DOCSLIB.ORG

The Proof Is in the Pudding a Look at the Changing Nature of Mathematical Proof

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Proof Is in the Pudding a Look at the Changing Nature of Mathematical Proof The Proof is in the Pudding A Look at the Changing Nature of Mathematical Proof Steven G. Krantz January 9, 2008 To Jerry Lyons, mentor and friend. Table of Contents Preface ix 0 What is a Proof and Why? 3 0.1 WhatisaMathematician? . 4 0.2 TheConceptofProof....................... 7 0.3 The Foundations of Logic . 14 0.3.1 The Law of the Excluded Middle . 16 0.3.2 Modus Ponendo Ponens andFriends . 17 0.4 WhatDoesaProofConsistOf? . 21 0.5 ThePurposeofProof.. .. ... .. .. .. .. ... .. .. 22 0.6 TheLogicalBasisforMathematics . 27 0.7 TheExperimentalNatureofMathematics . 29 0.8 TheRoleofConjectures . 30 0.8.1 AppliedMathematics. 32 0.9 MathematicalUncertainty . 36 0.10 ThePublicationofMathematics. 40 0.11 ClosingThoughts . 42 1 The Ancients 45 1.1 Eudoxus and the Concept of Theorem . 46 1.2 EuclidtheGeometer .. .. ... .. .. .. .. ... .. .. 47 1.2.1 EuclidtheNumberTheorist . 51 1.3 Pythagoras ............................ 53 2 The Middle Ages and Calculation 59 2.1 TheArabsandAlgebra. 60 2.2 TheDevelopmentofAlgebra. 60 iii iv 2.2.1 Al-Khwarizmi and the Basics of Algebra . 60 2.2.2 TheLifeofAl-Khwarizmi . 62 2.2.3 TheIdeasofAl-Khwarizmi. 66 2.2.4 Concluding Thoughts about the Arabs . 70 2.3 InvestigationsofZero. 71 2.4 TheIdeaofInfinity. ... .. .. .. .. ... .. .. .. .. 73 3 The Dawn of the Modern Age 75 3.1 Euler and the Profundity of Intuition . 76 3.2 DirichletandHeuristics. 77 3.3 ThePigeonholePrinciple. 81 3.4 TheGoldenAgeof theNineteenthCentury. 82 4 HilbertandtheTwentiethCentury 85 4.1 DavidHilbert ........................... 86 4.2 Birkhoff,Wiener,and AmericanMathematics . 87 4.3 L. E. J. Brouwer and Proof by Contradiction . 96 4.4 The Generalized Ham-Sandwich Theorem . 107 4.4.1 Classical Ham Sandwiches . 107 4.4.2 Generalized Ham Sandwiches . 109 4.5 Much Ado About Proofs by Contradiction . 111 4.6 ErrettBishopand ConstructiveAnalysis . 116 4.7 NicolasBourbaki .. ... .. .. .. .. ... .. .. .. .. .117 4.8 PerplexitiesandParadoxes . .129 4.8.1 Bertrand’sParadox . .130 4.8.2 The Banach-Tarski Paradox . 134 4.8.3 TheMontyHallProblem. .136 5 The Four-Color Theorem 141 5.1 HumbleBeginnings . .142 6 Computer-Generated Proofs 153 6.1 ABriefHistoryofComputing . .154 6.2 The Difference Between Mathematics and Computer Science . 162 6.3 HowtheComputerGeneratesaProof. 163 6.4 HowtheComputerGeneratesaProof. 166 v 7 The Computer as a Mathematical Aid 171 7.1 Geometer’sSketchpad . .172 7.2 Mathematica,Maple,andMatLab. 172 7.3 NumericalAnalysis . .175 7.4 ComputerImagingandProofs . .176 7.5 MathematicalCommunication . 178 8 The Sociology of Mathematical Proof 185 8.1 The Classification of the Finite, Simple groups . 186 8.2 de Branges andthe BieberbachConjecture . 193 8.3 Wu-Yi Hsiang and Kepler Sphere-Packing . 195 8.4 Thurston’sGeometrizationProgram. 201 8.5 Grisha Perelman and the Poincar´eConjecture . 209 9 A Legacy of Elusive Proofs 223 9.1 TheRiemannHypothesis. .224 9.2 The Goldbach Conjecture . 229 9.3 TheTwin-PrimeConjecture . .233 9.4 Stephen Wolfram and A New Kind of Science .........234 9.5 BenoitMandelbrotandFractals . 239 9.6 The P/NP Problem .......................241 9.6.1 The Complexity of a Problem . 242 9.6.2 Comparing Polynomial and Exponential Complexity . 243 9.6.3 PolynomialComplexity. .244 9.6.4 Assertions that Can Be Verified in Polynomial Time . 245 9.6.5 Nondeterministic Turing Machines . 246 9.6.6 Foundations of NP-Completeness . .246 9.6.7 PolynomialEquivalence . .247 9.6.8 Definition of NP-Completeness . .247 9.6.9 Intractable Problems and NP-Complete Problems . 247 9.6.10 Examples of NP-Complete Problems . 247 9.7 Andrew Wiles and Fermat’s Last Theorem . 249 9.8 TheElusiveInfinitesimal . .257 9.9 AMiscellanyofMisunderstood Proofs . 