may be unwittingly reinforcing social orders we wish</p><p>to change. As I will be arguing in the concluding part</p><p>of this editorial,</p><p>trying to change the game may be much more “empowering” than try-</p><p>ing to make everybody join in and play it well.</p><p>The Cultural Argument:</p><p>Mathematics Is a Necessary Ingredient</p><p>of Your Cultural Makeup</p><p>In the latest paragraph, I touched upon the issue of the place of math-</p><p>ematics in our culture and in an individual person’s identity. I will</p><p>now elaborate on this topic while presenting the cultural argument for</p><p>teaching mathematics to all.</p><p>Considering the fact that to think means to participate in some kind</p><p>of discourse, it is fair to say that our discourses, those discourses in</p><p>which each of us is able to participate, constitute who we are as so-</p><p>cial beings. In the society that appreciates intellectual skills and com-</p><p>munication, the greater and more diverse our discursive repertoire,</p><p>the richer, more valued, and more attractive our identities. However,</p><p>not all discourses are made equal, so the adjective “valued” must be</p><p>qualified. Some forms of communicating are considered to be good</p><p>for our identities, and some others much less so. As to mathematics,</p><p>136 Anna Sfard</p><p>many would say that it belongs to the former category. Considered as</p><p>a pinnacle of human intellectual achievement and thus as one of the</p><p>most precious cultural assets, it bestows some of its glory even on pe-</p><p>ripheral members of the mathematical community. Those who share</p><p>this view believe that mathematical competency makes you a better</p><p>person, if only because of the prestigious membership that it affords.</p><p>A good illustration of this claim comes from an Israeli study (Sfard</p><p>and Prusak 2005) in which 16-year-old immigrant students, originally</p><p>from the former Soviet Union, unanimously justified their choice of</p><p>the advanced mathematics program with claims that mathematics is an</p><p>indispensable ingredient of one’s identity. “Without mathematics, one</p><p>is not a complete human being,” they claimed.</p><p>But the truth is that the attitude demonstrated by those immigrant</p><p>students stands today as an exception rather than a rule. In the eyes</p><p>of today’s young people, at least those who come from cultural back-</p><p>grounds with which I am well acquainted, mathematics does not seem</p><p>to have the allure it had for my generation. Whereas this statement can</p><p>be supported with numbers that show a continuous decline in percent-</p><p>ages of graduates who choose to study mathematics (or science)—and</p><p>currently, this seems to be a general trend in the Western world3—I can</p><p>also present some firsthand evidence. In the same research in which the</p><p>immigrant students declared their need for mathematical competency</p><p>as a necessary ingredient of their identities, the Israeli-born partici-</p><p>pants spoke about mathematics solely as a stepping stone for whatever</p><p>else they would like to do in the future. Such an approach means that</p><p>one can dispose with mathematics once it has fulfilled its role as an en-</p><p>trance ticket to preferable places. For the Israeli-born participants, as</p><p>for many other young people these days, mathematical competency is</p><p>no longer a highly desired ingredient of one’s identity.</p><p>Considering the way the world has been changing in the past few</p><p>decades, it may not be too difficult to account for this drop in the</p><p>popularity of mathematics. One of the reasons may be the fact that</p><p>mathematical activity does not match the life experiences typical of our</p><p>postmodern communication-driven world. As aptly observed in a recent</p><p>book by Susan Cain (2012), the hero of our times is a vocal, assertive</p><p>extrovert with well-developed communicational skills and an insatiable</p><p>appetite for interpersonal contact. Although there is a clear tendency,</p><p>these days, to teach mathematics in collaborative groups—the type of</p><p>Why Mathematics? What Mathematics? 137</p><p>learning that is very much in tune with this general trend toward the</p><p>collective and the interpersonal—we need to remember that one can-</p><p>not turn mathematics into a discourse-for-oneself unless one also prac-</p><p>tices talking mathematics to oneself. And yet, as long as interpersonal</p><p>communication is the name of the game and a person with a preference</p><p>for the intrapersonal dialogue risks marginalization, few students may</p><p>be ready to suspend their intense exchanges with others for the sake of</p><p>well-focused, time consuming conversation with themselves.</p><p>In spite of all that has been said above, I must confess that the cul-</p><p>tural argument is particularly difficult for me to renounce. I have been</p><p>brought up to love mathematics for what it is. Born into the modernist</p><p>world ruled by logical positivism, I believed that mathematics must be</p><p>treated as a queen even when it acts as a servant. Like the immigrant</p><p>participants of Anna Prusak’s study, I have always felt that math ematics</p><p>is a valuable, indeed indispensable, ingredient of my identity—an ele-</p><p>ment to cherish and of which to be proud. But this belief is just a matter</p><p>of emotions. Rationally, there is little I can say in defense of this stance.</p><p>I am acutely aware of the fact that times change and that, these days,</p><p>modernist romanticism is at odds with postmodernist pragmatism. In</p><p>the end, I must concede that the designation of mathematics as a cul-</p><p>tural asset is not any different than that of poetry or art. Thus, however</p><p>we look at it, the cultural argument alone does not justify the promi-</p><p>nent presence of mathematics in school curricula.</p><p>The Unofficial Argument:</p><p>Mathematics Is a Perfect Selection Tool</p><p>My last argument harks back to the abuses of mathematics to which</p><p>I hinted while reflecting on the statement “mathematics empowers.”</p><p>I call it unofficial because no educational policy maker would admit</p><p>to its being the principal, if not the only, motive for his or her deci-</p><p>sions. I am talking here about the use of school mathematics as a basis</p><p>for the measuring- and-labeling practices mentioned above. In our</p><p>society, grades in mathematics serve as one of the main criteria for</p><p>selecting school graduates for their future careers. Justifiably or not,</p><p>mathematics is considered to be the lingua franca of our times, the uni-</p><p>versal language, less sensitive to culture than any other well-defined</p><p>discourse. No intellectual competency, therefore, seems as well suited</p><p>138 Anna Sfard</p><p>as mathematics for the role of a universal yardstick for evaluating and</p><p>comparing people. Add to this the common conviction that “Good in</p><p>math = generally brilliant” (with the negation being, illogically, “not</p><p>good in math = generally suspect”), and you begin realizing that teach-</p><p>ing mathematics and then assessing the results may be, above all, an</p><p>activity of classifying people with “price tags” that, once attached, will</p><p>have to be displayed whenever a person is trying to get access to one</p><p>career or another. I do not think that an elaborate argument is needed</p><p>to deconstruct this kind of motive. The very assertion that this harm-</p><p>ful practice is perhaps the only reason for requiring mathematics for all</p><p>should be enough to make us rethink our policies.</p><p>What Mathematics and Why? A Personal View</p><p>It is time for me to make a personal statement. Just in case I have been</p><p>misunderstood, let me make it clear: I do care for mathematics, and I</p><p>am as concerned as anybody about its future and the future of those who</p><p>are going to need it. All that I said above grew from this genuine con-</p><p>cern. By no means do I advocate discontinuing the practice of teaching</p><p>mathematics in school. All I am trying to say is that we should approach</p><p>the task in a more flexible, less authoritarian way, while giving more</p><p>thought to the question of how much should be required from all and</p><p>how much choice should be left to the learner. In other words, I propose</p><p>that we rethink school mathematics and revise it quite radically. As I</p><p>said before, if there is a doubt about the game being played, let us change</p><p>this game rather than just trying to play it well. These days, deep, far-</p><p>reaching change is needed in what we teach, to whom, and how.</p><p>I do have a concrete proposal with regard to what we can do, but let</p><p>me precede this discussion with two basic “don’t”s. First, let us not use</p><p>mathematics as a universal instrument for selection. This practice hurts</p><p>the student, and it spoils the mathematics that is being learned. Sec-</p><p>ond, let us not force the traditional school curriculum on everybody,</p><p>and, whatever mathematics we do decide to teach, let us teach it in a</p><p>different way.</p><p>In the rest of this editorial, let me elaborate on this latter issue,</p><p>which, in more constructive terms, can be stated as follows: Yes, let</p><p>us teach everybody some mathematics, the mathematics whose every-</p><p>day usefulness is beyond question. Arithmetic? Yes. Some geometry?</p><p>Why Mathematics? What Mathematics? 139</p><p>Definitely. Basic algebra? No doubt. Add to this some rudimentary</p><p>statistics, the extremely useful topic that is still only rarely taught in</p><p>schools, and the list of what I consider as “mathematics for all” is com-</p><p>plete. And what about trigonometry, calculus, liner algebra? Let us</p><p>leave these more advanced topic as electives, to be chosen by those who</p><p>want to study them.</p><p>But the proposed syllabus does not, per se, convey the idea of the</p><p>change I had in mind when claiming the need to rethink school math-</p><p>ematics. The question is not just of what to teach or to whom, but also</p><p>of how to conceptualize what is being taught so as to make it more con-</p><p>vincing and easier to learn. There are two tightly interrelated ways in</p><p>which mathematics could be framed in school as an object of learning:</p><p>We can think about mathematics as the art of communicating or as one of</p><p>the basic forms of literacy. Clearly, both these framings are predicated</p><p>on the vision of mathematics as a discourse. Moreover, a combination</p><p>of the two approaches could be found so that the student can benefit</p><p>from both. Let me briefly elaborate on each one of the two framings.</p><p>Mathematics as the Art of Communicating</p><p>As a discourse, mathematics offers special ways of communicating with</p><p>others and with oneself. When it comes to the effectiveness of com-</p><p>munication, mathematics is unrivaled: When at its best, it is ambiguity-</p><p>proof and has an unparalleled capacity for generalization. To put it</p><p>differently, mathematical discourse appears to be infallible—any two</p><p>people who follow its rules must eventually agree, that is, endorse the</p><p>same narratives; in addition, this discourse has an exceptional power of</p><p>expression, allowing us to say more with less.</p><p>I can see a number of reasons why teaching mathematics as the art</p><p>of communicating may be a good thing to do. First, it will bring to the</p><p>fore the interpersonal dimension of mathematics: The word communica-</p><p>tion reminds us that mathematics originates in a conversation between</p><p>mathematically minded thinkers, concerned about the quality of their</p><p>exchange at least as much as about what this exchange is all about. Sec-</p><p>ond, the importance of the communicational habits one develops when</p><p>motivated by the wish to prevent ambiguity and ensure consensus ex-</p><p>ceeds the boundaries of mathematics. I am prepared to go so far as to</p><p>claim that if some of the habits of mathematical communication were</p><p>140 Anna Sfard</p><p>regulating all human conversations, from those that take place between</p><p>married couples to those between politicians, our world would be a</p><p>happier place to live. Third, presenting mathematics as the art of in-</p><p>terpersonal communication is, potentially, a more effective educational</p><p>strategy than focusing exclusively on intrapersonal communication. The</p><p>interpersonal approach fits with today’s young people’s preferences. It</p><p>is also easier to implement. After all, shaping the ways that students</p><p>talk to each other is, for obvious reasons, a more straightforward job</p><p>than trying to mold their thinking directly. Fourth, framing the task of</p><p>learning mathematics as perfecting one’s ability to communicate with</p><p>others may be helpful, even if not sufficient, in overcoming the situat-</p><p>edness of mathematical learning. Challenging students to find solutions</p><p>that would convince the worst skeptic will likely help them develop the</p><p>lifelong habit of paying attention to the way they talk (and thus think!).</p><p>This kind of attention, being focused on one’s own actions, may bring</p><p>about discursive habits that are less context- dependent and more uni-</p><p>versal than those that develop when the learner is almost exclusively</p><p>preoccupied with mathematical objects. There may be more, but I</p><p>think these four reasons should suffice to explain why teaching math-</p><p>ematics as an art of communication appears to be a worthy endeavor.</p><p>Mathematics as a Basic Literacy</p><p>While teaching mathematics as an art of communicating, we stress the</p><p>question of how to talk. Fostering mathematical literacy completes the</p><p>picture by emphasizing the issues of when to talk mathematically and</p><p>what about.</p><p>Although, nowadays, mathematical literacy is a buzz phrase, a</p><p>cursory review of literature suffices to show that there is not much</p><p>agreement on how it should be used. For the sake of the present con-</p><p>versation, I define mathematical literacy as the ability to decide not</p><p>just about how to participate in mathematical discourse but also about</p><p>when to do so. The emphasis on the word when signals that mathemati-</p><p>cal literacy is different from the type of formal mathematical knowl-</p><p>edge that is being developed, in practice if not in principle, through the</p><p>majority of present-day curricula. These curricula offer mathematics</p><p>as, first and foremost, a self-sustained discourse that speaks about its</p><p>own unique objects and has few ties to anything external. Thus, they</p><p>Why Mathematics? What Mathematics? 141</p><p>stress the how of mathematics to the neglect of the when. Mathemati-</p><p>cal literacy, in contrast, means the ability to engage in mathematical</p><p>communication whenever this approach may help in understanding</p><p>and manipulating the world around us. It thus requires fostering the</p><p>how and the when of mathematical routines at the same time. To put it</p><p>in discursive terms, along with developing students’ participation in</p><p>mathematical discourse, we need to teach them how to combine this</p><p>discourse with other ones. Literacy instruction must stress students’</p><p>ability to switch to mathematical discourse from any other discourse</p><p>whenever appropriate and useful, and it has to foster the capacity for</p><p>incorporating some of the metamathematical rules of communication</p><p>into other discourses.</p><p>My proposal, therefore, is to replace the slogan “mathematics for</p><p>all” with the call for “mathematical literacy for all.” Arithmetic, geom-</p><p>etry, elementary algebra, the basics of statistics—these are mathemati-</p><p>cal discourses that, I believe, should become part and parcel of every</p><p>child’s literacy kit. This ideal is easier said than done, of course. Be-</p><p>cause of the inherent situatedness of learning, the call for mathematical</p><p>literacy presents educators with a major challenge. The question of how</p><p>to teach for mathematical literacy must be theoretically and empirically</p><p>studied. When we consider the urgency of the issue, we should make</p><p>sure that such research is given high priority.</p><p>In this editorial, I tried to make the case for a change in the way we</p><p>think about school mathematics. In spite of the constant talk about re-</p><p>form, the current mathematical curricula are almost the same in their</p><p>content (as opposed to pedagogy) as they were decades, if not centuries,</p><p>ago. Times change, but our general conception of school mathematics</p><p>remains invariant. As mathematics educators, we have a strong urge to</p><p>preserve the kind of mathematics that has been at the center of our lives</p><p>ever since our own days as school students. We want to make sure that</p><p>the new generation can have and enjoy all those things that our own</p><p>generation has seen as precious and enjoyable. But times do change, and</p><p>students’ needs and preferences change with them. With the advent</p><p>of knowledge technologies that allow</p><p>the future might even marginalize mathematics.</p><p>Prakash Gorroochurn surveys a collection of chance and statistics</p><p>problems that confused some of the brilliant mathematical minds of the</p><p>xx Introduction</p><p>past few centuries and initially were given erroneous solutions—but</p><p>had an important historical role in clarifying fine distinctions between</p><p>the theoretical notions involved.</p><p>Elie Ayache makes the logical-philosophical case that the prices</p><p>reached by traders in the marketplace take precedence over the intel-</p><p>lectual speculations intended to justify and to predict them; therefore,</p><p>the contingency of number prices supersedes the calculus of probabili-</p><p>ties meant to model it, not the other way around.</p><p>Finally, Kevin Hartnett reports on recent developments related to</p><p>the abc conjecture, a number theory result that, if indeed proven, will</p><p>have widespread implications for several branches of mathematics.</p><p>Other Notable Writings</p><p>This section of the introduction is intended for readers who want to</p><p>read more about mathematics. Most of the recent books I mention here</p><p>are nontechnical. I offer leads that can easily become paths to research</p><p>on various aspects of mathematics. The list that follows is not exhaus-</p><p>tive, of course—and I omit titles that appear elsewhere in this volume.</p><p>Every year I start by mentioning an outstanding recent work; this</p><p>time is no exception. The reader interested in the multitude of mo-</p><p>dalities available for conveying data (using graphs, charts, and other vi-</p><p>sual means) may relish the monumental encyclopedic work Information</p><p>Graphics by Sandra Rendgen and Julius Wiedemann.</p><p>An intriguing collection of essays on connections between mathe-</p><p>matics and the narrative is Circles Disturbed, edited by Apostolos Doxi-</p><p>adis and Barry Mazur. Another collection of essays, more technical but</p><p>still accessible in part to a general readership, is Math Unlimited, edited</p><p>by R. Sujatha and colleagues; it explores the relationship of mathemat-</p><p>ics with some of its many applications.</p><p>Among the ever more numerous popular books on mathematics I</p><p>mention Steven Strogatz’s The Joy of X, Ian Stewart’s In Pursuit of the</p><p>Unknown, Dana Mackenzie’s The Universe in Zero Words, Jeffrey Ben-</p><p>nett’s Math for Life, Lawrence Weinstein’s Guesstimation 2.0, Norbert</p><p>Hermann’s The Beauty of Everyday Mathematics, Keith Devlin’s Introduc-</p><p>tion to Mathematical Thinking and Leonard Wapner’s Unexpected Expec-</p><p>tations. Two successful older books that see new editions are Damned</p><p>Lies and Statistics by Joel Best and News and Numbers by Victor Cohn and</p><p>Introduction xxi</p><p>Lewis Cope. An eminently readable introduction to irrational numbers</p><p>is appropriately called The Irrationals, authored by Julian Havil. A much</p><p>needed book on mathematics on the wide screen is Math Goes to the Mov-</p><p>ies by Burkard Polster and Marty Ross.