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

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• Why Do This??

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

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Ashtanga First SeriesForward Bends
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First SeriesBend more
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One Drink Too Many.
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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

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Second Seriesbend back
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and waaay forwards
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and ???
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Layer 3 Manomayakosha
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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

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Third SeriesBalance
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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.
• Its understanding that conducts the sacrifice
  its understanding that performs the rites. Its
  understanding that all the gods venerate as the
  foremost brahman.

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Fourth Series
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Fourth Series
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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.

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Seated PracticesPranayama, Meditation
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Uttaravedi Agnicayana
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Vedic Ritual ReenactmentKerala, South India,
1975(courtesy IGNCA)
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The Sacrificial Enclosure
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Kalpa Computations
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Soma (Ephedra Dystachia)
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Agni Manthana
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Agni Manthin
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Agni Manthin

• Verbal root manth
• 1. To churn

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Agni Manthin

• Verbal root manth
• To churn
• To agitate, disturb

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Agni Manthin

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

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Churning the Ocean
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Rahu and Ketu
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Celestial Sphere(Gola)
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Rashis and Nakshatras
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Two Main Applications of Mathematics in India

• Kalpa (layout of sacrificial area, construction
  of ritual devices)

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

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Aryabhata

• Born 476 CE
• Astronomer, mathematician
• Author of the Aryabhatiya
• taught by Brahma
• Rotating earth

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Nilakantha Somayajin

• Nilakantha blue-throated
• Somayajin soma-sacrificer

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Nilakantha

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

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Jyotirmimamsa

• This is what some think
• Pleased by feats of asceticism, Brahma taught
  to Aryabhata the characteristics of the planets
  .. Because of Brahmas 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.

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The Aryabhatiyabhashya

• Composed late in his life
• Written on multiple levels
• Commentary on the Ganitapada alone is 180 pages
• Detailed proofs, but no diagrams!

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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 hes
  wrong)!

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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.

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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 thats how its commonly
  attempted. But in that case one doesnt get an
  even circumference, due to the lack of constant
  extension.

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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.

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Dissect the Circle, Rearrange
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• When all of them are placed together,
  tip-to-end in pairs, rectangle-ness occurs.

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You Get This
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Do Same Thing,With More Slices
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In the Limit
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But what about that volume?

• Aryabhata is saying

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But Really
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Aryabhata 17(ab)

• The square of the arm plus the square of the
  upright is the square of the diagonal.

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Connection to Areas
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Our StrategySlice, Dice and Rearrange
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