Mathematics without Apologies

Mathematics without Apologies: Portrait of a Problematic Vocation

michael harris
Copyright Date: 2015
Pages: 464
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  • Book Info
    Mathematics without Apologies
    Book Description:

    What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers-for the sake of truth, beauty, and practical applications-this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.

    Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party?

    Disarmingly candid, relentlessly intelligent, and richly entertaining,Mathematics without Apologiestakes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

    eISBN: 978-1-4008-5202-4
    Subjects: Mathematics, History

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. preface
    (pp. ix-xviii)
  4. acknowledgments
    (pp. xix-xxii)
  5. Part I
    • chapter 1 Introduction: The Veil
      (pp. 3-6)

      The next sentence of Hilbert’s famous lecture at the Paris International Congress of Mathematicians (ICM), in which he proposed twenty-three problems to guide research in the dawning century, claims that “History teaches the continuity of the development of science.”¹ We would still be glad to lift the veil, but we no longer believe in continuity. And we may no longer be sure that it’s enough to lift a veil to make our goals clear to ourselves, much less to outsiders.

      The standard wisdom is now that sciences undergo periodic ruptures so thorough that the generations of scientists on either side...

    • chapter 2 How I Acquired Charisma
      (pp. 7-40)

      My mathematical socialization began during the prodigious summer of 1968. While my future colleagues chanted in the Paris streets by day and ran the printing presses by night, helping to prepare the transition from structuralism to poststructuralism; while headlines screamed of upheavals—The Tet offensive! The Prague spring! Student demonstrations in Mexico City!—too varied and too numerous for my teenage imagination to put into any meaningful order; while cities across America burst into flames in reaction to the assassination of Martin Luther King and continued to smolder, I was enrolled in the Temple University summer program in mathematics for...

    • chapter α How to Explain Number Theory at a Dinner Party (First Session: Primes)
      (pp. 41-53)

      During the spring of 2008 I was invited by the Columbia University mathematics department to deliver the Samuel Eilenberg lectures—a perfect illustration of the Matthew Effect described in the previous chapter. The appointment involved living away from my family for several months. Working late in the department one Friday evening, I must have looked even more forlorn than usual, because a colleague passing my open door decided on the spot to invite me home to dinner. Several other mathematicians had been invited, along with a neighbor from another department and the neighbor’s visiting friend, a young British woman who...

    • chapter 3 Not Merely Good, True, and Beautiful
      (pp. 54-78)

      If it has become urgent for mathematicians to take up the “why” question, to which we are now ready to turn, it is because the professional autonomy to which we have grown attached is challenged by at least two proposals for reconfiguration. The first proposal is neither new nor specific to mathematics. The economic crisis that began in 2008 placed universities under enormous stress. Cutbacks have been especially severe in humanities departments, where the elimination of entire pro grams has become routine, but the crisis has also intensified existing pressure to subordinate scientific research in Europe and North America to...

    • chapter 4 Megaloprepeia
      (pp. 79-108)

      Economics originated in and has never completely broken with moral philosophy. A president of the American Economic Association recalled in 1968 that “when I was a student, economics was still part of the moral sciences tripos at Cambridge University.” On the grounds of the discipline’s history and its vocabulary (goods, utility, satisfaction) and because

      the fundamental principle that we should count all costs . . . and evaluate all rewards . . . is one which emerges squarely out of economics and which is at least a preliminary guideline in the formation of the moral judgment,

      “mathematical ethics” is a...

    • chapter β How to Explain Number Theory at a Dinner Party (Second Session: Equations)
      (pp. 109-127)

      Everyone who attends high school learns to solve simple algebraic equations. Alinear equationwith a single variablexhas exactly one solution;¹ for example,

      3x + 1 = 7

      has the unique solutionx= 2, by which we mean that (a) 3 × 2 + 1 = 7 and (b) if you put any other number in the place of 2, the two sides will no longer be equal.

      Aquadratic equationcan have two solutions, no solutions, or (more rarely) a single solution:

      x² – 4x + 3 = 0

      has the two solutionsx= 3 and...

    • bonus chapter 5 An Automorphic Reading of Thomas Pynchon’s Against the Day (Interrupted by Elliptical Reflections on Mason & Dixon)
      (pp. 128-138)

      Thomas Pynchon, postmodern author, is commonly said to have a non linear narrative style. Inger H. Dalsgaard suggests that “a novel likeAgainst the Daymay be read in non-linear fashion, in keeping with the operations of a time machine.”¹ No critic, however—not even the “seventeen of the foremost heavyweights from over forty years of Pynchon criticism”² who contributed to theCambridge Companion to Thomas Pynchon—seems to have taken seriously the possibility, to be explored in this chapter, that his narrative style might in fact bequadratic.

      Google gives no matches whatsoever for “quadratic narrative style,”³ and this...

  6. Part II
    • chapter 6 Further Investigations of the Mind-Body Problem
      (pp. 141-175)

      If, as French film critic Jean Mitry claimed, “a film is a mirror in which we recognize only what we present to it through what it reflects back to us,”² do we mathematicians recognize ourselves in the increasingly frequent cinematic images of our profession? When we look in our private mirrors, do we see Carlo Cecchi (Death of a Neapolitan Mathematician), Tilda Swinton (Conceiving Ada), Matt Damon (Good Will Hunting), Gwyneth Paltrow (Proof), Russell Crowe (A Beautiful Mind), Rachel Weisz (Agora), Sean Gullette and Mark Margolis (Pi), David Wenham (The Bank), or Béatrice Dalle (Domaine)? Or perhaps the autistic savants...

