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Biography Paper – Georg Cantor
Mike Garkie Math 4010 – History of Math UCD Denver 4/1/08 Biography Paper – Georg Cantor Few mathematicians are house-hold names; perhaps only Newton and Euclid would qualify. But there is a second tier of mathematicians, those whose names might not be familiar, but whose discoveries are part of everyday math. Examples here are Napier with logarithms, Cauchy with limits and Georg Cantor (1845 – 1918) with sets. In fact, those who superficially familier with Georg Cantor probably have two impressions of the man: First, as a consequence of thinking about sets, Cantor developed a theory of the actual infinite. And second, that Cantor was a troubled genius, crippled by Freudian conflict and mental illness. The first impression is fundamentally true. Cantor almost single-handedly overturned the Aristotle’s concept of the potential infinite by developing the concept of transfinite numbers. And, even though Bolzano and Frege made significant contributions, “Set theory … is the creation of one person, Georg Cantor.” [4] The second impression is mostly false. Cantor certainly did suffer from mental illness later in his life, but the other emotional baggage assigned to him is mostly due his early biographers, particularly the infamous E.T. Bell in Men Of Mathematics [7]. In the racially charged atmosphere of 1930’s Europe, the sensational story mathematician who turned the idea of infinity on its head and went crazy in the process, probably make for good reading. The drama of the controversy over Cantor’s ideas only added spice. 1 Fortunately, modern scholars have corrected the errors and biases in older biographies. -
Historical Background Italy, Due to the Threat There from Throw the Entire Balance Over in the Following the Defeat of Hasdrubal Barca Hannibal
• The numerical superiority they enjoyed with their new mercenaries; • The superior quality of their legions, probably the finest in the Roman army; and, • Overconfidence bred from seven years of campaigning without a serious defeat. Had the Scipios actually faced only 35,000 Carthaginians with over 50,000 legionnaires and mercenaries as they believed, their chances for success would have been good. But Hasdrubal Barca had two additional detachments: 3,000 Numidian cavalry under Masinissa and 7,500 warriors under Indibilis. And Hasdrubal Barca also were unable to obtain more troops from had a trick up his sleeve that was to Historical Background Italy, due to the threat there from throw the entire balance over in the Following the defeat of Hasdrubal Barca Hannibal. Instead, the Scipios hired on a favor of Carthage. at Dertosa (see issue Nr. 4 of C3i for large body of 20,000 Celt-Iberian Dertosa Battle Module) by the Scipio mercenaries. The Celt-Iberians were a While Hasdrubal Barca observed the brothers in 215 BC, Carthage responded mix of those two peoples, found mainly Romans from his position at Amtorgis, by sending reinforcements. Two armies in the wilds of central Spain. They had a he ordered the forces of Hasdrubal were dispatched, one under Hasdrubal's reputation for ferocity and fighting skill. Gisgo, Masinissa and Indibilis to younger brother Mago, and another Both sides confidently planned to take concentrate at Mago Barca's camp under a political rival of the Barca clan, the offensive in 211 BC. near Castulo. Once these forces were Hasdrubal Gisgo. For the next three united, it appears he intended to move years (214-212 BC), the three Publius and Gnaeus Scipio knew that north against the Romans with his Carthaginian armies battled the two Hasdrubal Barca was encamped north combined forces. -
Female Property Ownership and Status in Classical and Hellenistic Sparta
Female Property Ownership and Status in Classical and Hellenistic Sparta Stephen Hodkinson University of Manchester 1. Introduction The image of the liberated Spartiate woman, exempt from (at least some of) the social and behavioral controls which circumscribed the lives of her counterparts in other Greek poleis, has excited or horrified the imagination of commentators both ancient and modern.1 This image of liberation has sometimes carried with it the idea that women in Sparta exercised an unaccustomed influence over both domestic and political affairs.2 The source of that influence is ascribed by certain ancient writers, such as Euripides (Andromache 147-53, 211) and Aristotle (Politics 1269b12-1270a34), to female control over significant amounts of property. The male-centered perspectives of ancient writers, along with the well-known phenomenon of the “Spartan mirage” (the compound of distorted reality and sheer imaginative fiction regarding the character of Spartan society which is reflected in our overwhelmingly non-Spartan sources) mean