The Indian Mathematical Tradition with Special Reference to Kerala: Methodology and Motivation

Indian mathematics has its roots in Vedic literature. It is imperative to know about the ancient, medieval and modern time Indian mathematicians and their contribution in the field of Mathematics. Indian mathematicians made significant early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed here. Ancient Indian mathematicians have contributed immensely to the field of mathematics. The invention of zero is attributed to Indians and this contribution outweighs all other made by any other nation since it is the basis of the decimal number system, without which no advancement in mathematics would have been possible. The number system used today was invented by Indians and it is still called Indo-Arabic numerals because Indians invented them and the Arab merchants took them to the western world. It is also noticed that there is a distinct and inequitable neglect off the contributions of the sub-continent. Many of the developments of Indian mathematics remain almost completely ignored, or worse or attributed to the scholars of other nationalities, often European. However a few historians (mainly European) are reluctant to acknowledge the contributions of Indian mathematicians. They believe Indians borrowed the knowledge of mathematics from Greeks. Even after all those criticisms it is a great pleasure for us to have been born in such a country where so many great mathematicians like Aryabhata, Brahmagupta, Srinivasa Ramanujan, P.C. Mahalanobis , Calyampudi Radhakrishna Rao, D.R. Kaprekar, Satyendra Bose ,Bhaskara I, Mahavira, Bhaskara II and many others born.

I. Introductory Overview Mathematics in Ancient and Medieval India R. Sridharan II. Ancient Period Sanskrit Prosody, Pingala Sutras and Binary Arithmetic R. Sridharan. Shedding Some Localic and Linguistic Light on the Tetralemma Conundrums F. E. J. Linton III. Classical and Medieval Period Brahmagupta's Bhavana: Some Reflections Amartya Kumar Dattta The Karani: How to Use Integers to Make Accurate Calculations on Square Roots Francois Patte Relations between Approximations to the Sine in Kerala Mathematics Kim Plofker Algorithms in Indian Mathematics M. S. Sriram Algorithms in Indian Astronomy K. Ramasubramanian Proofs in Indian Mathematics M. D. Srinivas IV. Modern Period Ramanujan and Partial Fractions George E. Andrews Contributions of Indian Mathematicians to Quantum Statistics Gerard G. Emch.

It is a challenging task to engage in a philosophical reading of Indian mathematics, given its rich history, which itself does not present a unanimous account of the various mathematical achievements in ancient India. This brief essay is primarily a philosophical reading of the history of Indian mathematics, with respect to the concept of mathematical “proof.” A philosophical perspective on some of the issues that emerge in this tradition permits an analysis of its methodology and foundations; and sheds light on the validation of mathematical results and procedures. The analysis of the Indian mathematical tradition gives rise to several contentious issues. For instance, it is often presumed that the Indian mathematical achievements lacked rigorous, logical proof. However, many argue that the notion of upapatti (detailed demonstration) is significantly different from the notion of “proof” as understood in the Greco-European tradition of mathematics. Indian epistemological positions on the nature and validation of mathematical knowledge cannot be directly matched with those in the Western tradition. Various epistemological and metaphysical issues emerge not only from our analysis of these classical mathematical achievements but also from the rich philosophical theories/traditions in which Indian mathematicians were embedded.

Education is a very important Component of Society. It is the keystone of all the Socio- economic and Cultural developments in every society. In fact, the level of education reflects the Socio- cultural prosperity of every society & culture. The history of education in ancient India began with teaching of traditional elements such as Indian religions, Indian mathematics, Indian logic at early Hindu and Buddhist centres of learning such as Taxila (in modern ) and Nalanda (India) before the common era. Islamic education became ingrained with the establishment of the Islamic empires in the Indian subcontinent in the middle ages while the coming of the Europeans later bought western education to colonial India.

Foreword.- Prelude.- Indian Mathematics in the Medieval Islamic World.- Brahmagupta: the Ancient Indian Mathematician.- Indian Calendrical Calculations.- India's contributions to Chinese Mathematics up to the Eighth Century A.D..- Some Discussions about how Indian Trigonometry affected Chinese Calendar-Calculation in the Tang Dynasty.- On the Application of Areas in the Sulbasutra.- Indian Mathematical Tradition with special reference to Kerala: Methodology and Motivations.- Mainland South-East Asia as a Crossroad of Chinese and Indian Astronomy.- Mathematical Literature in the Regional Languages of India.- Pascal's Triangle in 500 BC.- Andre Weil: His Book on Number Theory and Indian References.- The Algorithm of Extraction in both Greek and Sino-Indian Mathematical Traditions.- Index.

