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The Best Writing on Mathematics 2014

The Best Writing on Mathematics 2014

Mircea Pitici Editor
Copyright Date: 2015
Pages: 360
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    The Best Writing on Mathematics 2014
    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 2014makes 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 John Conway presents examples of arithmetical statements that are almost certainly true but likely unprovable; Carlo Séquin explores, compares, and illustrates distinct types of one-sided surfaces known as Klein bottles; Keith Devlin asks what makes a video game good for learning mathematics and shows why many games fall short of that goal; Jordan Ellenberg reports on a recent breakthrough in the study of prime numbers; Stephen Pollard argues that mathematical practice, thinking, and experience transcend the utilitarian value of mathematics; and much, much more.

    In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes an introduction by editor Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us-and where it is headed.

    eISBN: 978-1-4008-6530-7
    Subjects: Mathematics
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Table of Contents

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  1. Front Matter (pp. i-vi)
  2. Table of Contents (pp. vii-viii)
  3. Introduction (pp. ix-xxiv)
    Mircea Pitici

    Welcome to reading the fifth anthology in our series of recent writings on mathematics. Almost all the pieces included here were first published during 2013 in periodicals, as book chapters, or online.

    Much is written about mathematics these days, and much of it is good, amid a fair amount of uninspired writing. What does and what doesn’t qualify as good is sometimes difficult to decide. I have learned that strictly normative criteria for what “good writing” on mathematics should be are of little utility, and in fact, might be counterproductive. Good writing comes in many styles. I know it when...

  4. Mathematics and the Good Life (pp. 1-19)
    Stephen Pollard

    A full account of mathematics will identify the distinctive contributions mathematics makes to our success as individuals and as a species. That, in turn, requires us to reflect both on what mathematicians do and on what it means for humanity to flourish. After all, we cannot say how mathematics contributes to our success unless we have a good sense of what mathematicians accomplish and a good idea of what it means for us to succeed. To give our discussion some focus, I offer the following proposition:

    Mathematics makes one substantial contribution to human prosperity: it enhances our instrumental control over...

  5. The Rise of Big Data: How It’s Changing the Way We Think about the World (pp. 20-32)
    Kenneth Cukier and Viktor Mayer-Schönberger

    Everyone knows that the Internet has changed how businesses operate, governments function, and people live. But a new, less visible technological trend is just as transformative: “big data.” Big data starts with the fact that there is a lot more information floating around these days than ever before, and it is being put to extraordinary new uses. Big data is distinct from the Internet, although the web makes it much easier to collect and share data. Big data is about more than just communication: the idea is that we can learn from a large body of information things that we...

  6. Conway’s Wizards (pp. 33-38)
    Tanya Khovanova

    This article is about a puzzle invented by John H. Conway. Conway e-mailed it to me in June 2009. I wrote about it on my blog [1]. I also invented and posted a simplified version [2] and a generalized version [3]. But I never published a solution or a discussion. Now the time has come to do so.

    John Conway sent me a puzzle about wizards, which he invented in the 1960s. Here it is:

    Last night I sat behind two wizards on a bus and overheard the following:

    A: “I have a positive integral number of children, whose ages...

  7. On Unsettleable Arithmetical Problems (pp. 39-48)
    John H. Conway

    Before Fermat’s Last Theorem was proved, there was some speculation that it might be unprovable. Many people noticed that the theorem and its negation have a different status. The negation asserts that for somen> 2 there is annth power that is the sum of two smaller ones. Exhibiting these numbers proves the negation and disproves the theorem itself. So if one shows that the theorem is not disprovable, then one also shows that there exist no such nth powers and therefore that the theorem is true.

    However, the theorem could conceivably be true without being provable. In this...