</p><p>Two venerable philosophers of science and mathematics have their</p><p>decades-long collections of short pieces republished in anthologies: Hil-</p><p>ary Putnam in Philosophy in an Age of Science and Philip Kitcher in Preludes</p><p>to Pragmatism. Other recent books in the philosophy of mathematics are</p><p>Logic and Knowledge edited by Carlo Cellucci, Emily Grosholz, and Emil-</p><p>iano Ippoloti; Introduction to Mathematical Thinking by Keith Devlin; From</p><p>Foundations to the Philosophy of Mathematics by Joan Roselló; Geometric</p><p>Possibility by Gordon Belot; and Mathematics and Scientific Representation</p><p>by Christopher Pincock. Among many works of broader philosophi-</p><p>cal scope that take mnemonic inspiration from mathematics, I mention</p><p>Spinoza’s Geometry of Power by Valtteri Viljanen, The Geometry of Desert by</p><p>Shelly Kagan, and The Politics of Logic by Paul Livingston. Two volumes</p><p>commemorating past logicians are Gödel’s Way by Gregori Chaitin and</p><p>his collaborators, and Hao Wang edited by Charles Persons and Mont-</p><p>gomery Link. And a compendious third edition of Michael Clark’s Para-</p><p>doxes from A to Z has just become available.</p><p>In the history of mathematics, a few books focus on particular</p><p>epochs —for instance, the massive History of Mathematical Proof in Ancient</p><p>Traditions edited by Karine Chemla and the concise History of the History</p><p>of Mathematics edited by Benjamin Wardhaugh; on the works and the</p><p>biography of important mathematicians, as in Henri Poincaré by Jeremy</p><p>Gray, Interpreting Newton, edited by Andrew Janiak and Eric Schliesser,</p><p>The King of Infinite Space [Euclid] by David Berlinski, and The Cult of Py-</p><p>thagoras by Alberto Martínez; on branches of mathe matics, for example</p><p>The Tangled Origins of the Leibnizian Calculus by Richard Brown, Calculus</p><p>and Its Origins by Richard Perkins, and Elliptic Tales by Avner Ash and</p><p>Robert Gross; or on mathematical word problems, as in Mathematical</p><p>Expeditions by Frank Swetz. An anthology rich in examples of popu-</p><p>lar writings on mathematics chosen from several centuries is Wealth in</p><p>Numbers edited by Benjamin Wardhaugh. Two collections of insightful</p><p>historical episodes are Israel Kleiner’s Excursions in the History of Math-</p><p>ematics and Alexander Ostermann’s Geometry by Its History. And an in-</p><p>triguing attempt to induce mathematical rigor in historical-religious</p><p>controversies is Proving History by Richard Carrier.</p><p>xxii Introduction</p><p>The books on mathematics education published every year are too</p><p>many to attempt a comprehensive survey; I only mention the few that</p><p>came to my attention. In the excellent series Developing Essential Under-</p><p>standing of the National Council of Teachers of Mathematics, two re-</p><p>cent substantial brochures written by Nathalie Sinclair, David Pimm,</p><p>and Melanie Skelin focus on middle school and high school geometry.</p><p>Also at the NCTM are Strength in Numbers by Ilana Horn; an anthol-</p><p>ogy of articles previously published in Mathematics Teacher edited by</p><p>Sarah Kasten and Jill Newton; and Teaching Mathematics for Social Jus-</p><p>tice, edited by Anita Wager and David Stinson. Other books on eq-</p><p>uity are Building Mathematics Learning Communities by Erika Walker and</p><p>Towards Equity in Mathematics Education edited by Helen Forgasz and</p><p>Ferdinand Rivera. In a niche of self-help books I would include Danica</p><p>Mc Kellar’s Girls Get Curves and Colin Pask’s Math for the Frightened. A</p><p>detailed ethnomathematics study is Geoffrey Saxe’s Cultural Develop-</p><p>ment of Mathematical Ideas [in Papua New Guinea]. And a second volume</p><p>on The Mathematics Education of Teachers was recently issued jointly by</p><p>the American Mathematical Society and the Mathematical Association</p><p>of America.</p><p>In a group I would loosely call applications of mathematics and con-</p><p>nections with other disciplines, notable are Fractal Architecture by James</p><p>Harris, Mathematical Excursions to the World’s Great Buildings by Alexander</p><p>Hahn, Proving Darwin by Gregory Chaitin, Visualizing Time by Graham</p><p>Wills, Evolution by the Numbers by James Wynn, and Mathematics and</p><p>Modern Art edited by Claude Bruter. Slightly technical but still widely</p><p>accessible are Introduction to Mathematical Sociology by Phillip Bonacich</p><p>and Philip Lu, The Essentials of Statistics [for social research] by Joseph</p><p>Healey, and Ways of Thinking, Ways of Seeing edited by Chris Bissell and</p><p>Chris Dillon. More technical are The Science of Cities and Regions by Alan</p><p>Wills, and Optimization by Jan Brinkhuis and Vladimir Tikhomirov.</p><p>I conclude by suggesting a few interesting websites. “Videos about</p><p>numbers & stuff ” is the deceptively self-deprecating subtitle of the</p><p>Numberphile (http://www.numberphile.com/) page, hosting many</p><p>instructive short videos on simple and not-so-simple mathematical</p><p>topics. A blog that keeps current with technology that helps teaching</p><p>mathematics is Mathematics and Multimedia (http://mathandmulti</p><p>media.com/); also a well-done educational site is Enrich Mathematics</p><p>(http://nrich.maths.org/frontpage) hosted by Cambridge University.</p><p>Introduction xxiii</p><p>Two popular sites for exchanging ideas and asking and answering</p><p>questions are the Math Overflow (http://mathoverflow.net/) and</p><p>an individual to be an agent of</p><p>her or his own learning, our ability to tell the learner what to study</p><p>changes as well. In this editorial, I proposed that we take a good look</p><p>at our reasons and then, rather than imposing one rigid model on all,</p><p>142 Anna Sfard</p><p>restrict our requirements to a basis from which many valuable variants</p><p>of mathematical competency may spring in the future.</p><p>Notes</p><p>1. I am talking about Western society because due to my personal background, this is the</p><p>only one I feel competent to talk about. The odds are, however, that in our globalized world</p><p>there is not much difference, in this respect, between Western society and all the others.</p><p>2. Just to make it clear, the former argument that mathematics is not necessarily useful in</p><p>every person’s life does not contradict the claim about its general usefulness!</p><p>3. As evidenced by numerous publications on the drop in enrollment to mathematics-</p><p>related university subjects (e.g., Garfunkel and Young 1998; ESRC 2006; and OECD 2006)</p><p>and by the frequent calls for research projects that examine ways to reverse this trend (see,</p><p>e.g., the Targeted Initiative on Science and Mathematics Education in the United Kingdom,</p><p>http://tisme-scienceandmaths.org/), the decline in young people’s interest in mathematics</p><p>and science is generally considered these days as one of the most serious educational prob-</p><p>lems, to be studied by educational researchers and dealt with by educators and policy makers.</p><p>References</p><p>Brown, J. S., Collins, A., and Duguid, P. (1989). Situated cognition and the culture of learn-</p><p>ing. Educational Researcher, 18(1), 32–42.</p><p>Cain, S. (2012). Quiet—The power of introverts in a world that can’t stop talking. New York:</p><p>Crown Publishers.</p><p>Economic and Social Research Council (ESRC), Teaching and Learning Research Pro-</p><p>gramme. (2006). Science education in schools: Issues, evidence and proposals. Retrieved from</p><p>http://www.tlrp.org/pub/documents/TLRP_Science_Commentary_FINAL.pdf.</p><p>Garfunkel, S. A., and Young, G. S. (1998). The Sky Is Falling. Notices of the AMS, 45, 256–257.</p><p>Lave, J. (1988). Cognition in practice. New York: Cambridge University Press.</p><p>Lave, J., and Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York:</p><p>Cambridge University Press.</p><p>Organization for Economic Co-operation and Development, Global Sciences Forum. (2006).</p><p>Evolution of student interest in science and technology studies. Retrieved from http://www</p><p>.oecd.org/dataoecd/16/30/36645825.pdf.</p><p>Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and</p><p>mathematizing. Cambridge, U.K.: Cambridge University Press.</p><p>Sfard, A., and Prusak, A. (2005). Telling identities: In search of an analytic tool for investi-</p><p>gating learning as a culturally shaped activity. Educational Researcher, 34(4), 14–22.</p><p>http://www.oecd.org/dataoecd/16/30/36645825.pdf</p><p>http://www.oecd.org/dataoecd/16/30/36645825.pdf</p><p>Math Anxiety: Who Has It, Why It</p><p>Develops, and How to Guard against It</p><p>Erin A. Maloney and Sian L. Beilock</p><p>Understanding Math Anxiety</p><p>For people with math anxiety, opening a math textbook or even enter-</p><p>ing a math classroom can trigger a negative emotional response, but it</p><p>does not stop there. Activities such as reading a cash register receipt can</p><p>send those with math anxiety into a panic. Math anxiety is an adverse</p><p>emotional reaction to math or the prospect of doing math [1]. Despite</p><p>normal performance in most thinking and reasoning tasks, people with</p><p>math anxiety perform poorly when numerical information is involved.</p><p>Why is math anxiety tied to poor math performance? One idea is</p><p>that math anxiety is simply a proxy for low math ability, meaning that</p><p>individuals with math anxiety are less skilled or practiced at math than</p><p>their nonanxious counterparts. After all, math anxious individuals tend</p><p>to stay away from math classes and learn less math in the courses they</p><p>do take [1]. Yet low math ability is not the entire explanation for why</p><p>math anxiety and poor math performance co-occur. It has been shown</p><p>that people’s anxiety about doing math—over and above their actual</p><p>math ability—is an impediment to math achievement [2]. When faced</p><p>with a math task, math anxious individuals tend to worry about the</p><p>situation and its consequences. These worries compromise cognitive</p><p>resources, such as working memory, a short-term system involved in</p><p>the regulation and control of information relevant to the task at hand</p><p>[3]. When the ability of working memory to maintain task focus is dis-</p><p>rupted, math performance often suffers.</p><p>Despite the progress made in understanding how math anxiety re-</p><p>lates to math performance, only limited attention has been devoted to</p><p>the antecedents of math anxiety. Determining who is most likely to</p><p>144 Maloney and Beilock</p><p>develop math anxiety, when they develop it, and why is essential for</p><p>gaining a full understanding of the math anxiety phenomenon and its</p><p>role in math achievement.</p><p>Math Anxiety: Antecedents and</p><p>Developmental Trajectory</p><p>The first years of elementary school are critical for learning basic math-</p><p>ematical skills. Yet until recently the dominant view among educators</p><p>and researchers alike was that math anxiety only arose in the context of</p><p>complex mathematics (e.g., algebra) and thus was not present in young</p><p>children. Math anxiety was thought to develop in junior high school,</p><p>coinciding with the increasing difficulty of the math curriculum to-</p><p>ward the end of elementary school [2]. Recent research challenges this</p><p>assumption. Not only do children as young as first grade report varying</p><p>levels of anxiety about math, which is inversely related to their math</p><p>achievement [4], but also this anxiety is also associated with a distinct</p><p>pattern of neural activity in brain regions associated with negative</p><p>emotions and numerical computations. When performing mathemati-</p><p>cal calculations, math anxious children, relative to their less anxious</p><p>counterparts, show hyperactivity in the right amygdala regions that are</p><p>important for processing negative emotions. This increased amygdala</p><p>activity is accompanied by reduced activity in brain regions known to</p><p>support working memory and numerical processing (e.g., the dorsolat-</p><p>eral prefrontal cortex and posterior parietal lobe) [5].</p><p>Both social influences and cognitive predispositions probably play a</p><p>role in the onset of math anxiety in early elementary school. In terms</p><p>of social influences, teachers who are anxious about their own math</p><p>abilities impart these negative attitudes to some of their students. Inter-</p><p>estingly, this transmission of negative math attitudes seems to fall along</p><p>gender lines. Beilock and colleagues found that it was only the female</p><p>students of highly math anxious female teachers (>90% of elementary</p><p>teachers in the United States are female) who tended to endorse the</p><p>stereotype that “boys are good at math, girls at reading” by the end of</p><p>a school year. Girls who endorsed this stereotype were also most likely</p><p>to be behind in math at the end of the school year [6]. Similar to how</p><p>social mores are passed down from one generation to another, negative</p><p>math attitudes seem to be transmitted from teacher to student.</p><p>Math Anxiety 145</p><p>Some children may also have a cognitive predisposition to develop</p><p>math anxiety. In adults, math anxiety is associated with deficits in one</p><p>or more of the fundamental building blocks of mathematics. For ex-</p><p>ample, adults who are math anxious are worse than their nonanxious</p><p>peers at counting objects, at deciding which of two numbers represents</p><p>a larger quantity, and at mentally rotating 3-D objects [7–9]. Similar to</p><p>how people who lack knowledge in a particular domain are often easily</p><p>swayed by negative messages [10], children who start formal schooling</p><p>with deficiencies in these mathematical building blocks may be espe-</p><p>cially predisposed to pick up on social cues (e.g., their teacher’s behav-</p><p>ior) that highlight math in negative terms.</p><p>Alleviating</p><p>Math Anxiety</p><p>Understanding the antecedents of math anxiety provides clues about</p><p>how to prevent its occurrence. For instance, bolstering basic numerical</p><p>and spatial processing skills may help to reduce the likelihood of devel-</p><p>oping math anxiety. If deficiencies in basic mathematical competencies</p><p>predispose students to becoming math anxious, then early identifica-</p><p>tion of at-risk students (coupled with targeted exercises designed to</p><p>boost their basic mathematical competencies and regulate their poten-</p><p>tial anxieties) may help to prevent children from developing math anxi-</p><p>ety in the first place.</p><p>Knowledge about the onset of math anxiety also sheds light on how</p><p>to weaken the link between math anxiety and poor math performance</p><p>in those who are already math anxious. If exposure to negative math</p><p>attitudes increases the likelihood of developing math anxiety, which</p><p>in turn adversely affects math learning and performance, then regula-</p><p>tion of the negativity associated with math situations may increase math</p><p>success, even for those individuals who are chronically math anxious.</p><p>Support for this idea comes from work showing that when simply an-</p><p>ticipating an upcoming math task, math anxious individuals who show</p><p>activation in a frontoparietal network known to be involved in the con-</p><p>trol of negative emotions perform almost as well as their nonanxious</p><p>counterparts on a difficult math test [11]. These neural findings suggest</p><p>that strategies that emphasize the regulation and control of negative</p><p>emotions—even before a math task begins—may enhance the math</p><p>performance of highly math anxious individuals.</p><p>146 Maloney and Beilock</p><p>One means by which people can regulate their negative emotions is</p><p>expressive writing in which people are asked to write freely about their</p><p>emotions for 10–15 minutes with respect to a specific situation (e.g.,</p><p>an upcoming math exam). Writing is thought to alleviate the burden</p><p>that negative thoughts place on working memory by affording people an</p><p>opportunity to re-evaluate the stressful experience in a manner that re-</p><p>duces the necessity to worry altogether. Demonstrating the benefits of</p><p>expressive writing, Ramirez and Beilock showed that having highly test</p><p>anxious high school students write about their worries before an up-</p><p>coming final exam boosted their scores from B– to B+ (even after tak-</p><p>ing into account grades across the school year) [12]. Similar effects have</p><p>been found specifically for math anxiety. Writing about math- related</p><p>worries boosts the math test scores of math anxious students [13].</p><p>Negative thoughts and worries can also be curtailed by reappraisal,</p><p>or reframing, techniques. Simply telling students that physiological re-</p><p>sponses often associated with anxious reactions (e.g., sweaty palms,</p><p>rapid heartbeat) are beneficial for thinking and reasoning can improve</p><p>test performance in stressful situations [14]. Having students think</p><p>positively about a testing situation can also help them to reinterpret</p><p>their arousal as advantageous rather than debilitating. For example,</p><p>when students view a math test as a challenge rather than a threat,</p><p>the stronger their physiological response to the testing situation (mea-</p><p>sured here in terms of salivary cortisol), the better, not worse, is their</p><p>performance [15].</p><p>Summing Up</p><p>Education, psychology, and neuroscience researchers have begun to un-</p><p>cover the antecedents of math anxiety. Not only is math anxiety pres-</p><p>ent at the beginning of formal schooling, which is much younger than</p><p>was previously assumed, but its development is also probably tied to</p><p>both social factors (e.g., a teacher’s anxiety about her own math ability)</p><p>and a student’s own basic numerical and spatial competencies—where</p><p>deficiencies may predispose students to pick up on negative environ-</p><p>mental cues about math. Perhaps most striking, many of the techniques</p><p>used to reduce or eliminate the link between math anxiety and poor</p><p>math performance involve addressing the anxiety rather than training</p><p>in math itself. When anxiety is regulated or reframed, students often</p><p>Math Anxiety 147</p><p>see a marked increase in their math performance. These findings un-</p><p>derscore the important role that affective factors play in situations that</p><p>require mathematical reasoning. Unfortunately, it is still quite rare that</p><p>numerical cognition research takes into account issues of math anxi-</p><p>ety when studying numerical and mathematical processing. By ignoring</p><p>the powerful role that anxiety plays in mathematical situations, we are</p><p>overlooking an important piece of the equation in terms of understand-</p><p>ing how people learn and perform mathematics.</p><p>Acknowledgments</p><p>This work was supported by U.S. Department of Education, Institute</p><p>of Education Sciences Grant R305A110682 and NSF CAREER Award</p><p>DRL-0746970 to Sian Beilock as well as the NSF Spatial Intelligence</p><p>and Learning Center (grant numbers SBE-0541957 and SBE-1041707).</p><p>References</p><p>[1] Hembree, R. (1990) The nature, effects, and relief of mathematics anxiety. J. Res. Math.</p><p>Educ. 21, 33–46.</p><p>[2] Ashcraft, M. H., et al. (2007) Is math anxiety a mathematical learning disability? In Why</p><p>Is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties</p><p>and Disabilities (Berch, D. B., and Mazzocco, M. M. M., eds.), pp. 329–348, Brookes.</p><p>[3] Beilock, S. L. (2010) Choke: What the Secrets of the Brain Reveal about Getting It Right When</p><p>You Have To, Simon & Schuster.</p><p>[4] Ramirez, G., et al. (2012) Math anxiety, working memory and math achievement in early</p><p>elementary school. J. Cogn. Dev. 14.2, 187–202.</p><p>[5] Young, C. B., et al. (2012) Neurodevelopmental basis of math anxiety. Psychol. Sci. 23,</p><p>492–501.</p><p>[6] Beilock, S. L., et al. (2010) Female teachers’ math anxiety affects girls’ math achieve-</p><p>ment. Proc. Natl. Acad. Sci. U.S.A. 107, 1060–1063.</p><p>[7] Maloney, E. A., et al. (2010) Mathematics anxiety affects counting but not subitizing dur-</p><p>ing visual enumeration. Cognition 114, 293–297.