DOCSLIB.ORG

Alfred Tarski. Life and Logic Reviewed by Hourya Benis Sinaceur

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Alfred Tarski. Life and Logic Reviewed by Hourya Benis Sinaceur Book Review Alfred Tarski. Life and Logic Reviewed by Hourya Benis Sinaceur Alfred Tarski. Life and Logic cal notions by mathematical means. For instance, Anita Burdman Feferman and Solomon Feferman his decision method for elementary algebra and Cambridge University Press, 2004 geometry is indeed a generalization of a Sturm’s US$34.99, 425 pages algorithm for counting the real roots of a poly- ISBN-13: 978-0-521-80240-6 nomial. Tarski was eager to bring to the fore new connections between mathematics and logic and Alfred Tarski (1901–1983) is one of the two to show how mathematized concepts of logic can greatest logicians of the twentieth century, the help to solve mathematical problems. Thus, Tarski other being Kurt Gödel (1906–1978). Each began initiated the shift from the foundational aims for his career in Europe, respectively in Warsaw and which various branches of modern logic were origi- Vienna, and came to America shortly before the nally developed to heuristic aims for which a new Second World War. In contrast to the otherworldly branch would turn out to be especially efficient. Gödel, Tarski was ambitious and practical. He Namely, Tarski is the father of model theory, the strove for, and succeeded at, building a school of results and tools of which are nowadays commonly logic at the University of California, Berkeley, that used in various mathematical disciplines (algebra, attracted students and distinguished researchers analysis, geometry, computer science, etc.). from all over the world. Tarski’s views and achievements have also Tarski was the leader of the “semantic turn” in changed the way we think about the nature, scope, mathematical logic. This means that he achieved and aims of logic. The rich and detailed biography a shift from a view focused on formal systems, of Tarski written by Anita and Solomon Fefer- axioms, and rules of deduction to a view focusing man show us the roots and the full extent of this on the relations between formal systems and their change. possible interpretations by usual mathematical The Fefermans’ biography is an enthralling theories such as real numbers or Cartesian geom- success story of a self-confident, enterprising, etry. Hence he gave precise definitions of semantic untiring, and entrepreneurial scientist, and a rich and scrupulous account of the numerous achieve- concepts that had been used informally before. ments accomplished by this powerful logician and The most important of those concepts are truth, his colleagues in philosophy of logic, semantics, satisfiability, and definability of a formula; logi- set theory, decision procedures, universal algebra, cal consequence; and model. Tarski was also the algebraic logic, axiomatic geometry, topology, and champion of the trend towards reconstructing logi- model theory. Hourya Benis Sinaceur is Directrice de recherche émérite It is particularly remarkable that the Fefer- at the Centre National de la Recherche Scientifique, mans were able to reconstruct so vividly Tarski’s Institut d’Histoire et de Philosophie des Sciences et childhood in Warsaw; the brilliant gymnasium des Techniques, Paris. Her email address is hourya. and university years; the change of name from [email protected]. Teitelbaum to Tarski (in 1924) and the conversion 986 NOTICES OF THE AMS VOLUME 54, NUMBER 8 to Christianity (motivated by the desire to apply Lvov, and Vienna to Harvard, Princeton, Berkeley, for an academic position); the leftist and socialist and Stanford, to cite only the main internationally bent; the opposition to Zionism and the search for known scientific centers. One meets “mythical” lo- assimilation despite the prejudices against Jews; gicians such as Gödel, with whom Tarski shared an the trips to Zacopane, a meeting place for Polish interest in completeness, definability, decidability, intellectuals in the Tatra mountains—in short, and recursivity, and many outstanding scholars, the scientific, political, cultural, and artistic atmo- such as: Stanisław Lesnievski, Jan Łukasiewicz, sphere that favored the development of Tarski’s Kazimierz Kuratowski, Wacław Sierpi´nski, Tadeusz multifaceted personality and help account for his Kotarbi´nski, Léon Chwistek, Adolf Lindenbaum, wide interests and capacities. No less remarkable is Andrzej Mostowski, Wanda Szmielew, Samuel Ei- the description of Tarski’s years in America, from lenberg, Bertrand Russell, Rudolf Carnap, Willard the time he arrived in 1939 to give invited lectures Van Orman Quine, Evert Beth, Karl Popper, John at Harvard University until his death in 1983, i.e., McKinsey, Paul Cohen, Alonzo Church, Barkley from his first uncertain steps in his profession to Rosser, Stephen Kleene, Bjarni Jónsson, Louise the heydays crowning the steadfast efforts to build Chin, Julia Robinson, Leon Henkin, Robert Vaught, a prestigious logic group at UC Berkeley. All of the Richard Montague, Dana Scott, Solomon Feferman, events are described in the same meticulous way. Jerome Keisler, Patrick Suppes, Georg