DOCSLIB.ORG

Nicole Oresme and .Hasdai Crescas on Many Worlds

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nicole Oresme and .Hasdai Crescas on Many Worlds NICOLE ORESME AND HASDAI. CRESCAS ON MANY WORLDS Warren Zev Harvey The two most creative thinkers in the new physics of the late fourteenth century were arguably Nicole Oresme (ca. –), grand maître of the College of Navarre at the University of Paris and later Bishop of Lisieux, and Hasdai. Crescas (ca. –/), Rabbi of the Jews of the Crown of Aragon and advisor to its kings. The direct or indirect influence of Oresme on Crescas was long ago noted by Pierre Duhem, and further explored by Shlomo Pines and others. This connection is particularly striking with regard to their notions of infinite space and eternal time, and their critique of Aristotle’s theory of natural places.1 Given Oresme’s connection to the Kingdom of Navarre, adjacent to the Crown of Aragon, it is likely that his works were known and available in Crescas’ vicinity. It is reasonable to speculate that Crescas knew scholars who had studied with Oresme in Paris, and it is not inconceivable that he met Oresme personally.2 In my following remarks, I shall compare the views of Oresme and Crescas on the problem of many worlds. Both philosophers discuss the problem primarily in response to Aristotle’s thesis in De caelo,I,– , a–b, that there is one and only one world. Although Oresme eventually accepts Aristotle’s thesis and Crescas explicitly rejects it, the approachesofthetwophilosopherstotheproblemareinmanyrespects similar. 1 P. D u h e m , Le Système du monde (Paris, –): :–, :–; S. Pines, “Scholasticism after Thomas Aquinas and the Teachings of Hasdai Crescas and his Predecessors,” Proceedings of the Israel Academy of Sciences and Humanities , no. (): –; reprinted in idem, Collected Works (Jerusalem, –): :–; and see my Physics and Metaphysics in Hasdai Crescas (Amsterdam, ): –, and also my Hasdai. Crescas (Heb.) (Jerusalem, ): –. 2 On the College of Navarre, see N. Gorochov, Le Collège de Navarre de sa fondation () au début de XVe siècle () (Paris, ). Cf. A. Albertos, R. Garcia-Alonso and J.M. Ortiz, “París : La fundación del Colegio de Navarra,” Príncipe de Viana (): –. warren zev harvey The two main discussions of Oresme’s on the problem of many worlds are found in his Quaestiones on Aristotle’s De caelo, I, qq. –,3 and in his Le livre du ciel et du monde, I, .4 The Quaestiones probably date from Oresme’s early teaching days at the College of Navarre (–), while the Du ciel et du monde was written much later, in , when Oresme was dean of the Cathedral of Rouen. The Du ciel et du monde is a French translation with commentary of Aristotle’s De caelo,andwas written at the behest of King Charles V of France, to whom Oresme was a close advisor. Crescas’ discussions of many worlds are found in his Light of the Lord, I, , ; I, , ; I, , ; I, , ; IIIa, , ; IV, –. The Light of the Lord was written over many years, and completed in .5 The discussion in I, , , although appearing early in the book, is manifestly a late interpolation by Crescas, appended to his arguments against the Aristotelian proposition that an infinite magnitude is impossible.6 ThediscussionsinI,, and , and I, , , are interconnected, and concern the possibility that different Gods rule different worlds.7 The passage in IIIa, , , concerns the doctrine of the eternal creation of successive worlds.8 IV,, treats the problem of eternity apartepostand successive worlds;9 and IV, , treats that of many worlds existing simultaneously.10 3 C. Kren, “The Questiones super De Celo of Nicole Oresme” (Ph.D. thesis, University of Wisconsin, ), Xerox Microfilms, Ann Arbor, no. –, pp. –. The thesis contains the Latin text and an annotated English translation. 4 Le Livre du ciel et du monde. Edited by A.D. Menut and A.J. Denomy, translated with an introduction by Menut (Madison, ): –. 5 Page references to the Light of the Lord will be to the useful vocalized edition by S. Fisher, Or ha-Semˇ (Jerusalem, ). The passages regarding many worlds are dis- cussed and translated in my Physics and Metaphysics, pp. –, –. In preparing the present study, I made use of two unpublished lectures on Crescas’ theory of many worlds: A. Ackerman, “Hasdai Crescas’ Discussion of the Possibility of Multiple Worlds,” Con- ference on Hasdai Crescas: Years after his Death, Zalman Shazar Center, Jerusalem, January ; and S. Feldman, “Plural Universes: A Debate in Late Medieval Jewish Phi- losophy,” Meeting of the Academy for Jewish Philosophy, American Philosophical Asso- ciation, Eastern Division, Washington, DC, December . 6 Or ha-Semˇ (ed. Fisher), p. See H.A. Wolfson, Crescas’ Critique of Aristotle (Cam- bridge, MA, ): .–., and p. n. ; and my Physics and Metaphysics, pp. –. 7 Or ha-Semˇ (ed. Fisher), pp. , , –. See my Physics and Metaphysics, pp. – . 8 Or ha-Semˇ (ed. Fisher), pp. –. See my Physics and Metaphysics, pp. –, –. 9 Or ha-Semˇ (ed. Fisher), pp. –. See my Physics and Metaphysics, pp. –. 10 Or ha-Semˇ (ed. Fisher), pp. –. See Wolfson, Crescas’ Critique, pp. – n. ; and my Physics and Metaphysics, pp. –..
