Saturday, March 27, 2010

Pictures from Machu Picchu

John Hemming provides a precise summary of the famous archaeological site in The Conquest of the Incas:
Most visitors to Peru see Machu Picchu, which is perched on a narrow saddle of rock high above a hairpin curve of the Urubamba [river]. The granite sugarloaf of Huayna Pichu towers above the ruin, and the surrounding forested hillsides are often gripped by shrouds of low clammy cloud. Such scenery makes Machu Picchu one of the world’s most eerily beautiful ruins.

I took these pictures on Christmas day, 2009. It was gloomy, rainy day, but this only enhanced the beauty of the place. Machu Picchu well and truly lives up to its hype. The animal in the first picture is a drenched llama. The fifth picture is of a turbulent Urubamba river, which is part of the Amazon system. The last shows the hairpin curve of the Urubamba observed from the perch of Machu Picchu. At the base of the mountain is the train station where visitors disembark. The final part of the journey is by bus, which switchbacks its way through the forested hillside before reaching the ruins.

Saturday, March 20, 2010

King Khan in Lima

My Name is Khan (MNIK) is one of the worst movies I’ve seen this year. The embarrassing sentimentality of its scenes, which promote a cheesy, highly simplistic race-and-religion-transcending solidarity, make it unwatchable in parts. Bollywood’s alpha males – the Khans and the Roshans – are no longer overtly alpha-male, but they still are superhuman in other ways: they deliver babies using vacuum cleaners (3 Idiots); they invent new devices and fix just about anything (3 Idiots and MNIK); and most importantly, they live the most ideal and moral lives, are tremendously compassionate, follow their own unique visions, and deliver telling truths.

What else is this if not poorly concealed narcissism?

But my post isn’t supposed to be a rant on the modern day Indian superhero. Rather, it’s a rambling account on how I came to know of the popularity of a certain Khan in an unlikely city: Lima, Peru. This shouldn’t surprise us, given how interconnected the world is today. In fact, most Africans and Central Asians I’ve spoken to are decently well versed with Bollywood.

And yet Lima? The Latin American capital is halfway across the world from India; besides Peruvians have as little clue about Indians as Indians about Peruvians. Still, Shahrukh has succeeded in establishing himself there, to the same extent that Machu Pichu – that most iconic and magical of archaeological sites – has succeeded in becoming a much sought after destination for Indians with money (when I landed in Lima, prominent among the names displayed on cutouts by the receiving parties at the airport, were “Mukherjee” and “Patel”).


Once, during a bus ride in Lima, I was seated across from the driver. He was a cheerful man. When he learned where I was from, he looked at me with a sort of awe that can only come from having discovered something immeasurably exotic. He announced my nationality a few times to the ticket collector, who wasn’t impressed. After we’d got past discussing Taj Mahal, he settled on Shahrukh Khan, with whom he was clearly besotted. Unfortunately, I was reduced to speaking in gestures and nodding sagely though I understood very little of what he was saying.

At a Peruvian-owned DVD shop a few blocks away from downtown Lima, near the Rimac river, I found an entire section dedicated to Bollywood. There was a massive poster of Jodhaa Akbar, with Hrithik and Aishwarya prominent. But if you look closely, a Khan poster lurks behind to the right, sidelined and only partly visible.

But Khan shouldn't feel slighted, for the most artistic of tributes to him in Lima comes from this illustrator, whom I found busy at work in a street not far from the DVD shop.


Just when Shahrukh seemed undisputed king in Lima, I learned that the immensely popular Thalaivar, who drives Tamil fans wild, has actually graced the land of the Incas for a song shoot at Machu Pichu in the Andean highlands (picture credits here). As can be seen, we have a feathered Aishwarya hopping with a bearded and macho looking Rajnikant. If the extras were local, then it follows that the Indians of India must have danced with the Indians of Peru.

I am sure the spirits of the dead Incas must have doubled in laughter upon watching this: “So these are the people we were mistaken for?”

Update: There is even a fan page on Facebook, called Bollywood Peru.

