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Here is a greatly simplified account of the book for the layperson.

Part I: The Nature of Mathematical Proof

Part II: The Calculus in India

Part III: The Transmission of the Calculus to Europe

Part IV: The Contemporary Relevance of the Revised History

 


 

Part I: The Nature of Mathematical Proof

The Indian work on the infinite series has been known for nearly a couple of centuries. However, under the onslaught of the Western narrative of the history of science, this was regarded as somehow suspect. It has often been claimed that the Indian series lacked proof. Hence tis book begins with a re-examination of the history and philosophy of what constitutes mathematical proof. The results are surprising.

First, history was falsified at Toledo in the 12th c. CE. After burning books for some 750 years,  the church turned towards books, to catch up with the Arabs, and financed the mass translations of Arabic books at the library of Toledo. However, during the Crusades it was galling for the church to admit learning from the hated Islamic enemy. So history was Hellenized by indiscriminately attributing all secular knowledge in those Arabic books to Greeks. (Euclid was one of the fictions constructed during this process.)

Secondly, the philosophy of mathematics too was changed. During the Inquisition anything (or person) to survive had to be theologically correct. Therefore, the philosophy of mathematics, and the understanding of mathematical proof was modified to bring it in line with the prevailing Christian theology (and especially the valuation of metaphysics over physics). A key idea here was the alleged universality and certainty of (a metaphysical notion of) mathematical proof.

However, formal mathematical proof depends upon logic, but there are an infinity of possible logics to choose from. If logic is decided culturally it is not universal, for Buddhists and Jains for instance have different logics. On the other hand, if the logic underlying proof is decided empirically, why shouldn't the empirical have a place in mathematical proof? (Incidentally, logic decided empirically need not be 2-valued, as natural language or quantum mechanics informs us.)

Part II: The Calculus in India

Next the book provides the first full account of the development of Indian infinite series over a thousand year period,  without appealing to formalism but instead explaining the proofs as given by the people who used the series. The development of these series is related to the bases of wealth in India: agriculture and overseas trade requiring navigation. The successful practice of agriculture in India required a good calendar, which could determine the monsoon or rainy season.  Recalibration of this calendar, and navigation both required knowledge of the shape and size of the earth, and means of determining latitude and longitude. A navigational instrument that has long been known is described---the new feature being the accuracy with which it can be used to measure of angles, thus providing also the instrumental basis of precise angle measurements.

Part III: The Transmission of the Calculus to Europe

Since the Indian infinite series precedes the appearance of the same series in Europe, and since the two were in contact, the onus of proof actually ought to be on those who claim that these identical series were independently rediscovered in Europe, when Europe had barely learnt to add and subtract without the abacus. This is especially the case since so must of post-Hellenic history, since Copernicus, depends upon this claim of independent rediscovery. Nevertheless, the third part of the  book takes up (a) the rules of evidence by which one can discriminate between transmission and independent rediscovery, and (b) how these rules of evidence can be applied to the case of the calculus.

The key motivation was the European navigation problem which required accurate trigonometric values, and an accurate calendar, for its solution. The agency was that of the missionaries who were present in Cochin, since 1500, and patronised by the same Raja who patronised the then-most-active people working on the Indian infinite series. The Jesuits took over the Cochin college in 1550 and started translating local knowledge and sending back these translations to Europe on the Toledo model. They were strongly motivated to learn about trigonometry and the calendar given the great practical importance of the European navigational problem, for the solution of which huge prized had been declared by various European governments. (That problem arose because Europeans did not know enough trigonometry to determine the size of the globe.)  The Jesuits had access to all literature they needed, because of their proximity to it, the support of the king, and also the full support of the local community of Syrian Christians until 1600.

There is ample circumstantial evidence that this very knowledge starts subsequently appearing in Europe, but its non-Christian origins could hardly be acknowledged in the days of the Inquisition, either by those high up in the hierarchy (like Clavius, Tycho Brahe, Kepler etc.) or by those who were threatened by it (like Mercator, Newton, etc.)

Part IV: The Contemporary Relevance of the Revised History

Why is math regarded as a difficult subject to learn? It is the theologification of mathematics that has made it hard to learn. It is impossible to teach numbers in a formally correct way without first teaching set theory, and this cannot be taught to children. Similarly, today it is taught that a point is not the dot that one sees on a paper, but something mysteriously different, leaving the child befuddled. So the way to make math teaching easy is to de-theologify math.  According to sunyavada, it is the dot on the piece of paper that is real, and the abstraction of a point which is erroneous and empty. This would also be in line with the developments in computer technology, which demand a better account of what can and what cannot be represented, because computers simply cannot pretend to understand something that they don't. 

The problem with infinities did not end with the formalisation of the real number system and limits, using sets, and the formalisation of set theory in the 1930's. The formulation of physics using differential equations assumes that physical quantities are differentiable, hence continuous on classical analysis. However, discontinuities and associated infinities continue to arise in physics, as in shock waves and the renormalization problem of quantum field theory. This shows that the calculus has not yet found a satisfactory formulation. How are these infinities to be handled? Either the number system must be changed further to allow infinities and infinitesimals, or the notion of non-representable must be accepted, as in computer arithmetic.

  

 

 

 

 

 

 

 

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