259 9.9.1 Frustration and Misunderstanding . 261 vi 10 “The Death of Proof?” 267 10.1 Horgan’sThesis. .268 10.2 Will“Proof” RemaintheBenchmark? . 271 11 Methods of Mathematical Proof 273 11.1 DirectProof. .. .. ... .. .. .. .. ... .. .. .. .. .274 11.2 ProofbyContradiction . .279 11.3 ProofbyInduction . .282 12 Closing Thoughts 287 12.1 WhyProofsareImportant . .288 12.2 WhyProofMustEvolve . .290 12.3 What Will Be Considered a Proof in 100 Years? . 292 References 295 viii Preface The title of this book is not entirely frivolous. There are many who will claim that the correct aphorism is “The proof of the pudding is in the eat- ing.” That it makes no sense to say, “The proof is in the pudding.” Yet people say it all the time, and the intended meaning is always clear. So it is with mathematical proof. A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no “unique” or “right” or “best” proof of any given result. A proof is part of a situational ethic. Situations change, mathematical values and standards develop and evolve, and thus the very way that we do mathematics will alter and grow. This is a book about the changing and growing nature of mathemati- cal proof. In the earliest days of mathematics, “truths” were established heuristically and/or empirically. There was a heavy emphasis on calculation. There was almost no theory, and there was little in the way of mathematical notation as we know it today. Those who wanted to consider mathemat- ical questions were thereby hindered: they had difficulty expressing their thoughts. They had particular trouble formulating general statements about mathematical ideas. Thus it was virtually impossible that they could state theorems and prove them. Although there are some indications of proofs even on ancient Babylo- nian tablets from 1000 B.C.E., it seems that it is in ancient Greece that we find the identifiable provenance of the concept of proof. The earliest math- ematical tablets contained numbers and elementary calculations. Because ix x of the paucity of texts that have survived, we do not know how it came about that someone decided that some of these mathematical procedures required logical justification. And we really do not know how the formal con- cept of proof evolved. The Republic of Plato contains a clear articulation of the proof concept. The Physics of Aristotle not only discusses proofs, but treats minute distinctions of proof methodology (see our Chapter 11). Many other of the ancient Greeks, including Eudoxus, Theaetetus, Thales, Euclid, and Pythagoras, either used proofs or referred to proofs. Protagoras was a sophist, whose work was recognized by Plato. His Antilogies were tightly knit logical arguments that could be thought of as the germs of proofs. But it must be acknowledged that Euclid was the first to systematically use precise definitions, axioms, and strict rules of logic. And to systemat- ically prove every statement (i.e., every theorem). Euclid’s formalism, and his methodology, has become the model—even to the present day—for es- tablishing mathematical facts. What is interesting is that a mathematical statement of fact is a free- standing entity with intrinsic merit and value. But a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it. In the physical sciences—chemistry, biology, or physics for example—the method for achieving this end is the re- producible experiment.1 For the mathematician, the reproducible experiment is a proof that others can read and understand and validate. Thus a “proof” can, in principle, take many different forms. To be ef- fective, it will have to depend on the language, training, and values of the “receiver” of the proof. A calculus student has little experience with rigor and formalism; thus a “proof” for a calculus student will take one form. A professional mathematician will have a different