Ancient Greek Mathematics
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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. -
A Centennial Celebration of Two Great Scholars: Heiberg's
A Centennial Celebration of Two Great Scholars: Heiberg’s Translation of the Lost Palimpsest of Archimedes—1907 Heath’s Publication on Euclid’s Elements—1908 Shirley B. Gray he 1998 auction of the “lost” palimp- tains four illuminated sest of Archimedes, followed by col- plates, presumably of laborative work centered at the Walters Matthew, Mark, Luke, Art Museum, the palimpsest’s newest and John. caretaker, remind Notices readers of Heiberg was emi- Tthe herculean contributions of two great classical nently qualified for scholars. Working one century ago, Johan Ludvig support from a foun- Heiberg and Sir Thomas Little Heath were busily dation. His stature as a engaged in virtually “running the table” of great scholar in the interna- mathematics bequeathed from antiquity. Only tional community was World War I and a depleted supply of manuscripts such that the University forced them to take a break. In 2008 we as math- of Oxford had awarded ematicians should honor their watershed efforts to him an honorary doc- make the cornerstones of our discipline available Johan Ludvig Heiberg. torate of literature in Photo courtesy of to even mathematically challenged readers. The Danish Royal Society. 1904. His background in languages and his pub- Heiberg lications were impressive. His first language was In 1906 the Carlsberg Foundation awarded 300 Danish but he frequently published in German. kroner to Johan Ludvig Heiberg (1854–1928), a He had publications in Latin as well as Arabic. But classical philologist at the University of Copenha- his true passion was classical Greek. In his first gen, to journey to Constantinople (present day Is- position as a schoolmaster and principal, Heiberg tanbul) to investigate a palimpsest that previously insisted that his students learn Greek and Greek had been in the library of the Metochion, i.e., the mathematics—in Greek. -
15 Famous Greek Mathematicians and Their Contributions 1. Euclid
15 Famous Greek Mathematicians and Their Contributions 1. Euclid He was also known as Euclid of Alexandria and referred as the father of geometry deduced the Euclidean geometry. The name has it all, which in Greek means “renowned, glorious”. He worked his entire life in the field of mathematics and made revolutionary contributions to geometry. 2. Pythagoras The famous ‘Pythagoras theorem’, yes the same one we have struggled through in our childhood during our challenging math classes. This genius achieved in his contributions in mathematics and become the father of the theorem of Pythagoras. Born is Samos, Greece and fled off to Egypt and maybe India. This great mathematician is most prominently known for, what else but, for his Pythagoras theorem. 3. Archimedes Archimedes is yet another great talent from the land of the Greek. He thrived for gaining knowledge in mathematical education and made various contributions. He is best known for antiquity and the invention of compound pulleys and screw pump. 4. Thales of Miletus He was the first individual to whom a mathematical discovery was attributed. He’s best known for his work in calculating the heights of pyramids and the distance of the ships from the shore using geometry. 5. Aristotle Aristotle had a diverse knowledge over various areas including mathematics, geology, physics, metaphysics, biology, medicine and psychology. He was a pupil of Plato therefore it’s not a surprise that he had a vast knowledge and made contributions towards Platonism. Tutored Alexander the Great and established a library which aided in the production of hundreds of books. -
THE PHILOSOPHY BOOK George Santayana (1863-1952)
Georg Hegel (1770-1831) ................................ 30 Arthur Schopenhauer (1788-1860) ................. 32 Ludwig Andreas Feuerbach (1804-1872) ...... 32 John Stuart Mill (1806-1873) .......................... 33 Soren Kierkegaard (1813-1855) ..................... 33 Karl Marx (1818-1883).................................... 34 Henry David Thoreau (1817-1862) ................ 35 Charles Sanders Peirce (1839-1914).............. 35 William James (1842-1910) ............................ 36 The Modern World 1900-1950 ............................. 36 Friedrich Nietzsche (1844-1900) .................... 37 Ahad Ha'am (1856-1927) ............................... 38 Ferdinand de Saussure (1857-1913) ............. 38 Edmund Husserl (1859–1938) ....................... 39 Henri Bergson (1859-1941) ............................ 39 Contents John Dewey (1859–1952) ............................... 39 Introduction....................................................... 1 THE PHILOSOPHY BOOK George Santayana (1863-1952) ..................... 40 The Ancient World 700 BCE-250 CE..................... 3 Miguel de Unamuno (1864-1936) ................... 40 Introduction Thales of Miletus (c.624-546 BCE)................... 3 William Du Bois (1868-1963) .......................... 41 Laozi (c.6th century BCE) ................................. 4 Philosophy is not just the preserve of brilliant Bertrand Russell (1872-1970) ........................ 41 Pythagoras (c.570-495 BCE) ............................ 4 but eccentric thinkers that it is popularly Max Scheler -