Saturday, March 13, 2010

On Kim Plofker's Mathematics in India

That the concept of zero and the now widely used, indispensable decimal numeric system (imprecisely called Arabic numerals) came from India is old news -- a piece of trivia circulated often enough to have little or no surprise value. There is more, much more, to ancient Indian mathematics than these iconic contributions. Though I am trained in the quantitative sciences, I learned all of it – linear algebra, the theory of probability, and combinatorics – from the classrooms and labs of US universities, and very little from India. So it is only appropriate – though also ironic in another way – that the knowledge of ancient India should be made available to me through the painstaking research of a Western historian of mathematics, Kim Plofker.

Plofker has plowed through a range of Indian texts, beginning from the Vedic period – that’s three thousand years ago – to the eighteenth century to produce Mathematics in India. Hers is an achievement as much of linguistics, translation and history as of mathematics. The book sits on my desk, and there it shall be for some years, gathering dust, as I dip in only occasionally. It’s not the type that can be easily read from cover to cover – no book of mathematics is – but Plofker tries her best to make terms and concepts accessible.

Luckily, David Mumford, winner of the Fields Medal – the most prestigious prize in mathematics – has written a six-page essay on the book, which captures, concisely, the major achievements of Indian mathematics. There is the use of Pythagoras’ famous theorem in the construction of pillars, some centuries before the Greek philosopher is said to have postulated it in fifth century BC; Pāṇini’s rules of Sanskrit grammar and recursion, which “without exaggeration…anticipated the basic ideas of modern computer science”; Pingala, whose study of Sanskrit verses led to the binary notation and the development of Pascal’s famous triangle, useful in the calculation of binomial coefficients (which, coincidentally, is what I am teaching now); and Madhava of Sangamagramma (circa 14th century), the genius of the Kerala School, who contributed along with others, to "the discoveries of the power series expansions of arctangent, sine, and cosine" (a text on this in Malayalam has survived).

But what interested me most was the applied orientation of Indian mathematics. Like the ancient Mexicans, who designed sophisticated calendars (and who also used the concept of zero, probably earlier than the Indians did, though such rat races about who did what first are useless beyond a point), the mathematicians of India were spurred by the questions of astronomy. But the applied orientation goes even further than that. Mumford writes:
"It is important to recognize two essential differences here between the Indian approach and that of the Greeks. First of all, whereas Eudoxus, Euclid, and many other Greek mathematicians created pure mathematics, devoid of any actual numbers and based especially on their invention of indirect reductio ad absurdum arguments, the Indians were primarily applied mathematicians focused on finding algorithms for astronomical predictions and philosophically predisposed to reject indirect arguments. In fact, Buddhists and Jains created what is now called Belnap’s four-valued logic claiming that assertions can be true, false, neither, or both. The Indian mathematics tradition consistently looked for constructive arguments and justifications and numerical algorithms [Mumford's italics]. So whereas Euclid’s Elements was embraced by Islamic mathematicians and by the Chinese when Matteo Ricci translated it in 1607, it simply didn’t fit with the Indian way of viewing math. In fact, there is no evidence that it reached India before the eighteenth century."
This is particularly pleasing to me, because my own research, coincidentally, is about using mathematics in computationally feasible ways for various applications, rather than basking in the beauty and abstraction of theorems. (I will admit, however, that I was once obsessed with writing a paper that had only theorems and proofs in it; I did manage something, but only by borrowing significantly from the work of others; and still, months of edits await the paper before it can find its way to print.)