set of values and experiences, and certainly different training; so a proof for the mathematician will take a different form. In today’s world there is considerable discussion—among mathematicians—about what constitutes a proof. And for physicists, who are our intellectual cousins, matters are even more confused. There are those workers in physics (such as Arthur Jaffe of Harvard, Charles Fefferman of Princeton, Ed Witten of the Institute for Advanced Study, Frank Wilczek of MIT, and Roger Penrose of Oxford) who believe that physical concepts 1More precisely, it is the reproducible experiment with control. For the careful scientist compares the results of his/her experiment with some standard or norm. That is the means of evaluating the result. xi should be derived from first principles, just like theorems. There are other physicists—probably in the majority—who reject such a theoretical approach and instead insist that physics is an empirical mode of discourse. These two camps are in a protracted and never-ending battle over the turf of their sub- ject. Roger Penrose’s new book The Road to Reality: A Complete Guide to the Laws of the Universe, and the vehement reviews of it that have appeared, is but one symptom of the ongoing battle. The idea of “proof” certainly appears in many aspects of life other than mathematics. In the courtroom, a lawyer (either for the prosecution or the defense) must establish his/her case by means of an accepted version of proof. For a criminal case this is “beyond a reasonable doubt” while for a civil case it is “the preponderance of evidence shows”. Neither of these is mathematical proof, nor anything like it. For the real world has no formal definitions and no axioms; there is no sense of establishing facts by strict logical exegesis. The lawyer certainly uses logic—such as “the defendant is blind so he could not have driven to Topanga Canyon on the night of March 23” or “the defendant has no education and therefore could not have built the atomic bomb that was used to .
Load more

Recommended publications
  • 2006 Annual Report
    Contents Clay Mathematics Institute 2006 James A. Carlson Letter from the President 2 Recognizing Achievement Fields Medal Winner Terence Tao 3 Persi Diaconis Mathematics & Magic Tricks 4 Annual Meeting Clay Lectures at Cambridge University 6 Researchers, Workshops & Conferences Summary of 2006 Research Activities 8 Profile Interview with Research Fellow Ben Green 10 Davar Khoshnevisan Normal Numbers are Normal 15 Feature Article CMI—Göttingen Library Project: 16 Eugene Chislenko The Felix Klein Protocols Digitized The Klein Protokolle 18 Summer School Arithmetic Geometry at the Mathematisches Institut, Göttingen, Germany 22 Program Overview The Ross Program at Ohio State University 24 PROMYS at Boston University Institute News Awards & Honors 26 Deadlines Nominations, Proposals and Applications 32 Publications Selected Articles by Research Fellows 33 Books & Videos Activities 2007 Institute Calendar 36 2006 Another major change this year concerns the editorial board for the Clay Mathematics Institute Monograph Series, published jointly with the American Mathematical Society. Simon Donaldson and Andrew Wiles will serve as editors-in-chief, while I will serve as managing editor. Associate editors are Brian Conrad, Ingrid Daubechies, Charles Fefferman, János Kollár, Andrei Okounkov, David Morrison, Cliff Taubes, Peter Ozsváth, and Karen Smith. The Monograph Series publishes Letter from the president selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. The next volume in the series will be Ricci Flow and the Poincaré Conjecture, by John Morgan and Gang Tian. Their book will appear in the summer of 2007. In related publishing news, the Institute has had the complete record of the Göttingen seminars of Felix Klein, 1872–1912, digitized and made available on James Carlson.
