Tuesday, June 25, 2024

A Glimpse into the Mathematical and Scientific Traditions of India

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Sri Vedanta Desika, a 14th century poet, wrote the pādukā sahasram, a 1008 verse Sanskrit poem in praise of Lord Ranganatha’s (Lord Ram’s) padukas at Srirangam. It is said that he composed the poem in just one night [1]. In addition, verse No. 929 and No. 930 are composed of the same 32 syllables. It means that verse No. 930 is derived from rearranging the order of the syllables in verse No. 929. Not only this, these two verses also provide a solution to the famous Knight’s tour problem [2] in mathematics. Knight’s tour [Fig. 1] is a sequence of steps a chess knight will take to visit every square on a chess board once and only once.

Knight's Tour
Fig 1: Example of Knight’s Tour

The two verses of Sri Desika are 32 syllables each, and can be laid out on a 8 × 4 chess board, each cell holding one syllable. When read in the order of the squares of the chess board, we get the first verse. But the order of the second verse is a knight’s tour.

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स्थिरागसां सदाराध्या विहताकततामता । सत्पादुके सरसा मा रङ्गराजपदं नय ॥                    – Verse 929

स्थिता समयराजत्पागतरा मोदके गवि । दुरंहसां सन्नतादा साध्यातापकरासरा ॥                    – Verse 930

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Fig 2: Chitra-kāvya from Pādukā Sahara

Fig. 2 shows the layout of the chess board with the 2 verses. Such poetry in Sanskrit, that can be laid out on a geometric shape is known as chitra-kāvya (picture poem). This style of poetry is quite well known in Sanskrit literature. It begins with one verse and derives another verse from this verse by performing a permutation of syllables. This by itself is an amazing feat. But incorporating in it a solution to the Knight’s tour problem is an astounding achievement, while keeping the verses meaningful and sublime. Technically, this is a half-tour, since only half the chess board is used, but two copies of this solution provide us with a full tour.

There are even earlier chitra-kāvyas providing solutions to the Knight’s tour problem. They are found in the works of Kashmiri poets Rudrata (815 AD) [4] and  Ratnākara (830 AD) [5]. The notable works are Kāvyānlankāra by Rudrata, Harvijayam by Ratnākara, Saraswati-kanthabharana by King Bhoja (1050 AD) [6] and Mānasollāsa by King Someshwara III (1130 AD) [6].

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What is remarkable is that the Knight’s tour problem in mathematics finds its origin to the 18th century, when the great Leonard Euler was the first mathematician to look at it, while the problem and its solution were known in India as early as the 9th century. It is considered as one of the great contributions to the field of combinatorics from India. This fact and the clever depiction of the solutions through poetry in chitra-kāvyas was noted and made famous [6] by Prof. Donald Knuth, arguably the greatest computer scientist in the world. He even wrote his own chitra-kāvyas [5] in English.

This example is from medieval India but illustrates the rich knowledge traditions that go back millennia. Be it Acharya Pāṇini, who’s grammar rules in Ashtādhyāyi (6th to 4th century BC) are used in modern computers [7] (Pāṇini–Backus form of  computer grammar for programming languages), or Acharya Pingala, who wrote Chandaḥśāstra (4th century BC),  a treatise on Sanskrit prosody [8] (metrical science of Sanskrit poetry), that includes the Fibonacci sequence, Pascal’s triangle, binary numbers (the basis for all of modern information technology), and the binomial theorem, concepts that we all believe were given to us by European scholars. Not only were these concepts quite well known in ancient India, but they were developed as part of developing Sanskrit poetry and were intricately assimilated into all Indian art forms, such as in Nātyaśāstra (100 BC to 350 AD) by Acharya Bharata. As such, amazing mathematical and scientific truths are embedded in sublime Sanskrit shlokas. Discovering such truths is one thing, expressing them in the form of poetry is quite another.

It is now well established that the Greek geometry was known in India centuries before the Greek.  One need not be reminded of the rich disciplines of medicine, logic, astronomy, economics etc. that have stood the test of times. [9]. Susruta (6th to 8th century BC), who is regarded as the father of surgery, was perhaps the first to describe various kinds of surgical instruments including endoscopes. Ancient astronomers could predict the occurrences of eclipses, knew many cosmic distances, the circumference of the earth etc. Kautilya’s Arthaśāstra is celebrated.  Vedanta, the pinnacle of Indian knowledge traditions, investigate the nature of the ultimate reality and the nature of consciousness.

Our traditions and knowledge have inspired numerous thinkers and scholars and have also left a profound impact on modern thought. Philosophers such as Schopenhauer, Huxley, Whitehead,  and Steiner, and scientists such as Schrödinger, Heisenberg, Oppenheimer, Tesla, Bohr,  and Bohm were deeply influenced by Indian thought. In his book What is life [10], Schrödinger acknowledges the wisdom of the Upanishads and describes अयम् आत्मा ब्रह्म, from the Mandukya Upanishad, as the grandest of all thoughts.

While by looking at all this we can safely say that Indian knowledge traditions have been profoundly infused with logic and science and many ideas have been far ahead of their time, this is merely a miniscule glimpse into what the Indic knowledge traditions encompass. It also stirs one to ponder if we are doing justice to our great intellectual heritage. Isaac Newton famously said that “If I have seen further it is by standing on the shoulders of Giants”. The least we could do is to acknowledge the enormous achievements of our ancestors, if not attempt to stand on their shoulders.

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