Kreisel, John The reader follows month by month or year by year Addison, Robert Solovay, Saul Kripke, and oth- the irresistible, though treacherous, ascension of ers who have left a mark on the fields of logic or a strong and passionate character. mathematics or philosophy. Everyone is more or “Life and Logic” is the meaningful subtitle of the less briefly described according to the closeness Fefermans’ book. As with the biography of Jean and depth of his connection to Tarski and in view van Heijenoort, which Anita published in 1993, of his more salient traits. the whole picture, or better the whole colorful and In addition, Tarski’s life story is linked with stirring movie about Tarski’s great adventure, is important intellectual movements. First of all, the composed by intertwining scenes of private life, extraordinary flourishing from about 1900 on- socio-political data, cultural customs and events, wards of mathematical, logical, and foundational academic affairs, and advances in logic. Thus, studies in Poland at the hands of the masters of fifteen chapters deal with the multidimensional the Warsaw-Lvov School (a thorough account of context of Tarski’s scientific research and achieve- this school and its emergence is to be found in ments, while six technical “Interludes”, spliced Jan Wole´nski’s Logic and Philosophy in the Lvov- between the chapters, give a synthetic account of Warsaw School, Kluwer, Dordrecht, 1989). The spe- the main results Tarski and his group obtained. A cific humus provided by the Polish in the interplay bibliography of selected works by and on Tarski of mathematics, logic, and philosophy constituted completes the book. the fertile ground upon which Tarski’s logical The benefits of reading this fascinating bi- masterpieces flourished. In writing them Tarski ography are at least threefold. First, one gets a was concerned on the one hand with showing the concrete image and a vivid portrait of one of the importance of logic for mathematics and, on the two logicians who changed the face of logic in the other hand, with articulating the methodology of twentieth century. We learn how high were his sci- deductive theories and the conceptual analysis of entific ambitions and how large was the realm of logical notions which would give instruments for his passions: for logic and mathematics naturally, rigorous philosophical work in this area. but also for philosophy (with a particular interest The second major movement that affected, and in classical Greek philosophy), for poetry, for the was affected by, Tarski was the Unity of Science arts, and also for climbing mountains and visiting movement, launched in 1934 by Otto Neurath, botanical gardens. Behind the demanding logician the coauthor (along with Rudolf Carnap and Hans one discovers the “bon vivant” who smoked im- Hahn) of the manifesto of the Vienna Circle. As is moderately; drank heavily; used drugs for keeping well known, the basic assumption of the manifesto awake and working until dawn; womanized all was the division of knowledge into empirical and throughout his life; liked parties; and was always logical statements, any other statement outside ready for hiking, traveling, meeting new people, of those categories being considered meaning- and exploring new sights of the world. less. The goal was then to obtain a scientific world The second important benefit is to become ac- conception by applying logical analysis to the quainted with the many people who played a role in empirical material. The brilliant Viennese math- Tarski’s life and career and built a large community ematician Karl Menger, who belonged to the Circle, devoted to various subfields of mathematical logic was invited to lecture in Warsaw in the autumn of (mainly model theory, set theory, and algebra of 1929. He was so impressed by the precise work logic). As in a Balzac novel, one is introduced to of the Polish logicians that he invited Tarski to a whole world, the world of logic and set theory Vienna. There were encounters that would become from around 1900 to the 1980s, and from Warsaw, important for Tarski’s influence outside Poland. SEPTEMBER 2007 NOTICES OF THE AMS 987 Most notably, Carnap attended Tarski’s lectures a scientific and intellectual movement of his own. and in the years following their first meeting in He was building “a school of thought” (p. 3). Inter- 1930, quickly grasped the intrinsic value of the se- national conferences organized in this perspective mantical approach, the importance of the theory of were not exclusively restricted to logicians; rather truth and of the Tarskian analysis of the concept of they included mathematicians, physicists, and logical consequence, and the possible instrumental philosophers of science. After the International contribution of semantics to the development of Symposium on the Axiomatic Method, with Spe- “scientific philosophy”. The 1935 Paris Congress cial Reference to Geometry and Physics, held at of the Unity of Science, where Tarski presented his Berkeley in December 1957/January 1958, Tarski concepts of truth and consequence, marked the planned the First International Congress for Logic, beginning of his international reputation.
