The year's finest writing on mathematics from around the world, with a foreword by Nobel Prize – winning physicist Roger Penrose
This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2013 makes available to a wide audience many articles not easily found anywhere else―and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here Philip Davis offers a panoramic view of mathematics in contemporary society; Terence Tao discusses aspects of universal mathematical laws in complex systems; Ian Stewart explains how in mathematics everything arises out of nothing; Erin Maloney and Sian Beilock consider the mathematical anxiety experienced by many students and suggest effective remedies; Elie Ayache argues that exchange prices reached in open market transactions transcend the common notion of probability; and much, much more.
In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a foreword by esteemed mathematical physicist Roger Penrose and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us―and where it is headed.
Review
"A marvelous selection of papers about mathematics written by the best. . . . Highly recommended to all with a broad interest in science, history, art, education, philosophy . . . which is almost anybody." ---A. Bultheel, European Mathematical Society Reviews
"In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a foreword by esteemed physicist and mathematician Freeman Dyson. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed." ― World Book Industry
"These selections provide a sampling of the state of the art through topics ranging from math anxiety to modern applications to the history of mathematics. With great emphasis on the interactions of mathematics with modern civilization, the arts, and philosophy, these articles can be a source of intellectual stimulation for educated lay readers and will provide, for working mathematicians or scientists, exposure to parts of the subject outside of their research range. . . . A well-chosen collection supplemented by an outstanding bibliography of suggested further reading." ---Harold D. Shane, Library Journal
"[T]his is a marvelous selection of papers about mathematics written by the best. They do not draw the reader into the mathematical jargon that is only of interest to the mathematical literate. In fact practically no mathematics is needed and formulas are almost completely absent. It is the best possible way of communicating mathematics to the non-mathematician and even the ones suffering from mathematical anxiety will enjoy reading the booklet. Of course this is only a relatively small selection but for the reader longing for more, Pitici gives in his introduction an even longer list of books, papers, websites and blogs that are equally worth reading. Pitici did once more an excellent job, and the result is highly recommended to all with a broad interest in science, history, art, education, philosophy . . . which is almost anybody." ---A. Bultheel, European Mathematical Society
"The list of titles of the essays reprinted in this volume would be almost enough to persuade many people that they will enjoy reading it, and indeed that they could profitably put it in front of anyone who wants to know what mathematics is about. It's a partial and selective picture, of course, but it's fresh, varied, and as its title might suggest, well written." ---Jeremy Gray, LMS Newsletter
"Most of the articles presented here are entertaining and enlightening, and the book should be recommended to everyone interested in mathematics." ― Zentralblatt MATH
"Praise for Princeton's previous editions: "A wonderful and varied bouquet of texts. . . . I highly recommend this book to everyone with an interest in mathematics."" ---Stephen Buckley, Irish Mathematical Society Bulletin
About the Author
Mircea Pitici holds a PhD mathematics education from Cornell University, where he teaches math and writing. He has edited The Best Writing on Mathematics since 2010. Roger Penrose is a Nobel Prize–winning physicist and the bestselling author, with Stephen Hawking, of The Nature of Space and Time (Princeton). He is the Rouse Ball Professor of Mathematics Emeritus at the University of Oxford.
Foreword Roger Penrose....................................................ixIntroduction Mircea Pitici................................................xvThe Prospects for Mathematics in a Multimedia Civilization Philip J. Davis......................................................................1Fearful Symmetry Ian Stewart..............................................23E pluribus unum: From Complexity, Universality Terence Tao................32Degrees of Separation Gregory Goth........................................47Randomness Charles Seife..................................................52Randomness in Music Donald E. Knuth.......................................56Playing the Odds Soren Johnson............................................62Machines of the Infinite John Pavlus......................................67Bridges, String Art, and Bézier Curves Renan Gross........................77Slicing a Cone for Art and Science Daniel S. Silver.......................90High Fashion Meets Higher Mathematics Kelly Delp..........................109The Jordan Curve Theorem Is Nontrivial Fiona Ross and William T. Ross.....120Why Mathematics? What Mathematics? Anna Sfard.............................130Math Anxiety: Who Has It, Why It Develops, and How to Guard against It Erin A. Maloney and Sian L. Beilock........................................143How Old Are the Platonic Solids? David R. Lloyd...........................149Early Modern Mathematical Instruments Jim Bennett.........................163A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today Frank Quinn.................................................175Errors of Probability in Historical Context Prakash Gorroochurn...........191The End of Probability Elie Ayache........................................213An abc Proof Too Tough Even for Mathematicians Kevin Hartnett.............225Contributors...............................................................231Notable Texts..............................................................237Acknowledgment.............................................................241Credits....................................................................243
CHAPTER 1
The Prospects for Mathematics in a Multimedia Civilization
Philip J. Davis
I. Multimedia Mathematics
First let me explain my use of the phrase "multimedia civilization." I mean it in two senses. In my first usage, it is simply a synonym for our contemporary digital world, our click-click world, our "press 1,2, or 3 world", a world with a diminishing number of flesh-and-blood servers to talk to. This is our world, now and for the indefinite future. It is a world that in some tiny measure most of us have helped make and foster.
