Mathematics in India

Mathematics in India

Kim Plofker
Copyright Date: 2009
Pages: 384
https://www.jstor.org/stable/j.ctt7s03d
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    Mathematics in India
    Book Description:

    Based on extensive research in Sanskrit sources,Mathematics in Indiachronicles the development of mathematical techniques and texts in South Asia from antiquity to the early modern period. Kim Plofker reexamines the few facts about Indian mathematics that have become common knowledge--such as the Indian origin of Arabic numerals--and she sets them in a larger textual and cultural framework. The book details aspects of the subject that have been largely passed over in the past, including the relationships between Indian mathematics and astronomy, and their cross-fertilizations with Islamic scientific traditions. Plofker shows that Indian mathematics appears not as a disconnected set of discoveries, but as a lively, diverse, yet strongly unified discipline, intimately linked to other Indian forms of learning.

    Far more than in other areas of the history of mathematics, the literature on Indian mathematics reveals huge discrepancies between what researchers generally agree on and what general readers pick up from popular ideas. This book explains with candor the chief controversies causing these discrepancies--both the flaws in many popular claims, and the uncertainties underlying many scholarly conclusions. Supplementing the main narrative are biographical resources for dozens of Indian mathematicians; a guide to key features of Sanskrit for the non-Indologist; and illustrations of manuscripts, inscriptions, and artifacts.Mathematics in Indiaprovides a rich and complex understanding of the Indian mathematical tradition.

    **Author's note: The concept of "computational positivism" in Indian mathematical science, mentioned on p. 120, is due to Prof. Roddam Narasimha and is explored in more detail in some of his works, including "The Indian half of Needham's question: some thoughts on axioms, models, algorithms, and computational positivism" (Interdisciplinary Science Reviews28, 2003, 1-13).

    eISBN: 978-1-4008-3407-5
    Subjects: Mathematics, History

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-xii)
  4. List of Abbreviations
    (pp. xiii-xiv)
  5. Chapter One Introduction
    (pp. 1-12)

    The mathematical heritage of the Indian subcontinent has long been recognized as extraordinarily rich. For well over 2500 years, Sanskrit texts have recorded the mathematical interests and achievements of Indian scholars, scientists, priests, and merchants. Hundreds of thousands of manuscripts in India and elsewhere attest to this tradition, and a few of its highlights decimal place value numerals, the use of negative numbers, solutions to indeterminate equations, power series in the Kerala school—have become standard episodes in the story told by general histories of mathematics. Unfortunately, owing mostly to various difficulties in working with the sources, the broader history...

  6. Chapter Two Mathematical Thought in Vedic India
    (pp. 13-42)

    As noted in section 1.3, the earliest extant Sanskrit texts are the ancient religious texts known as the Vedas, which are traditionally grouped into four sɑmhitᾱs or collections. Probably the oldest elements of these collections, based on comparisons of their vocabulary and grammatical and prosodic forms, are hymns to various deities in some sections of the Rg-veda or “Praise-Knowledge.” The standard model of ancient Indian historiography places their composition sometime in the second millennium BCE. Somewhat later than these Early Vedic hymns are Middle Vedic invocations or mantras used in rituals for performing religious sacrifices, recorded in the Yajur-veda (“Sacrifice-Knowledge”)....

  7. Chapter Three Mathematical Traces in the Early Classical Period
    (pp. 43-60)

    The middle of the first millennium BCE is generally considered the approximate end of the Vedic period, when the corpus of śruti or revealed scriptures was completed and Sanskrit started to shift from a primary language to a learned language.¹ As noted in chapter 1, at about the same time, Buddhism and Jainism emerged as distinct religious movements associated with the teachers Buddha and Mahāvīra, and the Persian Achaemenid empire reached to the northwest of the South Asian subcontinent. Within the next couple of centuries, Sanskrit was linguistically analyzed and standardized by the great grammarian Pānini in the form that...

