The Origin of Life:Ahistory of Ancient Greek Theories
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Fiboquadratic Sequences and Extensions of the Cassini Identity Raised from the Study of Rithmomachia
Fiboquadratic sequences and extensions of the Cassini identity raised from the study of rithmomachia Tom´asGuardia∗ Douglas Jim´enezy October 17, 2018 To David Eugene Smith, in memoriam. Mathematics Subject Classification: 01A20, 01A35, 11B39 and 97A20. Keywords: pythagoreanism, golden ratio, Boethius, Nicomachus, De Arithmetica, fiboquadratic sequences, Cassini's identity and rithmomachia. Abstract In this paper, we introduce fiboquadratic sequences as a consequence of an extension to infinity of the board of rithmomachia. Fiboquadratic sequences approach the golden ratio and provide extensions of Cassini's Identity. 1 Introduction Pythagoreanism was a philosophical tradition, that left a deep influence over the Greek mathematical thought. Its path can be traced until the Middle Ages, and even to present. Among medieval scholars, which expanded the practice of the pythagoreanism, we find Anicius Manlius Severinus Boethius (480-524 A.D.) whom by a free translation of De Institutione Arithmetica by Nicomachus of Gerasa, preserved the pythagorean teaching inside the first universities. In fact, Boethius' book became the guide of study for excellence during quadriv- ium teaching, almost for 1000 years. The learning of arithmetic during the arXiv:1509.03177v3 [math.HO] 22 Nov 2016 quadrivium, made necessary the practice of calculation and handling of basic mathematical operations. Surely, with the mixing of leisure with this exercise, Boethius' followers thought up a strategy game in which, besides the training of mind calculation, it was used to preserve pythagorean traditions coming from the Greeks and medieval philosophers. Maybe this was the origin of the philoso- phers' game or rithmomachia. Rithmomachia (RM, henceforward) became the ∗Department of Mathematics. -
Who Was Protagoras? • Born in Abdêra, an Ionian Pólis in Thrace
Recovering the wisdom of Protagoras from a reinterpretation of the Prometheia trilogy Prometheus (c.1933) by Paul Manship (1885-1966) By: Marty Sulek, Ph.D. Indiana University Lilly Family School of Philanthropy For: Workshop In Multidisciplinary Philanthropic Studies February 10, 2015 Composed for inclusion in a Festschrift in honour of Dr. Laurence Lampert, a Canadian philosopher and leading scholar in the field of Nietzsche studies, and a professor emeritus of Philosophy at IUPUI. Adult Content Warning • Nudity • Sex • Violence • And other inappropriate Prometheus Chained by Vulcan (1623) themes… by Dirck van Baburen (1595-1624) Nietzsche on Protagoras & the Sophists “The Greek culture of the Sophists had developed out of all the Greek instincts; it belongs to the culture of the Periclean age as necessarily as Plato does not: it has its predecessors in Heraclitus, in Democritus, in the scientific types of the old philosophy; it finds expression in, e.g., the high culture of Thucydides. And – it has ultimately shown itself to be right: every advance in epistemological and moral knowledge has reinstated the Sophists – Our contemporary way of thinking is to a great extent Heraclitean, Democritean, and Protagorean: it suffices to say it is Protagorean, because Protagoras represented a synthesis of Heraclitus and Democritus.” Nietzsche, The Will to Power, 2.428 Reappraisals of the authorship & dating of the Prometheia trilogy • Traditionally thought to have been composed by Aeschylus (c.525-c.456 BCE). • More recent scholarship has demonstrated the play to have been written by a later, lesser author sometime in the 430s. • This new dating raises many questions as to what contemporary events the trilogy may be referring. -
OUT of TOUCH: PHILOPONUS AS SOURCE for DEMOCRITUS Jaap
OUT OF TOUCH: PHILOPONUS AS SOURCE FOR DEMOCRITUS Jaap Mansfeld 1. Introduction The view that according to Democritus atoms cannot touch each other, or come into contact, has been argued by scholars, most recently and carefully by C.C.W. Taylor, especially in his very useful edition of the fragments of the early Atomists.1 I am not concerned here with the theoretical aspect of this issue, that is to say with the view that the early Atomists should have argued (or posthumously admitted) that atoms cannot touch each other because only the void that separates them prevents fusion. I wish to focus on the ancient evidence for this interpretation. Our only ancient source for this view happens to be Philoponus; to be more precise, one brief passage in his Commentary on Aristotle’s Physics (fr. 54c Taylor) and two brief passages in that on Aristotle’s On Generation and Corruption (54dand54eTaylor~67A7 DK). These commentaries, as is well known, are notes