Proof in Indian Mathematics

There is an incident in Ramaujan’s life in Cambridge, when he started working with Prof G H Hardy. Ramanujan did not think it was necessary for him to show the proofs of his discoveries. Hardy, who was steeped in the western mathematical tradition, had to explain to him the necessity of proof in the western tradition and train Ramanujan to document his proofs.

For a casual reader this incident may give an impression that Indian mathematicians did not value importance of proof.

Considering the sophisticated results propounded by Indian mathematicians, it is not possible that they did not have in their mind a clear understanding of the logic of their method. But they did not document these proofs. The reasons for this could be many.

1. The Indian tradition of learning was predominantly oral. Indians, in pre-modern times, also realised that any kind of documentation was not permanent as memory. Their belief in oral tradition has been vindicated by the fact that the way the Vedas have been recited have hardly changed in the last 3000 years. Even today, 2 vedapatis who have never met before can sit together and recite the vedas in perfect unison.

2. In ancient India the predominant method of schooling was in the Gurukuls where the teacher and students were in 24X7 contact and most learning happened through memorization & oral discussions. There was no need for a written source.

3. The teacher possibly left a lot of the lessons (like proofs) for self-study. Students were expected to introspect on their own, discuss and deepen their understanding and internalise the learning.

4. Much of the validation of scholarly competence took place through oral debates between scholars. Documentation was not considered necessary.

Another deeper reason could be the difference in the understanding of the role of mathematics. For Europeans, starting with the Greeks, mathematics was a discipline for developing critical thinking and understanding the world. Mathematics was to be studied for its own sake. Hence logic and proof were very important.

For Indians it was an applied subject which was useful for their religious rituals. They used it (as geometry) in constructing altars to specified dimensions and (as trigonometry) in studying movements of celestial bodies & working out the exact auspicious moments. Ganita was a part of Jyotisha. Hence the result and its application were more important than the proof.

What developed mathematics in Europe was the constant exchange of ideas, in written form, across countries and the patronage to learning from the royal families. The invention of the printing press also helped in the rapid & easy spread of ideas.

The modern technological & digital world is leaning more towards the western perception about mathematics. With an internal logic of its own, mathematics has grown broader and deeper to make an impact on almost every aspect of the physical & biological world.

The Indian attitude could be one of the reasons for the decline of the mathematical tradition in India in the recent centuries. Its tradition did not take math learning beyond a very small circle. Hence no large schools of mathematics emerged and the dissemination of the ideas to a larger audience did not happen.