148289397.Pdf

Total Page:16

File Type:pdf, Size:1020Kb

148289397.Pdf Through a mind darkly An empirically-informed philosophical perspective on systematic knowledge acquisition and cognitive limitations Helen De Cruz c Copyright by Helen De Cruz, 2011. All rights reserved. PhD-dissertation of University of Groningen Title: Through a mind darkly. An empirically-informed philosophical perspective on systematic knowledge acquisition and cognitive limita- tions. Author: H. L. De Cruz ISBN 978-90-367-5182-7 (ISBN print version: 978-90-367-5181-0) Publisher: University of Groningen, The Netherlands Printed by: Reproduct, Ghent, Belgium Cover illustration: The externalized thinker. This image is based on Au- gust Rodin’s bronze sculpture, The Thinker (Le Penseur, 1880). It il- lustrates one of the main theses of this dissertation: cognition is not a solitary and internalized process, but a collective, distributed and exter- nalized activity. RIJKSUNIVERSITEIT GRONINGEN Through a mind darkly An empirically-informed philosophical perspective on systematic knowledge acquisition and cognitive limitations Proefschrift ter verkrijging van het doctoraat in de Wijsbegeerte aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. E. Sterken in het openbaar te verdedigen op donderdag 27 oktober 2011 om 16.15 uur door Helen Lucretia De Cruz geboren op 1 september 1978 te Gent Promotor: Prof. dr. mr. I. E. Douven Beoordelingscommissie: Prof. dr. M. Muntersbjorn Prof. dr. J. Peijnenburg Prof. dr. L. Horsten CONTENTS List of figures ix List of tables xi List of papers xiii Acknowledgments xv Preface xvii 1 Introduction 1 1.1 Two puzzles on scientific knowledge ............ 1 1.1.1 Naturalism and knowledge acquisition ....... 1 1.1.2 Puzzle 1: The boundedness of human reasoning .. 3 1.1.3 Puzzle 2: Scientific reasoning ............ 8 1.2 The nature of scientific beliefs ................ 11 1.2.1 Core knowledge .................... 12 1.2.2 Intuitive and reflective beliefs ............ 15 1.2.3 Explanatory depth .................. 18 1.2.4 The role of testimony ................ 19 1.3 Discontinuity or continuity? ................. 26 1.3.1 Science as the result of deliberate practice ..... 27 1.3.2 Children as scientists ................. 31 1.3.3 Cognitive development as conceptual change ... 33 iii iv CONTENTS 1.3.4 Cognition in the wild ................ 38 I Mathematics 45 2 Innateness and mathematical concepts 47 2.1 Nativism and mathematical knowledge ........... 48 2.1.1 Historical claims ................... 48 2.1.2 Current positions in philosophy of mathematics .. 50 2.2 The case of arithmetic .................... 51 2.2.1 Premise 1: Infants’ looking time di↵ers between correct and incorrect arithmetical operations ... 55 2.2.2 Premise 2: This success is best explained by the infants’ conceptual knowledge of number ...... 56 2.2.3 Premise 3: Because the capacity arises early in de- velopment, it cannot have been learned through ex- perience ........................ 58 2.2.4 Conclusion: The property in question is probably innate ......................... 59 2.3 From intuitive to formal mathematical knowledge ..... 60 2.3.1 Characterizing intuitive numbers formally ..... 61 2.3.2 Learning natural numbers through axiomatic systems 65 2.4 Innate skills and mathematical practice .......... 67 2.5 Concluding remarks ..................... 71 3 Evolutionary perspective on number concepts 73 3.1 The cultural evolution of numerical concepts ....... 74 3.2 Cognitive modularity and culture .............. 75 3.2.1 Cognitive modularity ................. 75 3.2.2 Epidemiological approaches to culture ....... 81 3.3 Number as the proper domain of a conceptual module .. 85 3.3.1 Numerical competence in nonhuman animals, hu- man infants, and adults ............... 85 3.3.2 The neural architecture underlying numerical com- petence ........................ 