In my second usage, I refer to the widespread and increasing use of computers, fax, e-mail, the Internet, CD-ROMs, iPods, search engines, PowerPoint, and YouTube—in all mixtures. I mean the phrase to designate the cyberworld that embraces such terms as interface design, cybercash, cyberlaw, virtual-reality games, assisted learning , virtual medical procedures, cyberfeminism, teleimmersion, interactive literature, cinema, and animation, 3D conferencing, and spam, as well as certain nasty excrescences that are excused by the term "unforeseeable developments." The word (and combining form) "cyber" was introduced in the late 1940s by Norbert Wiener in the sense of feedback and control. Searching on the prefix "cyber" resulted in 304,000,000 hits, which, paradoxically, strikes me as a lack of control.
I personally cannot do without my word processor, my mathematical software, and yes, I must admit it, my search engines. I find I can check conjectures quickly and find phenomena accidentally. (It is also the case that I find too many trivialities!) As a writer, these tools are now indispensable for me.
Yes, the computer and all its ancillary spinoffs have become a medium, a universal forum, a method of communication, an aid both to productive work and to trouble making, from which none of us are able to escape. A mathematical engine, the computer is no longer the exclusive property of a few mathematicians and electrical engineers, as it was in the days of ENIAC et alia (late 1940s). Soon we will not be able to read anything or do any "brain work" without a screen in our lap and a mouse in our hands. And these tools, it is said, will soon be replaced by Google glasses and possibly a hyperGoogle brain. We have been seduced, we have become addicts, we have benefited, and we hardly recognize or care to admit that there is a downside.
What aspects of mathematics immersed in our cyberworld shall I consider? The logical chains from abstract hypotheses to conclusions? Other means of arriving at mathematical conclusions and suggesting actions? The semiotics of mathematics? Its applications (even to multimedia itself!)? The psychology of mathematical creation? The manner in which mathematics is done; is linked with itself and with other disciplines; is published, transmitted, disseminated, discussed, taught, supported financially, and applied? What will the job market be for its young practitioners? What will be the public's understanding and appreciation of mathematics? Ideally, I should like to consider all of these. But, of course, every topic that I've mentioned would deserve a week or more of special conferences and would result in a large book.
Poincaré's Predictions
We have now stepped into the new millennium, and inevitably this step suggests that I project forward in time. Although such projections, made in the past, have proved notoriously inadequate, I would be neglecting my duty if I did not make projections, even though it is guaranteed that they will become the objects of future humorous remarks.
Here's an example from the past. A century ago, at the Fourth International Congress of Mathematicians held at Rome in 1908, Henri Poincaré undertook such a task. In a talk entitled " The Future of Mathematics ," Poincaré mentioned 10 general areas of research and some specific problems within them, which he hoped the future would resolve. What strikes me now in reading his article is not the degree to which these areas have been developed—some have—but the inevitable omission of a multiplicity of areas that we now take for granted and that were then only in utero or not even conceived. Though the historian can always find the seeds of the present in the past, particularly in the thoughts of a mathematician as great as Poincaré, I might mention as omissions from Poincaré's prescriptive vision the intensification of the abstracting, generalizing, and structural tendencies; the developments in logic and set theory; pattern theory; and the emergence of new mathematics attendant upon thee physics of communication theory, fluids, materials, relativity, quantum theory, and cosmology. And of course, thee computer, in both its practical and theoretical aspects; the computer, which I believe is the most significant maaaathematical development of the 20th century; the computer, which has altered our lives almost as much as the "infernal" combustion engine and which may ultimately surpass it in influence.
Poincaré's omission of all problems relating immediately to the exterior world—with the sole exception (!) of Hill's theory of lunar motion—is also striking.
How then should the predictor with a clouded vision and limited experience proceed? Usually by extrapolating forward from current tendencies that are obvious even to the most imperceptive observer.
What Will Pull Mathematics into the Future?
