An enigma that is India

-details soon-

Indian mathematics and the Kerala School

India's greatest contribution to mathematics is zero - this has been accepted universally. But is it the only major contribution? If you think so, think again - your view is utterly wrong! Well, to prove that Indian contribution to mathematics and astronomy is infinitely more than what we all think, I've done some background work and presented some of the most astonishing findings here. But please don't be lead to believe that this is some figment of my imagination! I have done a proper literature research and this article features the excerpts from the findings of some recognised historians and mathematicians, who are, mostly, Europeans. The main objective of this article is to make known certain facts that, I am ashamed to say, almost all the Indians aren't aware of. While we have been reading and admiring the mathematical and scientific inventions/discoveries of the Europeans, especially the Greeks, developments commensurate if not greater, have happened in our own backyard.

It is said that the winners always write the history...well, sometimes re-write the history. We Indians have been under foreign occupation for a millennium, which has lead to a pathetic state of affairs. The Muslim invaders and the Europeans carried away not only immensely valuable things from India, but also the know-how. Now, we believe that great inventions and discoveries have been made elsewhere, when the truth of the matter is that our ancestors were the ones to have accomplished them!

To a typical historian of mathematics today, if there is one certainty, it is that Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) were the first to ‘invent’ a generalised system of infinitesimal calculus, an essential prelude to modern mathematics. Well, they wish! At least two hundred years earlier, the astronomer-mathematicians of Kerala, notably Madhava of Sangamagrama and his disciples, had discovered elements of that calculus, the forerunners of modern techniques used in mathematical analysis. Given the existence of a corridor of communication between Kerala and Europe, especially from the sixteenth century onwards, and the crucial importance of calculus in the growth of modern mathematics, one would have expected that the possibility of the transmission of the Kerala mathematics westwards would be
high on the agenda for historical investigation. That such an investigation has not yet been carried out may reflect the strength and the pervasive nature of Eurocentrism in the history of science.

It is not just calculus alone...Indian scholars made vast contributions to the field of mathematical astronomy and as a result contributed mightily to the developments of arithmetic, algebra, trigonometry and secondarily geometry (although this topic was well developed by the Greeks) and combinatorics. Perhaps most remarkable were developments in the fields of infinite series expansions of trigonometric expressions and differential calculus.

Surpassing all these achievements however was the development of decimal numeration and the place value system, which without doubt stand together as the most remarkable developments in the history of mathematics, and possibly one of the foremost developments in the history of humankind. The decimal place value system allowed the subject of mathematics to be developed in ways that simply would not have been possible otherwise. It also allowed numbers to be used more extensively and by vastly more people than ever before.

The earliest origins of Indian mathematics have been dated to around 3000 BC and this seems a sensible point at which to commence any discussion, while work of a significant nature was still being carried out in the south of India in the 16th century, following which there was an eventual decline. It is hence a vast time scale of almost 5000 years, and indeed it may be greater than that, the estimation of 3000 BC is a slightly crude approximation, and there remains much controversy with regards to the dating of many works prior to 400 AD. It is also worth pointing out that this lack of certainty has allowed several unscrupulous scholars to pick dates of choice for certain Indian discoveries so as to justify suggestions for Greek, Arab or other influences.

We can accurately claim that Aryabhata was born in 476 AD. He was 23 years old when he wrote his most significant mathematical work the Aryabhatiya (or Arya Bhateeya) in 499 AD. He was a member of the Kusuma Pura School, but is thought to have been a native of Kerala (in the extreme south of India), although unsurprisingly there is some debate. It is a concise astronomical treatise of 118 verses written in a poetic form, of which 33 verses are concerned with mathematical rules. It is important here to point out that no proofs are contained with his rules, and this is perhaps a primary reason for the neglect by western scholars. As Indian mathematics is (generally) devoid of proof it is not considered 'true' mathematics in its purest sense. However, adopting this stance is to deny the very origin of remarkable discoveries in mathematics, which may well have been the aim of Eurocentric scholars, as it allowed them to neglect the importance of Indian works in favour of European works. In the mathematical verses of the Aryabhatiya the following topics are covered:

Arithmetic: Method of inversion, various arithmetical operators, including the cube and cube root are though to have originated in Aryabhata's work. Aryabhata can also reliably be attributed with credit for using the relatively 'new' functions of squaring and square rooting.

Algebra: Formulae for finding the sum of several types of series; rules for finding the number of terms of an arithmetical progression; rules for solving examples on interest - which led to the quadratic equation; it is clear that Aryabhata knew the solution of a quadratic equation.

Trigonometry: Tables of sines, not copied from Greek works. The Aryabhatiya is the first historical work of the dated type that uses some of these (trigonometric) functions and contains a table of sines.
Spherical trigonometry: Some incorrect.

Geometry:Area of a triangle, similar triangles, volume rules.

