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A Century of Advancing Mathematics

A Century of Advancing Mathematics

Stephen F. Kennedy Editor
Donald J. Albers
Gerald L. Alexanderson
Della Dumbaugh
Frank A. Farris
Deanna B. Haunsperger
Paul Zorn
Series: Spectrum
Copyright Date: 2015
Edition: 1
Pages: 436
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  • Book Info
    A Century of Advancing Mathematics
    Book Description:

    The MAA was founded in 1915 to serve as a home for The American Mathematical Monthly. The mission of the Association—to advance mathematics, especially at the collegiate level—has, however, always been larger than merely publishing world class mathematical exposition. MAA members have explored more than just mathematics; we have, as this volume tries to make evident, investigated mathematical connections to pedagogy, history, the arts, technology, literature, every field of intellectual endeavor. Essays, all commissioned for this volume, include exposition by Bob Devaney, Robin Wilson, and Frank Morgan; history from Karen Parshall, Della Dumbaugh, and Bill Dunham; pedagogical discussion from Paul Zorn, Joe Gallian, and Michael Starbird, and cultural commentary from Bonnie Gold, Jon Borwein, and Steve Abbott. This volume contains 35 essays by all-star writers and expositors writing to celebrate an extraordinary century for mathematics—more mathematics has been created and published since 1915 than in all of previous recorded history. We’ve solved age-old mysteries, created entire new fields of study, and changed our conception of what mathematics is. Many of those stories are told in this volume as the contributors paint a portrait of the broad cultural sweep of mathematics during the MAA’s first century. Mathematics is the most thrilling, the most human, area of intellectual inquiry; you will find in this volume compelling proof of that claim.

    eISBN: 978-1-61444-522-7
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Preface
    (pp. ix-xii)
    The Editors
  4. Part I Mathematical Developments

    • The Hyperbolic Revolution: From Topology to Geometry, and Back
      (pp. 3-14)
      Francis Bonahon

      The late nineteen seventies and early eighties saw a surprising convergence between topology and rigid geometry. This followed the groundbreaking work of Bill Thurston on the geometrization of three-dimensional manifolds, but this was also part of a larger trend that resulted in a period of intense cross-fertilization between topology, geometry, dynamical systems, combinatorial group theory, and complex analysis.

      First, we should begin with the traditional difference between topology and geometry. Both fields consider geometric objects, but topologists allow themselves to deform these objects and stretch distances, whereas geometers tend to focus on the fine properties of these distances. As an...

    • A Century of Complex Dynamics
      (pp. 15-34)
      Daniel Alexander and Robert L. Devaney

      Like the MAA, the field of mathematics known as complex dynamics has been around for about one hundred years. Unlike the MAA, complex dynamics has had its ups and downs during this period. While the origins of complex dynamics stretch back into the late 1800s, the foundations of the contemporary study were established in the last years of World War I with the pioneering work of Gaston Julia and Pierre Fatou. Although one hundred years ago complex dynamics was a predominantly French field, there are some important American connections dating back to 1915, with some interesting historical connections to the...

    • Map-Coloring Problems
      (pp. 35-50)
      Robin Wilson

      In 1852 Augustus De Morgan was asked whether all plane maps can be colored with just four colors in such a way that neighboring countries are always colored differently. In 1976 Kenneth Appel and Wolfgang Haken answered this question in the affirmative. But why did this easily stated question take 124 years to be answered, and what was involved in its solution? And since maps drawn on a plane are equivalent to maps drawn on a sphere, can the problem be extended to the coloring of maps drawn on other surfaces? And once these problems were solved, what was there...

    • Six Milestones in Geometry
      (pp. 51-64)
      Frank Morgan

      What would you say are the six largest advances in geometry during your 100-year lifetime? You say you remember only the more recent ones? Me too. Here are my choices.

      An old theme in mathematics is that nice problems should have nice solutions, although this can be hard to prove and is sometimes false. The nicest problem about curves has the nicest possible solution: the shortest distance between two points is a straight line. Similarly the nicest problem about surfaces has a nice solution: given a smooth curve in R³, a least-area surface bounded by the curve as in Figure...

    • Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics
      (pp. 65-82)
      Eric S. Egge

      In 2005, at the Third International Conference on Permutation Patterns in Gainesville, Florida, Doron Zeilberger declared that “Not even God knowsa1000” Zeilberger’s claim raises thorny theological questions, which I am happy to ignore in this article, but it also raises mathematical questions. The quantitya1000.(1324) is the one-thousandth term in a certain sequencean.(1324). God may or may not be able to compute the thousandth term in this sequence, but how far can mortals get? If we can’t get beyond the fortieth or fiftieth term, can we at least approximate the one-thousandth term? How fast doesan.(1324) grow, anyway?...

