Leonhard Euler (1707–1783)
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Nonstandard Analysis (Math 649K) Spring 2008
Nonstandard Analysis (Math 649k) Spring 2008 David Ross, Department of Mathematics February 29, 2008 1 1 Introduction 1.1 Outline of course: 1. Introduction: motivation, history, and propoganda 2. Nonstandard models: definition, properties, some unavoidable logic 3. Very basic Calculus/Analysis 4. Applications of saturation 5. General Topology 6. Measure Theory 7. Functional Analysis 8. Probability 1.2 Some References: 1. Abraham Robinson (1966) Nonstandard Analysis North Holland, Amster- dam 2. Martin Davis and Reuben Hersh Nonstandard Analysis Scientific Ameri- can, June 1972 3. Davis, M. (1977) Applied Nonstandard Analysis Wiley, New York. 4. Sergio Albeverio, Jens Erik Fenstad, Raphael Høegh-Krohn, and Tom Lindstrøm (1986) Nonstandard Methods in Stochastic Analysis and Math- ematical Physics. Academic Press, New York. 5. Al Hurd and Peter Loeb (1985) An introduction to Nonstandard Real Anal- ysis Academic Press, New York. 6. Keith Stroyan and Wilhelminus Luxemburg (1976) Introduction to the Theory of Infinitesimals Academic Press, New York. 7. Keith Stroyan and Jose Bayod (1986) Foundations of Infinitesimal Stochas- tic Analysis North Holland, Amsterdam 8. Leif Arkeryd, Nigel Cutland, C. Ward Henson (eds) (1997) Nonstandard Analysis: Theory and Applications, Kluwer 9. Rob Goldblatt (1998) Lectures on the Hyperreals, Springer 2 1.3 Some history: • (xxxx) Archimedes • (1615) Kepler, Nova stereometria dolorium vinariorium • (1635) Cavalieri, Geometria indivisibilus • (1635) Excercitationes geometricae (”Rigor is the affair of philosophy rather -
Casting out Beams: Berkeley's Criticism of the Calculus
Casting Out Beams: Berkeley's Criticism of the Calculus Eugene C. Boman Penn State - Harrisburg First cast out the beam out of thine own eye; and thou shalt see clearly to cast out the mote out of thy brother's eye. King James Bible, Matthew 7:5 Calculus burst upon the scientific community in the late 17th century with unparalleled success. Nevertheless it took another 200 years and the combined efforts of many first rate mathematicians to finally "get it right." In particular the fundamental concept of a limit did not take on its final form until the 1800's with the work of Karl Weierstrass. The Method of Fluxions as it was known in Britain, was invented by Isaac Newton in the 1660's. For reasons known only to him, he did not publish his invention until much later. In the meantime Gottfried Wilhelm Leibniz published his own formulation of Calculus which he called Calculus Differentialis in 1684. Newton's approach was kinematic. Quantities were said to "flow" in time and the fluxion (derivative) of a flowing quantity was interpreted as a velocity. Leibniz's approach was based on the idea of infinitely small quantities: infinitesimals. Their public priority brawl is both legendary [11] and a stain on both of their reputations. A slightly less well known public argument, but mathematically far more important, began in 1734 when Bishop George Berkeley published a treatise entitled The Analyst. Berkeley derided the lack of rigor in the fundamentals of both renderings of the topic. To be sure, neither Newton's nor Leibniz's formulation of the Calculus would stand up to the modern standard of rigor. -
Utilitarianism in the Age of Enlightenment
UTILITARIANISM IN THE AGE OF ENLIGHTENMENT This is the first book-length study of one of the most influential traditions in eighteenth-century Anglophone moral and political thought, ‘theological utilitarianism’. Niall O’Flaherty charts its devel- opment from its formulation by Anglican disciples of Locke in the 1730s to its culmination in William Paley’s work. Few works of moral and political thought had such a profound impact on political dis- course as Paley’s Principles of Moral and Political Philosophy (1785). His arguments were at the forefront of debates about the constitution, the judicial system, slavery and poverty. By placing Paley’s moral thought in the context of theological debate, this book establishes his genuine commitment to a worldly theology and to a programme of human advancement. It thus raises serious doubts about histories