of Ammonius’ lectures, with additions by Philoponus himself. Bodnár has argued that this evidence is not good enough, because all other ancient sources, in the first place Aristotle, are eloquently silent on a ban on contact; what we have here therefore are ‘guesses of Philoponus, which are solely based on the text of Aristotle’.2 Taylor admits the force of this objection, but sticks to his guns: ‘perhaps’, 1 Taylor (1997) 222,(1999) 186–188, 192–193,(1999a) 184, cf. the discussion in Kline and Matheson (1987) and Godfrey (1990). On the problems related to the assumption that attraction plays an important part see further the cautious remarks of Morel (1996) 422–424, who however does not take into account the passages in Philoponus which suggest that atoms cannot touch each other. -
The “Atomic Theory” of Leucippus, and Its Impact on Medicine Before Hippocrates
Selected Brief Contribution The “atomic theory” of Leucippus, and its impact on medicine before Hippocrates Gregory Tsoucalas MD, MSc, MS(c), PhD, Konstantinos Laios PhD, Marianna Karamanou MD, MSc, PhD, George Androutsos MD History of Medicine Department, Medical School, University of Athens, Greece Gregory Tsoucalas, Kononos Str. 62-64, P.C. 11633, Pagrati, Athens, Greece. Mobile: 6945298205, Email: [email protected] Hell J Nucl Med 2013; 16(1): 68-69 Published on line: 10 April 2013 Leucippus, teacher of Democritus, was the first to describe Among Leucippus works stand the “Great Cosmology” (in the “atomic theory” in an attempt to explain the genesis of greek: Μέγας Διάκοσμος: About Great Cosmos) and On the the matter. Although only a fraction of his work was saved, Mind (in greek: Περί Νού). Of this second treatise, only one the “atom”, the smallest and indivisible particle of matter, but very significant fragment has survived, where we can enclosing a vacuum in which other particles move, still re- read: “nothing is in vain, but everything happens because mains the funtamental element of modern physics. of a reason and necessity”. This aphorism, not only encap- Leucippus (5th century BC, ca 430 BC) (Fig. 1), was born sulates the rationalism of the Ionian Greek School philoso- most probably in the city of Avdera, in Southern Thrace, phy, but also constitutes the basic axiom of ancient Greek Greece, where also Democritus was born. The island of Mi- medicine concerning the rational interpretation of diseases los, the Greek colony in Italy, Elea and the Greek city of Mile- and their therapy, which was later taught and described in tus in Asia Minor, were also mentioned as his birthplace [1- “Corpus Hippocraticum” [21]. -
An Introduction to Pythagorean Arithmetic
AN INTRODUCTION TO PYTHAGOREAN ARITHMETIC by JASON SCOTT NICHOLSON B.Sc. (Pure Mathematics), University of Calgary, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming tc^ the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1996 © Jason S. Nicholson, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my i department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Dale //W 39, If96. DE-6 (2/88) Abstract This thesis provides a look at some aspects of Pythagorean Arithmetic. The topic is intro• duced by looking at the historical context in which the Pythagoreans nourished, that is at the arithmetic known to the ancient Egyptians and Babylonians. The view of mathematics that the Pythagoreans held is introduced via a look at the extraordinary life of Pythagoras and a description of the mystical mathematical doctrine that he taught. His disciples, the Pythagore• ans, and their school and history are briefly mentioned. Also, the lives and works of some of the authors of the main sources for Pythagorean arithmetic and thought, namely Euclid and the Neo-Pythagoreans Nicomachus of Gerasa, Theon of Smyrna, and Proclus of Lycia, are looked i at in more detail. -
The Mathematical Anti-Atomism of Plato's Timaeus
The mathematical anti-atomism of Plato’s Timaeus Luc Brisson, Salomon Ofman To cite this version: Luc Brisson, Salomon Ofman. The mathematical anti-atomism of Plato’s Timaeus. Ancient Philoso- phy, Philosophy Documentation Center, In press. hal-02923266 HAL Id: hal-02923266 https://hal.archives-ouvertes.fr/hal-02923266 Submitted on 26 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The mathematical anti-atomism of Plato’s Timaeus Luc Brisson Salomon Ofman Centre Jean Pépin CNRS-Institut mathématique de Jussieu- CNRS-UMR 8230 Paris Rive Gauche École normale supérieure Sorbonne Université Paris Sciences Lettres Paris Université Abstract. In Plato’s eponymous dialogue, Timaeus, the main character presents the universe as an almost perfect sphere filled by tiny invisible particles having the form of four regular polyhedrons. At first glance, such a construction may seem close to an atomic theory. However, one does not find any text in Antiquity tying Timaeus’ cosmology to the atomists, and Aristotle made a clear distinction between Plato and the latter. Despite the cosmology in the Timaeus is so far apart from the one of the atomists, Plato is commonly presented in contemporary literature as some sort of atomist, sometimes as supporting a so-called form of ‘mathematical atomism’. -
