The Best Writing on Mathematics 2012

The Best Writing on Mathematics 2012

Mircea Pitici Editor
Copyright Date: November 2012
Pages: 376
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    The Best Writing on Mathematics 2012
    Book Description:

    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 2012 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 Robert Lang explains mathematical aspects of origami foldings; Terence Tao discusses the frequency and distribution of the prime numbers; Timothy Gowers and Mario Livio ponder whether mathematics is invented or discovered; Brian Hayes describes what is special about a ball in five dimensions; Mark Colyvan glosses on the mathematics of dating; 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 mathematician David Mumford 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.

    eISBN: 978-1-4008-4467-8
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Foreword: The Synergy of Pure and Applied Mathematics, of the Abstract and the Concrete
    (pp. ix-xvi)
    David Mumford

    All of us mathematicians have discovered a sad truth about our passion: It is pretty hard to tell anyone outside your field what you are so excited about! We all know the sinking feeling you get at a party when an attractive person of the opposite sex looks you in the eyes and asks—“What is it you do?” Oh, for a simple answer that moves the conversation along.

    Now Mircea Pitici has stepped up to the plate and for the third year running has assembled a terrific collection of answers to this query. He ranges over many aspects of...

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

    A little more than eight years ago I planned a series of “best writing” on mathematics with the sense that a sizable and important literature does not receive the notice, the consideration, and the exposure it deserves. Several years of thinking on such a project (for a while I did not find a publisher interested in my proposal) only strengthened my belief that the best of the nontechnical writings on mathematics have the potential to enhance the public reception of mathematics and to enrich the interdisciplinary and intradisciplinary dialogues so vital to the emergence of new ideas.

    The prevailing view...

  5. Why Math Works
    (pp. 1-7)
    Mario Livio

    Most of us take it for granted that math works—that scientists can devise formulas to describe subatomic events or that engineers can calculate paths for spacecraft. We accept the view, espoused by Galileo, that mathematics is the language of science and expect that its grammar explains experimental results and even predicts novel phenomena. The power of mathematics, though, is nothing short of astonishing. Consider, for example, Scottish physicist James Clerk Maxwell’s famed equations: not only do these four expressions summarize all that was known of electromagnetism in the 1860s, they also anticipated the existence of radio waves two decades...

  6. Is Mathematics Discovered or Invented?
    (pp. 8-20)
    Timothy Gowers

    The title of this chapter is a famous question. Indeed, perhaps it is a little too famous: It has been asked over and over again, and it is not clear what would constitute a satisfactory answer. However, I was asked to address it during the discussions that led to this volume, and since most of the participants in those discussions were not research mathematicians, I was in particular asked to give a mathematician’s perspective on it.

    One reason for the appeal of the question seems to be that people can use it to support their philosophical views. If mathematics is...

  7. The Unplanned Impact of Mathematics
    (pp. 21-29)
    Peter Rowlett

    As a child, I read a joke about someone who invented the electric plug and had to wait for the invention of a socket to put it in. Who would invent something so useful without knowing what purpose it would serve? Mathematics often displays this astonishing quality. Trying to solve real-world problems, researchers often discover that the tools they need were developed years, decades, or even centuries earlier by mathematicians with no prospect of, or care for, applicability. And the toolbox is vast, because once a mathematical result is proven to the satisfaction of the discipline, it doesn’t need to...

  8. An Adventure in theNth Dimension
    (pp. 30-42)
    Brian Hayes

    The area enclosed by a circle is πr² . The volume inside a sphere is 4/3πr³ . These are formulas I learned too early in life. Having committed them to memory as a schoolboy, I ceased to ask questions about their origin or meaning. In particular, it never occurred to me to wonder how the two formulas are related, or whether they could be extended beyond the familiar world of two-and three-dimensional objects to the geometry of higher-dimensional spaces. What’s the volume bounded by a four-dimensional sphere? Is there some master formula that gives the measure of a round object...