which treat the Enlightenment as an entirely secular enterprise, as well as those which see English thought as being markedly out of step with wider European intellectual developments. niall o’flaherty is a Lecturer in the History of European Political Thought at King’s College London. His research focuses on eighteenth- and nineteenth-century moral, political and religious thought in Britain. He has published articles on William Paley and Thomas Robert Malthus, and is currently writing a book entitled Malthus and the Discovery of Poverty. ideas in context Edited by David Armitage, Richard Bourke, Jennifer Pitts and John Robertson The books in this series will discuss the emergence of intellectual traditions and of related new disciplines. The procedures, aims and vocabularies that were generated will be set in the context of the alternatives available within the contemporary frameworks of ideas and institutions. -
Formation of a Newtonian Culture in New England, 1727--1779 Frances Herman Lord University of New Hampshire, Durham
University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Fall 2000 Piety, politeness, and power: Formation of a Newtonian culture in New England, 1727--1779 Frances Herman Lord University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Lord, Frances Herman, "Piety, politeness, and power: Formation of a Newtonian culture in New England, 1727--1779" (2000). Doctoral Dissertations. 2140. https://scholars.unh.edu/dissertation/2140 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has bean reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. -
Durham E-Theses
Durham E-Theses The Science and Logic of William Paley's Moral Philosophy WANG, CAN How to cite: WANG, CAN (2021) The Science and Logic of William Paley's Moral Philosophy , Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/14004/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk The Science and Logic of William Paley's Moral Philosophy Can Wang ABSTRACT William Paley's The Principles of Moral and Political Philosophy is one of the most influential modern works of theological utilitarianism. His views on moral philosophy, evidentialism and natural theology were required reading in English universities up until the 1850s. It is the purpose of this thesis to argue that Paley believed his moral philosophy to be a science that operated according to logical principles. Chapter 1 outlines the intellectual environment, religious context and secondary literature about Paley’s moral philosophy. -
Émilie Du Châtelets Institutions Physiques Über Die Grundlagen Der Physik
Émilie du Châtelets Institutions physiques Über die Grundlagen der Physik Dissertation zur Erlangung des akademischen Grades eines Doktors der Philosophie (Dr. phil.) im Fach Philosophie an der Fakultät für Kulturwissenschaften der Universität Paderborn vorgelegt von: Andrea Reichenberger Erstgutachter: Prof. Dr. Volker Peckhaus Zweitgutachter: Prof. Dr. Ruth Hagengruber Disputation: 14. Juli 2014 Note: summa cum laude 2 Inhaltsverzeichnis Vorwort 4 1 Einleitung 5 1.1 Thema und Forschungsstand . .5 1.2 Problemstellung und Ziel . 12 1.2.1 Zwischen Newton und Leibniz . 12 1.2.2 Zur Newton-Rezeption im 18. Jahrhundert . 16 1.2.3 Zur Leibniz-Rezeption im 18. Jahrhundert . 17 1.2.4 Leibniz contra Newton? . 21 1.2.5 Du Châtelets Rezeption . 25 2 Du Châtelet: Vita und Œuvre 31 2.1 Biographischer Abriss . 31 2.2 Werke und Editionsgeschichte . 33 2.2.1 Principes mathématiques . 33 2.2.2 Éléments de la philosophie de Newton . 37 2.2.3 Dissertation sur la nature et la propagation du feu . 38 2.2.4 Institutions physiques . 39 2.2.5 Essai sur l’optique . 40 2.2.6 Grammaire raisonnée . 40 2.2.7 De la liberté . 41 2.2.8 Discours sur les miracles & Examens de la Bible . 41 2.2.9 Discours sur le bonheur . 42 2.2.10 La Fable des abeilles de Mandeville . 44 2.2.11 Lettres . 45 3 Du Châtelets Institutions physiques im historischen Kontext 47 3.1 Die Editionsgeschichte . 47 3.2 Der Disput mit Jean-Jaques Dortous de Mairan . 48 3.3 Newton in Frankreich . 50 3.4 Der Vorwurf Johann Samuel Königs . -
Arxiv:0811.0164V8 [Math.HO] 24 Feb 2009 N H S Gat2006393)
