The Significance of Proofs

Justice Katju’s post, mentioned in my previous post, motivated me to get a copy of “The Aryabhatiya of Aryabhata” by Walter Eugene Clark.

Some Rules from the Book

The translation of rule 19 in the chapter titled Ganitapada, as given by Clark:

“19. The desired number of terms minus one, halved, plus the number of terms which precedes, multiplied by the common difference between the terms, plus the first term, is the middle term. This multiplied by the number of terms desired is the sum of the desired number of terms.

Or the sum of the first and last terms is multiplied by half the number of terms”

Clark then goes on to explain the above:

“This rule tells how to find the sum of any number of terms taken anywhere within an arithmetic progression. Let n be the number of terms extending from the (p + 1)th to the (p + n)th terms in an arithmetical progression, let d be the common difference between the terms, let a be the first term of the progression, and l the last term.

The second part of the rule applies only to the sum of the whole progression beginning with the first term.

S = n * [a + {((n -1)/2) + p} * d]

S = {(a + l) * n}/2

 “

The above is the sum of an arithmetic progression. Where is the proof/derivation of the rule?

Clark then gives Aryabhatta’s rule 20 of the Ganitapada as follows:

“20. Multiply the sum of the progression by eight times the common difference, add the square of the difference between twice the first term and the common difference, take the square root of this, subtract twice the first term, divide by the common difference, add one, divide by two. The result will be the number of terms.”

Really? Prove it!!

Using symbols the rule can be expressed as the algebraic formula:

N = [({sqrt{(8 * d * S) + (d – 2 * a)^2} – 2 * a}/d) + 1]

This is obtained from the first expression for S above, by taking the series from the first term onwards (p = 0). Since the expression contains the square of n, it means solving for n requires solving a quadratic equation. Why does the Aryabhattiya not have a derivation for the general formula?

The rules that follow are even more mind blowing! But, regrettably, no proofs/derivations.

Importance Attached to Memory

The rules (algorithms?), in the original Sanskrit, were in stanzas. Being in the form of poetic verse would make them easier to memorize. This was a handbook that could be memorized.

Memorizing formulae does not produce a trained mathematical mind. The discipline of proofs/derivation does.

Memorizing rules is good for speed. Memory, however, is fallible, Good practitioners should be able to at least derive the important formulae, which they use, from first principles – or at least have understood it once in their life.

Implication of Missing Proofs

Somehow I find it hard to believe that the derivations/proofs did not exist. What happened to them? Why were the proofs/derivations not passed on? Were they not valued? If they exist, why are they not given pride of place?

Without proofs/derivations what we have is not a textbook. What we have is a handbook of formulae. Handbooks are useful. But can the study of handbooks discipline the mind to be capable of making new discoveries and constructs? Computational rules/algorithm are Know-How. Proofs are Know-Why. Know-How is good enough for applying knowledge. Creating knowledge requires Know-Why. What does it say of us as a civilization if we celebrate formulae and disregard proofs?

A democracy requires debate. Debate requires ascertaining facts and applying logic. A democracy cannot exist on received wisdom (“Shabda Pramana”).  If as a society we blindly celebrate mathematical formulae without regard to proofs, will we celebrate rational discourse? Our preferred source of knowledge in all fields will be the word of authority (“Shabda Pramana”). Why then should we be surprised if our education system rewards rote learning?

Manjul Bhargava who got the Fields Prize this year, figured out the number of oranges that could be stacked on a triangular base whose sides had N oranges. The number equals N(N + 1)(N + 2)/6. Bhargava did it at the age of 8. The same formula is given as rule 21 of the Ganitapada. The proof can be seen here.  The proof depends upon a result proved on another webpage. That is how mathematics progresses – one proof built on one or more other proofs.

One proof built on other proofs: Is it conceptually identical to good engineering? I shall expand on that in my next post.