Plofker - Mathematics in India

Why is it so hard to find information about Indian math? Many researchers in the history of Indian mathematics have heard (or asked) this plaintive question at one time or another. Usually its posed by a frustrated non-Indologist colleague engaged in some attempt to integrate the Indian tradition into the history of mathematical sciences elsewhere in the world: for example, teaching a general history of math course or writing a general history of a mathematical topic.

Theres no denying that the Indian tradition presents some unique challenges for anyone interested in the history of mathematics. Its not that information about the subject isnt available, but its frequently difficult to separate reliable information from speculation or invention, or to extract from it a coherent and consistent overview of the historical development of Indian mathematical sciences. What other branch of history of math can show, for example, a pair of articles by two widely published researchers, appearing side by side in the same edited volume, whose estimates of their subjects approximate date of origin differ by as much as two thousand years?

As I explain in more detail in the following chapters, these difficulties are due in large part to the uncertainty of early Indian chronology, the absence of historical or biographical data in many Indian technical works, and the ways that Sanskrit literature deals with authority, intertextuality, and tradition. There are many missing links in the chain of historical fact tracing out the development of Indian mathematical sciences; some of these links will someday be uncovered by new research, while others may remain forever conjectural.

This does not mean that we cant construct a reasonable narrative for the history of Indian mathematics based on the available data and plausible inferences. The narrative currently accepted by most mainstream historians as consistent with the textual record, linguistic and archaeological evidence, and the known history of other mathematical traditions goes more or less like this: The earliest urban Indian cultures, centered in the river valleys in the northwest of the South Asian subcontinent around the third millennium BCE, have left no clear record of their mathematical knowledge, although we can infer from the complexity of their infrastructure and global trade that this knowledge must have been substantial. From the second millennium BCE onward, the northwestern region (and eventually the entire subcontinent) was dominated by Indo-European cultures whose language was an early form of Sanskrit. Their earliest surviving texts mostly reflect basic mathematical knowledge supporting a simple ritual calendar and the economy of a pastoral society. In the first millennium BCE, Sanskrit texts began to show more sophisticated techniques in geometry for religious ritual and in the computations of mathematical astronomy; the latter subject may have been influenced by knowledge of Mesopotamian astronomy transmitted from the Achaemenid empire. Mathematical methods for commerce and other purposes continued to develop in India through the start of the current era, and a mature decimal place value arithmetic was established well before the middle of the first millennium CE. Spurred by interest in the astrological doctrines learned from Greeks settling in western India, Indian scholars of this period incorporated into their own astronomy some of their underlying models and techniques, such as Hellenistic spherical geometry, celestial coordinate systems, and trigonometry of chords.

Over the next thousand years or so, the Indian mathematical sciences flourished as one of the richest and most fascinating scientific traditions ever known. Using rules composed mostly in Sanskrit verse and detailed prose commentaries on them, and without the formal deductive proof structure that we now routinely associate with mathematics, Indian mathematicians brilliantly explored topics in arithmetic, algebra, geometry, trigonometry, numerical approximations, combinatorics, series (including infinite series and infinitesimal methods), and a host of other fields. Mathematical subjects were closely linked with the discipline of mathematical astronomy; the professional lives of its practitioners were generally organized around family traditions of scholarship, court patronage, and informal collegial networks rather than official institutions of learning and formal credentials. Before the end of the first millennium CE, Indian mathematics and astronomy had influenced scientific traditions in Islamic West Asia, much of Southeast Asia, and China. In the second millennium, Indian exchanges with Islamic sciences significantly increased and direct encounters with European sciences followed. The Indian mathematical tradition remained active until it was displaced in the nineteenth and twentieth centuries by the heliocentric astronomy and Western mathematics promoted by European colonial powers.

Almost every one of the above statements is disputed by some historian (although we all seem to agree at least that Indian mathematics is brilliant and fascinating). The various objections to this mainstream narrative range from mere numerological fantasy to serious scholarly critique. In almost all cases, though, the contested issues ultimately depend on some debatable point of interpretation or speculation, for which neither the mainstream nor the revisionist view can point to incontrovertible documentary evidence to settle the question beyond a doubt. Many historians of mathematics, confronted with this uncertainty, have fallen back on temporizing or compromising between two opposing views, splitting the difference between their widely divergent estimates of dates or periods, or evading the difficulty by skimming over the history of Indian mathematics as briefly as possible. So readers go on wondering why discussions of Indian mathematics often seem sketchy or confusing.