Definitions and Nondefinability in Geometry: Pieri and the Tarski School 1
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Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing by ⌬ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding fixed, then increasing by (1). -
Foundations of Newtonian Dynamics: an Axiomatic Approach For
Foundations of Newtonian Dynamics: 1 An Axiomatic Approach for the Thinking Student C. J. Papachristou 2 Department of Physical Sciences, Hellenic Naval Academy, Piraeus 18539, Greece Abstract. Despite its apparent simplicity, Newtonian mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtle- ties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and deriva- tive theorems. This article attempts to clarify these issues for the benefit of the stu- dent by revisiting the foundations of Newtonian dynamics and by proposing a rigor- ous axiomatic approach to the subject. This theoretical scheme is built upon two fun- damental postulates, namely, conservation of momentum and superposition property for interactions. Newton’s laws, as well as all familiar theorems of mechanics, are shown to follow from these basic principles. 1. Introduction Teaching introductory mechanics can be a major challenge, especially in a class of students that are not willing to take anything for granted! The problem is that, even some of the most prestigious textbooks on the subject may leave the student with some degree of confusion, which manifests itself in questions like the following: • Is the law of inertia (Newton’s first law) a law of motion (of free bodies) or is it a statement of existence (of inertial reference frames)? • Are the first two of Newton’s laws independent of each other? It appears that -
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. -
Phoronomy: Space, Construction, and Mathematizing Motion Marius Stan
To appear in Bennett McNulty (ed.), Kant’s Metaphysical Foundations of Natural Science: A Critical Guide. Cambridge University Press. Phoronomy: space, construction, and mathematizing motion Marius Stan With his chapter, Phoronomy, Kant defies even the seasoned interpreter of his philosophy of physics.1 Exegetes have given it little attention, and un- derstandably so: his aims are opaque, his turns in argument little motivated, and his context mysterious, which makes his project there look alienating. I seek to illuminate here some of the darker corners in that chapter. Specifi- cally, I aim to clarify three notions in it: his concepts of velocity, of compo- site motion, and of the construction required to compose motions. I defend three theses about Kant. 1) his choice of velocity concept is ul- timately insufficient. 2) he sided with the rationalist faction in the early- modern debate on directed quantities. 3) it remains an open question if his algebra of motion is a priori, though he believed it was. I begin in § 1 by explaining Kant’s notion of phoronomy and its argu- ment structure in his chapter. In § 2, I present four pictures of velocity cur- rent in Kant’s century, and I assess the one he chose. My § 3 is in three parts: a historical account of why algebra of motion became a topic of early modern debate; a synopsis of the two sides that emerged then; and a brief account of his contribution to the debate. Finally, § 4 assesses how general his account of composite motion is, and if it counts as a priori knowledge. -
How to Study Mathematics – the Manual for Warsaw University 1St Year Students in the Interwar Period
TECHNICAL TRANSACTIONS CZASOPISMO TECHNICZNE FUNDAMENTAL SCIENCES NAUKI PODSTAWOWE 2-NP/2015 KALINA BARTNICKA* HOW TO STUDY MATHEMATICS – THE MANUAL FOR WARSAW UNIVERSITY 1ST YEAR STUDENTS IN THE INTERWAR PERIOD JAK STUDIOWAĆ MATEMATYKĘ – PORADNIK DLA STUDENTÓW PIERWSZEGO ROKU Z OKRESU MIĘDZYWOJENNEGO Abstract In 1926 and in 1930, members of Mathematics and Physics Students’ Club of the Warsaw University published the guidance for the first year students. These texts would help the freshers in constraction of the plans and course of theirs studies in the situation of so called “free study”. Keywords: Warsaw University, Interwar period, “Free study”, Study of Mathematics, Freshers, Students’ clubs, Guidance for students Streszczenie W 1926 r. i w 1930 r. Koło Naukowe Matematyków i Fizyków Studentów Uniwersytetu War- szawskiego opublikowało poradnik dla studentów pierwszego roku matematyki. Są to teksty, które pomagały pierwszoroczniakom w racjonalnym skonstruowaniu planu i toku ich studiów w warunkach tzw. „wolnego stadium”. Słowa kluczowe: Uniwersytet Warszawski, okres międzywojenny, „wolne stadium”, studio wanie matematyki, pierwszoroczniacy, studenckie koła naukowe, poradnik dla studentów DOI: 10.4467/2353737XCT.15.203.4408 * L. & A. Birkenmajetr Institute of History of Science, Polish Academy of Sciences, Warsaw, Poland; [email protected] 14 This paper is focused primarily on the departure from the “free study” in university learning in Poland after it regained its independence in 1918. The idea of the “free study” had been strongly cherished by professors and staff of the Philosophy Department of Warsaw University even though the majority of students (including the students of mathematics and physics) were not interested in pursuing an academic career. The concept of free study left to the students the decision about the choice of subjects they wished to study and about the plan of their work. -
Adolf Lindenbaum: Notes on His Life, with Bibliography and Selected References
