DOCSLIB.ORG

PEANO's PROFESSORS Enrico D'ovidio Was Only 33 Years Old When Peano Became a University Student, and He Was to Survive Peano. He

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
PEANO's PROFESSORS Enrico D'ovidio Was Only 33 Years Old When Peano Became a University Student, and He Was to Survive Peano. He APPENDIX 1 PEANO'S PROFESSORS Enrico D'Ovidio was only 33 years old when Peano became a university student, and he was to survive Peano. He was born on 11 August, 1842, in Campobasso, then in the Kingdom of the Two Sicilies. D'Ovidio studied in Naples with Achille Sannia in preparation for entrance into the School of Bridges and Roads, where he also studied for some time. He was influenced to take up an academic career, however, by the lectures of Battaglini, Fergola, Del Grosso, and Padula, and indeed the frrst volumes of Battaglini's Giornale di Matematiche (founded in 1863) contained a few notes by D'Ovidio on determinants and conics. He did not pursue a regular university course, but after teaching in secondary schools, he was granted a degree in mathematics ad honorem by the Facolta di Scienze di Napoli on his application. He was dispensed from any examination, it being considered that his scholarship was already sufficiently proven. In 1869 he published, with Sannia, a geometry text for use in the classical schools. The teaching of Euclid's Elements had recently been prescribed for these schools and this text remained important for some time. Although D'Ovidio did not wish to leave his family and native region, he was encouraged by Eugenio Beltrami to enter the competition in 1872 for the chair of Algebra and Analytic Geometry at the University of Turin, where he spent the remaining 46 years of his life. His most important scientific work occurred during his frrst 20 years there, but even after his retirement in 1918 he continued to take an active role in the life of the university and the Academy of Sciences. His presence in Turin, especially in the early days, contributed greatly to the improvement of North-South relations, both at the university and in the artistic circle presided over by Edmondo De Amicis and Giuseppe Giacosa. In the few years preceding 1872 the work of Lobachevskii, Bolyai, and Riemann in non-Euclidean geometry was becoming well known, as well as the Sixth Memoir of Cayley on projective metrics and Plucker's idea of considering lines as fundamental ideas in space. An important group of papers by D'Ovidio dealt with these ideas, especially with Euclidean and non­ Euclidean projective metrics, culminating in the memoir of 1877 on "The fundamental metric functions in spaces of arbitrarily many dimensions with 176 PEANO'S PROFESSORS 177 constant curvature." At the same time, D'Ovidio was investigating binary forms, which influencedPeano's early publications. Already in 1876,D'Ovidio had published the fust part (dealing with conics) of a three-part text in analytic geometry (the other parts were: quadrics in 1883 and the analytic theory of the fundamental geometric forms in 1885) which displayed his didactic ability. Because of D'Ovidio and his student Corrado Segre, an important school of geometry was developed at the University of Turin. D'Ovidio was instrumental in the establishment of the post of assistant for the exercises in algebra and analytic geometry. He always tried to have his best graduates for this position, so that it came to be a coveted position. His fust assistant was Fortunato Maglioli (1878-79). Then came Francesco Gerbaldi (1879-80). Peano was D'Ovidio's assistant from 1880 to 1883. He was followed by Corrado Segre (1883-84), when Peano became assistant to Angelo Genocchi. D'Ovidio was chairman of the Facolta di Scienze in 1879-80 and rector of the university from 1880 to 1885. He was again Science Chairman from 1893 to 1907, leaving this post (as well as that of Director of the Mathematics Library) when he was named a commissioner of the Poly technique of Turin, of which he was later director until 1922. He was a member of the principal Italian scientific societies (the Academy of Sciences of Turin from 1878, the Accademia dei Lincei from 1883, the Society of the Forty from 1884.) He was named a senator in March 1905, but there were rumors that this was due to a mix-up and that the nomination was intended for his brother Francesco, the noted philologist, who was in fact also named senator a few months later. D'Ovidio died in Turin on 21 March, 1933. Peano's professor of design, Count Carlo Giulio Ceppi, furnished a contrast to D'Ovidio. He was older (he was then 47), affected mannerisms, and was not at all a scientist - more a schoolteacher than a professor. He was born in Turin on 12 October, 1829, and died there on 9 November, 1921. He studied architecture with Carlo Promis at the University of Turin and received a degree in civil engineering and architecture on 12 August, 1851. Since he did not lack money, he decided to work on his own, but after the death of his mother his father urged him to take a regular position. To please him, Ceppi obtained in 1857 the vacant post of design tea<:her at the Royal Military Academy. There were probably other reasons for taking this post: the work required little preparation, there were many holidays, and he was dispensed from service in the National Guard. He became prominent in 1863 when he won fust prize for his project for a facade for the Church of Santa Maria del 178 APPENDIX 1 Fiore in Florence. (This project, however, was never carried out.) He taught at the Military Academy until 1870, but already in 1869 he succeeded Promis as teacher of architecture at the School of Engineering, which later became the Polytechnique. Shortly afterwards, for reasons not entirely clear, he transferred to the