Joy of Infinity : Mathematics in India
Every culture gave names to numbers. In India significantly, smaller numbers bore numerous names each, most of them derived from philosophical and spiritual concepts. A few examples will illustrate the point :
- Shunya(or Kha) for zero is a well-known case, but other terms for it , such as akasha, ambara, vyoman,(all meaning ‘sky’), ananta(‘infinite’), purna(‘full’), imply totality and wholeness rather than void, suggesting that the trwo concepts are simply two aspects of the same truth
- Number 1,eka suggests the notion of indivisibility : other names for it therefore include atman, Brahman, surya(the sun), adi(the beginning), akshara(the syllable OM), the moon(soma, indu), the earth and its many names, and many more symbols of unity
- Number 2, dvi, is also named after the Ashvins, the twice-born, Yama(as the primordial couple of the Rig Veda), the two eres, and other symbols of duality.
- Number 3,tri, after the three Vedas, Shiva’s three eyes, his trident(trishula), the three worlds, the three gunas, the triple Agni, and so on.
- Number 4, chatur, after the four ashramas or stages of human life, the four ages(yugas), the four Vedas, Vishnu’s arms or Brahma’s four faces.
- Number 5, pancha, after the five elements, the Pandavas or Rudra’s five faces
- Number 6, shat, after the siz rages, the six classical systems of philosophy(darshanas), Karttikeya’s six faces
- Number 7, sapta, after the seven Buddhas, the seven oceans(sagaras) and islands(dvipas), the seven seers(rishis), divine mothers, rivers, days of the week, horses of the Surya etc.
- Number 8, ashta, after the eight points of the compass or the eight mythical elephants upholding the world.
- Number 9, nava, after the planrts, the traditional nine jewels(ratna), the body’s orifices or Durga (celebarated during the nine nights of the Navaratri festival)
- Number 10, dasha, after Vishnu’s ten avatars, the Buddha’s ten powers and stages or Ravana’s ten heads!
The list goes on with 12 Adityas , 25 tattvas, 27 nakshatras, 33 Devas, 49 Vayus…and for larger numbers such terms would be attached together, leading to long words devoid of any meaning except the mathematical.
This device fell into disuse with the spread of the much more efficient decimal numeral system, but the association of spiritual and philosophical concepts with number names illustrates the absence of a rigid boundary between science and spirituality in the minds of Indian mathematicians. Perhaps that is why we find many of them reaching out to the infinite : long before Ramanujan, Indian savants shows “mania” for large numbers as French historian of number systems Georges Ifrah puts it in his monumental Universal History of Numbers. The Rig Veda makes frequent mentions of 100,000(foes, gifts, cattle herds) while the Yajur Veda goes up to a million million(1012), a number it calls parardha. This grew by leaps and bounds in Jaina literature(such as the Anuyugadvara Sutra), in which contemplation of than infinite and eternal universe lead to numbers exceeding 10250(in other words, 1 followed by 250 zeros), In the Lalitavistara Sutra, the Bodhisattva reels of endless series of multiples of 10 naming them up to 10145 and eventually conjuring up a number of equivalents to 10421.
Several texts propose different values for one of those colossal numbers; names asamkhyeya – or ‘innumerable’ which cannot be counted; the Lalitavistara Sutra, again, adds this poetic definition : it is the number of raindrops falling on all the worlds for ten thousand years!
Let us remembers, by contrast the the highest named number in ancient China and ancient Greece was 10,000(the ‘myriad’ in Greece) ; Arab names did no go beyond 1,000; and Europe had to wait until the thirteenth century before the French introduced the ‘million’ : only in the seventeenth century were the billions, trillion and the quadrillion conceived of. Ifrah repeatedly states his admiration for specific achievement of the Indian mind:
We have here, if need be , one more proof of the very clear Indian intellectual lead over all the contemporary Western thoughts and one more testimony to the great fertility of the Indian savants minds, …the Sanskrit numeral notation carried within itself the very seed of the discovery of the principle of the decimal place-value notation.