  8. Chapter Four The Mathematical Universe: Astronomy and Computation in the First Millennium
    (pp. 61-120)

    As far as we can tell from the surviving texts, the chief matrix of literate mathematical knowledge in Sanskrit around the middle of the first millennium CE was jyotisa, a mix of astronomy, calendrics, and astrology. Treatises on jyotis.a bring together in their astronomical computations a wide variety of mathematical knowledge whose complexity and maturity suggest the richness of the mostly unrecorded developments of preceding centuries. Some treatises also include chapters on general mathematical techniques covering a range of subjects that extends far beyond astronomy proper; these are discussed in the following chapter. But the history of Indian mathematics cannot...

  9. Chapter Five The Genre of Medieval Mathematics
    (pp. 121-172)

    In the previous chapter, we saw that the term “ganita” was used in Indian texts in a variety of ways, to refer to the computational algorithms of astronomy as opposed to its geometric models, or more broadly to mean quantitative astronomy as opposed to the more descriptive disciplines of astrology. However, ganita is also frequently used in an even wider sense, to mean any type of computational or quantitative practices: what we might call “mathematics” in general. It is this sense of ganita that now comes to the fore as we explore some chapters in medieval siddhᾱnta texts, and some...

  10. Chapter Six The Development of “Canonical” Mathematics
    (pp. 173-216)

    Mathematics and mathematical astronomy in the Indian tradition were constantly changing and growing throughout their history. But they also developed over time a certain degree of standardization in their textual format and content. The various classifications of mathematical topics apparently influencing early texts such as the Āryabhatiya, the Brᾱhma-sphutɑ-siddhᾱntɑ, and the Bakhshᾱlῑ Manuscript were mostly replaced by a more uniform structure for organizing mathematical knowledge.

    The closest thing to a universal canon of mathematical texts that emerged in Sanskrit was the output of the twelfth-century astronomer Bhᾱskara (II). His Siddhᾱnta-śiromani, Lῑlᾱvatῑ, and Bῑja-ganita become by the middle of the second...

  11. Chapter Seven The School of Mādhava in Kerala
    (pp. 217-254)

    Probably the most famous school in Indian mathematics, and the one that produced many of its most remarkable discoveries, is the guru-paramparā or “chain of teachers” originating with Mādhava in the late fourteenth century and continuing at least into the beginning of the seventeenth. These scholars lived in the region known as Kerala on the southwestern coast of India, in its central part between modern Kochi (or Cochin) and Kozhikode (or Calicut). What survives of their work includes writings in Sanskrit and in the local Dravidian vernacular called Malayalam. In astronomy they are generally considered to be followers of the...

  12. Chapter Eight Exchanges with the Islamic World
    (pp. 255-278)

    Prior to about the nineteenth century, most of the Indian mathematics that came to western Asia and Europe, as well as most of the Western mathematics that found its way to India, was transmitted by Muslim intermediaries in what became the vast empire of the caliphate. Consequently, the following brief sketch of the story of these interactions is called here “exchanges with Islam,” although it also covers some contacts with pre-Islamic cultures and non-Muslims in Islamic regions.

    After the heyday of the Indo-Greek kingdoms in northern and western India and the waning of Roman dominance, the Sasanian empire of pre-Islamic...

  13. Chapter Nine Continuity and Changes in the Modern Period
    (pp. 279-298)

    The interactions with Islamic mathematics and astronomy discussed in the previous chapter were by no means the only major developments affecting Sanskrit science during the second millennium. Until recently, though, this period was largely overshadowed in the historiography of India by the eras preceding and following it, namely, pre-Islamic India, on the one hand, and European colonization on the other. In the nineteenth and twentieth centuries, the previous half-millennium or so was often dismissed as a moribund interval of intellectual “stagnation” or “decay” for India, particularly in scientific thought.¹ This opinion had its conveniences for some European imperialists, who wanted...

  14. Appendix A. Some Basic Features of Sanskrit Language and Literature
    (pp. 299-316)
  15. Appendix B. Biographical Data on Indian Mathematicians
    (pp. 317-326)
  16. Bibliography
    (pp. 327-352)
  17. Index
    (pp. 353-357)