DOCSLIB.ORG

Download Thesis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Download Thesis This electronic thesis or dissertation has been downloaded from the King’s Research Portal at https://kclpure.kcl.ac.uk/portal/ Berkeley’s Analyst Rigour and Rhetoric Moriarty, Clare Marie Awarding institution: King's College London The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without proper acknowledgement. END USER LICENCE AGREEMENT Unless another licence is stated on the immediately following page this work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ You are free to copy, distribute and transmit the work Under the following conditions: Attribution: You must attribute the work in the manner specified by the author (but not in any way that suggests that they endorse you or your use of the work). Non Commercial: You may not use this work for commercial purposes. No Derivative Works - You may not alter, transform, or build upon this work. Any of these conditions can be waived if you receive permission from the author. Your fair dealings and other rights are in no way affected by the above. Take down policy If you believe that this document breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 26. Sep. 2021 Berkeley’s Analyst: Rigour and Rhetoric Clare Marie Moriarty A Thesis submitted for the degree of Doctor of Philosophy July 2018 Abstract of the Dissertation This thesis argues for a new interpretation of The Analyst based on Berkeley’s mature theory of meaning and its role in his views on religion and mathematics. I argue that we should read the Body of The Analyst as consisting in an argumentum ad hominem against ‘freethinkers’ who alleged that a mathematical/logicist criterion of intelligibility showed significant parts of religion to be unintelligible and irrational. By showing that the same standards (standards inspired by mathematics and logic) demonstrate the logical instability of calculus, he provides a reductio argument against this freethinking methodology. The text has typically been read as constituting a significant change in Berkeley’s position on the philosophy of mathematics—one involving a newly conciliatory outlook on the foundations and axioms of classical mathematics and an abandonment of the sweeping semantic pragmatism advanced by Euphranor at the end of Alciphron and the instrumentalism that defines much of his previous approach to mathematics and science. I argue that this ostensible endorsement of the foundations of traditional mathematics is merely a necessary condition of an internal argument Berkeley wishes to use to demonstrate that the calculus fails its own discipline’s tests of rigour. Further, by reading the text as I suggest, we can reconcile the arguments of The Analyst with the pragmatic theory of word meaning endorsed in the decisive argument of the final dialogue of Alciphron. 1 Table of Contents ABSTRACT OF THE DISSERTATION TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF ABBREVIATIONS GENERAL INTRODUCTION 1. BERKELEY’S EARLY VIEWS ON LANGUAGE AND MEANING 1.1 INTRODUCTION 1.2 EARLY THOUGHTS ON LANGUAGE AND MEANING: EDUCATIONAL INFLUENCES LOCKE’S PHILOSOPHY OF LANGUAGE: ‘OF WORDS’ 1.3 BERKELEY’S RECEPTION OF LOCKE ‘OF INFINITES’ AND BERKELEY’S NOTEBOOKS AN ESSAY TOWARDS A NEW THEORY OF VISION 1.4 LANGUAGE IN THE NEW THEORY OF VISION 2. THE PHILOSOPHY OF ALCIPHRON 2.1 INTRODUCTION 2.2 BERKELEY AND THE FREETHINKERS BERKELEY AT THE GUARDIAN UNDERSTANDING ALCIPHRON 2.3 ALCIPHRON’S FREETHINKERS 2.4 AGAINST SPECULATIVE PHILOSOPHY: BERKELEY’S ANTI-THEORETICAL ALCIPHRON PHILOSOPHY A CAVEAT CONCERNING PROTO-PRAGMATISM BARREN SPECULATION 2.5 DIALOGUE SEVEN 2 DEVELOPMENT MEANING IN DIALOGUE SEVEN 2.6 EXPERIMENTS IN LITERARY FORM: BERKELEY’S ‘SATYRICAL NATURE’ 3. BERKELEY’S MATHEMATICAL CONTEXT: HIS EARLY PHILOSOPHY AND THE EUCLIDEAN MATHEMATICAL LANDSCAPE 3.1 INTRODUCTION 3.2 EARLIEST MATHEMATICAL RHETORIC 3.3 PRESCRIPTIVE REMARKS: WHAT GEOMETRY SHOULD TREAT 3.4 PHILOSOPHICALLY OPPOSED: BERKELEY’S PHILOSOPHY AND EUCLIDEAN FOUNDATIONS 3.5 INFINITE DIVISIBILITY AND BERKELEY’S EARLY PHILOSOPHY 3.6 KEILL’S PROOF OF THE INFINITE DIVISIBILITY OF EXTENSION 3.7 CONCLUSION 4. THE PHILOSOPHY AND MATHEMATICS OF THE ANALYST 4.1 INTRODUCTION 4.2 THE ANALYST: A CRITICAL SUMMARY THE MATHEMATICAL CRITICISM FURTHER ARGUMENTS BERKELEY’S COMPENSATION OF ERRORS ATTEMPT AT RE-DESCRIBING THE METHOD DISPELLING PREVIOUS CRITICAL RESPONSES CONCLUSION OF THE BODY 4.3 THE QUERIES ANTI-MATHEMATICISM THE PROPER OBJECT OF GEOMETRY INFINITE DIVISIBILITY 4.4 THE ANALYST: INTERPRETATIVE VERDICT DEFENCE OF FREETHINKING IN MATHEMATICS AND REASONS FOR NOT REPLYING 4.5 CONCLUSION 5. THE EARLY MODERN AD HOMINEM ARGUMENTS AND THE ANALYST AD HOMINEMS 3 5.1 INTRODUCTION 5.2 MOTIVATION OF THE AD