Load more

Recommended publications
  • Poznań Studies in the Philosophy of the Sciences and the Humanities), 21Pp
    Forthcoming in: Uncovering Facts and Values, ed. A. Kuzniar and J. Odrowąż-Sypniewska (Poznań Studies in the Philosophy of the Sciences and the Humanities), 21pp. Abstract Alfred Tarski seems to endorse a partial conception of truth, the T-schema, which he believes might be clarified by the application of empirical methods, specifically citing the experimental results of Arne Næss (1938a). The aim of this paper is to argue that Næss’ empirical work confirmed Tarski’s semantic conception of truth, among others. In the first part, I lay out the case for believing that Tarski’s T-schema, while not the formal and generalizable Convention-T, provides a partial account of truth that may be buttressed by an examination of the ordinary person’s views of truth. Then, I address a concern raised by Tarski’s contemporaries who saw Næss’ results as refuting Tarski’s semantic conception. Following that, I summarize Næss’ results. Finally, I will contend with a few objections that suggest a strict interpretation of Næss’ results might recommend an overturning of Tarski’s theory. Keywords: truth, Alfred Tarski, Arne Næss, Vienna Circle, experimental philosophy Joseph Ulatowski ORDINARY TRUTH IN TARSKI AND NÆSS 1. Introduction Many of Alfred Tarski's better known papers on truth (e.g. 1944; 1983b), logical consequence (1983c), semantic concepts in general (1983a), or definability (1948) identify two conditions that successful definitions of “truth,” “logical consequence,” or “definition” require: formal correctness and material (or intuitive) adequacy.1 The first condition Tarski calls “formal correctness” because a definition of truth (for a given formal language) is formally correct when it is constructed in a manner that allows us to avoid both circular definition and semantic paradoxes.
    [Show full text]
  • Alfred Tarski His:Bio:Tar: Sec Alfred Tarski Was Born on January 14, 1901 in Warsaw, Poland (Then Part of the Russian Empire)
    bio.1 Alfred Tarski his:bio:tar: sec Alfred Tarski was born on January 14, 1901 in Warsaw, Poland (then part of the Russian Empire). Described as \Napoleonic," Tarski was boisterous, talkative, and intense. His energy was often reflected in his lectures|he once set fire to a wastebasket while disposing of a cigarette during a lecture, and was forbidden from lecturing in that building again. Tarski had a thirst for knowledge from a young age. Although later in life he would tell students that he stud- ied logic because it was the only class in which he got a B, his high school records show that he got A's across the board| even in logic. He studied at the Univer- sity of Warsaw from 1918 to 1924. Tarski Figure 1: Alfred Tarski first intended to study biology, but be- came interested in mathematics, philosophy, and logic, as the university was the center of the Warsaw School of Logic and Philosophy. Tarski earned his doctorate in 1924 under the supervision of Stanislaw Le´sniewski. Before emigrating to the United States in 1939, Tarski completed some of his most important work while working as a secondary school teacher in Warsaw. His work on logical consequence and logical truth were written during this time. In 1939, Tarski was visiting the United States for a lecture tour. During his visit, Germany invaded Poland, and because of his Jewish heritage, Tarski could not return. His wife and children remained in Poland until the end of the war, but were then able to emigrate to the United States as well.