  9. Structure and Randomness in the Prime Numbers
    (pp. 43-49)
    Terence Tao

    The prime numbers 2, 3, 5, 7, . . . are one of the oldest topics studied in mathematics. We now have a lot of intuition as to how the primes should behave, and a great deal of confidence in our conjectures about the primes . . . but we still have a great deal of difficulty in proving many of these conjectures! Ultimately, this difficulty occurs because the primes are believed to behave pseudorandomly in many ways, and not to follow any simple pattern. We have many ways of establishing that a pattern exists . . . but how...

  10. The Strangest Numbers in String Theory
    (pp. 50-60)
    John C. Baez and John Huerta

    As children, we all learn about numbers. We start with counting, followed by addition, subtraction, multiplication, and division. But mathematicians know that the number system we study in school is but one of many possibilities. Other kinds of numbers are important for understanding geometry and physics. One of the strangest alternatives is the octonions. Largely neglected since their discovery in 1843, in the past few decades they have assumed a curious importance in string theory. And indeed, if string theory is a correct representation of the universe, they may be part of the reason the universe has the number of...

  11. Mathematics Meets Photography: The Viewable Sphere
    (pp. 61-78)
    David Swart and Bruce Torrence

    Right now, without moving from your seat or from where you are standing, look at your surroundings. Not just left and right but all the way around to whatever is directly behind you too. Look at the ceiling or the sky. Look at the floor below, or maybe it’s a desk or a laptop below your nose. What you can see from your single point of view is a viewable sphere. Perhaps it helps to picture an imaginary sphere surrounding your head with imagery printed on it that matches your surroundings.

    Of course, we need to refine this idea slightly...

  12. Dancing Mathematics and the Mathematics of Dance
    (pp. 79-92)
    SARAH-MARIE BELCASTRO and Karl Schaffer

    If you’re not a dancer—and even if you are—you may be wondering how on earth mathematics and dance are related. We’re not talking about social or folk dances such as contra dance—a popular dance form among mathematicians—but about dance as an artistic endeavor. There are superficial links such as counting steps or noticing shapes, but also deeper connections, such as mathematical concepts arising naturally in dance, mathematics inspiring dance, or using mathematics to solve choreographic problems.

    We are both mathematicians and both dancers. For about 20 years, we have independently thought about ways in which dance...

  13. Can One Hear the Sound of a Theorem?
    (pp. 93-112)
    Rob Schneiderman

    Mathematics and music have been intertwined in a long-running drama that stretches back to ancient times and has featured contributions from many great minds, including Pythagoras, Euclid, Mersenne, Descartes, Galileo, Euler, Helmholtz, and many others (e.g., [1]). Applications of mathematics to music continue to develop in today’s digital world, which also supports active communities of musicologists and experimental composers who examine music methodically, often using mathematical elements. In light of the recent wave of musico-mathematical books, blogs, journals, and even articles in the Notices of the American Mathematical Society (Notices), this multifaceted side of the mathematical world deserves reexamination. Although...

  14. Flat-Unfoldability and Woven Origami Tessellations
    (pp. 113-128)
    Robert J. Lang

    The field of origami tessellations has seen explosive growth over the past 20 years. Interpreted broadly, an “origami tessellation” is a figure folded from a single sheet of paper in which the surface is divided up (tessellated) into a highly geometric pattern that is created by the folded edges and/or the transmission image of the varying layers (if folded from translucent paper and backlit), so that the pattern of the folded edges, rather than the outline of the figure, provides the dominant aesthetic. Though many origami tessellations are derived from regular tilings of the plane, the field of such 2D...

  15. A Continuous Path from High School Calculus to University Analysis
    (pp. 129-134)
    Timothy Gowers

    If I was asked to name the two most notable ways in which university-level mathematics differs from high school-level mathematics, then I would say that they were abstraction and rigor. Early courses at university in subjects such as group theory and linear algebra introduce students to the axiomatic way of thinking, while a first course in mathematical analysis introduces them to rigorous proofs of statements that they will hitherto have justified only informally, if at all. It is often claimed that mathematical analysis is difficult to learn because to understand it one must learn to think in a new way....