</p><p>[8] Maloney, E. A., et al. (2011) The effect of mathematics anxiety on the processing of nu-</p><p>merical magnitude. Q. J. Exp. Psychol. 64, 10–16.</p><p>[9] Maloney, E. A., et al. (2012) Reducing the sex difference in math anxiety: The role of</p><p>spatial processing ability. Learn. Individ. Diff. 22, 380–384.</p><p>[10] Petty, R. E., and Cacioppo, J. T. (1986) The elaboration likelihood model of persuasion.</p><p>Adv. Exp. Soc. Psychol. 19, 123–205.</p><p>[11] Lyons, I. M., and Beilock, S. L. (2011) Mathematics anxiety: Separating the math from</p><p>the anxiety. Cereb. Cortex http://dx.doi.org/10.1093/cercor/bhr289.</p><p>[12] Ramirez, G., and Beilock, S. L. (2011) Writing about testing worries boosts exam per-</p><p>formance in the classroom. Science 331, 211–213.</p><p>148 Maloney and Beilock</p><p>[13] Park, D., et al. (2011) Put your math burden down: expressive writing for the highly</p><p>math anxious. Paper presentation at the Midwestern Psychology Association, Chicago.</p><p>[14] Jamieson, J. P., et al. (2010) Turning the knots in your stomach into bows: Reappraising</p><p>arousal improves performance on the GRE. J. Exp. Soc. Psychol. 46, 208–212,</p><p>[15] Mattarella-Micke, A., et al. (2011) Choke or thrive? The relation between salivary cor-</p><p>tisol and math performance depends on individual differences in working memory and</p><p>math anxiety. Emotion 11, 1000–1005.</p><p>How Old Are the Platonic Solids?</p><p>David R. Lloyd</p><p>Recently a belief has spread that the set of five Platonic solids has been</p><p>known since prehistoric times, in the form of carved stone balls from</p><p>Scotland, dating from the Neolithic period. A photograph of a group of</p><p>these objects has even been claimed to show mathematical understand-</p><p>ing of the regular solids a millennium or so before Plato. I argue that</p><p>this is not so. The archaeological and statistical evidence do not sup-</p><p>port this idea, and it has been shown that there are problems with the</p><p>photograph. The high symmetry of many of these objects can readily be</p><p>explained without supposing any particular mathematical understand-</p><p>ing on</p><p>the part of the creators, and there seems to be no reason to</p><p>doubt that the discovery of the set of five regular solids is contemporary</p><p>with Plato.</p><p>Introduction</p><p>The flippant and rather too obvious answer to the question in the</p><p>title is “as old as Plato.” However, a little investigation shows that the</p><p>question is more complex and needs some firming up. If it is taken as</p><p>referring to any of the solids, then there is ancient testimony (Heath</p><p>1981, 162) that some of the five predate Plato. This tradition associ-</p><p>ates the tetrahedron, cube, and dodecahedron with the Pythagoreans.1</p><p>The same source (Heath 1981, 162) claims that the remaining two, the</p><p>octa hedron and icosahedron, were actually discoveries made by Plato’s</p><p>collaborator Theaetetus, and although there have been doubts about</p><p>this claim, particularly for the octahedron (Heath 1956, 438), there</p><p>is now substantial support for the idea that this ancient testimony, giv-</p><p>ing major credit to Theaetetus, is reliable (Waterhouse 1972; Artmann</p><p>1999, 249–251, 285, 296–298, 305–308).</p><p>150 David R. Lloyd</p><p>Nevertheless, Plato is responsible for giving us the first description</p><p>we have of the complete set of the five regular solids, in his dialogue</p><p>Timaeus (about 360 BCE), and probably this is the reason we speak of</p><p>Platonic solids. Plato was not claiming any originality here; rather he</p><p>seemed to be assuming that the hearers in his dialogue, and his readers,</p><p>would already be familiar with these solids. For Plato, their importance</p><p>was that these were the “most beautiful” (kallistos) structures possible,</p><p>and that from these he could construct a theory of the nature of matter.</p><p>It is probable that this kallistos description alludes to what Waterhouse</p><p>(1972) claims was then a relatively new discovery, that of the concept of</p><p>regularity, and to the demonstration, which is recorded later by Euclid</p><p>and which is probably also due to Theaetetus (Waterhouse 1972; Art-</p><p>mann 1999, 249–251), that there can only be five such structures.2</p><p>Thus, at least until recently, the consensus has been that although Plato</p><p>is not responsible for the solids themselves, the discovery of the set of</p><p>five, and of the fact that there are only five, are contemporary with him.</p><p>However, a belief has spread recently that the complete set of these</p><p>five regular solids has been known since much earlier times, in the form</p><p>of decorated stone balls from Scotland, all from the Neolithic period.</p><p>The origin of this belief is a photograph of five of these objects (Figure</p><p>1), which first appeared in a book published more than 30 years ago</p><p>(Critchlow 1979). In this book, it was alleged that the discovery of the</p><p>set of the five regular solids, and at least some of the associated theory,</p><p>predates Plato and his collaborators by a millennium3; other claims for</p><p>ancient mathematical knowledge also appear in this book.4</p><p>The claim that the five solids were known at this time was accepted</p><p>by Artmann, who comments that “All of the five regular solids appear in</p><p>these decorations” (Artmann 1999, 300–301), though he is very skepti-</p><p>cal of the other claims. He notes that a photograph of five of the ob-</p><p>jects had appeared in Mathematics Teaching (1985, p. 56). More recently,</p><p>Atiyah and Sutcliffe (2003) published a paper that reproduces the pho-</p><p>tograph shown by Critchlow, with a caption describing these objects as</p><p>“models” of the five solids, but they gave no attribution for the source</p><p>of the illustration. This paper has also appeared in the collected works</p><p>of Atiyah (2004), and in a review of this collection (Pekkonen 2006),</p><p>it was claimed that the stone balls “provide perfect models of the Pla-</p><p>tonic Solids one thousand years before Plato or Pythagoras.” The word</p><p>“models” suggests some sort of mathematical understanding on the</p><p>How Old Are the Platonic Solids? 151</p><p>part of the makers of the objects, and, almost certainly unintention-</p><p>ally, echoes Critchlow’s earlier claims. It is probably the appearance of</p><p>the photograph in the paper by Atiyah and Sutcliffe, and the authority</p><p>of the authors, which is responsible for the recent growth within the</p><p>scientific and mathematical communities of the idea that the set of the</p><p>five Platonic solids was known long before Plato. The photograph (see</p><p>Figure 1) now makes occasional appearances in lectures on aspects of</p><p>polyhedra and their symmetries, as an illustration of the idea, which</p><p>can also be found on various websites, that the solids are much older</p><p>than had been thought until recently.</p><p>Clearly the appearance of the set of all five solids at this early time,</p><p>if true, would be quite remarkable, and should the suggestion of some</p><p>sort of mathematical understanding be justifiable, there would be sub-</p><p>stantial implications for the history of mathematics. I examine some</p><p>aspects of the story in more detail here.</p><p>Archaeology and Statistical Frequencies</p><p>It is important to emphasize at the outset that the objects shown in Fig-</p><p>ure 1 are genuine Neolithic artifacts, almost certainly from the large</p><p>collection belonging to The National Museum of Scotland (NMS),5</p><p>which the museum dates to within the period 3200 bce to 2500 bce.</p><p>More than 400 such objects are known (Edmonds 1992), almost all</p><p>roughly spherical, though there are a few prolate spheroids, and most</p><p>are carved with approximately symmetric arrangements. With few ex-</p><p>ceptions, these objects are small enough to be held comfortably in one</p><p>Figure 1. Five stone balls, decorated with tapes that pick out some of the</p><p>symmetry elements (copyright G. Challifour).</p><p>152 David R. Lloyd</p><p>hand. The principal study of them, with a complete listing of those</p><p>known at the time, is by Marshall (1977, 40–72). On many but not</p><p>all of these objects, the carving is quite deep, so that there is a set</p><p>of protrusions or knobs with a common center, as in the examples in</p><p>Figure 1. They have been found in a variety of locations, but almost all</p><p>within the boundaries of present-day Scotland.</p><p>Of the five shown in Figure 1, only the second and fifth types (read-</p><p>ing from the left) have been discovered in large numbers. It is tempting</p><p>to classify these two types as “tetrahedral” and “octahedral.” However,</p><p>such terms are not used by archaeologists, who normally call these</p><p>“4-knob” and “6-knob” forms. This label is sometimes abbreviated to</p><p>“4K” and “6K,” and this convention is used here. The symmetries of the</p><p>objects are discussed below. Some 43 of the 4K objects, and 173 of the</p><p>6K, are known. The other numbers given by Marshall, up to 16K, are</p><p>3K (6), 5K (2),6 7K (18), 8K (10), 9K (3), 10K (4), 11K (1), 12K (8),</p><p>13K (0), 14K (5), 15K (1), and 16K (1). Although there were carvers</p><p>who were prepared to tackle spheres with larger numbers of knobs,7</p><p>perhaps to demonstrate their skill, most seem to have preferred to deal</p><p>with 14K or fewer. Given that these objects could only be created by</p><p>slow hand grinding, perhaps with the help of leather straps and abrasive</p><p>scraps of rock, this preference is not surprising; it must be difficult, if</p><p>not impossible, to achieve any precision when the knobs become small.</p><p>The first object in Figure 1, with 8K, is an example of a rare species.</p><p>Although 10 examples of 8K are known, by no means all of these have</p><p>the cube form shown here. Marshall, who had personally examined</p><p>almost all the known examples, notes that for the 8K objects, “This</p><p>group of balls has variety in the disposition of the eight knobs” (Mar-</p><p>shall 1977, 42). No such comment is made about any other group, and</p><p>examination of published illustrations of the 8K objects shows a total</p><p>of only three that have this approximately cubic shape.8 There are sub-</p><p>stantially more (18) examples known for the 7K version, so it does not</p><p>seem that the cube form of the 8K objects held a particular significance</p><p>for the makers or users.</p><p>There is a small peak in the numbers at 12K and 14K, suggesting</p><p>that these were interesting to the carvers. However, purely on statisti-</p><p>cal grounds, it seems</p><p>unlikely that the makers of such a wide variety</p><p>of objects would have made any connection to the particular selection</p><p>shown in Figure 1, since two of these are commonly found, while the</p><p>How Old Are the Platonic Solids? 153</p><p>others are all unusual. The choice of this particular set of five for the</p><p>photograph is probably dependent on knowledge that the carvers did</p><p>not possess.</p><p>The Provenance of the Objects in Figure 1</p><p>The illustration is taken from Critchlow (1979, 132, figure 114), who</p><p>believes that he can demonstrate a high level of mathematical skills</p><p>among the Neolithic peoples of the British Isles. The carved stone balls</p><p>are only part of this narrative. The original caption to the photograph</p><p>reads “A full set of Scottish Neolithic ‘Platonic solids’—a millennium</p><p>before Plato’s time.”</p><p>Oddly, the book gives no indication of which museum owns the five</p><p>objects shown, and because the text associated with this illustration</p><p>gives a detailed description of the five balls that belong to the Ash-</p><p>molean Museum at Oxford, some have concluded, incorrectly, that</p><p>Figure 1 shows objects from the Ashmolean collection (Lawlor 1982;</p><p>Atiyah and Sutclifffe 2003). It is evident from Critchlow’s text that he is</p><p>describing a completely different set, with different numbers of knobs,</p><p>from that shown in Figure 1. The Ashmolean set can be seen on the</p><p>web, together with the original drawings made by a curator at the time</p><p>of acquisition,9 and it is clearly different from Figure 1.</p><p>It is therefore of some interest to try to establish which objects were</p><p>photographed, if only to see whether they might have been found to-</p><p>gether, since that could imply that the carvers or owners saw some</p><p>sort of connection among these objects. Accompanying the original</p><p>of Figure 1 there is another photograph (Critchlow 1979, 132, figure</p><p>113), apparently taken on the same occasion, of a group of 4K objects,</p><p>several of which carry markings that identify them as belonging to what</p><p>was then the Scottish National Museum of Antiquities (NMA in the</p><p>Marshall list), now The National Museum of Scotland (NMS) in Edin-</p><p>burgh. The fourth object in Figure 1 is also shown in a close-up view</p><p>(Critchlow 1979, 149, figure 146), with the caption “on show in Edin-</p><p>burgh.” Comparison of this object with one of the illustrations on the</p><p>NMS website10 makes it very likely that these two photographs are of</p><p>the same object, recorded as having been discovered in Aberdeenshire.</p><p>Thus it seems almost certain that the five objects in Figure 1 are from</p><p>the NMS.11</p><p>154 David R. Lloyd</p><p>There are so many examples of 6K objects that it is not possible to</p><p>identify which particular one appears at the extreme right of the photo-</p><p>graph. However, inspection of the images of the 8K objects on the NMS</p><p>website shows that the leftmost object in Figure 1, the cube, is almost</p><p>certainly one that was found in Ardkeeling, Strypes, Moray.12</p><p>From comparison with the group of 4K objects shown by Critchlow</p><p>(1979, 132), and the museum’s own photograph, the second object in</p><p>Figure 1 (4K) is probably one13 that is described as of “unknown origin.”</p><p>Finally, the fourth object in Figure 1, like the third object, is recorded</p><p>as having been discovered in Aberdeenshire.14 However, unlike the</p><p>third object, this one is noted as having belonged to a John Rae (who</p><p>died in 1893).</p><p>The museum descriptions vary considerably for these objects and</p><p>indicate that they have been accumulated from four or five different</p><p>sources over the years. Thus the five could not have been found to-</p><p>gether; they were brought together in the photograph, probably for the</p><p>first time, in order to support a particular view.</p><p>How Many Platonic Solids?</p><p>Figure 1 was claimed, and has been widely accepted, as evidence that</p><p>the carvers were familiar with all five of the Platonic solids. However,</p><p>there is a problem with the tapes in this figure, which are not used</p><p>consistently. For the second, fourth, and fifth objects, they are used to</p><p>connect the knobs, though for the second one (the tetrahedron), con-</p><p>nections are also shown between the interstices. However, for the first</p><p>object, the tapes are connecting four-fold interstitial positions, and for</p><p>the third object, they connect the three-fold interstices. This incon-</p><p>sistency has tended to hide the fact, first pointed out by Hart (1998)</p><p>and later by le Bruyn (2009),15 that Figure 1 actually includes only four</p><p>of the Platonic structures. In his text, Critchlow defines the knobs as</p><p>vertices of the Platonic solids; this convention agrees with that used by</p><p>chemists in describing atom positions in molecules. According to this</p><p>convention, Figure 1 shows, from left to right, a cube, a tetrahedron,</p><p>two icosahedra, and an octahedron. Although on the third and fourth</p><p>objects the tapes are arranged differently, each is a 12K object. The</p><p>main difference between these two is in the form of the knobs, one pro-</p><p>nounced and well-rounded, the other flatter, so that the eye picks out</p><p>How Old Are the Platonic Solids? 155</p><p>pentagons, at least when helped by the tapes added to this object. If the</p><p>photographs on the museum’s website, without tapes, are compared,</p><p>then the difference between the two is much less obvious.</p><p>Thus there is no evidence here for a structure related to the fifth Pla-</p><p>tonic solid. If such a structure were to exist, it would be a 20K object</p><p>with the knobs at the corners of a dodecahedron. There is a reference</p><p>to “the dodecahedron on one specimen in the Museum of Edinburgh”</p><p>(Artmann 1999, 305), but since this occurs in a paragraph which men-</p><p>tions the photographs in Critchlow, this almost certainly means the</p><p>close-up photograph of the third object in Figure 1, with only 12K,</p><p>which was referred to above (Critchlow 1979, 149, figure 146).</p><p>According to the Marshall list, there are only two balls known that</p><p>have 20K; one of these balls is at the NMS. Alan Saville, senior curator</p><p>for earliest prehistory at this museum, has provided a photograph which</p><p>shows that this object is complex, and certainly not a dodecahedron. It</p><p>could be considered as a modified octahedron, with five large knobs</p><p>in the usual positions, but with the sixth octahedral position occupied</p><p>by 12 small knobs, and in addition there are also three small triangles</p><p>carved at some of the interstices, the three-fold positions of the “octa-</p><p>hedron.” These bumps make up a total of 20 “protrusions,” though the</p><p>word “knobs” is hard to justify.</p><p>The other 20K object is at the Kelvingrove Art Gallery and Museum</p><p>in Glasgow. Photographs taken by Tracey Hawkins, assistant curator,</p><p>show that this object also is far from being a dodecahedron, though this</p><p>time there are 20 clearly defined knobs of roughly the same size. The</p><p>shape is somewhat irregular, but two six-sided pyramids can be picked</p><p>out, and much of the structure, though not all, is deltahedral in form,</p><p>with sets of three balls at the corners of equilateral triangles.</p><p>No dodecahedral form with 20K has yet been found. There is there-</p><p>fore no evidence that the carvers were familiar with all five Platonic</p><p>solids. Even for the four that they did create, there is nothing to suggest</p><p>that they would have thought these in any way different from the multi-</p><p>tude of other shapes that they carved. Thus there is no evidence that the</p><p>concept of the set of five regular solids predates Plato. Nevertheless,</p><p>the carvers have come up with a variety of interesting shapes, almost</p><p>all with high symmetries, and I now turn to an examination of a wider</p><p>grouping of these objects and suggest how the symmetries may have</p><p>arisen.</p><p>156 David R. Lloyd</p><p>The High Symmetries of These Ancient Objects</p><p>The five objects in Figure 1 show, at least approximately, the symme-</p><p>tries of four of the Platonic solids. However, most of the other known</p><p>carved balls up to 14K, and many of the larger ones, also have approxi-</p><p>mations to high symmetries. Among the balls with relatively small</p><p>numbers of knobs, up to 14K, almost all have</p><p>deltahedral structures,</p><p>formed by the linking of equilateral triangles. (A few of the balls have</p><p>become worn or damaged, so the original form is not always obvious.)</p><p>Even though there is no evidence for the full set of Platonic solids</p><p>among these objects, it might still be claimed that their symmetries</p><p>suggest some sort of mathematical competence at this time. Thus it</p><p>seems worthwhile to enquire if there could be other, nonmathematical</p><p>reasons why such symmetric structures might have been created, but</p><p>any such attempt at “explanation” needs to be as simple as possible.