Peano Arithmetic
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Richard Dedekind English Version
RICHARD DEDEKIND (October 6, 1831 – February 12, 1916) by HEINZ KLAUS STRICK, Germany The biography of JULIUS WILHELM RICHARD DEDEKIND begins and ends in Braunschweig (Brunswick): The fourth child of a professor of law at the Collegium Carolinum, he attended the Martino-Katherineum, a traditional gymnasium (secondary school) in the city. At the age of 16, the boy, who was also a highly gifted musician, transferred to the Collegium Carolinum, an educational institution that would pave the way for him to enter the university after high school. There he prepared for future studies in mathematics. In 1850, he went to the University at Göttingen, where he enthusiastically attended lectures on experimental physics by WILHELM WEBER, and where he met CARL FRIEDRICH GAUSS when he attended a lecture given by the great mathematician on the method of least squares. GAUSS was nearing the end of his life and at the time was involved primarily in activities related to astronomy. After only four semesters, DEDEKIND had completed a doctoral dissertation on the theory of Eulerian integrals. He was GAUSS’s last doctoral student. (drawings © Andreas Strick) He then worked on his habilitation thesis, in parallel with BERNHARD RIEMANN, who had also received his doctoral degree under GAUSS’s direction not long before. In 1854, after obtaining the venia legendi (official permission allowing those completing their habilitation to lecture), he gave lectures on probability theory and geometry. Since the beginning of his stay in Göttingen, DEDEKIND had observed that the mathematics faculty, who at the time were mostly preparing students to become secondary-school teachers, had lost contact with current developments in mathematics; this in contrast to the University of Berlin, at which PETER GUSTAV LEJEUNE DIRICHLET taught. -
Hubert Kennedy Eight Mathematical Biographies
Hubert Kennedy Eight Mathematical Biographies Peremptory Publications San Francisco 2002 © 2002 by Hubert Kennedy Eight Mathematical Biographies is a Peremptory Publications ebook. It may be freely distributed, but no changes may be made in it. Comments and suggestions are welcome. Please write to [email protected] . 2 Contents Introduction 4 Maria Gaetana Agnesi 5 Cesare Burali-Forti 13 Alessandro Padoa 17 Marc-Antoine Parseval des Chênes 19 Giuseppe Peano 22 Mario Pieri 32 Emil Leon Post 35 Giovanni Vailati 40 3 Introduction When a Dictionary of Scientific Biography was planned, my special research interest was Giuseppe Peano, so I volunteered to write five entries on Peano and his friends/colleagues, whose work I was investigating. (The DSB was published in 14 vol- umes in 1970–76, edited by C. C. Gillispie, New York: Charles Scribner's Sons.) I was later asked to write two more entries: for Parseval and Emil Leon Post. The entry for Post had to be done very quickly, and I could not have finished it without the generous help of one of his relatives. By the time the last of these articles was published in 1976, that for Giovanni Vailati, I had come out publicly as a homosexual and was involved in the gay liberation movement. But my article on Vailati was still discreet. If I had written it later, I would probably have included evidence of his homosexuality. The seven articles for the Dictionary of Scientific Biography have a uniform appear- ance. (The exception is the article on Burali-Forti, which I present here as I originally wrote it—with reference footnotes. -
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 . -
Definitions and Nondefinability in Geometry 475 2
Definitions and Nondefinability in Geometry1 James T. Smith Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which han- dled some basic concepts awkwardly and imprecisely. They would introduce precision re- quired for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geo- metric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By fol- lowing this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics. 1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for general- ization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the ques- tion, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal. -
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. -
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. -
Models in Geometry and Logic: 1870-1920 ∗