While reaching out to the infinite, some of India’s common symbols of infinity were put to good use : numbers such as 109, 1014, 1021, 1023,1027, 1029,10105,10112,10119 were named after the lotus(padma, utpala, kumud….), others after the ocean, the earth, and of course ananta, the infinite itself.
Ultimately, mathematical infinity got a name of its own, khachheda or khahara. Khaccheda means ‘divided by kha ; kha being ‘space’ or ‘void’, one of the names for ‘zero’. ‘Division by zero’ gives an intuitive definition of infinity : any fraction increases in values as its denominator is reduced ; as the latter tends towards infinity. The term khachheda was introduced by the great mathematician Brahma-gupta in his Brahmasphuta Siddhanta(628 CE) ; khahara, with similar meaning, was used later by another celebrated scientist, Bhaskaracharya of 12th century CE.
Infinity is not found only at the colossal end of things : it also takes us to the infinitesimal. Thus the paramanu, or ‘supreme atom’, corresponded either to a length of about -.3 nanometre of to a weight of 0.614 microgram! We find the same phenomenon when we explore the mysteries of times : we are familiar with the almost infinite scale of the Yugas, but at the other end of it, in his Siddharth Shiromani, Bhaskaracharya defined the nimesha(literally, the blink of an eye) as one 972,000th of a day, or about 89 milliseconds, and went on dividing it further and further till he reached the truti, a unit of time equal to one 2,916,000,000th of a day – or about 30 microseconds! Of what use could such units of time been to the ancients?
India’s fascination with huge numbers is well illustrated by the legend of chaturanga, India’s early version of chess, a legend conveyed by Arabic sources. Sessa, a clever Brahmana, once, demonstrated this new games to a king, who was so pleased that the told him to ask for any reward. Sessa humble requested one grain of wheat on the first square of the board, two on the second, four on the third, eight on the fourth, and so on, doubling the number of grains on every square right up to the sixty fourth. The king thought the request was ridiculously modest and insisted on a more substantial one, but Sessa declined. The royal mathematicians set about calculating the amount, but after great labor could not proceed beyond a few squares. Embarrassed, the king had mathematicians called from a neighbouring kingdom, who were familiar with the new decimal place-value system and at home familiar with large numbers. They soon worked out the desired number of grains : 264- 1, a number that requires twenty digits to be written out in the conventional way. The savants explained to the king that even if the whole earth were sown with wheat, it would take over seventy harvests to get the desired quantity! To save the royal face, a wily minister suggested that he sould open his granary to the Brahamana and ask him to start counting the grains himself, never stopping till he reached the correct numbers. Sessa got the message and was never heard of again.
Leaving calculations aside, we find the astonishing mathematical concepts forming the canvas of ‘mythological’ stories. The Buddhist text Avatamshaka Sutra(often rendered as the ‘The Flower Ornament Sutra’ depicts a network of pearls placed in heavens by Indra in such a way that ‘in each pearl one can see the reflections of all the others, as well as the reflections within the reflections and so on.’ This might appear as mere poetic fancy and a practical impossibility, but three U.S mathematicians took it up as a challenge. What they found was a mathematical validation of the Buddhist image : Indra’s pearls precisely followed by arrangement of circles in a mathematical entity called a Schottky group, and they worked out several actual designs of pearls fulfilling the text’s apparently impossible conditions. But what is it that drove its authors to dictate them? Intuition? Or perhaps simply a poetic approach to mathematical truths, like the rain drops falling on all the worlds that defined the infinite.
It is fascinating to note that in 1957, the American physicist David Bohm, who was obsessed with unveiling the deeper reality of matter, had a ‘vision’ almost identical to that of the Avatamshaka Sutra. ‘The vision came to him,’ writes his biographer F. David Peat, ‘in the form of a large number of highly silvered spherical mirrors that reflected each other. The universe was composed of this infinity of reflections, and of reflections of reflections. Every atom was reflecting in this way, and the infinity of these reflections was reflected in each thing’ each was an infinite reflection of the whole.’ An eloquent remind that a mystical perception of reality is not India’s monopoly : it is a capacity inherent in human consciousness.
The above article has been adopted from the book “Indian Culture and India’s Future” by Michel Danino, released in January 2011
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