CORE Metadata, citation and similar papers at core.ac.uk Provided by Springer - Publisher Connector Log. Univers. 8 (2014), 285–320 c 2014 The Author(s). This article is published with open access at Springerlink.com 1661-8297/14/030285-36, published online December 3, 2014 DOI 10.1007/s11787-014-0108-2 Logica Universalis Adolf Lindenbaum: Notes on his Life, with Bibliography and Selected References Jan Zygmunt and Robert Purdy Abstract. Notes on the life of Adolf Lindenbaum, a complete bibliography of his published works, and selected references to his unpublished results. Mathematics Subject Classification. 01A60, 01A70, 01A73, 03-03. Keywords. Adolf Lindenbaum, biography, bibliography, unpublished works. This paper is dedicated to Adolf Lindenbaum (1904–1941)—Polish- Jewish mathematician and logician; a member of the Warsaw school of mathe- matics under Waclaw Sierpi´nski and Stefan Mazurkiewicz and school of math- ematical logic under JanLukasiewicz and Stanislaw Le´sniewski;1 and Alfred Tarski’s closest collaborator of the inter-war period. Our paper is divided into three main parts. The first part is biograph- ical and narrative in character. It gathers together what little is known of Lindenbaum’s short life. The second part is a bibliography of Lindenbaum’s published output, including his public lectures. Our aim there is to be complete and definitive. The third part is a list of selected references in the literature attesting to his unpublished results2 and delineating their extent. Just to confuse things, we name the second and third parts of our paper, respectively, “Bibliography Part One” and “Bibliography Part Two”. Why, we no longer remember. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Alice in the Wonderful Land of Logical Notions)
The Mystery of the Fifth Logical Notion (Alice in the Wonderful Land of Logical Notions) Jean-Yves Beziau University of Brazil, Rio de Janeiro, Brazilian Research Council and Brazilian Academy of Philosophy [email protected] Abstract We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of the four basic logical notions. This is what we are doing here, as well as introducing a challenging fifth logical notion. We first recall the context and origin of what are here called Tarski- Lindenbaum logical notions. In the second part, we present these notions in the simple case of a binary relation. In the third part, we examine in which sense these are considered as logical notions contrasting them with an example of a non-logical relation. In the fourth part, we discuss the formulations of the four logical notions in natural language and in first- order logic without equality, emphasizing the fact that two of the four logical notions cannot be expressed in this formal language. In the fifth part, we discuss the relations between these notions using the theory of the square of opposition. In the sixth part, we introduce the notion of variety corresponding to all non-logical notions and we argue that it can be considered as a logical notion because it is invariant, always referring to the same class of structures. -
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 . -
Geometry and Teaching
Andrew McFarland Joanna McFarland James T. Smith Editors Alfred Tarski Early Work in Poland – Geometry and Teaching This book is dedicated to Helen Marie Smith, in gratitude for her advice and support, and to Maria Anna McFarland, as she enters a world of new experiences. Andrew McFarland • Joanna McFarland James T. Smith Editors Alfred Tarski Early Work in Poland—Geometry and Teaching with a Bibliographic Supplement Foreword by Ivor Grattan-Guinness Editors Andrew McFarland Joanna McFarland Páock, Poland Páock, Poland James T. Smith Department of Mathematics San Francisco State University San Francisco, CA, USA ISBN 978-1-4939-1473-9 ISBN 978-1-4939-1474-6 (eB ook) DOI 10.1007/978-1-4939-1474-6 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2014945118 Mathematics Subject Classification (2010): 01A60, 01A70, 01A75, 03A10, 03B05, 03E75, 06A99, 28-03, 28A75, 43A07, 51M04, 51M25, 97B50, 97D40, 97G99, 97M30 © 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, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms 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 specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. -
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. -
Central Limit Theorems for the Brownian Motion on Large Unitary Groups
CENTRAL LIMIT THEOREMS FOR THE BROWNIAN MOTION ON LARGE UNITARY GROUPS FLORENT BENAYCH-GEORGES Abstract. In this paper, we are concerned with the large n limit of the distributions of linear combinations of the entries of a Brownian motion on the group of n × n unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of the asymptotic Gaussian feature of the entries of Haar distributed random unitary matrices, a result already proved by Diaconis et al. Abstract. Dans cet article, on consid`erela loi limite, lorsque n tend vers l’infini, de combinaisons lin´eairesdes coefficients d'un mouvement Brownien sur le groupe des matri- ces unitaires n × n. On prouve que le processus d'une telle combinaison lin´eaireconverge vers un processus gaussien. Diff´erentes ´echelles de temps et diff´erentes lois initiales sont consid´er´ees,donnant lieu `aplusieurs processus limites, li´es`ala construction g´eom´etrique du mouvement Brownien unitaire. En application, on propose une preuve tr`escourte du caract`ereasymptotiquement gaussien des coefficients d'une matrice unitaire distribu´ee selon la mesure de Haar, un r´esultatd´ej`aprouv´epar Diaconis et al. Introduction There is a natural definition of Brownian motion on any compact Lie group, whose distribution is sometimes called the heat kernel measure. Mainly due to its relations with the object from free probability theory called the free unitary Brownian motion and with the two-dimentional Yang-Mills theory, the Brownian motion on large unitary groups has appeared in several papers during the last decade.