Lecture 8. Eudoxus and the Avoidance of a Fundamental Conflict
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Skepticism and Pluralism Ways of Living a Life Of
SKEPTICISM AND PLURALISM WAYS OF LIVING A LIFE OF AWARENESS AS RECOMMENDED BY THE ZHUANGZI #±r A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI'I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN PHILOSOPHY AUGUST 2004 By John Trowbridge Dissertation Committee: Roger T. Ames, Chairperson Tamara Albertini Chung-ying Cheng James E. Tiles David R. McCraw © Copyright 2004 by John Trowbridge iii Dedicated to my wife, Jill iv ACKNOWLEDGEMENTS In completing this research, I would like to express my appreciation first and foremost to my wife, Jill, and our three children, James, Holly, and Henry for their support during this process. I would also like to express my gratitude to my entire dissertation committee for their insight and understanding ofthe topics at hand. Studying under Roger Ames has been a transformative experience. In particular, his commitment to taking the Chinese tradition on its own terms and avoiding the tendency among Western interpreters to overwrite traditional Chinese thought with the preoccupations ofWestern philosophy has enabled me to broaden my conception ofphilosophy itself. Roger's seminars on Confucianism and Daoism, and especially a seminar on writing a philosophical translation ofthe Zhongyong r:pJm (Achieving Equilibrium in the Everyday), have greatly influenced my own initial attempts to translate and interpret the seminal philosophical texts ofancient China. Tamara Albertini's expertise in ancient Greek philosophy was indispensable to this project, and a seminar I audited with her, comparing early Greek and ancient Chinese philosophy, was part ofthe inspiration for my choice ofresearch topic. I particularly valued the opportunity to study Daoism and the Yijing ~*~ with Chung-ying Cheng g\Gr:p~ and benefited greatly from his theory ofonto-cosmology as a means of understanding classical Chinese philosophy. -
Mathematicians
MATHEMATICIANS [MATHEMATICIANS] Authors: Oliver Knill: 2000 Literature: Started from a list of names with birthdates grabbed from mactutor in 2000. Abbe [Abbe] Abbe Ernst (1840-1909) Abel [Abel] Abel Niels Henrik (1802-1829) Norwegian mathematician. Significant contributions to algebra and anal- ysis, in particular the study of groups and series. Famous for proving the insolubility of the quintic equation at the age of 19. AbrahamMax [AbrahamMax] Abraham Max (1875-1922) Ackermann [Ackermann] Ackermann Wilhelm (1896-1962) AdamsFrank [AdamsFrank] Adams J Frank (1930-1989) Adams [Adams] Adams John Couch (1819-1892) Adelard [Adelard] Adelard of Bath (1075-1160) Adler [Adler] Adler August (1863-1923) Adrain [Adrain] Adrain Robert (1775-1843) Aepinus [Aepinus] Aepinus Franz (1724-1802) Agnesi [Agnesi] Agnesi Maria (1718-1799) Ahlfors [Ahlfors] Ahlfors Lars (1907-1996) Finnish mathematician working in complex analysis, was also professor at Harvard from 1946, retiring in 1977. Ahlfors won both the Fields medal in 1936 and the Wolf prize in 1981. Ahmes [Ahmes] Ahmes (1680BC-1620BC) Aida [Aida] Aida Yasuaki (1747-1817) Aiken [Aiken] Aiken Howard (1900-1973) Airy [Airy] Airy George (1801-1892) Aitken [Aitken] Aitken Alec (1895-1967) Ajima [Ajima] Ajima Naonobu (1732-1798) Akhiezer [Akhiezer] Akhiezer Naum Ilich (1901-1980) Albanese [Albanese] Albanese Giacomo (1890-1948) Albert [Albert] Albert of Saxony (1316-1390) AlbertAbraham [AlbertAbraham] Albert A Adrian (1905-1972) Alberti [Alberti] Alberti Leone (1404-1472) Albertus [Albertus] Albertus Magnus -
MONEY and the EARLY GREEK MIND: Homer, Philosophy, Tragedy