    [Show full text]
  • Alfred Tarski and a Watershed Meeting in Logic: Cornell, 1957 Solomon Feferman1
    Alfred Tarski and a watershed meeting in logic: Cornell, 1957 Solomon Feferman1 For Jan Wolenski, on the occasion of his 60th birthday2 In the summer of 1957 at Cornell University the first of a cavalcade of large-scale meetings partially or completely devoted to logic took place--the five-week long Summer Institute for Symbolic Logic. That meeting turned out to be a watershed event in the development of logic: it was unique in bringing together for such an extended period researchers at every level in all parts of the subject, and the synergetic connections established there would thenceforth change the face of mathematical logic both qualitatively and quantitatively. Prior to the Cornell meeting there had been nothing remotely like it for logicians. Previously, with the growing importance in the twentieth century of their subject both in mathematics and philosophy, it had been natural for many of the broadly representative meetings of mathematicians and of philosophers to include lectures by logicians or even have special sections devoted to logic. Only with the establishment of the Association for Symbolic Logic in 1936 did logicians begin to meet regularly by themselves, but until the 1950s these occasions were usually relatively short in duration, never more than a day or two. Alfred Tarski was one of the principal organizers of the Cornell institute and of some of the major meetings to follow on its heels. Before the outbreak of World War II, outside of Poland Tarski had primarily been involved in several Unity of Science Congresses, including the first, in Paris in 1935, and the fifth, at Harvard in September, 1939.
    [Show full text]
  • Alfred Tarski, Friend and Daemon Benjamin Wells
    Alfred Tarski, Friend and Daemon Benjamin Wells Engaging Tarski When I had passed doctoral exams in 1963 and Alfred Tarski’s name stayed with me after I read sounded Bob Vaught and John Addison on their about the Banach-Tarski paradox in [3] during availability for supervising my research, both said high school. I then discovered logic (and Tarski’s that the department recognized me as Tarski’s definition of truth) in the last year of college but student. News to me, despite the second opinion. still considered myself to be a topologist, not from So I called him to learn more about my fate. He love but from intimate contact in four courses as acknowledged the claim, “Good,” and invited me an undergraduate. So before arriving at UC Berke- for the first serious nighttime talk. ley in fall 1962 for graduate work, I thought only Source of the Title vaguely about taking a course with him. But my inclinations were shifting: dutifully registering Throughout cooperation and separation, Alfred for fall classes in topological groups and algebraic and I were friends, cordial and personable, but geometry, I added metamathematics and Tarski’s not really personal. He also established another general algebraic systems. The next fall, Tarski of- role, that of daemon in the sense of [4]: a leonine fered set theory. These two courses were my only externalized conscience, at least. The scope of Tarski lectures, but there were numerous seminars. this conscience was foremost mathematical, with a hope for political, a goal of cultural, a reserva- Visitors to Berkeley constantly identified Tarski tion on philosophical and moral, and a hint of with whatever topic occupied him so fruitfully and spiritual.
    [Show full text]
  • Download Download
    R E C E N Z J E ROCZNIKI FILOZOFICZNE Tom LIX, numer 1 – 2011 Anita Burdman-Feferman, Solomon Feferman, Alfred Tarski. aycie i logika , przeł. Joanna Goli Lska-Pilarek, Marian Srebrny, Warszawa: Wydaw- nictwa Akademickie i Profesjonalne 2009, ss. 475. ISBN 978-83-60501-94-8. Nie trzeba by 4 „rasowym” logikiem lub filozofem, by wiedzie 4, kim był Alfred Tarski (1901-1983). Powszechnie znany jest jako „człowiek, który zdefiniował praw- dB” lub – w Bb szemu gronu – jako współautor twierdzenia o paradoksalnym rozkładzie kuli. Wywarł znacz =cy wpływ na rozwój całej XX-wiecznej logiki i podstaw mate- matyki, a tak be – poprzez badania z zakresu semantyki formalnej i podstaw logiki – na epistemologi B, metodologi B nauk i filozofi B j Bzyka. Stworzył semantyk B logiczn =, przyczynił si B do rozwoju metamatematyki oraz teorii modeli, osi =gn =ł znacz =ce rezultaty w teorii mnogo Vci, topologii, geometrii i arytmetyce. Ksi =b ka Alfred Tarski. aycie i logika autorstwa Anity i Solomona Fefermanów została wydana przez Wydawnictwa Akademickie i Profesjonalne, w profesjonalnym przekładzie Joanny Goli Lskiej-Pilarek i Mariana Srebrnego. Szkoda jedynie, be polski przekład pojawił si B dopiero pi B4 lat po opublikowaniu oryginału ameryka Lskiego Alfred Tarski. Life and Logic przez University of Cambridge. Nie jest chyba przypadkiem, be identyczny podtytuł nosi biografia innego wiel- kiego uczonego, który wraz z Tarskim, lecz niezale bnie od niego, zmienił oblicze logiki XX wieku – Kurta Gödla (John Casti, Werner DePauli, Gödel. aycie i logika , tłum. P. Amsterdamski, Warszawa: Wyd. CiS 2003). Z jednej strony samo bycie, jak be ró bne w obu przypadkach: pełne pasji i nami Btno Vci pierwszego, wyobcowane i egocentryczne drugiego; z drugiej – logika, wielka miło V4 ich obu.