Mathematics grows from external pressures and from pressures internal to itself. I think the balance will definitely shift away from the internal and that there will be an increased emphasis on applications. Mathematicians require support; why should society support their activity? For the sake of pure art or knowledge? Alas, we are not classic Greeks, who scorned those who needed to profit from what they learned, or 18th century aristocrats, for whom science was a hobby. And even the material generated by these groups was pulled along by astronomy and astrology (for which a charge was made), geography, navigation, and mechanics. Society will now support mathematics generously only if it promises bottom-line benefits.
Now focus on the word "benefits." What is a benefit? Richard Hamming of the old Bell Telephone Laboratories said in a famous epigraph to his book on scientific computation, "The object of computation is not numbers but insight." Insight into a variety of physical and social processes, of course. But I perceive (40 years after Hamming's book and with a somewhat cynical eye) that the real object of computation, commercial and otherwise, is often neither numbers nor insight nor solutions to pressing problems, but worked on by physicists and mathematicians, to perfect money-making products. Often computer usages are then authorized by project managers who have little technical knowledge. The Descartesian precept cogito ergo sum has been replaced by producto ergo sum.
If, by chance, humanity benefits from this activity, then so much the better; everybody is happy. And if humanity suffers, the neo-Luddites will cry out and form chat groups on the Web, or the hackers will attack computer systems or humans. The techno-utopians will explain that you can't make omelets without breaking a few eggs. And pure mathematicians will follow along, moving closer to applications while justifying the purity of their pursuits to the administrators, politicians, and the public with considerable truth that one never knows in advance what products of pure imagination can be turned to society's benefit. The application of the theory of numbers to cryptography and the (Johann) Radon transform and its application to tomography have been displayed as shining examples of this. Using that most weasel of rhetorical expressions, "in principle," in principle, all mathematics is potentially useful.
I could use all my space describing many applications that seem now to be hot and are growing hotter. I will mention several and comment briefly on but a few of them. In selecting these few, I have ignored "pure" fields out of personal incompetence. I simply do not have the knowledge or authority to single out from a hundred expanding subfields the ones with particularly significant potential and how they have fared via multimedia. For more comprehensive and authoritative presentations, I recommend Mathematics: Frontiers and Perspectives and Mathematics Unlimited—2001 and Beyond.
Mathematics and the Physical and Engineering Sciences
These have been around since Galileo, but Newton's work was the great breakthrough. However, only in the past hundred years or so has theoretical mathematics been of any great use to technology. The pursuit of physical and engineering sciences is today unthinkable without significant computational power. The practice of aerodynamic design has altered significantly, but the "digital wind tunnel" has not yet arrived, and some have said it may never. Theories of turbulence are not yet in satisfactory shape—how to deal with widely differing simultaneous scales continues to perplex. Newtonians who deal with differential-integral systems must learn to share the stage with a host of probabilists with their stochastic equations. Withal, hurricane and tornado predictions continue to improve, perhaps more because of improvements in hardware (e.g., real-time data from aircraft or sondes and from nano-computers) than to the numerical algorithms used to deal with the numerous models that are in use. Predictions of earthquakes or of global warming are controversial and need work. Wavelet, chaos, and fractal theorists and multiresolution analysts are hard at work hoping to improve their predictions.
Mathematics and the Life Sciences
Mathematical biology and medicine are booming. There are automatic diagnoses. There are many models around in computational biology; most are untested. One of my old Ph.D. students has worked in biomolecular mathematics and designer drugs. He and numerous others are now attempting to model strokes via differential equations. Good luck!
I visited a large hospital recently and was struck by the extent that the aisles were absolutely clogged with specialized computers. Later, as a patient, I was all wired up and plugged into such equipment with discrete data and continuous waveforms displayed bedside and at the nurses' stations. Many areas of medical and psychological practice have gone or are going virtual. There is no doubt that we are now our own digital avatars and we are all living longer and healthier lives. In this development, mathematics, though way in the background and though not really understood by the resident physicians or nurses, has played a significant role.
Work on determining and analyzing the human genome sequences, with a variety of goals in mind and using essentially combinatorial and probabilistic methods, is a hot field. In the past decade, the cost of DNA sequencing has come down dramatically.
Genetic engineering on crops goes forward but has raised hackles and doomsday scenarios.
Mathematics and the Military Sciences
If mathematics contributes significantly to the life sciences, there is also mathematics that contributes to the "death sciences": war, both defensive and offensive. For the past 75 years, military problems have been a tremendous engine, supplying money and pulling both pure and applied formulations to new achievements: interior and exterior ballistics, creating bombs, missiles, rockets, antirocket rockets, drones, satellites, war-gaming strategies, and combat training in the form of realistic computer games. The use of mathematics in the service of war and defense will be around as long as aggression is a form of human and governmental behavior. Some authorities have claimed that aggression is built into the human brain. In any case, the psychology of aggression is an open field of study.