</p><p>The carvers are likely to have wanted to make their carved knobs</p><p>as distinct as possible, and since close grinding work is required, using</p><p>only the simplest of technologies, there would have been a need to keep</p><p>the knobs as far away from each other as possible, simply to have room</p><p>to work. Also, they would probably have seen the equilateral trian-</p><p>gles that are generated automatically by packing in a pile of roughly</p><p>spherical pebbles, or of fruits. They would have been familiar with seed</p><p>heads of plants, where similar packing effects can be seen over part of</p><p>a sphere, and it does not need much imagination to try to extend these</p><p>triangular patterns to cover a complete sphere, which would generate</p><p>a deltahedral structure.</p><p>If we assume that a guiding principle for the carvers was to keep the</p><p>carved knobs as far away from each other as possible, or, equivalently, to</p><p>produce an even spacing between these knobs, then the structures can</p><p>be modeled in terms of repulsion between points on a sphere. Within</p><p>chemistry, there is a well-known qualitative approach called “electron</p><p>pair repulsion theory.” This theory rationalizes the observed geometry</p><p>of simple molecules as the result of minimizing interactions between</p><p>bond (and other) electron pairs by maximizing the distances between</p><p>them, and in the commonest cases of four and six pairs, tetrahedral and</p><p>octahedral molecular geometries are often rationalized in these terms.</p><p>However, the numbers of knobs in these objects span a far greater range</p><p>than that found for electron pairs in molecules. A much more useful set</p><p>How Old Are the Platonic Solids? 157</p><p>of data for the present comparison is available from calculations on the</p><p>rather similar classical problem of the distribution of N electric charges</p><p>over the surface of a sphere (Erber and Hockney 1997; Atiyah and Sut-</p><p>cliffe 2003). This problem is sometimes referred to as the Thomson</p><p>problem (Atiyah and Sutcliffe 2003); an alternative name is the surface</p><p>Coulomb problem (Erber and Hockney 1997).</p><p>Encouragingly for chemists, the results of such calculations of the</p><p>minimum energy configuration of the N charges accurately reproduce</p><p>the predictions of electron pair repulsion theory, as N varies over the</p><p>normal range of numbers of electron pairs found in molecules. They</p><p>show an almost total preference for deltahedra as the minimum energy</p><p>configurations, and it is noticeable that the majority of the carved stone</p><p>balls are also deltahedra, though for balls with large numbers of knobs,</p><p>other patterns appear.</p><p>However, the only Platonic solids that are generated by minimiz-</p><p>ing interactions over the surface of a sphere are the three that are also</p><p>deltahedra. The cube is not a minimum energy configuration; N = 8</p><p>gives the square antiprism, in which opposite faces of a cube have been</p><p>rotated against each other by 45°. Also, the dodecahedron is not a mini-</p><p>mum; the calculation for 20 particles shows a deltahedron with three-</p><p>fold symmetry.</p><p>The calculations become quite difficult as N increases because the</p><p>number of false energy minima increases rapidly, particularly after</p><p>N = 12. For this reason alone, we should not expect the Neolithic stone</p><p>carvers to have discovered the optimum solutions for keeping the knobs</p><p>apart for larger numbers, but a comparison between what they achieved</p><p>and the calculations is interesting. A complete analysis would require</p><p>examination of the objects themselves, which are spread across several</p><p>museums; here only a partial analysis, based on published illustrations</p><p>and descriptions, is attempted.16</p><p>The least number of knobs recorded is three, and there are five ex-</p><p>amples of this. The Marshall description of the set reads, “Two of the</p><p>balls are atypical, having rounded projecting knobs making a more or</p><p>less triangular object which is oval in section. The others have clear</p><p>cut knobs.” It seems likely from this description that all the balls are</p><p>more or less of planar trigonal symmetry, as expected from the sur-</p><p>face Coulomb calculations. Many 4K balls have been illustrated, and</p><p>all are clearly approximations to the expected tetrahedral symmetry.</p><p>158 David R. Lloyd</p><p>However, the execution is of variable quality; the one shown on the</p><p>Ashmolean Museum website has one of the “three-fold” axes at least</p><p>20° away from the expected position. Tetrahedral geometry is difficult</p><p>to work with since there is no direction from which a carver could</p><p>check his or her work as it proceeded by “looking for opposites” (Ed-</p><p>monds 1992, 179), as in the next example.</p><p>In contrast to the geometrical variations with the tetrahedral 4K, the</p><p>octahedral 6K balls seem to be more accurately executed, as well as</p><p>being by far the commonest type. This situation may be due to the fact</p><p>that a pair of poles and an equator are relatively easy to mark on a sphere</p><p>by eye. After this step, two perpendicular polar great circles can be</p><p>added, and there is one beautifully decorated ball, with no deep carv-</p><p>ing, on which exactly this pattern of three great circles is seen.17 Such a</p><p>marking with perpendicular great circles allows the construction of oc-</p><p>tahedral 6K objects by using the intersections as markers for the centers</p><p>of the knobs to be carved, but they could also provide a route to the 4K</p><p>objects by using four of the centers of the spherical triangles as markers.</p><p>After the 4K and 6K balls, there are more examples of 7K than of</p><p>any other.</p><p>Electron pair repulsion, and the surface Coulomb calculations, both</p><p>predict a pentagonal bipyramidal structure, with five knobs in a plane,</p><p>and most carvers seem to have discovered this structure. Seven of</p><p>those illustrated show this; particularly good examples are at the Ash-</p><p>molean Museum18 and at the Hunterian Museum.19 The latter museum</p><p>also has an example of 7K that has an asymmetric structure with very</p><p>oval shapes for the knobs20; this shaping might be a consequence of the</p><p>carver having to compensate after missing the symmetric structure.</p><p>The 8K objects clearly presented the carvers with some difficulty,</p><p>since they seem to have been uncertain about the best form to use.</p><p>Because this object is the only small unit that does not work as a delta-</p><p>hedron, it is hardly surprising that they did not discover the optimum</p><p>square antiprismatic structure. However, there is an apparently clear</p><p>path to the cube through the polar circle construction described above,</p><p>which divides the sphere into eight spherical triangles, as illustrated by</p><p>the tapes on the 8K object at the left in Figure 1. The fact that so many</p><p>tetrahedra but so few cubes have been found almost suggests an avoid-</p><p>ance of the cube structure; more equal numbers would have been ex-</p><p>pected if the set of Platonic solids had any significance for the carvers.</p><p>How Old Are the Platonic Solids? 159</p><p>There is only one 9K object for which an illustration is available,21</p><p>and this object is badly worn. There is a l0K object in the collection</p><p>of the Ashmolean Museum, which can be seen on their website. A de-</p><p>tailed description of this particular object by Critchlow (1979, 147)</p><p>makes it very clear that the form is essentially a pair of square pyramids</p><p>on a common four-fold axis in a “staggered” configuration (that is, ro-</p><p>tated against each other by 45°). This method creates a deltahedron</p><p>that has the same symmetry as that calculated (D4d in the Schoenflies</p><p>convention used by Atiyah and Sutcliffe</p><p>2003). There is a similar object</p><p>in the Hunterian collection.22</p><p>From the carver’s point of view, the form they chose for 12K might</p><p>well have seemed to be an extension of the 10K structure, a staggered</p><p>pair of pentagonal pyramids, but the result has the much higher sym-</p><p>metry of the icosahedron, the Platonic structure predicted by the cal-</p><p>culations. There are two examples of this in Figure 1, and there is a</p><p>third in the Dundee museum (Critchlow 1979, 145).</p><p>Some of these suggested structural principles outlined still apply</p><p>to the two 14K objects that have been illustrated. Both are pairs of</p><p>hexagonal pyramids, and the one in the Ashmolean Museum has the</p><p>pyramids in a staggered configuration, as in the calculated structure</p><p>(D6d symmetry).23 The other example has the pyramids related by a</p><p>reflection plane,24 an “eclipsed” geometry. The simple “keep the knobs</p><p>apart” principle seems to be joined by other influences now, but as</p><p>noted earlier, even modern calculations of structure become difficult</p><p>after N = 12. For higher numbers, the predicted deltahedral structures</p><p>become less common among the carved spheres; one possible reason</p><p>for this phenomenon may be that eclipsed structures are easier to con-</p><p>struct. The knobs have all been generated from spheres by grinding</p><p>away grooves, and a reflection plane allows two or more knobs to be</p><p>ground together with a single longer groove.</p><p>It seems clear that the high symmetries of the objects with relatively</p><p>small numbers of knobs arise quite naturally from the simple principle</p><p>of keeping knobs away from each other, or, equivalently, maintaining</p><p>even spacing. Three of the Platonic solids are generated automatically</p><p>in this way, and the large numbers found for the tetrahedron and octa-</p><p>hedron suggest that the carvers, or their sponsors, found something</p><p>attractive about these structures, quite possibly connected to what we</p><p>would call “aesthetic” considerations. However, the gross disparity in</p><p>160 David R. Lloyd</p><p>numbers between the octahedra and the cubes is a fairly clear indica-</p><p>tion that neither the concept of regularity, nor any of the other math-</p><p>ematical aspects of the Platonic solids, were understood at this time.</p><p>Conclusions</p><p>Although there is high-quality craftsmanship in these Neolithic objects,</p><p>there seems to be little or no evidence for what we would recognize as</p><p>mathematical ideas behind them. In particular, there is no evidence for</p><p>a prehistoric knowledge of the set of five Platonic solids, and it seems</p><p>inappropriate to describe the objects as “models” of anything. The con-</p><p>ventional historical view that the discovery of the concept of the set of</p><p>five regular solids was contemporary with Plato can still be taught, and</p><p>it is unchallenged by the existence of these beautiful objects. Reasons</p><p>for producing them should be sought in disciplines such as aesthetics</p><p>and anthropology rather than in mathematics.</p><p>Postscript</p><p>There is no evidence that the Neolithic sculptors ever made a stone</p><p>dodecahedron. However, the set of five stone Platonic solids has now</p><p>been completed, after a delay of five millennia, by the contemporary</p><p>British sculptor Peter Randall-Page. His “Solid Air II” is shown in Fig-</p><p>ure 2. The base is one of the pentagonal groups of balls, so one of the</p><p>Figure 2. “Solid Air II” by Peter Randall-Page. Used by permission of the artist.</p><p>How Old Are the Platonic Solids? 161</p><p>five-fold axes of this dodecahedron is vertical. The first photograph is</p><p>taken close to one of the other five-fold axes, and shows a pentagonal</p><p>face with five others surrounding it.</p><p>Acknowledgments</p><p>I thank Graham Challifour for permission to reproduce his photo-</p><p>graph (Figure 1) and Alan Saville of The National Museum of Scotland,</p><p>Tracey Hawkins of the Glasgow Museums, and Sally-Ann Coupar of</p><p>the Hunterian Museum of Glasgow, for help with various objects in the</p><p>collections of their museums.</p><p>Notes</p><p>1. Certainly the tetrahedron, in the form of an early gaming die, is much older. For an</p><p>entertaining account of this and other aspects of the history of the regular solids, see du</p><p>Sautoy (2008, 40–58).</p><p>2. It can also be argued that the description “most beautiful” suggests that Plato had some</p><p>idea of what we call symmetry (Lloyd 2010).</p><p>3. The original claim was “by a millennium.” In fact, given the current dating of the ob-</p><p>jects, this claim could be extended to “more than two millennia.”</p><p>4. The claims in Critchlow’s book were repeated by Lawlor (1982, 97), whose book</p><p>includes a better reproduction of the photograph. Figure 1 is taken from Lawlor’s version.</p><p>5. The National Museum of Scotland, Edinburgh. Items from this collection are referred</p><p>to by their NMS numbers in subsequent notes. Illustrations can be found on their website:</p><p>http://nms.scran.ac.uk/search/?PHPSESSID=nnog07544f585b9idc2sftdc13.</p><p>6. The 5K objects are atypical, in that these two are heavily decorated with additional</p><p>carvings, and a third is described as “oval” rather than spherical; these objects are not con-</p><p>sidered further.</p><p>7. There are examples with much higher numbers of knobs, extending to well over 100,</p><p>but for almost all of these examples, only one, two, or zero are known (three each for 27K</p><p>and for 50K).</p><p>8. NMS 000-180-001-392-C, 000-180-001-369-C, and 000-180-001-328-C.</p><p>9. http://www.ashmolean.org/ash/britarch/highlights/ stone-balls.html.</p><p>10. NMS 000-180-001-363-C.</p><p>11. According to a personal communication from G. Challifour, all the objects in his</p><p>photograph (Figure 1) were from the same museum.</p><p>12. NMS 000-180-001-392-C.</p><p>13. NMS 000-180-001-719-C.</p><p>14. NMS 000-180-001-368-C.</p><p>15. The title of this web page includes the word “hoax.” In the correspondence on this</p><p>page, I have pointed out that there is no evidence to support any such suggestion; see also</p><p>Hart (1998, Addendum 2009).</p><p>16. In addition to those in the NMS (see footnote 5), several useful illustrations can be</p><p>found in Marshall 1977, and a few in Critchlow 1979; five more are available at the Ashmolean</p><p>162 David R. Lloyd</p><p>site (footnote 9). The Hunterian Museum and Art Gallery in Glasgow also has a large collec-</p><p>tion, but access to the illustrations requires the museum reference number.</p><p>17. Marshall 1977, Figure 9:3; the object is AS 16 at the Carnegie Inverurie Museum.</p><p>18. Note 9; the curator’s drawing shows this structure rather better than the photograph.</p><p>19. Two views of this item, GLAHM B.1951.245d, are available at http://tinyurl.com</p><p>/yhqhlxj.</p><p>20. Hunterian Museum GLAHM B.1951.112.</p><p>21. Hunterian Museum GLAHM B.1914 349.</p><p>22. Hunterian Museum GLAHM B.1951.876.</p><p>23. There is another illustration of this phenomenon in Critchlow (1979, 147) with a</p><p>description that confirms this symmetry.</p><p>24. As photographed, the six-fold axis is almost vertical and enters through the left-hand</p><p>top knob (see Hunterian Museum GLAHM B.1951.245a).</p><p>References</p><p>Artmann, B., Euclid–the creation of mathematics, Springer, 1999.</p><p>Atiyah, Michael, Collected works, Vol. 6, Oxford University Press, 2004.</p><p>Atiyah, Michael, and Sutcliffe, P., “Polyhedra in physics, chemistry and geometry,” Milan</p><p>Journal of Mathematics, 71 (2003), 33–58, http://arxiv.org/abs/math-ph/0303071.</p><p>Critchlow, Keith, Time stands still: new light on megalithic science, Gordon Fraser, 1979.</p><p>du Sautoy, Marcus, Finding moonshine, Fourth Estate, 2008.</p><p>Edmonds, M. R., “Their use is wholly unknown,” in Vessels for the ancestors, N. Sharples and</p><p>A. Sheridan (eds.), Edinburgh University Press, 1992.</p><p>Erber, T., and Hockney, G. M., “Complex systems: equilibrium configurations of n equal</p><p>charges on a sphere (2</p><p>http://www.neverendingbooks.org/index.php/the-scottish-solids-hoax</p><p>.html. Accessed February 2012.</p><p>Lloyd, D. R., “Symmetry and Beauty in Plato,” Symmetry, 2 (2010), 455–465.</p><p>Marshall, D. N., “Carved stone balls,” Proceedings of the Society of Antiquaries of Scotland, 108</p><p>(1977).</p><p>Pekkonen, O., “Atiyah, M., Collected works, vol. 6” (review), The Mathematical Intelligencer,</p><p>28 (2006), 61–62.</p><p>Waterhouse, W. C., “The discovery of the regular solids,” Archive for the History of Exact Sci-</p><p>ences, 9 (1972), 212–221.</p><p>http://tinyurl.com/yhqhlxj</p><p>http://tinyurl.com/yhqhlxj</p><p>http://www.neverendingbooks.org/index.php/the-scottish-solids-hoax.html</p><p>http://www.neverendingbooks.org/index.php/the-scottish-solids-hoax.html</p><p>Early Modern Mathematical Instruments</p><p>Jim Bennett</p><p>For the purpose of reviewing the history of mathematical instruments</p><p>and the place the subject might command in the history of science, if</p><p>we take “early modern” to cover the period from the 16th to the 18th</p><p>century, a first impression may well be one of a change from vigorous</p><p>development in the 16th century to relatively mundane stability in the</p><p>18th. More careful scrutiny suggests that this perception is relative and</p><p>depends more on our priorities as historians than on the interests of the</p><p>instrument manufacturers or users. In fact, the role of mathematical in-</p><p>struments and the agenda of their designers and makers were fairly steady</p><p>over the period, and what changed was how these might be viewed in</p><p>the broader context of natural philosophy and its instruments. Follow-</p><p>ing the mathematical thread in instrumentation consistently through</p><p>this time has not been a popular option, even for instrument historians</p><p>and even though it has a demonstrable, categorical presence. It is all too</p><p>easy to be diverted by telescopes, microscopes, barometers, air pumps,</p><p>electrical machines, and the like, once these seductive newcomers ar-</p><p>rive on the scene, parading their novel and challenging entanglement</p><p>with natural knowledge. Yet, as a category, “mathematical instruments”</p><p>retained its currency and meaning throughout the period, while com-</p><p>mercial, professional, and bureaucratic engagement with such objects</p><p>continued to expand. It remained more common to want to know the</p><p>time than the barometric pressure, and, despite the growing popularity</p><p>of orreries, there was still more money in octants.</p><p>When historians encounter mathematical instruments, such as as-</p><p>trolabes, sundials, quadrants, surveyors’ theodolites, or gunners’ sights</p><p>and rules, it is generally not in the secondary histories where they first</p><p>learn their trade but, instead, in museums. Coming from book-learned</p><p>history, with its recent material turn in the history of science, and</p><p>164 Jim Bennett</p><p>looking for a material culture from the 16th and 17th centuries can</p><p>be a dispiriting and perplexing experience. Collections rich in mate-</p><p>rial from the period present uncompromising rows of instruments that</p><p>are clearly challenging in their technical content but seem obsessed</p><p>with the wrong kinds of questions. Sundials are plentiful, whereas tele-</p><p>scopes and microscopes are fabulously rare. Horary quadrants and gun-</p><p>ners’ sights present themselves in baffling varieties, but not a single air</p><p>pump seems to survive.</p><p>The dominant instrument culture up to the end of the 17th century</p><p>characterized itself as “mathematical.” By then it incorporated a range</p><p>of applications of mainly geometrical techniques to an array of what had</p><p>become, at least in aspects of their practice, mathematical arts. Astron-</p><p>omy had set the pace for instrumentation. By the 16th century, instru-</p><p>mentation had long been an integral part of its practice—instruments</p><p>for the immediate requirement of positional measurement, leading on</p><p>to those with more particular functionality, such as astrolabes, sundi-</p><p>als, and horary quadrants. The armillary sphere (Figure 1) may well</p><p>have begun as a measuring instrument, but because its arrangement of</p><p>circles reflected those used by astronomers for registering the heavenly</p><p>motions, it could also be used for teaching the practice of astronomy</p><p>and for a limited amount of calculation.