A STRICT NON-STANDARD INEQUALITY .999 ...< 1 KARIN USADI KATZ AND MIKHAIL G. KATZ∗ Abstract. Is .999 ... equal to 1? A. Lightstone’s decimal expan- sions yield an infinity of numbers in [0, 1] whose expansion starts with an unbounded number of repeated digits “9”. We present some non-standard thoughts on the ambiguity of the ellipsis, mod- eling the cognitive concept of generic limit of B. Cornu and D. Tall. A choice of a non-standard hyperinteger H specifies an H-infinite extended decimal string of 9s, corresponding to an infinitesimally diminished hyperreal value (11.5). In our model, the student re- sistance to the unital evaluation of .999 ... is directed against an unspoken and unacknowledged application of the standard part function, namely the stripping away of a ghost of an infinitesimal, to echo George Berkeley. So long as the number system has not been specified, the students’ hunch that .999 ... can fall infinites- imally short of 1, can be justified in a mathematically rigorous fashion. Contents 1. The problem of unital evaluation 2 2. A geometric sum 3 3.Arguingby“Itoldyouso” 4 4. Coming clean 4 5. Squaring .999 ...< 1 with reality 5 6. Hyperreals under magnifying glass 7 7. Zooming in on slope of tangent line 8 arXiv:0811.0164v8 [math.HO] 24 Feb 2009 8. Hypercalculator returns .999 ... 8 9. Generic limit and precise meaning of infinity 10 10. Limits, generic limits, and Flatland 11 11. Anon-standardglossary 12 Date: October 22, 2018. 2000 Mathematics Subject Classification. Primary 26E35; Secondary 97A20, 97C30 . Key words and phrases. -
Contributions to the History of Number Theory in the 20Th Century Author
Peter Roquette, Oberwolfach, March 2006 Peter Roquette Contributions to the History of Number Theory in the 20th Century Author: Peter Roquette Ruprecht-Karls-Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail: [email protected] 2010 Mathematics Subject Classification (primary; secondary): 01-02, 03-03, 11-03, 12-03 , 16-03, 20-03; 01A60, 01A70, 01A75, 11E04, 11E88, 11R18 11R37, 11U10 ISBN 978-3-03719-113-2 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2013 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 To my friend Günther Frei who introduced me to and kindled my interest in the history of number theory Preface This volume contains my articles on the history of number theory except those which are already included in my “Collected Papers”. -
The Project Gutenberg Ebook #31061: a History of Mathematics
The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A History of Mathematics Author: Florian Cajori Release Date: January 24, 2010 [EBook #31061] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber's note Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for screen viewing, but may be easily formatted for printing. Please consult the preamble of the LATEX source file for instructions. A HISTORY OF MATHEMATICS A HISTORY OF MATHEMATICS BY FLORIAN CAJORI, Ph.D. Formerly Professor of Applied Mathematics in the Tulane University of Louisiana; now Professor of Physics in Colorado College \I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."|J. W. L. Glaisher New York THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1909 All rights reserved Copyright, 1893, By MACMILLAN AND CO. Set up and electrotyped January, 1894. Reprinted March, 1895; October, 1897; November, 1901; January, 1906; July, 1909. Norwood Pre&: J. S. Cushing & Co.|Berwick & Smith. -
0521418542Pre.Tex Copy
P1: JZZ 0521418542agg.xml CY509-Haakonssen 052141854 2 October 6, 2005 16:19 THE CAMBRIDGE HISTORY OF EIGHTEENTH-CENTURY PHILOSOPHY More than thirty eminent scholars from nine different countries have contributed to The Cambridge History of Eighteenth-Century Philosophy – the most comprehensive and up-to-date history of the subject available in English. In contrast with most histories of philosophy and in keeping with preceding Cambridge volumes in the series, the subject is treated systematically by topic, not by individual thinker, school, or movement, thus enabling a much more historically nuanced picture of the period to be painted. As in previous titles in the series, the volume has extensive biographical and bibliographical research materials. During the eighteenth century,the dominant concept in philosophy was human nature, and so it is around this concept that the present work is centered. This