Bookstores are filled with titles like “Extreme Indian Arithmetic Drills” and “The Unknown Secrets of the Indians.” Newspapers carry reports of Indian children memorizing multiplication tables far beyond nine times nine, the standard for young elementary students in Japan.

And Japan’s few Indian international schools are reporting a surge in applications from Japanese families.

At the Little Angels English Academy & International Kindergarten, the textbooks are from India, most of the teachers are South Asian, and classroom posters depict animals out of Indian tales. The kindergarten students even color maps of India in the green and saffron of its flag.

Little Angels is located in this Tokyo suburb, where only one of its 45 students is Indian. Most are Japanese. [link]

Wow. Never thought I would thank my sixth standard (that’s “sixth grade” in Americanese) class teacher, the late Mrs. Sachdev, but thank you, Mrs. Sachdev for teaching me to go beyond nine times nine. I am now cool in Japan.

The first wave of slicing up services work and sending it abroad has been all about business operations. Computer programming, call centers, product design and back-office jobs like accounting and billing have to some degree migrated abroad, mainly to India. The Internet, of course, makes it possible, while lower wages in developing nations make outsourcing attractive to corporate America.

The second wave, according to some entrepreneurs, venture capitalists and offshoring veterans, will be the globalization of consumer services. People like Ms. Yamaki and Mr. Tham, they predict, are the early customers in a market that will one day include millions of households in the United States and other nations. [link]

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So it looks like the theory that calculus in India predated Newton’s and Liebnitz’s calculus is back in the news again. I blogged about it in a series of three posts some time back, and it’s good to see that it’s getting some traction. I seriously doubt it will enter the mainstream any time soon, though. Well, not until India dominates the world, that is. Rest assured, the probability of that happening in any of our lifetimes is about the same as it snowing in San Francisco in July.

This is interesting because it adds support to the theory that mathematical developments in Kerala were exported to the West through this trade route. I’m referring to my two earlier posts [here and here] about how much of mathematics is purported to have been developed in India, and of course, how little credit is actually given. In that post, I mentioned that there is some evidence to show that calculus (amongst other mathematical concepts) was transmitted to Europe after being developed in Kerala. This was well before Liebniz and Newton were said to have developed it. In fact, the world’s first calculus text was developed in India, when Jyeshtadeva compiled the findings of the Kerala school in a text sometime in the 16th century. Given that this is the case, one might even speculate as to whether its a coincidence that calculus began making great strides at around this time in Europe. Well, maybe you can write that off as a crazy conspiracy theory, but I think its definitely worth taking a deeper look at exactly how much of calculus was developed in Kerala.

Math Teachers: This should be, at the very least, mentioned in any calculus class – college or high school. Most of the usual talk is about how there was a running feud between Newton and Liebniz, and so forth. Certainly, credit should go to them if they indeed developed these concepts on their own, but Kerala’s contributions should definitely be mentioned.

Desis: Instead of wasting your time on changing so-called “controversial” texts in school text books both, in India and the US (California) over so-called religious and colonial (non)issues, I think this is a more worthy endeavour to undertake. Wouldn’t you agree?

So, where does one begin looking for these kinds of things? Of course, the only places to start are Wikipedia and Google. As it turns out, there are quite a few resources out there about the subject. Furthermore, there are tremendous contributions by Indians to mathematics, so it seems. So, what are they? I'll get into that, but first, a disclaimer: I am, by no means, an expert on the history of math, and furthermore, I haven't actually verified any of the stuff I found.

Ok, so as I was saying, it seems that Indian contribution to mathematics was tremendous. Einstein, himself, said,

We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made.

Even Laplace (the 19th century French mathematician, whose known to me for his contributions to probability theory) said,

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

According to Wikipedia, one of the earliest acknowledgements to contributions by Indians to mathematics was made (or rather not made – see the phrasing of the quote) in 662 AD or CE by a Syrian bishop by the name of Severus Sebokt (also spelled Sebokht) whose main claim to fame was to translate Aristotle's Logic and, among other things, play some role in bringing the Indian numbering system to the Arab nations. The proof of that, lies in this passage written by him:

I shall not speak here of the science of the Hindus, who are not even Syrians, and not of their subtle discoveries in astronomy that are more inventive than those of the Greeks and of the Babylonians; not of their eloquent ways of counting nor of their art of calculation, which cannot be described in words — I only want to mention those calculations that are done with nine numerals. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value.

