Ideas of Mathematical Proof in Classical India:

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Ideas of Mathematical Proof in Classical India:. A Reading of the Aryabhatiyabhashya of Nilakantha Somayajin. Why Do This??. The Five Sheaths of the Human Person (Taittiriya Upanishad). Layer 1: Annamayakosha.

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## Ideas of Mathematical Proof in Classical India:

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### Ideas of Mathematical Proof in Classical India:

of Nilakantha Somayajin

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.

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 …”
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…”
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.”
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.
Agni Manthin

Verbal root manth:

1. To churn

Agni Manthin

Verbal root manth:

• To churn
• To agitate, disturb
Agni Manthin

Verbal root manth:

• To churn
• To agitate, disturb
• To apply friction to any part of a body in order to produce offspring

Jyotisha (“star science”, began as timing of Vedic rituals, eventually turned to astronomy/astrology)

Two Main Applications of Mathematics in India
Aryabhata
• Born 476 CE
• Astronomer, mathematician
• Author of the Aryabhatiya
• “taught by Brahma”
• Rotating earth
Nilakantha Somayajin
• Nilakantha = “blue-throated”
• Somayajin = “soma-sacrificer”
Nilakantha
• 1444 – 1545 CE
• Kerala, South India
• Namputiri Brahmin
• Learned in Nyaya logic, Vedanta philosophy
• Investigations in the scientific method, “science and religion” issues
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.

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