HOMINEM EXEGETICAL COMPLEXITY AND MATHEMATICAL LEGACY 5.3 AD HOMINEM ARGUMENTS: CONTEMPORARY AND EARLY MODERN 5.4 THE ANALYST AD HOMINEM ARGUMENTS AND EVALUATION CRITERIA A TU QUOQUE AD HOMINEM OF PRAGMATIC INCONSISTENCY A CIRCUMSTANTIAL AD HOMINEM CONCERNING TESTIMONIAL AUTHORITY CONCLUSION BIBLIOGRAPHY 4 Acknowledgements Please excuse the length of this section where I try and balance the equation expressing the amount of support I have received against the gratitude I have expressed. Or, as Berkeley wrote to Pope, ‘As I take ingratitude to be a greater crime than impertinence, I chose rather to run the risk of being thought the latter, than not to return you my thanks (…).’1 I am very grateful for the support I received from the Arts and Humanities Research Council. The doctoral grant, bursary and funding of a semester at UC Berkeley, made this PhD possible. I am grateful to the Royal Institute of Philosophy for supporting my final year and for the Susan Stebbing bursary for women at KCL. Thanks to KCL Philosophy and the Graduate School for countless travel grants that allowed me to present this work to new groups of people in interesting places. Jasper Reid, primary supervisor—thanks for your critical eye and near-omniscient command of modern philosophy. I’m sure it’s been tricky working with someone who cries so frequently and unpredictably (I will reflect on the mutual compatibility of these claims applied to this act when I have time). I’ve learned a huge amount and am really thankful for the time you’ve given me. David Papineau—thanks for all of the humorous philosophical chatter and general support. I’ve found your encouragement very buoying in times of stress, and I hope that there is a paper in our mutual interest in Flann O’Brien’s De Selby. John Callanan—you deserve particular thanks for the tricky job of trying to drag my tired body over the line. As reward, I promise to read some Kant. Thank you so much for your kindness and support. Your enthusiasm for the project was 1 (Hight 2012, 116) 5 appreciated more in the last months than you could know. I really have tried to tame the prose. I began studying philosophy at UCD when, having failed some French exams, I realised it wasn’t possible to transfer into any other subject in final year without having completed the previous years. I kept at it because of enormous support from the faculty—especially Jim O’Shea, Roland Stout and Dermot Moran. Particularly, though, Maria Baghramian: thank you so much for seeing something in me, and for making me feel I have the support and care of an amazing person. I’ve had a lot of help from faculty and fellow graduate students at KCL. Sam, Dave, Matt, Fintan, Chris, Peter, John, Caspar, Mike, Nils thanks for always making yourselves available to discuss all things philosophical. Owen: thanks for singing with me. John: I’ll see you soon for when our roots in constant conference planning flower into the party planning business that will make us rich. Alex: thanks for making philosophy fun to discuss again, and for a really incredible Donegal trip. Kimi: thanks for being my loyal pool partner and for our many victories! To Emma, the other woman, thanks for the laughs and shared frustrations... Thanks to Ben Davies for systematically removing ‘probably’, ‘might’, ‘potentially not so ridiculous, all things considered’ etc. from my application documents over the years. Ben Jeffery, thanks for being the person to whom I take my most extravagant neuroses. To departmental bff Paul Doody, thanks for your patient encouragement, empathy and each and every weird picture you have sent me—you are a total philosophical wizard. To the history of philosophy women: Ana, Emily, Rosa, Sarah and Jen: thanks for all the great discussions and papers that have added hugely to how I think about my own work. Jen: thanks for being the best friend imaginable and an amazing example to follow. Thanks for teaching me to climb and for being so magnificent that my mother talks constantly of adult adoption. Cambridge parents, Beth and Chris: you two have been a splendid source of fun and distraction in a tricky time! Beth: I can’t thank you enough for all the support, abstract-reading, drinks, laughs etc. you have given me, but I intend to find a way to do so eventually! Seriously, thanks. 6 My home pals: receiving such support has made me feel the deepest sense of gratitude. Fay, Nikki, Nina, Karen, Krisy, Cloe, Michelle, Sonia, Lisa, Ruth, Kim, Ciara, Babs: thank you so much! Thanks also to a newer school friend, Féaron, for swimming with me in the very coldest water, and combining complaints about academia with a much improved front crawl! Clem, excitement about your pregnancy was a welcome excitement and distraction—thanks for all the encouragement.