This page intentionally left blank MONEY AND THE EARLY GREEK MIND How were the Greeks of the sixth century bc able to invent philosophy and tragedy? In this book Richard Seaford argues that a large part of the answer can be found in another momentous development, the invention and rapid spread of coinage, which produced the first ever thoroughly monetised society. By transforming social relations, monetisation contributed to the ideas of the universe as an impersonal system (presocratic philosophy) and of the individual alienated from his own kin and from the gods (in tragedy). Seaford argues that an important precondition for this monetisation was the Greek practice of animal sacrifice, as represented in Homeric epic, which describes a premonetary world on the point of producing money. This book combines social history, economic anthropology, numismatics and the close reading of literary, inscriptional, and philosophical texts. Questioning the origins and shaping force of Greek philosophy, this is a major book with wide appeal. richard seaford is Professor of Greek Literature at the University of Exeter. He is the author of commentaries on Euripides’ Cyclops (1984) and Bacchae (1996) and of Reciprocity and Ritual: Homer and Tragedy in the Developing City-State (1994). MONEY AND THE EARLY GREEK MIND Homer, Philosophy, Tragedy RICHARD SEAFORD cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521832281 © Richard Seaford 2004 This publication is in copyright. -
Meet the Philosophers of Ancient Greece
Meet the Philosophers of Ancient Greece Everything You Always Wanted to Know About Ancient Greek Philosophy but didn’t Know Who to Ask Edited by Patricia F. O’Grady MEET THE PHILOSOPHERS OF ANCIENT GREECE Dedicated to the memory of Panagiotis, a humble man, who found pleasure when reading about the philosophers of Ancient Greece Meet the Philosophers of Ancient Greece Everything you always wanted to know about Ancient Greek philosophy but didn’t know who to ask Edited by PATRICIA F. O’GRADY Flinders University of South Australia © Patricia F. O’Grady 2005 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. Patricia F. O’Grady has asserted her right under the Copyright, Designs and Patents Act, 1988, to be identi.ed as the editor of this work. Published by Ashgate Publishing Limited Ashgate Publishing Company Wey Court East Suite 420 Union Road 101 Cherry Street Farnham Burlington Surrey, GU9 7PT VT 05401-4405 England USA Ashgate website: http://www.ashgate.com British Library Cataloguing in Publication Data Meet the philosophers of ancient Greece: everything you always wanted to know about ancient Greek philosophy but didn’t know who to ask 1. Philosophy, Ancient 2. Philosophers – Greece 3. Greece – Intellectual life – To 146 B.C. I. O’Grady, Patricia F. 180 Library of Congress Cataloging-in-Publication Data Meet the philosophers of ancient Greece: everything you always wanted to know about ancient Greek philosophy but didn’t know who to ask / Patricia F. -
Lecture 4. Pythagoras' Theorem and the Pythagoreans
Lecture 4. Pythagoras' Theorem and the Pythagoreans Figure 4.1 Pythagoras was born on the Island of Samos Pythagoras Pythagoras (born between 580 and 572 B.C., died about 497 B.C.) was born on the island of Samos, just off the coast of Asia Minor1. He was the earliest important philosopher who made important contributions in mathematics, astronomy, and the theory of music. Pythagoras is credited with introduction the words \philosophy," meaning love of wis- dom, and \mathematics" - the learned disciplines.2 Pythagoras is famous for establishing the theorem under his name: Pythagoras' theorem, which is well known to every educated per- son. Pythagoras' theorem was known to the Babylonians 1000 years earlier but Pythagoras may have been the first to prove it. Pythagoras did spend some time with Thales in Miletus. Probably by the suggestion of Thales, Pythagoras traveled to other places, including Egypt and Babylon, where he may have learned some mathematics. He eventually settled in Groton, a Greek settlement in southern Italy. In fact, he spent most of his life in the Greek colonies in Sicily and 1Asia Minor is a region of Western Asia, comprising most of the modern Republic of Turkey. 2Is God a mathematician ? Mario Livio, Simon & Schuster Paperbacks, New York-London-Toronto- Sydney, 2010, p. 15. 23 southern Italy. In Groton, he founded a religious, scientific, and philosophical organization. It was a formal school, in some sense a brotherhood, and in some sense a monastery. In that organization, membership was limited and very secretive; members learned from leaders and received education in religion. -
