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Objectiones Qvintae Y Disqvisitio Metaphysica De Pierre Gassendi
OBJECTIONES QVINTAE Y DISQVISITIO METAPHYSICA DE PIERRE GASSENDI . TESIS DOCTORAL. REALIZADA POR JESÚS DEL VALLE CORTÉS. DIRIGIDA POR EL CATEDRÁTICO DOCTOR DON MARCELINO RODRÍGUEZ DONÍS. A la memoria de Fernando del Valle Vázquez, de su hijo agradecido. Y a Silvia Aguirrezábal Guerrero. LIBRO PRIMERO. EXHORDIVM. A MODO DE INTRODUCCIÓN. ería necesario un largo estudio para establecer con detalle las condiciones en las cuales fueron compuestas, en primer lugar, las S Quintas Objeciones de Pierre Gassendi, Prepósito de la Iglesia de Digne y agudísimo Filósofo, designadas de esta manera en un Índice redactado por Mersenne, y después las Instancias 1, fechadas el 15 de marzo de 1642, pero que seguramente comenzaron a difundirse durante el invierno anterior . Son palabras de Bernard Rochot 2 y con ellas comenzaba la introducción de su traducción francesa de la Petri Gassendi Disquisitio metaphysica . Esta tesis no es sino el cumplimiento de este anhelo de Rochot, que ha ya mucho convertimos en propio, si bien hemos ampliado los límites de la empresa, toda vez que este trabajo no se contenta con plasmar los acontecimientos que dieron lugar a la aparición de ambos escritos, sino que incluye el análisis pormenorizado de ellos, e incluso el de las escuetas réplicas de Descartes a las Instancias . Es la primera vez que se recorre el ciclo completo de los argumentos de la querella, hasta su cierre con la Carta a Clerselier . Eran años de carrera cuando estudiábamos, bajo la tutela del Doctor D. Marcelino Rodríguez Donís, las Meditationes de prima philosophia , con sus siete Objeciones y Respuestas y, de entre ellas, las agrias Objectiones quintae sobresalían, a nuestro juicio, muy por encima de las demás, a pesar de que uno de los siete objetores fuese el ínclito Thomas Hobbes 3. -
Pascal's and Huygens's Game-Theoretic Foundations For
Pascal's and Huygens's game-theoretic foundations for probability Glenn Shafer Rutgers University [email protected] The Game-Theoretic Probability and Finance Project Working Paper #53 First posted December 28, 2018. Last revised December 28, 2018. Project web site: http://www.probabilityandfinance.com Abstract Blaise Pascal and Christiaan Huygens developed game-theoretic foundations for the calculus of chances | foundations that replaced appeals to frequency with arguments based on a game's temporal structure. Pascal argued for equal division when chances are equal. Huygens extended the argument by considering strategies for a player who can make any bet with any opponent so long as its terms are equal. These game-theoretic foundations were disregarded by Pascal's and Huy- gens's 18th century successors, who found the already established foundation of equally frequent cases more conceptually relevant and mathematically fruit- ful. But the game-theoretic foundations can be developed in ways that merit attention in the 21st century. 1 The calculus of chances before Pascal and Fermat 1 1.1 Counting chances . .2 1.2 Fixing stakes and bets . .3 2 The division problem 5 2.1 Pascal's solution of the division problem . .6 2.2 Published antecedents . .7 2.3 Unpublished antecedents . .8 3 Pascal's game-theoretic foundation 9 3.1 Enter the Chevalier de M´er´e. .9 3.2 Carrying its demonstration in itself . 11 4 Huygens's game-theoretic foundation 12 4.1 What did Huygens learn in Paris? . 13 4.2 Only games of pure chance? . 15 4.3 Using algebra . 16 5 Back to frequency 18 5.1 Montmort . -
Nominalism and Constructivism in Seventeenth-Century Mathematical Philosophy
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematica 32 (2005) 33–59 www.elsevier.com/locate/hm Nominalism and constructivism in seventeenth-century mathematical philosophy David Sepkoski Department of History, Oberlin College, Oberlin, OH 44074, USA Available online 27 November 2003 Abstract This paper argues that the philosophical tradition of nominalism, as evident in the works of Pierre Gassendi, Thomas Hobbes, Isaac Barrow, and Isaac Newton, played an important role in the history of mathematics during the 17th century. I will argue that nominalist philosophy of mathematics offers new clarification of the development of a “constructivist” tradition in mathematical philosophy. This nominalist and constructivist tradition offered a way for contemporary mathematicians to discuss mathematical objects and magnitudes that did not assume these entities were real in a Platonic sense, and helped lay the groundwork for formalist and instrumentalist approaches in modern mathematics. 2003 Elsevier Inc. All rights reserved. Résumé Cet article soutient que la tradition philosophique du nominalisme, évidente dans les travaux de Pierre Gassendi, Thomas Hobbes, Isaac Barrow et Isaac Newton, a joué un rôle important dans l’histoire des mathématiques pendant le dix-septième siècle. L’argument princicipal est que la philosophie nominaliste des mathématiques est à la base du développement d’une tradition « constructiviste » en philosophie mathématique. Cette tradition nominaliste et constructiviste a permis aux mathématiciens contemporains de pouvoir discuter d’objets et quantités mathématiques sans présupposer leur réalité au sens Platonique du terme, et a contribué au developpement desétudes formalistes et instrumentalistes des mathématiques modernes. -