Load more

Recommended publications
  • Arxiv:1812.00226V2 [Math.HO] 11 Feb 2019 2010 EBI’ ELFUDDFCIN N THEIR and FICTIONS WELL-FOUNDED LEIBNIZ’S ..Bso W Prahs5 Approaches Two on Bos 130 1.2
    LEIBNIZ’S WELL-FOUNDED FICTIONS AND THEIR INTERPRETATIONS JACQUES BAIR, PIOTR BLASZCZYK, ROBERT ELY, PETER HEINIG, AND MIKHAIL G. KATZ Abstract. Leibniz used the term fiction in conjunction with in- finitesimals. What kind of fictions they were exactly is a subject of scholarly dispute. The position of Bos and Mancosu contrasts with that of Ishiguro and Arthur. Leibniz’s own views, expressed in his published articles and correspondence, led Bos to distinguish between two methods in Leibniz’s work: (A) one exploiting clas- sical ‘exhaustion’ arguments, and (B) one exploiting inassignable infinitesimals together with a law of continuity. Of particular interest is evidence stemming from Leibniz’s work Nouveaux Essais sur l’Entendement Humain as well as from his correspondence with Arnauld, Bignon, Dagincourt, Des Bosses, and Varignon. A careful examination of the evidence leads us to the opposite conclusion from Arthur’s. We analyze a hitherto unnoticed objection of Rolle’s concern- ing the lack of justification for extending axioms and operations in geometry and analysis from the ordinary domain to that of infini- tesimal calculus, and reactions to it by Saurin and Leibniz. A newly released 1705 manuscript by Leibniz (Puisque des per- sonnes. ) currently in the process of digitalisation, sheds light on the nature of Leibnizian inassignable infinitesimals. In a pair of 1695 texts Leibniz made it clear that his incompa- rable magnitudes violate Euclid’s Definition V.4, a.k.a. the Archi- medean property, corroborating the non-Archimedean construal of the Leibnizian calculus. Keywords: Archimedean property; assignable vs inassignable quantity; Euclid’s Definition V.4; infinitesimal; law of continuity; arXiv:1812.00226v2 [math.HO] 11 Feb 2019 law of homogeneity; logical fiction; Nouveaux Essais; pure fiction; quantifier-assisted paraphrase; syncategorematic; transfer princi- ple; Arnauld; Bignon; Des Bosses; Rolle; Saurin; Varignon Contents 1.
    [Show full text]
  • Hermeneutics of the Differential Calculus in Eighteenth- Century Europe
    Hermeneutics of the differential calculus in eighteenth- century Europe: from the Analyse des infiniment petits by L’Hôpital (1696) to the Traité élémentaire de calcul différentiel et de calcul intégral by Lacroix (1802)12 Mónica Blanco Abellán Departament de Matemàtica Aplicada III Universitat Politècnica de Catalunya In the history of mathematics it has not been unusual to assume that the communication of mathematical knowledge among countries flowed without constraints, partly because mathematics has often been considered as “universal knowledge”. Schubring,3 however, does not agree with this view and prefers referring to the basic units of communication, which enable the common understanding of knowledge. The basic unit should be constituted by a common language and a common culture, both interacting within a common national or state context. Insofar this interaction occurs within a national educational system, communication is here potentially possible. Consequently Schubring proposes comparative analysis of textbooks as a means to examine the differences between countries with regard to style, meaning and epistemology, since they emerge from a specific educational context. Taking Schubring’s views as starting point, the aim of this paper is to analyze and compare the mathematical development of the differential calculus in France, Germany, Italy and Britain through a number of specific works on the subject, and within their corresponding educational systems. The paper opens with an outline of the institutional framework of mathematical education in these countries. In order to assess the mathematical development of the works to be analyzed, the paper proceeds with a sketch of the epistelomogical aspects of the differential calculus in the eighteenth century.