    [Show full text]
  • The Development of Mathematical Logic from Russell to Tarski: 1900–1935
    The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives .
    [Show full text]
  • Lecture 1: Tarski on Truth Philosophy of Logic and Language — HT 2016-17
    Lecture 1: Tarski on Truth Philosophy of Logic and Language — HT 2016-17 Jonny McIntosh [email protected] Alfred Tarski (1901-1983) was a Polish (and later, American) mathematician, logician, and philosopher.1 In the 1930s, he published two classic papers: ‘The Concept of Truth in Formalized Languages’ (1933) and ‘On the Concept of Logical Consequence’ (1936). He gives a definition of truth for formal languages of logic and mathematics in the first paper, and the essentials of the model-theoretic definition of logical consequence in the second. Over the course of the next few lectures, we’ll look at each of these in turn. 1 Background The notion of truth seems to lie at the centre of a range of other notions that are central to theorising about the formal languages of logic and mathematics: e.g. validity, con- sistency, and completeness. But the notion of truth seems to give rise to contradiction: Let sentence (1) = ‘sentence (1) is not true’. Then: 1. ‘sentence (1) is not true’ is true IFF sentence (1) is not true 2. sentence (1) = ‘sentence (1) is not true’ 3. So, sentence (1) is true IFF sentence (1) is not true 4. So, sentence (1) is true and sentence (1) is not true Tarski is worried that, unless the paradox can be resolved, metatheoretical results in- voking the notion of truth or other notions that depend on it will be remain suspect. But how can it be resolved? The second premise is undeniable, and the first premise is an instance of a schema that seems central to the concept of truth, namely the following: ‘S’ is true IFF S, 1The eventful story of Tarski’s life is told by Anita and Solomon Feferman in their wonderful biography, Alfred Tarski: Life and Logic (CUP, 2004).
    [Show full text]
  • Carnap's Contribution to Tarski's Truth
    JOURNAL FOR THE HISTORY OF ANALYTICAL PHILOSOPHY CARNAP’S CONTRIBUTION TO TARSKI’S TRUTH VOLUME 3, NUMBER 10 MONIKA GrUBER EDITOR IN CHIEF KEVIN C. KLEMENt, UnIVERSITY OF MASSACHUSETTS In his seminal work “The Concept of Truth in Formalized Lan- guages” (1933), Alfred Tarski showed how to construct a for- EDITORIAL BOARD mally correct and materially adequate definition of true sentence GaRY EBBS, INDIANA UnIVERSITY BLOOMINGTON for certain formalized languages. These results have, eventu- GrEG FROSt-ARNOLD, HOBART AND WILLIAM SMITH COLLEGES ally, been accepted and applauded by philosophers and logi- HENRY JACKMAN, YORK UnIVERSITY cians nearly in unison. Its Postscript, written two years later, SANDRA LaPOINte, MCMASTER UnIVERSITY however, has given rise to a considerable amount of contro- LyDIA PATTON, VIRGINIA TECH versy. There is an ongoing debate on what Tarski really said MARCUS ROSSBERG, UnIVERSITY OF CONNECTICUT in the postscript. These discussions often regard Tarski as pu- MARK TEXTOR, KING’S COLLEGE LonDON tatively changing his logical framework from type theory to set AUDREY YAP, UnIVERSITY OF VICTORIA theory. RICHARD ZACH, UnIVERSITY OF CALGARY In what follows, we will compare the original results with those REVIEW EDITORS presented two years later. After a brief outline of Carnap’s pro- JULIET FLOYD, BOSTON UnIVERSITY gram in The Logical Syntax of Language we will determine its sig- CHRIS PINCOCK, OHIO STATE UnIVERSITY nificance for Tarski’s final results. ASSISTANT REVIEW EDITOR SEAN MORRIS, METROPOLITAN STATE UnIVERSITY OF DenVER DESIGN DaNIEL HARRIS, HUNTER COLLEGE C 2015 MONIKA GrUBER CARNAP’S CONTRIBUTION TO TARSKI’S TRUTH couple of years ago. In particular, in addition to the previously studied languages, he decides to investigate the languages the MONIKA GrUBER structure of which cannot be brought into harmony with the principles of the theory of semantical categories.
