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The Best Writing on Mathematics 2011

The Best Writing on Mathematics 2011

Mircea Pitici Editor
Copyright Date: November 2011
Pages: 414
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  • Book Info
    The Best Writing on Mathematics 2011
    Book Description:

    This 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 2011 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 Ian Hacking discusses the salient features that distinguish mathematics from other disciplines of the mind; Doris Schattschneider identifies some of the mathematical inspirations of M. C. Escher's art; Jordan Ellenberg describes compressed sensing, a mathematical field that is reshaping the way people use large sets of data; Erica Klarreich reports on the use of algorithms in the job market for doctors; 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 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.

    eISBN: 978-1-4008-3954-4
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-x)
  3. Foreword: Recreational Mathematics
    (pp. xi-xvi)
    Freeman Dyson

    Hobbies are the spice of life. Recreational mathematics is a splendid hobby which young and old can equally enjoy. The popularity of Sudoku shows that an aptitude for recreational mathematics is widespread in the population. From Sudoku it is easy to ascend to mathematical pursuits that offer more scope for imagination and originality. To enjoy recreational mathematics you do not need to be an expert. You do not need to know the modern abstract mathematical jargon. You do not need to know the difference between homology and homotopy. You need only the good old nineteenth-century mathematics that is taught in...

  4. Introduction
    (pp. xvii-xxx)
    Mircea Pitici

    This new volume in the series of The Best Writing on Mathematics brings together a collection of remarkable texts, originally printed during 2010 in publications from several countries. A few exceptions from the strict timeframe are inevitable, due to the time required for the distribution, surveying, reading, and selection of a vast literature, part of it coming from afar.

    Over the past decade or so, writing about mathematics has become a genre, with its own professional practitioners—some highly talented, some struggling to be relevant, some well established, some newcomers. Every year these authors, considered together, publish many books. This...

  5. What Is Mathematics For?
    (pp. 1-12)
    Underwood Dudley

    A more accurate title is “What is mathematics education for?” but the shorter one is more attention-getting and allows me more generality. My answer will become apparent soon, as will my answer to the sub-question of why the public supports mathematics education as much as it does.

    So that there is no confusion, let me say that by “mathematics” I mean algebra, trigonometry, calculus, linear algebra, and so on: all those subjects beyond arithmetic. There is no question about what arithmetic is for or why it is supported. Society cannot proceed without it. Addition, subtraction, multiplication, division, percentages: though not...

  6. A Tisket, a Tasket, an Apollonian Gasket
    (pp. 13-26)
    Dana Mackenzie

    In the spring of 2007 I had the good fortune to spend a semester at the Mathematical Sciences Research Institute in Berkeley, an institution of higher learning that takes “higher” to a whole new extreme. Perched precariously on a ridge far above the University of California at Berkeley campus, the building offers postcard-perfect vistas of the San Francisco Bay, 1,200 feet below. That’s on the west side. Rather sensibly, the institute assigned me an office on the east side, with a view of nothing much but my computer screen. Otherwise I might not have gotten any work done.

    However, there...

  7. The Quest for God’s Number
    (pp. 27-34)
    Rik van Grol

    The Rubik’s cube triggered one of the largest puzzle crazes in the world. The small mechanical puzzle, invented by Ernő Rubik in Hungary, has sold more than 350 million copies. Although it has existed since 1974, the popularity of the cube skyrocketed around 1980, when the cube was introduced outside of Hungary. In the early days, simply solving the puzzle was the main issue, especially because no solution books were available and there was no Internet. But solving the puzzle in the shortest amount of time was also hot news. In the early 1980s the best times were on the...

  8. Meta-morphism: From Graduate Student to Networked Mathematician
    (pp. 35-42)
    Andrew Schultz

    While the stereotypical mathematician is a hermit locked alone in his office, the typical mathematician is far from a solitary explorer. A great amount of the mathematics produced today is created collaboratively, spurred into existence during those quintessentially mathematical social interactions: on chalkboards following a seminar talk, on napkins during a coffee break at a conference, on the back of a coaster at a pub. Though it often isn’t clear to those wading through graduate programs, one of the key metamathematical skills one should develop while working on a master’s or Ph.D. is the ability to participate in this social...

  9. One, Two, Many: Individuality and Collectivity in Mathematics
    (pp. 43-50)
    Melvyn B. Nathanson

    “Fermat’s last theorem” is famous because it is old and easily understood, but it is not particularly interesting. Many, perhaps most, mathematicians would agree with this statement, though they might add that it is, nonetheless, important because of the new mathematics created in the attempt to solve the problem. By solving Fermat, Andrew Wiles became one of the world’s best known mathematicians, along with John Nash, who achieved fame by being crazy, and Theodore Kaczynski, the Unabomber, by killing people.