The work of Aryabhata also affords a proof that the decimal system was well in vogue. Of the mathematics contained within the Aryabhatiya the most remarkable is an approximation for p, which is surprisingly accurate. The value given is: p = 3.1416. With little doubt this is the most accurate approximation that had been given up to this point in the history of mathematics. He found it from the circle with circumference 62832 and diameter 20000. Critics have tried to suggest that this approximation is of Greek origin. However with confidence it can be argued that the Greeks only used p = sqrt(10) and p = 22/7 and that no other values can be found in Greek texts.

Aryabhata's work on astronomy was also pioneering, and was far less tinged with a mythological flavour. Among many theories he was the first to suggest that diurnal motion of the 'heavens' is due to rotation of the earth about its axis, which is incredibly insightful (unsurprisingly he was criticised for this).

In the period between 14th and 16th centuries A.D., the south-western tip of India escaped the majority of the political upheaval, which engulfed the rest of the country, allowing a generally peaceful existence to continue. Thus the pursuit of scientific development was able to continue 'uninterrupted'. It has only recently come to light that mathematics (and astronomy) continued to flourish in this area for several hundred years. Kerala mathematics was strongly influenced by astronomy, but this led to the derivation of mathematical results of huge importance. As a result of the recency of these discoveries it is quite probable that there are still further discoveries of 'Kerala mathematics' to be made, and a full analysis has yet to be carried out. However several findings have already been made that show several major concepts of renaissance European mathematics were first developed in India. European scholars may have had first hand knowledge of some Kerala mathematics, as the area was a focal point for trading with many parts of the world, including Europe. There is also some evidence of a transfer of technology between Europe and Kerala. It is true that Kerala was in continuous contact with China, Arabia, and at the turn of the 16th century, Europe, thus transmission might well have been possible. However the current theory is that Keralese calculus remained localised until its discovery by Charles Whish in the late 19th century. There is no evidence of direct transmission by way of relevant manuscripts but there is evidence of methodological similarities, communication routes and a suitable chronology for transmission. Events also suggest it is quite possible that Jesuits (Christian missionaries) in Kerala were 'encouraged' to acquire mathematical knowledge while there.

It is important to bear in mind that the conceptual and epistemological bases of Madhava’s (the most famous Kerala mathematician) mathematics had little affinity with those of early Greek mathematics. Instead, they were founded on the principles elaborated in Aryabhatiya.

......................TO BE CONTINUED.......................

Jews - A unique group of gifted people

In my opinion, the Jewish people are the leading group of people in terms of accomplishments. For a relatively small number of followers of this faith, the Jews have produced some remarkable talents - Albert Einstein, John von Neumann, Gary Kasparov, Steven Spielberg, to name a few. It is astonishing to note the percentage of Jewish Nobel Prize winners - in the scientific research fields of Physics, Chemistry, Medicine and Economics, about 26% of all the winners are Jewish. It is even more astonishing to note that the Jews make-up only 0.25% of the world's population!!

But life has not been all rosy for the Jews - far from it. For centuries, they have suffered hardships and discrimination, due to a host of reasons. The purpose of this article is to present a condensed form of the evolution of Jews into what we see today. In addition, it is also intended to present a chronology of events in the Jewish history; upon reading this article, the reader, if not already aware, will have quite a clear picture of the turbulent past of the Jews. There is a popular opinion especially in the west that the Jews are money-minded and that hoarding money is one of their prime aims. A probable explanation for that is furnished here.

The origin of the Jews is traditionally dated around 1800 BC, when a monotheistic religion developed in the land of Israel. As a result of foreign conquests and expulsions starting in 8th century BC, especially due to the Romans, a Jewish diaspora was formed, which, unfortunately for the Jews, continued to haunt them as late as 20th century AD. Most of the Jewish population were expelled from the land of Israel and sold as slaves/settled in Europe and various parts of the Middle East. Did you know that Jesus Christ was Jewish by birth?

The violence and other activities of hatred, termed as Anti-Semitism, followed the Jews whereever they went. Since Judaism was generally the largest minority religion in Christian Europe and much of the Islamic world, Jews were often the primary targets of religiously-motivated violence and persecution from Christian and, to a lesser degree, Islamic rulers. Though the beginning of anti-semitic sentiments and acts of hatred pre-dated the life of Jesus Christ, they magnified and multiplied after his crucifiction. The reason is the alleged role of the Jews in the crucifiction.

During the middle ages, the Jews settled in Europe during the time of the Roman Empire, but the rise of the Catholic Church resulted in frequent expulsions and persecutions. The Crusades routinely attacked Jewish communities, and increasingly harsh laws restricted them from most economic activities and land ownership, leaving open only money-lending and a few other trades. This is the main reason why the Jewish traditionally have a very stronghold on issues pertaining to finance and pawnbrokerage. This stronghold has lead to the conclusion that they are money-minded. (In my opinion, when a person is deprived of opportunities, he/she will desperately cling on to the available chances; that is precisely what the Jews have been doing for centuries. So, it is rather shallow to heap accusations on them). This is also a probable reason why the Jews figure prominently in the academia, as their fields of activities were restricted for centuries, enabling them to channalise their energy into scientific research fields.