    [Show full text]
  • The History of Logic
    c Peter King & Stewart Shapiro, The Oxford Companion to Philosophy (OUP 1995), 496–500. THE HISTORY OF LOGIC Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and nega- tion in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (uni- versal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’).
    [Show full text]
  • 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.
    [Show full text]
  • Biographies.Pdf
    Chapter udf Biographies bio.1 Georg Cantor his:bio:can: An early biography of Georg Cantor sec (gay-org kahn-tor) claimed that he was born and found on a ship that was sailing for Saint Petersburg, Russia, and that his parents were unknown. This, however, is not true; although he was born in Saint Petersburg in 1845. Cantor received his doctorate in mathematics at the University of Berlin in 1867. He is known for his work in set theory, and is credited with founding set theory as a distinctive research discipline. He was the first to prove that there are infinite sets of different sizes. His theo- ries, and especially his theory of infini- ties, caused much debate among mathe- maticians at the time, and his work was controversial. Cantor's religious beliefs and his Figure bio.1: Georg Cantor mathematical work were inextricably tied; he even claimed that the theory of transfinite numbers had been communicated to him directly by God. Inlater life, Cantor suffered from mental illness. Beginning in 1894, and more fre- quently towards his later years, Cantor was hospitalized. The heavy criticism of his work, including a falling out with the mathematician Leopold Kronecker, led to depression and a lack of interest in mathematics. During depressive episodes, Cantor would turn to philosophy and literature, and even published a theory that Francis Bacon was the author of Shakespeare's plays. Cantor died on January 6, 1918, in a sanatorium in Halle. 1 Further Reading For full biographies of Cantor, see Dauben(1990) and Grattan-Guinness(1971).
    [Show full text]
  • Tarski's Conception of Logic
    Tarski’s Conception of Logic1 Solomon Feferman In its widest scope, Tarski thought the aims of logic should be the creation of “a unified conceptual apparatus which would supply a common basis for the whole of human knowledge.” Those were his very words in the Preface to the first English edition of the Introduction to Logic (1940). Toward that grand end, in the post-war years when the institutional and financial resources became available, with extraordinary persistence and determination Tarski campaigned vigorously on behalf of logic on several fronts from his increasingly powerful base at the University of California in Berkeley. The first order of business was to build up a school in logic bridging the university’s Mathematics and Philosophy Departments, and the opening wedge in that was the hiring of Leon Henkin as Professor of Mathematics in 1953. From then on, Henkin was Tarski’s right-hand man in the logic campaigns, locally, nationally and internationally, but he had other allies, both in Mathematics and in Philosophy. The first goal was to increase the proportion of logicians on the mathematics faculty to 10% of the whole; that took a number of years, eventually achieved with the appointment of Addison, Vaught, Solovay, Scott, Silver, Harrington and McKenzie. Through his influence in Philosophy, he succeeded in recruiting Myhill, Craig, Chihara and Sluga. Hans Sluga tells a story which gives a vivid picture of how Tarski operated: they met in 1966 when Tarski was in London to give the Shearman Lectures at Bedford College. Sluga, then a young faculty member interested in the philosophy of logic, was delegated to show him around.
    [Show full text]