Load more

Recommended publications
  • E D I T O R I a L
    E D I T O R I A L “RETRO-PROGRESSING” As one browses through the history section of a library, one of the volumes that is likely to catch one’s attention is The Discoverers by Daniel Boorstin.1 It is an impressive, 700-page volume. Published in 1983, it chronicles in a semi-popular style selected aspects of man’s discoveries. Two chapters entitled “The Prison of Christian Dogma” and “A Flat Earth Returns” deal with the outlandish con- cept of an earth that is flat instead of spherical. Boorstin has impressive academic credentials from Harvard and Yale, and has held prestigious positions such as the Librarian of Congress, Director of the National Museum of History and Tech- nology, and Senior Historian of the Smithsonian Institution. In The Discoverers he reflects the popular view that the ancient Greeks, including Aristotle and Plato, believed Earth to be a sphere; however, after the rise of Christianity a period of “scholarly amnesia” set in which lasted from around 300 to 1300 A.D. During this time, according to Boorstin, “Christian faith and dogma replaced the useful [spherical] image of the world that had been so slowly, so painfully, and so scrupulously drawn by ancient geographers.” The “spherical” view was replaced by the concept of a flat earth, which Boorstin characterizes as “pious caricatures.”2 Boorstin bolsters his case by mentioning a couple of minor writers — Lactantius and Cosmas — who believed in a flat earth and lived during this “dark age.” He also implicates the powerful authority of St. Augustine of Hippo (354-430), who “heartily agreed” that antipodes “could not exist.”3 Antipodes represented lands on the opposite side of a spherical earth where trees and men, if present, would be upside down, with their feet above their heads; hence, the “antipodes” (opposite-feet) designation.
    [Show full text]
  • 7. Mathematical Revival in Western Europe
    7. Mathematical revival in Western Europe (Burton , 6.2 – 6.4, 7.1 ) Mathematical studies and discoveries during the early Dark Ages in Europe were extremely limited . One illustration of this fact is the chronology of the highly esteemed English scholar Bede (The Venerable Bede , 672 /673 – 735) , which set the convention that the year 1 B.C.E. immediately precedes the year 1 A.D. ; this remains unchanged even though 0 is now generally accepted to be part of the number system . However , there were contributors to the subject during the period from the Latin commentator Boëthius (Anicius Manlius Severinus Boëthius , c. 475 – 524) shortly after the fall of the Western Roman Empire and continuing through to the end of the 12 th century ; Boëthius has been described as the last of the Romans and the first of the scholastic philosophers (predating the latter by many centuries) , and his mathematical texts of were still widely used 600 years after they were written . Several other names from this period are mentioned in Sections 5.4 and 6.1 of Burton , and the latter ’s exercises also mention Alcuin of York (735 – 804) and Gerbert d ’Aurillac (940 – 1003) , who later became Pope Sylvester I I (997 – 1003) . During the second half of the 11 th century some important political developments helped raise European ’s consciousness of ancient Greek mathematical work and the more recent advances by Indian and Arabic mathematicians. Many of these involved Christian conquests of territory that had been in Muslim hands for long periods of time . Specific examples of particular importance for mathematics were the Norman conquest of Sicily in 1072 , the Spanish reconquista during which extensive and important territories in the Iberian Peninsula changed from Muslim to Christian hands , and the start of the Crusades in 1095 .