Calculation and Controversy
calculation and controversy The young Newton owed his greatest intellectual debt to the French mathematician and natural philosopher, René Descartes. He was influ- enced by both English and Continental commentators on Descartes’ work. Problems derived from the writings of the Oxford mathematician, John Wallis, also featured strongly in Newton’s development as a mathe- matician capable of handling infinite series and the complexities of calcula- tions involving curved lines. The ‘Waste Book’ that Newton used for much of his mathematical working in the 1660s demonstrates how quickly his talents surpassed those of most of his contemporaries. Nevertheless, the evolution of Newton’s thought was only possible through consideration of what his immediate predecessors had already achieved. Once Newton had become a public figure, however, he became increasingly concerned to ensure proper recognition for his own ideas. In the quarrels that resulted with mathematicians like Gottfried Wilhelm Leibniz (1646–1716) or Johann Bernoulli (1667–1748), Newton supervised his disciples in the reconstruction of the historical record of his discoveries. One of those followers was William Jones, tutor to the future Earl of Macclesfield, who acquired or copied many letters and papers relating to Newton’s early career. These formed the heart of the Macclesfield Collection, which has recently been purchased by Cambridge University Library. 31 rené descartes, Geometria ed. and trans. frans van schooten 2 parts (Amsterdam, 1659–61) 4o: -2 4, a-3t4, g-3g4; π2, -2 4, a-f4 Trinity* * College, Cambridge,* shelfmark* nq 16/203 Newton acquired this book ‘a little before Christmas’ 1664, having read an earlier edition of Descartes’ Geometry by van Schooten earlier in the year. -
Isaac Barrow
^be ©pen Court A MONTHLY MAGAZINE S>et»otc^ to tbe Science of l^eUdion, tbe l^elfdfon ot Science, anb tbc BXiCnsion ot tbe Itelidioud parliament Ibea Founded by Edwabo C. Hegeuol VOL. XXX. (No. 2) FEBRUARY, 1916 NO. 717 CONTENTS: MM Frontispiece. Isaac Barrow. Isaac Barrow: The Drawer of Tangents. J. M. Child 65 ''Ati Orgy of Cant.'' Paul Carus 70 A Chippewa Tomahawk. An Indian Heirloom with a History (Illus- trated). W. Thornton Parker 80 War Topics. —In Reply to My Critics. Paul Carus 87 Portraits of Isaac Barrow 126 American Bahaism and Persia 126 A Correction 126 A Crucifix After Battle (With Illustration) ^ 128 ^be i^pen Court publiebing CompaniS i CHICAGO Per copy, 10 cents (sixpence). Yearly, $1.00 (in the U.P.U., 5i. 6d). Entered w Secoad-QaM Matter March j6, 1897, at the Post Office at Chicago. III., under Act of Marck 3, \\j% Copyright by The Open Court Publishing Company, 1916 tTbe ©pen Court A MONTHLY MAGAZINE S>et»otc^ to tbe Science ot l^eUdion, tbe l^elfdfon ot Science* anb tbe BXiCndion ot tbe 1{eliaiou0 parliament Ibea Founded by Edwabo C Hegeueb. VOL. XXX. (No. 2) FEBRUARY, 1916 NO. 717 CONTENTS: Frontispiece. Isaac Barrow. Isaa£ Barrow: The Drawer of Tangents. J. M. Child 65 ''A7i Orgy of Cant.'" Paul Carus 70 A Chippewa Totnahawk. An Indian Heirloom with a History (Illus- trated). W. Thornton Parker 80 War Topics. —In Reply to My Critics. Paul Carus 87 Portraits of Isaac Barrow • 126 American Bahaism and Persia 126 A Correction 126 A Crucifix After Battle (With Illustration) 128 ^be ®pen Court publisbino CompaniS I CHICAGO Per copy, 10 cents (sixpence). -
A Tale of the Cycloid in Four Acts
A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries. -
Cycloid Article(Final04)
The Helen of Geometry John Martin The seventeenth century is one of the most exciting periods in the history of mathematics. The first half of the century saw the invention of analytic geometry and the discovery of new methods for finding tangents, areas, and volumes. These results set the stage for the development of the calculus during the second half. One curve played a central role in this drama and was used by nearly every mathematician of the time as an example for demonstrating new techniques. That curve was the cycloid. The cycloid is the curve traced out by a point on the circumference of a circle, called the generating circle, which rolls along a straight line without slipping (see Figure 1). It has been called it the “Helen of Geometry,” not just because of its many beautiful properties but also for the conflicts it engendered. Figure 1. The cycloid. This article recounts the history of the cycloid, showing how it inspired a generation of great mathematicians to create some outstanding mathematics. This is also a story of how pride, pettiness, and jealousy led to bitter disagreements among those men. Early history Since the wheel was invented around 3000 B.C., it seems that the cycloid might have been discovered at an early date. There is no evidence that this was the case. The earliest mention of a curve generated by a -1-(Final) point on a moving circle appears in 1501, when Charles de Bouvelles [7] used such a curve in his mechanical solution to the problem of squaring the circle. -
Marin Mersenne As Mathematical Intelligencer
Hist. Sci., li (2013) SMall SkIllS, BIG NETwORkS: MaRIN MERSENNE aS MaTHEMaTICal INTEllIGENCER Justin Grosslight Independent Scholar Writing in August 1634, the French polymath Nicolas-Claude Fabri de Peiresc reflected upon Marin Mersenne’s endeavours. Mersenne, Peiresc asserted, had forced himself into “frontiers that are a little more in fashion of the times than these prolix treaties of the schools that so few men handle outside of the colleges”.1 Peiresc was quite prescient to realize the novel claims that Mersenne was promoting. From the middle of the 1620s until his death in 1648, Mersenne encouraged discussion on a number of new mathematical concepts, ideas that lacked a secure home within the Aristotelian university curriculum.2 By mathematics, I am referring to mixed mathematics, the application of arithmetic and geometry often to physical proc- esses, and related topics in natural philosophy: examples include Galilean mechan- ics, the question of whether a void exists in nature, and analysis of conic sections and their spatial counterparts.3 For Peiresc and others, Mersenne was an exemplar of new, heterodox ideas: though he was a member of the religious Minim order, he was respected widely as a mathematician, he lived a gregarious life in cosmopoli- tan Paris, and he was an intimate friend of the famed French mathematician rené Descartes as well as a correspondent with the Italian Galileo Galilei and the Dutch Christiaan huygens.4 But while Mersenne is still remembered as a mathematician and correspondent of famous mathematicians, Peiresc’s description of Mersenne as trend-conscious or even a trendsetter has been effectively forgotten. -
The History of Newton' S Apple Tree
Contemporary Physics, 1998, volume 39, number 5, pages 377 ± 391 The history of Newton’s apple tree (Being an investigation of the story of Newton and the apple and the history of Newton’s apple tree and its propagation from the time of Newton to the present day) R. G. KEESING This article contains a brief introduction to Newton’s early life to put into context the subsequent events in this narrative. It is followed by a summary of accounts of Newton’s famous story of his discovery of universal gravitation which was occasioned by the fall of an apple in the year 1665/6. Evidence of Newton’s friendship with a prosperous Yorkshire family who planted an apple tree arbour in the early years of the eighteenth century to celebrate his discovery is presented. A considerable amount of new and unpublished pictorial and documentary material is included relating to a particular apple tree which grew in the garden of Woolsthorpe Manor (Newton’s birthplace) and which blew down in a storm before the year 1816. Evidence is then presented which describes how this tree was chosen to be the focus of Newton’s account. Details of the propagation of the apple tree growing in the garden at Woolsthorpe in the early part of the last century are then discussed, and the results of a dendrochronological study of two of these trees is presented. It is then pointed out that there is considerable evidence to show that the apple tree presently growing at Woolsthorpe and known as `Newton’s apple tree’ is in fact the same specimen which was identi®ed in the middle of the eighteenth century and which may now be 350 years old. -
Isaac Barrow on God, Geometry, and the Nature of Space Antoni Malet