allows the contributors to offer both detailed explorations of the epistemological, metaphysical, and ethical themes that continue to stand at the forefront of philoso- phy and to voice a critical attitude toward the historiography behind this emphasis in philosophical thought. At the same time, due attention is paid to historical con- text, with particular emphasis on the connections among philosophy, science, and theology. This judiciously balanced, systematic, and comprehensive account of the whole of Western philosophy during the period will be an invaluable resource for philoso- phers, intellectual historians, theologians, political -
Slopes, Rates of Change, and Derivatives
2.2 Slopes, Rates of Change, and Derivatives APPLICATION PREVIEW The Confused Creation of Calculus* Calculus evolved from consideration of four major problems that were current in 17th century Europe: how to define instantaneous velocity, how to define tangent lines to curves, how to find maxi- mum and minimum values of functions, and how to calculate areas and volumes, all of which will be discussed in this and the follow- ing chapters. While calculus is now presented as a logical mathe- matical system, it began instead as a collection of self-contradictory ideas and poorly understood techniques, accompanied by much controversy and criticism. It was developed by two people of dia- metrically opposed temperaments, Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716), each while in his twenties (Newton at age 22 and Leibniz at about 29). Newton attended mediocre schools in England, entered Cam- bridge University (with a deficiency in Euclidean geometry), im- mersed himself in solitary studies, and graduated with no particular distinction. He then returned to his family’s small farm where, during a 2-year period, he singlehandedly developed calcu- lus, in addition to the science of optics and the theory of universal gravitation, all of which he kept to himself. He then returned to Cambridge for a master’s degree and secured an appointment as a professor, after which he became even more solitary and intro- verted. A contemporary described him as never taking “any recre- ation or pastime either in riding out to take the air, walking, bowling, or any other exercise whatever, thinking all hours lost that was not spent in his studies.” He was often seen “with shoes down at heels, stockings untied,..., and his head scarcely combed.” He was not a popular teacher, sometimes lecturing to empty class- rooms, and his mathematical writings were very difficult to under- stand (he told one friend that he wrote this way intentionally “to avoid being bated by little smatterers in mathematics”). -
The Early Criticisms of the Calculus of Newton and Leibniz
Ghosts of Departed Errors: The Early Criticisms of the Calculus of Newton and Leibniz Eugene Boman Mathematics Seminar, February 15, 2017 Eugene Boman is Associate Professor of Mathematics at Penn State Harrisburg. alculus is often taught as if it is a pristine thing, emerging Athena-like, complete and whole, from C the head of Zeus. It is not. In the early days the logical foundations of Calculus were highly suspect. It seemed to work well and effortlessly, but why it did so was mysterious to say the least and until this foundational question was answered the entire enterprise was suspect, despite its remarkable successes. It took nearly two hundred years to develop a coherent foundational theory and, when completed, the Calculus that emerged was fundamentally different from the Calculus of the founders. To illustrate this difference I would like to compare two proofs of an elementary result from Calculus: That the derivative of sin(푥) is cos(푥). The first is a modern proof both in content and style. The second is in the style of the Seventeenth and Eighteenth centuries. A MODERN PROOF THAT THE DERIVATIVE OF 푠푛(푥) IS 푐표푠(푥) 푑(sin(푥)) A modern proof that = cos(푥) is not a simple thing. It depends on the theory of limits, which we 푑푥 will simply assume, and in particular on the following lemma, which we will prove. sin(푥) Lemma: lim = 1. 푥→0 푥 Proof of Lemma: Since in Figure 1 the radius of the circle is one we see that radian measure of the angle and the length of the subtended arc are equal.