Mathematics and Entertainment
There is mathematics and entertainment through animation, simulation, and computer graphics. The ex-executive of Silicon Graphics opined some years ago that the future of the United States lay not in manufacturing nor in the production of food, but in producing a steady flow of entertainment for the rest of the world. Imagine this: a future president of the United States may have to warn us against the media-entertainment complex as Eisenhower did with the military- industrial complex.
But there is more! Through animation and simulation, the world of defense joins up with the world of entertainment and the world of medical technology. These worlds find common problems and can often share computer software. (There have been conferences on this topic.) Mickey Mouse flies the stealth bomber, and virtual surgery can be performed via the same sort of software products. There are now university departments devoted to the design of new video and simulation games, using humans as players, and with a wide variety of applications, including pure research. Young mathematicians have begun to offer their talents not just to university departments of mathematics but also to Hollywood and TV producers.
Mathematics and Money
Marriages of business and mathematics are booming. Property, business, and trade have always been tremendous consumers of low-level mathematics. In deep antiquity, they were probably the generators of such mathematics. But now it is no longer low-level. Zebra stripes (i.e., product identification, or UPC codes) have an interesting mathematical basis. Mathematics and business is a possible major in numerous universities, with professorial chairs in the subject. Software is sold for portfolio management and to automate income tax returns. Wall Street is totally computerized, nanotrading is almost continuous, and millions are playing the market using clever statistical strategies often of their own personal devising. The practice of arbitrage has generated theorems, textbooks, and software.
Mathematics and the Graphic Arts
Graphic art is being revolutionized along mathematical lines, a tendency—would you believe it?—that was present 3,000 years ago in the art of Egypt when some of their art was pixelized. Computer art and op art are now commonplace on gallery walls if a bit ho-hum. Is such art a kind of "soft mathematics"? When I consider the progress that computer art has made from the theoretical approximation theory of my research years, from the elementary paint programs developed in such places as the University of Utah a generation and a half ago, to the sophisticated productions of today's Pixar Animation Studios, my mind boggles. Three-dimensional printing, which makes a solid object from a digital model, advances the older programmed lathes. Two, three, and higher dimensional graphical presentations are more prosaic as art but are also important scientifically; they play an increasing role in presentation and interpretation.
Mathematics, Law, Legislation, and Politics
Law is just beginning to feel the effect of mathematization. Leibniz and Christian Wolff talked about this three centuries ago. Nicholas Bernoulli talked about it. Read Bernoulli's 1709 inaugural dissertation On the Use of Probability (Artis conjectandi) in Law.
There are now DNA identifications. The conjectured (or proved) liaison between Thomas Jefferson and Sally Hemings made the front pages. But can statisticians be trusted? "Experts" are often found testifying on both sides of a question.
Statistics are more and more entering the courts as evidence, and courts may soon require a resident statistician to interpret things for the judges, even as my university department has a full-time computer maven to resolve the questions and glitches that arise constantly. There are class action and discrimination suits based on statistical evidence. Multiple regression enters into the picture strongly. Mathematical algorithms themselves have been scrutinized and may be subject to litigation as part of intellectual property. In the burgeoning field of "jurimath" or "jurimetrics," there are now texts for lawyers and a number of journals. This field should be added to the roster of applications of mathematics and should be taught in colleges and law schools.
Consider polls. We spend millions and millions of dollars polling voters, polling consumers, asking people how they feel about anything at all. Consider the census. How should one count? Counting, the simplest, most basic of all mathematical operations, in the sharp theoretical sense, turns out to be a practical impossibility. Sampling is recommended; it reduces the variance but increases the discrepancy. It has been conjectured that sampling will increase the power of the minority party, hence the majority party is against it. As early as Nov. 30, 1998, the case was argued before the U.S. Supreme Court. Despite all these developments, we are far from Leibniz' dream of settling human disputes by computation.
Mathematics in the Service of Cross- or Trans-Media
Here are a few instances:
Music <-> score
Oral or audio <-> hard copy
Motion has been captured from output from a wired-up human, then analyzed and synthesized. (The animation of Gollum in Lord of the Rings was produced in this way.)
Balletic motion <-> Choreographic notation
Voice -> action. Following the instructions of an electronic voice, I push a few buttons and a ticket gets printed out, all paid for. But the buttons are becoming fewer as voice interpretation improves daily.
