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Biography Paper – Georg Cantor
Mike Garkie Math 4010 – History of Math UCD Denver 4/1/08 Biography Paper – Georg Cantor Few mathematicians are house-hold names; perhaps only Newton and Euclid would qualify. But there is a second tier of mathematicians, those whose names might not be familiar, but whose discoveries are part of everyday math. Examples here are Napier with logarithms, Cauchy with limits and Georg Cantor (1845 – 1918) with sets. In fact, those who superficially familier with Georg Cantor probably have two impressions of the man: First, as a consequence of thinking about sets, Cantor developed a theory of the actual infinite. And second, that Cantor was a troubled genius, crippled by Freudian conflict and mental illness. The first impression is fundamentally true. Cantor almost single-handedly overturned the Aristotle’s concept of the potential infinite by developing the concept of transfinite numbers. And, even though Bolzano and Frege made significant contributions, “Set theory … is the creation of one person, Georg Cantor.” [4] The second impression is mostly false. Cantor certainly did suffer from mental illness later in his life, but the other emotional baggage assigned to him is mostly due his early biographers, particularly the infamous E.T. Bell in Men Of Mathematics [7]. In the racially charged atmosphere of 1930’s Europe, the sensational story mathematician who turned the idea of infinity on its head and went crazy in the process, probably make for good reading. The drama of the controversy over Cantor’s ideas only added spice. 1 Fortunately, modern scholars have corrected the errors and biases in older biographies. -
Georg Cantor English Version
GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle. -
Cantor and Continuity
Cantor and Continuity Akihiro Kanamori May 1, 2018 Georg Cantor (1845-1919), with his seminal work on sets and number, brought forth a new field of inquiry, set theory, and ushered in a way of proceeding in mathematics, one at base infinitary, topological, and combinatorial. While this was the thrust, his work at the beginning was embedded in issues and concerns of real analysis and contributed fundamentally to its 19th Century rigorization, a development turning on limits and continuity. And a continuing engagement with limits and continuity would be very much part of Cantor's mathematical journey, even as dramatically new conceptualizations emerged. Evolutionary accounts of Cantor's work mostly underscore his progressive ascent through set- theoretic constructs to transfinite number, this as the storied beginnings of set theory. In this article, we consider Cantor's work with a steady focus on con- tinuity, putting it first into the context of rigorization and then pursuing the increasingly set-theoretic constructs leading to its further elucidations. Beyond providing a narrative through the historical record about Cantor's progress, we will bring out three aspectual motifs bearing on the history and na- ture of mathematics. First, with Cantor the first mathematician to be engaged with limits and continuity through progressive activity over many years, one can see how incipiently metaphysical conceptualizations can become systemati- cally transmuted through mathematical formulations and results so that one can chart progress on the understanding of concepts. Second, with counterweight put on Cantor's early career, one can see the drive of mathematical necessity pressing through Cantor's work toward extensional mathematics, the increasing objectification of concepts compelled, and compelled only by, his mathematical investigation of aspects of continuity and culminating in the transfinite numbers and set theory. -
Infinitesimals Via Cauchy Sequences: Refining the Classical Equivalence
INFINITESIMALS VIA CAUCHY SEQUENCES: REFINING THE CLASSICAL EQUIVALENCE EMANUELE BOTTAZZI AND MIKHAIL G. KATZ Abstract. A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number sys- tem. Such an approach can be seen as formalizing Cauchy’s senti- ment that a null sequence “becomes” an infinitesimal. We signal a little-noticed construction of a system with infinitesimals in a 1910 publication by Giuseppe Peano, reversing his earlier endorsement of Cantor’s belittling of infinitesimals. June 2, 2021 1. Historical background Robinson developed his framework for analysis with infinitesimals in his 1966 book [1]. There have been various attempts either (1) to obtain clarity and uniformity in developing analysis with in- finitesimals by working in a version of set theory with a concept of infinitesimal built in syntactically (see e.g., [2], [3]), or (2) to define nonstandard universes in a simplified way by providing