What might be an example of this “applied” orientation? Mumford writes of the “the discovery of the formula for the area and volume of the sphere by Bhaskara II” (circa 12th century):
“Essentially, he [Bhaskara II] rediscovered the derivation found in Archimedes’ On the Sphere and the Cylinder I. That is, he sliced the surface of the sphere by equally spaced lines of latitude and, using this, reduced the calculation of the area to the integral of sine. Now, he knew that cosine differences were sines but, startlingly, he integrates sine by summing his tables! He seems well aware that this is approximate and that a limiting argument is needed but this is implicit in his work. My belief is that, given his applied orientation, this was the more convincing argument. In any case, the argument using the discrete fundamental theorem of calculus is given a few centuries later by the Kerala school, where one also finds explicit statements on the need for a limiting process, like: “The greater the number [of subdivisions of an arc], the more accurate the circumference [given by the length of the inscribed polygon]” and “Here the arc segment has to be imagined to be as small as one wants. . . [but] since one has to explain [it] in a certain [definite] way, [I] have said [so far] that a quadrant has twenty-four chords.”
One of the delightful bits in the excerpt is the choice of “twenty-four chords” – it implies a casual, intuitive approach, yet the same is preceded by knowledge of what needs to be done to achieve exactness. That’s the essence of a “heuristic” or an approximation, which engineers use all the time.

Finally, the closing paragraph of Mumford’s essay:
It is high time that the full story of Indian mathematics from Vedic times through 1600 became generally known. I am not minimizing the genius of the Greeks and their wonderful invention of pure mathematics, but other peoples have been doing math in different ways, and they have often attained the same goals independently. Rigorous mathematics in the Greek style should not be seen as the only way to gain mathematical knowledge. In India, where concrete applications were never far from theory, justifications were more informal and mostly verbal rather than written. One should also recall that the European Enlightenment was an orgy of correct and important but semirigorous math in which Greek ideals were forgotten. The recent episodes with deep mathematics flowing from quantum field and string theory teach us the same lesson: that the muse of mathematics can be wooed in many different ways and her secrets teased out of her. And so they were in India: read this book to learn more of this wonderful story!

Sunday, March 07, 2010

The conquest story revealed. And coming up...

Alright -- it’s time to get more direct. February was full of strange posts that had to do with a famous conquest in history. Which conquest was this and why wasn’t I referring to it more directly? The reason is a decidedly vain one. Just as the hero in an Indian movie has to make a special, memorable entry, so I too was looking for some grand way in which to begin writing about my travels of last December and January. And if my attempt only made you yawn and put you to sleep -- well, at least it served some purpose.

So let me get it out of the way: the story I was referring to was the conquest of the Incas. The mountain range along whose length the kingdom had its domain is the Andes; HC, the king in the first part, is Huayna Capac; the warriors from the ocean were Spanish conquistadors, the battle-efficient beasts they ride are horses (which the Incas had never seen before, hence the belief that the men were half-men, half-beasts); the captain of the invading warrior army, FP, is Francisco Pizzaro; HS, one of Huayna Capac’s sons, is Huascar; the other son, AH -- the protagonist of the story, who was kidnapped by the Spaniards -- is Atahualpa; the capital city C, is the 11000-feet high Andean city of Cuzco; the northern city, Q, is Quito; M, where the history-changing, woefully one-sided battle was fought and lost by the Incas, is the city of Cajamarca. Finally, the symbol with the intersecting pieces is the Christian cross, and the box with the creased pads and strange printed symbols, which Atahualpa threw in irritation, is the Bible.

I hid the names deliberately to tickle your curiosity, but by not talking about the horses, the beards of the Spaniards, the cross and the Bible directly, I was also trying to capture the disorientation of the Incas who had never seen such things before. (They did not have a writing system but that did not impede them from running an efficiently administered empire; they did have complex knotted strings, called qhipu, which clearly had some sort of accounting function, if not more).

The Andean empire of the Incas was the last completely isolated empire in the world. That makes their achievements even more special and original, but it also made them vulnerable to the Spaniards who came with the knowledge and resources of both Europe and Asia, a much larger, more contiguous, more diverse and better connected landmass than the Americas.

My twenty-day visit to Peru and Bolivia inspired the posts, as did John Hemming's magisterial and impeccably researched The Conquest of the Incas. In the next weeks and months, I will write about the history of these countries, provide photographs, excerpts from books, and discuss current affairs. Bolivia, where Evo Morales was recently reelected, presents a fascinating study in popular left wing movements. That is precisely the reason I visited the city of La Paz, whose politics is as dramatic as its impossibly high setting (at over 12,000 feet, La Paz is the highest capital city in the world).

Watch this space, then. The posts won’t come all at once or regularly, but come they will.