The year's finest writing on mathematics from around the world, with a foreword by Nobel Prize – winning physicist Roger Penrose
This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2013 makes available to a wide audience many articles not easily found anywhere else―and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here Philip Davis offers a panoramic view of mathematics in contemporary society; Terence Tao discusses aspects of universal mathematical laws in complex systems; Ian Stewart explains how in mathematics everything arises out of nothing; Erin Maloney and Sian Beilock consider the mathematical anxiety experienced by many students and suggest effective remedies; Elie Ayache argues that exchange prices reached in open market transactions transcend the common notion of probability; and much, much more.
In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a foreword by esteemed mathematical physicist Roger Penrose and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us―and where it is headed.
Review
"A marvelous selection of papers about mathematics written by the best. . . . Highly recommended to all with a broad interest in science, history, art, education, philosophy . . . which is almost anybody." ---A. Bultheel, European Mathematical Society Reviews
"In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a foreword by esteemed physicist and mathematician Freeman Dyson. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed." ― World Book Industry
"These selections provide a sampling of the state of the art through topics ranging from math anxiety to modern applications to the history of mathematics. With great emphasis on the interactions of mathematics with modern civilization, the arts, and philosophy, these articles can be a source of intellectual stimulation for educated lay readers and will provide, for working mathematicians or scientists, exposure to parts of the subject outside of their research range. . . . A well-chosen collection supplemented by an outstanding bibliography of suggested further reading." ---Harold D. Shane, Library Journal
"[T]his is a marvelous selection of papers about mathematics written by the best. They do not draw the reader into the mathematical jargon that is only of interest to the mathematical literate. In fact practically no mathematics is needed and formulas are almost completely absent. It is the best possible way of communicating mathematics to the non-mathematician and even the ones suffering from mathematical anxiety will enjoy reading the booklet. Of course this is only a relatively small selection but for the reader longing for more, Pitici gives in his introduction an even longer list of books, papers, websites and blogs that are equally worth reading. Pitici did once more an excellent job, and the result is highly recommended to all with a broad interest in science, history, art, education, philosophy . . . which is almost anybody." ---A. Bultheel, European Mathematical Society
"The list of titles of the essays reprinted in this volume would be almost enough to persuade many people that they will enjoy reading it, and indeed that they could profitably put it in front of anyone who wants to know what mathematics is about. It's a partial and selective picture, of course, but it's fresh, varied, and as its title might suggest, well written." ---Jeremy Gray, LMS Newsletter
"Most of the articles presented here are entertaining and enlightening, and the book should be recommended to everyone interested in mathematics." ― Zentralblatt MATH
"Praise for Princeton's previous editions: "A wonderful and varied bouquet of texts. . . . I highly recommend this book to everyone with an interest in mathematics."" ---Stephen Buckley, Irish Mathematical Society Bulletin
About the Author
Mircea Pitici holds a PhD mathematics education from Cornell University, where he teaches math and writing. He has edited The Best Writing on Mathematics since 2010. Roger Penrose is a Nobel Prize–winning physicist and the bestselling author, with Stephen Hawking, of The Nature of Space and Time (Princeton). He is the Rouse Ball Professor of Mathematics Emeritus at the University of Oxford.
Excerpt. © Reprinted by permission. All rights reserved.
The BEST WRITING on MATHEMATICS 2013
By Mircea Pitici
PRINCETON UNIVERSITY PRESS
Copyright © 2014 Princeton University Press
All rights reserved.
ISBN: 978-0-691-16041-2
Contents
Foreword Roger Penrose....................................................ixIntroduction Mircea Pitici................................................xvThe Prospects for Mathematics in a Multimedia Civilization Philip J.
Davis......................................................................1Fearful Symmetry Ian Stewart..............................................23E pluribus unum: From Complexity, Universality Terence Tao................32Degrees of Separation Gregory Goth........................................47Randomness Charles Seife..................................................52Randomness in Music Donald E. Knuth.......................................56Playing the Odds Soren Johnson............................................62Machines of the Infinite John Pavlus......................................67Bridges, String Art, and Bézier Curves Renan Gross........................77Slicing a Cone for Art and Science Daniel S. Silver.......................90High Fashion Meets Higher Mathematics Kelly Delp..........................109The Jordan Curve Theorem Is Nontrivial Fiona Ross and William T. Ross.....120Why Mathematics? What Mathematics? Anna Sfard.............................130Math Anxiety: Who Has It, Why It Develops, and How to Guard against It
Erin A. Maloney and Sian L. Beilock........................................143How Old Are the Platonic Solids? David R. Lloyd...........................149Early Modern Mathematical Instruments Jim Bennett.........................163A Revolution in Mathematics? What Really Happened a Century Ago and Why It
Matters Today Frank Quinn.................................................175Errors of Probability in Historical Context Prakash Gorroochurn...........191The End of Probability Elie Ayache........................................213An abc Proof Too Tough Even for Mathematicians Kevin Hartnett.............225Contributors...............................................................231Notable Texts..............................................................237Acknowledgment.............................................................241Credits....................................................................243
CHAPTER 1
The Prospects for Mathematics in a Multimedia Civilization
Philip J. Davis
I. Multimedia Mathematics
First let me explain my use of the phrase "multimedia civilization." I mean it in two senses. In my first usage, it is simply a synonym for our contemporary digital world, our click-click world, our "press 1,2, or 3 world", a world with a diminishing number of flesh-and-blood servers to talk to. This is our world, now and for the indefinite future. It is a world that in some tiny measure most of us have helped make and foster.