88 3.4 The epidemiology of numerical concepts .......... 92 3.4.1 The positive integers ................. 92 3.4.2 Zero .......................... 96 CONTENTS v 3.4.3 Negative numbers .................. 101 3.5 Conclusion .......................... 103 4 The extended mind and natural numbers 105 4.1 The bounds of cognition ................... 106 4.2 Domain-specificity in numerical representations ...... 108 4.3 Two cognitive routes for number words ........... 112 4.3.1 Counting ....................... 112 4.3.2 Approximate number words ............. 113 4.4 External media and cognitive processes .......... 115 4.4.1 Number words and language ............ 116 4.4.2 Body parts ...................... 120 4.4.3 Tallies and tokens .................. 122 4.4.4 Numerical notation systems ............. 125 4.4.5 Gestures ........................ 127 4.5 Discussion and concluding remarks ............. 128 5 Mathematical symbols as epistemic actions 131 5.1 Introduction .......................... 132 5.2 How we acquire mathematical knowledge ......... 132 5.3 Elementary numerical knowledge .............. 134 5.4 Symbols and mathematical cognition ............ 136 5.4.1 External media in mathematical cognition ..... 136 5.4.2 Costs and benefits of symbol use .......... 140 5.5 Specific properties of mathematical symbols ........ 143 5.5.1 Negative numbers .................. 145 5.5.2 Algebra ........................ 146 5.6 The opacity of mathematical symbols ........... 151 II Science 153 6 Intuitive ontologies in scientific understanding 155 6.1 Introduction .......................... 156 6.2 Intuitive ontologies ...................... 157 6.2.1 What are intuitive ontologies? ........... 157 6.2.2 Neural underpinnings of intuitive ontologies .... 163 6.3 Intuitive ontologies and folk theories ............ 165 6.4 Epistemological limitations to intuitive ontologies ..... 166 vi CONTENTS 6.5 Intuitive ontologies and scientific understanding ...... 167 6.6 Theories on human evolution ................ 170 6.6.1 Pruning and straightening the bushy tree of human evolution ....................... 171 6.6.2 Essentialism and humanized apes .......... 175 6.7 Concluding reflections .................... 179 7 Science as structured imagination 181 7.1 Introduction .......................... 182 7.2 Structured imagination .................... 182 7.2.1 Creativity is structured ............... 182 7.2.2 Analogies in everyday creative thought ....... 184 7.3 Intuitive ontologies and scientific reasoning ........ 185 7.3.1 Intelligibility ..................... 185 7.3.2 Analogies and scientific creativity .......... 186 7.4 Distant analogies as a source of creativity ......... 189 7.4.1 Early modern physiology .............. 190 7.4.2 Early evolutionary biology .............. 193 7.4.3 The evolution of the human mind ......... 195 7.5 Concluding reflections .................... 199 8 The epistemic status of scientific beliefs 201 8.1 Introduction .......................... 202 8.2 Cognitive biases and the perception of reality ....... 203 8.3 Evolutionary arguments ................... 207 8.4 Evolutionary debunking arguments ............. 209 8.5 Cultural transmission of scientific knowledge ........ 213 8.6 18th- and 19th-century transmutation theories ...... 218 9 Evolution and justification 225 9.1 The evolved mind and epistemic justification ....... 226 9.2 Cartesian God or Cartesian demon ............. 227 9.2.1 Evolutionary arguments generalized ........ 229 9.2.2 Evolutionary debunking arguments generalized .. 230 9.3 Responses to the circularity charge ............. 236 9.3.1 Dodging the bullet .................. 236 9.3.2 Biting the bullet ................... 237 9.4 Extended cognition and evolved cognitive biases ..... 