IAS Oral History Project Interview Transcript Atle Selberg
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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, -
“Gödel's Modernism: on Set-Theoretic Incompleteness,” Revisited
“G¨odel'sModernism: on Set-Theoretic Incompleteness," revisited∗ Mark van Atten and Juliette Kennedy As to problems with the answer Yes or No, the con- viction that they are always decidable remains un- touched by these results. —G¨odel Contents 1 Introduction 1.1 Questions of incompleteness On Friday, November 15, 1940, Kurt G¨odelgave a talk on set theory at Brown University.1 The topic was his recent proof of the consistency of Cantor's Con- tinuum Hypothesis, henceforth CH,2 with the axiomatic system for set theory ZFC.3 His friend from their days in Vienna, Rudolf Carnap, was in the audience, and afterward wrote a note to himself in which he raised a number of questions on incompleteness:4 (Remarks I planned to make, but did not) Discussion on G¨odel'slecture on the Continuum Hypothesis, November 14,5 1940 There seems to be a difference: between the undecidable propo- sitions of the kind of his example [i.e., 1931] and propositions such as the Axiom of Choice, and the Axiom of the Continuum [CH ]. We used to ask: \When these two have been decided, is then everything decided?" (The Poles, Tarski I think, suspected that this would be the case.) Now we know that (on the basis of the usual finitary rules) there will always remain undecided propositions. ∗An earlier version of this paper appeared as ‘G¨odel'smodernism: on set-theoretic incom- pleteness', Graduate Faculty Philosophy Journal, 25(2), 2004, pp.289{349. Erratum facing page of contents in 26(1), 2005. 1 1. Can we nevertheless still ask an analogous question? I.e. -
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. -
Georg Kreisel Papers SC0136
http://oac.cdlib.org/findaid/ark:/13030/kt4k403759 No online items Guide to the Georg Kreisel Papers SC0136 Daniel Hartwig & Jenny Johnson Department of Special Collections and University Archives October 2010 Green Library 557 Escondido Mall Stanford 94305-6064 [email protected] URL: http://library.stanford.edu/spc Note This encoded finding aid is compliant with Stanford EAD Best Practice Guidelines, Version 1.0.This encoded finding aid is compliant with Stanford EAD Best Practice Guidelines, Version 1.0. Guide to the Georg Kreisel Papers SC0136 1 SC0136 Language of Material: English Contributing Institution: Department of Special Collections and University Archives Title: Georg Kreisel papers creator: Kreisel, Georg Identifier/Call Number: SC0136 Physical Description: 24.75 Linear Feet Date (inclusive): 1957-1984 Language of Material: English Language of Material: English Abstract: Correspondence with professional colleagues, collaborators, students, and others, primarily from 1962 to 1984, lecture notes, manuscripts and other writings. Ownership & Copyright All requests to reproduce, publish, quote from, or otherwise use collection materials must be submitted in writing to the Head of Special Collections and University Archives, Stanford University Libraries, Stanford, California 94304-6064. Consent is given on behalf of Special Collections as the owner of the physical items and is not intended to include or imply permission from the copyright owner. Such permission must be obtained from the copyright owner, heir(s) or assigns. See: http://library.stanford.edu/depts/spc/pubserv/permissions.html. Restrictions also apply to digital representations of the original materials. Use of digital files is restricted to research and educational purposes. Biographical/Historical Sketch Professor of Logic and the Foundations of Mathematics at Stanford University. -
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]. -
Kreisel and Wittgenstein
Kreisel and Wittgenstein Akihiro Kanamori September 17, 2018 Georg Kreisel (15 September 1923 { 1 March 2015) was a formidable math- ematical logician during a formative period when the subject was becoming a sophisticated field at the crossing of mathematics and logic. Both with his technical sophistication for his time and his dialectical engagement with man- dates, aspirations and goals, he inspired wide-ranging investigation in the meta- mathematics of constructivity, proof theory and generalized recursion theory. Kreisel's mathematics and interactions with colleagues and students have been memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of interpersonal conceptual interaction, Kreisel during his life time had extended engagement with two celebrated logicians, the mathematical Kurt G¨odeland the philosophical Ludwig Wittgenstein. About G¨odel,with modern mathemat- ical logic palpably emanating from his work, Kreisel has reflected and written over a wide mathematical landscape. About