New literary and documentary evidence on powder technology in ancient and medieval India is presented and discussed. It is shown that the genesis of metal powder in ancient India is related to the discovery of naturally occurring gold powder in the form of alluvial placer deposits commonly found in river beds. Philological explanation of some ancient Sanskrit words used for denoting gold confirms the antiquity of alluvial placer gold powder. The ‘Pipīlaka’ (ant) gold powder of the ‘Mahābhārata’, a typical placer gold powder obtained by washing auriferous sands dug and collected by ants in ancient India, is discussed. This was of great interest in ancient India because of its relatively high purity. It is shown that the ant gold powder reported by Herodotus and others has a different source from the Pipīlaka gold powder of the Mahābhārata: it was obtained by washing auriferous sands dug and collected by some animals, probably marmots, in the desert part of Ladakh in north western India. Literary evidence indirectly throwing light on the compaction of metal powder is also reported. The various applications of metal powder in ancient and medieval India are discussed in detail. New evidence indicates that the foremost application of prepared fine metal powder or metal dust was in various medicinal preparations dating back as far as the Vedic period. The application of metal powders in preparing explosive mixtures and as a currency material is also described. PM/0583

This paper is an effort to assemble in a synchronized manner, the brilliant contribution of the illustrious Keralian Mathematician and Astronomers, since Vararuchi (fourteenth century C.E) to Medieval Periods Madhava (C.1340- 1425), Paramesvara (C.1360- 1455), Nilakantha Somayaji (C. 1444 -1545) and his followers. It would entail that Indian Mathematics was well ahead by 350 to 400 years in comparison to European contributors Newton, Leibnitz, Taylor and others.

Indian mathematicians developed some of the most important concepts in mathematics, including place-value numeration and zero. By developing new techniques in arithmetic, algebra, and trigonometry, medieval Indian mathematicians helped make modern science and technology possible. This paper explains how the mathematics developed and how to use in daily life.

Mathematics is an international subject, and it has no geographical boundary for its development. Many countries have contributed for the development of the discipline of mathematics. The development of mathematical concepts started dates back to more than thousand years. For the development of the discipline mathematics, many countries mathematicians have contributed such as the Babylonian, Egyptian, Greek, Chinese and Indians etc. The concept of mathematics have come to us only for the need of calculations in daily life and commercial activities, measurement of land around for crop and to predict astronomical events happening around us by observing day and night. Therefore, mathematics is a science, because it is based on observation and recorded systematically in logical order. The contribution of Indian mathematicians in the field of mathematics is far more than western mathematicians. Hence, this study is suggesting that, there is a need to introduce among us a good number of Indian Mathematician contributions to develop positive attitude among the learners and to appreciate the mathematics subject and contribution of Indian mathematicians in the field of mathematics. Ancient Indian mathematicians have contributed immensely to the field of mathematics. The invention of zero is attributed to Indians and this contribution outweighs all other made by any other nation since it is the basis of the decimal number system, without which no advancement in mathematics would have been possible. The number system used today was invented by Indians and it is still called Indo-Arabic numerals because Indians invented them and the Arab merchants took them to the western world. It is also noticed that there is a distinct and inequitable neglect off the contributions of the sub-continent. Many of the developments of Indian mathematics remain almost completely ignored, or worse or attributed to the scholars of other nationalities, often European. However a few historians (mainly European) are reluctant to acknowledge the contributions of Indian mathematicians. Some of the great mathematicians are Pythagoras, Fibonacci, Euclid, Archimedes, Rene Descartes, and Aryabhata, who have contributed significant inputs in the field of mathematics in ancient times.