    [Show full text]
  • FIELDS MEDAL for Mathematical Efforts R
    Recognizing the Real and the Potential: FIELDS MEDAL for Mathematical Efforts R Fields Medal recipients since inception Year Winners 1936 Lars Valerian Ahlfors (Harvard University) (April 18, 1907 – October 11, 1996) Jesse Douglas (Massachusetts Institute of Technology) (July 3, 1897 – September 7, 1965) 1950 Atle Selberg (Institute for Advanced Study, Princeton) (June 14, 1917 – August 6, 2007) 1954 Kunihiko Kodaira (Princeton University) (March 16, 1915 – July 26, 1997) 1962 John Willard Milnor (Princeton University) (born February 20, 1931) The Fields Medal 1966 Paul Joseph Cohen (Stanford University) (April 2, 1934 – March 23, 2007) Stephen Smale (University of California, Berkeley) (born July 15, 1930) is awarded 1970 Heisuke Hironaka (Harvard University) (born April 9, 1931) every four years 1974 David Bryant Mumford (Harvard University) (born June 11, 1937) 1978 Charles Louis Fefferman (Princeton University) (born April 18, 1949) on the occasion of the Daniel G. Quillen (Massachusetts Institute of Technology) (June 22, 1940 – April 30, 2011) International Congress 1982 William P. Thurston (Princeton University) (October 30, 1946 – August 21, 2012) Shing-Tung Yau (Institute for Advanced Study, Princeton) (born April 4, 1949) of Mathematicians 1986 Gerd Faltings (Princeton University) (born July 28, 1954) to recognize Michael Freedman (University of California, San Diego) (born April 21, 1951) 1990 Vaughan Jones (University of California, Berkeley) (born December 31, 1952) outstanding Edward Witten (Institute for Advanced Study,
    [Show full text]
  • The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
    The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series.
    [Show full text]
  • Complex Analysis (Princeton Lectures in Analysis, Volume
    COMPLEX ANALYSIS Ibookroot October 20, 2007 Princeton Lectures in Analysis I Fourier Analysis: An Introduction II Complex Analysis III Real Analysis: Measure Theory, Integration, and Hilbert Spaces Princeton Lectures in Analysis II COMPLEX ANALYSIS Elias M. Stein & Rami Shakarchi PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright © 2003 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Control Number 2005274996 ISBN 978-0-691-11385-2 British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 5 7 9 10 8 6 To my grandchildren Carolyn, Alison, Jason E.M.S. To my parents Mohamed & Mireille and my brother Karim R.S. Foreword Beginning in the spring of 2000, a series of four one-semester courses were taught at Princeton University whose purpose was to present, in an integrated manner, the core areas of analysis. The objective was to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. The present series of books is an elaboration of the lectures that were given. While there are a number of excellent texts dealing with individual parts of what we cover, our exposition aims at a different goal: pre- senting the various sub-areas of analysis not as separate disciplines, but rather as highly interconnected.
    [Show full text]
  • Fundamental Theorems in Mathematics
    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
    [Show full text]
  • Fefferman and Schoen Awarded 2017 Wolf Prize in Mathematics
    COMMUNICATION Fefferman and Schoen Awarded 2017 Wolf Prize in Mathematics Biographical Sketch: Charles Fefferman Charles Fefferman was born in Washington, DC, in 1949. Showing exceptional ability in mathematics as a child, he entered the University of Maryland in 1963, at the age of fourteen, having bypassed high school. He published his first mathematics paper in a journal at the age of fifteen. In 1966, at the age of seventeen, he received his BS in mathematics and physics and was awarded a three-year NSF fellowship for research. He received his PhD from Princeton University in 1969 under the direction of Elias Stein. After spending the year 1969–1970 as a lecturer at Princeton, he accepted an assistant professorship at the University of Chicago. He was promoted to full professor in 1971—the youngest full professor ever appointed in Charles Fefferman Richard Schoen the United States. He returned to Princeton in 1973. He has been the recipient of a Sloan Foundation Fellowship Charles Fefferman of Princeton University and Richard (1970) and a NATO Postdoctoral Fellowship (1971). He Schoen of the University of California, Irvine, have been was awarded the Fields Medal in 1978. His many awards awarded the 2017 Wolf Prize in Mathematics by the Wolf and prizes include the Salem Prize (1971); the inaugural Foundation. Alan T. Waterman Award (1976); the Bergman Prize (1992); The prize citation reads: “Charles Fefferman has made and the Bôcher Memorial Prize of the AMS (2008). He was major contributions to several fields, including several elected to the American Academy of Arts and Sciences in complex variables, partial differential equations and 1972, the National Academy of Sciences in 1979, and the subelliptic problems.