    Wiles is known not only because of the problem he solved, but also because of how he solved it. He...

  10. Reflections on the Decline of Mathematical Tables
    (pp. 51-54)
    Martin Campbell-Kelly

    For some people it’s typewriters. For other people it’s mechanical calculating machines that bring a nostalgic tear to the eye. For me it’s mathematical tables. The sight—even the smell—of a set of four-figure tables transports me to my distant school and college days. You can still find mathematical tables—their yellowed pages filled with decimal digits and not much else—in secondhand bookstores and occasionally on eBay. I once thought I might like to collect mathematical tables, but then I discovered from the Index to Mathematical Tables¹ that many hundreds of tables have been published. Even a selective...

  11. Under-Represented Then Over-Represented: A Memoir of Jews in American Mathematics
    (pp. 55-66)
    Reuben Hersh

    When I studied at the Courant Institute of NYU from 1957 to 1962, its Jewish (specifically, Ashkenazi) flavor was impossible to miss. Of course, it was in large part the creation of Richard Courant, who came to New York in 1934 as a Jewish refugee expelled by Adolf Hitler from his post as the leader of the great and famous mathematical school at Göttingen in Germany. Two of NYU’s most important professors, Kurt Friedrichs and Fritz John, had been Courant’s students at Göttingen [13]. Lipman Bers was also a refugee. Of the younger members of the brilliant faculty, Peter Lax...

  12. Did Over-Reliance on Mathematical Models for Risk Assessment Create the Financial Crisis?
    (pp. 67-74)
    David J. Hand

    The German chemist Baron Justus von Liebig said, “We are too much accustomed to attribute to a single cause that which is the product of several” (von Liebig, 1872). It seems to me that this is perfectly reasonable. So that our limited human brains can cope with the awesome complexity of the natural world, and the corresponding complexity of the artificial globalized society and economy we have constructed, we naturally seek to reduce them to simple components. And then we try to identify the most important of these components and focus our attention on this.

    An illustration of this single...

  13. Fill in the Blanks: Using Math to Turn Lo-Res Datasets into Hi-Res Samples
    (pp. 75-81)
    Jordan Ellenberg

    In the early spring of 2009, a team of doctors at the Lucile Packard Children’s Hospital at Stanford University lifted a 2-year-old into an MRI scanner. The boy, whom I’ll call Bryce, looked tiny and forlorn inside the cavernous metal device. The stuffed monkey dangling from the entrance to the scanner did little to cheer up the scene. Bryce couldn’t see it, in any case; he was under general anesthesia, with a tube snaking from his throat to a ventilator beside the scanner. Ten months earlier, Bryce had received a portion of a donor’s liver to replace his own failing...

  14. The Great Principles of Computing
    (pp. 82-92)
    Peter J. Denning

    Computing is integral to science—not just as a tool for analyzing data, but as an agent of thought and discovery.

    It has not always been this way. Computing is a relatively young discipline. It started as an academic field of study in the 1930s with a cluster of remarkable papers by Kurt Gödel, Alonzo Church, Emil Post, and Alan Turing. The papers laid the mathematical foundations that would answer the question “what is computation?” and discussed schemes for its implementation. These men saw the importance of automatic computation and sought its precise mathematical foundation. The various schemes they each...

  15. Computer Generation of Ribbed Sculptures
    (pp. 93-114)
    James Hamlin and Carlo H. Séquin

    The 28-foot-tall Solstice sculpture by Charles Perry (Figure 1), located in downtown Tampa, Florida, is a prime example of the “ribbed sculptures” to be discussed here. Ribbed sculptures offer a translucent, “airy” presence in indoor as well as outdoor settings. Because of the substantial open space between the ribs, they do not cast harsh shadows or block views completely. Moreover, they are reasonably cost effective to be constructed at a large scale—much less expensive than large free-form bronze sculptures, investment cast from many individual molds.

    These ribbed sculptures may trace their roots to the pioneering work of some constructivist...