Wittgenstein on the other hand, with an early personal connection established Kreisel would return as if with an anxiety of influence to their ways of thinking about logic and mathematics, ever in a sort of dialectic interplay. In what follows we draw this out through his published essays|and one letter|both to elicit aspects of influence in his own terms and to set out a picture of Kreisel's evolving thinking about logic and mathematics in comparative relief.1 As a conceit, we divide Kreisel's engagements with Wittgenstein into the \early", \middle", and \later" Kreisel, and account for each in successive sec- tions. x1 has the \early" Kreisel directly interacting with Wittgenstein in the 1940s and initial work on constructive content of proofs. -
Georg Kreisel Correspondence with Jean Van Heijenoort
http://oac.cdlib.org/findaid/ark:/13030/c84q7wxh No online items Guide to the Georg Kreisel Correspondence with Jean van Heijenoort Daniel Hartwig Stanford University. Libraries.Department of Special Collections and University Archives Stanford, California October 2010 Copyright © 2015 The Board of Trustees of the Leland Stanford Junior University. All rights reserved. Note This encoded finding aid is compliant with Stanford EAD Best Practice Guidelines, Version 1.0. Guide to the Georg Kreisel SC0233 1 Correspondence with Jean van Heijenoort Overview Call Number: SC0233 Creator: Kreisel, Georg, 1923- Title: Georg Kreisel correspondence with Jean van Heijenoort Dates: 1949-1981 Physical Description: 6 Linear feet Summary: Correspondence, notes, memoranda, articles and other materials by Professor Georg Kreisel, sent to his colleague, Professor J. van Heijenoort of Harvard University. Includes some correspondence with other colleagues. Language(s): The materials are in English. Repository: Department of Special Collections and University Archives Green Library 557 Escondido Mall Stanford, CA 94305-6064 Email: [email protected] Phone: (650) 725-1022 URL: http://library.stanford.edu/spc Gift of J. van Heijenoort, 1981. Information about Access This collection is open for research. Ownership & Copyright All requests to reproduce, publish, quote from, or otherwise use collection materials must be submitted in writing to the Head of Special Collections and University Archives, Stanford University Libraries, Stanford, California 94304-6064. Consent is given on behalf of Special Collections as the owner of the physical items and is not intended to include or imply permission from the copyright owner. Such permission must be obtained from the copyright owner, heir(s) or assigns. -
An Interview with Martin Davis
Notices of the American Mathematical Society ISSN 0002-9920 ABCD springer.com New and Noteworthy from Springer Geometry Ramanujan‘s Lost Notebook An Introduction to Mathematical of the American Mathematical Society Selected Topics in Plane and Solid Part II Cryptography May 2008 Volume 55, Number 5 Geometry G. E. Andrews, Penn State University, University J. Hoffstein, J. Pipher, J. Silverman, Brown J. Aarts, Delft University of Technology, Park, PA, USA; B. C. Berndt, University of Illinois University, Providence, RI, USA Mediamatics, The Netherlands at Urbana, IL, USA This self-contained introduction to modern This is a book on Euclidean geometry that covers The “lost notebook” contains considerable cryptography emphasizes the mathematics the standard material in a completely new way, material on mock theta functions—undoubtedly behind the theory of public key cryptosystems while also introducing a number of new topics emanating from the last year of Ramanujan’s life. and digital signature schemes. The book focuses Interview with Martin Davis that would be suitable as a junior-senior level It should be emphasized that the material on on these key topics while developing the undergraduate textbook. The author does not mock theta functions is perhaps Ramanujan’s mathematical tools needed for the construction page 560 begin in the traditional manner with abstract deepest work more than half of the material in and security analysis of diverse cryptosystems. geometric axioms. Instead, he assumes the real the book is on q- series, including mock theta Only basic linear algebra is required of the numbers, and begins his treatment by functions; the remaining part deals with theta reader; techniques from algebra, number theory, introducing such modern concepts as a metric function identities, modular equations, and probability are introduced and developed as space, vector space notation, and groups, and incomplete elliptic integrals of the first kind and required. -
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. -
Major Awards Winners in Mathematics: A
