An Indian History of Numbers

OPINION NATURE|Vol 459|4 June 2009

and supervisors were all cleared of misconduct in the final report, they could have been more Buddha is said to have wooed curious about his results. Similarly, when Schön his future wife by reeling off a applied for patents, he was not required to show huge number series. or have others sign his laboratory notebook — his use of which is questionable.

It must also be noted that an accusation of SOLTAN/SYGMA/CORBIS F. misconduct is not a finding. And an incorrect or false accusation can ruin reputations or destroy careers, even if the accusation is later recognized as wrong. This puts a burden on institutions that requires thought and investigation before public disclosure, which slows the process. Reich opines that the memory of this particular affair will fade. But the possibility of scientific misconduct cannot be eradicated. Many people are involved in educating scientists at all stages of their careers to keep the possibility of scientific fraud, plagiarism, conflict of interest and other unethical behaviour before them. For example, at a 2003 workshop that I convened in London, more than 70 international partici- pants, including publishers, editors and working scientists, discussed aspects of unethical behaviour, as well as the Schön affair. The result was a An Indian history of numbers set of guidelines that was adopted, after minor modification, by the council of the International In a world divided by culture, politics, religion sciences. Another reason is that many ancient Union of Pure and Applied Physics (see http:// and race, it is a relief to know one thing that Indian mathematical texts have long been tinyurl.com/iupap-ethics). stands above them — mathematics. The diver- extinct; often, the only indication that they Scientific associations, such as the American sity among today’s mathematicians shows that it existed comes from scholars who refer to the Physical Society (APS) and the Council of Sci- scarcely matters who invents concepts or proves work of their predecessors. As Plofker wryly ence Editors, have developed and updated codes theorems; cold logic is immune to prejudice, notes, two historians of Indian maths recently of behaviour and educational programmes whim and historical accident. And yet, through- published articles in the same edited volume, before and since this landmark case. The APS out history, different families of humans have wherein the estimates of their subject’s origins policy (http://tinyurl.com/aps-policy) includes distilled the essence of the cosmos to capture differed by about 2,000 years. a report on ethics education, which is vital to the magic of numbers in many ways. Still, surviving Sanskrit texts reveal a rich define and promote ethical behaviour by all Mathematics in India shows just how tradition of Indian mathematical discoveries scientists. Reich herself references the Beasley different one of these ways was, and how cul- lasting more than 2,500 years. In the Early Vedic report using a link on the website of the APS. ture and mathematical period (1200–600 bc), a I was responsible for posting it there, with per- development are intimately Mathematics In India decimal system of numbers mission from Bell Labs’ owners Lucent Tech- connected. This carefully by Kim Plofker was already established in nologies, following the retraction of six articles researched chronicle of Princeton University Press: 2009. India, together with rules co-authored by Schön in the APS journals for the principal contributions 394 pp. £28.95 for arithmetical operations which I was then editor-in-chief. It is fortunate made by a great civiliza- (ganita) and geometry that this stable link keeps the report accessible tion covers the earliest days of Indian history (rekha-ganita). These were encoded in a years later (see http://publish.aps.org/reports). through to the beginning of the modern period. complex system of chants, prayers, hymns, The prevention and uncovering of scientific Regrettably, it stops short of the legendary curses, charms and other religious rituals. Cryp- misconduct requires vigilance, education and mathematician Srinivasa Ramanujan (born tic phrases called sutras contained arithmetical humility on the part of scientists, managers, 1887), whose name is still seen in today’s rules for activities such as laying out a temple or editors, journals and journalists to maximize research papers. arranging a sequence of sacrificial fires. the probability of correction. In the Schön affair, Kim Plofker’s book fulfils an important need Large numbers held immense fascination. several of these elements combined to raise that in a world where mathematical historiography Acclamations of praise to the air, sky, times probability enough so that correction occurred. has been shaped by the dominance of the of day or heavenly bodies were expressed in It is unfortunate that Reich did not draw fairer Greco-Christian view and the Enlightenment powers of ten that went to a trillion or more. conclusions, despite her hard work in reporting period. Too little has been written on the math- Reputedly, the young Prince Buddha success- the facts. ■ ematical contributions of other cultures. One fully competed for the hand of Princess Gopa Martin Blume is editor-in-chief emeritus at the reason for the neglect of Indian mathematics by reciting a number table that included names American Physical Society and senior physicist was Eurocentrism — British colonial historians for the powers of ten beyond the twentieth emeritus at Brookhaven National Laboratory, paid it little attention, assuming that Indians decimal place. Upton, New York, USA. had been too preoccupied with spiritual matters As in other early agricultural civilizations, e-mail: [email protected] to make significant contributions to the exact Indian mathematics probably emerged in

