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A Course on Mathematics in India (From Vedic Period to Modern Times)

In 662 CE, a Syrian bishop named Severus Sebokht wrote

When Ibn Sina (980 – 1037 CE) was about ten years old, a group of missionaries belonging to an Islamic sect came to Bukhara from Egypt  and he writes that it is from them that he learned Indian arithmetic. This, George Gheverghese Joseph, writes in The Crest of the Peacock: Non-European Roots of Mathematics shows that Indian math was being used from the borders of central Asia to North Africa and Egypt.

Though there is a such a rich history, we rarely learn about the greatness of Indian mathematicians in schools. Even our intellectuals are careful to glorify the West and ignore the great traditions of India. A prime example of that was an article by P. Govindapillai, the Communist Party ideologue, in which he lamented that the world did not know about the contributions of the Arab scientist al-Hassan. In response, I wrote an Op-Ed in Mail Today in 2009.

Thus it is indeed great to see that NPTEL ran a course on Mathematics in India – From Vedic Period to Modern Times. The entire series of around 40 lectures is available online. It is there on YouTube as well. It starts with Mathematics in ancient India with the Śulbasūtras and goes past the period of Ramanujam. It goes through various regional scientists including the members of the Kerala School of Astronomy and covers the difference between the Greco-Roman system of proofs and how Indian mathematicians did it. Kudos to Prof. M. D. Srinivas, Prof. M. S. Sriram and Prof.K. Ramasubramanian for making this available to the general public.

PS: @sundeeprao points to this course on Ayurvedic Inhertance of India

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In Pragati: Where is the Indo-European homeland these days?

(This article was originally published in Pragati)

The discovery of the relation between Sanskrit and European languages by Sir William Jones resulted not only in the birth of comparative philology, but it also initiated the search for the Indo-European homeland. There was no consensus on homeland location. In The Quest for the Origins of Vedic Culture: The Indo-Aryan Migration Debate, Edwin Bryant writes, “The Indo-European homeland has been located and relocated everywhere from the North Pole to South Pole, to China. It has been placed in South India, Central India, North India, Tibet, Bactria, Iran, the Aral Sea, the Caspian Sea, the Black Sea, Lithuania, the Caucasus, the Urals, the Volga Mountains, South Rusia, the steppes of Central Asia, Asia Minor, Anatolia, Scandinavia, Finland, Sweden, the Baltic, western Europe, northern Europe, central Europe and eastern Europe.”

During the time of William Jones, scholarship was focussed on reconciling Indian history with theBible. In 1790, Jones reaffirmed the “sanctity of the venerable books (of Genesis)” and put the origins of Indian empire within the safe confines of Bishop Usher’s creation date of 4004 BCE. Under Max Muller, who claimed that Genesis was historical, the biblical heritage survived with the narrative that superior civilisations of Europe, Persia and India had one language family. Though India was initially considered as the homeland, by the 19th century that was no longer the case. Bryant writes, “The Indomania of the early British Orientalists did not die of natural causes; it was killed off and replaced by an Indophobia initiated by Evangelism and Utilitarianism epitomised by Charles Grant and James Mill respectively.”

The 19th century was also a period of racial science and it was encouraged by Orientalists in Madras who discovered that South Indian languages were not derived from Sanskrit. Following this discovery, Vedic texts were interpreted to read that white-skinned Aryans subdued dark-skinned and snub-nosed dasas. The similarity of Indo-European languages along with such heroic conquests led the search for the mysterious homeland from where these aristocrats set forth. With this the British could explain their presence in India as yet another wave of Aryan invasion, similar to the many waves that happened before. Once scholars started searching for the homeland, it turned out that you could throw a dart at a world map and there was a theory of origin from that place.

As of 2013, there are three homeland theories that are prominent. The first one — the Anatolian-Neolithic — proposes that Indo-European originated in Anatolia and spread through Europe along with the spread of farming. The spread of the language towards India was explained using two models. The first one proposed that the language spread eastward from Anatolia to India and the second one suggested that it was a later southward migration from Central Asia that bought the language to India. After going back and forth between these two models, the present version argues that Indo-European spread symmetrically westward to Europe and eastward to India. The second theory suggests that the homeland was not in Anatolia, but to the south of the Caucasus. The spread of the language did not happen with the spread of farming, but at a much later date. This theory also posits a secondary homeland located north of the Black and Caspian seas. The third one suggests that the homeland was located between the Volga and Dnieper (The Pontic-Caspian) during 4500–3000 BCE.