    • What Is the Best Approach to Counting Primes?
      (pp. 83-116)
      Andrew Granville

      As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann’s seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann’s theory with one that is significantly simpler.

      You have probably seen a proof that there are...

    • A Century of Elliptic Curves
      (pp. 117-132)
      Joseph H. Silverman

      Elliptic curves appear in many places and in many guises throughout mathematics, including algebraic geometry, algebra, number theory, topology, cryptography, real and complex analysis, and mathematical physics, to name just a few [25]. My aim in this brief essay is to provide an overview of the study of the number-theoretic side of elliptic curves during the past century (plus epsilon). Considerations of space and taste have determined the particular topics covered. Those seeking further information will find many paths to follow. For a few suggestions, see the list of references preceding the bibliography.

      There are many ways to define an...

  5. Part II Historical Developments

    • The Mathematical Association of America: Its First 100 Years
      (pp. 135-157)
      David E. Zitarelli

      This article presents an overview of the history of the Mathematical Association of America as part of the celebration of its centennial in 2015. It describes events this author regards as the most important over the century but the account is certainly not exhaustive; for example, it makes little mention of competitions conducted under the aegis of the Association or of the expanded book publication program. Our account begins with the founding of the MAA and then describes its sections, governance, and meetings. Overarching activities are outlined in two distinct periods, 1916–1955 and 1955–2014, with an explanation for...

    • The Stratification of the American Mathematical Community: The Mathematical Association of America and the American Mathematical Society, 1915–1925
      (pp. 159-175)
      Karen Hunger Parshall

      The Mathematical Association of America (MAA) officially came into existence over the course of a two-day-long meeting held on 30–31 December, 1915 in Columbus, Ohio. Some 450 people nationwide had answered a written call in support of the creation of a new organization that would specifically foster collegiate mathematics, and over 100 of them opted to spend their New Year’s Eve in Ohio in order to discuss and ratify a constitution and bylaws for the new society. There was, of course, already a national association for mathematicians. The American Mathematical Society (AMS) had been founded in 1888 as the...

    • Time and Place: Sustaining the American Mathematical Community
      (pp. 177-190)
      Della Dumbaugh

      The “common interest” of mathematicians in the last quarter of the nineteenth century contributed to the emergence of an American mathematical community. What “standards and traditions” evolved as mathematicians strengthened this community in the opening decades of the twentieth century? The private correspondence of Leonard Dickson and Oswald Veblen provides a loose frame to explore the views of these leaders as they not only sustained, but also advanced this young community.

      It is almost an oxymoron to see the words “Leonard Dickson” and “correspondence” in the same sentence. The gruff Dickson worked tirelessly to advance mathematics and preserve his privacy,...

    • Abstract (Modern) Algebra in America 1870–1950: A Brief Account
      (pp. 191-216)
      Israel Kleiner

      Algebra in the US did not develop in isolation. American mathematicians were in contact with, and drew inspiration from, researchers in algebra in Germany, and to a lesser extent in England and France. As Eric Temple Bell notes, “the debt of American algebra to the Germany of the late 1880s and early 1890s is very great” [7, p. 3]; see also [59]. Very significant in the development of American algebra was also the seven-year stay at Johns Hopkins (1876–1883) of the distinguished English algebraist James Joseph Sylvester [59]. And “as late as 1904... 20% of the members of the...

  6. Part III Pedagogical Developments

    • The History of the Undergraduate Program in Mathematics in the United States
      (pp. 219-238)
      Alan Tucker

      The undergraduate mathematics program in America has had a punctuated evolution. The Mathematical Association of America was organized 100 years ago, at the end of a period of dramatic rethinking of American education at all levels, one product of which was the introduction of academic majors. The mathematics major was static in its first forty years, followed by great changes from 1955 to 1975, and then a period of relative stability to the present.

      The educational concerns of the Mathematical Association of America also changed in the 1950s. Initially, its educational recommendations focused on preparing high school students for college...