mathematical structures that would not have all the features of a Hewitt–Luxemburg-style ultrapower [4], [5], but nevertheless would suffice for basic application and possibly could be helpful in teaching. The second approach has been carried out, for example, by James Henle arXiv:2106.00229v1 [math.LO] 1 Jun 2021 in his non-nonstandard Analysis [6] (based on Schmieden–Laugwitz [7]; see also [8]), as well as by Terry Tao in his “cheap nonstandard analysis” [9]. These frameworks are based upon the identification of real sequences whenever they are eventually equal.1 A precursor of this approach is found in the work of Giuseppe Peano. In his 1910 paper [10] (see also [11]) he introduced the notion of the end (fine; pl. -
Introduction: the 1930S Revolution
PROPERTY OF MIT PRESS: FOR PROOFREADING AND INDEXING PURPOSES ONLY Introduction: The 1930s Revolution The theory of computability was launched in the 1930s by a group of young math- ematicians and logicians who proposed new, exact, characterizations of the idea of algorithmic computability. The most prominent of these young iconoclasts were Kurt Gödel, Alonzo Church, and Alan Turing. Others also contributed to the new field, most notably Jacques Herbrand, Emil Post, Stephen Kleene, and J. Barkley Rosser. This seminal research not only established the theoretical basis for computability: these key thinkers revolutionized and reshaped the mathematical world—a revolu- tion that culminated in the Information Age. Their motive, however, was not to pioneer the discipline that we now know as theoretical computer science, although with hindsight this is indeed what they did. Nor was their motive to design electronic digital computers, although Turing did go on to do so (in fact producing the first complete paper design that the world had seen for an electronic stored-program universal computer). Their work was rather the continuation of decades of intensive investigation into that most abstract of subjects, the foundations of mathematics—investigations carried out by such great thinkers as Leopold Kronecker, Richard Dedekind, Gottlob Frege, Bertrand Russell, David Hilbert, L. E. J. Brouwer, Paul Bernays, and John von Neumann. The concept of an algorithm, or an effective or computable procedure, was central during these decades of foundational study, although for a long time no attempt was made to characterize the intuitive concept formally. This changed when Hilbert’s foundation- alist program, and especially the issue of decidability, made it imperative to provide an exact characterization of the idea of a computable function—or algorithmically calculable function, or effectively calculable function, or decidable predicate. -
1 CS Peirce's Classification of Dyadic Relations: Exploring The
C.S. Peirce’s Classification of Dyadic Relations: Exploring the Relevance for Logic and Mathematics Jeffrey Downard1 In “The Logic of Mathematics; an Attempt to Develop my Categories from Within” [CP 1.417-520] Charles S. Peirce develops a scheme for classifying different kinds of monadic, dyadic and triadic relations. His account of these different classes of relations figures prominently in many parts of his philosophical system, including the phenomenological account of the categories of experience, the semiotic account of the relations between signs, objects and the metaphysical explanations of the nature of such things as chance, brute existence, law-governed regularities and the making and breaking of habits. Our aim in this essay is to reconstruct and examine central features of the classificatory system that he develops in this essay. Given the complexity of the system, we will focus our attention in this essay mainly on different classes of degenerate and genuine dyadic relations, and we will take up the discussion of triadic relations in a companion piece. One of our reasons for wanting to explore Peirce’s philosophical account of relations is to better understand how it might have informed the later development of relations as 1 Associate Professor, Department of Philosophy, Northern Arizona University. 2 In what follows, I will refer to Peirce’s essay “The Logic of Mathematics; An Attempt to Develop my Categories from Within” using the abbreviated title “The Logic of Mathematics.” 3 One reason to retain all of the parts of the figures—including the diamonds that represent the saturated bonds—is that it makes it possible to study more closely the combinatorial possibilities of the system. -
Peano 441 Peano