university. Beginning around 1867 a number of buildings were erected after his design, perhaps the best of these being the Palazzo Ceriana, designed in 1878. Ceppi was noted for never wearing a topcoat and it is said that from the age of 20 he slept, summer and winter, without covers except for a single sheet. Donato Levi was five years younger than Ceppi and, like Ceppi, a graduate in civil engineering and architecture (1856). From the age of 18 he was an assistant at the Astronomical Observatory of Turin, where he was principally concerned with meteorological observations. Before this he had already started his teaching career. After five years as a secondary school teacher of mathematics, he taught privately and in 1866 he won the competitive post of dottore aggregato in science at the University of Turin (a rank carrying no salary or duties.) From 1875 he was assistant in the school of projective and descriptive geometry. His publisheq works are few and not very important, but they display that orderliness, clarity, and precision of language which was valued in his classroom lessons. Levi was precise and punctual in all his duties until December 1883, when the sickness began that, after several improvements and relapses, carried him off on 21 August, 1885, at the age of 51. Hugo Schiff, like Levi, was 43 years old when Peano attended his first class in chemistry. This was also Schiff's first class in Turin. He was born on 3 April, 1834, in Frankfurt am Main, into an old Jewish family that traced its origin to Spain. He received his chemical training in Gottingen, but transferred to Berne in 1856 for political reasons. He returned to Gottingen to obtain a Ph.D. in 1857, but returned again to Berne, where he began teaching chemistry in 1858. In 1862 he and his brother Moritz went to Italy where he worked first in a laboratory in Pisa, then was professor of chemistry at the Museum of Natural History in Florence, then professor at the University of Turin (1876-79), and fmally professor at the University of Florence from 1880. Schiff liked to tell about the sacrifices he made during the first years of his career, especially in Switzerland, but he was very economical and after 50 years he was able to leave a large inheritance. Politically, he was a socialist and one of the founders of the newspaper Avanti. An accomplished linguist, PEANO'S PROFESSORS 179 he knew several modern European languages, as well as Greek, Latin, Arabic, and Hebrew. He was a member of the Academy of Sciences of Turin from 1898. Schiff had notable success in his study of organic chemistry. In his labora­ tory teaching he was very serious, taking special care in his teaching of chemical analysis. He also invented a number of useful laboratory instruments and published an excellent introductory text in chemistry in 1876. Schiff enjoyed good health and worked hard until December 1899, when he fell in the street and was run over by a loaded cart. He never recovered completely and died in Florence on 8 September, 1915. Michele Lessona was born on the outskirts of Turin on 20 September, 1823, the son of Carlo Lessona, a professor of veterinary science. He enrolled in the medical program of the University of Turin, obtained a post in the Collegio delle Provincie, and was graduated on 12 August, 1846, with a degree in medicine and surgery. For a brief period he practiced medicine in Turin, but he soon fell in love with one of his sisters' teachers, with whom he fled to Egypt to escape the anger of her parents, who were opposed to their marriage. There followed a period of wandering, but after stops in Malta and Cairo, he was named director of the hospital in Khankah. He left Cairo, with his wife and baby only a few months old on 13 May, 1848, but after his arrival in Khankah, a cholera epidemic broke out that took his wife, leaving him alone with his daughter.
Load more

Recommended publications
  • 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.
    [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]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • Writing the History of the Exact Sciences in Nineteenth-Century Rome
    ORE Open Research Exeter TITLE For science and for the Pope-king: writing the history of the exact sciences in nineteenth-century Rome AUTHORS Mazzotti, Massimo JOURNAL The British Journal for the History of Science DEPOSITED IN ORE 01 August 2008 This version available at http://hdl.handle.net/10036/33812 COPYRIGHT AND REUSE Open Research Exeter makes this work available in accordance with publisher policies. A NOTE ON VERSIONS The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of publication BJHS, 2000, 33, 257–282 For science and for the Pope-king: writing the history of the exact sciences in nineteenth-century Rome MASSIMO MAZZOTTI* Abstract. This paper analyses the contents and the style of the Bullettino di bibliografia e di storia delle scienze matematiche e fisiche (1868–1887), the first journal entirely devoted to the history of mathematics. It is argued that its innovative and controversial methodological approach cannot be properly understood without considering the cultural conditions in which the journal was conceived and realized. The style of the Bullettino was far from being the mere outcome of the eccentric personality of its editor, Prince Baldassarre Boncompagni. Rather, it reflected in many ways, at the level of historiography of science, the struggle of the official Roman Catholic culture against the growing secularization of knowledge and society. While history of mathematics was a well-established discipline by the mid-nineteenth century, no periodical which was entirely devoted to such a discipline existed until 1868.
    [Show full text]
  • 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.