    • Inquiry-Based Learning Through the Life of the MAA
      (pp. 239-252)
      Michael Starbird

      Robert Lee Moore was one of the most famous mathematics teachers in the mid-twentieth century. He conducted his classes in a distinctive manner. He never lectured. Instead, he posed questions and his students were required to discover answers to these difficult questions independently without any outside help from books, teachers, or each other. In class, Professor Moore started with the student he considered the least able and asked, “Mr.— can you answer the next question?” If that student could not answer the question, Professor Moore asked the same question to the next weakest student and so on until he found...

    • A Passport to Pleasure
      (pp. 253-256)
      Bob Kaplan and Ellen Kaplan

      Math cuts you no slack. You can argue the merits ofCrime and Punishmentwith experts because beauty is in the eye of the beholder, but doubt what V – E + F add up to and proof will send you sprawling. It is this absolutism that by turns appeals and appalls. It can make a lifelong devotee like Bertrand Russell conclude that what he thought would be as beautiful as Dante’sParadisowas no more than a series of hollow tautologies. But it can set each of us dancing at 2 AM when the doors to this paradise all at...

    • Strength in Numbers: Broadening the View of the Mathematics Major
      (pp. 257-262)
      Rhonda Hughes

      The prevailing ethos in many mathematics departments is that only the “best and brightest” should be encouraged to major in mathematics. Many students are quickly dismissed as not good enough or not cut out for mathematics. The conventional view of potential majors is based on stereotypes that rely on inaccurate and narrow views about who can and should do mathematics. These assessments are subjective and often based on expectations derived from one’s own educational and cultural experiences.

      The February 2012 Presidential ReportEngage to Excel: Producing One Million Additional College Graduates with Degrees in Science, Technology, Engineering, and Mathematicsidentified...

    • A History of Undergraduate Research in Mathematics
      (pp. 263-274)
      Joseph A. Gallian

      Research in mathematics by undergraduates is now commonplace. Summer undergraduate research programs abound. Many institutions fund undergraduate student research. Senior theses routinely include original results. There are numerous conferences where the focus is on presentations by undergraduates. There are mathematics journals that specialize in publishing papers with undergraduate authors. Research experience is expected for admission to leading graduate programs. The annual Joint Mathematics Meetings (JMM) of the American Mathematical Society (AMS) and the Mathematical Association of American (MAA) and the annual summer MAA MathFest are attended by large numbers of undergraduates. These are recent developments. In this article we identify...

    • The Calculus Reform Movement: A Personal Account
      (pp. 275-282)
      Paul Zorn

      What follows is a limited, personal, and opinionated account, not a comprehensive history, of some aspects of the calculus reform movement of the 1980s and 1990s. The author was himself closely involved, and therefore by no means disinterested. He thanks his friend Deborah Hughes Hallett for useful conversation and ideas, but all opinions and errors here are his.

      The era of calculus reform began “officially” in early 1986, when the Tulane Conference on Calculus Instruction, organized by Ronald Douglas, met in New Orleans. The conversation caught on. A follow-up gathering in 1987 attracted over 600 mathematicians and scientists to a...

    • Introducing ex
      (pp. 283-294)
      Gilbert Strang

      The day whenexappears is important in teaching and learning calculus. This is the great new function—but how to present it? The presentation decides whether the chance to connect with key ideas (past and future) is taken or missed. Textbooks offer four main approaches toex:

      1. Use the derivative ofxn/n!Add those terms to matchdy/dxwithy:

      2. Take the nth power of (1 +x/n) as in compound interest. Letnapproach infinity.

      3. The slope ofbxisCtimesbx: Chooseeas the value ofbthat makesC= 1....

  7. Part IV Computational Developments

    • Computational Experiences in the Pre-Electronic Days
      (pp. 297-300)
      Philip J. Davis

      The events recorded here took place almost seventy years ago. This article is therefore what Benjamin Disraeli, novelist and twice Prime Minister of England, has called an instance of one’s “anecdotage.”

      My first knowledge of the details of scientific computation came from a book discarded by the MIT Library and brought home by my elder brother, then an undergraduate at MIT [1]. This book derived from the computation laboratory of the University of Edinburgh run by Sir Edmund Whittaker (1873–1956). It is interesting and amusing to read how the individual computer’s desks were outfitted. (Incidentally, in those years a...

    • A Century of Visualization: One Geometer’s View
      (pp. 301-312)
      Thomas F. Banchoff

      Here is a collection of stories about the dramatic changes in the role of visualization in mathematics, particularly in geometry and topology, during the first hundred years of the MAA. I have lived three-quarters of that century and I have been teaching in colleges and universities for the past fifty years. I myself am a very visual person and I have met a number of other mathematicians with that proclivity over the years. Of course I have met a number of mathematicians who don’t see a need to accompany arguments with picture or models, and some who completely disdain “visual...