    [Show full text]
  • Doing Hard Time: Is God the Prisoner of the Oldest Dimension?
    Doing Hard Time: Is God the Prisoner of the Oldest Dimension? R.T. Mullins University of Notre Dame Abstract: In this paper I shall consider an objection to divine temporality called “The Prisoner of Time” objection. I shall begin by distinguishing divine timelessness from divine temporality in order to clear up common misunderstandings and caricatures of divine temporality. From there I shall examine the prisoner of time objection and explain why the prisoner of time objection fails to be a problem for the Christian divine temporalist. During the 20th Century, various Christian philosophers and theologians have rejected divine timelessness. This is not an entirely new phenomena; one can find philosophers and theologians from the Reformation on rejecting divine timelessness. Thinkers like Pierre Gassendi, Isaac Newton, Samuel Clarke, and John Tillotson all rejected the doctrine of divine timelessness long ago for philosophical, theological, scientific, and biblical reasons.1 For these thinkers, time is a necessary concomitant of God’s eternal nature. Despite the efforts of these thinkers, the doctrine of divine timelessness continued to play a central role in Christian theology until the 20th Century. During the 20th Century, and the beginning of this century, one can find a more wide spread rejection of divine timelessness amongst Christian philosophers and systematic theologians. Some of these rejections have been quite rigorous, while others have been misguided. The more recent widespread rejection of the atemporal God has led many into the loving arms of the temporal God. However, divine temporality is not without its critics. Divine atemporality refuses to lay down and die.
    [Show full text]
  • Semiotic Reflections on Medieval and Contemporary Graphic Representations of Motion
    Working paper, presented at the History and Pedagogy of Mathematics Conference, 14-18 July 2008, Mexico City. SEMIOTIC REFLECTIONS ON MEDIEVAL AND CONTEMPORARY GRAPHIC REPRESENTATIONS OF MOTION Luis Radford Université Laurentienne Ontario, Canada Introduction In this paper I will present an ongoing semiotic investigation of motion and some 14th century attempts at representing it in geometrical terms. The semiotic approach that I will follow focuses on the manner in which time, space and velocity were signified in certain historical periods. I take as my starting point the idea that the conceptualisation of motion and its key conceptual elements—speed, time and velocity— can only be understood within the scope of the epistemological configurations that make motion thinkable in a certain way. In other words, any attempt at investigating conceptualizations of time, speed, and velocity needs to take into account the cultural mental structures that underpin them. These cultural mental structures, or épistèmes, to use Michel Foucault’s term, are not mere spiritual or intellectual forms evolving on their own. They arise, acquire shape and evolve under the evolution of social practices, as well as their material conditions, forms of production and social relations. Thus, in the perspective that I am advocating here, to ask questions about time, speed and velocity is also to ask questions about the cultural and social conditions that made these theoretical constructs possible. Time, Speed and Velocity in the Middle Ages It is very well known that the investigation of motion became a prominent field of research only in the early years of modernity. However, that does not mean that motion was not an object of reflection in previous historical periods.
    [Show full text]
  • Oelze (Humboldt University of Berlin)
    Summary of Talk Given at the Conference “Immateriality, Thinking, and the Self in the Long Middle Ages” University of Cambridge, Philosophy Faculty, 14 July, 2015 Material Souls, Material Thoughts? Some Medieval Views on Rational Operations in Non-Rational Animals Anselm Oelze (Humboldt University of Berlin) Introductory Remarks Medieval theories of animal cognition by and large start from the assumption that only human animals are endowed with intellect (intellectus) and reason (ratio). That is, ac- cording to the standard medieval view, dogs, cats, donkeys, and other nonhuman anim- als have souls, yet their souls are not immaterial intellectual souls but material sensory souls (including usually five external senses as well as a number of so-called internal senses, e.g., common sense, imagination, estimation, and memory). Consequently, they are incapable of engaging in rational operations such as concept formation, judging, reasoning, and thinking, for only the intellect has access to the realm of universals (or concepts) which are the building blocks of judgments, syllogisms, and thoughts. Nevertheless, on various occasions medieval philosophers do note that in many cases nonhuman animals seem to judge (iudicare), to reason (ratiocinari, sillogizari), or to think (cogitare), too, insofar as their behaviour seems to be brought about by rational processes. In the following, an overview will be given of what six (later) medieval au- thors, namely Thomas Aquinas, Gregory of Rimini, Roger Bacon, Albertus Magnus, John Buridan, and Nicole Oresme, say on how there can be rational processes in non-ra- tional animals, especially reasoning and thinking (note that ʻreasoningʼ and ʻthinkingʼ are not necessarily synonymous here; rather, reasoning is taken to be some type of thinking; for more on that see Section III).