In a 2013 paper titled Twenty-first century clouds over Indo-European homelands, J P Mallory used the common notion that Anatolian was one of the first languages to split away in the proto-Indo-European framework to evaluate the three homeland theories. In the Pontic-Caspian model, the ancestors of Anatolians leave the region north of the Black Sea and move to Anatolia. Indo-European develops later in the Pontic-Caspian region and the speakers disperse both east and west in the Bronze age. The Near Eastern model presents crazy travel plans. First the Anatolians move out of Anatolia into the Balkans and Indo-European develops in that space. Before the Anatolians move back to their homeland, the Indo-Europeans move out requiring carefully choreographed movements of peoples. In the Anatolian Neolithic model, the Anatolians do not travel back and forth to the Balkans, but stay put. Instead the Indo-Europeans disperse around the world.

For each of the homeland theories and their paths of dispersal there are sufficient counter arguments that make it untenable or look ridiculous. To give an example, a theory presented in 2012, required two linguistic groups, and separated geographically for 2500 years to have similar linguistic changes. Mallroy writes, “the statisticians who devised this model seem to require some form of mutual contact at a distance, one of the stranger aspects of quantum theory that Einstein once dismissed as Spukhaftige Fernwirkung (“spooky action at a distance”) ”

A second problem with all these migration theories is this:  If agriculture was the source of language expansion, did the region from Anatolia to the Indus speak the same language at some point? Very often historians tell us that the invading/migrating Aryans changed the linguistic landscape of North-West India. If the agricultural spread theory is true, then it was not just the Indus languages that were changed. In the 2500 km distance from Anatolia to the farming community of Mehrgarh in Balochistan, there were four other non-Indo-European speaking regions (Hurrian, Semitic, Sumerian and Elamite) and the migration model requires major language shift in all these areas. Mallroy writes, “In any event, all three models require some form of major language shift despite there being no credible archaeological evidence to demonstrate, through elite dominance or any other mechanism, the type of language shift required to explain, for example, the arrival and dominance of the Indo-Aryans in India.”

One possibility is that the language did not spread through invasion or the current favourite — migration — or due to elite dominance, but due to demic diffusion. Peter Bellwood looked at the farming hypothesis and coupled it with new archaeological discoveries in the Gangetic plains, and proposed last year that Indo-European speakers arrived in North-West two millenia earlier than expected. This gave possibility to the development of Vedic language in the region and not in Central Asia. It also provided the ability for the language to spread slowly rather than suddenly. Later in his life Max Müller questioned the concept of a single Proto-Indo-European language. Martin Lewis, a historical geographer at Stanford University, writes, “He [ Müller] further contended that speakers of these dialects might have spread their tongues not by way of massive invasions but rather through the gradual infiltration of relatively small numbers of people out of their Asian homeland.”

To paraphrase an Indo-European scholar, the right question to ask these days is not where the Indo-European homeland is, but rather, where do they put it now?  Since Indian history has been deeply tied to the movement of Indo-European people, it is important to understand the debate that is going on. If one has to be cynical of the whole enterprise, it has to come from an understanding of the complexity and not through a simplistic denial of the theory. In her book History of Ancient and Early Medeival India: From the Stone Age to the 12th Century Upinder Singh wrote that most historians have abandoned the idea of an Aryan invasion for a ‘several waves of migration’ theory. Though no one knows where they came from or which path they followed, Indian history is still firmly rooted in these external origins.

The problems with two centuries of linguistics do not end with diverse homelands or inconclusive paths of migration. There are fundamental issues on what languages belong to the Indo-European tree.  Where does Graeco-Armenian or Italo-Celtic belong? Is Tokharian an orphan or should it be associated with the German branch? A debate which is going on this year is if Basque, the ancestral language spoken by people living in the region spanning northeastern Spain and southwestern France, is an Indo-European language or not. These doubts, (See  An earlier date for Indo-Europeans in Northwest India) which exist in Indo-European linguistics, is absent in Indian history narratives. There is not an iota of scepticism and a simplistic model still seems to be the norm.