Going back to the continuation of the Jewish diaspora, during the Middle Ages, Jews in Islamic lands generally had more rights than under Christian rule, with a golden age of coexistence in Islamic Spain from about 900 to 1200 AD, when Spain became the center of the richest, most populous, and most influential Jewish community of the time. However, by the thirteenth century, Jews were soon expelled from Spain after the Christian re-conquest. Many of these Jews found refuge in the Ottoman empire (centered around modern-day Turkey), which remained tolerant of its Jewish population for much of its history.

In the late middle ages, a devastating pandemic "Black death" struck Europe, killing upto a third of the Europe's population. In many parts of Europe, rumours circulated that Jews caused the plague by deliberately poisoning wells. Comparatively fewer Jews died from the Black Death, in part due to Rabbinical laws, that called for a lifestyle that was, in general, cleaner than that of a typical medieval villager, and because of their isolation in Jewish ghettos; this raised the suspicion of people unable to attribute the difference in death rates to any natural cause (The original reason was the Bubonic Plague). As a result, the Jewish were subjected to massacres, further depleting their population.

Their struggle for existence continued, until the traumatic Holocaust, a systematic and (German) state-sponsored persecution and massacre of Jews by the Nazis during the World War - II. This changed their destiny completely, leading to them being massacred and driven-out of Germany, Poland and other European countries. About six million Jews were massacred. The aftermath of World War - II left millions of Jews to look for a place to live. That they settled in their ancient native land of Israel, much to the dislike of the Islamic people who were living there till the arrival of the Jewish people, and the subsequent tensions between the Jewish state of Israel and the neighbouring Islamic countries are very well known and need no recitation. (I am not in a position to state whether their act of settling down in Israel, thereby displacing the then inhabitants is right or wrong. I have my own views concerning this, but I would rather not list them here). In addition to Israel, a large Jewish population found refuge in the U.S.A.

To conclude this article on a philosophical note, the history of the Jews is a lesson by itself. Their continued existence and excellence in the face of so many obstacles is admirable indeed, and in the Darwin's game of "survival of the fittest", the Jews, though not the fittest (in terms of numbers, wealth etc), have managed to survive and have made (and still making) their mark.

-more details soon-

Taj Mahal or Tejo Mahalaya?

This is a rather sensitive issue. In fact, some historians claim that the archaeological survey of India has remained mum on this issue, inspite of its' knowledge pertaining to the real identity of Taj Mahal, to maintain the communal harmony prevalent in our country (India) and to avoid unwarrented incidents. It is to be noted that the details furnished here are excerpts from the works of some historians; the knowledge of yours truly is the result of studying these works - no more and no less.

Indian historian P.N.Oak authored a book titled 'Taj Mahal - the true story', in which he has provided more than 100 arguements as to why the story of the Taj Mahal that most of us have known about may not be the real truth. Contrary to what visitors are made to believe the Tajmahal is not an Islamic mausoleum but an ancient Shiva temple known as 'Tejo Mahalaya' which the 5th generation Moghul emperor Shahjahan commandeered from the then Maharaja of Jaipur. The Taj Mahal, should therefore, be viewed as a temple palace and not as a tomb. The arguements presented in this book are compelling and difficult to ignore. These opinions have found a lot of supporters, who are, by the way, not religious fanatics trying to denounce the Islamic identity of the greatest monument in India - one of the supporters is Stephen Knapp, a foreigner who grew up in a Christian family, and who has dedicated himself to spreading the deepest and most practical levels of spiritual knowledge about the soul-our real identity. He has been researching Vedic spirituality and comparative religious study for over 30 years in a variety of settings. Knapp aside, Professor Marvin Mills of New York has also published his views in this regard.

Since it is nearly impossible to list all the reasons here in my blog and since it will just be a redundance to do so, I recommend the following books/websites for further research. Keeping up with the aim of my blog, this article is aimed at presenting the hidden truth. After reading a number of publications, I think there is an outside chance that Taj Mahal might have been Tejo Mahalaya. At the same time, there are no conclusive evidences that prove this theory beyond doubt.

Opinions are subjective and bound to differ. You judge the real identity of the famous monument for yourself, after a comprehensive research. I request the readers to refrain from any religious comments that may provoke untoward incidents. It is possible to rewrite history but one has to keep in mind, the price to be paid in doing so.

Click on the links to follow them:

http://www.stephen-knapp.com/was_the_taj_mahal_a_vedic_temple.htm

http://www.stephen-knapp.com/true_story_of_the_taj_mahal.htm

http://www.stephen-knapp.com/an_architect_looks_at_the_taj_mahal_legend.htm

Tajmahal The True Story, authored by P.N. Oak can be ordered from :
A. Ghosh Publisher, 5720 W. Little York # 216, Houston, Texas 77091