  8. Color illustration section (pp. None)
  9. Crinkly Curves (pp. 49-63)
    Hayes Hayes

    In 1877, the German mathematician Georg Cantor made a shocking discovery. He found that a two-dimensional surface contains no more points than a one-dimensional line. Cantor compared the set of all points forming the area of a square with the set of points along one of the line segments on the perimeter of the square. He showed that the two sets are the same size. Intuition rebels against this notion. Inside a square you could draw infinitely many parallel line segments side by side. Surely an area with room for such an infinite array of lines must include more points...

  10. Why Do We Perceive Logarithmically? (pp. 64-73)
    Lav R. Varshney and John Z. Sun

    Whether it is hearing a predator approaching or a family member talking, whether it is perceiving the size of a cartload of chimpanzees or a pride of lions, whether it is seeing a ripe fruit or an explosion, the ability to sense and respond to the natural world is critical to our survival. Indeed, the world is a random place where all intensities of sensory stimuli can arise, from smallest to largest. We must register them all, from the buzz of an insect to the boom of an avalanche or an erupting volcano: our sensory perception must allow us to...

  11. The Music of Math Games (pp. 74-86)
    Keith Devlin

    Search online for video games and apps that claim to help your children (or yourself) learn mathematics, and you will be presented with an impressively large inventory of hundreds of titles. Yet hardly any survive an initial filtering based on seven very basic pedagogic “no-nos” that any game developer should follow if the goal is to use what is potentially an extremely powerful educational medium to help people learn math. A good math learning game or app should avoid:

    Confusing mathematics itself (which is really a way of thinking) with its representation (usually in symbols) on a flat, static surface....

  12. The Fundamental Theorem of Algebra for Artists (pp. 87-95)
    Bahman Kalantari and Bruce Torrence

    It’s simple. It’s beautiful. And it doesn’t get more fundamental: Every polynomial factors completely over the complex numbers. So says the fundamental theorem of algebra.

    If you are like us, you’ve been factoring polynomials more or less since puberty. What may be less clear is why. The short answer is that polynomials are the most basic and ubiquitous functions in existence. They are used to model all manner of natural phenomena, so solving polynomial equations is a fundamental skill for understanding our world. And thanks to the fundamental theorem, solving polynomial equations comes down to factoring. However, finding the factors...

  13. The Arts—Digitized, Quantified, and Analyzed (pp. 96-104)
    Nicole Lazar

    The inspiration for this column is an article I came across on the website ofFinancial Timesfrom June 15, 2013, titled “Big Data Meets the Bard.” The focus of the article is Stanford University’s “Literary Lab” run by Franco Moretti. In the lab, Moretti, his colleagues, and his students use data analysis techniques—and especially tools that we have come to associate with Big Data—to analyze plays, books, authors, and entire genres of literature. This got me thinking more broadly about the intersections between statistics and the arts, and between Big Data and the arts, which here I’ll...

  14. On the Number of Klein Bottle Types (pp. 105-127)
    Carlo H. Séquin

    A Klein bottle is a closed, single-sided mathematical surface of genus 2, sometimes described as a closed bottle for which there is no distinction between inside and outside. A canonical example of this surface is depicted in Figure 1A, with its characteristic “mouth” at the bottom where the “inside” of the surface connects to its “outside.” However, if one consults Wikipedia or some appropriate math text, one can find Klein bottles that look radically different, such as the twisted loop with a figure-8 profile (Figure 1B) or Lawson’s [9] single-sided surface of genus 2 (Figure 1C), which has been conjectured...

  15. Adventures in Mathematical Knitting (pp. 128-143)

    I have known how to knit since elementary school, but I can’t quite remember when I first started knitting mathematical objects. At the latest, it was during my first year of graduate school. I knitted a lot that year because I never got enough sleep and needed to keep myself awake during class. During the fall term I made a sweater for my dad, finishing the seams right after my last final, and in the spring I completed a sweater for my mom. Also that spring, during topology class, I knitted a Klein bottle, a mathematical surface that is infinitely...