So, clearly, I am (as usual) a little late in catching on to these things. However, there are several periods in the Indian contribution to mathematics, including the Harappan period, the Vedic period described mainly in the Sulba Sutras, the Jain period, the Classical period and the more recent Kerala mathematics. One could probably write a book on each of the individual periods, so sufficeth to say that each of these periods have made some contribution or another. These include topics as diverse as (of course) the number system, the calculation of pi, the concept of infinity, what is known as the Pythagorean Theorem (a primitive form of which was originally devised in the Sulba Sutras to construct sacrificial altars, of all things), logarithms, differential calculus and much much more. I should also mention that there were contibutions to mathematical logic as well, and one of the main contributions to this area comes from Panini who contributed through his work on linguistics and grammar. On his work, Wikipedia has said that:

A consequence of his grammar's focus on brevity is its highly unintuitive structure, reminiscent of contemporary "machine language" (as opposed to "human readable" programming languages).

This, of course, completely explains why Indians are so good at anything IT-related (note: facetious tone).
Also, as I have come to realize, that the charges of Eurocentrism are quite well-known and backed up by people like Florian Cajori a 20th century Swiss math historian who taught at several places in the US, including UC Berkeley. Here's what Wikipedia had to say about the Eurocentrism:

Unfortunately, Indian contributions have not been given due acknowledgement in modern history, with many discoveries/inventions by Indian mathematicians now attributed to their western counterparts, due to Eurocentrism.

Even Ian G. Pearce, a student of mathematics who has written about the history of Indian mathematics, has a chapter on the "Possible transmission of Keralese mathematics to Europe." He says that Indians may have been instrumental in European navigation techniques.

There were significant financial rewards for 'anyone' who could 'assist' in the improvement of navigation techniques. It is thought 'information' was sought from India in particular due to the influence of 11th century Arabic translations of earlier Indian navigational methods.

Does anyone see the irony here? We may well have led to the improvement of the very techniques that were used to get to India and eventually colonize us! A final note on Eurocentrism from Wikipedia:

More recently, evidence has been unearthed that reveals that the foundations of calculus were laid in India, at the Kerala School. Some scholars have suggested that calculus and other mathematics of India were transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China, Arabia, and from around 1500, Europe as well, thus transmission would have been possible. There is no direct evidence by way of relevant manuscripts but the evidence of methodological similarities, communication routes and a suitable chronology for transmission is hard to dismiss.

Interesting stuff, to say the least…

Firstly, to establish some kind of credibility, he says that

[T]his is not a rhetorical proclamation of some overzealous Indian chauvinists. Two India-born American university professors, V. Lakshmikantham and S. Leela, have documented extensive new data on ancient Indian mathematics and on the bankruptcy of the theory of Aryan invasion of India from the northern-central plains in Asia.Along with their own meticulous research of original Sanskrit texts and related vernacular literature, the authors draw upon the works of a few European scholars. With the publication of this amazing monograph on Indian mathematics, the cloud of ignorance and deliberate misrepresentation of the many achievements in ancient India is beginning to lift.

Apparently, there was a mix-up between Chandragupta Maurya of the Mauryan dynasty and Chandragupta of the Gupta dynasty. Moreover, it seems that this mix-up was deliberate. I don’t follow exactly what Joshi is saying here, but it seems that the names were deliberately “confused” to bring all the ancient Indian scriptures into the Christian era. In his words:

Thanks to the inventive and resourceful William Jones of the IHS, the entire chronology of events was summarily shortened by more than 1,200 years. Consequently, the times of ancient astronomers and mathematicians had to be moved into the Christian era.

It seems that even the famous indologist Max Mueller got things screwed up.

Max Mueller, concocted the age of the Rig Veda to be 1200 B.C., with the stipulation it was written by nomadic Aryans (riding on horseback, presumably with a mobile library). Actually, the Rig Veda was compiled well before 3000 B.C.

A few other interesting facts that follow from this re-dating of things:

Contrary to popular belief, Gautam Buddha lived during 1887–1807 B.C., and the short but remarkable life’s mission of Adi Shankaracharya was accomplished between 509 and 477 B.C. The first known mathematician and astronomer from India, Aryabhatta, was born in 2765 B.C., and the Sulvasutras, heralding the discipline of geometric algebra, were completed before his birth. But in the occidental “scholarship,” Aryabhatta’s year of birth was changed to 476 C.E. with the misreading of his epoch-making Aryabhatteeum.

