The Forgotten Story of Indian Calculus - Numberphile

Here's a detailed summary of the YouTube video transcript, maintaining the original language and technical precision:

Key Concepts

• Professor P.P. Divakaran: A renowned physicist and mathematician, retired from the Tata Institute of Fundamental Research, who passed away at 89. He was instrumental in bringing attention to 14th and 15th-century Indian Calculus.
• Indian Calculus (Kerala School of Mathematics): A school of mathematical thought in Kerala, India, active from the 14th to 16th centuries, which developed calculus concepts independently of European mathematicians.
• Madhava of Sangamagrama: The founder of the Kerala School of Mathematics, born around 1340.
• Yuktibhasa: A significant 16th-century text written in Malayalam, meaning "language of logic and proofs," which documented the calculus developed by the Kerala School.
• Gurukula Culture: A traditional Indian system of education where students live with and learn from a teacher.
• Series Expansion: Expressing a function as an infinite sum of terms, such as power series.
• Limits: A fundamental concept in calculus that describes the value a function approaches as its input approaches some value.
• Trigonometric Functions: Functions of an angle, such as sine, cosine, and tangent.
• Decimal System: A positional numeral system with a base of 10.

Professor P.P. Divakaran and His Work

The video begins by commemorating the passing of Professor P.P. Divakaran, a distinguished physicist and mathematician from the Tata Institute of Fundamental Research, at the age of 89 on August 23rd. Professor Divakaran, a close friend of the speaker, made significant contributions to particle physics and, more recently, to bringing the achievements of 14th and 15th-century Indian Calculus to global recognition. He published extensively on this topic and authored a book on it with Springer.

Transition from Particle Physics to Indian Mathematics

The speaker addresses the apparent shift in Professor Divakaran's focus from particle physics to ancient Indian mathematics. Professor Divakaran was primarily a particle physicist throughout his career, having been recognized by the Indian National Talent Search in the 1950s, earning his PhD in Chicago, and working with Professor Dalitz at Oxford. Upon returning to India, he became a professor of particle physics, specializing in quantum field theory. After retiring in the late 1990s, he received a fellowship that allowed him to visit various institutions. It was during a visit to the Chennai Mathematical Institute that he encountered historians of mathematics researching Indian Calculus. His background as a mathematical physicist enabled him to read and appreciate these historical texts more deeply than a pure historian might.

Personal Connection and Interaction

The speaker shares a personal connection with Professor Divakaran, having met him twice. The first meeting was when the speaker was a first-year PhD student, and Professor Divakaran was visiting the speaker's institute to give talks on physics. The second, more recent meeting, involved spending two months with Professor Divakaran daily at the Inter-University Centre for Astronomy and Astrophysics in India, where the speaker was "kind of taking care of him." They maintained regular contact via email, discussing physics and mathematics, making his passing particularly sad for the speaker.

The Kerala School of Mathematics: An Independent Development of Calculus

The discussion then shifts to Indian Calculus, specifically the Kerala School of Mathematics, contrasting it with the well-known discoveries of Leibniz and Isaac Newton.

Origins and Founders

• No Direct Connection to Newton/Leibniz: The speaker emphasizes that the Indian development of calculus is independent of Newton and Leibniz, acknowledging their significant contributions.
• Geographical and Historical Context: The Kerala School flourished in the southern Indian state of Kerala, which was part of the Vijayanagara kingdom during the 14th century.
• Madhava of Sangamagrama: The central figure of this school, Madhava, is believed to have immigrated around 1350 AD and established a mathematics school. While his life details are scarce, his birth year is estimated to be 1340.
• The "Nila School": Professor Divakaran referred to it as the "Nila School of Mathematics," named after a river around which the mathematicians lived. This school continued for approximately 300 years.

The Gurukula System and Cultural Impact

• Educational Structure: The school operated under a Gurukula system, where disciples lived with and learned from a master. It was likely centered around the teacher's residence rather than a large campus.
• Intellectual Hub: The school became a significant center for mathematics and a catalyst for cultural, linguistic, and even Ayurvedic advancements in Kerala.