    • The Future of Mathematics: 1965 to 2065
      (pp. 313-330)
      Jonathan M. Borwein

      William Gibson, the science fiction writer, who coined the termcyberspacewell before he purchased his first personal computer has commented that the future is already here but that it is very poorly distributed. Mindful of the dangers of futurology, I shall look forward and back fifty years while where possible eschewing the unknowable.

      It’s generally the way with progress that it looks much greater than it really is. (Ludwig Wittgenstein, 1889–1951, “whereof one cannot speak, thereof one must be silent”)

      The world will change. It will probably change for the better. It won’t seem better to me. ....

  8. Part V Culture and Communities

    • Philosophy of Mathematics: What Has Happened Since Gödel’s Results?
      (pp. 333-350)
      Bonnie Gold

      When the MAA was founded in 1915, the mathematical community was in the midst of its “foundational crisis.” Cantor had recently developed his theory of transfinite numbers (which led to assorted paradoxes). Frege’s attempt via logicism to put mathematics on a sound foundation had foundered on the Russell paradox. Brouwer’s intuitionism had introduced a completely different approach to mathematics and philosophy of mathematics. And Hilbert was about to begin (in 1917) his program to set mathematics on a sound foundation via formalism and finitism, by finding a finitary consistency proof for arithmetic and analysis. There are those who say that...

    • Twelve Classics People who Love Mathematics Should Know; or, “What do you mean, you haven’t read E. T. Bell?”
      (pp. 351-364)
      Gerald L. Alexanderson

      I’m old enough to think that everyone in our field must be familiar withMen of Mathematics, but I find that some of my younger colleagues have never heard of it. I hope to remedy this sad example of a generation gap. The following list of twelve classics in mathematics would be appropriate as recommended reading in a general mathematics course for non-majors, but I would go further: they should be required reading even for undergraduate mathematics majors. We should never neglect our majors.

      Yes, the title may offend some people these days, but in 1937 readers probably thought it...

    • The Dramatic Life of Mathematics: A Centennial History of the Intersection of Mathematics and Theater in a Prologue, Three Acts, and an Epilogue
      (pp. 365-378)
      Stephen D. Abbott

      Tom Stoppard’sArcadiaopened at the National Theatre in London on April 13, 1993. Two months later, on June 23 at the Isaac Newton Institute in nearby Cambridge, Andrew Wiles went public with his initial proof of Fermat’s Last Theorem. This was almost certainly the first time a Cambridge mathematics conference had a direct bearing on events at the National, but indeed, Wiles’s surprise announcement sent Stoppard scrambling to update the program. In the first scene of Stoppard’s new play, thirteen-year-old Thomasina Coverly has been assigned the task of proving FLT but, having overheard some house gossip earlier in that...

    • 2007: The Year of Euler
      (pp. 379-386)
      William Dunham

      Many years ago, on my first visit to the offices of the Mathematical Association of America, I entered what seemed to be a generic lobby, with a receptionist at the main desk and portraits of past dignitaries lining the walls. It looked more or less like any other organizational headquarters in the Dupont Circle area of Washington, DC.

      But then I noticed an old book resting upon a pedestal in the lobby’s center. This volume, dense with Latin words and mathematical symbols, was Leonhard Euler’s great text,Introductio in analysin infinitorum. Its presence confirmed that this was amathematicsorganization,...

    • The Putnam Competition: Origin, Lore, Structure
      (pp. 387-392)
      Leonard F. Klosinski

      George Csicsery, the noted documentary filmmaker, recently released a film about the 2006 American team that participated in the International Mathematical Olympiad, called “Hard Problems: the Road to the World’s Toughest Math Contest.” Well, maybe it is the toughest mathematical contest for the students participating. But there’s another mathematical competition for more experienced students of mathematics, one with an even longer history and with problems that stump even the professors. The William Lowell Putnam Mathematical Competition, long administered by the Mathematical Association of America (MAA), just celebrated its seventy-fifth anniversary. Its list of winners, like that of the International Mathematical...

    • Getting Involved with the MAA: A Path Less Traveled
      (pp. 393-396)
      Ezra “Bud” Brown

      Many of us in the MAA are given student memberships as undergraduates, or graduate students, or high-school students. Here is a member with quite a different story.