response to the need to measure land areas quite differently, with the former emphasizing by religion, by the need to build temples of and keep track of financial transactions, proof and the latter, result. Probably because of specific proportions and to meet astrological incomes and taxation. A rigid caste and class Islamic influence, Indian ideas on the nature of imperatives. Similarly, it could be argued that hierarchy reserved the mystery of numbers mathematical proof moved in the direction of Islamic mathematics was religiously motivated for elite Brahmins. To maintain personal greater rigour. — for example, by the need to know the pre- power, mathematical knowledge was jealously The book carefully separates fact from hyper- cise times of daily prayers, and to determine guarded. Its communication was deliberately bole, copiously quoting formulae. This makes the direction of the holy Kaaba (the Qibla). made difficult, such as in the perplexing rhyth- for heavy reading in places, and one wishes that But a quadratic equation solved by whoever, mic chant of mathematician Aryabhatta in it had been interspersed with vignettes and light by whatever means and for whatever pur- the fifth century ad: “makhi-bhakhi-phakhi- anecdotes. It is more of a research monograph pose must give exactly the same solutions. dhaki-nakhi-nakhi-nakhi-hasjha-skaki-kisga- than a popular book. But that is the price that Ultimately, mathematics is mathematics. ■ sghaki-kighva-ghaki…” This recital of values scholarship exacts. Pervez Hoodbhoy is professor of physics at of sine differences in arc minutes would be Mathematics in India explains how the early Quaid-e-Azam University, Islamabad, Pakistan. memorized by aspiring mathematicians in development of Indian maths was influenced e-mail: [email protected] much the same way as verses of the sacred text Bhagavadgita. The book details the impressive achievements of Indian mathematicians, from Aryabhatta through Brahmagupta, Mahavira, Bhaskara and Averting environmental crisis Madhava, until the Sanskrit tradition became One of the most important questions of our Nature and Power is a very different work, irrelevant with the invasion of modern math- age is when will humankind take action to written by an environmental historian who ematics from Europe in the nineteenth century. offset the impact of climate change? Will we refuses to follow convention unless he finds Major discoveries include finding the solution do something before the inexorable rise in sea compelling reason to do so. Joachim Radkau’s to indeterminate equations and the develop- level or wait until the effect is cataclysmic? Can vision in this broad-reaching history of the state ment of infinite series for trigonometric quan- the history of previous environmental crises of the environment — in particular, its soils, tities. Discovered in the fourteenth century by help us to predict the course of this one? forests and waters — is from three perspec- the Kerala school founded by Madhava, these Two books address these challenges. In the tives, namely German, continental European series built on the work of Bhaskara II and grew first, Becoming Good Ancestors, David Ehren- and global. His unrelenting focus on detail may from the ingenious computation of a circle’s feld, a zoologist and the founding editor of frustrate the reader who seeks straightforward circumference. By breaking up the circle into the journal Conservation Biology, examines narrative. Many writers can be said to miss polygons, Madhava was able to calculate the the destructive tendencies the forest for the trees, value of pi correct to 11 decimal places. Some of humankind. He asks if Becoming Good Ancestors: How We but Radkau is extreme, at developments preceded those in Europe. For we can “move ourselves Balance Nature, Community, and times abandoning the tree example, Reuben Burrow — a British math- and our society toward a Technology for the twig, bud, leaf or ematician posted to Bengal as an instructor in more stable, less frantic, by David Ehrenfeld abscission scar. the engineers corps — was intrigued by rules more responsible, and far Oxford University Press: 2008. Radkau’s guiding lights he discovered in an unnamed Sanskrit text, and more satisfying life”. The 320 pp. $19.95, £10.99 are demographer Thomas wrote a paper in 1790 entitled ‘A Proof that the book is an expanded and Malthus, sociologist Max Hindoos had the Binomial Theorem’. revised collection of some Nature and Power: A Global History Weber and the nine- But how peculiarly Indian was early Indian three dozen essays that of the Environment teenth-century agricul- mathematics? Did it evolve in isolation or did Ehrenfeld published first by Joachim Radkau tural chemist Justus von it absorb ideas and knowledge from elsewhere? in Orion magazine and Cambridge University Press: 2008. Liebig. Radkau draws on Cultural pride in their recently reinvigorated then in Swimming Les- 448 pp. $24.99, £14.99 von Liebig in his discus- country causes some Indians to claim that all sons: Keeping Afloat in the sions on the chemistry, worthwhile mathematics originated in ancient Age of Technology (Oxford University Press, well-being and degradation of soils; on Weber India. But this book will not please them. 2002). The result is a very good read. in his acknowledgement of the crucial role of Plofker is not ready to certify that the concept Ehrenfeld believes that we must jettison culture and power in the course of environ- of zero was an Indian invention; it could well our arrogant assumption of being able to fix mental history; and on Malthus for his case that have been conveyed by Chinese Buddhist anything through technology, that we forget population pressure on resources is the enemy pilgrims. Nor is she willing to believe that at our peril what worked well in the past and of sustainability. Radkau doesn’t just apply differential and integral calculus were antici- that economics is a cloudy lens through which these and many other ideas, but tests, refines pated in India ahead of the work of Gottfried to view human behaviour. He says that we and refutes them in chapters that range through Leibniz and Isaac Newton. must and can reconnect with a nature that is time and across the globe. The book often The chapter entitled ‘Exchanges with the resilient and that, despite globalization, local startles. Insights come thick and fast with Rad- Islamic World’ is of particular significance. communities will never completely disappear. kau’s ironic and unexpected turns of phrase. The Muslim conquest of India brought with it The essays retain the qualities that made them True to form, he refuses to take the easy the Islamic mathematical tradition, which was appealing when they were first published — route. He comes close to declaring that envi- founded on Greek mathematics. Muslims made brevity, passion and accessibility. However, as ronmental history is mostly about decline, important advances in maths between the satisfying as they are as self-contained medi- and that human population control is the key ninth and thirteenth centuries. Greco-Islamic tations, they do not hang together well, either to reducing resource pressure, yet he pulls and Indian mathematics were structured as a sequence or as a complete analysis. back from whatever precipice he is nearing