Eudoxus Erin Sondgeroth 9/1/14
Eudoxus Erin Sondgeroth 9/1/14 There were many brilliant mathematicians from the time period of 500 BC until 300 BC, including Pythagoras and Euclid. Another fundamental figure from that era was Greek mathematician Eudoxus of Cnidus. Eudoxus was born in Cnidus (present-day Turkey) in approximately 408 BC and died in 355 BC at the age of 53. Eudoxus studied math under Archytus in Tarentum, medicine under Philistium in Sicily, astronomy in Egytpt, and philosophy and rhetoric under Plato in Athens. After his many years of studying, Eudoxus established his own school at Cyzicus, where had many pupils. In 365 BC Eudoxus moved his school to Athens in order to work as a colleague of Plato. It was during this time that Eudoxus completed some of his best work and the reasoning for why he is considered the leading mathematician and astronomer of his day. In the field of astronomy, Eudoxus`s best-known work was his planetary model. This model had a spherical Earth that was at rest in the center of the universe and twenty-seven concentric spheres that held the fixed stars, sun, moon, and other planets that moved in orbit around the Earth. While this model has clearly been disproved, it was a prominent model for at least fifty years because it explained many astronomical phenomenon of that time, including the sunrise and sunset, the fixed constellations, and movement of other planets in orbits. Luckily, Eudoxus`s theories had much more success in the field of mathemat- ics. Many of his proposition, theories, and proofs are common knowledge in the math world today. -
Nicolaus Copernicus: the Loss of Centrality
I Nicolaus Copernicus: The Loss of Centrality The mathematician who studies the motions of the stars is surely like a blind man who, with only a staff to guide him, must make a great, endless, hazardous journey that winds through innumerable desolate places. [Rheticus, Narratio Prima (1540), 163] 1 Ptolemy and Copernicus The German playwright Bertold Brecht wrote his play Life of Galileo in exile in 1938–9. It was first performed in Zurich in 1943. In Brecht’s play two worldviews collide. There is the geocentric worldview, which holds that the Earth is at the center of a closed universe. Among its many proponents were Aristotle (384–322 BC), Ptolemy (AD 85–165), and Martin Luther (1483–1546). Opposed to geocentrism is the heliocentric worldview. Heliocentrism teaches that the sun occupies the center of an open universe. Among its many proponents were Copernicus (1473–1543), Kepler (1571–1630), Galileo (1564–1642), and Newton (1643–1727). In Act One the Italian mathematician and physicist Galileo Galilei shows his assistant Andrea a model of the Ptolemaic system. In the middle sits the Earth, sur- rounded by eight rings. The rings represent the crystal spheres, which carry the planets and the fixed stars. Galileo scowls at this model. “Yes, walls and spheres and immobility,” he complains. “For two thousand years people have believed that the sun and all the stars of heaven rotate around mankind.” And everybody believed that “they were sitting motionless inside this crystal sphere.” The Earth was motionless, everything else rotated around it. “But now we are breaking out of it,” Galileo assures his assistant. -
The Planetary Increase of Brightness During Retrograde Motion: an Explanandum Constructed Ad Explanantem
Studies in History and Philosophy of Science 54 (2015) 90e101 Contents lists available at ScienceDirect Studies in History and Philosophy of Science journal homepage: www.elsevier.com/locate/shpsa The planetary increase of brightness during retrograde motion: An explanandum constructed ad explanantem Christián Carlos Carman National University of Quilmes/CONICET, Roque Sáenz Peña 352, B1876BXD Bernal, Buenos Aires, Argentina article info abstract Article history: In Ancient Greek two models were proposed for explaining the planetary motion: the homocentric Received 5 May 2015 spheres of Eudoxus and the Epicycle and Deferent System. At least in a qualitative way, both models Received in