  16. Mathematics Teachers’ Subtle, Complex Disciplinary Knowledge
    (pp. 135-140)
    Brent Davis

    What mathematical competencies must a teacher have to teach the subject well? This question has proven difficult to investigate (1). A current view is that teachers’ knowledge of mathematics “remains inert in the classroom unless accompanied by a rich repertoire of mathematical knowledge and skills relating directly to the curriculum, instruction, and student learning” (2). Unfortunately, there is no consensus on which “knowledge and skills” might activate teachers’ inert knowledge. Two perspectives prevail, neither with a research base that enables strong claims about practice. The majority of current studies focus on explicit knowledge of curriculum content and instructional strategies. Such...

  17. How to Be a Good Teacher Is an Undecidable Problem
    (pp. 141-148)
    Erica Flapan

    I began teaching my own classes when I was in graduate school. At that time, I never gave much thought to the question of how to be a good teacher. I lectured, following the book, interacting with the students, explaining the material step-by-step, and working out sample problems. The students seemed to appreciate my energy, enthusiasm, clarity, and willingness to answer their questions, and that’s all there was to it. I continued teaching quite happily in this manner throughout graduate school and two postdoctoral appointments.

    Then I got a tenure-track job at a liberal arts college and suddenly began getting...

  18. How Your Philosophy of Mathematics Impacts Your Teaching
    (pp. 149-162)
    Bonnie Gold

    “My philosophy of mathematics? I don’t have one! I’m a mathematician, not a philosopher. I leave philosophical questions to the philosophers.” Maybe. Or perhaps you are among those mathematicians who are interested in the philosophy of mathematics. Whatever your attitude toward the philosophy of mathematics, when you teach mathematics, you do in fact take, and teach your students, positions on philosophical issues concerning mathematics. If you do not think about them, then you probably acquired your positions from your teachers when they imposed them on you without discussion. Furthermore, you may find, if you do examine the positions you are...

  19. Variables in Mathematics Education
    (pp. 163-172)
    Susanna S. Epp

    Variables are of critical importance in mathematics. For instance, Felix Klein wrote in 1908 that “one may well declare that real mathematics begins with operations with letters” [3], and Alfred Tarski wrote in 1941 that “the invention of variables constitutes a turning point in the history of mathematics” [5]. In 1911, A. N. Whitehead expressly linked the concepts of variables and quantification to their expressions in informal English when he wrote, “The ideas of ‘any’ and ‘some’ are introduced to algebra by the use of letters. . . . it was not till within the last few years that it...

  20. Bottom Line on Mathematics Education
    (pp. 173-175)
    David Mumford and Sol Garfunkel

    First some axioms: Mathematics is honestly useful for all citizens. It can help them in school, at work, as citizens and in their daily lives. This is the reason we teach mathematics every year from kindergarten through the end of high school. The mathematical education of the general public is a priority of our educational system above and beyond the education of future mathematicians and scientists.

    What follows from these axioms is that we need a system of mathematics education that seeks first and foremost to recognize the mathematical needs of average citizens and is designed to ensure that those...

  21. History of Mathematics and History of Science Reunited?
    (pp. 176-185)
    Jeremy Gray

    How to write the history of modern mathematics? This question, in itself no harder or less capable of an answer than the broader question of how to write the history of modern science, should be part of that broader question, but it has become separated. Recent initiatives, however, suggest that these questions can once again be raised and discussed together. There are in each case several fundamental latent issues. The question of how we should do something invites us to consider who “we” are and for whom we are doing it. I duck the first of these considerations and note...

  22. Augustus De Morgan behind the Scenes
    (pp. 186-196)
    Charlotte Simmons

    Augustus De Morgan (1806– 1871) was a nineteenth century mathematician and prolific writer, author of more than 160 papers and 18 textbooks on algebra, arithmetic, trigonometry, probability, logic, and calculus, plus 850 articles in the popular, working-class oriented Penny Cyclopedia [7]. Here, however, we explore his contributions from behind the scenes, as a mentor to other mathematicians. Both Sir William Rowan Hamilton and George Boole, for example, two of the greatest algebraists of the nineteenth century, were close friends of De Morgan. During the period in which they produced some of their greatest work, both were in regular correspondence with...