    [Show full text]
  • Sources and Studies in the History of Mathematics and Physical Sciences
    Sources and Studies in the History of Mathematics and Physical Sciences Editorial Board J.Z. Buchwald J. Lu¨tzen G.J. Toomer Advisory Board P.J. Davis T. Hawkins A.E. Shapiro D. Whiteside Gert Schubring Conflicts between Generalization, Rigor, and Intuition Number Concepts Underlying the Development of Analysis in 17–19th Century France and Germany With 21 Illustrations Gert Schubring Institut fu¨r Didaktik der Mathematik Universita¨t Bielefeld Universita¨tstraße 25 D-33615 Bielefeld Germany [email protected] Sources and Studies Editor: Jed Buchwald Division of the Humanities and Social Sciences 228-77 California Institute of Technology Pasadena, CA 91125 USA Library of Congress Cataloging-in-Publication Data Schubring, Gert. Conflicts between generalization, rigor, and intuition / Gert Schubring. p. cm. — (Sources and studies in the history of mathematics and physical sciences) Includes bibliographical references and index. ISBN 0-387-22836-5 (acid-free paper) 1. Mathematical analysis—History—18th century. 2. Mathematical analysis—History—19th century. 3. Numbers, Negative—History. 4. Calculus—History. I. Title. II. Series. QA300.S377 2005 515′.09—dc22 2004058918 ISBN-10: 0-387-22836-5 ISBN-13: 978-0387-22836-5 Printed on acid-free paper. © 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring St., New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, com- puter software, or by similar or dissimilar methodology now known or hereafter developed is for- bidden.
    [Show full text]
  • Berkeley Bibliography
    Berkeley Bibliography (1979-2019) Abad, Juan Vázques. “Observaciones sobre la noción de causa en el opusculo sobre el movimiento de Berkeley.” Analisis Filosofico 6 (1986): 35-44. Abelove, H. “George Berkeley’s Attitude to John Wesley: the Evidence of a Lost Letter.” Harvard Theological Review 70 (1977): 175-76. Ablondi, Fred. “Berkeley, Archetypes, and Errors.” Southern Journal of Philosophy 43 (2005): 493- 504. _____. “Absolute Beginners: Learning Philosophy by Learning Descartes and Berkeley.” Metascience 19 (2010): 385-89. _____. “On the Ghosts of Departed Quantities.” Metascience 21 (2012): 681-83. _____. “Hutcheson, Perception, and the Sceptic’s Challenge.” British Journal for the History of Philosophy 20 (2012): 269-81. Ackel, Helen. Über den Prozess der menschlichen Erkenntnis bei John Locke und George Berkeley. München und Ravensburg: Grin, 2008. Adamczykowa, Izabella. “The Role of the Subject in the Cognitive Process after George Berkeley: Passive for Active Subject?” Annales Universitatis Mariae Curie Sklodowska, Sectio 1 Philosophia-Sociologia 6 (1981): 43-57. Adar, Einat. “From Irish Philosophy to Irish Theatre: The Blind (Wo)Man Made to See.” Estudios Irlandeses 12 (2017): 1-11. Addyman, David and Feldman, Matthew. “Samuel Beckett, Wilhelm Windelband, and the Interwar ‘Philosophy Notes’.” Modernism/Modernity 18 (2011): 755-70. Agassi, Joseph. “The Future of Berkeley’s Instrumentalism.” International Studies in Philosophy 7 (1975), 167-78. Aichele, Alexander. “Ich Denke Was, Was Du Nicht Denkst, Und Das Ist Rot. John Locke Und George Berkeley Über Abstrakte Ideen Und Kants Logischer Abstraktionismus.” Kant- Studien 103 (2012): 25-46. Airaksinen, Timo. “Berkeley and the Justification of Beliefs [Abstract].” Berkeley Newsletter 8 (1985), 9.