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. -
Theon of Alexandria and Hypatia
CREATIVE MATH. 12 (2003), 111 - 115 Theon of Alexandria and Hypatia Michael Lambrou Abstract. In this paper we present the story of the most famous ancient female math- ematician, Hypatia, and her father Theon of Alexandria. The mathematician and philosopher Hypatia flourished in Alexandria from the second part of the 4th century until her violent death incurred by a mob in 415. She was the daughter of Theon of Alexandria, a math- ematician and astronomer, who flourished in Alexandria during the second part of the fourth century. Information on Theon’s life is only brief, coming mainly from a note in the Suda (Suida’s Lexicon, written about 1000 AD) stating that he lived in Alexandria in the times of Theodosius I (who reigned AD 379-395) and taught at the Museum. He is, in fact, the Museum’s last attested member. Descriptions of two eclipses he observed in Alexandria included in his commentary to Ptolemy’s Mathematical Syntaxis (Almagest) and elsewhere have been dated as the eclipses that occurred in AD 364, which is consistent with Suda. Although originality in Theon’s works cannot be claimed, he was certainly immensely influential in the preservation, dissemination and editing of clas- sic texts of previous generations. Indeed, with the exception of Vaticanus Graecus 190 all surviving Greek manuscripts of Euclid’s Elements stem from Theon’s edition. A comparison to Vaticanus Graecus 190 reveals that Theon did not actually change the mathematical content of the Elements except in minor points, but rather re-wrote it in Koini and in a form more suitable for the students he taught (some manuscripts refer to Theon’s sinousiai). -
Chapter Two Democritus and the Different Limits to Divisibility
CHAPTER TWO DEMOCRITUS AND THE DIFFERENT LIMITS TO DIVISIBILITY § 0. Introduction In the previous chapter I tried to give an extensive analysis of the reasoning in and behind the first arguments in the history of philosophy in which problems of continuity and infinite divisibility emerged. The impact of these arguments must have been enormous. Designed to show that rationally speaking one was better off with an Eleatic universe without plurality and without motion, Zeno’s paradoxes were a challenge to everyone who wanted to salvage at least those two basic features of the world of common sense. On the other hand, sceptics, for whatever reason weary of common sense, could employ Zeno-style arguments to keep up the pressure. The most notable representative of the latter group is Gorgias, who in his book On not-being or On nature referred to ‘Zeno’s argument’, presumably in a demonstration that what is without body and does not have parts, is not. It is possible that this followed an earlier argument of his that whatever is one, must be without body.1 We recognize here what Aristotle calls Zeno’s principle, that what does not have bulk or size, is not. Also in the following we meet familiar Zenonian themes: Further, if it moves and shifts [as] one, what is, is divided, not being continuous, and there [it is] not something. Hence, if it moves everywhere, it is divided everywhere. But if that is the case, then everywhere it is not. For it is there deprived of being, he says, where it is divided, instead of ‘void’ using ‘being divided’.2 Gorgias is talking here about the situation that there is motion within what is. -
Epicurus and the Epicureans on Socrates and the Socratics
chapter 8 Epicurus and the Epicureans on Socrates and the Socratics F. Javier Campos-Daroca 1 Socrates vs. Epicurus: Ancients and Moderns According to an ancient genre of philosophical historiography, Epicurus and Socrates belonged to distinct traditions or “successions” (diadochai) of philosophers. Diogenes Laertius schematizes this arrangement in the following way: Socrates is placed in the middle of the Ionian tradition, which starts with Thales and Anaximander and subsequently branches out into different schools of thought. The “Italian succession” initiated by Pherecydes and Pythagoras comes to an end with Epicurus.1 Modern views, on the other hand, stress the contrast between Socrates and Epicurus as one between two opposing “ideal types” of philosopher, especially in their ways of teaching and claims about the good life.2 Both ancients and moderns appear to agree on the convenience of keeping Socrates and Epicurus apart, this difference being considered a fundamental lineament of ancient philosophy. As for the ancients, however, we know that other arrangements of philosophical traditions, ones that allowed certain connections between Epicureanism and Socratism, circulated in Hellenistic times.3 Modern outlooks on the issue are also becoming more nuanced. 