Fields Medal Winner Manjul Bhargava On the Pythagorean Theorem Controversy

prajendran writes There were a lot of controversies generated at the Indian Science Congress earlier this month, including claims of ancient aircraft in India, the use of plastic surgery there, and ways to divine underground water sources using herbal paste on the feet. One argument that could be tested using some form of evidence was the assertion by Science Minister Harsh Vardhan that the Pythagorean theorem was discovered in India. Manjul Bhargava, a Princeton University professor of mathematics and a Fields Medal winner describes why the question is not defined well.

Comments

Divergent creation theory

[Anonymous Coward]

It could have been created in both places, it being a relatively simple law of mathematics that anyone pondering triangles is bound to discover soon enough.

There are other examples of things being invented in two separate places at roughly the same time. Why the need for bragging rights? Let the evidence do that.

Re: Divergent creation theory

[Anonymous Coward]

"Why the need for bragging rights?"

Because humans are petty little prideful bitches who use nationalism to compensate for insecurities and a lack of any real sense of self worth. This is the paleo-mathematical version of "whose dick is bigger".

Re: Divergent creation theory

[ArmoredDragon]

I know parent is trolling, but that's actually a real problem with real world consequences that they've been running into:

http://news.bbc.co.uk/2/hi/sou...

Re:Divergent creation theory

[queBurro]

Newton and Leibniz... http://xkcd.com/626/

Re:UFC Fight Night

[Headw1nd]

I almost modded this funny, thinking it was a parody. Is the spam becoming aware of it's surroundings?

Does it really matter now?

[SeaFox]

Pythagorean theorem was discovered in India

Nobody's going to change the name of it now, and there's no copyright royalties to be had on it.

he made a very good point

[onepoint]

He made a very good point, it's all about perspective.

With that said, does it not sound like India reading a page from the book "stranger in a strange land"

Re:he made a very good point

[swillden]

The only good point he made is that by mathematical standards the question is who proved the theorem.

I disagree. Proofs aren't the only important element of mathematical creation/discovery. Conjectures are also crucial, and there are lots of important conjectures which are notable long before they're proved. The Pythagorean theorem is clearly one such, because it's extremely useful even if you can't prove it. For that matter, as noted by the article, the Egyptians found it very useful, and they not only didn't have a proof, they didn't even fully understand the relation. They merely knew that some certain combinations of proportions made right triangles... and then used that fact all over the place. The Babylonians also probably understood the principle, and the Pythagoreans likely learned it from them or the Egyptians.

In addition, even a proof is irrelevant if it just gets lost, or buried. Communication of proofs, especially as part of a systematic theory is even more important and -- as you correctly noted -- that achievement is indisputably Greek. How much of it was due to the Pythagorean mystics and how much to Euclid is a matter of much debate; some historians of mathematics argue that the Pythagoreans discovered essentially everything in the first two books of Elements. Euclid's main achievement with respect to the theorem may well have been mostly just to record it and remove all of the references to beans and the rest of the Pythagorean mysticism. What the truth is we'll likely never know, but the Greeks attributed the knowledge of the theorem to Pythagoras, which I think is quite meaningful.

All of these stages in the development, proof, formalization and dissemination of important ideas are crucial. The best point to be made here is that the question is inherently meaningless. Any attempt to pick an "origin" must fail because the theorem originated over millenia, and was likely independently discovered in different regions at different times. Even if it's a Chinese manuscript that contains the earliest proof, it seems unlikely that the Greeks got it from the Chinese, and it appears that the Chinese proof in question had little effect on history, Eastern or Western, while the Greek proof, alongside the rest of Elements, fundamentally shaped Western civilization.

That last claim may seem a little too strong, but it's not. Greek Mathematics didn't so much influence Greek philosophy as create it, and Greek philosophy similarly founded Western philosophy as a whole.

Plato's philosophy in particular, was essentially mathematical, and his notion of Forms, the central element of his ideas, is clearly an attempt to relate the pure, abstract beauty of geometry to the world as a whole, and to use it as a vehicle for understanding reality and man's relationship with it. Aristotle was, in many ways, the anti-Plato, but he also deeply honored mathematics. All of the rest of Western philosophy, including its deep influence on social and political structures, can be viewed, as Russell said, as a series of footnotes to Plato and Aristotle, they were that important. And a large part of the powerful influence of Greek ideas on Roman, medieval Christian, Renaissance and modern philosophy derived from the elegance and power of Greek mathematics. Although it wasn't often stated so clearly, the indisputable clarity and power of Greek mathematics impressed later generations and convinced them that the rest of Greek wisdom might well be equally profound.

The Pythagorean version, as presented by Euclid, mattered.