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Tipu Sultan- The Tyrant of Mysore by Sandeep Balakrishna

In A Survey of Kerala History, Sreedhara Menon summarizes the impact of Tipu Sultan’s brutal raids on Kerala and concludes that it introduced modern and progressive ideas to Kerala. These progressive ideas include collecting taxes directly from the peasants and building roads which connected various remote parts of Kerala. Menon also credits Tipu Sultan for creating a social revolution in Kerala by attacking Brahmins, Kshatriyas and Nairs. By declaring Nairs as the lowest caste and by forcefully converting them into Islam gave the lower castes a sense of prestige and position. As to the drawbacks to Tipu’s wars in Kerala, Menon writes that the wars halted black pepper cultivation and thus destroyed the economy. The ports became empty and unused and the foreign currency that came from trade relations which spanned a millennia came to a halt pushing people into poverty.

Fortunately Menon does not call him a freedom fighter, but many many historians and politicians and literary types like Girish Karnad and Bhagwan Gidwani consider him so. They also consider him to be a great warrior, a humanist, the son of Kannada and a tolerant ruler. Sandeep, by going through copious amounts of primary sources on Tipu finds that a fictional narrative has been constructed regarding every aspect of Tipu’s life and the tale which was spread by bards who relied on bakshish, now is spread by modern bards for various nefarious reasons.

The part that Sreedhara Menon whitewashed s expanded by Sandeep and that is not for the faint of heart. Tipu hunted down the Nairs who rebelled against him and forced them to surrender. Here is the what happened next:

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Another account of Tipu’s march through Calicut records that both men, women and children were hanged; churches and temples were desecrated; women were forced to marry Muslim men. Proud of his accomplishments in Calicut, he wrote a letter gloating about this massive conversion. He also congratulated his commanding officer for circumcising the captives and converting the others. As he marched to Travancore, burning towns and villages, he was halted by the Nairs and extreme cruelty by Tipu caused an exodus of people from Malabar. This is the level of progressiveness that is attributed to Tipu by a writer who had a good idea of what really happened as he was the former editor of the Gazetteer of Kerala. The trend seems to be to not let facts get in the way of a progressive interpretation.

Another myth that prevails is that Tipu was a freedom fighter because he fought against the British. Less mentioned is the fact that he working on replacing one colonial power for the other for his own personal gain. In various letters written to the French,he conveyed the notion that he was friends with the local Muslim rulers and with the combined French army, they could rout the British. To entice the French, he promised half the territory that would be taken away from the British and he had correspondence with Napoleon himself. Napoleon was not the only foreigner with whom he bargained. He wrote letters to the Caliph, to Zaman Shah of Afghanistan, and to other foreign Muslim rulers, inviting them to wage the battle against the infidels.

Besides revealing such less mentioned facts, the book begins with the crux of the problem which is the problem with historiography in India. These narratives are not written with a focus on revealing the truth, but for subverting certain truths. There is a revealing conversation between S L Bhyrappa and G. Parthasarathy, a Nehru-Gandhi family acolyte, who lead a committee to foster national integration through education. Parthasarathy tells Bhyrappa, who at that time was a philosophy lecturer, that teaching about the iconoclasm of Aurangzeb and Mahmud of Ghazni would poison the minds of the students, offend the minorities and “cleave the society”. Hence it was important to use “maturity and discrimination” in selecting the narrative.

In 2009, I wrote a piece for Pragati about these biases and one of the solutions was for us not to leave the history to historians.

Lawsuits, protests, activism—these can be an effective tools, but there is also a need to popularise the discourse. Stephen Ambrose, David McCullough are masters of the popular history genre in the West. Barring a few honourable exceptions, in the Indian context this genre consists of writing more biographies of Nehru and Gandhi. There is a need to add more voices to this discourse—to explain how the invasion theory evolved to migration theory to Aryan trickle down theory—because this Aryan-Dravidian race theory still has serious social and political implications in India.[Op-Ed in Pragati: Getting Objective about it]

Sandeep’s book is a good step in that direction.

