While a beautiful thing in itself, knowledge generates many different types of rewards, from productive use of inventions to the creation of new bonds among people. The 17th century French writer Rabutin, Comte de Bussy, famously remarked, “Love comes from blindness, friendship from knowledge”. Love may well result from the inability to see what one is getting into. However, it has certainly enriched the world in many different ways — particularly through the creation of great literature, such as Romeo and Juliet, Abhijnana Shakuntala, and Layla and Majnun. But what does friendship produce — whether or not knowledge generates it (as Bussy-Rabutin claimed)?
I want to concentrate particularly on the opposite direction of influence emphasised by Bussy-Rabutin — not on how knowledge produces friendship, but on how friendship generates knowledge. The understanding that friendship helps the creation of knowledge is particularly important in the philosophy and history of science. Nationalist sentiments may make a country claim a secluded flowering of science and mathematics, detached from the rest of the world (unrelated to what we can learn from others — from our friends), but that is not the way science and mathematics — and ultimately culture, too — proceed. For example, the view of ancient India as an island, making its discoveries and inventions in splendid isolation — detached from the rest of the world — is pleasing to intellectual nationalists in India, but it is fundamentally mistaken.
We learn from each other and our intellectual horizons are expanded by being in touch with what others know. Once acquired, our newly learned knowledge expands under its own dynamics and we can give to the outside world much more than we received from it.
Consider the golden age of Indian mathematics. This was not the Vedic period, contrary to what is often claimed (exaggerated claims about Vedic mathematics have tended to generate a world of fantasy in parts of university education in India today). The golden age of mathematics in India was, rather, the classical period in the first millennium, quite close in time to the flowering of the great literature of Kalidasa, Sudraka and other writers. The great mathematical revolution in India was led particularly by Aryabhata, born in 476 AD, and developed by Varahamihira, Brahmagupta, Bhaskara and others. Aryabhata’s departures had sophistication and extraordinary reach that were quite uncommon in the mathematics of his time. There is much evidence that while deeply original, Aryabhata’s mathematics was substantially influenced by mathematical developments in Greece, Babylon and Rome. There was outside influence, and yet in Aryabhata’s hands, mathematics in India — and astronomy too — took gigantic leaps that were pioneering contributions for the whole world. India learned something but gave to the world enormously more than what it had learned from outside.
And as new understandings were born in India, they spread abroad, not only to Greece and Rome, but particularly to China, where they played a central role in the extraordinary progress in Chinese astronomical work (even the head of the official Chinese Board of Astronomy in the critically important 8th century was an Indian mathematician, Gautama), and to the Arab-speaking world which would become the most important vehicle of mathematical progress in the 8th to the 11th centuries. What began with India learning something from others soon became India teaching a lot to others, and these others, in turn, made huge contributions to the world of mathematics. Friendship, in the broadest sense (including the ability to learn from each other), played a central role in this interactive process, each step reinforcing the next, across national boundaries.
Emerging in a primitive form in Sumeria and Babylon, trigonometric ideas received the attention of Euclid and Archimedes in Greek mathematics in the 3rd century BC and Hipparchus in Asia Minor a century later. In the first century BC, Surya Siddhanta in India aired trigonometric constructions with further sophistication. The Greek influence was clearly present in Indian mathematics, but Surya Siddhanta had more developed trigonometry, particularly applied to astronomy, than what Alexander and the Greek settlers brought to India. To consider one example, when, towards the end of the 5th century AD, Aryabhata produced his comprehensive account of advances in mathematics, the concept of the sine, which is still perhaps the most widely used trigonometric notion, found its definitive exploration.
But how did this Aryabhatian concept come to be called “sine”, which is not a word in Sanskrit or any other Indian language? This bit of linguistic history, which I have discussed in The Argumentative Indian, is worth recollecting. Aryabhata called the sine by the Sanskrit name “jya-ardha” — half-chord — making use of the geometric basis of trigonometry, and often referred to it as “jya” for short. When the Arab mathematicians translated this concept into Arabic, they called it “jiba” — a corruption of jya. Arabic is written only with consonants, omitting the vowels, and so Aryabhata’s jya was represented as “j, b” — the two consonants in jiba. The sound jiba has no meaning in Arabic, but the same representation “j, b” can also be pronounced as “jaib”, which is a fine Arabic word, meaning a cove or bay.
When the Arab texts on sophisticated trigonometry, on the lines derived from Aryabhata, were ultimately translated into Latin (Gherardo of Cremona, an Italian working in Toledo, did the translation in 1150 AD), the word jaib, meaning a cove or a bay, was translated into the corresponding Latin word “sinus”, which is Latin for a cove or bay. And from there — from the word sinus — comes the modern trigonometric term “sine”. The much-used mathematical term sine carries within it the memory of Aryabhata’s Sanskrit term jya, and its sequential Arabic and Latin translations. What came to India from Europe in a somewhat simple form, went back to the world as a more developed tool of mathematics and astronomy.
The separatist outlook in the development of science, mathematics and culture is seriously misleading. Indeed, the role of friendship applies not only across national borders, but also within borders. Divisions, tensions and violence between groups and sects that political separatists like promoting (even within a nation), not only damage our social lives, but also work as barriers to intellectual progress within as well as across nations.
Indeed, the isolationist view of the progress of knowledge is fundamentally defective — no matter how appealing it may be to the nationalist and the sectarian. Friendship is important for our intellectual pursuits. Of course, it has many other rewards as well, but the advancement of science and mathematics — and of knowledge in general — is an important part of the beautiful impact of friendship.
This article first appeared in the print edition of January 8, 2020, under the title “Friendship and progress”. The writer, a Nobel laureate in economics, is Thomas W. Lamont University Professor and professor of economics and philosophy at Harvard University. Excerpted from the keynote lecture at the Infosys Prize ceremony, 2020.
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