There may have been a half-dozen proofs of the Pythagorean theorem created, recorded and lost, in many locations around the world, perhaps long before Pythagoras. But none of them mattered. The one that did is the Greek proof, and the Greeks credited the Pythagoreans.

Re:Umm, no.

[ColdWetDog]

Or you could just RTFA and discover that the nice Indian mathematician had some cogent and logical things to say.

TL;DR - it's complicated.

Re:India? I don't think so...

[fustakrakich]

Pythagoras hacked Sony to suppress the truth

Pythagoras imposed his quasi-religious philosophies... about never urinating towards the sun...

when Pythagoras’s student Hippasus tried to calculate the value of [square root of] 2, he found that it was not possible to express it as a fraction, thereby indicating the potential existence of a whole new world of numbers, the irrational numbers (numbers that can not be expressed as simple fractions of integers). This discovery rather shattered the elegant mathematical world built up by Pythagoras and his followers, and the existence of a number that could not be expressed as the ratio of two of God's creations (which is how they thought of the integers) jeopardized the cult's entire belief system.

Poor Hippasus was apparently drowned by the secretive Pythagoreans for broadcasting this important discovery to the outside world.

Re:Umm, no.

[Anonymous Coward]

The nice Indian mathematician does bring up some nice cogent and logical things.

But he also leaves out some points which are fairly damning to the argument that the Indians had much to do with this. Many/most non-Indian historians of mathematics seem to believe that the key Indian document here was very likely based on earlier (non-Indian) traditions. In other words, it was just a copy of stuff from Mesopotamia.

I'll quote the wikipedia article on the Theorem (which in turn supplies full quotes from the scholarly document if you hate wikipedia):

"Van der Waerden believed that "it was certainly based on earlier traditions". Boyer (1991) thinks the elements found in the ulba-stram may be of Mesopotamian derivation."

That makes any claims that India "discovered" the theorem really really weak by any definition I would think.

Re:I dno't know.

[peragrin]

If you are going to nominate for lifetime achievement in pedantry I would go Slashdot.org instead if just one article.

Re:India? I don't think so...

[Anonymous Coward]

Pythagoras imposed his quasi-religious philosophies... about never urinating towards the sun...

That was a translation error. What he actually said is you don't piss against the wind.

Re:Umm, no.

[crunchygranola]

The nice Indian mathematician does bring up some nice cogent and logical things.

But he also leaves out some points which are fairly damning to the argument that the Indians had much to do with this. Many/most non-Indian historians of mathematics seem to believe that the key Indian document here was very likely based on earlier (non-Indian) traditions. In other words, it was just a copy of stuff from Mesopotamia.

I'll quote the wikipedia article on the Theorem (which in turn supplies full quotes from the scholarly document if you hate wikipedia):

"Van der Waerden believed that "it was certainly based on earlier traditions". Boyer (1991) thinks the elements found in the ulba-stram may be of Mesopotamian derivation."

That makes any claims that India "discovered" the theorem really really weak by any definition I would think.

I have actually read Van der Waerden's books on Mespotamian mathematics and astronomy (I have copies of them at hand). His "belief" is not evidence of any kind. He is simply supposing, without any supporting evidence.

And Boyer, who wrote his history of mathematics 50 years ago (1991 is a reprint, he died in 1976), was no expert in ancient mathematics. He has been called the "Gibbon of Mathematics" which is a very good analogy, since Gibbon's work represents a compilation of everything known and believed about the Romans, written from the perspective of an 18th century European, complete with moral interpretations drawn from contemporary cultural viewpoints. It was a work that says at least as much about Gibbon and Europe of the time, as it does about the Romans. Similarly Boyer's beliefs represent the assumptions of a western scholar trained in the 1930s.

No one has yet shown any evidence at all that the suryas actually draw from Mesopotamian sources. Saying it doesn't make it true.

Re:Umm, no.

[Anonymous Coward]

It's complicated, but the arguments for each can be summarized:

First known indication of knowledge of the relation for integer-sided triangles (Pythagorean triples) - 2,500 BC in Egypt
First known general statement of relation - ca. 1,800 BC in Mesopotamia
First known general statement of relation with respect to right triangles - ca. 800 BC in India
First known rigourous proof of the relation - ca. 1046 to 256 BC in China
(Pythagoras, who may or may not have had a proof - ca. 570 to 495 BC in Greece)

British!

[LostMyBeaver]

It was created by God and brought to Earth by his son Jesus and he's British!

You heathen bastards!

Re:The theorem part

[sconeu]

That's not how I RTFA'ed. I saw it as

• * Egypt had (undocumented) knowledge of a few right triangles
• * Mespotamia had documented knowledge of many right triangles, -- including large ones, which would indicate knowledge of the theorem
• * India has the first documented statement of the theorem
• * China has the first documented proof of the theorem

Re: Umm, no.