A Short History of Greek Mathematics
Cambridge Library Co ll e C t i o n Books of enduring scholarly value Classics From the Renaissance to the nineteenth century, Latin and Greek were compulsory subjects in almost all European universities, and most early modern scholars published their research and conducted international correspondence in Latin. Latin had continued in use in Western Europe long after the fall of the Roman empire as the lingua franca of the educated classes and of law, diplomacy, religion and university teaching. The flight of Greek scholars to the West after the fall of Constantinople in 1453 gave impetus to the study of ancient Greek literature and the Greek New Testament. Eventually, just as nineteenth-century reforms of university curricula were beginning to erode this ascendancy, developments in textual criticism and linguistic analysis, and new ways of studying ancient societies, especially archaeology, led to renewed enthusiasm for the Classics. This collection offers works of criticism, interpretation and synthesis by the outstanding scholars of the nineteenth century. A Short History of Greek Mathematics James Gow’s Short History of Greek Mathematics (1884) provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. -
Pythagoras and the Pythagoreans1
Pythagoras and the Pythagoreans1 Historically, the name Pythagoras meansmuchmorethanthe familiar namesake of the famous theorem about right triangles. The philosophy of Pythagoras and his school has become a part of the very fiber of mathematics, physics, and even the western tradition of liberal education, no matter what the discipline. The stamp above depicts a coin issued by Greece on August 20, 1955, to commemorate the 2500th anniversary of the founding of the first school of philosophy by Pythagoras. Pythagorean philosophy was the prime source of inspiration for Plato and Aristotle whose influence on western thought is without question and is immeasurable. 1 c G. Donald Allen, 1999 ° Pythagoras and the Pythagoreans 2 1 Pythagoras and the Pythagoreans Of his life, little is known. Pythagoras (fl 580-500, BC) was born in Samos on the western coast of what is now Turkey. He was reportedly the son of a substantial citizen, Mnesarchos. He met Thales, likely as a young man, who recommended he travel to Egypt. It seems certain that he gained much of his knowledge from the Egyptians, as had Thales before him. He had a reputation of having a wide range of knowledge over many subjects, though to one author as having little wisdom (Her- aclitus) and to another as profoundly wise (Empedocles). Like Thales, there are no extant written works by Pythagoras or the Pythagoreans. Our knowledge about the Pythagoreans comes from others, including Aristotle, Theon of Smyrna, Plato, Herodotus, Philolaus of Tarentum, and others. Samos Miletus Cnidus Pythagoras lived on Samos for many years under the rule of the tyrant Polycrates, who had a tendency to switch alliances in times of conflict — which were frequent. -
Women in Early Pythagoreanism
Women in Early Pythagoreanism Caterina Pellò Faculty of Classics University of Cambridge Clare Hall February 2018 This dissertation is submitted for the degree of Doctor of Philosophy Alla nonna Ninni, che mi ha insegnato a leggere e scrivere Abstract Women in Early Pythagoreanism Caterina Pellò The sixth-century-BCE Pythagorean communities included both male and female members. This thesis focuses on the Pythagorean women and aims to explore what reasons lie behind the prominence of women in Pythagoreanism and what roles women played in early Pythagorean societies and thought. In the first chapter, I analyse the social conditions of women in Southern Italy, where the first Pythagorean communities were founded. In the second chapter, I compare Pythagorean societies with ancient Greek political clubs and religious sects. Compared to mainland Greece, South Italian women enjoyed higher legal and socio-political status. Similarly, religious groups included female initiates, assigning them authoritative roles. Consequently, the fact that the Pythagoreans founded their communities in Croton and further afield, and that in some respects these communities resembled ancient sects helps to explain why they opened their doors to the female gender to begin with. The third chapter discusses Pythagoras’ teachings to and about women. Pythagorean doctrines did not exclusively affect the followers’ way of thinking and public activities, but also their private way of living. Thus, they also regulated key aspects of the female everyday life, such as marriage and motherhood. I argue that the Pythagorean women entered the communities as wives, mothers and daughters. Nonetheless, some of them were able to gain authority over their fellow Pythagoreans and engage in intellectual activities, thus overcoming the female traditional domestic roles. -