241 CONTENTS vii 9.5 Conclusion: A defeasible evolutionary account ...... 245 10 Conclusion 247 10.1 General conclusion ...................... 247 10.2 Normative implications ................... 248 10.3 Metaphysical implications .................. 253 10.4 Naturalism as philosophy of the gaps? ........... 254 Summary 261 Samenvatting 263 Bibliography 267 LIST OF FIGURES 1.1 Dual Earth .......................... 35 2.1 Simple addition and subtraction tasks ........... 54 2.2 Two views of intuitive numerosity ............. 65 3.1 Patterns of brain activity involved in calculation ..... 91 3.2 Linguistic representation of numbers ............ 96 3.3 Historical spread of the number zero with its symbolic rep- resentations throughout the Old World. .......... 99 4.1 The Oksapmin counting system ............... 121 4.2 Notched bone from Ishango, Congo ............. 123 4.3 Brain areas involved in calculation in Chinese and English speakers ............................ 124 5.1 Representing simultaneous equations with counting rods . 148 7.1 Anatomical drawing ..................... 191 8.1 Common yellow wood-sorrel (Oxalis stricta) under normal lighting conditions and photographed using an ultraviolet- sensitive camera ....................... 205 8.2 Relationship between values of N and the complexity of biological theory that can be maintained .......... 221 ix LIST OF TABLES 4.1 The Greek alphabetic system of numerals ......... 126 5.1 Number of operations required to solve a system of n equa- tions with n unknowns .................... 150 8.1 Costs and benefits of agency detection ........... 210 9.1 Generalized forms of EA and EDA. ............. 228 xi LIST OF PAPERS INCORPORATED IN THE TEXT De Cruz, H. & De Smedt, J. (2010a). The innateness hypothesis and mathematical concepts. Topoi, 29, 3–13. De Cruz, H. (2006). Why are some numerical concepts more successful than others? An evolutionary perspective on the history of number concepts. Evolution and Human Behavior, 27,
Recommended publications
  • Some Interesting Facts, Myths and History of Mathematics
    International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 4 Issue 6 || August. 2016 || PP-54-68 Some Interesting Facts, Myths and History of Mathematics Singh Prashant1 1(Department of Computer Science, Institute of Science, Banaras Hindu University) ABSTRACT : This paper deals with primary concepts and fallacies of mathematics which many a times students and even teachers ignore. Also this paper comprises of history of mathematical symbols, notations and methods of calculating time. I have also included some ancient techniques of solving mathematical real time problems. This paper is a confluence of various traditional mathematical techniques and their implementation in modern mathematics. I. INTRODUCTION I have heard my father saying that ―Mathematics is the only genuine subject as it does not change with boundary of countries‖. It is lucrative just because of its simplicity. Galileo once said, ―Mathematics is the language with which God wrote the Universe.‖ He was precise in calling mathematics a language, because like any dialect, mathematics has its own rubrics, formulas, and nuances. In precise, the symbols used in mathematics are quite unique to its field and are profoundly engrained in history. The following will give an ephemeral history of some of the greatest well-known symbols employed by mathematics. Categorized by discipline within the subject, each section has its own interesting subculture surrounding it. Arithmetic is the most rudimentary part of mathematics and covers addition, subtraction, multiplication, and the division of numbers. One category of numbers are the integers, -n,…-3,-2,-1,0,1,2,3,…n , where we say that n is in .The capital letter Z is written to represent integers and comes from the German word, Zahlen, meaning numbers.