In a manuscript which is being studied here for the first time, al-Samaw'al (12th century) quotes a paragraph from al-Bīrūnī (11th century) which shows that the latter knew not only of Brahmagupta's method of quadratic interpolation, but also of another Indian method (called sankalt). Al-Samaw'al examines these methods, as well as linear interpolation, compares them, and evaluates their respective results. He also tries to improve them. In this article the author shows that al-Bīrūnī had used four methods of interpolation, two of which were of Indian origin; and that al-Samaw'al explicitly introduced a new way of evaluating these different methods. He also throws light on the active movement of research on numerical methods that constitutes the background to al-Bīrūmī's and al-Samaw'al's work, and uses modern means to evaluate the different methods and to justify the mathematicians' choices. The author has edited and translated al-Samaw'al's text in order to make it available to historians of Arabic and Indian mathematics. This allows them to follow the history and the mathematical arguments, and deepens our understanding of the importance to the development of mathematics of certain astronomical work. It also highlights the contribution of Indian mathematicians to the development of Arabic mathematics.

Differential Power Analysis (DPA) presents a major challenge to mathematically secure cryptographic protocols. Attacks can break the encryption by measuring the energy consumed in the working digital circuit. To prevent this types of attack, this paper proposes the use of reversible logic for designing a high speed complex multiplier (ASIC) based on Vedic mathematics in cryptosystem. Reversible logic is gaining significance in the context of emerging technology such as quantum computing. Ideally, reversible circuits do not loose information during computation. Thus, it would be of great significance to apply reversible logic to design for secure cryptosystem. The idea for designing the multiplier unit is adopted from ancient Indian mathematics “Vedas”. On account of these formulas, the partial products and sums are generated in one step which reduces the carry propagation from LSB to MSB. The implementation of the vedic mathematics & their application to the complex multiplier ensure substantial reduction of propagation delay in comparison with DA based architecture (distributed arithmetic) & parallel adder based implementation which are commonly used. The functionality of these circuits was checked & performance parameters like propagation delay and dynamic power consumption were calculated by spice specter using standard 90nm cmos technology.

Several books have been written on the history of Indian tradition in mathematics.2 In addition, many books on history of mathematics devote a section, sometimes even a chapter, to the discussion of Indian mathematics. Many of the results and algorithms discovered by the Indian mathematicians have been studied in some detail. But, little attention has been paid to the methodology and foundations of Indian mathematics. There is hardly any discussion of the processes by which Indian mathematicians arrive at and justify their results and procedures. And, almost no attention is paid to the philosophical foundations of Indian mathematics, and the Indian understanding of the nature of mathematical objects, and validation of mathematical results and procedures.

Vedic Mathematics is the ancient methodology of Indian mathematics which has a unique technique of calculations based on 16 Sutras (Formulae). A high speed complex multiplier design (ASIC) using Vedic Mathematics is presented in this paper. The idea for designing the multiplier and adder/sub-tractor unit is adopted from ancient Indian mathematics “Vedas”. On account of those formulas, the partial products and sums are generated in one step which reduces the carry propagation from LSB to MSB. The implementation of the Vedic mathematics and their application to the complex multiplier ensure substantial reduction of propagation delay in comparison with DA based architecture and parallel adder based implementation which are most commonly used architectures. The functionality of these circuits was checked and performance parameters like propagation delay and dynamic power consumption were calculated by spice spectre using standard 90nm CMOS technology. The propagation delay of the resulting (16, 16)×(16, 16) complex multiplier is only 4ns and consume 6.5 mW power. We achieved almost 25% improvement in speed from earlier reported complex multipliers, e.g. parallel adder and DA based architectures.

While there has been an awareness of ancient Indian mathematics in the West since the sixteenth century, historians discussed the Indian mathematical tradition only after the publication of the first translations by Colebrooke in 1817. Its reception cannot be comprehended without accounting for the way that new European mathematics was shaped by Renaissance humanist writings. We sketch this background and show with one case study on algebraic solutions to a linear problem, how the understanding and appreciation of Indian mathematics was deeply influenced by the humanist prejudice that all higher intellectual culture, in particular all science, had risen from Greek soil.

This chapter discusses what has been achieved so far and what implications this has for the foundations of mathematics. Among others, two questions are treated: First, mathematical proofs were carried out before the notion of proof was made precise, a vicious circle? Second, in view of the deficiencies in expressive power of first-order logic (as shown in the previous chapter), what effect does the restriction to first-order logic have on the scope of the investigations? Answering these questions, the development of present-day mathematics in the framework of first-order logic is outlined, leading to the introduction of the Zermelo-Fraenkel axioms for set theory.