revised form could explain the retrograde motion, the most challenging phenomenon to be explained using circular 17 September 2015 motions. Nevertheless, there is another explanandum: during retrograde motion the planets increase their brightness. It is natural to interpret a change of brightness, i.e., of apparent size, as a change in Keywords: distance. Now, while according to the Eudoxian model the planet is always equidistant from the earth, Epicycle; according to the epicycle and deferent system, the planet changes its distance from the earth, Deferent; fi Homocentric spheres; approaching to it during retrograde motion, just as observed. So, it is usually af rmed that the main ’ Brightness change reason for the rejection of Eudoxus homocentric spheres in favor of the epicycle and deferent system was that the first cannot explain the manifest planetary increase of brightness during retrograde motion, while the second can. In this paper I will show that this historical hypothesis is not as firmly founded as it is usually believed to be. -
``Mathematics'' and ``Physics'' in the Science of Harmonics
NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 4 no. 3-4 2016 Mathematics and Mechanics of Complex Systems STEFANO ISOLA “MATHEMATICS” AND “PHYSICS” IN THE SCIENCE OF HARMONICS msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 4, No. 3-4, 2016 dx.doi.org/10.2140/memocs.2016.4.213 ∩ MM “MATHEMATICS” AND “PHYSICS” IN THE SCIENCE OF HARMONICS STEFANO ISOLA Some aspects of the role that the science of harmonics has played in the history of science are discussed in light of Russo’s investigation of the history of the concepts of “mathematics” and “physics”. 1. The rambling route of the ancient scientific method In several places in Russo’s writings on the history of science, one can find en- lightening discussions about the meanings of the concepts of “physics” and “math- ematics”, along with the particular notions of truth involved in them; see, e.g., [58, Chapter 6.6; 60, Chapter 15; 56; 57]. Both terms derive from the Greek: the original meaning of the former was the investigation of everything that lives, grows or, more generally, comes into existence, whereas the latter referred to all that is studied, thus deriving its meaning not from its content but from its method. In the Hellenistic period, the term “physics” continued to be used to indicate that sector of philosophy that addressed nature (the other sectors being ethics and logic), thus corresponding to what came to be called “natural philosophy” in modern times. On the other hand, the term “mathematics” was used to indicate all the disciplines (including geometry, arithmetic, harmonics, astronomy, optics, mechanics, hydro- statics, pneumatics, geodesy and mathematical geography) that shared the same method of investigation, based on the construction of theories by which “theorems” are proved, leaning on explicitly stated initial assumptions. -
“Point at Infinity Hape of the World
“Point at Infinity hape of the World Last week, we began a series of posts dedicated to thinking about immortality. If we want to even pretend to think precisely about immortality, we will have to consider some fundamental questions. What does it mean to be immortal? What does it mean to live forever? Are these the same thing? And since immortality is inextricably tied up in one’s relationship with time, we must think about the nature of time itself. Is there a difference between external time and personal time? What is the shape of time? Is time linear? Circular? Finite? Infinite? Of course, we exist not just across time but across space as well, so the same questions become relevant when asked about space. What is the shape of space? Is it finite? Infinite? It is not hard to see how this question would have a significant bearing on our thinking about immortality. In a finite universe (or, more precisely, a universe in which only finitely many different configurations of maer are possible), an immortal being would encounter the same situations over and over again, would think the same thoughts over and over again, would have the same conversations over and over again. Would such a life be desirable? (It is not clear that this repetition would be avoidable even in an infinite universe, but more on that later.) Today, we are going to take a lile historical detour to look at the shape of the universe, a trip that will take us from Ptolemy to Dante to Einstein, a trip that will uncover a remarkable confluence of poetry and physics. -