</p><p>The armillary sphere perfectly indicates the ambiguity of some</p><p>mathematical instruments, in turn reflecting tensions in the discipline</p><p>of astronomy itself: Are the circles and motions of the instrument in-</p><p>tended primarily to correspond to the heavens or to the geometrical</p><p>practice of astronomers? Many books in the “sphaera” tradition, popu-</p><p>larized in the work of Sacrobosco,1 can be read equally easily with ref-</p><p>erence to the sky or to an artificial instrument, the armillary sphere.</p><p>The emergence of instruments as a prominent instantiation of disciplin-</p><p>ary practice may be more profound than has generally been allowed, to</p><p>be acknowledged alongside instantiation through treatises, terminolo-</p><p>gies, diagrams, constructions, and rules. An important character of</p><p>mathematical instruments is that they face more toward disciplinary</p><p>practice than toward the natural world. That is where they derive their</p><p>regulation and legitimacy, while the discipline in turn is partly charac-</p><p>terized through its engagement with artificial instruments.</p><p>In seeking to characterize the role of instruments in disciplin-</p><p>ary practice, we can look at how they appear in published treatises.</p><p>Mathematical Instruments 165</p><p>Descriptions are found within more general astronomical works, when</p><p>some account is needed of the nature of measurement, and an impor-</p><p>tant example of this would be Ptolemy’s Almagest itself. But even before</p><p>the first printed edition (1515) of Almagest, an independent literature of</p><p>printed treatises on mathematical instruments was launched in 1513 by</p><p>Johann Stoeffler with his Elucidatio fabricae ususque astrolabii.2 The for-</p><p>mula of “construction and use” was adopted for many of the accounts</p><p>of particular mathematical instruments that were published throughout</p><p>the century. Stoeffler first tells his readers how to make an astrolabe,</p><p>then, in a series of worked examples, how to use it. The treatise pres-</p><p>ents itself not in the context of an overarching discipline but as an inde-</p><p>pendent manual for a particular instrument.</p><p>Through the 16th century, many accounts of individual instruments</p><p>were published, alongside treatises on sets or selections of instruments,</p><p>Figure 1. Armillary sphere by Carlo Plato, Rome, 1598. (Museum of the</p><p>History of Science, Oxford, inventory no. 45453.)</p><p>166 Jim Bennett</p><p>whether closely related in type, such as in Sebastian Münster’s books on</p><p>sundials, or covering as broad a range as possible, as in Giovanni Paolo</p><p>Gallucci’s Della fabrica et uso di diversi stromenti di astronomia et cosmographia</p><p>(1597).3 This relative detachment from what might be considered the</p><p>parent or foundational discipline, seen in the publication of books de-</p><p>voted solely to instruments and their use, is worth a moment’s thought.</p><p>Instrument development had a narrative that ran alongside the parent</p><p>discipline but was not dependent on changes or advances in astronomy.</p><p>The originality of astronomers and other mathematicians could be ex-</p><p>ercised through the design and improvement of instruments; a network</p><p>of instrumental relations could spread among separate disciplines (as-</p><p>tronomy, navigation, surveying, architecture, warfare) based on practi-</p><p>cal techniques deployed in instruments in different fields of practice; a</p><p>community of makers promoted the development of these links and of</p><p>instrument design and use as a commercial imperative, while a growing</p><p>number of practitioners did the same in the hope of professional advan-</p><p>tage. In short, instruments had a life that was not entirely dependent on</p><p>an academic mathematical discipline, and this phenomenon is reflected</p><p>in the range of 16th century publications.</p><p>At the end of the century, the rise of the independent treatise on</p><p>one or more mathematical instruments reflects back into astronomical</p><p>measurement itself through the</p><p>publication of Tycho Brahe’s Astronomiae</p><p>instauratae mechanica of 1598.4 This detailed account of his observatory</p><p>and its individual instruments, set out one after another in a thorough</p><p>and comprehensive manner, established an influential precedent for</p><p>ambitious observatory astronomers; later examples in our period in-</p><p>clude Johannes Hevelius, John Flamsteed, and Ole Roemer.</p><p>Before moving the narrative into the 17th century, and as a narrative</p><p>thread to lead us there, we might briefly consider the subject of dialing</p><p>and the division of instruments—sundials and horary quadrants—that</p><p>characterized the subject. If there is a mismatch between the interests</p><p>of early modernists and those of their actors with regard to mathemati-</p><p>cal instruments generally, the disparity is even greater when it comes</p><p>to dialing. Not only might the problem of finding time from the sun,</p><p>mostly with portable rather than fixed instruments, be solved in a great</p><p>variety of imaginative and technically challenging ways, be applicable</p><p>to different conventions of dividing the day or night and numbering the</p><p>hours, be made relevant to the whole Earth instead of a single latitude,</p><p>Mathematical Instruments 167</p><p>and so on; but also, the ambitions of such instruments could extend far</p><p>beyond telling the time “here and now”— or even “there and then.”</p><p>Their functionality could extend outside time telling into other areas</p><p>of astronomical problem solving.</p><p>In the 16th century, dialing was a vibrant, challenging, even com-</p><p>petitive astronomical and geometrical discipline, with a large following,</p><p>many new designs of instruments, and a correspondingly healthy output</p><p>of publications. It was related closely to the popular contemporary dis-</p><p>cipline of cosmography—the mathematical, or part-mathematical, dis-</p><p>cipline dealing with the geometrical relationships between the heavens</p><p>and the Earth. Leading mathematicians, such as Regiomontanus, Peter</p><p>Apianus, Gemma Frisius, and Oronce Fine, were centrally involved,</p><p>seeing dialing as an integral part of their disciplinary practice.5 Yet dial-</p><p>ing is now all but ignored by historians of science. Extending our appre-</p><p>ciation to this aspect of contemporary astronomy broadens our grasp of</p><p>its scope and methodology. Planetary theory is important, of course, but</p><p>to achieve a rounded account of the discipline we cannot afford to ne-</p><p>glect an aspect that was clearly significant for many of its practitioners.</p><p>Following the development of dials and quadrants takes us seamlessly</p><p>into the 17th century and the rise in importance of English mathemati-</p><p>cians and makers. The institutional mission of Gresham College in Lon-</p><p>don, with its recognition of the practical mathematical arts and sciences</p><p>as a subject for the kind of systematic treatment that might underpin a</p><p>series of weekly lectures, was important for bringing the English into a</p><p>European movement—one that combined learning, technical innova-</p><p>tion, practical application, publication, manufacture, and commerce.</p><p>The professor of astronomy from 1619 to 1626, Edmund Gunter, had a</p><p>particular commitment to, and success with, instrumentation, and his</p><p>eponymous quadrant (Figure 2) established itself as a standard portable</p><p>astronomical instrument through the 17th and 18th centuries.6 Charac-</p><p>teristic of the sundial’s functionality being extended beyond time tell-</p><p>ing, by restricting his projection of hour lines to a particular latitude,</p><p>Gunter was able to include other lines, such as the horizon, the ecliptic,</p><p>and lines of solar azimuth, so that his quadrant could be applied to</p><p>a range of astronomical calculations. Gunter worked closely with the</p><p>instrument maker Elias Allen, whose shop near St. Clement’s Church7</p><p>in the Strand was a place of resort and exchange for many interested in</p><p>practical mathematics.</p><p>168 Jim Bennett</p><p>Another of Allen’s mathematical patrons was William Oughtred,</p><p>whose design of the universal equinoctial ring dial, like the Gunter</p><p>quadrant, was popular through the 18th century. Heir to the dialing</p><p>literature from 16th century continental mathematicians, Oughtred’s</p><p>design was developed from the astronomer’s rings of Gemma Frisius.</p><p>A design of his own was his “horizontal instrument,” a dial based on a</p><p>projection onto the horizon of the sun’s daily paths throughout the year,</p><p>the ecliptic, and the equator; like Gunter’s quadrant, it had a range</p><p>of astronomical functionality. There were fixed and portable versions,</p><p>and the latter were combined with one of the most important achieve-</p><p>ments of the new English school of mathematical instrumentation, the</p><p>delivery of logarithmic calculation as a practical technique by means</p><p>of an instrument. The close links between publication and instru-</p><p>ment manufacture are maintained: The horizontal instrument and the</p><p>“circles of proportion” (Oughtred’s circular logarithmic rule) occupied</p><p>either side of a portable instrument (that could be bought in the shop</p><p>of Elias Allen) and appeared together in Oughtred’s book of 1632 (pub-</p><p>lished and sold by Allen).8</p><p>Figure 2. Gunter quadrant by Elias Allen, circa 1630. Courtesy of Whipple</p><p>Museum of the History of Science, Wh. 1764.</p><p>Mathematical Instruments 169</p><p>The development of specialized or professionalized calculation</p><p>around this time is an instructive episode for understanding the nature</p><p>of mathematical instruments. Imperatives from the world beyond math-</p><p>ematics—in warfare and navigation—were creating opportunities for</p><p>ambitious mathematicians, adaptable practitioners, and entrepreneurial</p><p>makers. The kinds of proportional calculations required of the gunner</p><p>in working out the weight of shot or the appropriate charge of pow-</p><p>der were amenable to geometrical handling using similar triangles and</p><p>could readily be rendered instrumental by a “sector” (or “compass of</p><p>proportion”), where a pair of scales on the faces of flat rules connected</p><p>by a compass joint could be opened to different angles, according to the</p><p>proportionality required. Galileo’s “geometrical and military compass”</p><p>is only the best known of a range of such instruments, where combina-</p><p>tions of different pairs of lines might handle not just the direct propor-</p><p>tions needed, say, by surveyors drawing plans to scale, but also lines</p><p>for areas (useful again to surveyors as well as carpenters) and volumes</p><p>(used by, for example, masons, architects, and gunners).9</p><p>The sector embodied a very adaptable technique, and, after the</p><p>mathematics of the Mercator chart for sailing had been explained by</p><p>another mathematician in the Gresham circle, Edward Wright, Gunter</p><p>designed a specialized version for managing the kind of trigonometrical</p><p>calculations required when working with a chart where the scale varied</p><p>as the secant of the latitude. Gunter went on to devise a rule with lines</p><p>carrying logarithmic and trigonometric functions so that multiplica-</p><p>tions and divisions could be carried out by the straightforward addition</p><p>or subtraction of lengths transferred by means of a pair of dividers. Two</p><p>such rules (dispensing with the dividers) created the once-familiar loga-</p><p>rithmic slide rule. Logarithms had only recently been invented by John</p><p>Napier, and their rapid application in Gunter’s rule and Oughtred’s cir-</p><p>cles indicates the continuing significance of the role played by instru-</p><p>ments in contemporary mathematics.</p><p>The practical discipline that had introduced such technical novel-</p><p>ties as these maintained a strong commercial presence throughout the</p><p>17th century, now with the addition of English workshops to those</p><p>of continental Europe. Indeed, as latecomers to the field, the English</p><p>(which meant largely London) makers seemed particularly active. It is</p><p>worth stressing that mathematical instrument makers were specialist</p><p>craftsmen who formed an identifiable and readily understood trade.</p><p>170 Jim Bennett</p><p>In some continental cities there was a measure of guild regulation,</p><p>but in London, although the makers had to belong to some London</p><p>company, it</p><p>did not matter which, and there was no nominated home</p><p>for these mathematical workers. So, they are found in the lists of the</p><p>Grocers, the Drapers, and many others. In the 17th century, the arts</p><p>of war became somewhat less evident in the literature and surveying</p><p>somewhat more so; the two fields are linked in the technique of tri-</p><p>angulation and the instruments designed to facilitate it. In surveying,</p><p>these methods are used for drawing maps; in a military context, they</p><p>are used in range finders.</p><p>Mathematical instrument makers did not become involved in the pro-</p><p>duction of telescopes or microscopes. Makers of optical instruments, if</p><p>they were not astronomers themselves turning their hands to practical</p><p>work, emerged from among the most skilled and ambitious of the spec-</p><p>tacle makers, while mathematical instrument makers continued their</p><p>independent and customary trade, applying their engraving skills to</p><p>practical ends in the mathematical arts. They were not concerned with</p><p>discoveries in the far away or the very small. But there was one area of</p><p>practical optics that came to impinge on their world—namely, in mea-</p><p>suring instruments for astronomy, where telescopic sights were applied</p><p>to divided instruments to increase the accuracy of the alignment of</p><p>the index on the distant target. In a superficial sense, this development</p><p>witnessed a connection between the two separate areas of work: An</p><p>optical instrument was combined with an astronomical measuring in-</p><p>strument, and the division between optical and mathematical had been</p><p>breached. But the added telescope was intended simply to improve the</p><p>accuracy of the measurement; it did not alter the fundamental function</p><p>of the instrument. The same conjunctions were made in surveying and</p><p>navigation, with the addition of telescopic sights to designs of theodo-</p><p>lite and sextant, but more effectively in the 18th century.</p><p>Retailing (if not manufacturing) across the division between math-</p><p>ematical and optical instruments came in a more profound way toward</p><p>the end of the 17th century; early instances were found in the practices</p><p>of the London makers Edmund Culpeper and John Rowley. It was here</p><p>that the relatively loose regulation of the city companies in control-</p><p>ling manufacturing boundaries was a significant advantage over more</p><p>restrictive regimes. So the 18th century opened with a commercial if</p><p>not a conceptual link between the two areas of instrumentation, and,</p><p>Mathematical Instruments 171</p><p>as the popularity of experimental natural philosophy grew, a further</p><p>commercial opportunity arose through the potential for a trade in the</p><p>apparatus of natural philosophy, such as air pumps, electrical machines,</p><p>and all the burgeoning content of the “cabinet of physics.”</p><p>Much of the staple fare of the mathematical instrument maker re-</p><p>mained in his repertoire through the 18th century, while again new</p><p>designs were added. Sundials and quadrants were as popular as ever—</p><p>before being tempted to add “despite increasing numbers of pocket</p><p>watches,” remember that watches had to be set to time. Though with</p><p>roots in designs by Hooke and Newton, angular measurement by the</p><p>principle of reflection now found original and successful applications in</p><p>octants, sextants, and circles, used in navigation and surveying, while</p><p>more ambitious theodolites, perhaps adding an altitude arc or circle and</p><p>a telescope to the azimuth function (Figure 3), made a significant im-</p><p>pression on the practice of surveyors. Professional mathematical prac-</p><p>titioners such as these came to be associated even more than previously</p><p>with their instrumentation, while the numbers of mathematical navi-</p><p>gators and surveyors continued to grow. An informal professionaliza-</p><p>tion was underway, not regulated by institutes but encouraged through</p><p>academies, more often private than public, through textbooks, and</p><p>through the efforts of instrument makers.</p><p>There are three general reasons why the trade in mathematical in-</p><p>struments comes back from the margins of vision and into the range</p><p>of historians of science. One reason arises from the entrepreneurial</p><p>efforts of a number of makers to deal across all classes of instrument:</p><p>in mathematics, optics, and natural philosophy. Prominent examples</p><p>in London would be George Adams and Benjamin Martin—neither</p><p>primarily identified with mathematical instruments but both including</p><p>such instruments in their comprehensive range. They were also promi-</p><p>nent in the rise of public, commercial natural philosophy and the grow-</p><p>ing fashion for attending lectures and buying books and instruments.</p><p>This popularity drew octants and theodolites into the same commercial</p><p>project as orreries and microscopes.</p><p>Second, the leading mathematical instrument makers became re-</p><p>sponsible for building the major instruments in the growing number</p><p>of astronomical observatories, whose chief concern was astronomi-</p><p>cal measurement. It was in the 18th century that it first became nor-</p><p>mal to turn to commercial makers with commissions for observatory</p><p>172 Jim Bennett</p><p>instruments. The recipients of such commissions were, of necessity,</p><p>the leading mathematical makers, and, for reasons of status if noth-</p><p>ing else, they retained their “mathematical” identity. Status was par-</p><p>ticularly evident in London, where these elite makers were patronized</p><p>by the Board of Longitude and the Royal Observatory and could be-</p><p>come Fellows of the Royal Society and be published in the Philosophical</p><p>Transactions. Toward the end of the century, mathematical instrument</p><p>makers such as Jesse Ramsden and Edward Troughton came to occupy</p><p>Figure 3. Theodolite by George Adams, London, late 18th century.</p><p>( Museum of the History of Science, Oxford, inventory no. 71425.)</p><p>Mathematical Instruments 173</p><p>positions of such respect and influence that they were bound to find</p><p>their places in the history of the period’s science, yet there are sectors</p><p>signed “Ramsden” and sundials by “Troughton.”10</p><p>Third, the growing intellectual investment in accuracy of observa-</p><p>tion and measurement in natural investigation could be realized only</p><p>through the mechanical knowledge and skill that had built up through</p><p>centuries and now lay in the hands and workshops of mathematical in-</p><p>strument makers.</p><p>Although a greater preoccupation with optical and experimental in-</p><p>struments is understandable, historians should not neglect the math-</p><p>ematical. Without understanding that category, and the categorization</p><p>of instrument making as a whole in the period, we cannot properly ap-</p><p>preciate other, more compelling, aspects of the instrument narrative.</p><p>Mathematical instruments have a much longer history than optical and</p><p>natural philosophical instruments, and although their established exis-</p><p>tence contributes to the possibility of instruments with other ambitions,</p><p>the later types pursue separate developments, in different workshops</p><p>and areas of practice, fully merging only later in post hoc commer-</p><p>cial convenience. It is from this commercial union that the “scientific</p><p>instrument” emerged. Furthermore, it is the practical mathematical</p><p>work that engages with the worlds of commerce, bureaucracy, war,</p><p>and empire, all of which speak to the breadth of the history of science</p><p>as practiced today. Finally, if we want to bring the likes of Ramsden and</p><p>Troughton into our mainstream narrative, as we surely must given the</p><p>importance ascribed to them by their contemporaries, that can be done</p><p>only with an appreciation of the disciplinary tradition they recognized</p><p>and through which they understood and defined their own work—that</p><p>of the mathematical instrument.