      You had a talent and a taste for mathematics, went through school and somehow or other ended up with a PhD in mathematics with a dissertation in number theory and a tenure-track job as a college professor at a university that was “on the way up.” This was in the day when research mathematicians talked about teaching behind closed doors—if they talked about teaching at all. Although you always enjoyed teaching, research...

    • Henry L. Alder
      (pp. 397-400)
      Donald J. Albers and Gerald L. Alexanderson

      Through the 1940s and into the late 1950s the administrative work of the MAA was handled at offices on the campus of the University of Buffalo. From 1943 to 1947 W. B. Carver of Cornell was the MAA Secretary-Treasurer, succeeded in 1948 by Harry M. Gehman who served as Secretary-Treasurer until 1959 and then as Treasurer from 1960 to 1967. He also served as Executive Director from 1960 to 1968. So, to some extent, the MAA was a one-man operation during that time [4]. The job of Secretary-Treasurer was split in 1960 when Henry L. Alder of the University of...

    • Lida K. Barrett
      (pp. 401-404)
      Kenneth A. Ross

      As a project for the 2015 centennial of the MAA, members of the history subcommittee of the MAA centennial committee have been interviewing prominent members of the mathematical community. The excerpts here are based on an interview that took place August 11, 2006, in Knoxville, Tennessee.

      In the fifth grade. I was bored in school and acted up, so I had to stay after school and do long-division problems. I got really skilled at arithmetic in the process of doing problems as rapidly as possible. Later in junior high I was encouraged by Miss Emma French to try for the...

    • Ralph P. Boas
      (pp. 405-406)
      Daniel Zelinsky

      Ralph Philip Boas was a prolific analyst and a powerful contributor to the mathematics community. He wrote nearly two hundred papers on real and complex variables and served as president of the MAA, vice president of the AMS, and editor of several journals. He was connected withMathematical Reviewsfrom its founding by Neugebauer in 1940 and served as its executive editor from 1945 to 1950. He was editor ofThe American Mathematical Monthlyfrom 1977 to 1980 and was the recipient of the Association’s Gung and Hu Distinguished Service to Mathematics Award in 1981. He was my close friend...

    • Leonard Gillman—Reminiscences
      (pp. 407-410)
      Martha J. Siegel

      My acquaintance with Len Gillman began when we arrived, separately but simultaneously, at the University of Rochester in September 1960. He came to Rochester from Purdue to be department chair. I came to begin graduate school. He immediately made the graduate students feel welcome as part of the professional life of the department. As I had an undergraduate degree in mathematics education and some classroom experience as a student teacher, I quickly became the “teaching” expert among the graduate students. Most of them were men; all had no experience in front of a classroom and almost all of us were...

    • Paul Halmos: No Apologies
      (pp. 411-414)
      John Ewing

      Hardy’s famous lament was written near the end of his life when he felt his research career was over. It is a sentiment often felt by aging mathematicians, even when they don’t admit it. But it was not a sentiment shared by Paul Halmos. His abiding interest in writing and speaking about mathematics—in communicating its elegance, beauty, mystery, and fascination—was a hallmark of his life. Paul loved mathematics; he loved being a mathematician; and he was unapologetic about his passion.

      This passion was evident in his written exposition, which was prolific. More than half his nearly 200 published...

    • Ivan Niven
      (pp. 415-420)
      Kenneth A. Ross

      Ivan Morton Niven was born in Vancouver, B. C. on October 25, 1915, and lived there until he was 21. His working-class parents had emigrated from Scotland. Ivan grew up “in a school system that was tightly structured and disciplined by today’s standards. (There was little concern about whether we felt good about ourselves.) The purpose was to stuff our empty heads as full of knowledge as possible” [8].

      Ivan earned his Bachelor’s and Master’s degrees (1934 and 1936) at the University of British Columbia, and he was awarded his PhD in 1938 at the University of Chicago. He worked...

    • George Pólya and the MAA
      (pp. 421-423)
      Gerald L. Alexanderson

      In an earlier time mathematicians in academe were expected to be members of both the American Mathematical Society and the Mathematical Association of America, even when they may have thought of themselves mainly as researchers or, on the other hand, as teachers, depending in part on the kind of institution they were affiliated with. For George Pólya (1887–1985) there was no doubt about his stature as a research mathematician, but his early interest in problem solving made him a natural participant in the MAA. So he retained an active membership in both.

      His principal MAA involvement was his participation...