  16. Lorenz System Offers Manifold Possibilities for Art
    (pp. 115-120)
    Barry A. Cipra

    The Lorenz attractor has been a favorite of mathematical lepidopterists ever since chaos theory took off in the 1970s. First described by MIT meteorologist Edward Lorenz in 1963, the delicate butterfly wings that unfurl from an unassuming cocoon of simple equations have graced the pages of innumerable papers in dynamical systems. But for Bernd Krauskopf and Hinke Osinga, there’s an equally attractive geometric counterpart to Lorenz’s eponymous point set: a smoothly convoluted surface known as the Lorenz manifold.

    Krauskopf and Osinga, longstanding collaborators in the Department of Engineering Mathematics at the University of Bristol in the U.K., presented recent results...

  17. The Mathematical Side of M. C. Escher
    (pp. 121-149)
    Doris Schattschneider

    While the mathematical side of Dutch graphic artist M. C. Escher (1898–1972) is often acknowledged, few of his admirers are aware of the mathematical depth of his work. Probably not since the Renaissance has an artist engaged in mathematics to the extent that Escher did, with the sole purpose of understanding mathematical ideas in order to employ them in his art. Escher consulted mathematical publications and interacted with mathematicians. He used mathematics (especially geometry) in creating many of his drawings and prints. Several of his prints celebrate mathematical forms. Many prints provide visual metaphors for abstract mathematical concepts; in...

  18. Celebrating Mathematics in Stone and Bronze
    (pp. 150-168)
    Helaman Ferguson and Claire Ferguson

    I celebrate mathematics with sculpture and sculpture with mathematics. Eons-old stone strikes me as a perfect medium through which to celebrate timeless mathematics.

    I used to simplify life by putting science in one room and art in another. This avoided exposing my entangled soul, lest someone think me not sufficiently dedicated to one discipline or the other. Keep science and art separated, my generation was informed. “You can’t do both.” Parents advised, “If you can do science and have a lick of sense, you’d better. Artists starve.”

    Now we live in a golden age of both art and science. More...

  19. Mathematics Education: Theory, Practice, and Memories over 50 Years
    (pp. 169-187)
    John Mason

    Having been invited to look back over my life in mathematics education, I take the liberty of recalling some of the most stimulating moments as they come back to me, in an attempt to analyze what mathematics education has been about for me.* In particular I want to suggest that while the field has maintained and even widened the gap between theory and practice, it is incumbent upon us to remain steadfast that the purpose of our work is to understand and contribute to student learning of mathematics. One way I have consistently attempted to do this is to try...

  20. Thinking and Comprehending in the Mathematics Classroom
    (pp. 188-202)
    Douglas Fisher, Nancy Frey and Heather Anderson

    Literacy—reading, writing, speaking, and listening—is a critical foundational skill that provides individuals access to information in all other disciplines and domains. As teachers and researchers, we know that literacy impacts every aspect of a person’s life, from success in school and work to living a productive life. As Shanahan (2007) noted in his International Reading Association keynote, low levels of literacy put people at risk in all kinds of ways, from being taken advantage of by scam artists to not understanding health information.

    Literacy involves more than learning how to read. Breaking the code and developing fluency are...

  21. Teaching Research: Encouraging Discoveries
    (pp. 203-218)
    Francis Edward Su

    What does teaching have to do with research or discovery?* What does it take to turn a learner into a discoverer? Or to turn a teacher into a co-adventurer? A handful of experiences—from teaching a middle school math class to doing research with undergraduates—have changed the way that I would answer these questions. Some of the lessons I’ve learned have surprised me.

    The title of this article may seem a little puzzling. After all, the words teaching and research usually only appear in the same sentence when separated by the word and, on a list of a faculty...

  22. Reflections of an Accidental Theorist
    (pp. 219-235)
    Alan H. Schoenfeld

    Many years ago, David Wheeler asked me to write “Confessions of an Accidental Theorist” (Schoenfeld, 1987), in which I described how I had come to examine the research issues on which I had focused.* The SIG/ RME Senior Scholar Award provides me with a wonderful opportunity to reflect once again on those and related issues. I am truly grateful for the opportunity.

    The word accidental in the articles’ titles refers to the fact that when I began doing both mathematical and educational research, I was “theory-neutral.” In mathematics, unless one worries about foundations (logic), one just goes about one’s work:...

  23. The Conjoint Origin of Proof and Theoretical Physics
    (pp. 236-256)
    Hans Niels Jahnke

    Historians of science and mathematics have proposed three different answers to the question of why the Greeks invented proof and the axiomatic-deductive organization of mathematics (see Szabó1960, 356 ff.).*

    The socio-political thesis claims a connection between the origin of mathematical proof and the freedom of speech provided by Greek democracy, a political and social system in which different parties fought for their interests by way of argument. According to this thesis, everyday political argumentation constituted a model for mathematical proof.

