This work explores a possible course of evolution of mathematics in ancient times in India when there was no script, no place-value system, and no zero. Reviewing examples of time-reckoning, large numbers, sacrificial altar-making, and astronomy, it investigates the role of concrete objects, natural events, rituals and names in context-dependent arithmetic, revealing its limited scope confined to counting, addition and subtraction. Higher operations, namely, multiplication, division and fractional calculations had to wait until the advent of symbolic numerals and procedures for computation. It is argued that the impression of these higher operations in a period usually known as the Vedic times is caused by inadvertent interpolation of present knowledge of mathematics in modern readings of the ancient texts.
The Jain text claims that 296 is the population of human beings in the world. And Lalitavistara says that tallaksana (1053) is the basic unit for calculating the size of Mount Meru. Furthermore, according to bodhisattva, the next larger number beyond tallaksana is dhvajagravat (1055), using which as a unit, it is possible to calculate the grains of sand in the river Ganges. Mount Meru is a mythical mountain measuring the height of which is as meaningless as counting the sand grains in the river Ganges. Similarly, an assumption of the human population of 296 (~ 1029) on earth in the Anuyogdvara sutra, composed around 100 BCE (Datta and Singh, 2004), is equally fantastic and unreal; one wonders how the author reached that conclusion. Therefore, although the reference to sand grains, Mount Meru or to the world population provide a context for the mind to grasp these numbers, the best that can be claimed in the light of their unreality and sheer extravagant size is that by applying a for
id: 93f7fb25121b99397d1c8731353527d5 - page: 7
Perhaps such exercises were indulged in only to establish that starting from a small number one can reach, in a limited number of steps, such immense numbers that are simply incomprehensible and beyond human imagination and experience, conveying a sense of infinitude. Generation of long numbers starting from eka or 1 to parrdha (1,000,000,000,000) in YV17.2 was perhaps one of the earliest such attempts. As Vedic philosophy is well-known to be imbued with the thoughts of infinite and infinity, it may be conjectured that the sequence of long numbers in a religious, ritualistic text like Yajurveda was perhaps meant to guide ones mind out of the practical world, stepwise, into the metaphysical world. The last comment does not mean that such sequences were confined solely to the Vedic religious texts. As already discussed, the Jain and the Buddhist traditions too had similar large numbers. 7
id: 55fbfd3f15a9618b50f8489a60088855 - page: 7
Fascination for long numbers persisted in mathematical works well after the first millennium of the Common Era. A notable point about these number names is that often they dont seem to represent a fixed value. While ten-, a hundredand a thousand-millions are arbuda, nyarbuda and samudra in Yajurveda17.2, in the 5th century Aryabhatia (2.2), the same values are named koti, arbuda and vranda, respectively (Shukla and Sarma, 1976). Similarly, while an ayuta is tenthousand in the Yajurveda, in Lalitavistara it is a billion. Even after Indian mathematics had already made major advances in the first millennium of this era, the lists of numbers quoted by Sridhara (8th century), Mahavira (9th century), Bhaskar II (12th century), and Narayana (14th century) are often different in their names and values (Datta and Singh, 2004, pp. 12-13). Al-Biruni in his book on India also draws attention to such variations in names of numbers (Sachau, 1910, pp. 175-177). Thus, while the tradition of gener
id: 253b3b8fb2ff14a8b2ea5029b57db67c - page: 8
It is therefore possible that the names of large numbers in the Vedic period were not supposed to have any significance other than being the tokens for memorization of mechanically generated numbers.
id: a2871f441e3d3c17a74b1e942f053cf6 - page: 8