Yuktibhasa: The Key Text

• Content and Significance: The primary source of modern knowledge about the Kerala School is the book Yuktibhasa (meaning "language of logic and proofs"), written around the 1520s. This text details the calculus developed over the preceding 150 years.
• Timeline: Yuktibhasa was written more than 200 years before Newton and Leibniz. While it is now recognized in mathematical circles and on platforms like Wikipedia, there is no evidence that Newton or Leibniz had access to it. Professor Divakaran, after reviewing Newton's works, found no such evidence, and the distinct Indian cultural thinking made a direct influence unlikely.

Core Concepts of Indian Calculus

The discussion delves into the specific mathematical achievements of the Kerala School.

Measuring Arc Length and Series Expansions

• Motivation: A fundamental problem addressed by calculus is measuring the length of a curve or arc. While Newton approached this through motion and changing velocity, Indian mathematicians were primarily concerned with circles and changes with respect to angles.
• Methodology: The core technique involved dividing a curved object into infinitesimally small segments, treating each as a straight line in the limit as the segment length approaches zero, and then summing these segments to find the total arc length. This is the foundational concept of integration.
• Key Discoveries: Using this methodology, they derived series expansions for:
  • sin(x) (similar to the Maclaurin series)
  • cos(x)
  • arctan(x) (the Gregory series)

Similarities and Differences with Western Calculus

• Fundamental Techniques: The underlying techniques of splitting objects into small parts and taking limits were fundamentally the same as those used by Newton and Leibniz. This marked the first time in mathematical history that limits were handled with such rigor.
• Scope of Functions: A key difference lies in the scope of functions addressed.
  • Kerala School: Primarily focused on trigonometric functions (sine, cosine, tangent) due to their work on circles.
  • Newton/Leibniz: Developed calculus for a broader range of general functions, including exponential and logarithmic functions, which were not explored by the Indian school.
• Focus on Foundations: The Indian mathematicians concentrated on the foundational issues of series expansion, carefully adding corrections and approximations, rather than generalizing to all function types.

Reasons for Limited Global Spread

Several factors contributed to the Kerala School's discoveries not becoming widely known globally before Newton and Leibniz.

• Language Barrier: Yuktibhasa was written in Malayalam, the local language of Kerala. Most scholars at the time, including Western scholars, were more familiar with Sanskrit. This linguistic barrier prevented wider access to the detailed proofs.
• Political and Social Instability: Kerala was under Portuguese colonial influence, leading to conflicts that contributed to the decline of the school and its mathematical culture. This decline lasted for centuries until the work was rediscovered about 50-60 years ago.
• Comparison with Earlier Works: A preceding text, Aryabhatiya Bhasya by Nilakantha, which discussed the work of the older Indian mathematician Aryabhata, was written in Sanskrit and was more widely known. The reason for Yuktibhasa being written in Malayalam remains unclear, but it significantly hindered its dissemination, especially given the advanced and formal nature of its proofs (e.g., power series expansions), which were not even conceived in northern India at the time.

Professor Divakaran's Perspective and Legacy

Professor Divakaran's views on the recognition of Indian Calculus are shared.

• No Sadness or Injustice: He did not express sadness or a sense of injustice regarding the delayed recognition. He understood the complex reasons behind it and did not attribute it to malice.
• Efforts in Dissemination: He was happy with his own efforts to bring this knowledge to light and delved deeply into the history, tracing mathematical developments back to 1500 BC, through Aryabhata (500 AD), and culminating in the Kerala School, which he considered a pinnacle of mathematics.
• Madhava's Prominence: Madhava is considered the most prominent mathematician of the medieval ages globally.
• Potential Impact of Early Dissemination: The speaker speculates that if this knowledge had spread to the Western world earlier, it could have significantly accelerated mathematical and physical advancements. The widespread adoption of the decimal system, which was already in use in India for millennia, would have facilitated this dissemination.

The Speaker's Personal Feelings

The speaker expresses pride in this Indian mathematical heritage, not just as an Indian but as a scholar. They are not surprised by mathematical achievements originating from India, as mathematics was deeply embedded in its culture. However, they would be intrigued by discoveries like the law of gravitation. The speaker finds it remarkable that modern concepts of calculus, function expansion, and approximation existed so long ago, reflecting a very contemporary way of thinking.