MATHEMATICAL HERITAGE OF INDIA: SOME REMEDIAL ISSUES: MATHEMATICS HISTORY IN TEACHING
Vinod Mishra
Sant Longowal Institute of Engineering & Technology, India
Abstract. Indian mathematics has its glorious root in the civilization. This article explores the various causes of lacking of its expansion during ancient and medieval periods. Its pedagogical advantages in teaching are well known and, therefore, necessary steps for upliftment of history of Indian mathematics have been suggested. At the end, utility of history of mathematics in mathematical teaching has been experimented and a course on history of Indian mathematics has been proposed.
1. Introduction
Mathematics has the largest history of more than 5000 years developed in isolation in the beginning of different cultures-later exchanged the ideas somehow till the emergence of modern era-and what we observe today is the synthesis of fruitful concepts of ancient, medieval and, of course, later modern periods in its major share.
Post independent India has paid special attention towards the better understanding of history of Indian science by initiation of Government of India by constituting a National Commission (1969) functional under the Indian National Science Academy (INSA, New Delhi) and establishing National Institute of Science, Technology and Development Studies (New Delhi). In this context it is worthy to mention that the first major activity in the form of a symposium was held at University of Delhi in 1950. Since then a number of theme oriented programs (lectures/seminars/symposia/workshops) on history of mathematics in global perspective have been organized. Doubtless to say that the first International Congress on History of Science (July 8-14, 1929), Mexico City, under the auspices of International Commission on History of Science proved to be a pace-setter for the whole world.
INSA has brought record publications of books and project based monographs on history of mathematics. It also brings a journal Indian Journal on History of Science. Yet another organization Indian Society for History of Mathematics constituted in 1978, successfully brings out journal Ganita Bharati.
2. Mathematics in India-Some Remarks
In Indian subcontinent, mathematics flourished under the banner of Hindu astronomy in the early centuries and after the emergence of Jaina philosophy in about 300 B.C., Indian mathematics grew interacting each other. Interfusion of Indian mathematics with other cultural mathematics after the siddhantic age cannot be ruled out, in fact on small scale. The concept of permutation and combination, infinity among others are of course some of the notable contributions of the Jaina sects.
The ancient civilization of India differs from those of Egypt, Mesopotamia and Greece, in that of its traditions has been preserved without a break down to the present day. Until the advent of the archaeologist, the peasant of Egypt or Iraq had no knowledge of the culture of his forefathers, and it is doubtful whether his Greek counterpart had any but the vagvest ideas about the glory of Peri-clean Athens. In each case there had been an almost complete break with the past. On the other hand, the earliest Europeans to visit which indeed exaggerated that antiquity, and claimed not to have fundamentally changed for many thousands of years. To this day legends known to the humblest India recall the names of shadowy chieftains who lived nearly a thousand years before Christ, and the orthodox Brahman in his daily worship repeats hymns composed earlier. India and China have, in fact, the oldest continuous cultural traditions in the world [2, p. 4].
Regarding transmission of Buddhist culture and Indian scientific culture outside India, Gupta's view is worth noticeable (cf. Ganita Bharati 11 (1989), 38-39):
The rock edicts of king Asoka (3rd century B.C.) show that he had already paved the way for the expansion of Buddhism outside India. Subsequently, Buddhist missionaries took Buddhism to Central Asia, China, Korea, Japan and Tibet in the North, and to Burma, Ceylon, Thailand, Cambodia and other countries of the South. This helped in spreading Indian culture to these countries. It is well said that “Buddhism was, in fact, a spring wind flowing from one end of the garden of Asia to the other and causing to bloom not only the lotus of India, but the rose of Persia, the temple flower of Ceylon, the zebina of Tibet, the chrysanthemum of China and the cherry of Japan. It is also said that Asian culture is, as a whole, Buddhist culture”. Moreover, some of these countries received with Buddhism not only their region but practically the whole of their civilization and culture.
The tradition of Buddhist education system gave birth to large-scale monastic universities. Some of these famous universities were Nalanda, Valabhi, Vikramsila, Jagaddala and Odantapuri. They attracted students and scholars from all parts of Asia. Of these, the Nalanda University was most famous with about 10000 students and 1500 teachers. The range of studies covered both sacred and secular subjects of Buddhist as well as Brahminical learning. The monks eagerly studied, besides Buddhist works (including Abhidharmakosa), the Vedas, medicine, arithmetic, occult sciences and other popular subjects. There was a special provision for the study of astronomy and an astronomical observatory is said to be a part thereof [op. cit, p. 39].
Today we know more about western contribution than Indian. What are the reasons behind? Why did they impose their superiority? and put stamps even on the problems / ideas which were well-known and well-defined in India hundreds year ago. A few instances are:
- The theorem “the diagonal rope of an oblong produces both (areas) which its side and length produce separately” (cf. [7]) was the contribution of Baudhayana of India who lived in about 800 B.C. It was dedicated to Pythagoras (ca. 540 A.D.) of Greek which had been little known to him.
- Needham’s observation that the “Rule of Three though generally attributed to India is found in the Han Chiu Chang earlier than any Sanskrit texts” ([11, p. 146], cf. [10, p.1]). Very recently, after a lengthy discussion, Maiti on the basis of T.S. Kuppana Shastri’s work tried to trace that the antiquity of the Rule of Three’ (trairasika) dates back to Vedangajyotisa (ca. 600 B.C.).
- The equivalent of the so-called Leibniz’s series (Leibniz, 1646-1716 A.D.)
had been obtained by Madhava (fl. 1360-1425 A.D.).
- The result
for a cyclic quadrilateral, wherein r = circum-radius, a, b, c, d, the lengths of sides and s = (a+b+c+d)/2, has been given by Paramesvara (fl. 1360-1455) more than two centuries earlier than what is stated to have been rediscovered in Europe by S.A.J. L’Huilier (1782 A.D.).
Others record correctly. A few quotes are:
1. “I shall not now speak of the knowledge of the Hindus,------of their subtle discoveries in the science of astronomy –discoveries even more ingenious than those of the Greaks and Babylonians – of their method of calculation which no words can praise strongly enough –I mean the system using nine symbols. If these things were known by the people who think that they alone have mastered the sciences because they speak Greek they would perhaps, be convinced, though little late in the day, that other folk, not only Greeks, but men of a different tongue, know something as well as they”.
The Syrian astronomer–monk Severus Sebokht (writing A.D.662) (cf. [2, p. VI ])
2. “Most of the popular history of mathematics texts gives the impression that contemporary mathematical though stems directly from resurgence of western creativity after the dark age. However, mathematical ideas continued to develop in India, the Middle East and Orient from their ancient beginnings. There is evidence that some of modern mathematics travelled from these sources.
Examples of Hindu mathematics appear in extant manuscripts from 8th and 9th centuries. Many of the permutation and combination formulas attributed to Cardano, Tartaglia and Pascal were known to the Hindus-----. It is clear that the algebra of combinations in the 12th century and perhaps earlier was considerably more advanced in Hindu mathematics than in Western Circles”.
Sharon Kunoff (1990/91)
3. “The development of our system of notation for integers was one of the two most influential contributions of India to the history of mathematics. The other was the introduction of an equivalent of the sine function in trigonometry to replace the Greek tables of chords.”
C.B. Boyer [p. 241]
4. “The Rule of Three which originated among the Hindus is a device used by oriental merchants to secure results to certain numerical problems.”
Lam Lay Yong (p.329)(cf. Ganita Bharati 18(1996), p.7)
Other few examples are the (so-called) Pascal’s triangle, the notion of differential calculus, the basic trigonometry, Pell’s equation etc. Behind these discoveries they sometimes claim that Indians took inspiration from a common source, though it might not be.
3. Root Cause of Lacking of Indian Mathematics Expansion
Let us see the root cause of lacking of Indian mathematics expansion during ancient and medieval periods.