Democritus and Protagoras
“Man is the measure of cosmos” Democritus and Protagoras: Values and Beliefs on Naturalism By Anastasia Theologou Submitted to Central European University Department of Philosophy In partial fulfillment of the requirements for the degree of Master of Arts Supervisor: Istvan Bodnar CEU eTD Collection Budapest, Hungary 2015 Table of contents ABSTRACT ....................................................................................................................................... iii ACKNOWLEDGEMENTS ................................................................................................................ iv Introduction .......................................................................................................................................... 1 1. Democritus’ philosophy in Diodorus Siculus .......................................................................... 4 1.1. Democritus’ cosmogony ................................................................................................. 6 1.2. The primary state of human beings ................................................................................. 8 1.3. Living the bestial life .................................................................................................... 10 1.4. Fear................................................................................................................................ 10 1.5. Language ....................................................................................................................... 11 -
Diogenes Laërtius: a Moderate Skeptic in the History of Philosophy (Book IX)∗
Philosophy Study, April 2021, Vol. 11, No. 4, 293-302 doi: 10.17265/2159-5313/2021.04.005 D D AV I D PUBLISHING Diogenes Laërtius: A Moderate Skeptic in the History of Philosophy (Book IX)∗ Ramón Román-Alcalá University of Córdoba, Córdoba, Spain This paper presents the keys and reasons for Diogenes Laërtius’ alleged scepticism, based on an analysis of the general design of his work The Lives and Opinions of Eminent Philosophers. I believe that it would be manifestly erroneous to seek confirmation of this scepticism solely in Book IX without taking into account the overall structure of the work. A convincing explanation is also provided of one of the most enigmatic and most studied phrases in this work. What did Diogenes mean when he said that Apollonides of Nicaea was ὁ παρ’ ἡμῶν (“one of us”)? Keywords: scepticism, Diogenes Laërtius, sextus, Pyrrho, The Lives of Philosophers The Singular Book IX of “The Lives” Diogenes Laërtius’ work Lives and Opinions of Eminent Philosophers was already relatively well-known in antiquity, due not only to the author’s popularity but also to the singularity of the work, in which he had succeeded in faithfully characterising much of ancient historiography. Moreover, and this is important to highlight in our introduction, taking into account the structure of many of the books and sources mentioned by Diogenes, this work began a new and highly original historiographic genre, which could be called “successions”. Diogenes’ Lives of Philosophers is a singular work. It starts with an introduction focusing on the origins of philosophy (which were undoubtedly Greek), the different classifications of philosophy and philosophers. -
The Project Gutenberg Ebook #31061: a History of Mathematics
The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A History of Mathematics Author: Florian Cajori Release Date: January 24, 2010 [EBook #31061] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber's note Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for screen viewing, but may be easily formatted for printing. Please consult the preamble of the LATEX source file for instructions. A HISTORY OF MATHEMATICS A HISTORY OF MATHEMATICS BY FLORIAN CAJORI, Ph.D. Formerly Professor of Applied Mathematics in the Tulane University of Louisiana; now Professor of Physics in Colorado College \I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."|J. W. L. Glaisher New York THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1909 All rights reserved Copyright, 1893, By MACMILLAN AND CO. Set up and electrotyped January, 1894. Reprinted March, 1895; October, 1897; November, 1901; January, 1906; July, 1909. Norwood Pre&: J. S. Cushing & Co.|Berwick & Smith.