that we must treat ancient images of women with caution. Nevertheless, ancient perceptions of their position as significant holders of property have been affirmed in recent modern studies.3 The issue at the heart of my paper is to what extent female property-holding really did translate into enhanced status and influence. In Sections 2-4 of this paper I shall approach this question from three main angles. What was the status of female possession of property, and what power did women have directly to manage and make use of their property? What impact did actual or potential ownership of property by Spartiate women have upon their status and influence? And what role did female property-ownership and status, as a collective phenomenon, play within the crisis of Spartiate society? First, however, in view of the inter-disciplinary audience of this volume, it is necessary to a give a brief outline of the historical context of my discussion. -
The 1761 Transit of Venus and Hell's Transition to Fame
Chapter 3 A New Node of Science in Action: The 1761 Transit of Venus and Hell’s Transition to Fame I reckon there is no one interested in astronomy who does not wait impa- tiently to learn what was observed during the recent meeting of Venus with the Sun, especially since there is no other encounter between celes- tial bodies from which we are able to ascertain with a greater degree of exactness the still unknown, or not yet sufficiently well defined, paral- laxes of the Sun and Venus. Eustachio Zanotti 17611 1 A Golden Opportunity The above assessment of Hell’s Bolognese colleague Eustachio Zanotti could hardly have been more to the point. The passage (or transit) of Venus in front of the Sun as seen from the Earth is a rare astronomical phenomenon: it comes in pairs separated by eight years, after which it does not take place for more than a whole century. The first transit of Venus observed by means of astro- nomical equipment was in 1639. Since then, transits of Venus have occurred in the years 1761 and 1769, 1874 and 1882, and 2004 and 2012—but they will not happen again until 2117 and 2125. The 1639 transit of Venus made no immediate impact and (as far is known) was only observed by two amateur astronomers in the English countryside.2 By contrast, the pre-calculated transits of 1761 and 1 Eustachio Zanotti, De Veneris ac Solis Congressu Observatio habita in Astronomico Specula Bononiensis Scientiarum Instituti Die 5 Junii mdcclxi (Bologna: Laelii e Vulpe, 1761), 1. -
An Analysis of Foreign Policy Motivation in the Peloponnesian War
Review of International Studies (2001), 27, 69–90 Copyright © British International Studies Association ‘Chiefly for fear, next for honour, and lastly for profit’: an analysis of foreign policy motivation in the Peloponnesian War WILLIAM O. CHITTICK AND ANNETTE FREYBERG-INAN Abstract. This article applies a three-dimensional framework for the analysis of the role of motivation in foreign policy decision-making to the foreign policy decisions of individuals and cities in Thucydides’ History of the Peloponnesian War. First, the authors briefly intro- duce their framework for analysis. Using the speeches in Thucydides to explicate the motives and goals of individuals and cities, the authors then trace the relationships between the motivational dispositions of foreign policy actors and their foreign policy behaviour. In so doing, they demonstrate both the relevance of a concern with individual motivation for foreign policy analysis and the usefulness of their analytical framework for studying the impact of the relevant motives. The authors also show how ideological statements can be ana- lysed to determine the relative salience of individual motives and collective goals, suggesting a relationship between ideological reasoning and motivational imbalance which can adversely affect the policymaking process. In conclusion, they briefly assess the theoretical and norma- tive as well as practical policy implications of their observations. Introduction The study of foreign policy entails the analysis of human reactions to the threats, challenges, and opportunities presented by the international environment. However, foreign policy decision-makers do not react directly to situations or events. Instead, they react according to the ways in which they perceive and interpret those situations and events. -
A Glimpse Into the Roman Finances of the Second Punic War Through