In my second usage, I refer to the widespread and increasing use of computers, fax, e-mail, the Internet, CD-ROMs, iPods, search engines, PowerPoint, and YouTube—in all mixtures. I mean the phrase to designate the cyberworld that embraces such terms as interface design, cybercash, cyberlaw, virtual-reality games, assisted learning , virtual medical procedures, cyberfeminism, teleimmersion, interactive literature, cinema, and animation, 3D conferencing, and spam, as well as certain nasty excrescences that are excused by the term "unforeseeable developments." The word (and combining form) "cyber" was introduced in the late 1940s by Norbert Wiener in the sense of feedback and control. Searching on the prefix "cyber" resulted in 304,000,000 hits, which, paradoxically, strikes me as a lack of control.
I personally cannot do without my word processor, my mathematical software, and yes, I must admit it, my search engines. I find I can check conjectures quickly and find phenomena accidentally. (It is also the case that I find too many trivialities!) As a writer, these tools are now indispensable for me.
Yes, the computer and all its ancillary spinoffs have become a medium, a universal forum, a method of communication, an aid both to productive work and to trouble making, from which none of us are able to escape. A mathematical engine, the computer is no longer the exclusive property of a few mathematicians and electrical engineers, as it was in the days of ENIAC et alia (late 1940s). Soon we will not be able to read anything or do any "brain work" without a screen in our lap and a mouse in our hands. And these tools, it is said, will soon be replaced by Google glasses and possibly a hyperGoogle brain. We have been seduced, we have become addicts, we have benefited, and we hardly recognize or care to admit that there is a downside.
What aspects of mathematics immersed in our cyberworld shall I consider? The logical chains from abstract hypotheses to conclusions? Other means of arriving at mathematical conclusions and suggesting actions? The semiotics of mathematics? Its applications (even to multimedia itself!)? The psychology of mathematical creation? The manner in which mathematics is done; is linked with itself and with other disciplines; is published, transmitted, disseminated, discussed, taught, supported financially, and applied? What will the job market be for its young practitioners? What will be the public's understanding and appreciation of mathematics? Ideally, I should like to consider all of these. But, of course, every topic that I've mentioned would deserve a week or more of special conferences and would result in a large book.
Poincaré's Predictions
We have now stepped into the new millennium, and inevitably this step suggests that I project forward in time. Although such projections, made in the past, have proved notoriously inadequate, I would be neglecting my duty if I did not make projections, even though it is guaranteed that they will become the objects of future humorous remarks.
Here's an example from the past. A century ago, at the Fourth International Congress of Mathematicians held at Rome in 1908, Henri Poincaré undertook such a task. In a talk entitled " The Future of Mathematics ," Poincaré mentioned 10 general areas of research and some specific problems within them, which he hoped the future would resolve. What strikes me now in reading his article is not the degree to which these areas have been developed—some have—but the inevitable omission of a multiplicity of areas that we now take for granted and that were then only in utero or not even conceived. Though the historian can always find the seeds of the present in the past, particularly in the thoughts of a mathematician as great as Poincaré, I might mention as omissions from Poincaré's prescriptive vision the intensification of the abstracting, generalizing, and structural tendencies; the developments in logic and set theory; pattern theory; and the emergence of new mathematics attendant upon thee physics of communication theory, fluids, materials, relativity, quantum theory, and cosmology. And of course, thee computer, in both its practical and theoretical aspects; the computer, which I believe is the most significant maaaathematical development of the 20th century; the computer, which has altered our lives almost as much as the "infernal" combustion engine and which may ultimately surpass it in influence.
Poincaré's omission of all problems relating immediately to the exterior world—with the sole exception (!) of Hill's theory of lunar motion—is also striking.
How then should the predictor with a clouded vision and limited experience proceed? Usually by extrapolating forward from current tendencies that are obvious even to the most imperceptive observer.
What Will Pull Mathematics into the Future?