PEANO PEANO On his life and work, see M . Berthelot, "Necrologie," his position at the military academy but retained his in Bulletin . Societe chindque de France, A5 (1863), 226- professorship at the university until his death in 1932, 227 ; J. R. Partington, A History of Chemistry, IV (London, having transferred in 1931 to the chair of complemen- 1964), 584-585 ; and F. Szabadvary, History of Analytical tary mathematics. He was elected to a number of Chemistry (Oxford, 1966), 251 . scientific societies, among them the Academy of F . SZABADVARY Sciences of Turin, in which he played a very active role . He was also a knight of the Order of the Crown of Italy and of the Order of Saint Maurizio and PEANO, GIUSEPPE (b . Spinetta, near Cuneo, Italy, Saint Lazzaro. Although lie was not active politically, 27 August 1858 ; d. Turin, Italy, 20 April 1932), his views tended toward socialism ; and lie once invited mathematics, logic . a group of striking textile workers to a party at his Giuseppe Peano was the second of the five children home. During World War I he advocated a closer of Bartolomeo Peano and Rosa Cavallo . His brother federation of the allied countries, to better prosecute Michele was seven years older . There were two the war and, after the peace, to form the nucleus of a younger brothers, Francesco and Bartolomeo, and a world federation . Peano was a nonpracticing Roman sister, Rosa . Peano's first home was the farm Tetto Catholic . Galant, near the village of Spinetta, three miles from Peano's father died in 1888 ; his mother, in 1910. -
2.5. INFINITE SETS Now That We Have Covered the Basics of Elementary
2.5. INFINITE SETS Now that we have covered the basics of elementary set theory in the previous sections, we are ready to turn to infinite sets and some more advanced concepts in this area. Shortly after Georg Cantor laid out the core principles of his new theory of sets in the late 19th century, his work led him to a trove of controversial and groundbreaking results related to the cardinalities of infinite sets. We will explore some of these extraordinary findings, including Cantor’s eponymous theorem on power sets and his famous diagonal argument, both of which imply that infinite sets come in different “sizes.” We also present one of the grandest problems in all of mathematics – the Continuum Hypothesis, which posits that the cardinality of the continuum (i.e. the set of all points on a line) is equal to that of the power set of the set of natural numbers. Lastly, we conclude this section with a foray into transfinite arithmetic, an extension of the usual arithmetic with finite numbers that includes operations with so-called aleph numbers – the cardinal numbers of infinite sets. If all of this sounds rather outlandish at the moment, don’t be surprised. The properties of infinite sets can be highly counter-intuitive and you may likely be in total disbelief after encountering some of Cantor’s theorems for the first time. Cantor himself said it best: after deducing that there are just as many points on the unit interval (0,1) as there are in n-dimensional space1, he wrote to his friend and colleague Richard Dedekind: “I see it, but I don’t believe it!” The Tricky Nature of Infinity Throughout the ages, human beings have always wondered about infinity and the notion of uncountability. -
Project Gutenberg's Essays on the Theory of Numbers, by Richard
Project Gutenberg’s Essays on the Theory of Numbers, by Richard Dedekind This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net Title: Essays on the Theory of Numbers Author: Richard Dedekind Translator: Wooster Woodruff Beman Release Date: April 8, 2007 [EBook #21016] Language: English Character set encoding: TeX *** START OF THE PROJECT GUTENBERG EBOOK THEORY OF NUMBERS *** Produced by Jonathan Ingram, Keith Edkins and the Online Distributed Proofreading Team at http://www.pgdp.net Transcriber’s Note: The symbol 3 is used as an approximation to the au- thor’s Part-of symbol, not to be confused with the digit 3. Internal page ref- erences have been been adjusted to fit the pagination of this edition. A few typographical errors have been corrected - these are noted at the very end of the text. IN THE SAME SERIES. ON CONTINUITY AND IRRATIONAL NUMBERS, and ON THE NATURE AND MEANING OF NUMBERS. By R. Dedekind. From the German by W. W. Beman. Pages, 115. Cloth, 75 cents net (3s. 6d. net). GEOMETRIC EXERCISES IN PAPER-FOLDING. By T. Sundara Row. Edited and revised by W. W. Beman and D. E. Smith. With many half-tone engravings from photographs of actual exercises, and a package of papers for folding. Pages, circa 200. Cloth, $1.00. net (4s. 6d. net). (In Preparation.) ON THE STUDY AND DIFFICULTIES OF MATHEMATICS. -
Giuseppe Peano and His School: Axiomatics, Symbolism and Rigor
Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 25-1 | 2021 The Peano School: Logic, Epistemology and Didactics Giuseppe Peano and his School: Axiomatics, Symbolism and Rigor Paola Cantù and Erika Luciano Electronic version URL: http://journals.openedition.org/philosophiascientiae/2788 DOI: 10.4000/philosophiascientiae.2788 ISSN: 1775-4283 Publisher Éditions Kimé Printed version Date of publication: 25 February 2021 Number of pages: 3-14 ISBN: 978-2-38072-000-6 ISSN: 1281-2463 Electronic reference Paola Cantù and Erika Luciano, “Giuseppe Peano and his School: Axiomatics, Symbolism and Rigor”, Philosophia Scientiæ [Online], 25-1 | 2021, Online since 01 March 2021, connection on 30 March 2021. URL: http://journals.openedition.org/philosophiascientiae/2788 ; DOI: https://doi.org/10.4000/ philosophiascientiae.2788 Tous droits réservés Giuseppe Peano and his School: Axiomatics, Symbolism and Rigor Paola Cantù Aix-Marseille Université, CNRS, Centre Gilles-Gaston-Granger, Aix-en-Provence (France) Erika Luciano Università degli Studi di Torino, Dipartimento di Matematica, Torino (Italy) Peano’s axioms for arithmetic, published in 1889, are ubiquitously cited in writings on modern axiomatics, and his Formulario is often quoted as the precursor of Russell’s Principia Mathematica. Yet, a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics remains to be made. In line with increased interest in the philosophy of mathematics for the investigation of mathematical practices, this thematic issue adds some contributions to a possible reconstruction of the philosophical views of the Peano School. These derive from logical, mathematical, linguistic, and educational works1, and also interactions with contemporary scholars in Italy and abroad (Cantor, Dedekind, Frege, Russell, Hilbert, Bernays, Wilson, Amaldi, Enriques, Veronese, Vivanti and Bettazzi). -
Peano Arithmetic
CSC 438F/2404F Notes (S. Cook) Fall, 2008 Peano Arithmetic Goals Now 1) We will introduce a standard set of axioms for the language LA. The theory generated by these axioms is denoted PA and called Peano Arithmetic. Since PA is a sound, axiomatizable theory, it follows by the corollaries to Tarski's Theorem that it is in- complete. Nevertheless, it appears to be strong enough to prove all of the standard results in the field of number theory (including such things as the prime number theo- rem, whose standard proofs use analysis). Even Andrew Wiles' proof of Fermat's Last Theorem has been claimed to be formalizable in PA. 2) We know that PA is sound and incomplete, so there are true sentences in the lan- guage LA which are not theorems of PA. We will outline a proof of G¨odel'sSecond Incompleteness Theorem, which states that a specific true sentence, asserting that PA is consistent, is not a theorem of PA. This theorem can be generalized to show that any consistent theory satisfying general conditions cannot prove its own consistency. 3) We will introduce a finitely axiomatized subtheory RA (\Robinson Arithmetic") of PA and prove that every consistent extension of RA (including PA) is \undecidable" (meaning not recursive). As corollaries, we get a stronger form of G¨odel'sfirst incom- pleteness theorem, as well as Church's Theorem: The set of valid sentences of LA is not recursive. The Theory PA (Peano Arithmetic) The so-called Peano postulates for the natural numbers were introduced by Giuseppe Peano in 1889. -
Study on Sets
International Journal of Research (IJR) Vol-1, Issue-10 November 2014 ISSN 2348-6848 Study on Sets Sujeet Kumar & Ashish Kumar Gupta Department of Information and technology Dronacharya College of Engineering,Gurgaon-122001, India Email:[email protected], Email:[email protected] Set difference of U and A, denoted U \ A, Abstract- is the set of all members of Uthat are not members of A. The set difference {1,2,3} \ Set theory is the branch of mathematical logic that studies sets, which {2,3,4} is {1} , while, conversely, the set are collections of objects. Although any type of object can be collected difference {2,3,4} \ {1,2,3} is {4} . into a set, set theory is applied most often to objects that are relevant to mathematics.In this research paper we studied about Basic concepts and When A is a subset of U, the set notation, some ontology and applications. We have also study about difference U \ A is also called combinational set theory, forcing, cardinal invariants, fuzzy set theory. the complement of A inU. In this case, if We have described all the basic concepts of Set Theory. the choice of U is clear from the context, the notation Acis sometimes used instead Keywords- of U \ A, particularly if U is a universal set as in the study of Venn diagrams. Combinational ;fuzzy ; forcing; cardinals; ontology 1. INTRODUCTION Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set of all Set theory is the branch of mathematical logic that studies objects that are a member of exactly one sets, which are collections of objects.