    [Show full text]
  • Preserving the Unity of Mathematics CONTENTS
    ICMAT Newsletter #13 Autumn 2016 CSIC - UAM - UC3M - UCM EDITORIAL After the last European Congress of Mathematics, held between July 18th and July 22nd in Berlin, Marta Sanz-Solé, professor at the University of Barcelona, speaks about the European Mathematical Society, of which she was the Chair person between 2010 a 2014. The ECM: Preserving the Unity of Mathematics Less than one year ago, in October, 2015, the European Mathe- matical Society (EMS) celebrated its 25th anniversary. The place chosen to commemorate the event was the Institut Henri Poin- caré, the home of mathematics and theoretical physics in Paris since 1928. In that same year, the London Mathematical Soci- ety, one of the oldest mathematical societies in Europe, cele- brated its 150th anniversary, while many other societies have recently commemorated their 100th anniversary or are about to Rascon Image: Fernando do so. This difference in age may give rise to certain complexes, although there is no reason why this should happen. The EMS has enjoyed great success during its short existence, thanks to the well-defined space it occupies in a world that is advancing rapidly towards globalization, and the exclusively international scenario in which it carries out its functions. The conception and creation of the EMS in 1990 is better under- stood in the political context of Europe that arose after the tragic experience of World War II. In 1958, the communitary structure that currently underpins the Europan Union (EU) was estab- CONTENTS lished, on the basis of which the identity of Europe has been con- solidated, and the project, initially focused on the economy, has evolved along all fronts that affect the lives of those belonging Editorial: Marta Sanz-Solé (University of to the member states; in particular, education, research and in- Barcelona)..............................................................1 novation.
    [Show full text]
  • The Legacy of Norbert Wiener: a Centennial Symposium
    http://dx.doi.org/10.1090/pspum/060 Selected Titles in This Series 60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994) 59 William Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. Parshall, Editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, Editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, Editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, Editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, Editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, Editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R.
    [Show full text]
  • Annual Report for the Fiscal Year July 1, 1980
    The Institute for Advanced Study .nnual Report 1980 This Annual Report has been made possible by a generous grant from the Union Carbide Corporation. ! The Institute for Advanced Study Annual Report for the Fiscal Year July 1, 1980-June 30, 1981 The Institute for Advanced Study Olden Lane Princeton, New Jersey 08540 U.S.A. Printed by Princeton University Press Designed by Bruce Campbell x4S36 /98I It is fundamental to our purpose, and our Extract from the letter addressed by the express desire, that in the appointments to Founders to the Institute's Trustees, the staff and faculty, as well as in the dated June 6, 1930, Newark, New Jersey. admission of workers and students, no account shall be taken, directly or indirectly, of race, religion or sex. We feel strongly that the spirit characteristic of America at its noblest, above all, the pursuit of higher learning, cannot admit of any conditions as to personnel other than those designed to promote the objects for which this institution is established, and particularly with no regard whatever to accidents of race, creed or sex. /r2- S39 Table of Contents Trustees and Officers Founders Caroline Bamberger Fuld Louis Bamberger Board of Trustees Daniel Bell Howard C. Kauffmann Professor of Sociology President Harvard University Exxon Corporation Charles L. Brown John R. Opel Chairman the Board of and Chief President and Chief Executive Officer Executive Officer IBM Corporation American Telephone and Telegraph Company Howard C. Petersen Philadelphia, Pennsylvania Fletcher L. Byrom Chairman of the Board Martin E. Segal Koppers Company, Inc. Partner, Wertheim & Co.; Chairman, Martin E.