Isaac Barrow on God, Geometry, and the Nature of Space Antoni Malet Universitat Pompeu Fabra, Barcelona INTRODUCTION Historians used to portray Isaac Barrow (1630-1677) as an obscure figure in Newton's background. He was a Janus-faced figure mainly of interest —one of his biographers once famously said— because he was “a link between the old and the new philosophy,” the old being his and the new Newton's.1 This theme became commonplace in dealing with Barrow. For decades, and with the exception of Barrow’s role in the philosophical articulation of the idea of absolute time, Barrow’s contributions were found to be hopelessly out of tune with contemporary modern intellectual tendencies. Consequently, attempts to trace any kind of substantial influence from Barrow on Newton always found an unsympathetic reception.2 The most important attempt to 1 P.H. Osmond, Isaac Barrow, His Life and Times (London: Soc. for the Promotion of Christian Knowledge, 1944), p. 1. 2 Barrow’s articulation of the notion of absolute time is particularly well analyzed in E.A. Burtt’s old but still useful The Metaphysical Foundations of Modern Physical Science (Atlantic Highlands, N.J.: Humanities Press, 1989, first ed. 1924), p. 150-61. On Barrow’s notion of space, see E. Grant, Much ado about nothing. Theories of space and vacuum from the Middle Ages to the Scientific Revolution (Cambridge: Cambridge University Press, 1981), p. 236-238. See also E.W. Strong, 'Barrow and Newton', Journal of the History of Philosophy (1970), 8, 155-72; A. R. Hall, Henry More. -
The Geometrical Lectures of Isaac Barrow, Translated, with Notes And
Open Court Classics of Science and Philosophy, iOMETRICAL LECTURES OP ISAAC BARROW J. M, CHILD CO GEOMETRICAL LECTURES ISAAC BARROW Court Series of Classics of Science and Thilosophy, 3\(o. 3 THE GEOMETRICAL LECTURES OF ISAAC BARROW . HI TRANSLATED, WITH NOTES AND PROOFS, AND A DISCUSSION ON THE ADVANCE MADE THEREIN ON THE WORK OF HIS PREDECESSORS IN THE INFINITESIMAL CALCULUS BY B.A. (CANTAB.), B.Sc. (LOND.) 546664 CHICAGO AND LONDON. THE OPEN COURT PUBLISHING COMPANY 1916 Copyright in Great Britain under the Ad of 191 1 33 LECTI ONES Geometric^; In quibus (praffertim) GEHERALJA Cttr varum L iacaruw STMPTOUATA D E. C L A II 7^ A T^ T !(, Audore IsAACoBARRow Collegii S S. Trhiitatn in Acad. Cant ah. SociO, & Socictatif lie- ti< Sodale. Oi H< TO. <fvftt Aoj/r/ito; iruiv]<t nxSHfufla, at tr& wVi^C) o%*( OITTI a* i r>riu- GfetJfft , rttrv -mufAttli <iow it, yupi mt , *'Ao &.'> n$uw, O 6)f TO fi>1/Si' H tr;t ^urt^i oTi a'uT&i/yi'jrf **ir. Plato de Repub. VII. L ON D IN I, Cttlielmi Godkd & venales Typis , proftant apud OR.ivi.innm Jh-tnitf>t Tt-wmsre, & Pullejn Juniorcm. UW. D C L X X. " Note the absence of the usual words Habitae Cantabrigioe," which on the title-pages of his other works indicate that the latter were delivered as Lucasian Lectures. J. M. C. PREFACE ISAAC BARROW was Ike first inventor of the Infinitesimal Calculus ; Newton got the main idea of it from Barrow by personal communication ; and Leibniz also was in some measure indebted to Barrow's work, obtaining confirmation of his oivn original ideas, and suggestions for their further development, from the copy of JJarrow's book (hat he purchased in 1673. -
Introduction: 'Mind Almost Divine'
Cambridge University Press 978-0-521-66310-6 - From Newton to Hawking: A History of Cambridge University’s Lucasian Professors of Mathematics Edited by Kevin C. Knox and Richard Noakes Excerpt More information Introduction: ‘Mind almost divine’ Kevin C. Knox1 and Richard Noakes2 1Institute Archives, California Institute of Technology, Pasadena, CA, USA 2Department of History of Philosophy of Science, Cambridge, UK Whosoever to the utmost of his finite capacity would see truth as it has actually existed in the mind of God from all eternity, he must study Mathematics more than Metaphysics. Nicholas Saunderson, The Elements of Algebra1 As we enter the twenty-first century it might be possible to imag- ine the world without Cambridge University’s Lucasian professors of mathematics. It is, however, impossible to imagine our world with- out their profound discoveries and inventions. Unquestionably, the work of the Lucasian professors has ‘revolutionized’ the way we think about and engage with the world: Newton has given us universal grav- itation and the calculus, Charles Babbage is touted as the ‘father of the computer’, Paul Dirac is revered for knitting together quantum mechanics and special relativity and Stephen Hawking has provided us with startling new theories about the origin and fate of the uni- verse. Indeed, Newton, Babbage, Dirac and Hawking have made the Lucasian professorship the most famous academic chair in the world. While these Lucasian professors have been deified and placed in the pantheon of scientific immortals, eponymity testifies to the eminence of the chair’s other occupants. Accompanying ‘Newton’s laws of motion’, ‘Babbage’s principle’ of political economy, the ‘Dirac delta function’ and ‘Hawking radiation’, we have, among other things, From Newton to Hawking: A History of Cambridge University’s Lucasian Professors of Math- ematics, ed.