Tipu Sultan – Tyrant of Mysore, Rare Publications; 1st Edition 2013 edition (November 30, 2013), 186 pages [Kindle Edition | Flipkart| Amazon India]

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In Pragati: Book Review of Riddle of the Labyrinth by Margalit Fox

Margalit Fox reveals the life and struggles of the people behind the decipherment of Linear B, an unknown language in an unknown script, similar to Indus-Saraswati writing. 

One of the most puzzling unsolved mysteries of the ancient world is the writing system of the Indus-Saraswati civilisation. There are over 4200 inscriptions, on seals, on tablets and on pottery; of the 400 signs, only 200 have been used more than five times. Decoding this writing would not only reveal details of life during that period, but also put an end to various debates over the identity of the residents of the Indus region. There has been no dearth of decipherments: many have read proto-Dravidian into the script and others Sanskrit. In The Lost River, Michel Danino writes that the only safe statement that can be made is that the seals played an important part in trade and permitted the identification of either traders or their goods.

Linear B Image

A sample of Linear B script

Decipherment of Indus writing is hard because it falls into the most difficult category in the relation between script and language. The easiest one is where a known language is written in a known script, like English written using Roman alphabets. A difficult case is where a known language is written in an unknown script like when Rongorongo is used to write Rapa Nui. Equally difficult is the case where a known script is used to write an unknown language like when Greek alphabets are used to write Etruscan. The most difficult one is when the language and script are unknown and there is no help for the decipherer. The Indus writing falls into this category.

Linear B, the writing found on the island of Crete, belonged to this category as well but was decoded half a century after it was discovered. The writing was used by Minoans who flourished during the Bronze age following the decline of the Indus-Saraswati civilisation. Found by an English digger named Arthur Evans, it was named Linear Script Class B. The decipherment story of Linear B might have turned into a dry academic discussion on the difference between proto-writing and writing or on if it was a memory aid for rituals or just meaningless visual art, but Margalit Fox makes it a fascinating tale as the decipherment is tied to the life of three unusual people: a glorified English tomb robber, a largely forgotten American classicist and a gifted English architect; it is the human element that adds depth to the mystery.

The three decipherers

The challenges that faced the decipherers looked insurmountable. When Evans found the tablets in 1900, there were no computers that could detect patterns or do statistical analysis. It looked as if we would never find out if the tablets would reveal a Western epic like the Iliad or just bland accounting records. Though there were no external clues from a Rosetta Stone, some information could be gleaned from the tablets. Evans, for example, figured out the direction of writing and the word breaks because they were separated by tick marks. He also figured out the numerical system used by the Cretan scribes. Some tablets, which had arrow signs on them, were found near a chest filled with arrows; the context gave an idea of what those tablets represented. Some of the tablets were found in the palace complex in boxes with pictograms representing the contents

Nothing beyond this was known when Alice Kober started work on the script in the United States of America. She was an assistant professor of classics at Brooklyn College, who taught introductory Latin and Classics during the day and worked on deciphering the secrets of the Cretans by night. In preparation for the work, Kober learned many fields such as archaeology, linguistics, statistics. Since she was not sure about the language of the Linear B writing, she spent fifteen years studying languages from Chinese to Akkadian to Sanskrit. Without seeing the tablets and by looking the two hundred inscriptions that were available, she worked on them methodically and came close to solving the mystery. Not much credit was given to her in other books about Linear B decipherment and the book tries to correct that by detailing her contributions.

She solved many mysteries, which Arthur Evans or other scholars could not solve; this included figuring out which signs depicted male and female animals as well the sign for boy and girl.  A major breakthrough for Kober was the discovery that Linear B was inflected, which meant that they depended on word endings like adding -ed to denote past tense and -s to indicate plural. With this discovery, she was able to eliminate many languages, which don’t use inflection and focus on languages, which did. She was also able to figure out what was known as a bridging character, which enabled her to figure out the relative relationship between the characters in the script.

The last person mentioned in the book is the one who finally deciphered it. Michael Ventris, is a person who would have been dismissed by modern scholars as a quack for he was not a linguist or a classicist or a scholar in any other field of humanities; he worked as an architect. Like Kober, he too was obsessed with Linear B, even publishing a paper when he was 19. Unlike Kober, Ventris had access to larger number of Linear B symbols. As he sorted the characters based on their frequency and position, he found certain characters appeared at the beginning of the words.