    [Show full text]
  • The Kriyākramakarī's Integrative Approach to Mathematical Knowledge
    History of Science in South Asia A journal for the history of all forms of scientific thought and action, ancient and modern, in all regions of South Asia The Kriyākramakarī’s Integrative Approach to Mathematical Knowledge Roy Wagner Eidgenössische Technische Hochschule Zürich MLA style citation form: Roy Wagner. “The Kriyākramakarī’s Integrative Approach to Mathematical Knowl- edge.” History of Science in South Asia, 6 (2018): 84–126. doi: 10.18732/hssa.v6i0.23. Online version available at: http://hssa-journal.org HISTORY OF SCIENCE IN SOUTH ASIA A journal for the history of all forms of scientific thought and action, ancient and modern, inall regions of South Asia, published online at http://hssa-journal.org ISSN 2369-775X Editorial Board: • Dominik Wujastyk, University of Alberta, Edmonton, Canada • Kim Plofker, Union College, Schenectady, United States • Dhruv Raina, Jawaharlal Nehru University, New Delhi, India • Sreeramula Rajeswara Sarma, formerly Aligarh Muslim University, Düsseldorf, Germany • Fabrizio Speziale, Université Sorbonne Nouvelle – CNRS, Paris, France • Michio Yano, Kyoto Sangyo University, Kyoto, Japan Publisher: History of Science in South Asia Principal Contact: Dominik Wujastyk, Editor, University of Alberta Email: ⟨[email protected]⟩ Mailing Address: History of Science in South Asia, Department of History and Classics, 2–81 HM Tory Building, University of Alberta, Edmonton, AB, T6G 2H4 Canada This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. Copyrights of all the articles rest with the respective authors and published under the provisions of Creative Commons Attribution-ShareAlike 4.0 License.
    [Show full text]
  • Numbers 1 to 100
    Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number)
    [Show full text]
  • Development of Infinite Series in India
    International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2015): 78.96 | Impact Factor (2015): 6.391 Development of Infinite Series in India Sanjay M. Deshpande1, Dr. Anant W. Vyawahare2 1Professor, Department of Mathematics, Bhawabhuti Mahavidyalaya, Amgaon, Dist. Gondia, Maharashtra, India, Pin 441 902 249 Gajanan Nagar , Wardha Road, NAGPUR, Maharashtra, India, Pin 440 015 Abstract: India has a long tradition of study of Mathematics. In India, all Mathematicians and mathematics were deeply respected. The development of mathematical ideas in India should be studied from the history of Indian Mathematicians and their texts. Indian Mathematicians made significant contribution in the field of mathematics and produced a large contribution in which some concepts of calculus are present. The idea of calculus in India starts with the development of decimal number system, concept of zero and infinity, irrational numbers, trigonometry, arithmetic and geometric progression, finding the value of π and calculating the first order and second order differences of sine values. Now our aim is to discuss the background of Indian Mathematicians in the field of calculus. The most important core concept of calculus is infinite series. Indian mathematicians developed the concept of infinite series without using the concept of set theory and functions. We discuss how the concept of infinite series is development in India. Keywords: Arithmetic Progression, Geometric Progression, Infinite geometric series, Infinite series for π, Sine series and Cosine series 1. Introduction 2. Arithmetic Progression Indian mathematicians studied arithmetic and geometric Āryabhata – I (499 A.D.) stated following results on series at a very early date.
    [Show full text]
  • Aestimatio 13 (2016–2018) 119–140 120 Aestimatio
    Les mathématiques de l’autel védique. Le Baudhāyana Śulbasūtra et son commentaire Śulbadīpikā, edited by Jean-Michel Delire École Pratique des Hautes Études: Sciences Historiques et Philologiques II. Hautes Études Orientales — Extrême-Orient 54.15. Geneva: Librairie Droz, 2016. Pp. xiv + 626. ISBN 978–2–600–01382–6. Paper. USD $144.00 Reviewed by Satyanad Kichenassamy Université de Reims Champagne–Ardenne [email protected] The work under review is devoted to the translation from Sanskrit into French of a late commentary on an ancient Indian mathematical text, the Baudhāyana Śulvasūtra1 (BŚl). This text is dated ca 800–400 bc.2 It opens with a discourse on geometry, possibly the earliest mathematical discourse3 from India still extant. It continues with applications to the building of structures of very specific shapes required for ‘solemn’ ritual purposes, by arranging and stacking bricks according to elaborate rules: these are the Vedic altar(s) of the title.There are mathematical constraints on the shapes of the bricks, on the overall shape of the structures, on the number of bricks and the total area that they cover, and on the relation between consecutive layers. The area-constraint in particular requires the elaborate tools described in the opening discourse. Among Indian texts of the same class, BŚl is the most complete and systematic, and in it we recognize ideas that were developed in later Indian mathematics. P.-S. Filliozat states in his preface that ‘[n]o text, in the immense mathematical 1 Also spelled ‘Śulbasūtra’. Thibaut’s sectioning of the text into three parts will be used, following established usage.