Models in Geometry and Logic: 1870-1920 ∗ Patricia Blanchette University of Notre Dame 1 Abstract Questions concerning the consistency of theories, and of the independence of axioms and theorems one from another, have been at the heart of logical investigation since the beginning of logic. It is well known that our own contemporary methods for proving independence and consistency owe a good deal to developments in geometry in the middle of the nineteenth century, and especially to the role of models in estab- lishing the consistency of non-Euclidean geometries. What is less well understood, and is the topic of this essay, is the extent to which the concepts of consistency and of independence that we take for granted today, and for which we have clear techniques of proof, have been shaped in the intervening years by the gradual de- velopment of those techniques. It is argued here that this technical development has brought about significant conceptual change, and that the kinds of consistency- and independence-questions we can decisively answer today are not those that first motivated metatheoretical work. The purpose of this investigation is to help clarify what it is, and importantly what it is not, that we can claim to have shown with a modern demonstration of consistency or independence.1 ∗This is the nearly-final version of the paper whose official version appears in Logic, Methodology, and Philosophy of Science - proceedings of the 15th International Congress, edited by Niiniluoto, Sepp¨al¨a,Sober; College Publications 2017, pp 41-61 1Many thanks to the organizers of CLMPS 2015 in Helsinki, and also to audience members for helpful comments. -
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. -
An Introduction to Elementary Set Theory
An Introduction to Elementary Set Theory Guram Bezhanishvili and Eachan Landreth∗ 1 Introduction In this project we will learn elementary set theory from the original historical sources by two key figures in the development of set theory, Georg Cantor (1845{1918) and Richard Dedekind (1831{ 1916). We will learn the basic properties of sets, how to define the size of a set, and how to compare different sizes of sets. This will enable us to give precise definitions of finite and infinite sets. We will conclude the project by exploring a rather unusual world of infinite sets. Georg Cantor, the founder of set theory, considered by many as one of the most original minds in the history of mathematics, was born in St. Petersburg, Russia in 1845. His parents moved the family to Frankfurt, Germany in 1856. Cantor entered the Wiesbaden Gymnasium at the age of 15, and two years later began his university career in Z¨urich, Switzerland. In 1863 he moved to the University of Berlin, which during Cantor's time was considered the world's leading center of mathematical research. Four years later Cantor received his doctorate under the supervision of the great Karl Weierstrass (1815{1897). In 1869 Cantor obtained an unpaid lecturing post at the University of Halle. Ten years later he was promoted to a full professor. However, Cantor never achieved his dream of holding a Chair of Mathematics at Berlin. It is believed that one of the main reasons was the nonacceptance of his theories of infinite sets by the leading mathematicians of that time, most noticeably by Leopold Kronecker (1823{1891), a professor at the University of Berlin and a very influential figure in German mathematics, both mathematically and politically. -
La Scuola Di Giuseppe Peano
AperTO - Archivio Istituzionale Open Access dell'Università di Torino La Scuola di Giuseppe Peano This is the author's manuscript Original Citation: Availability: This version is available http://hdl.handle.net/2318/75035 since Publisher: Deputazione Subalpina di Storia Patria Terms of use: Open Access Anyone can freely access the full text of works made available as "Open Access". Works made available under a Creative Commons license can be used according to the terms and conditions of said license. Use of all other works requires consent of the right holder (author or publisher) if not exempted from copyright protection by the applicable law. (Article begins on next page) 06 October 2021 Erika Luciano, Clara Silvia Roero * LA SCUOLA DI GIUSEPPE PEANO Lungo tutto l’arco della sua vita universitaria, dal 1880 al 1932, Peano amò circondarsi di allievi, assistenti, colleghi e in- segnanti, cui chiedeva di prendere parte alle iniziative cultura- li o di ricerca che egli stava realizzando: la Rivista di Matema- tica, il Formulario, il Dizionario di Matematica, le Conferenze Matematiche Torinesi, l’Academia pro Interlingua e il periodi- co Schola et Vita. Non stupisce dunque che fin dagli anni No- vanta dell’Ottocento alcuni contemporanei, in lettere private o in sede di congressi internazionali e in articoli, facessero espli- citamente riferimento ad un preciso gruppo di ricercatori, qua- lificandolo come la ‘Scuola italiana’ o la ‘Scuola di Peano’. Dal- le confidenze di G. Castelnuovo a F. Amodeo, ad esempio, sappiamo che nel 1891 la cerchia dei giovani matematici, so- prannominata la Pitareide, che a Torino soleva riunirsi a di - scutere all’American Bar, si era frantumata in due compagini, * Desideriamo ringraziare Paola Novaria, Laura Garbolino, Giuseppe Semeraro, Margherita Bongiovanni, Giuliano Moreschi, Stefania Chiavero e Francesco Barbieri che in vario modo hanno facilitato le nostre ricerche ar- chivistiche e bibliografiche.