1 DL 1.13 (Dorandi 2013). An overview of ancient genres of philosophical history can be seen in Mansfeld 1999, 16–25. On Diogenes’ scheme of successions and other late antique versions of it, see Kienle 1961, 3–39. 2 According to Riley 1980, 56, “there was a fundamental difference of opinion concerning the role of the philosopher and his behavior towards his students.” In Riley’s view, “Epicurean criticism was concerned mainly with philosophical style,” not with doctrines (which would explain why Plato was not as heavily attacked as Socrates). -
Thales of Miletus Sources and Interpretations Miletli Thales Kaynaklar Ve Yorumlar
Thales of Miletus Sources and Interpretations Miletli Thales Kaynaklar ve Yorumlar David Pierce October , Matematics Department Mimar Sinan Fine Arts University Istanbul http://mat.msgsu.edu.tr/~dpierce/ Preface Here are notes of what I have been able to find or figure out about Thales of Miletus. They may be useful for anybody interested in Thales. They are not an essay, though they may lead to one. I focus mainly on the ancient sources that we have, and on the mathematics of Thales. I began this work in preparation to give one of several - minute talks at the Thales Meeting (Thales Buluşması) at the ruins of Miletus, now Milet, September , . The talks were in Turkish; the audience were from the general popu- lation. I chose for my title “Thales as the originator of the concept of proof” (Kanıt kavramının öncüsü olarak Thales). An English draft is in an appendix. The Thales Meeting was arranged by the Tourism Research Society (Turizm Araştırmaları Derneği, TURAD) and the office of the mayor of Didim. Part of Aydın province, the district of Didim encompasses the ancient cities of Priene and Miletus, along with the temple of Didyma. The temple was linked to Miletus, and Herodotus refers to it under the name of the family of priests, the Branchidae. I first visited Priene, Didyma, and Miletus in , when teaching at the Nesin Mathematics Village in Şirince, Selçuk, İzmir. The district of Selçuk contains also the ruins of Eph- esus, home town of Heraclitus. In , I drafted my Miletus talk in the Math Village. Since then, I have edited and added to these notes. -
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. -
Hypatia of Alexandria
Hypathia of Alexandria Doina Ionescu Astronomical Institute of the Romanian Academy, E–mail: [email protected] Introduction - Born in 350-355/370; - Lived and learned in Alexandria, Roman Egypt; - The daughter of Theon, the last director of the Museum of Alexandria; - Trained by her father in physical education, mathematics, astronomy, philosophy, arts, literature, the principles of teaching, oratory; - Died in 415, killed by a Christian mob who blamed her for religious turmoil. The Musaeum of Alexandria - Founded in the 3rd century BCE by Ptolemy I Soter or his son Ptolemy II; - Comprised gardens, a room for shared dining, a reading room, lecture halls, meetings rooms and a library; - The Library of Alexandria: an acquisitions department and a cataloguing department; - The Mouseion (“The House of the Muses”) – an institution that brought together the best scholars of the Hellenistic world, a university; - Destruction of the Mouseion and Library of Alexandria: 1. Julius Caesar’s Fire in the Alexandrian War, 48 BC; 2. The attack of Aurelian in the 3rd century AD; 3. The decree of Theophilus in AD 391; 4. The Muslim conquest in AD 642 and thereafter. Theon (b: 335 – d. early 5th century) - Most of the references on him and on Hypathia : Suda, the 10th century Byzantine encyclopedia; - Highly educated scholar, mathematician and astronomer; - A member and possibly the last director of the Alexandrian Museion, on public payroll. - Devoted his scholarship to the study of his predecessors Euclid and Ptolemy; his recensions were designed for students; - Euclid’s Elements; - Thirteen books of Ptolemy’ Almagest ; Handy Tables : The Great Commentary, in five books, and The Little Commentary , in one; - He worked together with scholar and student associates and with his daughter, Hypathia - A treatise “On the Small Astrolabe” ; - On Signs and the examination of Birds and the Croaking of Ravens: two essays on the function of the star Syrius and the influence of the planetary spheres on the Nile; - 364 AD: predicted eclipses of the Sun and Moon in Alexandria. -
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.