    [Show full text]
  • The D'alembert Paradox, 1900
    1 The d’Alembert Paradox, 1900-1914: Levi-Civita and his Italian and French Followers Rossana Tazzioli Université de Lille 1, Sciences et Technologies UFR de Mathématiques, Laboratoire Paul Painlevé Cité Scientifique 59655 Villeneuve d’Ascq Cedex (France) Tel. 06 29 12 33 50 e-mail address: [email protected] Abstract Before the First World War, Tullio Levi-Civita (1873-1941) was already a well-known mathematician in ItalY and abroad, in particular in France. Professor at the UniversitY of Padua since 1898, he had published important contributions to tensor calculus, theory of relativity, differential GeometrY, hYdrodYnamics, and the three-bodY problem. In 1918, when he moved to the University of Rome, he engaged himself in creatinG an international school of mathematics. In this paper, we focus on d’Alembert’s paradox to which Levi- Civita and some of his Italian and French followers contributed remarkable solutions. This case-study is used to illustrate Levi-Civita’s approach to hYdrodYnamics and its influence in ItalY and France, especially in the period 1910-1914. Key-words Tullio Levi-Civita, Henri Villat, d’Alembert’s paradox, hYdrodYnamics 1. Introduction Since ancient times it had been known that a certain resistance is opposed to a bodY movinG in a fluid. However, in 1768 Jean le Rond d’Alembert published a treatise entitled “Paradoxe proposé aux Géomètres sur la résistance des fluides” in which for incompressible and inviscid potential flow he obtained the result of zero drag on head-tail sYmmetric two-dimensional bodies movinG with constant velocity relative to the fluid.1 The contradiction to experimental 1 See [1] Appendix A for a modern statement of the d’Alembert paradox.
    [Show full text]
  • Alexander Karp · Gert Schubring Editors Handbook on the History of Mathematics Education Handbook on the History of Mathematics Education
    Alexander Karp · Gert Schubring Editors Handbook on the History of Mathematics Education Handbook on the History of Mathematics Education Alexander Karp • Gert Schubring Editors Handbook on the History of Mathematics Education Editors Alexander Karp Gert Schubring Teachers College Universität Bielefeld Columbia University Institut für Didaktik der Mathematik New York , NY , USA Bielefeld , Germany ISBN 978-1-4614-9154-5 ISBN 978-1-4614-9155-2 (eBook) DOI 10.1007/978-1-4614-9155-2 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013949144 © Springer Science+Business Media New York 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfi lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifi cally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.
    [Show full text]
  • Peano Arithmetic
    Peano Arithmetic Guram Bezhanishvili∗ In this project we will learn the first-order theory of arithmetic, known as Peano Arithmetic. The formal development of arithmetic goes all the way back to ancient Greek mathematics. But the modern theory of arithmetic was developed only in the second half of the nineteenth century with the work of Hermann Grassmann (1809{1877), Richard Dedekind (1833{1916), and Gottlob Frege (1848{1925). However, it was not until Giuseppe Peano's (1858{1932) treatise \Arithmetices principia, nova methodo exposita" (The principles of arithmetic, presented by a new method) that the axiomatic theory of arithmetic was devised as we know it today.1 Giuseppe Peano was born on 27 August of 1858 near Turin, in the village of Spinetta. He went to village school first in Spinetta and then in nearby Cuneo. In the early seventies he continued his studies in Turin. In 1876 Giuseppe enrolled in the University of Turin, from which he graduated in 1880 with `high honors'. Peano spent his entire academic career at the University of Turin, starting as an assistant in 1880, and steadily progressing through the ranks all the way until he reached the rank of a full chair, which he held from 1895 until his death in 1932. In 1886{1901 he also held a position of professor at the Royal Military Academy of Turin. Peano's main contributions were in analysis, axiomatization of mathematics, and most impor- tantly in mathematical logic. In fact, in the late nineteenth and early twentieth centuries he was considered the leading figure in mathematical logic alongside Frege and Bertrand Russell (1872{ 1970).
    [Show full text]
  • Definitions and Nondefinability in Geometry: Pieri and the Tarski School 1
    Definitions and Nondefinability in Geometry: Pieri and the Tarski School 1 James T. Smith San Francisco State University Introduction In 1886 Mario Pieri became professor of projective and descriptive geometry at the Royal Military Academy in Turin and, in 1888, assistant at the University of Turin. By the early 1890s he and Giuseppe Peano, colleagues at both, were researching related ques- tions about the foundations of geometry. During the next two decades, Pieri used, refined, and publicized Peano’s logical methods in several major studies in this area. The present paper focuses on the history of two of them, Pieri’s 1900 Point and Motion and 1908 Point and Sphere memoirs, and on their influence as the root of later work of Alfred Tarski and his followers. It emphasizes Pieri’s achievements in expressing Euclidean geometry with a minimal family of undefined notions, and in requiring set-theoretic constructs only in his treatment of continuity. It is adapted from and expands on material in the 2007 study of Pieri by Elena Anne Marchisotto and the present author. Although Pieri 1908 had received little explicit attention, during the 1920s Tarski noticed its minimal set of undefined notions, its extreme logical precision, and its use of only a restricted variety of logical methods. Those features permitted Tarski to adapt and reformulate Pieri’s system in the context of first-order logic, which was only then emerging as a coherent framework for logical studies. Tarski’s theory was much simpler, and encouraged deeper investigations into the metamathematics of geometry. In particular, Tarski and Adolf Lindenbaum pursued the study of definability, extend- ing earlier work by the Peano School.
    [Show full text]