    [Show full text]
  • Science Lives: Video Portraits of Great Mathematicians
    Science Lives: Video Portraits of Great Mathematicians accompanied by narrative profiles written by noted In mathematics, beauty is a very impor- mathematics biographers. tant ingredient… The aim of a math- Hugo Rossi, director of the Science Lives project, ematician is to encapsulate as much as said that the first criterion for choosing a person you possibly can in small packages—a to profile is the significance of his or her contribu- high density of truth per unit word. tions in “creating new pathways in mathematics, And beauty is a criterion. If you’ve got a theoretical physics, and computer science.” A beautiful result, it means you’ve got an secondary criterion is an engaging personality. awful lot identified in a small compass. With two exceptions (Atiyah and Isadore Singer), the Science Lives videos are not interviews; rather, —Michael Atiyah they are conversations between the subject of the video and a “listener”, typically a close friend or colleague who is knowledgeable about the sub- Hearing Michael Atiyah discuss the role of beauty ject’s impact in mathematics. The listener works in mathematics is akin to reading Euclid in the together with Rossi and the person being profiled original: You are going straight to the source. The to develop a list of topics and a suggested order in quotation above is taken from a video of Atiyah which they might be discussed. “But, as is the case made available on the Web through the Science with all conversations, there usually is a significant Lives project of the Simons Foundation. Science amount of wandering in and out of interconnected Lives aims to build an archive of information topics, which is desirable,” said Rossi.
    [Show full text]
  • John Forbes Nash Jr
    John Forbes Nash Jr. (1928–2015) Camillo De Lellis, Coordinating Editor John Forbes Nash Jr. was born in Bluefield, West Virginia, on June 13, 1928 and was named after his father, who was an electrical engineer. His mother, Margaret Virginia (née Martin), was a school teacher before her marriage, teaching English and sometimes Latin. After attending the standard schools in Bluefield, Nash entered the Carnegie Institute of Technology in Pittsburgh (now Carnegie Mel- lon University) with a George Westinghouse Scholarship. He spent one semester as a student of chemical engi- neering, switched momentarily to chemistry and finally decided to major in mathematics. After graduating in 1948 with a BS and a MS at the same time, Nash was of- fered a scholarship to enter as a graduate student at either Harvard or Princeton. He decided for Princeton, where in 1950 he earned a PhD de- of John D. Stier. gree with his celebrated work on noncooperative Courtesy games, which won him the John and Alicia Nash on the day of their wedding. Nobel Prize in Economics thirty-four years later. In the summer of 1950 groundbreaking paper “Real algebraic manifolds”, cf. [39], he worked at the RAND (Re- much of which was indeed conceived at the end of his search and Development) graduate studies: According to his autobiographical notes, Corporation, and although cf. [44], Nash was prepared for the possibility that the he went back to Princeton game theory work would not be regarded as acceptable during the autumn of the as a thesis at the Princeton mathematics department. same year, he remained a Around this time Nash met Eleanor Stier, with whom he consultant and occasion- had his first son, John David Stier, in 1953.
    [Show full text]
  • Mathematical Perspectives
    MATHEMATICAL PERSPECTIVES BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 4, October 2009, Pages 661–666 S 0273-0979(09)01264-6 Article electronically published on May 14, 2009 ANDREWEIL´ ARMAND BOREL The mathematician Andr´e Weil died in Princeton at the age of ninety-two, after a gradual decline in his physical, but not mental, capacities. In January 1999, a conference on his work and its influence took place at the Institute for Advanced AndreWeil´ 6 May 1906–6 August 1998 This article is from the Proceedings of the American Philosophical Society, vol. 145, no. 1, March 2001, pp. 108–114. Reprinted with permission. c 2009 American Mathematical Society 661 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 662 ARMAND BOREL Study, the poster of which gave this capsule description of him: A man of formidable intellectual power, moved by a global view and knowledge of mathematics, of its history and a strong belief in its unity, Andr´e Weil has profoundly influenced the course of mathe- matics by the breadth and depth of his publications, his correspon- dence and his leading contributions to the work of N. Bourbaki, a statement of which this text is basically just an elaboration. In attempting to give an impression of Weil’s work and thought processes, I shall have to navigate between the Charybdis of undefined mathematical jargon and the Scylla of vague, seemingly but not necessarily more understandable, statements. I hope the reader will bear with me! Andr´e Weil was born in Paris in a Jewish family, to an Alsatian physician and his Russian-born wife, of Austrian origin.
    [Show full text]