</p><p>Notes</p><p>1. Johannes de Sacrobosco wrote his basic textbook on astronomy in c. 1230, and it was</p><p>known from manuscript and printed versions (variously titled De sphaera mundi, or Tractatus de</p><p>sphaera, or simply De sphaera) for some four centuries.</p><p>2. Johann Stoeffler, Elucidatio fabricae ususque astrolabii (Oppenheim, 1513).</p><p>3. Sebastian Münster, Compositio horologiorum (Basel, 1531);</p><p>Münster, Horologiographia</p><p>(Basel, 1533); and Giovanni Paolo Gallucci, Della fabrica et uso di diversi stromenti di astronomia</p><p>et cosmographia (Venice,1597).</p><p>4. Tycho Brahe, Astronomiae instauratae mechanica (Wandsbeck, 1598).</p><p>174 Jim Bennett</p><p>5. Jim Bennett, “Sundials and the Rise and Decline of Cosmography in the Long 16th</p><p>Century,” Bulletin of the Scientific Instrument Society, 2009, no. 101, pp. 4–9. More generally,</p><p>see Hester Higton, Sundials at Greenwich (Oxford: Oxford Univ. Press, 2002).</p><p>6. Edmund Gunter, The Description and Use of the Sector, the Crosse-Staffe, and Other Instru-</p><p>ments (London,1624).</p><p>7. St. Clement’s Church was the name commonly used in the 17th century for the church</p><p>now known as St. Clement Danes.</p><p>8. William Oughtred, The Circles of Proportion and the Horizontall Instrument (London,</p><p>1632).</p><p>9. Filippo Camerota, Il compasso di Fabrizio Mordente, per la storia del compasso di proporzione</p><p>(Florence: Olschki, 2000).</p><p>10. Anita McConnell, Jesse Ramsden (1735–1800): London’s Leading Scientific Instrument</p><p>Maker (Aldershot: Ashgate, 2007).</p><p>A Revolution in Mathematics?</p><p>What Really Happened a Century Ago</p><p>and Why It Matters Today</p><p>Frank Quinn</p><p>The physical sciences all went through ‘‘revolutions”: wrenching tran-</p><p>sitions in which methods changed radically and became much more</p><p>powerful. It is not widely realized, but there was a similar transition</p><p>in mathematics between about 1890 and 1930. The first section briefly</p><p>describes the changes that took place and why they qualify as a “revolu-</p><p>tion,” and the second section describes turmoil and resistance to the</p><p>changes at the time.</p><p>The mathematical event was different from those in science, how-</p><p>ever. In science, most of the older material was wrong and discarded,</p><p>whereas old mathematics needed precision upgrades but was mostly</p><p>correct. The sciences were completely transformed while mathemat-</p><p>ics split, with the core changing profoundly but many applied areas,</p><p>and mathematical science outside the core, relatively unchanged. The</p><p>strangest difference is that the scientific revolutions were highly vis-</p><p>ible, whereas the significance of the mathematical event is essentially</p><p>unrecognized. The section “Obscurity” explores factors contributing</p><p>to this situation and suggests historical turning points that might have</p><p>changed it.</p><p>The main point of this article is not that a revolution occurred, but</p><p>that there are penalties for not being aware of it. First, precollege math-</p><p>ematics education is still based on 19th century methodology, and it</p><p>seems to me that we will not get satisfactory outcomes until this ap-</p><p>proach changes [9]. Second, the mathematical community is adapted</p><p>to the social and intellectual environment of the mid- and late 20th</p><p>century, and this environment is changing in ways likely to marginalize</p><p>176 Frank Quinn</p><p>core mathematics. But core mathematics provides the skeleton that</p><p>supports the muscles and sinews of science and technology; margin-</p><p>alization will lead to a scientific analogue of osteoporosis. Deliberate</p><p>management [2] might avoid this, but only if the disease is recognized.</p><p>The Revolution</p><p>This section describes the changes that took place in 1890–1930, draw-</p><p>backs, objections, and why the change remains almost invisible. In spite</p><p>of the resistance, it was incredibly successful. Young mathematicians</p><p>voted with their feet, and, over the strong objections of some of the</p><p>old guard, most of the community switched within a few generations.</p><p>Contemporary Core Methodology</p><p>To a first approximation, the method of science is “find an explanation</p><p>and test it thoroughly,” whereas modern core mathematics is “find an</p><p>explanation without rule violations.” The criteria for validity are radi-</p><p>cally different: Science depends on comparison with external reality,</p><p>whereas mathematics is internal.</p><p>The conventional wisdom is that mathematics has always depended</p><p>on error-free logical argument, but this is not completely true. It is quite</p><p>easy to make mistakes with infinitesimals, infinite series, continuity,</p><p>differentiability, and so forth, and even possible to get erroneous con-</p><p>clusions about triangles in Euclidean geometry. When intuitive formula-</p><p>tions are used, there are no reliable rule-based ways to see that these are</p><p>wrong, so in practice, ambiguity and mistakes used to be resolved with</p><p>external criteria, including testing against accepted conclusions, feed-</p><p>back from authorities, and comparison with physical reality. In other</p><p>words, before the transition, mathematics was to some degree scientific.</p><p>The breakthrough was the development of a system of rules and pro-</p><p>cedures that really worked, in the sense that, if they are followed care-</p><p>fully, then arguments without rule violations give completely reliable</p><p>conclusions. It became possible, for instance, to see that some intui-</p><p>tively outrageous things are nonetheless true. Weierstrass’s nowhere-</p><p>differentiable function (1872) and Peano’s horrifying space-filling</p><p>curve (1890) were early examples, and we have seen much stranger</p><p>things since. There is no abstract reason (i.e., apparently no proof) that</p><p>A Revolution in Mathematics? 177</p><p>such a useful system of rules exists and no assurance that we would find</p><p>it. However, it does exist and, after thousands of years of tinkering and</p><p>under intense pressure from the sciences for substantial progress, we</p><p>did find it. Major components of the new methods are the following:</p><p>precise definitions: Old definitions usually described what</p><p>things are supposed to be and what they mean, and extraction of</p><p>properties relied to some degree on intuition and physical experi-</p><p>ence. Modern definitions are completely self-contained, and the</p><p>only properties that can be ascribed to an object are those that can</p><p>be rigorously deduced from the definition.</p><p>logically complete proofs: Old proofs could include appeals</p><p>to physical intuition (e.g., about continuity and real numbers),</p><p>authority (e.g., “Euler did this, so it must be okay”), and casual</p><p>establishment of alternatives (“these must be all the possibilities</p><p>because I can’t imagine any others”). Modern proofs require each</p><p>step to be carefully justified.</p><p>Definitions that are modern in this sense were developed in the late</p><p>1800s. It took a while to learn to use them: to see how to pack wis-</p><p>dom and experience into a list of axioms, how to fine-tune them to</p><p>optimize their properties, and how to see opportunities where a new</p><p>definition might organize a body of material. Well-optimized modern</p><p>definitions have unexpected advantages. They give access to material</p><p>that is not (as far as we know) reflected in the physical world. A really</p><p>“good” definition often has logical consequences that are unanticipated</p><p>or counterintuitive. A great deal of modern mathematics is built on</p><p>these unexpected bonuses, but they would have been rejected in the</p><p>old, more scientific approach. Finally, modern definitions are more ac-</p><p>cessible to new users. Intuitions can be developed by working directly</p><p>with definitions, and this method is faster and more reliable than trying</p><p>to contrive a link to physical experience.</p><p>Logically complete proofs were developed by Frege and others be-</p><p>ginning in the 1880s, and by Hilbert after 1890 and (it seems to me)</p><p>rounded out by Gödel around 1930. Again it took a while to learn to</p><p>use these: The “official” description as a sequence of statements ob-</p><p>tained by logical operations, and so forth, is cumbersome and opaque,</p><p>but ways were developed to compress and streamline proofs without</p><p>178 Frank Quinn</p><p>losing reliability. It is hard to describe precisely what is acceptable as</p><p>a modern proof because the key criterion, “without losing reliability,”</p><p>depends heavily on background and experience. It is clearer and per-</p><p>haps more important what is not acceptable: no appeals to authority or</p><p>physical intuition, no “proof by example,” and no leaps of faith, no mat-</p><p>ter how reasonable they might seem. As with definitions,</p><p>the Mathematics StackExchange (http://math.stackexchange.com/).</p><p>The Parameterized Complexity (http://fpt.wikidot.com/) page offers</p><p>a useful forum of information and instruction for those interested in</p><p>mathematical connections to computing, psychology, and cognitive</p><p>sciences. A brief history of mathematics is available on the Story of</p><p>Mathematics (http://www.storyofmathematics.com/index.html) site.</p><p>Many institutions host a wealth of materials on their internet sites; I</p><p>mention here the Harvey Mudd College (http://www.math.hmc.edu</p><p>/funfacts/), the Clay Mathematics Institute (http://www.claymath.org</p><p>/index.php), and the Cornell Mathematics Library (http://mathematics</p><p>.library.cornell.edu/).</p><p>I hope you, the reader, find the same value and excitement in reading</p><p>the texts in this volume as I found while searching, reading, and se-</p><p>lecting them. For comments on this book and to suggest materials for</p><p>consideration in preparing future volumes, I encourage you to send cor-</p><p>respondence to me at Mircea Pitici, P.O. Box 4671, Ithaca, NY 14852.</p><p>Works Mentioned</p><p>Ash, Avner, and Robert Gross. Elliptic Tales: Curves, Counting, and Number Theory. Princeton,</p><p>NJ: Princeton University Press, 2012.</p><p>Barnsley, Michael F. Fractals Everywhere. Mineola, NY: Dover, 2012.</p><p>Belot, Gordon. Geometric Possibility. Oxford, UK: Oxford University Press, 2011.</p><p>Bennett, Jeffrey. Math for Life: Crucial Ideas You Didn’t Learn in School. Greenwood Village, CO:</p><p>Roberts & Co. Publishers, 2012.</p><p>Berlinski, David. The King of Infinite Space: Euclid and His Elements. New York: Basic Books, 2012.</p><p>Best, Joel. Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists,</p><p>updated edition. Berkeley, CA: University of California Press, 2012.</p><p>Bissell, Chris, and Chris Dillon. (Eds.) Ways of Thinking, Ways of Seeing: Mathematical and Other</p><p>Modeling in Engineering and Technology. Heidelberg, Germany: Springer Verlag, 2012.</p><p>Bonacich, Phillip, and Philip Lu. Introduction to Mathematical Biology. Princeton, NJ: Princeton</p><p>University Press, 2012.</p><p>Brinkhuis, Jan, and Vladimir Tikhomirov. Optimization: Insights and Applications. Princeton,</p><p>NJ: Princeton University Press, 2012.</p><p>Brown, Richard C. The Tangled Origins of Leibnizian Calculus: A Case Study of Mathematical Revo-</p><p>lution. Singapore: World Scientific, 2012.</p><p>Bruter, Claude. (Ed.) Mathematics and Modern Art. New York: Springer-Verlag, 2012.</p><p>Carrier, Richard C. Proving History: Bayes’s Theorem and the Quest for the Historical Jesus. Am-</p><p>herst, MA: Prometheus Books, 2012.</p><p>xxiv Introduction</p><p>Cellucci, Carlo, Emily Grosholz, and Emiliano Ippoloti. (Eds.) Logic and Knowledge. New-</p><p>castle upon Tyne, UK: Cambridge Scholars Publishing, 2011.</p><p>Chaitin, Gregory. Proving Darwin: Making Biology Mathematical. New York: Pantheon, 2012.</p><p>Chaitin, Gregory, Newton da Costa, and Francisco Antonio Doria. Gödel’s Way: Exploits into</p><p>an Undecidable World. Leiden, Netherlands: CRC Press, 2012.</p><p>Chemla, Karine. (Ed.) The History of Mathematical Proof in Ancient Traditions. Cambridge, UK:</p><p>Cambridge University Press, 2012.</p><p>Clark, Michael. Paradoxes from A to Z, 3rd ed. New York: Routledge, 2012.</p><p>Cohn, Victor, and Lewis Cope. News & Numbers: A Writer’s Guide to Statistics, 3rd ed. Oxford,</p><p>UK: Wiley-Blackwell, 2012.</p><p>Devlin, Keith. Introduction to Mathematical Thinking. Stanford, CA: Keith Devlin, 2012.</p><p>Doxiadis, Apostolos, and Barry Mazur. (Eds.) Circles Disturbed: The Interplay of Mathematics and</p><p>Narrative. Princeton, NJ: Princeton University Press, 2012.</p><p>Forgasz, Helen, and Ferdinand Rivera. (Eds.) Towards Equity in Mathematics Education: Gender,</p><p>Culture, and Diversity. Heidelberg, Germany: Springer Verlag, 2012.</p><p>Frappier, Mélanie, Derek H. Brown, and Robert DiSalle. (Eds) Analysis and Interpretation in</p><p>the Exact Sciences. Heidelberg, Germany: Springer Verlag, 2012.</p><p>Gray, Jeremy. Henri Poincaré: A Scientific Biography. Princeton, NJ: Princeton University Press,</p><p>2013.</p><p>Hahn, Alexander J. Mathematical Excursions to the World’s Great Buildings. Princeton, NJ:</p><p>Princeton University Press, 2012.</p><p>Harris, James. Fractal Architecture: Organic Design Philosophy in Theory and Practice. Albuquer-</p><p>que, NM: University of New Mexico Press, 2012.</p><p>Havil, Julian. The Irrationals. Princeton, NJ: Princeton University Press, 2012.</p><p>Healey, Joseph F. The Essentials of Statistics: A Tool for Social Research. Belmont, CA: Wads-</p><p>worth, 2013.</p><p>Hermann, Norbert. The Beauty of Everyday Mathematics. Heidelberg, Germany: Springer Verlag,</p><p>2012.</p><p>Horn, Ilana Seidel. Strength in Numbers: Collaborative Learning in Secondary Mathematics. Reston,</p><p>VA: The National Council of Teachers of Mathematics, 2012.</p><p>Janiak, Andrew, and Eric Schliesser. (Eds.) Interpreting Newton: Critical Essays. Cambridge,</p><p>UK: Cambridge University Press, 2012.</p><p>Kagan, Shelly. The Geometry of Desert. Oxford, UK: Oxford University Press, 2012.</p><p>Kasten, Sarah, and Jill Newton. (Eds.) Reasoning and Sense-Making Activities for High School</p><p>Mathematics. Reston, VA: The National Council of Teachers of Mathematics, 2012.</p><p>Kitcher, Philip. Preludes to Pragmatism: Toward a Reconstruction of Philosophy. Oxford, UK: Ox-</p><p>ford University Press, 2012.</p><p>Kleiner, Israel. Excursions in the History of Mathematics. Heidelberg, Germany: Springer Verlag,</p><p>2012.</p><p>Livingston, Paul M. The Politics of Logic: Badiou, Wittgenstein, and the Consequences of Formalism.</p><p>New York: Routledge, 2012.</p><p>Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations.</p><p>Princeton, NJ: Princeton University Press, 2012.</p><p>Martínez, Alberto A. The Cult of Pythagoras: Math and Myth. Pittsburg, PA: Pittsburg Univer-</p><p>sity Press, 2012.</p><p>The Mathematical Education of Teachers, II. Providence, RI: American Mathematical Society,</p><p>2012.</p><p>McKellar, Danica. Girls Get Curves: Geometry Takes Shape. New York: Hudson Street Press, 2012.</p><p>Introduction xxv</p><p>Ostermann, Alexander. Geometry by Its History. Heidelberg, Germany: Springer Verlag,</p><p>2012.</p><p>Parsons, Charles, and Montgomery Link. (Eds.) Hao Wang: Logician and Philosopher. London,</p><p>UK: College Publications, 2011.</p><p>Pask, Colin. Math for the Frightened: Facing Scary Symbols and Everything Else That Freaks You Out</p><p>about Mathematics. Amherst, NY: Prometheus Books, 2012.</p><p>Perkins, David. Calculus and Its Origins. Washington, DC: Mathematical Association of</p><p>America, 2012.</p><p>Pincock, Christopher. Mathematics and Scientific Representation. Oxford, UK: Oxford Univer-</p><p>sity Press, 2012.</p><p>Polster, Burkard, and Marty Ross. Mathematics Goes to the Movies. Baltimore, MD: The Johns</p><p>Hopkins University Press, 2012.</p><p>Putnam, Hilary. Philosophy in an Age of Science: Physics, Mathematics, and Skepticism. Cambridge,</p><p>MA: Harvard University Press, 2012.</p><p>Rendgen, Sandra, and Julius Wiedemann. Information Graphics. Cologne, Germany: Taschen,</p><p>2012.</p><p>Roselló, Joan. From Foundations to Philosophy of Mathematics: An Historical Account of their Develop-</p><p>ment in XX Century and Beyond. Cambridge, UK: Cambridge University Press, 2012.</p><p>Saxe, Geoffrey B. Cultural Development of Mathematical Ideas: Papua New Guinea Studies. New</p><p>York: Cambridge University Press, 2012.</p><p>Sinclair, Nathalie, David Pimm, and Melanie Skelin. Developing Essential Understanding of Ge-</p><p>ometry for Teaching Mathematics in Grades 6–8. Reston, VA: The National Council of Teach-</p><p>ers of Mathematics, 2012.</p><p>Sinclair, Nathalie, David Pimm, and Melanie Skelin. Developing Essential Understanding of Ge-</p><p>ometry for Teaching Mathematics in Grades 9–12. Reston, VA: The National Council of Teach-</p><p>ers of Mathematics, 2012.</p><p>Stewart, Ian. In Pursuit of the Unknown: 17 Equations that Changed the World. New York: Basic</p><p>Books, 2012.</p><p>Strogatz, Steven. The Joy of X: A Guided Tour of Math, from One to Infinity. Boston: Houghton</p><p>Mifflin Harcourt, 2012.</p><p>Sujatha, R., H. M. Ramaswamy, and C. S. Yogananda. (Eds.) Math Unlimited: Essays in Math-</p><p>ematics. St. Helier, British Channel Islands: Science Publishers, 2012.</p><p>Swetz, Frank J. Mathematical Expeditions: Exploring</p><p>this approach</p><p>has unexpected advantages. Trying to fix gaps in first approximations</p><p>to proofs can lead to conclusions we could not have imagined and would</p><p>not have dared conjecture. They also make research more accessible:</p><p>Rank-and-file mathematicians can use the new methods confidently</p><p>and effectively, whereas success with older methods was mostly limited</p><p>to the elite.</p><p>Drawbacks</p><p>As mathematical practice became better adapted to the subject, it lost</p><p>features that were important to many people.</p><p>The new methodology is less accessible to nonusers. Old-style defi-</p><p>nitions, for instance, usually related things to physical experience, so</p><p>many people could connect with them in some way. Users found these</p><p>connections dysfunctional, and they can derive effective intuition much</p><p>faster from precise definitions. But modern definitions have to be used</p><p>to be understood, so they are opaque to nonusers. The drawback here</p><p>is that nonusers only saw a loss: The old dysfunctionality was invisible,</p><p>whereas the new opacity is obvious.</p><p>The new methodology is less connected to physical reality. For</p><p>example, nothing in the physical world can be described with com-</p><p>plete precision, so completely rule-based reasoning is not appropriate.</p><p>In fact, the history of science is replete with embarrassing blunders</p><p>caused by excessively deductive reasoning; see the section “Hilbert’s</p><p>Missed Opportunities” for context and illustrations. Professional prac-</p><p>tice now accommodates this problem through the use of mathematical</p><p>models: Mathematics applies to the model but no longer even pretends</p><p>to say anything about the fit between model and reality. The earlier</p><p>connection to reality may have been an illusion, but people saw it as a</p><p>drawback that had to be abandoned. In the other direction, core math-</p><p>ematics no longer accepts externally verified (experimental) results as</p><p>“known” because this method would bring with it the same limitations</p><p>A Revolution in Mathematics? 179</p><p>on deductive reasoning that are necessary in science. Even the most</p><p>seemingly minor flaw sooner or later causes proof by contradiction and</p><p>similar methods to collapse.