Cultural foundations of mathematics : the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE

The volumes of the PROJECT OF HISTORY OF SCIENCE, PHILOSOPHY AND CULTURE IN INDIAN CIVILIZATION aim to discover the central aspects of India’s heritage and present them in an interrelated manner. In spite of their unitary look, these volumes recognize the difference between the areas of material civilization and those of ideational culture. The Project is not being executed by a single group of thinkers, methodologically uniform or ideologically identical in their commitments. Rather, contributions are made by different scholars of diverse ideological persuasions and methodological approaches. The Project is marked by what may be called *methodological pluralism". In spite of its primarily historical character, the Project, both in its conceptualization and execution, has been shaped by scholars drawn from different disciplines. It 1s the first time that an endeavour of such unique and comprehensive character has been undertaken to study critically a major world civilization. This volume examines, in depth, the implications of Indian history and philosophy for contemporary mathematics and science. The conclusions challenge current formal mathematics and its basis in the Western dogma that deduction is infallible (or that it 1s less fallible than induction). The development of the calculus in India, over a thousand years, is exhaustively documented in this volume, along with novel insights, and is related to the key sources of wealth—monsoon-dependent agriculture and navigation required for overseas trade—and the corresponding requirement of timekeeping. Rejecting the usual double standard of evidence used to construct Eurocentric history, a single, new standard of evidence for transmissions is proposed. Using this, it 1s pointed out that Jesuits in Cochin, following the Toledo model of translation, had long-term opportunity to transmit Indian calculus texts to Europe. The European navigational problem of determining latitude, longitude, and loxodromes, and the 1582 Gregorian calendar-reform, provided ample motivation. The mathematics in these earlier Indian texts suddenly starts appearing in European works from the mid-16th century onwards, providing compelling circumstantial evidence. While the calculus in India had valid pramāna, this differed from Western notions of proof, and the Indian (algorismus) notion of number differed from the European (abacus) notion. Hence, like their earlier difficulties with the algorismus, Europeans had difficulties in understanding the calculus, which, like computer technology, enhanced the ability to calculate, albeit in a way regarded as epistemologically insecure. Present-day difficulties in learning mathematics are related, via “phylogeny is ontogeny”, to these historical difficulties in assimilating imported mathematics. An appendix takes up further contemporary implications of the new philosophy of mathematics for the extension of the calculus, which is needed to handle the infinities arising in the study of shock waves and the renormalization problem of quantum field theory. D. P. CHATTOPADHYAYA has studied, researched on law, philosophy, and history, and has taught at various universities in India, Asia, Europe, and the USA from 1954 to 1994. Founder Chairman of the Indian Council of Philosophical Research (1981—90) and President-cum-Chairman of the Indian Institute of Advanced Studies, Shimla (1984—91), Chattopadhyaya is currently the Project Director of the multidisciplinary ninety-six-volume Project of History of Indian Science, Philosophy and Culture, and Chairman of the Centre for Studies in Civilizations. He is a life member of the Russian Academy of Sciences and a member of the International Institute of Philosophy, Paris. Among his thirty-five books, of which he has authored eighteen and edited seventeen, are Individuals and Societies; Individuals and Worlds; Sri Aurobindo and Karl Marx; Anthropology and Historiography of Science; Induction, Probability and Skepticism; Sociology, Ideology and Utopia; Societies, Cultures and Ideologies; Interdisciplinary Studies in Science, Society, Value and Civilizational Dialogue, and Philosophy of Science, Phenomenology and Other Essays. Besides, he has held high public offices, namely, of Union cabinet minister and state governor. C. K. RAJU, Ph.D. (Indian Statistical Institute) has taught in both mathematics and statistics departments of the University of Poona for several years (1981-88), before joining C-DAC (Centre for Development of Advanced Computing) to play a key role in building the first Indian supercomputer Param (1988-95). He has proposed a new mathematical basis for physics (using functional differential equations or a “tilt” in the arrow of time), leading to a new understanding of quantum mechanics, in Time: Towards a Consistent