Letter Geochemical Perspectives Letters the history of the western world. Carthage was a colony founded next to modern Tunis in the 8th century BC by Phoenician merchants. During the 3rd century BC its empire expanded westward into southern Spain and Sardinia, two major silver producers of the West Mediterranean. Meanwhile, Rome’s grip had tight- © 2016 European Association of Geochemistry ened over the central and southern Italian peninsula. The Punic Wars marked the beginning of Rome’s imperial expansion and ended the time of Carthage. A glimpse into the Roman finances The First Punic War (264 BC–241 BC), conducted by a network of alliances in Sicily, ended up with Rome prevailing over Carthage. A consequence of this of the Second Punic War conflict was the Mercenary War (240 BC–237 BC) between Carthage and its through silver isotopes unpaid mercenaries, which Rome helped to quell, again at great cost to Carthage. Hostilities between the two cities resumed in 219 BC when Hannibal seized the F. Albarède1,2*, J. Blichert-Toft1,2, M. Rivoal1, P. Telouk1 Spanish city of Saguntum, a Roman ally. At the outbreak of the Second Punic War, Hannibal crossed the Alps into the Po plain and inflicted devastating mili- tary defeats on the Roman legions in a quick sequence of major battles, the Trebia (December 218 BC), Lake Trasimene (June 217 BC), and Cannae (August 216 BC). As a measure of the extent of the disaster, it was claimed that more than 100,000 Abstract doi: 10.7185/geochemlet.1613 Roman soldiers and Italian allies lost their lives in these three battles, including The defeat of Hannibal’s armies at the culmination of the Second Punic War (218 BC–201 three consuls. -
Bernhard Riemann 1826-1866
Modern Birkh~user Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkh~iuser in recent decades have been groundbreaking and have come to be regarded as foun- dational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain ac- cessible to new generations of students, scholars, and researchers. BERNHARD RIEMANN (1826-1866) Bernhard R~emanno 1826 1866 Turning Points in the Conception of Mathematics Detlef Laugwitz Translated by Abe Shenitzer With the Editorial Assistance of the Author, Hardy Grant, and Sarah Shenitzer Reprint of the 1999 Edition Birkh~iuser Boston 9Basel 9Berlin Abe Shendtzer (translator) Detlef Laugwitz (Deceased) Department of Mathematics Department of Mathematics and Statistics Technische Hochschule York University Darmstadt D-64289 Toronto, Ontario M3J 1P3 Gernmany Canada Originally published as a monograph ISBN-13:978-0-8176-4776-6 e-ISBN-13:978-0-8176-4777-3 DOI: 10.1007/978-0-8176-4777-3 Library of Congress Control Number: 2007940671 Mathematics Subject Classification (2000): 01Axx, 00A30, 03A05, 51-03, 14C40 9 Birkh~iuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkh~user Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. -
The Coins from the Necropolis "Metlata" Near the Village of Rupite
margarita ANDONOVA the coins from the necropolis "metlata" near the village of rupite... THE COINS FROM THE NECROPOLIS METLATA NEAR THE VILLAGE "OF RUPITE" (F. MULETAROVO), MUNICIPALITY OF PETRICH by Margarita ANDONOVA, Regional Museum of History– Blagoevgrad This article sets to describe and introduce known as Charon's fee was registered through the in scholarly debate the numismatic data findspots of the coins on the skeleton; specifically, generated during the 1985-1988 archaeological these coins were found near the head, the pelvis, excavations at one of the necropoleis situated in the left arm and the legs. In cremations in situ, the locality "Metlata" near the village of Rupite. coins were placed either inside the grave or in The necropolis belongs to the long-known urns made of stone or clay, as well as in bowls "urban settlement" situated on the southern placed next to them. It is noteworthy that out of slopes of Kozhuh hill, at the confluence of 167 graves, coins were registered only in 52, thus the Strumeshnitsa and Struma Rivers, and accounting for less than 50%. The absence of now identified with Heraclea Sintica. The coins in some graves can probably be attributed archaeological excavations were conducted by to the fact that "in Greek society, there was no Yulia Bozhinova from the Regional Museum of established dogma about the way in which the History, Blagoevgrad. souls of the dead travelled to the realm of Hades" The graves number 167 and are located (Зубарь 1982, 108). According to written sources, within an area of 750 m². Coins were found mainly Euripides, it is clear that the deceased in 52 graves, both Hellenistic and Roman, may be accompanied to the underworld not only and 10 coins originate from areas (squares) by Charon, but also by Hermes or Thanatos. -
Names of Botanical Genera Inspired by Mythology