</p><p>In practice, this issue led to a division into “core” mathematics and</p><p>“mathematical science.” For instance, if numerical approximations of</p><p>fluid flow seem to reproduce experimental observations, then this</p><p>could be taken as evidence that the approximation scheme converges.</p><p>This conclusion does not have the certainty of modern proof and cannot</p><p>be accepted as “known” in the core sense. However, it is a reasonable</p><p>scientific conclusion and appropriate for mathematical science. Similarly,</p><p>the Riemann hypothesis is incredibly well tested. For scientific pur-</p><p>poses it is a solid fact, but it is unproved and remains dangerous for core</p><p>use. Another view of this development is that, as mathematical meth-</p><p>ods diverged from those of science, mathematics divided into a core</p><p>branch that separated from physical science in order to exploit these</p><p>methods and a mathematical science branch that accepted the limita-</p><p>tions in order to remain connected. The drawback here is that the new</p><p>power in the core and the support it gives to applied areas are invisible</p><p>to outsiders, whereas the separation from science is obvious. People</p><p>wonder: Is core mathematics a pointless academic exercise and math-</p><p>ematical science the real thing?</p><p>Opposition</p><p>Henri Poincaré was the most visible and articulate opponent of the new</p><p>methods; cf. [6]. He felt that Dedekind’s derivation of the real numbers</p><p>from the integers was a particularly grievous conceptual error because</p><p>it damaged connections to reality and intuitive understanding of con-</p><p>tinuity. Some of the arguments were quite heated; the graphic novel</p><p>Logicomix [1] dramatically illustrates the turmoil (though it muddles the</p><p>issues a bit). Scholarly works [3] are more dignified but give the same</p><p>picture.</p><p>As the transition progressed, the arguments became more heated</p><p>but more confined. At the beginning, traditionalists were deeply of-</p><p>fended but not threatened. But because modern methods lack external</p><p>checks, they depend heavily on fully reliable inputs. Older material</p><p>was filtered to support this, and as the transition gained momentum,</p><p>some old theorems were reclassified as “unproved,” some methods</p><p>180 Frank Quinn</p><p>became unacceptable for publication, and quite a few ways of look-</p><p>ing at things were rejected as dangerously imprecise. Understandably,</p><p>many eminent late 19th century mathematicians were outraged by</p><p>these reassessments. These battles were fought by proxy, however. For</p><p>instance, Poincaré’s monumental development of the theory of mani-</p><p>folds was quite intuitive, and we now know that some of his basic in-</p><p>tuitions were wrong. But, in the early 20th century, only a fool would</p><p>have openly criticized Poincaré, and he could not respond to implicit</p><p>reproaches. As a result, the arguments usually concerned abstrac-</p><p>tions, such as “creativity” and “understanding,” often in the context</p><p>of education.</p><p>On a more general level, scientific concerns about the new methods</p><p>were reasonable. The crucial importance of external reality checks in</p><p>physics had been a hard-won lesson, and analogous revolutions in bi-</p><p>ology and chemistry were still in progress (Darwin’s Origin of Species</p><p>appeared in 1859, and Mendeleev’s periodic table in 1869). How could</p><p>mathematical use of the discredited “pure reason” approach possibly be</p><p>a good thing?</p><p>Most of the various schools of philosophy were, and remain, uncon-</p><p>vinced by the new methods. Philosophers controlled words such as “re-</p><p>ality,” “knowledge,” “infinite,” “meaning,” “truth,” and even “number,”</p><p>and these were interpreted in ways unfriendly to the new mathematics.</p><p>For example, if a mathematical idea is not clearly manifested in the</p><p>physical world, how can it be “real”? And if it is not real, how can it have</p><p>“meaning,” and how can it make sense to claim to “know” something</p><p>about it? In practice, mathematicians do find that their world has mean-</p><p>ing and at least a psychological reality. If philosophy were a science, then</p><p>this would qualify as a challenge for a better interpretation of “real.”</p><p>But philosophy is not a science. The arguments are plagued by ambigu-</p><p>ity and cultural and linguistic biases. “Validation” is mostly a matter of</p><p>conviction and belief, not functionality, so there are few mechanisms to</p><p>correct or even expose the flaws. Thus, rather than refine the meaning</p><p>of “reality” to accommodate what people actually do, philosophers split</p><p>into Platonists and non-Platonists, depending on whether they believed</p><p>that mathematics fit their own interpretation. The Platonic view is hard</p><p>to defend because mathematics honestly does not fit the usual mean-</p><p>ings of “real” (see the confusion in Linnebo’s overview [5]). The non-</p><p>Platonic view is essentially that mathematicians are deluded. Neither</p><p>A Revolution in Mathematics? 181</p><p>view is useful for mathematics. To make real progress, mathematics</p><p>had to break with philosophy and, as usual in a divorce, there are bad</p><p>feelings on both sides.1</p><p>The precollege-education community was, and remains, antagonis-</p><p>tic to the new methodology. One reason is that traditional mathema-</p><p>ticians, most notably Felix Klein, were extremely influential in early</p><p>20th century educational reform. Klein founded the International</p><p>Commission on Mathematical Instruction (ICMI) [4], the education</p><p>arm of the International Mathematical Union. His 1908 book Elemen-</p><p>tary Mathematics from an Advanced Viewpoint was a virtuoso example of</p><p>19th century methods and did a lot to cement their place in education.</p><p>The “Klein project” [4] is a contemporary international effort to update</p><p>the topics in Klein’s book but has no plan to update the methodology.2 In</p><p>brief, traditionalists lost the battle in the professional community but</p><p>won in education. The failure of “new math” in the 1960s and ’70s is</p><p>taken as further confirmation that modern mathematics is unsuitable</p><p>Word Problems across the Ages. Baltimore,</p><p>MD: The Johns Hopkins University Press, 2012.</p><p>Viljanen, Valtteri. Spinoza’s Geometry of Power. Cambridge, UK: Cambridge University Press,</p><p>2012.</p><p>Wager, Anita A., and David W. Stinson. (Eds.) Teaching Mathematics for Social Justice: Conversa-</p><p>tions with Educators. Reston, VA: The National Council of Teachers of Mathematics, 2012.</p><p>Walker, Erica N. Building Mathematics Learning Communities: Improving Outcomes in Urban High</p><p>Schools. New York: Teachers College, Columbia University Press, 2012.</p><p>Wapner, Leonard M. Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball. New</p><p>York: Taylor and Francis Group, 2012.</p><p>Wardhaugh, Benjamin. (Ed.) Wealth of Numbers: An Anthology of 500 Years of Popular Mathemat-</p><p>ics. Princeton, NJ: Princeton University Press, 2012.</p><p>Wardhaugh, Benjamin. (Ed.) The History of the History of Mathematics: Case Studies for the Seven-</p><p>teenth, Eighteenth, and Nineteenth Centuries. Bern, Switzerland: Peter Lang, 2012.</p><p>Weinstein, Lawrence. Guesstimation 2.0: Solving Today’s Problems on the Back of a Napkin. Prince-</p><p>ton, NJ: Princeton University Press, 2012.</p><p>xxvi Introduction</p><p>Wills, Graham. Visualizing Time. Heidelberg, Germany: Springer Verlag, 2012.</p><p>Wilson, Alan. The Science of Cities and Regions: Lectures on Mathematical Model Design. New York:</p><p>Springer Science+Business Media, 2012.</p><p>Wynn, James. Evolution by the Numbers: The Origins of Mathematical Argument in Biology. Ander-</p><p>son, SC: Parlor Press, 2012.</p><p>The BEST</p><p>WRITING on</p><p>MATHEMATICS</p><p>2013</p><p>The Prospects for Mathematics</p><p>in a Multimedia Civilization</p><p>Philip J. Davis</p><p>I. Multimedia Mathematics</p><p>First let me explain my use of the phrase “multimedia civilization.”* I</p><p>mean it in two senses. In my first usage, it is simply a synonym for our</p><p>contemporary digital world, our click-click world, our “press 1,2, or 3</p><p>world”, a world with a diminishing number of flesh-and-blood servers</p><p>to talk to. This is our world, now and for the indefinite future. It is</p><p>a world that in some tiny measure most of us have helped make and</p><p>foster.</p><p>In my second usage, I refer to the widespread and increasing use</p><p>of computers, fax, e-mail, the Internet, CD-ROMs, iPods, search en-</p><p>gines, PowerPoint, and YouTube—in all mixtures. I mean the phrase</p><p>to designate the cyberworld that embraces such terms as interface de-</p><p>sign, cybercash, cyberlaw, virtual-reality games, assisted learning , vir-</p><p>tual medical procedures, cyberfeminism, teleimmersion, interactive</p><p>literature, cinema, and animation, 3D conferencing, and spam, as well</p><p>as certain nasty excrescences that are excused by the term “unforesee-</p><p>able developments.” The word (and combining form) “cyber” was intro-</p><p>duced in the late 1940s by Norbert Wiener in the sense of feedback and</p><p>control. Searching on the prefix “cyber” resulted in 304,000,000 hits,</p><p>which, paradoxically, strikes me as a lack of control.</p><p>I personally cannot do without my word processor, my mathemati-</p><p>cal software, and yes, I must admit it, my search engines. I find I can</p><p>check conjectures quickly and find phenomena accidentally. (It is also</p><p>* This article is an expanded and newly updated version of a Urania Theater talk given</p><p>as part of the International Congress of Mathematicians, Berlin, Germany, Aug. 21, 1998.</p><p>2 Philip J. Davis</p><p>the case that I find too many trivialities!) As a writer, these tools are</p><p>now indispensable for me.</p><p>Yes, the computer and all its ancillary spinoffs have become a me-</p><p>dium, a universal forum, a method of communication, an aid both to</p><p>productive work and to trouble making, from which none of us are</p><p>able to escape. A mathematical engine, the computer is no longer the</p><p>exclusive property of a few mathematicians and electrical engineers,</p><p>as it was in the days of ENIAC et alia (late 1940s). Soon we will not</p><p>be able to read anything or do any “brain work” without a screen in</p><p>our lap and a mouse in our hands. And these tools, it is said, will soon</p><p>be replaced by Google glasses and possibly a hyperGoogle brain. We</p><p>have been seduced, we have become addicts, we have benefited, and we</p><p>hardly recognize or care to admit that there is a downside.</p><p>What aspects of mathematics immersed in our cyberworld shall I</p><p>consider? The logical chains from abstract hypotheses to conclusions?</p><p>Other means of arriving at mathematical conclusions and suggesting</p><p>actions? The semiotics of mathematics? Its applications (even to mul-</p><p>timedia itself!)? The psychology of mathematical creation? The man-</p><p>ner in which mathematics is done; is linked with itself and with other</p><p>disciplines; is published, transmitted, disseminated, discussed, taught,</p><p>supported financially, and applied? What will the job market be for its</p><p>young practitioners? What will be the public’s understanding and ap-</p><p>preciation of mathematics? Ideally, I should like to consider all of these.</p><p>But, of course, every topic that I’ve mentioned would deserve a week or</p><p>more of special conferences and would result in a large book.</p><p>Poincaré’s Predictions</p><p>We have now stepped into the new millennium, and inevitably this</p><p>step suggests that I project forward in time. Although such projections,</p><p>made in the past, have proved notoriously inadequate, I would be ne-</p><p>glecting my duty if I did not make projections, even though it is guar-</p><p>anteed that they will become the objects of future humorous remarks.</p><p>Here’s an example from the past. A century ago, at the Fourth In-</p><p>ternational Congress of Mathematicians held at Rome in 1908, Henri</p><p>Poincaré undertook such a task. In a talk entitled “The Future of Mathe-</p><p>matics,” Poincaré mentioned 10 general areas of research and some spe-</p><p>cific problems within them, which he hoped the future would resolve.</p><p>Mathematics in a Multimedia Civilization 3</p><p>What strikes me now in reading his article is not the degree to which</p><p>these areas have been developed—some have—but the inevitable</p><p>omission of a multiplicity of areas that we now take for granted and</p><p>that were then only in utero or not even conceived. Though the histo-</p><p>rian can always find the seeds of the present in the past, particularly in</p><p>the thoughts of a mathematician as great as Poincaré, I might mention</p><p>as omissions from Poincaré’s prescriptive vision the intensification of</p><p>the abstracting, generalizing, and structural tendencies; the develop-</p><p>ments in logic and set theory; pattern theory; and the emergence of</p><p>new mathematics attendant upon the physics of communication theory,</p><p>fluids, materials, relativity, quantum theory, and cosmology. And of</p><p>course, the computer, in both its practical and theoretical aspects; the</p><p>computer, which I believe is the most significant mathematical devel-</p><p>opment of the 20th century; the computer, which has altered our lives</p><p>almost as much as the “infernal” combustion engine and which may</p><p>ultimately surpass it in influence.</p><p>Poincaré’s omission of all problems relating immediately to the ex-</p><p>terior world—with the sole exception (!) of Hill’s theory of lunar mo-</p><p>tion—is also striking.</p><p>How then should the predictor with a clouded vision and limited</p><p>experience proceed? Usually by extrapolating forward from current</p><p>tendencies that are obvious even to the most imperceptive observer.</p><p>What Will Pull Mathematics into the Future?</p><p>Mathematics grows from external pressures and from pressures internal</p><p>to itself. I think the balance will definitely shift away from the internal</p><p>and that there will be an increased emphasis on applications. Math-</p><p>ematicians require support; why should society support their activity?</p><p>For the sake of pure art or knowledge? Alas, we are not classic Greeks,</p><p>who scorned those who needed to profit from what they learned, or</p><p>18th century aristocrats, for whom science was a hobby. And even the</p><p>material generated by these groups was pulled along by astronomy and</p><p>astrology (for which a charge was made), geography, navigation, and</p><p>mechanics. Society will now support mathematics generously only if it</p><p>promises bottom-line benefits.</p><p>Now focus on the word “benefits.” What is a benefit? Richard Ham-</p><p>ming of the old Bell Telephone Laboratories said in a famous epigraph</p><p>4 Philip J. Davis</p><p>to his book on scientific computation, “The object of computation is</p><p>not numbers but insight.” Insight into a variety of physical and social</p><p>processes, of course. But I perceive (40 years after Hamming’s book</p><p>and with a somewhat cynical eye) that the real object of computation,</p><p>commercial and otherwise, is often neither numbers nor insight nor</p><p>solutions to pressing problems, but worked on by physicists and math-</p><p>ematicians, to perfect money-making products. Often computer us-</p><p>ages are then authorized by project managers who have little technical</p><p>knowledge. The Descartesian precept cogito ergo sum has been replaced</p><p>by producto ergo sum.</p><p>If, by chance, humanity benefits from this activity, then so much the</p><p>better; everybody is happy. And if humanity suffers, the neo-Luddites</p><p>will cry out and form chat groups on the Web, or the hackers will at-</p><p>tack computer systems or humans. The techno-utopians will explain</p><p>that you can’t make omelets without breaking a few eggs. And pure</p><p>mathematicians will follow along, moving closer to applications while</p><p>justifying the purity of their pursuits to the administrators, politi-</p><p>cians, and the public with considerable truth that one never knows in</p><p>advance what products of pure imagination can be turned to society’s</p><p>benefit. The application of the theory of numbers to cryptography and</p><p>the (Johann) Radon transform and its application to tomography have</p><p>been displayed as shining examples of this. Using that most weasel of</p><p>rhetorical expressions, “in principle,” in principle, all mathematics is</p><p>potentially useful.</p><p>I could use all my space describing many applications that seem now</p><p>to be hot and are growing hotter. I will mention several and comment</p><p>briefly on but a few of them. In selecting these few, I have ignored</p><p>“pure” fields out of personal incompetence. I simply do not have the</p><p>knowledge or authority to single out from a hundred expanding sub-</p><p>fields the ones with particularly significant potential and how they have</p><p>fared via multimedia. For more comprehensive and authoritative pre-</p><p>sentations, I recommend Mathematics: Frontiers and Perspectives and Math-</p><p>ematics Unlimited—2001 and Beyond.</p><p>Mathematics and the Physical and Engineering Sciences</p><p>These have been around since Galileo, but Newton’s work was the great</p><p>breakthrough. However, only in the past hundred years or so has theo-</p><p>retical mathematics been of any great use to technology. The pursuit</p><p>Mathematics in a Multimedia Civilization 5</p><p>of physical and engineering sciences is today unthinkable without sig-</p><p>nificant computational power. The practice of aerodynamic design has</p><p>altered significantly, but the “digital wind tunnel” has not yet arrived,</p><p>and some have said it may never. Theories of turbulence are not yet</p><p>in satisfactory shape—how to deal with widely differing simultaneous</p><p>scales continues to perplex. Newtonians who deal with differential-</p><p>integral systems must learn to share the stage with a host of probabilists</p><p>with their stochastic equations. Withal, hurricane and tornado predic-</p><p>tions continue to improve, perhaps more because of improvements in</p><p>hardware (e.g., real-time data from aircraft or sondes and from nano-</p><p>computers) than to the numerical algorithms used to deal with the nu-</p><p>merous models that are in use. Predictions of earthquakes or of global</p><p>warming are controversial and need work. Wavelet, chaos, and fractal</p><p>theorists and multiresolution analysts are hard at work hoping to im-</p><p>prove their predictions.</p><p>Mathematics and the Life Sciences</p><p>Mathematical biology and medicine are booming. There are automatic</p><p>diagnoses. There are many models around in computational biology;</p><p>most are untested. One of my old Ph.D. students has worked in bio-</p><p>molecular mathematics and designer drugs. He and numerous others are</p><p>now attempting to model strokes via differential equations. Good luck!</p><p>I visited a large hospital recently and was struck by the extent that</p><p>the aisles were absolutely clogged with specialized computers. Later,</p><p>as a patient, I was all wired up and plugged into such equipment with</p><p>discrete data and continuous waveforms displayed bedside and at the</p><p>nurses’ stations. Many areas of medical and psychological practice have</p><p>gone or are going virtual. There is no doubt that we are now our own</p><p>digital avatars and we are all living longer and healthier lives. In this</p><p>development, mathematics, though way in the background and though</p><p>not really understood by the resident physicians or nurses, has played a</p><p>significant role.</p><p>Work on determining and analyzing the human genome sequences,</p><p>with a variety of goals in mind and using essentially combinatorial and</p><p>probabilistic methods, is a hot field. In the past decade, the cost of DNA</p><p>sequencing has come down dramatically.</p><p>Genetic engineering on crops goes forward but has raised hackles</p><p>and doomsday scenarios.</p><p>6 Philip J. Davis</p><p>Mathematics and the Military Sciences</p><p>If mathematics contributes significantly to the life sciences, there is</p><p>also mathematics that contributes to the “death sciences”: war, both</p><p>defensive and offensive. For the past 75 years, military problems have</p><p>been a tremendous engine, supplying money and pulling both pure and</p><p>applied formulations to new achievements: interior and exterior bal-</p><p>listics, creating bombs, missiles, rockets, antirocket rockets, drones,</p><p>satellites, war-gaming strategies, and combat training in the form of</p><p>realistic computer games. The use of mathematics in the service of war</p><p>and defense will be around as long as aggression is a form of human and</p><p>governmental behavior. Some authorities have claimed that aggression</p><p>is built into the human brain. In any case, the psychology of aggression</p><p>is an open field of study.</p><p>Mathematics and Entertainment</p><p>There is mathematics and entertainment through animation, simulation,</p><p>and computer graphics. The ex-executive of Silicon Graphics opined</p><p>some years ago that the future of the United States lay not in manufac-</p><p>turing nor in the production of food, but in producing a steady flow of</p><p>entertainment for the rest of the world. Imagine this: a future president</p><p>of the United States may have to warn us against the media-entertain-</p><p>ment complex as Eisenhower did with the military- industrial complex.</p><p>But there is more! Through animation and simulation, the world</p><p>of defense joins up with the world of entertainment and the world of</p><p>medical technology. These worlds find common problems and can</p><p>often share computer software. (There have been conferences on this</p><p>topic.) Mickey Mouse flies the stealth bomber, and virtual surgery can</p><p>be performed via the same sort of software products. There are now</p><p>university departments devoted to the design of new video and simula-</p><p>tion games, using humans as players, and with a wide variety of appli-</p><p>cations, including pure research. Young mathematicians have begun to</p><p>offer their talents not just to university departments of mathematics but</p><p>also to Hollywood and TV producers.</p><p>Mathematics and Money</p><p>Marriages of business and mathematics are booming. Property, busi-</p><p>ness, and trade have always been tremendous consumers of low-level</p><p>mathematics. In deep antiquity, they were probably the generators of</p><p>Mathematics in a Multimedia Civilization 7</p><p>such mathematics. But now it is no longer low-level. Zebra stripes (i.e.,</p><p>product identification, or UPC codes) have an interesting mathemati-</p><p>cal basis. Mathematics and business is a possible major in numerous</p><p>universities, with professorial chairs in the subject. Software is sold</p><p>for portfolio management and to automate income tax returns. Wall</p><p>Street is totally computerized, nanotrading is almost continuous, and</p><p>millions are playing the market using clever statistical strategies often</p><p>of their own personal devising. The practice of arbitrage has generated</p><p>theorems,</p><p>textbooks, and software.</p><p>Mathematics and the Graphic Arts</p><p>Graphic art is being revolutionized along mathematical lines, a</p><p>tendency —would you believe it?—that was present 3,000 years ago</p><p>in the art of Egypt when some of their art was pixelized. Computer</p><p>art and op art are now commonplace on gallery walls if a bit ho-hum.</p><p>Is such art a kind of “soft mathematics”? When I consider the progress</p><p>that computer art has made from the theoretical approximation theory</p><p>of my research years, from the elementary paint programs developed in</p><p>such places as the University of Utah a generation and a half ago, to the</p><p>sophisticated productions of today’s Pixar Animation Studios, my mind</p><p>boggles. Three-dimensional printing, which makes a solid object from</p><p>a digital model, advances the older programmed lathes. Two, three,</p><p>and higher dimensional graphical presentations are more prosaic as art</p><p>but are also important scientifically; they play an increasing role in pre-</p><p>sentation and interpretation.</p><p>Mathematics, Law, Legislation, and Politics</p><p>Law is just beginning to feel the effect of mathematization. Leibniz and</p><p>Christian Wolff talked about this three centuries ago. Nicholas Ber-</p><p>noulli talked about it. Read Bernoulli’s 1709 inaugural dissertation On</p><p>the Use of Probability (Artis conjectandi) in Law.</p><p>There are now DNA identifications. The conjectured (or proved)</p><p>liaison between Thomas Jefferson and Sally Hemings made the front</p><p>pages. But can statisticians be trusted? “Experts” are often found testi-</p><p>fying on both sides of a question.</p><p>Statistics are more and more entering the courts as evidence, and</p><p>courts may soon require a resident statistician to interpret things for</p><p>the judges, even as my university department has a full-time computer</p><p>maven to resolve the questions and glitches that arise constantly. There</p><p>8 Philip J. Davis</p><p>are class action and discrimination suits based on statistical evidence.</p><p>Multiple regression enters into the picture strongly. Mathematical algo-</p><p>rithms themselves have been scrutinized and may be subject to litigation</p><p>as part of intellectual property. In the burgeoning field of “jurimath” or</p><p>“jurimetrics,” there are now texts for lawyers and a number of journals.</p><p>This field should be added to the roster of applications of mathematics</p><p>and should be taught in colleges and law schools.</p><p>Consider polls. We spend millions and millions of dollars polling</p><p>voters, polling consumers, asking people how they feel about any-</p><p>thing at all. Consider the census. How should one count? Counting,</p><p>the simplest, most basic of all mathematical operations, in the sharp</p><p>theoretical sense, turns out to be a practical impossibility. Sampling is</p><p>recommended; it reduces the variance but increases the discrepancy. It</p><p>has been conjectured that sampling will increase the power of the mi-</p><p>nority party, hence the majority party is against it. As early as Nov. 30,</p><p>1998, the case was argued before the U.S. Supreme Court. Despite all</p><p>these developments, we are far from Leibniz’ dream of settling human</p><p>disputes by computation.</p><p>Mathematics in the Service of Cross- or Trans-Media</p><p>Here are a few instances:</p><p>Music  score</p><p>Oral or audio  hard copy</p><p>Motion has been captured from output from a wired-up human,</p><p>then analyzed and synthesized. (The animation of Gollum in</p><p>Lord of the Rings was produced in this way.)</p><p>Balletic motion  Choreographic notation</p><p>Voice  action. Following the instructions of an electronic voice,</p><p>I push a few buttons and a ticket gets printed out, all paid for.</p><p>But the buttons are becoming fewer as voice interpretation im-</p><p>proves daily.</p><p>Under this rubric, I would also place advanced search methods that</p><p>go beyond the “simple” Google type. Image recognition and feature</p><p>identification, often using statistics, lead to the lively field of searching</p><p>for image or audio content (e.g., find me a Vermont landscape with a</p><p>Mathematics in a Multimedia Civilization 9</p><p>red barn and a large pile of pumpkins). IDs and person identification</p><p>schemes are products here.</p><p>Mathematics and Education</p><p>Though computer science departments declared their independence</p><p>of mathematics or electrical engineering departments in the mid-</p><p>’70’s, undergraduate majors in computer science are usually required</p><p>to have a semester or two of calculus and a semester of discrete math.</p><p>Depending on what kind of work they intend to do, it may be use-</p><p>ful for them to have additionally linear algebra, probability, geom-</p><p>etry, numerical analysis, or mathematical logic. Thus, mathematics</p><p>or the spirit of mathematics, though it may be hidden from view,</p><p>lurks behind most theoretical or commercial developments in the</p><p>field.</p><p>Regarding the changes in classroom education, one of my colleagues</p><p>wrote me as follows:</p><p>My teaching has already changed a great deal. Assignments, etc.</p><p>go on the web page. Students use e-mail to ask questions which I</p><p>then bring up in class. They find information for their papers out</p><p>there on the web. We spend one day a week doing pretty serious</p><p>computing, producing wonderful graphics, setting up the math-</p><p>ematical part of it and dumping the whole mess into documents</p><p>that can be placed on a web page. I am having more fun than I</p><p>used to, and the students appear to be having a pretty good time</p><p>while learning a lot. Can all this be bad?</p><p>A distinguished applied mathematician of my acquaintance is spend-</p><p>ing part of his time producing CD-ROMs to publicize his theories and</p><p>experiences. The classic modes of elementary and advanced teaching</p><p>have been amplified and sometimes displaced by computer products.</p><p>Thousands of lectures, often illustrated and interactive, on all conceiv-</p><p>able mathematical topics, can be downloaded. Automated correction</p><p>of papers has been going on for decades. There is software to detect</p><p>plagiarism.</p><p>A good university computer store has more of these products for sale</p><p>than there are brands of cheese in the gourmet market. Correspon-</p><p>dence courses via such things as e-mail are flourishing. “Web universi-</p><p>ties” that have up to 100,000 students per class have sprung up all over</p><p>10 Philip J. Davis</p><p>the world. Will the vast armies of flesh-and-blood teachers become</p><p>obsolete?</p><p>Working Habits and the Working Environment</p><p>A colleague wrote to me</p><p>On balance, I believe that science will suffer in the multi-media</p><p>age. My experience is that true thinking now goes against the</p><p>grain. I feel I have to be rude to arrange for a few peaceful hours a</p><p>day for real work. Saying no to too many invitations, writing short</p><p>answers to too many e-mail questions about research. A letter</p><p>comes to me from a far corner of the world: “Please explain line</p><p>six of your 1987 paper.” I stay home in the mornings hiding from</p><p>my office equipment. Big science projects, interdisciplinary proj-</p><p>ects, big pushes, aided and abetted by multi-media and easy trans-</p><p>portation have diminished my available time for real thought. I am</p><p>also human and succumb to the glamour of today’s technoglitz.</p><p>Another colleague said of his mathematical research experiences,</p><p>Publication on line is easy. But there is too much of it, often relat-</p><p>ing trivial advances. What is out there often lacks depth and goes</p><p>unrefereed.</p><p>The book world, or more generally the world of information produc-</p><p>tion, dissemination, credit, and copyright royalties is in an absolute</p><p>turmoil. Is all this really for the better? Qui vivra verra.</p><p>Dissemination</p><p>A half century ago, mathematicians used to relegate certain jobs to</p><p>other disciplines or crafts or professions. We have now become our</p><p>own typists, typesetters, drafters, library scientists, book designers,</p><p>publishers, jobbers, public relations agents, and sales agents. For all that</p><p>the computer is rapid, these activities absorb substantial blocks of time</p><p>and energy that were formerly devoted to thinking about problems.</p><p>Occasionally, in the past, scientists had to leave pure creation and</p><p>discovery and worry about dissemination. In 1597, when Tycho Brahe</p><p>moved from</p><p>Copenhagen to Prague to take a job there, he lugged</p><p>along his own heavy printing press. Journals? Who needs them now</p><p>when you can download papers by the hundreds? The dissemination</p><p>Mathematics in a Multimedia Civilization 11</p><p>of mathematics through textbooks and learned journals is threatened</p><p>with obsolescence in favor of online electronic publishing. Every man</p><p>and woman is now a journal “publisher” or a mathematical blogger. I</p><p>exchange PDF files with a colleague 7,000 miles away.</p><p>Whereas people such as Copernicus and Newton waited years before</p><p>they published, today’s scientists, under a variety of pressures, go on</p><p>line with electronic publishing before their ideas are out of the oven,</p><p>often unchecked; and they can receive equally rapid and equally half-</p><p>baked feedback. Refereeing has diminished. The authority that once</p><p>attached to the published page has vanished.</p><p>Are books obsolete? A collaborator working with me and much in love</p><p>with mathematical databases, picked up (even after advanced filtering)</p><p>more than 100,000 references to a key word that was relevant to our</p><p>work. This information overdose produced an immediate blockage or at-</p><p>rophy of the spirit in him. How can we afford the time to assess this raw,</p><p>unassimilated information overload ? Should we then simply plow ahead</p><p>on our own as best we can and hope that we will come up with something</p><p>new and of interest ? Semioticist and novelist Umberto Eco wrote, in How</p><p>to Travel with a Salmon, “. . . the whole information industry runs the risk</p><p>of no longer communicating anything because it tells too much.”</p><p>My eyebrows were raised recently when I learned that as part of a</p><p>large grant application to the National Science Foundation, the appli-</p><p>cants were advised to include a detailed plan for the dissemination of</p><p>their work. In the multimedia age, mathematics is being transformed</p><p>into a product to be marketed as other products. Just as, for a price,</p><p>a person seeking employment can get a company to produce a fancy</p><p>CV, there are or there will be, I feel sure, companies that, for a price,</p><p>will undertake to subcontract the dissemination of mathematical</p><p>applications.</p><p>The Public Understanding of Mathematics</p><p>Every aspect of our lives is increasingly being mathematized. We are</p><p>dominated by and we are accommodating to mathematical theories,</p><p>engines, and the arrangements they prescribe. Yet, paradoxically, the</p><p>nature of technology makes it possible, through chipification, for the</p><p>mathematics itself to disappear into the background and for the public</p><p>to be totally unaware of it. It is probably the case that despite the claims</p><p>of teachers and educational administrators, the general population</p><p>12 Philip J. Davis</p><p>needs now to know less mathematics than at any time in the last several</p><p>hundred years even as they cause it to be used.</p><p>When mathematics hits the front page of newspapers or on TV, what</p><p>we hear is the solution to some difficult, long-standing problem associ-</p><p>ated with large prizes and prestige. Without in any way minimizing</p><p>such intellectual achievements, these accomplishments are not what the</p><p>professional mathematical community devotes the bulk of its time to,</p><p>nor do they enter into areas of social utility.</p><p>What civilization needs is a critical education that brings an aware-</p><p>ness and judgment of the mathematics that dominates it and is able</p><p>to react with some force to evaluate, accept, reject, slow down, re-</p><p>direct, and reformulate these abstract symbols that are affecting our</p><p>lives. Technology is not neutral; mathematics is not neutral; they foster</p><p>certain kinds of behavior. Individual mathematicians are aware of this.</p><p>Groups calling themselves “technorealists” maintain websites. But with</p><p>some exceptions, the awareness has not yet penetrated the educational</p><p>process nor the general public, and the latter only when a new (math-</p><p>ematical, obviously) tax scheme is proposed.</p><p>Benefits</p><p>From all these developments, there have been and surely there will be</p><p>future benefits, and the advancing waves of new technologies that are</p><p>sweeping over us hardly need a public relations agent to trumpet the ben-</p><p>efits or thrills. The size of personal fortunes made from the sale of the</p><p>latest hardware or software derives from the public saying “Yes. Bring on</p><p>more.” Mental gridlock and backlash have only begun to appear.</p><p>II. The Inner Texture (or Soul) of Mathematics</p><p>Let me now go up a meta-level and ask, “How will multimedia affect</p><p>the conceptualization, the imagery, and the methodology of future</p><p>mathematics? How will the metaphysics or the philosophy of the sub-</p><p>ject be affected ? What is advanced mathematics going to be doing other</p><p>than presenting displays of its own narcissism?” These questions for me</p><p>are tremendously interesting but difficult. Here again, all I can do is to</p><p>describe what I see and to project forward. I shall make the attempt in</p><p>terms of what I call the tensions of texture.</p><p>Mathematics in a Multimedia Civilization 13</p><p>The Discrete vs. the Continuous</p><p>This is an old dichotomy going back to the Greek atomists. Metaphysi-</p><p>cians for centuries have argued the adage: Natura non facit saltum.</p><p>The point of William Everdell’s book The First Moderns is that ap-</p><p>parently to be up to date is to be discrete, or discontinuous. Everdell</p><p>shows how this concept has operated in literature and art as well as in</p><p>science and mathematics. Not so long ago, there was a movement afoot</p><p>in the United States, asserting that continuous mathematics ought to</p><p>give way in education and in philosophy to discrete mathematics. After</p><p>all, it was said, the world is built up from discrete elements. Moreover,</p><p>every digital computer implementation of continuous theories first re-</p><p>duces the continuous to the discrete. The movement seems now to have</p><p>quieted down a bit. After all, the discrete, in many respects, is more</p><p>difficult to handle than the continuous.</p><p>The Deterministic vs. the Probabilistic</p><p>This split applies not only to the modeling of the exterior world but</p><p>also resides interior to mathematics itself. Internally, it relates to such</p><p>questions as “Are the truths of mathematics probabilistic?” externally</p><p>to the question of the extent to which we are able to live with such</p><p>truths.</p><p>In a different direction, the knowledge that one little old lady resid-</p><p>ing in Rhode Island has won $17 million in the Powerball lottery does</p><p>more to question and destroy the relevance of probability theory in</p><p>the public’s mind than all the skepticism that the theorems imply to</p><p>such as myself. States now use lotteries in increasing measure as a form</p><p>of taxation. GTECH, located in downtown Providence, Rhode Island,</p><p>with world prominence, provides software and hardware for all sorts of</p><p>gaming devices and opportunities from online betting to casinos. For</p><p>employment in this company, strong skills in mathematics and statistics</p><p>are advised.</p><p>Despite that fact, probability theory is a relative newcomer on the</p><p>mathematical scene (from the 16th century). The split between the de-</p><p>terministic and the probabilistic is old, going back surely as far back as</p><p>the philosophic discussions of free will, fate, kismet, and the seemingly</p><p>arbitrary actions ascribed to the gods and the devil.</p><p>14 Philip J. Davis</p><p>Despite Einstein’s quip that God doesn’t play dice with the universe,</p><p>the relevance of probability theory is now strong and becoming stron-</p><p>ger. Niels Bohr rapped Einstein’s knuckles: “Albert, don’t tell God what</p><p>he should do with his dice.” David Mumford opines that probability</p><p>theory and statistical reasoning</p><p>will emerge as better foundations for scientific models . . . and as</p><p>essential ingredients of theoretical mathematics, even the founda-</p><p>tions of mathematics itself.</p><p>Thinking vs. Clicking</p><p>I have heard over and over again from observers that thinking increas-</p><p>ingly goes against the grain. Is thinking obsolete or becoming more ob-</p><p>solete? To think is to click. To click is to think. Are these the equations</p><p>for the</p>