  16. The Mathematics of Fountain Design: A Multiple-Centers Activity (pp. 144-155)
    Marshall Gordon

    Teachers of mathematics recognize the difficulty of reaching every student when the range of student abilities puts a considerable strain on the classroom discussion and time. In response to the problem, students are grouped so that those with greater mathematical aptitude help those who have difficulties. Although this approach is to be appreciated, it tends to mean that the more able students have less opportunity to explore further their own initiatives in mathematics, while those who have more difficulties find themselves on the receiving end with little opportunity to be in the role of enriching the mathematics experience for everyone,...

  17. Food for (Mathematical) Thought (pp. 156-164)
    Penelope Dunham

    To paraphrase an old line, “Waiter, what’s this potato chip doing in my calculus class?” The answer is, “It’s a concrete model for a mathematical concept, and it generates extra interest for students as they learn that concept.” In other words, the potato chip is an example of using food as a (tasty) manipulative to help students “touch it, feel it, learn it” in math class.

    Moyer defines manipulative materials as “objects designed to represent explicitly and concretely mathematical ideas that are abstract” [6, p. 176]. The use of manipulatives in activities at all levels of mathematics instruction is endorsed...

  18. Wondering about Wonder in Mathematics (pp. 165-187)
    Dov Zazkis and Rina Zazkis

    We start with recreating a personal encounter.

    A professor at a small university begins a lesson on fractals. He writes a simple-looking equationZ_{n+1}=(Z_{n})^{2}+C. He then describes an iterative process in which numbers (that correspond to points on the complex plane) are placed into the equation, yielding new numbers that are then placed back into the equation. At the same time, he quickly types an algorithm that tests the behavior of a grid of points. He finishes writing the code that accompanies his explanation, pauses for a second, says, “Here we go,” and presses . The image in Figure...

  19. The Lesson of Grace in Teaching (pp. 188-197)
    Francis Edward Su

    I’m honored, but I’m also really humbled to be giving this talk to a room full of great teachers because I know that each of you has a rich and unique perspective on teaching.* I had to ask myself: Could I really tell you anything significant about teaching?

    So I decided instead to talk about something else, something that at first may appear to have nothing to do with teaching, and yet it has everything to do with teaching.

    I want to talk about the biggest life lesson that I have learned and that I continue to learn over and...

  20. Generic Proving: Reflections on Scope and Method (pp. 198-215)
    Uri Leron and Orit Zaslavsky

    A generic proof is, roughly, a proof carried out on a generic example. We introduce the termgeneric provingto denote any mathematical or educational activity surrounding a generic proof. The notions of generic example, generic proof, and proof by generic example have been discussed by a number of scholars (Mason and Pimm 1984; Balacheff 1988; Rowland 1998; Malek and Movshovitz-Hadar 2011). All acknowledge the role of proof not only in terms of validating the conclusion of a theorem but, just as importantly, as a means of gaining insights about why the theorem is true. In particular, we support and...

  21. Extreme Proofs I: The Irrationality of \sqrt{2} (pp. 216-227)
    John H. Conway and Joseph Shipman

    Mathematicians often ask, “What is the best proof” of something, and indeed Erdős used to speak of “Proofs from the Book,” meaning, of course, God’s book. Aigner and Ziegler (1998) have attempted to reconstruct some of this Book.

    Here we take a different and more tolerant approach. We shouldn’t speak of “the best” proof because different people value proofs in different ways. Indeed one person’s value might oppose another’s. For example, a proof that quotes well-known results from Galois theory is valued negatively by someone who knows nothing of that theory but positively by the instructor in a course on...

  22. Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor (pp. 228-238)
    Michael J. Barany

    With the restoration of King Louis XVIII of France in 1814, one revolution had come to an end, but another was just beginning. Historians often describe the French Revolution of 1789, along with its reactions and repercussions, as the start of the modern era. For many historians of mathematics, however, the modern era began with the Bourbon Restoration and the mathematics of Augustin-Louis Cauchy.