    The internalist thesis holds that mathematical proof emerged from the necessity to identify and eliminate incorrect statements from the corpus...

  24. What Makes Mathematics Mathematics?
    (pp. 257-285)
    Ian Hacking

    We have seldom paused to ask what counts as mathematics. Certainly we have thought a good deal about the nature of mathematics—in my case, for example, about the logicist claim that mathematics is logic.¹ But we took “mathematics” for granted, and seldom reflected on why we so readily recognize a conjecture, a fact, a proof idea, a piece of reasoning, or a sub-discipline, as mathematical. We asked sophisticated questions about which parts of mathematics are constructive, or about set theory. But we shied away from the naïve question of why so many diverse topics addressed by real-life mathematicians are...

  25. What Anti-realism in Philosophy of Mathematics Must Offer
    (pp. 286-311)
    Feng Ye

    This article proposes a new approach to anti-realism in the philosophy of mathematics. The realism versus anti-realism debate in the philosophy of mathematics comes from some conflicting intuitions regarding mathematics. The basic intuition favoring realism (or anti-anti-realism) is (Burgess 2004; Rosen and Burgess 2005; Colyvan 1999, 2002; Baker 2001, 2005):¹ as long as mathematicians and scientists attempt to refer to mathematical entities and assert mathematical theorems in their best theories, we already have our best reason to believe that mathematical entities exist and mathematical theorems are true, because we should respect scientists’ understanding of their own theories and take their...

  26. Seeing Numbers
    (pp. 312-329)
    Ivan M. Havel

    In his influential book The Principles of Psychology (1890) the well-known psychologist and philosopher William James listed seven “elementary mental categories” that he postulated as having a natural origin [Jam07, p. 629]. In an alleged order of genesis he listed, in the third place, after the ideas of time and space, the idea of number.

    Mentioning the idea of number along with the ideas of time and space as something natural is interesting both philosophically and from the viewpoint of cognitive science. In this respect it is worth noting that one of the symptomatic features of mainstream cognitive science is...

  27. Autism and Mathematical Talent
    (pp. 330-335)
    Ioan James

    Autism is a developmental or personality disorder, not an illness, but autism can coexist with mental illnesses such as schizophrenia and manic-depression. It shows itself in early childhood and is present throughout life; sometimes it becomes milder in old age. Nowadays it is recognized as a wide spectrum of disorders, with classical autism, where the individual is wrapped up in his or her own private world, at one extreme. It is estimated that in the United Kingdom slightly under one percent of the population, about half a million people, have a disorder on the autism spectrum. The corresponding figure for...

  28. How Much Math Is Too Much Math?
    (pp. 336-346)
    Chris J. Budd and Rob Eastaway

    Mathematics is a difficult subject to communicate to the general public, for reasons that we will explore in this article. It also takes time and energy to communicate it well. So why do we bother communicating in the first place, and what do we hope to achieve when we attempt to communicate math to any audience, whether it is a primary school class, bouncing off the walls with enthusiasm, or a bored class of teenagers on the last lesson of the afternoon? We always have to tread a narrow line between boring our audience with technicalities at one end, and...

  29. Hidden Dimensions
    (pp. 347-355)
    Marianne Freiberger

    That geometry should be relevant to physics is no surprise—after all, space is the arena in which physics happens. What is surprising, though, is the extent to which the geometry of space actually determines physics and just how exotic the geometric structure of our Universe appears to be.

    One mathematician who’s got first-hand experience of the fascinating interplay between physics and geometry is Shing-Tung Yau. In a new book called The Shape of Inner Space (co-authored by Steve Nadis) Yau describes how the strange geometrical spaces he discovered turned out to be just what theoretical physicists needed in their...

  30. Playing with Matches
    (pp. 356-366)
    Erica Klarreich

    The mass entry of women into the American workforce in the 20th century has, proverbially, turned family life into a juggling act, with couples coordinating work schedules, childcare, and chores in days that never seem quite long enough.

    Imagine, then, how difficult it is for professional couples—such as medical students and academics—who have to relocate just to find jobs in the first place. For these couples, solving the “two-body problem” can be far from trivial. All too often, one member of a couple has to decide about a job offer before the other has lined up any prospects....

  31. Notable Texts
    (pp. 367-370)
  32. Contributors
    (pp. 371-378)
  33. Acknowledgments
    (pp. 379-380)
  34. Credits
    (pp. 381-383)