Names of botanical genera inspired by mythology Iliana Ilieva * University of Forestry, Sofia, Bulgaria. GSC Biological and Pharmaceutical Sciences, 2021, 14(03), 008–018 Publication history: Received on 16 January 2021; revised on 15 February 2021; accepted on 17 February 2021 Article DOI: https://doi.org/10.30574/gscbps.2021.14.3.0050 Abstract The present article is a part of the project "Linguistic structure of binomial botanical denominations". It explores the denominations of botanical genera that originate from the names of different mythological characters – deities, heroes as well as some gods’ attributes. The examined names are picked based on “Conspectus of the Bulgarian vascular flora”, Sofia, 2012. The names of the plants are arranged in alphabetical order. Beside each Latin name is indicated its English common name and the family that the particular genus belongs to. The article examines the etymology of each name, adding a short account of the myth based on which the name itself is created. An index of ancient authors at the end of the article includes the writers whose works have been used to clarify the etymology of botanical genera names. Keywords: Botanical genera names; Etymology; Mythology 1. Introduction The present research is a part of the larger project "Linguistic structure of binomial botanical denominations", based on “Conspectus of the Bulgarian vascular flora”, Sofia, 2012 [1]. The article deals with the botanical genera appellations that originate from the names of different mythological figures – deities, heroes as well as some gods’ attributes. According to ICBN (International Code of Botanical Nomenclature), "The name of a genus is a noun in the nominative singular, or a word treated as such, and is written with an initial capital letter (see Art. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Queen Arsinoë II, the Maritime Aphrodite and Early Ptolemaic Ruler Cult
ΑΡΣΙΝΟΗ ΕΥΠΛΟΙΑ Queen Arsinoë II, the Maritime Aphrodite and Early Ptolemaic Ruler Cult Carlos Francis Robinson Bachelor of Arts (Hons. 1) A thesis submitted for the degree of Master of Philosophy at The University of Queensland in 2019 Historical and Philosophical Inquiry Abstract Queen Arsinoë II, the Maritime Aphrodite and Early Ptolemaic Ruler Cult By the early Hellenistic period a trend was emerging in which royal women were deified as Aphrodite. In a unique innovation, Queen Arsinoë II of Egypt (c. 316 – 270 BC) was deified as the maritime Aphrodite, and was associated with the cult titles Euploia, Akraia, and Galenaië. It was the important study of Robert (1966) which identified that the poets Posidippus and Callimachus were honouring Arsinoë II as the maritime Aphrodite. This thesis examines how this new third-century BC cult of ‘Arsinoë Aphrodite’ adopted aspects of Greek cults of the maritime Aphrodite, creating a new derivative cult. The main historical sources for this cult are the epigrams of Posidippus and Callimachus, including a relatively new epigram (Posidippus AB 39) published in 2001. This thesis demonstrates that the new cult of Arsinoë Aphrodite utilised existing traditions, such as: Aphrodite’s role as patron of fleets, the practice of dedications to Aphrodite by admirals, the use of invocations before sailing, and the practice of marine dedications such as shells. In this way the Ptolemies incorporated existing religious traditions into a new form of ruler cult. This study is the first attempt to trace the direct relationship between Ptolemaic ruler cult and existing traditions of the maritime Aphrodite, and deepens our understanding of the strategies of ruler cult adopted in the early Hellenistic period. -
Math 126 Lecture 4. Basic Facts in Representation Theory
Math 126 Lecture 4. Basic facts in representation theory. Notice. Definition of a representation of a group. The theory of group representations is the creation of Frobenius: Georg Frobenius lived from 1849 to 1917 Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups, the representation theory of groups and the character theory of groups. Find out more at: http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Frobenius.html Matrix form of a representation. Equivalence of two representations. Invariant subspaces. Irreducible representations. One dimensional representations. Representations of cyclic groups. Direct sums. Tensor product. Unitary representations. Averaging over the group. Maschke’s theorem. Heinrich Maschke 1853 - 1908 Schur’s lemma. Issai Schur Biography of Schur. Issai Schur Born: 10 Jan 1875 in Mogilyov, Mogilyov province, Russian Empire (now Belarus) Died: 10 Jan 1941 in Tel Aviv, Palestine (now Israel) Although Issai Schur was born in Mogilyov on the Dnieper, he spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He went to Latvia at the age of 13 and there he attended the Gymnasium in Libau, now called Liepaja. In 1894 Schur entered the University of Berlin to read mathematics and physics. Frobenius was one of his teachers and he was to greatly influence Schur and later to direct his doctoral studies. Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices. This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius.