Indian mathematics

"The Indian system of counting is probably the most successful intellectual innovation ever devised by human beings. It has been universally adopted. ...It is the nearest thing we have to a universal language."

"Medieval Indian mathematicians, such as Brahmagupta (7th century), Mahavira (9th century) and Bhüskara (19th century), made several discoveries which in Europe were not known until the Renaissance or later, They understood the import of positive and negative quantities, evolved sound systems of extracting square and cube roots, and could solve quadratic and certain types of indeterminate equations."

"Through the necessity of accurately laying out the open-air site of a sacrifice Indians very early evolved a simple system of geometry, but in the sphere of practical knowledge the-world owes most to India in the realm of mathematics, which were developed in Gupta times to a stage more advanced than that reached by any other nation of antiquity. ‘The success of Indian mathematics was mainly due to the fact that the Indians had a clear conception of abstract number, as distinct from numerical quantity of objects or spatial extension. While Greek mathematical science was largely based on mensuration and geometry, India transcended these conceptions quite early, and, with the aid of a simple numeral notation, devised a rudimentary algebra which allowed more complicated calculations than were possible to the Greeks, and led to the study of number for its own sake."

"The cord stretched in the diagonal of an oblong produces both [areas] which the cords forming the longer and shorter sides of an oblong produce separately"

"It is India that gave us the ingenious method of expressing all numbers by ten symbols, each receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit."

"In the Surya Siddhanta is contained a system of trigonometry which not only goes beyond anything known to the Greeks, but involves theorem which were not discovered in Europe till two centuries ago."

"We owe a lot to the Indians who taught us how to count, without which no worthwhile scientific discovery could have been made."

"In order to instill a proper and well-founded pride in Hindus, it is (once more) most important to restore the truth about Hindu history, especially about Hindu society's glorious achievements. In technology, it cannot match China, which was the world leader until a mere three, four centuries ago. But in abstract sciences like linguistics, logic, mathematics, Hindu culture has been the chief pioneer."

"In the Vedic Age, India was very religious, but it was also ahead of the rest in mathematics and astronomy. Thus, the geometry of the Shulba Sutras, geometrical appendices to the manuals of ritual (Shrauta Sutras), include the oldest known formulation of the theorem named after Pythagoras, developed in the context of Vedic altar-building. Modern Hindus are fond of recalling this scientific element in their tradition, e.g. by quoting Carl Sagan: “Hindu cosmology gives a time-scale for the earth and the universe which is consonant with that of modern scientific cosmology”, as opposed to the limited Biblical-Quranic cosmology, which was protected against more far-sighted alternatives by a vigilant religious orthodoxy."

"In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.”"

"In the whole history of Mathematics, there has been no more revolutionary step than the one which the Indian made when they invented the sign ‘0’ for the empty column of the counting frame."

"Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. ...2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. ...3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians."

"It is clear how much we owe to this brilliant civilization, and not only in the field of arithmetic; by opening the way to the generalization of the concept of the number, the Indian scholars enabled the rapid development of mathematics and exact sciences. The discoveries of these men doubtless required much time and imagination and, above all, a great ability for abstract thinking. These major discoveries took place within an environment which was at once mystical, philosophical, religious, cosmological, mythological and metaphysical."

"Ancient Indian culture has regarded the science of numbers as the noblest of its arts … A thousand years ahead of Europeans, Indian savants knew that zero and infinity were mutually inverse notions. In short, Indian science was born out of a mystical and religious culture and the etymology of the Sanskrit word used to describe numbers and the science of numbers bears witn The early passion which Indian civilization had for high numbers was a significant factor contributing to the discovery of the place-value system, and not only offered the Indians the incentive to go beyond the calculable physical world, but also led to an understanding much earlier than in our civilization of the notion of mathematical infinity itself."

"Because the Greeks were ‘ingenious and inventive’, they must have developed mathematics, and that is what we must assume even in the absence of evidence. And because the Hindus were backward and primitive, they could not have created mathematics, and that is what we must assume even in the presence of evidence."

"C'est de l'Inde que nous vient l'ingénieuse méthode d'exprimer tous les nombres avec dix caractères, en leur donnant à la fois, une valeur absolue et une valeur de position; idée fine et importante, qui nous paraît maîntenant si simple, que nous en sentons à peine, le mérite. Mais cette simplicité même, et l'extrême facilité qui en résulte pour tous les calculs, placent notre système d'arithmétique au premier rang des inventions utiles; et l'on appréciera la difficulté d'y parvenir, si l'on considère qu'il a échappé au génie d'Archimède et d'Apollonius, deux des plus grands hommes dont l'antiquité s'honore."

"My confidence in our shared future is grounded in my respect for India’s treasured past—a civilization that has been shaping the world for thousands of years. Indians unlocked the intricacies of the human body and the vastness of our universe. And it is no exaggeration to say that our information age is rooted in Indian innovations—including the number zero."

"The science of numbers is used in all transactions whether earthly or Vedic, meditation or business, in the science of love, in the art of cooking, in economics and statecraft, in music and medicine, in theatre, architecture and all such things. (9-10) In prosody, in poetics and poetry, in logic, grammar and such like, in the presentation of the qualities of various arts, mathematics is held in esteem. For the movements of the sun and stars, for eclipses and planetary motion, for the three significant questions and the course of the moon, for all these connections it is used (11-12). The number, diameter and perimeter of islands, of oceans and mountains, for the dimensions of rows of buildings of the earth’s inhabitants, of the space between the worlds, of the world of light, for the worlds of devas, for the dimensions of the dwellers in hell; for measurements of all sort and voluminous evidence, all these are made out by means of computation (13- 14). The organization of living beings, the length of their lives, their eight attributes and other similar things, their travel and cohabitation and such other things, all depend on mathematics. What more can be said? All there is in the three worlds, moving or unmoving, all that cannot be described without mathematics (15-16)."

"To the Hindus is due the invention of algebra and geometry, and their application to astronomy."

"I once heard, and I think it is true, that only one man in the world—some Indian mathematician—understood the mathematics of string theory in eleven-dimensional space, and he dreamed it."

"We know that the trigonometric sine is not mentioned by Greek mathematicians and astronomers, that it was used in India from the Gupta period onwards (third century).... The only conclusion possible is that the use of sines is an Indian development and not a Greek one. But Tannery, persuaded that the Indians could not have made any mathematical inventions, preferred to assume that the sine was a Greek idea not adopted by Hipparchus, who gave only a cable of chords. For Tannery, the fact that the Indians knew of sines was sufficient proof that they must have heard about them from the Greeks."

"It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo" "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras."

"Many of the key results of mathematics, some of which are at the very “core” of modern day mathematics, are of Indian origin. ...‘I wish to conclude initially by simply saying that the work of Indian mathematicians has been severely neglected by western historians.’...‘it is likely European scholars are resistant due to the way in which the inclusion of non-European, including Indian, contributions shakes up views that have been held for hundredsof years, and challenges the very foundations of the Eurocentric ideology."

"The Hindus no less than the Greeks have shared in the work of constructing scientific concepts and methods in the investigation of physical phenomena , as well as of building up a body of positive knowledge which has been applied to industrial technique; and Hindu scientific ideas and methodology (e.g. the inductive method or methods of algebraic analysis) have deeply influenced the course of natural philosophy in Asia—in the East as well as the West—in China and Japan, as well as in the Saracen empire."

"However, Seidenberg was told by the indologists that these Sutras, or any Vedic text for that matter, were definitely written later than 1700 BC. But mathematical data cannot be manipulated just like that, and Seidenberg remained convinced of his case: “Whatever the difficulty there may be [concerning chronology], it is small in comparison with the difficulty of deriving the Vedic ritual application of the theorem from Babylonia. (The reverse derivation is easy)… the application involves geometric algebra, and there is no evidence of geometric algebra from Babylonia. And the geometry of Babylonia is already secondary whereas in India it is primary.” [To satisfy the indologists, he said that the Shulba Sutra had conserved an older tradition, and that it is from this one that the Babylonians had learned their mathematics:] “Hence we do not hesitate to place the Vedic (…) rituals, or more exactly, rituals exactly like them, far back of 1700 BC. (…) elements of geometry found in Egypt and Babylonia stem from a ritual system of the kind described in the Sulvasutras.”"

"There is not, and cannot be, number as such. There are several number worlds as there are several Cultures. We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number — each type fundamentally peculiar and unique, an expression of a specific world feeling, a symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one. ... and so it is understandable that even negative numbers, which to us offer no conceptual difficulty, were impossible in the Classical mathematic, let alone zero as a number, that refined creation of a wonderful abstractive power which, for the Indian soul that conceived it as base for a positional numeration, was nothing more nor less than the key to the meaning of existence. Negative magnitudes have no existence.... But when we are told that probably (it is at best a doubtful venture to meditate upon so alien an expression of Being) the Indians conceived numbers which according to our ideas possessed neither value nor magnitude nor relativity, and which only became positive and negative, great or small units in virtue of position, we have to admit that it is impossible for us exactly to re-experience what spiritually underlies this kind of number. For us, 3 is always something, be it positive or negative; for the Greeks it was unconditionally a positive magnitude, +3; but for the Indian it indicates a possibility without existence, to which the word “something” is not yet applicable, outside both existence and non-existence which are properties to be introduced into it. +3, -3, 1/3, are thus emanating actualities of subordinate rank which reside in the mysterious substance (3) in some way that is entirely hidden from us. It takes a Brahmanic soul to perceive these numbers as self-evident, as ideal emblems of a self-complete world form; to us they are as unintelligible as is the Brahman Nirvana, for which, as lying beyond life and death, sleep and waking, passion, compassion and dispassion and yet somehow actual, words entirely fail us. Only this spirituality could originate the grand conception of nothingness as a true number, zero, and even then this zero is the Indian zero for which existent and non-existent are equally external designations."

"I shall not now speak of the knowledge of the Hindus … of their suitable discoveries in the science of astronomy—discoveries even more ingenious than those of the Greeks and Babylonians, of their rational system of mathematics, or of their method of calculation which no words can praise strongly enough; I mean the system using nine symbols."

"I will omit all discussion of the science of the Hindus, a people not the same as the Syrians, their subtle discoveries in the science of astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; their computing that surpasses description. I wish only to say that this computation is done by means of nine signs. If those who believe because they speak Greek, that they have reaced the limits of science should know these things, they would be convinced that there are also others who know something."

"Like the crests on the heads of peacocks, like the gems on the hoods of the cobras, mathematics is at the top of the Vedanga sastras."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."