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What Math Can Tell Us About the Nature of the Universe
On Monday, September 13, 2013, the New York Times published my opinion piece “How to Fall in Love with Math.” I awoke to find my email inbox overflowing with messages, not just from acquaintances but also from a bewildering number of strangers.
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Many more responses poured in online on the Times website—some enthusiastic about math, some scathing, all impassioned—clearly, I’d struck a nerve. By mid-afternoon, the number of posts had reached 360, and the paper closed the comments section. The article quickly climbed to the top of the Times’s most-emailed list and remained there for much of the next day.
The aim of my piece was to challenge the popular notion that mathematics is synonymous with calculation. Starting with arithmetic and proceeding through algebra and beyond, the message drummed into our heads as students is that we do math to “get the right answer.” The drill of multiplication tables, the drudgery of long division, the quadratic formula and its memorization—these are the dreary memories many of us carry around from school as a result.
But what if we liberated ourselves from the stress of finding “the right answer”? What would math look like if delinked from this calculation-driven motivation? What, if anything, would remain of the subject?
What would math look like if delinked from this calculation-driven motivation?The answer is ideas. That’s what mathematics is truly about, the realm where it really comes alive. Ideas that engage and intrigue us as humans, that help us understand the universe. Ideas about the perfection of numbers, the nature of space and geometry, the spontaneous formation of patterns, the origins of randomness and infinity. The neat thing is that such ideas can be enjoyed without needing any special mathematical knowledge or being a computation whiz.
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The Social Brain: Sociological Foundations 1666927058, 9781666927054
Table of contents :
Dedication
Contents
Acknowledgments
Preface: Caveat Emptor
Introduction
1. The Sociological Imagination
2. Sociology Comes into View
3. Evolution Invents the Social
4. Individual and Society
5. The Social Body in Society and Politics
6. Genius Incorporated
7. Improvisation Incorporated
8. From the Matrix to Reality
9. Robots, AI, Brains, and Bodies in the Information Age
10. The Sociology of Consciousness
11. The Social Life of the Brain
12. The Social Brain in Health and Illness
13. Final Considerations
Bibliography
Index
About the Author
Citation preview
The Social Brain Aravind Asok We discuss elements of a social history of the theory of projective modules over commutative rings. We attempt to study the question: how did the theory of projective modules become one of “mainstream” focus in mathematics? To do this, we begin in what one might call the pre-history of projective modules, describing the mathematical culture into which the notion of projective module was released. These recollections involve four pieces: (a) analyzing aspects of the theory of fiber bundles, as it impinges on algebraic geometry, (b) understanding the rise of homological techniques in algebraic topology, (c) describing the influence of category-theoretic ideas in topology and algebra and (d) revisiting the story of the percolation of sheaf-theoretic ideas through algebraic geometry. We will then argue that it was this unique confluence of mathematical events that allowed projective modules to emerge as objects of central mathematical importance. More precisely, we will first argue that, in the context of social currents of the time, projective modules initially were isolated as objects of purely technical convenience reflecting the aesthetic sensibilities of the creators of the fledgling theory of homological algebra. Only later did they transcend this limited role to become objects of “mainstream importance” due to influence from the theory of algebraic fiber bundles and the theory of sheaves. Along the way, we aim to show how strong personal ties emanating from the Bourbaki movement and its connections in mathematical centers including Paris, Princeton and Chicago were essential to the entrance, propagation and mainstream mathematical acceptance of the theory. Guidance by abstract ideas is a dangerous business when not controlled by strong personal relations. Paul Feyerabend, The Tyranny of Science This text grew from my attempt first to justify, bu Chinese astronomer and mathematician (1231–1316) Guo Shoujing Xingtai, Hebei province Guo Shoujing (Chinese: 郭守敬, 1231–1316), courtesy name Ruosi (若思), was a Chinese astronomer,hydraulic engineer, mathematician, and politician of the Yuan dynasty. The later Johann Adam Schall von Bell (1591–1666) was so impressed with the preserved astronomical instruments of Guo that he called him "the Tycho Brahe of China."Jamal ad-Din cooperated with him. In 1231, in Xingtai, Hebei province, China, Guo Shoujing was born into a poor family. He was raised primarily by his paternal grandfather, Guo Yong, who was famous throughout China for his expertise in a wide variety of topics, ranging from the study of the Five Classics to astronomy, mathematics, and hydraulics. Guo Shoujing was a child prodigy, showing exceptional intellectual promise. By his teens, he obtained a blueprint for a water clock which his grandfather was working on, and realized its principles of operation. He improved the design of a type of water clock called a lotus clepsydra, a water clock with a bowl shaped like a lotus flower on the top into which the water dripped. After he had mastered the construction of such water clocks, he began to study mathematics at the age of 16. From mathematics, he began to understand hydraulics, as well as astronomy. At 20, Guo became a hydraulic engineer. In 1251, as a government official, he helped repair a bridge over the Dahuoquan River. Kublai realized the importance of hydraulic engineering, irrigation, and water transport, which he believed could help alleviate uprisings within the empire, and sent
The Social Brain Sociological Foundations Sal Restivo
LEXINGTON BOOKS
Lanham • Boulder • New York • London
Published by Lexington Books An imprint of The Rowman & Littlefield Publishing Group, Inc. 4501 Forbes Boulevard, Suite 200, Lanham, Maryland 20706 www.rowman.com 86-90 Paul Street, London EC2A 4NE Copyright © 2023 by The Rowman & Littlefield Publishing Group, Inc. All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without written permission from the publisher, except by a reviewer who may quote passages in a review. British Library Cataloguing in Publication Information Available Library of Congress Cataloging-in-Publication Data Available ISBN 978-1-66692-705-4 (cloth ; alk. paper) ISBN 978-1-66692-706-1 (electronic) The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI/NISO Z39.48-1992.
Dedicated to my wife Ya Wen
Contents
Acknowledgments ix Preface: Caveat Emptor
xi
Introduction xv Chapter 1: The Sociological Imagination
1
Chapter 2: Sociology Comes into View
13
Chapter 3: Evolution Invents the Social
27
Chapter 4: Constructing projective modules
elements of a social historyAbstract
1 Prolegomena
Guo Shoujing
Born 1231 Died 1314 or 1316 Known for Shòushí Calendar (授时曆; 'Season-Granting Calendar') Scientific career Fields Astronomy, hydraulic engineering, mathematics Institutions Gaocheng Astronomical Observatory Early life
Career