    [Show full text]
  • History of Calculus
    History of Calculus Let’s start with the definition of mathematics and calculus. Mathematics – from the Greek knowledge, study learning. Is the study of quantity, structure, space and change. Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Is the mathematics of instantaneous rates of change – how rapidly is some particular quantity changing at this very instant? Calculus has two main branches. Differential calculus provides methods for calculating rates of change and it has many geometric applications, in particular finding tangents to curves. Integral calculus does the opposite: give the rates of chance of some quantity, it specifies the quantity itself. Geometric applications of integral calculus include the computation of areas and volumes. Perhaps the most significant is this unexpected connection between two apparently unrelated classical geometric questions: finding tangents to a curve and finding areas. Why do we have to learn mathematics? Where mathematics came from? Mathematics came from our imagination; it is a complete invention of the human brain. With the invention of mathematics we create science. Mathematics makes up human, because it’s a distinction that we have from all others animals in the animal’s kingdom. The important of mathematics and science is so great that no person living in the twentieth century can claim to be educated if they are unaware of the modern vision of the physical universe and the history of the magnificent concepts that it embodies. Science is not rigid. Dogmas are rigid. We should understand that science is the business of measuring things.
    [Show full text]
  • Gallois Galois
    GALLOIS GALOIS Individual works include Grundzfige der schlesischen geometer with Michel Rolle and Pierre Varignon . He Klinratologie (Breslau, 1857); Uher die Verbesserung der stated his intention of publishing a critical translation Plan eten elem en te (Breslau, 1858) ; Uber eine Bestimmung of Pappus, but nothing came of this project . Instead, der Sonnenparallaxe arts korrespondierenden Beohachtungen he stimulated the quarrel between his two colleagues des Planeten Flora (Breslau, 1875) ; and Mitteilungen der concerning differential calculus and impeded its set- Koniglichen Universitdts-Stern warte Breslau fiber hier hisher tlement until 1706 . gewonnene Resultate fur die geographischen and klimato- Despite this negative attitude, the consequences of logischen Ortsverhdltnisse (Breslau, 1879). If . SECONDARY LITERATURE . Works on Galle and his which might have been disastrous, Gallois deserves work are W . Foerster, "J . G. Galle," in Vierteljahrsschrift recognition by historians of science for his activities der Astronornischen Gesellschaft, 46 (1911), 17-22 ; and D . as a publicist. Although he wrote somewhat fancifully Wattenberg, J. G. Galle (Leipzig, 1963). and with little concern for coherence, he was of H . C . FREILSLEBEN service in his time as a disseminator of ideas and his work is still valuable as an historical source . GALLOIS, JEAN (b. Paris, France, 11 June 1632 ; d. Paris, 19 April 1707), history of science. The son of a counsel to the Parlement of Paris, BIBLIOGRAPHY Gallois seems to have distinguished himself in that city around 1664 by the breadth of his learning, I . ORIGINAL WORKS . Gallois's translations and other works include Traduction latine du traite de paix des by his knowledge of Hebrew and of both living and Pyrenees (Paris, 1659) ; Breviannn Colhertinum (Paris, classical languages, by his interest in the sciences, and 1679) : "Extrait du livre intitule : Observations physiques et by a genuine literary talent .
    [Show full text]
  • A Calendar of Mathematical Dates January
    A CALENDAR OF MATHEMATICAL DATES V. Frederick Rickey Department of Mathematical Sciences United States Military Academy West Point, NY 10996-1786 USA Email: fred-rickey @ usma.edu JANUARY 1 January 4713 B.C. This is Julian day 1 and begins at noon Greenwich or Universal Time (U.T.). It provides a convenient way to keep track of the number of days between events. Noon, January 1, 1984, begins Julian Day 2,445,336. For the use of the Chinese remainder theorem in determining this date, see American Journal of Physics, 49(1981), 658{661. 46 B.C. The first day of the first year of the Julian calendar. It remained in effect until October 4, 1582. The previous year, \the last year of confusion," was the longest year on record|it contained 445 days. [Encyclopedia Brittanica, 13th edition, vol. 4, p. 990] 1618 La Salle's expedition reached the present site of Peoria, Illinois, birthplace of the author of this calendar. 1800 Cauchy's father was elected Secretary of the Senate in France. The young Cauchy used a corner of his father's office in Luxembourg Palace for his own desk. LaGrange and Laplace frequently stopped in on business and so took an interest in the boys mathematical talent. One day, in the presence of numerous dignitaries, Lagrange pointed to the young Cauchy and said \You see that little young man? Well! He will supplant all of us in so far as we are mathematicians." [E. T. Bell, Men of Mathematics, p. 274] 1801 Giuseppe Piazzi (1746{1826) discovered the first asteroid, Ceres, but lost it in the sun 41 days later, after only a few observations.
