Ideas of Mathematical Proof in Classical India:

1. Ideas of Mathematical Proof in Classical India: A Reading of the Aryabhatiyabhashya of Nilakantha Somayajin

2. Why Do This??

3. The Five Sheaths of the Human Person (Taittiriya Upanishad)

4. Layer 1: Annamayakosha Now a man here is formed from the essence of food. This here is his head. This is his right wing; this is his left wing. This is his torso, and this is his tail on which he rests.

5. Ashtanga First Series:Forward Bends

6. First Series:Bend more …

7. …One Drink Too Many.

8. Layer 2: Pranamayakosha • Different from and lying within this self formed from the essence of food is the self consisting of life-breath … . The head is simply the out-breath, the right wing is the inter-breath, the left wing is the in-breath; the torso is space, and the tail on which it rests is the earth. • “Life-breath – gods breathe along with it, as do men and beasts …”

9. Second Series:bend back …

10. … and waaay forwards …

11. … and ???

12. Layer 3: Manomayakosha

13. Manomayakosha • Different from and lying within this self consisting of life-breath is the self consisting of mind, which suffuses this other self completely … . The head is simply the Yajus formulas, the right wing is the Rig verses; the left wing is the Saman chants; the torso is the rule of substitution, and the tail is the Atharva-Angirases. • “Before they reach it, words turn back, together with the mind…”

14. Third Series:Balance

15. Layer 4: Vijnanamayakosha • Different from and lying within this self consisting of mind is the self consisting of understanding … The head is simply faith; the right wing is the truth; the left wing is the real; the torso is the performance, and the tail on which it rests is the celebration. • “It’s understanding that conducts the sacrifice; it’s understanding that performs the rites. It’s understanding that all the gods venerate as the foremost brahman.”

16. Fourth Series

17. Fourth Series

18. Layer 5: Anandamayakosha • Different from and lying within this self consisting of understanding is the self consisting of bliss, which suffuses this other self completely … Of this self, the head is simply the pleasure; the right wing is the delight; the left wing is the thrill; the torso is the bliss, and the tail on which it rests is the brahman.

19. Seated Practices:Pranayama, Meditation

20. Uttaravedi Agnicayana

21. Vedic Ritual ReenactmentKerala, South India, 1975(courtesy IGNCA)

22. The Sacrificial Enclosure

23. Kalpa Computations

24. Soma (Ephedra Dystachia)

25. Agni Manthana

26. Agni Manthin

27. Agni Manthin Verbal root manth: 1. To churn

28. Agni Manthin Verbal root manth: • To churn • To agitate, disturb

29. Agni Manthin Verbal root manth: • To churn • To agitate, disturb • To apply friction to any part of a body in order to produce offspring

30. Churning the Ocean

31. Rahu and Ketu

32. Celestial Sphere(Gola)

33. Rashis and Nakshatras

34. Kalpa (layout of sacrificial area, construction of ritual devices) Jyotisha (“star science”, began as timing of Vedic rituals, eventually turned to astronomy/astrology) Two Main Applications of Mathematics in India

35. Aryabhata • Born 476 CE • Astronomer, mathematician • Author of the Aryabhatiya • “taught by Brahma” • Rotating earth

36. Nilakantha Somayajin • Nilakantha = “blue-throated” • Somayajin = “soma-sacrificer”

37. Nilakantha • 1444 – 1545 CE • Kerala, South India • Namputiri Brahmin • Learned in Nyaya logic, Vedanta philosophy • Investigations in the scientific method, “science and religion” issues

38. Jyotirmimamsa This is what some think: “Pleased by feats of asceticism, Brahma taught to Aryabhata [the characteristics] of the planets .. Because of Brahma’s omniscience, freedom from passion, etc., how can there be criticism of Aryabhata?” Stupid! It is not thus. The favor of a deity causes mental clarity only. Neither Brahma nor the Sun-god taught it – only Aryabhata.

39. The Aryabhatiyabhashya • Composed “late in his life;” • Written on multiple levels; • Commentary on the Ganitapada alone is 180 pages; • Detailed proofs, but no diagrams!

40. Typical Commentarial Tasks • Grammatical analysis of the current verse • Argue why the rule is best stated this way • Show that the rule is useful • Show that the verse is pretty poetry • Justify its placement in the sequence of verses • Prove the rule • Never, ever embarrass the Teacher (even if he’s wrong)!

41. Aryabhatiya 7 The area of a circle is just half its circumference multiplied by the radius. That [area], multiplied by its own square root, is exactly the volume of a sphere.

42. Nila comments: How then is a circle made? Scratching it out using just the tip of a paint brush with some heated lamp-black – let that be done by skilled artists only! At least that’s how it’s commonly attempted. But in that case one doesn’t get an even circumference, due to the lack of [constant] extension.

43. How to get the area formula: The portions of the area of a circular figure all have the form of needles, because, all around [the center], the portions are wheel-spokes, rounded and separated at the ends. Because of their infinity, you can make [the wide ends] as small as you like. Their needle-form is due to the fact that all of the cut-up portions touch the center [of the circle].

44. Dissect the Circle, Rearrange

45. When all of them are placed together, tip-to-end in pairs, “rectangle-ness” occurs.

46. You Get This:

47. Do Same Thing,With More Slices

48. In the Limit …

49. But what about that volume? • Aryabhata is saying:

50. But Really