Science Networks. Historical Studies Science Networks. Historical Studies Founded by Erwin Hiebert and Hans Wußing Vo l u m e 4 6 Edited by Eberhard Knobloch, Helge Kragh and Volker Remmert Editorial Board: K. Andersen, Amsterdam S. Hildebrandt, Bonn H.J.M. Bos, Amsterdam D. Kormos Buchwald, Pasadena U. Bottazzini, Roma Ch. Meinel, Regensburg J.Z. Buchwald, Pasadena J. Peiffer, Paris K. Chemla, Paris W. Purkert, Bonn S.S. Demidov, Moskva D. Rowe, Mainz M. Folkerts, München A.I. Sabra, Cambridge, Mass. P. Galison, Cambridge, Mass. Ch. Sasaki, Tokyo I. Grattan-Guinness, London R.H. Stuewer, Minneapolis J. Gray, Milton Keynes V.P. Vizgin, Moskva R. Halleux, Liége Tito M. Tonietti And Yet It Is Heard Musical, Multilingual and Multicultural History of the Mathematical Sciences — Volume 1 Tito M. Tonietti Dipartimento di matematica University of Pisa Pisa Italy ISBN 978-3-0348-0671-8 ISBN 978-3-0348-0672-5 (eBook) DOI 10.1007/978-3-0348-0672-5 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2014935966 © Springer Basel 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com) The history of the sciences is a grand fugue, in which the voices of various peoples chime in, each in their turn. It is as if an eternal harmony conversed with itself as it may have done in the bosom of God, before the creation of the world Wolfgang Goethe Foreword Musica nihil aliud est, quam omnium ordinem scire. Music is nothing but to know the order of all things. Trismegistus in Asclepius, cited by Athanasius Kircher, Musurgia universalis, Rome 1650, vol. II, title page Tito Tonietti has certainly written a very ambitious, extraordinary book in many respects. Its subtitle precisely describes his scientific aims and objectives. His goal here is to present a musical, multilingual, and multicultural history of the mathematical sciences, since ancient times up to the twentieth century. To the best of my knowledge, this is the first serious, comprehensive attempt to do justice to the essential role music played in the development of these sciences. This musical aspect is usually ignored or dramatically underestimated in descrip- tions of the evolution of sciences. Tonietti stresses this issue continually. He states his conviction at the very beginning of the book: Music was one of the primeval mathematical models for natural sciences in the West. “By means of music, it is easier to understand how many and what kinds of obstacles the Greek and Roman natural philosophers had created between mathematical sciences and the world of senses.” Yet, also in China, it is possible to narrate the mathematical sciences by means of music, as Tonietti demonstrates in Chap. 3. Even in India, certain ideas would seem to connect music with mathematics. Narrating history through music remains his principle and style when he speaks about the Arabic culture. Tonietti emphasizes throughout the role of languages and the existence of cultural differences and various scientific traditions, thus explicitly extending the famous Sapir-Whorf hypothesis to the mathematical sciences. He emphatically rejects Eurocentric prejudices and pleads for the acceptance of cultural variety. Every culture generates its own science so that there are independent inventions in different contexts. For him, even the texts of mathematicians acquire sense only if they are set in their context: “The Indian brahwana and the Greek philosophers developed their mathematical cultures in a relative autonomy, maintaining their own characteristics.” vii viii Foreword To mention another of Tonietti’s examples: The Greek and Latin scientific cultures, the Chinese scientific culture cannot be reduced to some general charac- teristics. Chinese books offered different proofs from those of Euclid. He draws a crucial conclusion: Such differences should not be transformed into inferiority or exclusion. For the Chinese, as well as for the Indians, the Pythagorean distinction between integers – or ratios between them – and other, especially irrational numbers does not seem to make sense. The Chinese mathematical theory of music was invented through solid pipes. Tonietti does not conceal another matter of fact: In his perspective of history, harmony is not only the daughter of Venus, but also of a father like Mars. For good reasons he dedicates a long chapter to Kepler’s world harmony, which indeed deserves more attention. He disagrees with the many modern historians of science who transformed Kepler’s diversity into inferiority “with the aggravating circumstances of those intolerable nationalistic veins from which we particularly desire to stay at a good distance.” Tonietti’s original approach enables him to gain many essential new insights: The true achievements of Aristoxenus, Vincenzo Galilei, Stevin (equable temperament), Lucretius’s contributions to the history of science overlooked up to now, the reasons the prohibition of irrational numbers was eclipsed during the seventeenth century, and the understanding of the reappearance of mathematics as the language essential to express the new science in this century, to mention some of them. Or, as he puts it: “The question has become rather how to interpret the musical language of the spheres and not whether it came from God.” Tonietti emphatically refuses corruptions, discriminations, distortions, simplifi- cations, anachronisms, nationalisms of authors, and cultures trying to show that “even the mathematical sciences are neither neutral nor universal nor eternal and depend on the historical and cultural contexts that invent them.” He places music in the foreground, he has not written a history of music with just hints to acoustic theories. In spite of all his efforts and the more than thousand pages of his book, Tonietti calls his attempt a modest proposal, a beginning. It is certainly a provocative book that is worth diligently studying and continuing even if not every modern scholar will accept all of its statements and conclusions. Berlin, Germany Eberhard Knobloch February 2014 Contents for Volume I 1 Introduction .................................................................. 1 Part I In the Ancient World 2 Above All with the Greek Alphabet........................................ 9 2.1 The Most Ancient of All the Quantitative Physical Laws ............ 9 2.2 ThePythagoreans....................................................... 11 2.3 Plato..................................................................... 19 2.4 Euclid ................................................................... 25 2.5 Aristoxenus ............................................................. 35 2.6 ClaudiusPtolemaeus ................................................... 41 2.7 Archimedesanda FewOthers ......................................... 54 2.8 TheLatinLucretius..................................................... 64 2.9 TextsandContexts...................................................... 82 3 In Chinese Characters ....................................................... 97 3.1 MusicinChina,Yuejing,Confucius................................... 97 3.2 TuningReed-Pipes ..................................................... 100 3.3 TheFigureoftheString................................................ 111 3.4 CalculatinginNineWays .............................................. 117 3.5 TheQi................................................................... 124 3.6 Rules,RelationshipsandMovements.................................. 140 3.6.1 CharactersandLiteraryDiscourse ............................ 141 3.6.2 A LivingOrganismonEarth................................... 143 3.6.3 Rules,ModelsinMovementandValues...................... 147 3.6.4 The Geometry of the Continuum in Language................ 151