    [Show full text]
  • Arxiv:1210.7750V1 [Math.HO] 29 Oct 2012 Tsml Ehdo Aiaadmnm;Rfato;Selslaw
    ALMOST EQUAL: THE METHOD OF ADEQUALITY FROM DIOPHANTUS TO FERMAT AND BEYOND MIKHAIL G. KATZ, DAVID M. SCHAPS, AND STEVEN SHNIDER Abstract. We analyze some of the main approaches in the liter- ature to the method of ‘adequality’ with which Fermat approached the problems of the calculus, as well as its source in the παρισoτης´ of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermat’s method of finding max- ima, minima, tangents, and solving other problems that a mod- ern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat’s col- lected works [66, p. 133-172]. We show that at least some of the manifestations of adequality amount to variational techniques ex- ploiting a small, or infinitesimal, variation e. Fermat’s treatment of geometric and physical applications sug- gests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the rel- evance to understanding Fermat of 19th century dictionary defini- tions of παρισoτης´ and adaequare, cited by Breger, and take issue with his interpretation of adequality, including his novel reading of Diophantus, and his hypothesis concerning alleged tampering with Fermat’s texts by Carcavy. We argue that Fermat relied on Bachet’s reading of Diophantus. Diophantus coined the term παρισoτης´ for mathematical pur- poses and used it to refer to the way in which 1321/711 is ap- proximately equal to 11/6. Bachet performed a semantic calque in passing from pariso¯o to adaequo.
    [Show full text]
  • Sources in the Development of Mathematics
    This page intentionally left blank Sources in the Development of Mathematics The discovery of infinite products by Wallis and infinite series by Newton marked the beginning of the modern mathematical era. The use of series allowed Newton to find the area under a curve defined by any algebraic equation, an achievement completely beyond the earlier methods of Torricelli, Fermat, and Pascal. The work of Newton and his contemporaries, including Leibniz and the Bernoullis, was concentrated in math- ematical analysis and physics. Euler’s prodigious mathematical accomplishments dramatically extended the scope of series and products to algebra, combinatorics, and number theory. Series and products proved pivotal in the work of Gauss, Abel, and Jacobi in elliptic functions; in Boole and Lagrange’s operator calculus; and in Cayley, Sylvester, and Hilbert’s invariant theory. Series and products still play a critical role in the mathematics of today. Consider the conjectures of Langlands, including that of Shimura-Taniyama, leading to Wiles’s proof of Fermat’s last theorem. Drawing on the original work of mathematicians from Europe, Asia, and America, Ranjan Roy discusses many facets of the discovery and use of infinite series and products. He gives context and motivation for these discoveries, including original notation and diagrams when practical. He presents multiple derivations for many important theorems and formulas and provides interesting exercises, supplementing the results of each chapter. Roy deals with numerous results, theorems, and methods used by students, mathematicians, engineers, and physicists. Moreover, since he presents original math- ematical insights often omitted from textbooks, his work may be very helpful to mathematics teachers and researchers.