    In a 1972 talk at a Mathematical Association of America sectional meeting, Judith Grabiner offered an important interpretation of this critical juncture in the history of mathematics [5]. Asking whether mathematical truth was time-dependent, Grabiner argued...

  23. Plato, Poincaré, and the Enchanted Dodecahedron: Is the Universe Shaped Like the Poincaré Homology Sphere? (pp. 239-250)
    Lawrence Brenton

    A staple of fantasy/horror stories goes like this. You are imprisoned in an enchanted room. You try to escape by walking out the door. But as you step out through the door—you step in through the window! Is that weird, or what?! Could the universe be like that?

    Well, I am not sure about the whole universe, but the surface of the Earth is certainly “like that.”

    In Figure 1A, suppose we want to escape the cruel world by exiting through the door stage right. In real life, when we step off the eastern edge of the world, we...

  24. Computing with Real Numbers, from Archimedes to Turing and Beyond (pp. 251-269)
    Mark Braverman

    We are so immersed in numbers in our daily lives that it is difficult to imagine that humans once got by without them. When numbers were finally introduced in ancient times, they were used to represent specific quantities (such as commodities, land, and time); for example, “four apples” is just a convenient way to rephrase “an apple and an apple and an apple and an apple”; that is, numbers had algorithmic meaning millennia before computers and algorithmic thinking became as pervasive as they are today. The natural numbers 1, 2, 3, … are the easiest to define and “algorithmize.” Given...

  25. Chaos at Fifty (pp. 270-287)
    Adilson E. Motter and David K. Campbell

    In classical physics, one is taught that given the initial state of a system, all of its future states can be calculated. In the celebrated words of Pierre Simon Laplace, “An intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it—an intelligence sufficiently vast to submit these data to analysis … for it, nothing would be uncertain and the future, as the past, would be present to its eyes” (Laplace 1902). Or, put another way, the clockwork universe holds true.

    Herein lies the rub:Exactknowledge of...

  26. Twenty-Five Analogies for Explaining Statistical Concepts (pp. 288-298)
    Roberto Behar, Pere Grima and Lluís Marco-Almagro

    Well-told analogies, anecdotes, and jokes are resources that teachers use to reinforce the transmission of ideas and concepts. What is more, when used correctly, they bring an informal tone to the class that becomes more enjoyable and sparks student interest.

    Recognizing the importance of analogies in the teaching and learning process is not new. Donnelly and McDaniel (1993, 2000) highlighted the possibility of using analogies for teaching in general, and the excellent article of Martin (2003) documents their use in teaching statistics. There are articles with compilations of analogies (Chanter 1983; Brewer 1989) and articles dealing with one specific analogy...

  27. College Admissions and the Stability of Marriage (pp. 299-307)
    David Gale and Lloyd S. Shapley

    The problem with which we shall be concerned relates to the following typical situation: A college is considering a set ofnapplicants of which it can admit a quota of onlyq. Having evaluated their qualifications, the admissions office must decide which ones to admit. The procedure of offering admission only to theqbest-qualified applicants is not generally satisfactory, for it cannot be assumed that all who are offered admission will accept. Accordingly, in order for a college to receiveqacceptances, it generally has to offer to admit more thanqapplicants. The problem of determining how...

  28. The Beauty of Bounded Gaps (pp. 308-314)
    Jordan Ellenberg

    Last May, Yitang “Tom” Zhang, a popular math professor at the University of New Hampshire, stunned the world of pure mathematics when he announced that he had proven the “bounded gaps” conjecture about the distribution of prime numbers—a crucial milestone on the way to the even more elusive twin primes conjecture, and a major achievement in itself.

    The stereotype, outmoded though it is, is that new mathematical discoveries emerge from the minds of dewy young geniuses. But Zhang is over 50. What’s more, he hasn’t published a paper since 2001. Some of the world’s most prominent number theorists have...

  29. Contributors (pp. 315-324)
  30. Notable Writings (pp. 325-332)
  31. Acknowledgments (pp. 333-334)
  32. Credits (pp. 335-336)