    [Show full text]
  • Mathematics in India
    TRADITIONSKnowledg & PRACTICES OF INDIA e Textbook for Class XI Module 7 Mathematics in India CENTRAL BOARD OF SECONDARY EDUCATION Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India TRADITIONSKnowledg & PRACTICESe OF INDIA Textbook for Class XI Module 7 Mathematics in India CENTRAL BOARD OF SECONDARY EDUCATION Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India No part of this publication may be reproduced or 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 Central Board of Secondary Education (CBSE). Preface India has a rich tradition of intellectual inquiry and a textual heritage that goes back to several hundreds of years. India was magnificently advanced in knowledge traditions and practices during the ancient and medieval times. The intellectual achievements of Indian thought are found across several fields of study in ancient Indian texts ranging from the Vedas and the Upanishads to a whole range of scriptural, philosophical, scientific, technical and artistic sources. As knowledge of India's traditions and practices has become restricted to a few erudite scholars who have worked in isolation, CBSE seeks to introduce a course in which an effort is made to make it common knowledge once again. Moreover, during its academic interactions and debates at key meetings with scholars and experts, it was decided that CBSE may introduce a course titled ‘Knowledge Traditions and Practices of India’ as a new Elective for classes XI - XII from the year 2012-13. It has been felt that there are many advantages of introducing such a course in our education system.
    [Show full text]
  • Jantar Mantar.Pdf
    J M 2 J M GROUP 07 Iswen Vellu U024101H [email protected] Lim Tiong Yeow U014082R [email protected] Chen Youlin, Felicia U010804A [email protected] Tan Miao Nong U022231E [email protected] 3 J M CONTENTS INTRODUCTION SECTION I: P OLITICAL S TAGE IN 18TH C ENTURY I NDIA M AHARAJA S AWAI J AI S INGH II J AI S INGH’ S P OLITICAL C AREER SECTION II: T HE D ELHI O BSERVATORY B LUEPRINTS OF AN A STRONOMICAL O BSERVATORY T HE S AWAI’ S M AGNUM O PUS C ONCEPTION OF A N EW C ITY T HE V ASTU S HILPA S HASTRA F UNDAMENTAL A RCHITECTURAL T ENETS M YSTICAL C OMPONENT OF THE S HILPA S HASTRA I NTEGRATION INTO THE T OWN P LAN OF J AIPUR J AI S INGH’ S A SSISTANTS C OMPLETION OF J AIPUR AND THE J ANTAR M ANTAR 4 J M I NSTRUMENT N OMENCLATURE AND THE M ANDALA C ORRELATION R ITUALS AND C EREMONIES OF THE S OLAR K ING SECTION III: T HE I NSTRUMENTS OF THE J ANTAR M ANTAR T HE S AMRAT Y ANTRA I NSTRUMENT I NACCURACIES: T HE C ONUNDRUM OF S HADE D IFFRACTION T HE M ERIT OF S IZE R ATIOCINATION OF THE M AHARAJA SECTION IV: T HE O THER O BSERVATORIES C ONCLUSION E NDNOTE P HOTOGRAPHIC C REDITS B IBLIOGRAPHY 5 J M Introduction Clearly visible from afar, the structures of the Jantar Mantar in Jaipur are striking to behold.