    [Show full text]
  • Berkeley Bibliography
    Berkeley Bibliography (1979-2011) Abad, Juan Vázques. “Observaciones sobre la noción de causa en el opusculo sobre el movimiento de Berkeley.” Analisis Filosofico 6 (1986): 35-44. Abelove, H. “George Berkeley’s Attitude to John Wesley: the Evidence of a Lost Letter.” Harvard Theological Review 70 (1977): 175-76. Ablondi, Fred. “Berkeley, Archetypes, and Errors.” Southern Journal of Philosophy 43 (2005): 493- 504. _____. “Absolute Beginners: Learning Philosophy by Learning Descartes and Berkeley.” Metascience 19 (2010): 385-89. Ackel, Helen. Über den Prozess der menschlichen Erkenntnis bei John Locke und George Berkeley. München und Ravensburg: Grin, 2008. Adamczykowa, Izabella. “The Role of the Subject in the Cognitive Process after George Berkeley: Passive for Active Subject?” Annales Universitatis Mariae Curie Sklodowska, Sectio 1 Philosophia-Sociologia 6 (1981): 43-57. Agassi, Joseph. “The Future of Berkeley’s Instrumentalism.” International Studies in Philosophy 7 (1975), 167-78. Airaksinen, Timo. “Berkeley and the Justification of Beliefs [Abstract].” Berkeley Newsletter 8 (1985), 9. _____. “Berkeley and the Justification of Beliefs.” Philosophy and Phenomenological Research 48 (1987), 235-56. _____. “The Chain and the Animal: Idealism in Berkeley’s Siris.” In Gersh and Moran (2006), 224- 43. _____. “The Path of Fire: The Meaning and Interpretation of Berkeley’s Siris.” In Daniel, New Interpretations (2007), 261-81. _____. “Berkeley and Newton on Gravity in Siris.” In Parigi (2010b), 87-106. _____. “Active Principles and Trinities in Berkeley’s Siris.” Revue Philosophique de la France et de l’Ėtranger 135 (2010): 57-70. _____. “Rhetoric and Corpuscularism in Berkeley’s Siris.” History of European Ideas 37 (2011): 23-34.
    [Show full text]
  • One of Berkeley’S Arguments on Compensating Errors in The
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Historia Mathematica 38 (2011) 219–231 www.elsevier.com/locate/yhmat One of Berkeley’s arguments on compensating errors in the calculus Kirsti Andersen * Department of Sciences Studies, Aarhus University, C.F. Moellers Allé 8, Building 1110, DK 8000 Aarhus, Denmark Abstract This paper addresses three questions related to George Berkeley’s theory of compensating errors in the calculus published in 1734. The first is how did Berkeley conceive of Leibnizian differentials? The second and most central question concerns Berkeley’s procedure which consisted in identifying two quantities as errors and proving that they are equal. The question is how was this possible? The answer is that this was not possible, because in his calculations Berkeley misguided himself by employing a result equivalent to what he wished to prove. In 1797 Lazare Carnot published the expression “a compensation of errors” in an attempt to explain why the calculus functions. The third question is: did Carnot by this expression mean the same as Berkeley? Ó 2010 Elsevier Inc. All rights reserved. Riassunto Questo articolo affronta tre domande concernenti la teoria di Berkeley sulla compensazione degli errori nel calcolo pubblicata nel 1734. La prima è, come concepiva Berkeley i differenziali leibniziani? La seconda domanda, quella più importante, concerne la procedura di Berkeley consistente nell’identificare due quantità come errori per poi provare che esse sono uguali. La domanda è, come è possibile questo? La risposta è che ciò non è possibile dato che nei suoi calcoli Berkeley si ingannò utilizzando un risultato equivalente a quello che voleva provare.
    [Show full text]
  • Rolle's Theorem and Bolzano-Cauchy Theorem : a View
    Rolle’s Theorem and Bolzano-Cauchy Theorem : A View from the End of the 17th Century until K. Weierstrass’ epoch G.I. SINKEVICH ABSTRACT We discuss the history of the famous Rolle’s theorem “If a function is continuous at [a, b], differentiable in (a, b), and f (a) = f (b), then there exists a point c in (a, b) such that f’(c) = 0”, and that of the related theorem on the root interval, “If a function is continuous on [a, b] and has different signs at the ends of the interval, then there exists a point c in (a, b) such that f (c) = 0”. Key words : Rolle’s Theorem, Bolzano–Cauchy Theorem. INTRODUCTION Contemplating over the 19th century reforms in analysis bring to mind mainly two names, A. Cauchy and K. Weierstrass. Actually, quite a few mathematicians form part of any meaningful history of the topic during the era. Some of them lived before, while others were their contemporaries, and without due reference to them the history of the topic would not only lack certain important colours, but also the thoroughness needed in a historical analysis. Mathematician and theologist Bernard Bolzano was one of the ingenious messengers of the ideas that brought the reform. Many fundamental ideas which were later developed by Cauchy, Weierstrass, Cantor and Dedekind, were originated by him. These include the notion of the least upper bound (1817), realization of the need for developing a theory of real numbers (1830s) and set-theoretic understanding of numbers.1 1 Bolzano’s ideas gained popularity in Germany thanks to Hermann Hankel who in 1870 published the above mentioned work of Bolzano in Tubingen and popularized his other works.
    [Show full text]