    [Show full text]
  • The Earliest Translations of Aristotle's Politics and The
    ECKART SCHÜTRUMPF THE EARLIEST TRANSLATIONS OF ARISTOTle’S POLITICS AND THE CREATION OF POLITICAL TERMINOLOGY 8 MORPHOMATA LECTURES COLOGNE MORPHOMATA LECTURES COLOGNE 8 HERAUSGEGEBEN VON GÜNTER BLAMBERGER UND DIETRICH BOSCHUNG ECKART SCHÜTRUMPF THE EARLIEST TRANSLATIONS OF ARISTOTLe’s POLITICS AND THE CREATION OF POLITICAL TERMINOLOGY WILHELM FINK unter dem Förderkennzeichen 01UK0905. Die Verantwortung für den Inhalt der Veröffentlichung liegt bei den Autoren. Bibliografische Informationen der Deutschen Nationalbibliothek: Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen National­ biblio grafie; detaillierte Daten sind im Internet über www.dnb.d­nb.de abrufbar. Alle Rechte, auch die des auszugweisen Nachdrucks, der fotomechanischen Wieder gabe und der Übersetzung vorbehalten. Dies betrifft auch die Vervielfälti­ gung und Übertragung einzelner Textabschnitte, Zeichnungen oder Bilder durch alle Verfahren wie Speicherung und Übertragung auf Papier, Transparente, Filme, Bänder, Platten und andere Medien, soweit es nicht § 53 und 54 UrhG ausdrücklich gestatten. © 2014 Wilhelm Fink, Paderborn Wilhelm Fink GmbH & Co. Verlags­KG, Jühenplatz 1, D­33098 Paderborn Internet: www.fink.de Lektorat: Sidonie Kellerer, Thierry Greub Gestaltung und Satz: Kathrin Roussel, Sichtvermerk Printed in Germany Herstellung: Ferdinand Schöningh GmbH & Co. KG, Paderborn ISBN 978-3-7705-5685-4 CONTENT 1. The earliest Latin translations of Aristotle— William of Moerbeke 9 2. Nicole Oresme 25 3. Leonardo Bruni’s principles of translation 28 4. Bruni’s translation of Aristotle’s Politics 33 5. The political terminology in Bruni’s translation— a new Humanist concept of res publica? 39 6. The controversy over Bruni’s translation— contemporary and modern 65 Appendix 77 Bibliography 78 This study goes back ultimately to a response I gave on two pa­ pers presented on “Translating Aristotle’s Politics in Medieval and Renaissance Europe” at the “International Conference on Translation.
    [Show full text]
  • REBIRTH, REFORM and RESILIENCE Universities in Transition 1300-1700
    REBIRTH, REFORM AND RESILIENCE Universities in Transition 1300-1700 Edited by James M. Kittelson and Pamela J. Transue $25.00 REBIRTH, REFORM, AND RESILIENCE Universities in Transition, 1300-1700 Edited by James M. Kittelson and Pamela]. Transue In his Introduction to this collection of original essays, Professor Kittelson notes that the university is one of the few institutions that medieval Latin Christendom contributed directly to modern Western civilization. An export wherever else it is found, it is unique to Western culture. All cultures, to be sure, have had their intellec­ tuals—those men and women whose task it has been to learn, to know, and to teach. But only in Latin Christendom were scholars—the company of masters and students—found gathered together into the universitas whose entire purpose was to develop and disseminate knowledge in a continu­ ous and systematic fashion with little regard for the consequences of their activities. The studies in this volume treat the history of the universities from the late Middle Ages through the Reformation; that is, from the time of their secure founding, through the period in which they were posed the challenges of humanism and con­ fessionalism, but before the explosion of knowl­ edge that marked the emergence of modern science and the advent of the Enlightenment. The essays and their authors are: "University and Society on the Threshold of Modern Times: The German Connection," by Heiko A. Ober­ man; "The Importance of the Reformation for the Universities: Culture and Confessions in the Criti­ cal Years," by Lewis W. Spitz; "Science and the Medieval University," by Edward Grant; "The Role of English Thought in the Transformation of University Education in the Late Middle Ages," by William J.
    [Show full text]
  • De Sphaera of Johannes De Sacrobosco in the Early Modern
    Matteo Valleriani Editor De sphaera of Johannes de Sacrobosco in the Early Modern Period The Authors of the Commentaries De sphaera of Johannes de Sacrobosco in the Early Modern Period Matteo Valleriani Editor De sphaera of Johannes de Sacrobosco in the Early Modern Period The Authors of the Commentaries Editor Matteo Valleriani Max Planck Institute for the History of Science Berlin, Germany Technische Universität Berlin Berlin, Germany University of Tel Aviv Tel Aviv, Israel ISBN 978-3-030-30832-2 ISBN 978-3-030-30833-9 (eBook) https://doi.org/10.1007/978-3-030-30833-9 © The Editor(s) (if applicable) and The Author(s) 2020. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc.
    [Show full text]
  • Averroes' Natural Philosophy and Its Reception in the Latin West Http
    Averroes’ Natural Philosophy and its Reception in the Latin West Edited by Paul J.J.M. Bakker With contributions from: Jean-Baptiste Brenet (Universite de Paris I – Pantheon-Sorbonne), Cristina Cerami (cnrs, umr 7219:sphere/chspam), Silvia Donati (Albertus-Magnus-Institut, Bonn), Dag Nikolaus Hasse (Julius-Maximilians-Universitat Wurzburg), Craig Martin (Oakland University), Edith Dudley Sylla (North Carolina State University), Cecilia Trifogli (All Souls College, Oxford) © 2015 by Leuven University Press / Presses Universitaires de Louvain / Universitaire Pers Leuven, Minderbroedersstraat 4, B-3000 Leuven (Belgium) ISBN 978 94 6270 046 8 D / 2015 / 1869 / 48 Distributed by Leuven University Press http://upers.kuleuven.be/nl/book/9789462700468 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License: http://creativecommons.org/licenses/by-nc-nd/3.0/ 2015167 [Bakker] 001-Prelims-print [date 1510191536 : version 1509110900] page -3 AVERROES’ NATURAL PHILOSOPHY AND ITS RECEPTION IN THE LATIN WEST Edited by Paul J.J.M. Bakker LEUVEN UNIVERSITY PRESS Reprint from Averroes’ Natural Philosophy and its Reception in the Latin West - ISBN 978 94 6270 046 8 - © Leuven University Press, 2015 2015167 [Bakker] 001-Prelims-print [date 1510191536 : version 1509110900] page -4 © 2015 by De Wulf-Mansioncentrum – De Wulf-Mansion Centre Leuven University Press / Presses Universitaires de Louvain / Universitaire Pers Leuven Minderbroedersstraat 4, B-3000 Leuven (Belgium) All rights reserved. Except in those cases expressly determined by law, no part of this publication may be multiplied, saved in an automated datafile or made public in any way whatsoever without the express prior written consent of the publishers. ISBN 978 94 6270 046 8 D/2015/1869/48 NUR: 732 Reprint from Averroes’ Natural Philosophy and its Reception in the Latin West - ISBN 978 94 6270 046 8 - © Leuven University Press, 2015 2015167 [Bakker] 001-Prelims-print [date 1510191536 : version 1509110900] page -5 ..