    • chapter β.5 How to Explain Number Theory at a Dinner Party (Impromptu Minisession: Transcendental Numbers)
      (pp. 175-180)

      We have been talking about roots of polynomial equations, and I think you are now willing to admit that the root of a polynomialf(x) is an answer to a question about numbers and is, therefore, a number. For example, if

      f(x) = x³ – px + q,

      then the question

      For what α is f(α) = 0?

      has three answers. You remember we gave formulas for these answers and called them Olga, Masha, and Irina. You also remember that there is usually no formula whenfis a polynomial of degree 5 or more; nevertheless, the roots are answers to...

    • chapter 7 The Habit of Clinging to an Ultimate Ground
      (pp. 181-221)

      The notions of real interest to mathematicians like myself are not on the printed page. They lurk behind the doors of conception. It is believed² that they will some day emerge and shed so much light on earlier concepts that the latter will disintegrate into marginalia. By their very nature, they elude precise definition, so that on the conventional account they are scarcely mathematical at all. Coming to grips with them is not to be compared with attempting to solve an intractable problem, an experience that drives most narratives of mathematical discovery. What I have in mind harbors a more...

    • chapter 8 The Science of Tricks
      (pp. 222-256)

      In January 2010 it was revealed that I am a trickster. Toby Gee, a young English mathematician, broke the story. Before a packed Paris auditorium, Gee explained how he and two even-younger colleagues had found a new way to exploit what he called “Harris’s tensor product trick”—implicitly allowing that it might not be the only item in my bag of tricks, that I’m not necessarily a one-trick pony—to improve on my most recent work. I had first employed the trick in a solo paper, but it had been recycled to much greater effect in my joint paper with...

  7. Part III
    • chapter γ How to Explain Number Theory at a Dinner Party (Third Session: Congruences)
      (pp. 259-264)

      The proof of Gauss’ two square theorem, as well as its statement, is based on the distinction between even and odd numbers. But the statement also distinguishes between odd numbers of the form 4k+ 1 and those of the form 4k+ 3. One should imagine a cyclic classification of numbers into four classes, calledcongruence classes modulo 4:

      0, 4, 8, 12, . . .

      1, 5, 9, 13, . . .

      2, 6, 10, 12, . . .

      3, 7, 11, 15, . . .

      These classes can be labeled by their first members: the class of...

    • chapter 9 A Mathematical Dream and Its Interpretation
      (pp. 265-278)

      On sabbatical from my position as professor at Brandeis, I spent the 1992–1993 academic year in France, visiting colleagues and teaching courses at two universities—Université Paris 7, in the center of Paris, and Université Paris-Sud, in Orsay, a halfhour’s train ride to the south—in preparation for a possible move to Paris. Boston was then and still is one of the world’s great mathematical centers, and by attending Harvard’s number theory seminar and the MIT representation theory seminar, I kept in touch with all the most important developments relevant to my own work in automorphic forms, on the...

    • chapter 10 No Apologies
      (pp. 279-310)

      When as teenagers we began our initiation into the values and aspirations of research in pure mathematics, we never tired of quoting to each other—and to the uninitiated—from G.H. Hardy’sA Mathematician’s Apology, especially the parts where he insisted that “I have never done anything ‘useful.’” It was our cliché that he had chosen number theory exactly “because of its supreme uselessness” and indeed that “[n]o one has yet discovered any warlike purpose to be served by the theory of numbers. . . .”¹ Nowadays the cliché opens with the same Hardy quota tions but immediately veers off...

    • chapter δ How to Explain Number Theory at a Dinner Party (Fourth Session: Order and Randomness)
      (pp. 311-320)

      In 1801 Gauss published theDisquisitiones Arithmeticae, considered the founding text of modern number theory. Much of theDisquisitionesis concerned with the study of solutions to equations of degree 2 (quadratic equations) in two variables, such as x² + y² = 3 or x² + xy + y² = 11: equations of ellipses or hyperbolas. For our purposes here, quadratic equations (especially equations of ellipses) with integer coefficients are completely understood, thanks to Gauss. This does not mean that one cannot still ask questions about quadratic equations in two variables whose answers are unknown. In this as in all...

  8. afterword: The Veil of Maya
    (pp. 321-326)

    Scholarship seems not to have considered which metaphorical veil, orSchleier, Hilbert was inviting his listeners to lift, among the many on offer to an educated German in the late nineteenth century. Was it the “veil of poetry” [der Dichtung Schleier] a radiant goddess handed to Goethe, revealing herself as Truth; or was it one of the seven veils of Oscar Wilde’sSalomé, which Richard Strauss orchestrated, a few years after Hilbert’s lecture, in theSchleiertanzthat concludes his opera?

    Rather than assuming Hilbert sought to awaken the prurient inclinations of coming generations, it is safest to assume that he...

  9. notes
    (pp. 327-396)
  10. bibliography
    (pp. 397-422)
  11. index of mathematicians
    (pp. 423-426)
  12. subject index
    (pp. 427-438)