    [Show full text]
  • From Vedic Square to the Digital Root ‘Clockworks’ of Modulo 90 Factorization
    From Vedic Square to the Digital Root ‘Clockworks’ of Modulo 90 Factorization "Vedic Mathematics is...a sophisticated pedagogic and research tool ... ." – Dr. L.M. Singhvi, former UK High Commissioner for India, from his forward to Cosmic Calculator, by Williams & Gaskell Mathematicians from India’s ancient past are credited with inventing the ‘Vedic Square’ (pictured above). As anyone who has studied Vedic mathematics is aware, this deceptively simple multiplication matrix is a powerful analytical and computational tool. The numbers in the table’s column and row headings are cross-multiplied, and their products converted to their digital roots, distributed across 92 cells. [Digital root (aka repeated digit sum, iterated sum-of-digits, or modulus 9 function) refers to the single digit (1 - 9) that results when all digit sums are summed in turn; this reflects the fact that any natural number is congruent modulo 9 to its digital root. For example: 89 = 8+9 = 17 = 1+7 = 8, or dr(89) = 8.] When it comes to The Digital Root, we completely agree with the importance attributed to it by Talal Ghannam, Ph.D. (in physics): “It was through my search for meaning that I discovered how important the digital root space is, which in return rewarded me with a look at how elegant and beautiful the world really is.” from The Mystery of Numbers Revealed Through Their Digital Root. In the color-coded matrix below we apply the Vedic Square concept to the period-24 digital root of natural numbers not divisible by 2, 3, or 5─otherwise known as n ≡ {1, 7, 11, 13, 17, 19, 23, 29} modulo 30.
    [Show full text]
  • THE WORLD of CONQUERORS the History, Literature, Religion and Culture of the Jains
    SUBJECT: THE WORLD OF CONQUERORS The history, literature, religion and culture of the Jains A thesis submitted to The Intercultural Open University Opeinde, The Netherlands By Dr Natubhai Shah For the doctoral degree In Jain Religion DEDICATION AS A MARK OF RESPECT AND DEVOTION TO AACAARYA VIJAY VALLABHSURISVARJI WHOSE VISION FOR THE JAIN COMMUNITY FOR EMPIRICAL AND SPIRITUAL EDUCATION HAS TRANSFORMED THOUSANDS OF YOUNG JAINS AS SUCCESSFUL CITIZENS OF THE WORLD 2 ABSTRACT ainism is the oldest extant religion in Eurasia but it is the least known in J the West. Although its teachings are as relevant in our own day as they were in the days of Mahavira who revived it more than two and half millennia ago, why this should be is almost certainly due to its small number of adherents in India: four millions plus (Jain leaders estimate twelve million and claim that it was much larger in earlier centuries of Common Era) out of a total population of nearly a billion. Jainism possesses a unique all-embracing precept from which all else flows: ahimsaa. Ahimsaa means ‘non-violence and reverence for all life’ a precept that forms the core of Jain theology; for Jains, both ascetic and lay, and it is the fundamental belief that governs their behaviour. This is supplemented by aparigraha (non-attachment to worldly possessions) and anenkaantavaada (multiplicity of views) This dissertation aims to analyse the role of Jain beliefs from their evolution in the mists of antiquity, through their reformulation by Mahavira, the last of the twenty four luminaries of Jainism in the sixth century BCE, and their historical influence on Jains and beyond up to our own times.
    [Show full text]
  • Meetings 351
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Meetings 351 MEETINGS In this department every effort is made to publish timely announcements of future meetings, as well as reports of events and papers presented at past meetings. For this we rely on the organizers of meetings to send us announcements as early as pos- sible, and on colleagues in each country to send us reports of local activities in the history of mathematics. Unnecessary duplication may be avoided by checking with the editor or the nearest member of the International Commission listed on the back cover of Historia Mathematics before submitting reports to this department. FOURTH ANNUAL CONFERENCE OF THE INDIAN SOCIETY FOR HISTORY OF MATHEMATICS By R. C. Gupta Birla Institute of Technology, P-0. Mesra, Ranchi, India The Fourth Annual Conference of the Indian Society for History of Mathematics (ISHM) was held at Delhi, December 7-8, 1980, under the auspices of the Faculty of Mathematics, University of Delhi. In his Welcoming Address, Professor Gurbaksh Singh, Vice- Chancellor of Delhi University, pointed out that "Mathematics has formed the basis of all development of science and technology, and continues to be modern man's most powerful intellectual tool," adding that "the History of Mathematics is a mirror of human civilization and its importance cannot be overemphasized." He announced the creation of a chair in the History of Science/Math- ematics at Delhi University. The Inaugural Address was delivered by Mr. P. N. Haksar, who warned against dogmatism in science, pointing to Galileo as one of its victims.