International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.3, No.10, October 2014 DOI:10.15693/ijaist/2014.v3i10.81-92 Major Awards Winners in Mathematics: A Bibliometric Study Rajani. S Dr. Ravi. B Research Scholar, Rani Channamma University, Deputy Librarian Belagavi & Professional Assistant, Bangalore University of Hyderabad, Hyderabad University Library, Bangalore II. MATHEMATICS AS A DISCIPLINE Abstract— The purpose of this paper is to study the bibliometric analysis of major awards like Fields Medal, Wolf Prize and Abel Mathematics is the discipline; it deals with concepts such Prize in Mathematics, as a discipline since 1936 to 2014. Totally as quantity, structure, space and change. It is use of abstraction there are 120 nominees of major awards are received in these and logical reasoning, from counting, calculation, honors. The data allow us to observe the evolution of the profiles of measurement and the study of the shapes and motions of winners and nominations during every year. The analysis shows physical objects. According to the Aristotle defined that top ranking of the author’s productivity in mathematics mathematics as "the science of quantity", and this definition discipline and also that would be the highest nominees received the prevailed until the 18th century. Benjamin Peirce called it "the award at Institutional wise and Country wise. competitors. The science that draws necessary conclusions". United States of America awardees got the highest percentage of about 50% in mathematics prize. According to David Hilbert said that "We are not speaking Index terms –Bibliometric, Mathematics, Awards and Nobel here of arbitrariness in any sense. -
“YOU MUST REMEMBER THIS” Abraham (“Abe”) Edel
MATERIAL FOR “YOU MUST REMEMBER THIS” Abraham (“Abe”) Edel (6 December 1908 – 22 June 2007) “Twenty-Seven Uses of Science in Ethics,” 7/2/67 Abraham Edel, In Memoriam, by Peter Hare and Guy Stroh Abraham Edel, 1908-2007 Abraham Edel was born in Pittsburgh, Pennsylvania on December 6, 1908. Raised in Yorkton, Canada with his older brother Leon who would become a biographer of Henry James, Edel studied Classics and Philosophy at McGill University, earning a BA in 1927 and an MA in 1928. He continued his education at Oxford where, as he recalled, “W.D. Ross and H.A. Prichard were lecturing in ethics, H.W.B. Joseph on Plato, and the influence of G. E. Moore and Bertrand Russell extended from Cambridge. Controversy on moral theory was high. The same was true of epistemology, where Prichard posed realistic epistemology against Harold Joachim who was defending Bradley and Bosanquet against the metaphysical realism of Cook Wilson.” He received a BA in Litterae Humaniores from Oxford in 1930. In that year he moved to New York City for doctoral studies at Columbia University, and in 1931 began teaching at City College, first as an assistant to Morris Raphael Cohen. F.J.E. Woodbridge directed his Columbia dissertation, Aristotle’s Theory of the Infinite (1934). This monograph and two subsequent books on Aristotle were influenced by Woodbridge’s interpretation of Aristotle as a philosophical naturalist. Although his dissertation concerned ancient Greek philosophy, he was much impressed by research in the social sciences at Columbia, and the teaching of Cohen at City College showed him how philosophical issues lay at the root of the disciplines of psychology, sociology, history, as well as the natural sciences. -
THE ELEMENTARY PROOF of the PRIME NUMBER THEOREM: an HISTORICAL PERSPECTIVE (By D
THE ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM: AN HISTORICAL PERSPECTIVE (by D. Goldfeld) The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x>1, let π(x) denote the number of primes less than x. The prime number theorem is the assertion that x lim π(x) =1. x→∞ log(x) This theorem was conjectured independently by Legendre and Gauss. The approximation x π(x)= A log(x)+B was formulated by Legendre in 1798 [Le1] and made more precise in [Le2] where he provided the values A =1,B = −1.08366. On August 4, 1823 (see [La1], page 6) Abel, in a letter to Holmboe, characterizes the prime number theorem (referring to Legendre) as perhaps the most remarkable theorem in all mathematics. Gauss, in his well known letter to the astronomer Encke, (see [La1], page 37) written on Christmas eve 1849 remarks that his attention to the problem of finding an asymptotic formula for π(x) dates back to 1792 or 1793 (when he was fifteen or sixteen), and at that time noticed that the density of primes in a chiliad (i.e. [x, x + 1000]) decreased approximately as 1/ log(x) leading to the approximation x dt π(x) ≈ Li(x)= . 2 log(t) The remarkable part is the continuation of this letter, in which he said (referring to x Legendre’s log(x)−A(x) approximation and Legendre’s value A(x)=1.08366) that whether the quantity A(x) tends to 1 or to a limit close to 1, he does not dare conjecture.