And this is where I get confused:

These were not accidental errors, but were the result of a carefully planned alteration of manuscript copies. Notice that the four Vedas preceded the Sulvasutras. Note also none of the Vedangas, the Upangas, the Brahmanas, the Aranyakas, and the Upanishads could possibly have been written later than the second millennium B.C.

Methinks Mr. Joshi has fallen prey to the very over-zealous chauvinism that he criticized in the beginning. However, some more interesting facts that come to light through the review are that the Indians invented (or discovered) quite a few mathematical concepts that are attributed to Egyptions and Greeks.

For example, we are told the Gregory-Leibniz series for p/4 was first discovered by Nilkanta and was clearly stated in his Tantra Sangraha (1500 C.E.). The so-called Pythagoras’s Theorem (sixth century B.C.) and its converse was known to the Indian sages of the third millennium B.C. The general principle of trigonometric functions was enunciated in the Surya Siddhanta, preceding even the Sulvasutras period. Brahmagupta (30 B.C.) solved the second order indeterminate equation Nx^2 + 1 = y^2, and foresaw Newton’s Law of Gravitation. The authors also demonstrate that Bhaskara II (486 C.E.) had the expertise in the area that was re-invented and, of course, systematized as Differential Calculus by Newton and Leibniz in the late 17th century. The Greeks got their plane geometry from India and their language was derived from Sanskrit. Incidentally, the Greeks “themselves had supposed or conjectured, that they had received their intellectual capital, especially in geometry” either from China or from India.

Now that is interesting: Indians discovered Pythagoras’ Theorem, trigonometric functions and a whole lot more. Well, Mr. Joshi then goes into a rant towards western civilization in general, so I’m not going to get into all of that. If you’re interested in reading it, then go to the link above by all means. However, I think the book would make an interesting read. Definitely worth the $26 (which just about puts it into the Free SuperSaver Shipping category at Amazon).

Update: I followed up on this posting here and here.

But wait. To attain nirvana, one must be tuned into the cosmic consciousness! And how can one expect to get tuned into the cosmic consciousness, when there's no machine to do that!! Well, that's all in the past now. An NYU student Rob Seward, has created a Consciousness Field Resonator for his Master's thesis to tell us about the ebbs and flows of the universal (ok, not universal, but global – you got to start somewhere!) consciousness.

So, the story begins with all the various random number generators at various research institutions across the world. It seems that every time a major earth-shattering event occurs (say, something like 9/11 or Princess Di's funeral) there are statistical anomalies within the numbers generated by them. We know that because they all feed into this one server at Princeton which looks for and catches these statistical anomalies. The reason, they say, for these statistical anomalies is that there are changes in the global consciousness.

Rob Seward's creation has a random number generator and hardware designed to catch anomalies in generation of these random numbers. Which means that if the generator produces numbers which have some kind of discernible pattern (which they really shouldn't – but will only have when there are changes in the global consciousness level) then the machine will alert the owner to these changes.

So, the bottom-line? You have a companion piece to the Buddha Machine. You can buy enlightenment and a machine that tells you about the crests and troughs of global consciousness. I'm reserving my comments about how the *&%# anyone can spend valuable dollars of research funds on this stuff for another post…

Step 1: Create some pi ambiance. Just like people display a tree and mistletoe, wear Christmas gear, and sing Christmas songs around Christmas, there is a lot of room for making the environment around you reminiscent of pi. Probably the favorite of most is to wear a pi t-shirt–this idea can be taken further to include pi jewelry (maybe a necklace whose beads represent the numbers in pi), a pi mug or clock, or other pi paraphernalia.

Step 2: Convert things into pi. This step is absolutely necessary for two reasons: To utterly confuse people who have no idea what you are talking about (thus opening the door for enlightenment) and to have fun seeing how many things can be referenced with pi.

And my favorite:

Step 4: Eat pi foods. Many creative ways exist to do this. First, there’s the punny approach, like eating pineapple, pizza, or pine nuts and drinking pina coladas or pineapple juice. Second, there’s the shape approach, like making cookies or pancakes shaped like pi or making a pie with a pi cut out of the center of the crust.

More here.