  23. Routing Problems: A Historical Perspective
    (pp. 197-208)
    Giuseppe Bruno, Andrea Genovese and Gennaro Improta

    In 1741, Leonhard Euler published (in the Commentarii of the Saint Petersburg Academy) a paper presenting some results related to the so-called Seven Bridges of Königsberg Problem. The Pregel river (Pregolja in Russian), coming from the east, crosses Lithuania and enters a Russian enclave (once named Eastern Prussia, between Lithuania and Poland) whose main city is Kaliningrad (the ancient Königsberg). The two branches of the river (Novaya Pregolja and Staraya Pregolja) cross Königsberg, forming an island in the heart of the city before merging and leading to Vistula Lagoon and then to the Baltic Sea.¹ Königsberg city center was composed...

  24. The Cycloid and Jean Bernoulli
    (pp. 209-215)
    Gerald L. Alexanderson

    Johann (Jean) Bernoulli (1667– 1748), the younger brother of Jacob (Jacques) Bernoulli, was a member of a large family of respected spice traders and scholars in Basel. Originally Flemish, they had fled to Switzerland to avoid religious persecution at the hands of the Spanish, who then occupied the Low Countries.

    The 1742 Opera Omnia of Johann is a four-volume set that includes not only his mathematics but also work on fermentation and on the design of naval vessels. Handsomely produced by the Swiss firm of Bousquet, the first volume opens with a frontispiece portrait of Bernoulli (Figure 1) followed by...

  25. Was Cantor Surprised?
    (pp. 216-233)
    Fernando Q. Gouvêa

    Mathematicians love to tell each other stories. We tell them to our students, too, and they eventually pass them on. One of our favorites, and one that I heard as an undergraduate, is the story that Cantor was so surprised when he discovered one of his theorems that he said, “I see it, but I don’t believe it!” The suggestion was that sometimes we might have a proof, and therefore know that something is true, but nevertheless still find it hard to believe.

    That sentence can be found in Cantor’s extended correspondence with Dedekind about ideas that he was just...

  26. Why Is There Philosophy of Mathematics at All?
    (pp. 234-254)
    Ian Hacking

    Mathematics is the only specialist branch of human knowledge that has consistently obsessed many dead great men in the Western philosophical canon. Not all, for sure, but Plato, Descartes, Leibniz, Kant, Husserl, and Wittgenstein form a daunting array. And that list omits the angry skeptics about the significance of mathematical knowledge, such as Berkeley and Mill, and the logicians, such as Aristotle and Russell.

    Why has mathematics mattered to so many famous philosophers? And why does it infect, in many cases, their entire philosophies? Aside from the naysayers, such as Mill, it is first of all because they have experienced...

  27. Ultimate Logic: To Infinity and Beyond
    (pp. 255-261)
    Richard Elwes

    When David Hilbert left the podium at the Sorbonne in Paris, France, on August 8, 1900, few of the assembled delegates seemed overly impressed. According to one contemporary report, the discussion following his address to the second International Congress of Mathematicians was “rather desultory.” Passions seem to have been more inflamed by a subsequent debate on whether Esperanto should be adopted as mathematics’ working language.

    Yet Hilbert’s address set the mathematical agenda for the twentieth century. It crystallized into a list of 23 crucial unanswered questions, including how to pack spheres to make best use of the available space, and...

  28. Mating, Dating, and Mathematics: It’s All in the Game
    (pp. 262-272)
    Mark Colyvan

    Why do people stay together in monogamous relationships? Love? Fear? Habit? Ethics? Integrity? Desperation? In this chapter, I consider a rather surprising answer that comes from mathematics. It turns out that cooperative behavior, such as mutually faithful marriages, can be given a firm basis in a mathematical theory known as game theory. I suggest that faithfulness in relationships is fully accounted for by narrow self-interest in the appropriate game theory setting. This is a surprising answer because faithful behavior is usually thought to involve love, ethics, and caring about the well-being of your partner. It seems that the game theory...

  29. Contributors
    (pp. 273-280)
  30. Notable Texts
    (pp. 281-284)
  31. Acknowledgments
    (pp. 285-286)
  32. Credits
    (pp. 287-288)