    [Show full text]
  • A Medieval Hebrew Treatise on Obstetrics
    Medical History, 1988, 33: 96-119. A MEDIEVAL HEBREW TREATISE ON OBSTETRICS by RON BARKAI* Recent years have shown an increasing interest in the history ofmedicine. This is no longer a distant, estranged branch ofstudy, reserved for a handful ofspecialists, but an important part of the framework encompassing the study of the past. An especially vital facet ofthis discipline is the history ofwomen's medicine, from which much can be learned concerning the ways in which our forebears viewed the physiological and mental differences between men and women, the formation of life within the womb, and the birth process itself.1 As in other areas ofmedieval science, researchers in this field must consider not only the ancient Greek and Latin sources, but the mutual influences and exchange of medical information among the Muslim, Jewish, and Christian cultures. The gynaecological literature written in Latin Europe during the Middle Ages has, to date, been given considerable coverage in academic studies. This refers mainly to the two most popular treatises of the Middle Ages and the Renaissance, both attributed to Trotula, a woman doctor and lecturer at the Salerno school ofmedicine: Trotula minor and Trotula major.2 The Muslim gynaecological literature has thus far not been studied nearly so widely as the Latin. Nevertheless, the most important treatise in this area, written in the tenth century by 'Arib ibn Sa'id of Cordova, has been published and translated into French * Dr Ron Barkai, Department of History, Tel Aviv University, Ramat-Aviv, 69 978 Tel Aviv, Israel. A striking example ofthis type ofstudy is Danielle Jacquart and Claude Thomasset, Sexualite et savoir medicale au moyen dge, Paris, Presses universitaires de France, 1985.
    [Show full text]
  • Time in Literature: a Study of Shakespeare and Dr. Iqbal
    www.ijcrt.org © 2018 IJCRT | Volume 6, Issue 1 February 2018 | ISSN: 2320-2882 Time in Literature: A study of Shakespeare and Dr. Iqbal Dr. Irshad Ahmad Itoo1 & Dr. Bashir Ahmad Gulzarpuri2 1 Faculty Department of English 2 Head Department of Urdu Govt Degree College Bijbehara Govt Degree College Bijbehara Jammu and Kashmir. Jammu and Kashmir. Abstract There are several scientific and philosophical concepts of the universe whose epistemology has not yet been defined successfully. Although, the stalwarts, philosophers, scientists, and anthropologists, who since times immemorial, have tried their best to unlock the knots that are embedded in these Metaphysical concepts like true nature of “Time”. The concept has also become a subject of discourse for writers of literature in its diverse fields like poetry and prose. The present study will concentrate on the poetic side and will analyze the verse of two great literary figures viz William Shakespeare and Sir Muhammad Iqbal, who like that of the scientists and anthropologists had made tireless effort to reveal the mystery of time concept and have pondered over the subject-matter in the literary field. Key words: - Absolute Time, Absolute Space, Dimensions of Time, Human Consciousness and Impact of Time. Introduction Time has been the subject of discourse from the ancient Greek civilisation and we find a threadbare converse on the theme relating to the metaphysical and transcendental world. The main questions put forward that have been discussed are: 1. Exactly what is time? 2. What is it made of? 3. Does time have a beginning and an end? 4. Is it a straight line or a circle? So far as the concept of time is concerned, the ancient Greek philosopher namely Plato identified time with the period of motion of the heavenly bodies (Weinert 9).
    [Show full text]
  • Boethius (480 – 524)
    The Saga of Mathematics A Brief History Boethius (480 – 524) Boethius became an orphan when he was Europe Smells the Coffee seven years old. He was extremely well educated. Boethius was a Chapter 6 philosopher, poet, mathematician, statesman, and (perhaps) martyr. Lewinter & Widulski The Saga of Mathematics 1 Lewinter & Widulski The Saga of Mathematics 2 Boethius (480 – 524) Boethius (480 – 524) He is best known as a translator of and This shows Boethius commentator on Greek writings on logic and calculating with mathematics (Plato, Aristotle, Nichomachus). Arabic numerals His mathematics texts were the best available competing with and were used for many centuries at a time Pythagoras using an when mathematical achievement in Europe abacus. was at a low point. It is from G. Reisch, Boethius’ Arithmetic taught medieval scholars Margarita about Pythagorean number theory. Philosophica (1508). Lewinter & Widulski The Saga of Mathematics 3 Lewinter & Widulski The Saga of Mathematics 4 Boethius (480 – 524) Boethius (480 – 524) Boethius was a main source of material for One of the first musical works to be printed the quadrivium, which was introduced into was Boethius's De institutione musica, written monasteries and consisted of arithmetic, in the early sixth century. geometry, astronomy, and the theory of It was for medieval authors, from around the music. ninth century on, the authoritative document on Greek music-theoretical thought and Boethius wrote about the relation of music systems. and science, suggesting that the pitch of a For example, Franchino Gaffurio in Theorica note one hears is related to the frequency of musica (1492) acknowledged Boethius as the sound.
    [Show full text]