[Crazy Taco]

I was in Mumbai just over a year ago and went to the Nehru planetarium. They had a diorama there of the first moon landing. Everything looked perfect, from the Apollo spacecraft to the little astronaut in a space suit standing on the Lunar surface. There was one blatant problem though... they replaced the American flag with the Indian flag! My boss (also American) and I had a good time laughing about that.

Fields Medal Winner, not just the politician

[billstewart]

I'll give the Indian politician the amount of credit it was due, along with mystical spacecraft flying to other planets and such. But this article by a guy who won the bloody Fields Medal not only deserves a lot more credibility before reading it, but also after - he talks about the discoveries of various parts of the idea in different parts of the world. And Indian and Arab mathematicians did contribute a huge amount to culture and civilization; you can't even claim they made zero contributions without using the zero they contributed,

In a nutshell...

[gwstuff]

-- In India there is an undeniable and strong tendency to construct narratives of how everything good in the world was discovered in India. All Indians don't share this perspective, in fact it is shared by a minority, but er, that amounts to 150 million people or something.

-- This tendency is inward, not outward looking. This politician Harsh Vardhan is a fuck up, like a lot of Indian politicians. But generally this thinking is not directed at bragging to the rest of the world about how great India is, rather it is to nurse, heal, revive people's connections with their own trampled culture and history -- one that in recent times is increasingly being supplanted by a pseudo-western culture and western lifestyle. It's a way of telling people in India to give their intellectual heritage another chance.

-- Honestly, most rational people don't give a damn about where the Pythagorus theorem was invented. I mean if it were an easily provable fact, then it might be an interesting piece of historical information, but given that it's ambiguous who cares, unless to stoke one's nationalist ego.

-- The Princeton mathematician who won the Fields Medal... which is like a Nobel prize except that it's given once every 4 years... is a reference because of his grasp of mathematics, not because he's Indian. If you think of him as "some Indian guy trying to pocket a laurel for his fatherland" then that's a strong statement about you, not about him.

Re:Don't need theory to get right angles

[billstewart]

One of the real values of Euclid's Elements is the insistence on proof of everything, which is part of what differentiates it from much of Classical Greek "science"; assertions like Aristotle's claim that heavy objects fall faster than light ones weren't good enough. And it's not like the Pythagoreans weren't mystics either; there's a story that one of their deep dark secrets was the irrationality of sqrt(2), which really annoyed them because it showed that their mathematically perfect universe wasn't.

Knowing that a 3-4-5 triangle has a right angle isn't the same as being able to prove it, or as knowing the general principle behind why it's true. It's the kind of thing you can find by trial and error, and that (both the successful and unsuccessful trials) may be a starting place for reasoning about the general principles.

Re: Umm, no.

[iluvcapra]

"The USA was a major force in dismantling colonialization. The exceptions to this seem to have been through the involvement of Wall Street anglophiles (or just plain agents of the British) "

Hmm. James Monroe, Teddy Roosevelt, William McKinley, William Randolph Hearst, General Pershing and Commodore Perry, all Anglophiles...

Ask the Spanish-speakers of the Western Hemisphere about the US commitment to anti-colonialism.

Re: Umm, no.

[Crudely_Indecent]

That's the problem I've seen with Indian workers as well, and several people I've worked with have seen it too. It seems to be a real problem, because they will not say no to their bosses - about anything. If they're given a task beyond their skillset, they say yes anyway because saying no would be disrespectful (or so I understand).

On one occasion, I was hired to spend a day working with the IT manager of a company in Dallas - to find what was happening with their network performance. I found (poorly configured) routers everywhere. Triple, quadruple, quintuple NAT, cross linked networks - dueling DHCP servers. It was a mess. It turned out that their IT manager managed to graduate his Indian university with a computer science degree and yet knew virtually nothing about anything. When his boss said add another router - he said yes.

I left after turning those routers into switches and restoring the performance they were missing, but not the performance they could have had if they'd put it together with the right parts to begin with.

I was paid in cash, by the IT manager - so I suspect that I was paid out of his pocket to save his job.

Re:Watch out for Disney

[jabuzz]

Actually Disney would never argue for that, because they would be on the hook for billions of dollars in back copyright payments for all the works that they have "used" out of copyright.

Personally I feel that if a firm or body wants to make use of a copyright extension, then back payments would be applicable to people who's copyright would not have expired had that extension been in place when they made use of the work. So Disney for example would need to payout on Pinocchio as Carlo Collodi only died in 1890, so in 1940 it would still have been under copyright by modern standards.

Re:Watch out for Disney

[toddestan]

So how is extending the copyright on already existing works not an ex post facto law? Changing the terms for new copyrighted works would not be an ex post facto law, But a retroactive blanket change of the terms on already copyrighted works, many of which are decades old?