    [Show full text]
  • 1 Introduction
    Babylonian Pythagoras’ theorem, the early history of Zero and a polemic on the study of the History of Science Rahul Roy, Indian Statistical Institute, New Delhi. A difference between fancy and fact is that fancies may be as you please but facts are as the universe pleases. Robert Kaplan, The Nothing that is – A natural history of zero. 1 Introduction An irksome feature in the study of the history of science, mathematics, and society formation is encountering ‘Eurocentrism’ at practically every level. The notion that everything ‘civilised’ originated in Europe is an enterprise which began around two hundred years ago, at a time when the world was divided into the ‘dark’ continents and their ‘enlightened’ colonial masters. Besides strengthening the view that the ‘dark’ continents were indeed pitch dark, Eurocentrism was a reflex of the colonizers quest for legitimacy as the font of all things civilised. Many sociologists find in this Eurocentrism the seed which later bore the bitter fruit of Hitler’s Aryan supremacy theory. Lately, however historians cutting across the North-South and East-West divides have come to realize the folly of Eurocentrism and the ways it has hindered the development and a critical study of history, be it of science or otherwise1. Unfortunately, some of us in India have not learned from these mistakes, and stubbornly continue to engage in an ‘Indocentric’ view of science, mathe- matics, society, language, etc. Thus we are told that “Sanskrit is the mother of all languages”, “Indus valley civilisation is Vedic in origin”, “India gave the world zero”, or more ludicrously as Mukherjee [1991] writes “the math- ematical conception of zero ..
    [Show full text]
  • Novel Proofs for Summations in the Nisrstarthaduti
    Indian Journal of History of Science, 52.2 (2017) 124-137 DOI: 10.16943/ijhs/2017/v52i2/49051 Novel Proofs for Summations in the Nisṛṣṭārthadūtī Aditya Kolachana∗ , K Mahesh∗∗ and K Ramasubramanian∗ ∗ ∗ (Received 05 January 2017) Abstract Recent scholarly investigations have brought to light various mathematical proofs given by the astronomers and mathematicians of the Kerala school. However, not much is known regarding the proofs discussed by mathematicians outside this school. The present paper aims to partially fill this gap by presenting the interesting proofs of various summations discussed by Munīśvara—a 17th century astronomer from Vārāṇasī—in his Nisṛṣṭārthadūtī, a hitherto unpublished commentary on Bhāskara’s Līlāvatī. Key words: Līlāvatī, Munīśvara, Nisṛṣṭārthadūtī, Proofs, Saṅkalita, Summations, Upapatti 1. INTRODUCTION given by Nīlakaṇṭha and Śaṅkara in their texts Āryabhaṭīya-bhāṣya (a commentary on Āryab- Starting with Āryabhaṭa, Indian mathematicians haṭīya) and Kriyākramakarī (a commentary on have dealt with a wide range of topics ranging Līlāvatī) respectively. Similarly, Sarma et al. from basic arithmetic to advanced geometry and (2008) show that Jyeṣṭhadeva in his Gaṇita-yukti- algebra in their works. Among the many topics, bhāṣā discusses some ingenious proofs in connec- most works include a chapter describing summa- tion with the infinite series expression for 휋 and tions and progressions, which gives relations for other trigonometric functions given by Mādhava. the sum of first 푛 natural numbers, their squares, Such extensive and perceptive proofs as discussed cubes etc., apart from various rules pertaining to by these authors are not generally known to have arithmetic and geometric progressions. However, been presented by mathematicians outside of the no primary text of Indian mathematics presents the Kerala school.
    [Show full text]