Theory (1994; Fundamental Theories of Physics, vol. 65). In The Eleven Pictures of ‘Time (2003) he examined time as the critical link between science and various religions, in the context of the “clash of civilizations”, and formulated a new ethical principle. He has been a Fellow of the Indian Institute of Advanced Study (1991-93), the National Institute of Science Technology and Development Studies (1995-98), and an Affiliated Fellow of the Nehru Memorial Museum and Library, where he coordinated a project of the Indian National Science Academy (1998-2001) on the historical origin of the calculus. He has been on the editorial board of the Journal of Indian Council of Philosophical Research. For the past several years (since 2000) he worked as a Professor of Computer Science, has also taught computer courses on television, and has authored texts and software for educational and industrial use. He has been associated with the Project of History of Indian Science, Philosophy and Culture (PHISPC) since its early days, and has contributed many articles to PHISPC volumes since 1991. **From the backpage** (http://ckraju.net/IndianCalculus/back_page.htm) The volumes of the Project of History of Science, Philosophy and Culture in Indian Civilization aim to discover the central aspects of India’s heritage and present them in an interrelated manner. In spite of their unitary look, these volumes recognize the difference between the areas of material civilization and those of ideational culture. The Project is not being executed by a single group of thinkers, methodologically uniform or ideologically identical in their commitments. Rather, contributions are made by different scholars of diverse ideological persuasions and methodological approaches. The Project is marked by what may be called ‘methodological pluralism’. In spite of its primarily historical character, this project, both in its conceptualization and execution, has been shaped by scholars drawn from different disciplines. It is the first time that an endeavour of such unique and comprehensive character has been undertaken to study critically a major world civilization. This volume examines in depth the implications of Indian history and philosophy for contemporary mathematics and science. The conclusions challenge current formal mathematics and its basis in the Western dogma that deduction is infallible (or that it is less fallible than induction). The development of the calculus in India, over a thousand years, is exhaustively documented in the volume, along with novel insights, and is related to the key sources of wealth—monsoon-dependent agriculture and navigation required for overseas trade—and the corresponding requirement of timekeeping. Rejecting the usual double standard of evidence used to construct Eurocentric history, a single, new standard of evidence for transmissions is proposed. Using this, it is pointed out that Jesuits in Cochin, following the Toledo model of translation, had long-term opportunity to transmit Indian calculus texts to Europe. The European navigational problem of determining latitude, longitude, and loxodromes, and the 1582 Gregorian calendar-reform, provided ample motivation. The mathematics in these earlier Indian texts suddenly starts appearing in European works from the mid 16th c. onwards, providing compelling circumstantial evidence. While the calculus in India had valid pramāna, this differed from Western notions of proof, and the Indian (algorismus) notion of number differed from the European (abacus) notion. Hence, like their earlier difficulties with the algorismus, Europeans had difficulties in understanding the calculus, which, like computer technology, enhanced the ability to calculate, albeit in a way regarded as epistemologically insecure. Present-day difficulties in learning mathematics are related, via “phylogeny is ontogeny”, to these historical difficulties in assimilating imported mathematics. An appendix takes up further contemporary implications of the new philosophy of mathematics for the extension of the calculus needed to handle the infinities arising in the study of shock waves and the renormalization problem of quantum field theory. Cover Half title Imprint Contents List of Boxes, Tables, and Figures Synoptic Contents General Editor and Author General Introduction Preface Introduction I The Nature of Mathematical Proof 1 Euclid and Hilbert 2 Proof vs Pramāna II The Calculus in India 3 Infinite Series and 7 4 Time, Latitude, Longitude, and the Globe 5 Navigation: Kamāl or Rāpalagai III Transmission of the Calculus to Europe 6 Models of Information Transmission 7 How and Why the Calculus Was Imported into Europe 8 Numbers in Calculus, Algorismus, and Computers IV The Contemporary Relevance of the Revised History 9 Math Wars and the Epistemic Divide in Mathematics A Distributions, Renormalization, and Shocks Index