The Arithmetic of the Spheres
The Arithmetic of the Spheres Je↵ Lagarias, University of Michigan Ann Arbor, MI, USA MAA Mathfest (Washington, D. C.) August 6, 2015 Topics Covered Part 1. The Harmony of the Spheres • Part 2. Lester Ford and Ford Circles • Part 3. The Farey Tree and Minkowski ?-Function • Part 4. Farey Fractions • Part 5. Products of Farey Fractions • 1 Part I. The Harmony of the Spheres Pythagoras (c. 570–c. 495 BCE) To Pythagoras and followers is attributed: pitch of note of • vibrating string related to length and tension of string producing the tone. Small integer ratios give pleasing harmonics. Pythagoras or his mentor Thales had the idea to explain • phenomena by mathematical relationships. “All is number.” A fly in the ointment: Irrational numbers, for example p2. • 2 Harmony of the Spheres-2 Q. “Why did the Gods create us?” • A. “To study the heavens.”. Celestial Sphere: The universe is spherical: Celestial • spheres. There are concentric spheres of objects in the sky; some move, some do not. Harmony of the Spheres. Each planet emits its own unique • (musical) tone based on the period of its orbital revolution. Also: These tones, imperceptible to our hearing, a↵ect the quality of life on earth. 3 Democritus (c. 460–c. 370 BCE) Democritus was a pre-Socratic philosopher, some say a disciple of Leucippus. Born in Abdera, Thrace. Everything consists of moving atoms. These are geometrically• indivisible and indestructible. Between lies empty space: the void. • Evidence for the void: Irreversible decay of things over a long time,• things get mixed up. (But other processes purify things!) “By convention hot, by convention cold, but in reality atoms and• void, and also in reality we know nothing, since the truth is at bottom.” Summary: everything is a dynamical system! • 4 Democritus-2 The earth is round (spherical). -
A Acerbi, F., 45 Adam, C., 166–169, 171, 175, 176, 178–181, 183–185
Index A 80, 81, 84, 86, 87, 93, 97, 98, 107, 112, Acerbi, F., 45 137, 198, 201, 212, 233, 235, 236 Adam, C., 166–169, 171, 175, 176, 178–181, Arius Didymus, 35 183–185, 187, 188, 200, 201 Arnauld, 217, 227, 233, 239–242, 245 Adams, R.M., 234, 245, 306 Arnzen, R., 55 Adorno, T.W., 147 Arriaga, R.de, 6 Adrastos, 77–81, 89, 96 Arthur, R, 254 Aetius, 35 Athenaeus, 39 Agapius, 56 Atherton, M., 170, 174 Aglietta, M., 151 Aujac, G., 145 Aichelin, J., 232 Autolycus, 15, 20, 21, 23, 24 Aiken, J.A., 148 Ayers, M., 229 Aime, M., 145 Aksamija, N., 152 Alberti, L.B., 9, 148, 150 B Alembert, J.d', 70 Bacon, R., 165, 166 Alexander of Aphrodisias, 16, 35, 43, 54, 56, Banu Musa, 53 58, 61, 77, 78, 81 Barnes, J., 38, 212, 235 Algra, K., 34, 35, 38 Barozzi, F., 115, 119, 120 Al-Haytham, 164–167 Barrow, I., 6, 220 Alhazen, see Ibn Al-Haytham Basileides of Tyre, 37 Al-Khwārizmī, 92 Baudrillard, J., 144 Al-Kindī, 161, 162, 164, 165 Bayle, P., 237 Al-Nayrizi, 55–57, 59 Beauchamp, T.L., 274 Al-Sijzī, 110 Bechtle, G., 105 Allison, H.E., 283 Belting, H., 145, 148 Andronicus of Rhodes, 102 Benatouïl, T., 114 Apollodorus, 41, 42 Benjamin, W., 144, 145, 148 Apollonios of Perge (Apollonius), 15, 19–21, Bentley, R., 223–226 23–27, 37, 38, 52, 53, 63, 81, 106, 123 Berggren, J.L., 94 Apostle Thomas, 147 Berkeley, G., 11, 160, 161, 166, 170, 186 Aratus, 24 Bernadete, J., 212, 213 Arbini, R., 170 Bernanos, G., 137 Archedemus, 41 Bernoulli, J., 233 Archimedes, 15, 19–21, 23, 24, 27, 29, 38, 39, Bessel, F.W., 72 44, 45, 47, 68, 106, 123 Bioesmat-Martagon, L., 72 Ariew, R., 110, 212, 214 Blumenberg, H., 143 Aristaeus, 27 Boer, E., 93 Ariston, 39 Bolyai, J., 11 Aristotle, 3, 5, 6, 15, 16, 18, 19, 24–34, 36, Bonola, R., 70 40–45, 47–50, 55, 58, 60, 61, 68, 75, Borelli, G.A., 6 © Springer International Publishing Switzerland 2015 311 V.