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Cultural Foundations of Mathematics:
The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. ce


Author: C. K. Raju


Project of History of Science, Philosophy and Culture in Indian Civilization Volume X Part 4

General Editor: D.P. Chattopadhyaya



The volumes of the Project of History of Science, Philosophy and Culture in Indian Civilization aim to discover the central aspects of India痴 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 僧ethodological 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洋onsoon-dependent agriculture and navigation required for overseas trade預nd 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 菟hylogeny 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.



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