Mathematics grows from external pressures and from pressures internal to itself. I think the balance will definitely shift away from the internal and that there will be an increased emphasis on applications. Mathematicians require support; why should society support their activity? For the sake of pure art or knowledge? Alas, we are not classic Greeks, who scorned those who needed to profit from what they learned, or 18th century aristocrats, for whom science was a hobby. And even the material generated by these groups was pulled along by astronomy and astrology (for which a charge was made), geography, navigation, and mechanics. Society will now support mathematics generously only if it promises bottom-line benefits.
Now focus on the word "benefits." What is a benefit? Richard Hamming of the old Bell Telephone Laboratories said in a famous epigraph to his book on scientific computation, "The object of computation is not numbers but insight." Insight into a variety of physical and social processes, of course. But I perceive (40 years after Hamming's book and with a somewhat cynical eye) that the real object of computation, commercial and otherwise, is often neither numbers nor insight nor solutions to pressing problems, but worked on by physicists and mathematicians, to perfect money-making products. Often computer usages are then authorized by project managers who have little technical knowledge. The Descartesian precept cogito ergo sum has been replaced by producto ergo sum.
If, by chance, humanity benefits from this activity, then so much the better; everybody is happy. And if humanity suffers, the neo-Luddites will cry out and form chat groups on the Web, or the hackers will attack computer systems or humans. The techno-utopians will explain that you can't make omelets without breaking a few eggs. And pure mathematicians will follow along, moving closer to applications while justifying the purity of their pursuits to the administrators, politicians, and the public with considerable truth that one never knows in advance what products of pure imagination can be turned to society's benefit. The application of the theory of numbers to cryptography and the (Johann) Radon transform and its application to tomography have been displayed as shining examples of this. Using that most weasel of rhetorical expressions, "in principle," in principle, all mathematics is potentially useful.
I could use all my space describing many applications that seem now to be hot and are growing hotter. I will mention several and comment briefly on but a few of them. In selecting these few, I have ignored "pure" fields out of personal incompetence. I simply do not have the knowledge or authority to single out from a hundred expanding subfields the ones with particularly significant potential and how they have fared via multimedia. For more comprehensive and authoritative presentations, I recommend Mathematics: Frontiers and Perspectives and Mathematics Unlimited—2001 and Beyond.
Mathematics and the Physical and Engineering Sciences
These have been around since Galileo, but Newton's work was the great breakthrough. However, only in the past hundred years or so has theoretical mathematics been of any great use to technology. The pursuit of physical and engineering sciences is today unthinkable without significant computational power. The practice of aerodynamic design has altered significantly, but the "digital wind tunnel" has not yet arrived, and some have said it may never. Theories of turbulence are not yet in satisfactory shape—how to deal with widely differing simultaneous scales continues to perplex. Newtonians who deal with differential-integral systems must learn to share the stage with a host of probabilists with their stochastic equations. Withal, hurricane and tornado predictions continue to improve, perhaps more because of improvements in hardware (e.g., real-time data from aircraft or sondes and from nano-computers) than to the numerical algorithms used to deal with the numerous models that are in use. Predictions of earthquakes or of global warming are controversial and need work. Wavelet, chaos, and fractal theorists and multiresolution analysts are hard at work hoping to improve their predictions.
Mathematics and the Life Sciences
Mathematical biology and medicine are booming. There are automatic diagnoses. There are many models around in computational biology; most are untested. One of my old Ph.D. students has worked in biomolecular mathematics and designer drugs. He and numerous others are now attempting to model strokes via differential equations. Good luck!
I visited a large hospital recently and was struck by the extent that the aisles were absolutely clogged with specialized computers. Later, as a patient, I was all wired up and plugged into such equipment with discrete data and continuous waveforms displayed bedside and at the nurses' stations. Many areas of medical and psychological practice have gone or are going virtual. There is no doubt that we are now our own digital avatars and we are all living longer and healthier lives. In this development, mathematics, though way in the background and though not really understood by the resident physicians or nurses, has played a significant role.
Work on determining and analyzing the human genome sequences, with a variety of goals in mind and using essentially combinatorial and probabilistic methods, is a hot field. In the past decade, the cost of DNA sequencing has come down dramatically.
Genetic engineering on crops goes forward but has raised hackles and doomsday scenarios.
Mathematics and the Military Sciences
If mathematics contributes significantly to the life sciences, there is also mathematics that contributes to the "death sciences": war, both defensive and offensive. For the past 75 years, military problems have been a tremendous engine, supplying money and pulling both pure and applied formulations to new achievements: interior and exterior ballistics, creating bombs, missiles, rockets, antirocket rockets, drones, satellites, war-gaming strategies, and combat training in the form of realistic computer games. The use of mathematics in the service of war and defense will be around as long as aggression is a form of human and governmental behavior. Some authorities have claimed that aggression is built into the human brain. In any case, the psychology of aggression is an open field of study.