1) The proofs were not explicitly mentioned through they were understood well. In Hayashi’s view, “Neither the Aryabhatiya nor the Brahmasphutasiddhanta contains proof of their mathematical rules, but this does not necessarily mean that their authors did not prove them. It was probably a matter of the style of exposition. In fact, later prose commentaries contain a number of demonstrations or derivations, together with underlying principles ------. The recognition of the importance of proofs dates back at least to the time of Bhaskara I (around A.D. 600), who, in this commentary on the Aryabhatiya, rejected the Jain value of ,, saying that it was only a tradition and there was no derivation of it. He also emphasized the importance of verifications of solutions to mathematical problems. ------verifications are still found in Simhatilakasuri’s commentary (in the thirteenth century) on Sripati's Ganitatilaka, but occur very rarely thereafter, in contradiction to the growing popularity of derivations”
2) Chronological order had not been strictly adhered to. “Chronology is the back bone of history and its knowledge is essential for a historian dealing with any period, culture-area or subject. There cannot be a coherent history without a chronological order. Proper historical writing is not possible unless there is a sound chronology”, says Gupta (1989). For the case of India he further emphasis that “the problem of chronology continues to be very serious especially with regard to the prehistoric and ancient periods. The dates of most of the important events and literacy sources are full of serious controversies and divergent opinions. What to say about the absolute chronology, even a relative chronology is not free from challenges.” Chronology problem has arisen because whatever Indian writers wrote they dedicated to God and did not want credit by mentioning authorship, date etc. and more so in some cases they generally attributed as if the things (matters) were enunciated in antiquity.
3) There was no tradition of educating enmasses through proper writings. While writing proto- historic period of India, Balshaw (p. 30) argues that “among the many people who entered India in the 2nd millennium B.C.# was a group of related tribes whose priests has perfected a very advanced poetic technique, which they used for the composition of hymns to be sung in praise of their gods at sacrifices. These tribes, chief of which was that of the Bharatas, settled mainly in East Punjab and in the region between the Satlaj and the Jamna which later became known as Brahmavarta. The hymns composed by their priests in their new home were carefully handed down by word of mouth, and early in the first millennium B.C. were collected and arranged. They were still not committed to writing, but by now they were looked on as so sacred that even minor alternations in their text were not permitted, and the priestly schools which preserved then devise the most remarkable and effective system of checks and counter checks to ensure their purity. Even when the art of writing was widely known in India the hymns were rarely written, but, thanks to the brilliant feats of memory of many generations of Brahmans and the extreme sanctity which the hymns were thought to possess, they have survived to the present day in a form which, from internal evidence, appears not to have been seriously tampered with for nearly three thousand years. This great collection of hymns is Rg Veda, still in theory the most sacred of the numerous sacred texts of the Hindus.”
In connection with “sutra period, an age of specialization”, Bag [p. 4] writes “------that the study of mathematics started with the sutra period. At first, the study was strictly surbservient to the primary needs and education meant only the transmission of traditions from the teacher to the pupil and the committing to memory the sacred texts. In course of time, however, the contents of this education began to widen out and each one of the several angas (limbs) of the Veda began to develop. It is in this connection with the construction of sacrificial altars of proper size and shape that the problems of geometry and perhaps also of arithmetic and algebra were evolved. The study of astronomy arose out of the necessity for fixing the proper time for sacrifices.” It will be worth noticing that the Aryabhatiya is the oldest Indian work on astronomy with a chapter on mathematics having mentioned the date of compilation 499 A.D.
4) Till the middle of eighteenth century A.D. Europeans were unaware of Indian culture and its contributions, though they knew it little from various sources. The main reason was the lack of knowledge of Sanskrit, the principal Indian language, in which most of the oriental books had been written. “Until the last half of the 18th century Europeans made no real attempt to study India’s ancient past, and her early history was known only from brief passages in the works of Greek and Latin authors. A few devoted missionaries in the Pennisula gained deep understanding of contemporary Indian life, and a brilliant mastery of the vernaculars, but they made no real attempt to understand the historical background of the culture of the people among whom they worked. They accepted that culture at its face value, as very ancient and unchanging, and their only studies of Indian’s past were in the nature of speculations linking the Indians with the descendants of Noah and the vanished empires of the Bible” [2, p. 4]. Meanwhile Father Hanxleden was one of a few Jesuits who succeeded in mastering Sanskrit in the year from 1699 to 1732 and compiled the first ever Sanskrit grammar in European language. Since then more scholars including Father Coeurdoux (1767) (a Jesuit) and Sir William Jones (1783) (an Englishman) took keen interest and recognized the kinship of Sanskrit. [2, pp. 4-5].