Mathematics and Entertainment
There is mathematics and entertainment through animation, simulation, and computer graphics. The ex-executive of Silicon Graphics opined some years ago that the future of the United States lay not in manufacturing nor in the production of food, but in producing a steady flow of entertainment for the rest of the world. Imagine this: a future president of the United States may have to warn us against the media-entertainment complex as Eisenhower did with the military- industrial complex.
But there is more! Through animation and simulation, the world of defense joins up with the world of entertainment and the world of medical technology. These worlds find common problems and can often share computer software. (There have been conferences on this topic.) Mickey Mouse flies the stealth bomber, and virtual surgery can be performed via the same sort of software products. There are now university departments devoted to the design of new video and simulation games, using humans as players, and with a wide variety of applications, including pure research. Young mathematicians have begun to offer their talents not just to university departments of mathematics but also to Hollywood and TV producers.
Mathematics and Money
Marriages of business and mathematics are booming. Property, business, and trade have always been tremendous consumers of low-level mathematics. In deep antiquity, they were probably the generators of such mathematics. But now it is no longer low-level. Zebra stripes (i.e., product identification, or UPC codes) have an interesting mathematical basis. Mathematics and business is a possible major in numerous universities, with professorial chairs in the subject. Software is sold for portfolio management and to automate income tax returns. Wall Street is totally computerized, nanotrading is almost continuous, and millions are playing the market using clever statistical strategies often of their own personal devising. The practice of arbitrage has generated theorems, textbooks, and software.
Mathematics and the Graphic Arts
Graphic art is being revolutionized along mathematical lines, a tendency—would you believe it?—that was present 3,000 years ago in the art of Egypt when some of their art was pixelized. Computer art and op art are now commonplace on gallery walls if a bit ho-hum. Is such art a kind of "soft mathematics"? When I consider the progress that computer art has made from the theoretical approximation theory of my research years, from the elementary paint programs developed in such places as the University of Utah a generation and a half ago, to the sophisticated productions of today's Pixar Animation Studios, my mind boggles. Three-dimensional printing, which makes a solid object from a digital model, advances the older programmed lathes. Two, three, and higher dimensional graphical presentations are more prosaic as art but are also important scientifically; they play an increasing role in presentation and interpretation.
Mathematics, Law, Legislation, and Politics
Law is just beginning to feel the effect of mathematization. Leibniz and Christian Wolff talked about this three centuries ago. Nicholas Bernoulli talked about it. Read Bernoulli's 1709 inaugural dissertation On the Use of Probability (Artis conjectandi) in Law.
There are now DNA identifications. The conjectured (or proved) liaison between Thomas Jefferson and Sally Hemings made the front pages. But can statisticians be trusted? "Experts" are often found testifying on both sides of a question.
Statistics are more and more entering the courts as evidence, and courts may soon require a resident statistician to interpret things for the judges, even as my university department has a full-time computer maven to resolve the questions and glitches that arise constantly. There are class action and discrimination suits based on statistical evidence. Multiple regression enters into the picture strongly. Mathematical algorithms themselves have been scrutinized and may be subject to litigation as part of intellectual property. In the burgeoning field of "jurimath" or "jurimetrics," there are now texts for lawyers and a number of journals. This field should be added to the roster of applications of mathematics and should be taught in colleges and law schools.
Consider polls. We spend millions and millions of dollars polling voters, polling consumers, asking people how they feel about anything at all. Consider the census. How should one count? Counting, the simplest, most basic of all mathematical operations, in the sharp theoretical sense, turns out to be a practical impossibility. Sampling is recommended; it reduces the variance but increases the discrepancy. It has been conjectured that sampling will increase the power of the minority party, hence the majority party is against it. As early as Nov. 30, 1998, the case was argued before the U.S. Supreme Court. Despite all these developments, we are far from Leibniz' dream of settling human disputes by computation.
Mathematics in the Service of Cross- or Trans-Media
Here are a few instances:
Music <-> score
Oral or audio <-> hard copy
Motion has been captured from output from a wired-up human, then analyzed and synthesized. (The animation of Gollum in Lord of the Rings was produced in this way.)
Balletic motion <-> Choreographic notation
Voice -> action. Following the instructions of an electronic voice, I push a few buttons and a ticket gets printed out, all paid for. But the buttons are becoming fewer as voice interpretation improves daily.
(Continues...) Excerpted from The BEST WRITING on MATHEMATICS 2013 by Mircea Pitici. Copyright © 2014 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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