He made one major intuitive leap, which Alice Kober failed to do, and with that he was able to solve the mystery. There were codes, which differed only in the last character and they were found only in Knossos which meant that those characters represented the name of the place. Now it was time to figure out what the words actually meant. Since Kober had figured out that it was an inflected language, he discarded Etruscan and considered other options like Greek. According to the wisdom at that time, Greek speakers arrived much later and so this would have been unacceptable. Ventris, then performed a second leap. During the Iron Age, a writing system called the Cypriot script existed. While the language remained a mystery, the sounds of the symbols were known as the script was used to write Greek following the Hellenization of Cyprus. As those sound values were substituted, the words began to make sense. By the time he was 30, he had solved Linear B.

Lessons for the Indus Decipherers

ten indus glyphs

Ten Indus glyphs discovered near the northern gate of Dholavira 

The people who worked on the decipherment faced great challenges, but they had some qualities that helped them makes progress. They were intelligent, had great memory and were single mindedly focussed on the issue. Margalit Fox explains in detail the tremendous skills that are required to find success in an impossible task like decoding an unknown script of an unknown language. You need rigour of the mind, ferocity of determination, a deliberate way of working, along with a flair for languages. When you are stuck with such a problem, where no external help is available and the problem looks unsolvable, you have to look closely for sometimes the clues lie in the puzzle itself revealing itself to the careful observer.

There is a certain orthodoxy in the Indus politics, which prevents scholars from considering that an Indo-European language was spoken in the region. Solely based on linguistics, it has been argued that Indo-European speakers arrived in North-West India following the decline of the Indus civilisation and hence the language should not even be considered as a possibility. Thus most decipherments argue that the language spoken in Indus Valley was non Indo-Aryan. Similarly, for decoding Linear B, there was intense speculation on the language of the tablets, but Greek was ruled out because Greek speakers were known to have arrived later. Evans thought that the Minoan culture was different from the later Greek culture and there was no relation between the two. Alice Kober refused to play that game, refused to give sound values to the characters, and firmly said that the script had to be analysed based on the internal evidence devoid of the decipherer’s  prejudice. She was highly against starting with a preconceived idea and then trying to prove it.

Now even in Indus studies the data is pointing to interesting possibilities. A 2012 paper by Peter Bellwood, Professor of Archaeology at the School of Archaeology and Anthropology of the Australian National University suggests that Indo-European speakers may have been present in Northwest India much earlier, maybe even two millennia earlier than previously assumed.  According to Bellwood, the urban Harappan civilisation had a large number of Indo-European speakers alongside the speakers of other languages which may have included Dravidian.

In a 2010 paper, Professor Jonathan Mark Kenoyer, who has been excavating at Harappa for three decades wrote that even though the Indus script has not been deciphered, Dravidian, Austro-Asiatic, Sino-Tibetan and Indo-Aryan co-existed in the region. Paul Heggarty, a linguist at the Max Planck Institute for Evolutionary Anthropology, in a 2013 paper writes that Indo-European speakers may have reached Mehrgarh much earlier than 4000 BCE. With such information, there is a need to break away from the orthodoxy. (See “An earlier date for Indo-Europeans in Northwest India”)

Maybe there is a scholar or an amateur who is as meticulous as Kober or as gifted as Ventris to whom the secrets of the Indus would be revealed, but the  decipherment of Indus script is hard because of the brevity of the seals; the average is five symbols. To figure out anything from seals averaging just five signs is an impossible task. Another possibility that could help is the existence of  a bilingual inscription in one of the regions with which the Indus people traded. For example, there existed an Indus colony in ancient Mesopotamia. This village was located in an area called Lagash in southwestern Mesopotamia which had cities like Girsu, Nina, and a port city and area called Guabba. Scholars also found a  reference to a personal seal of a Meluhhan (assumed to be a person from the Indus region) translator — Shu-ilishu — who lived in Mesopotamia.

Thus 4000 years back, there was a man in Mesopotamia who could speak Meluhhan as well as Sumerian or Akkadian. He could read those Indus tablets. This is not surprising since the Meluhhan merchants would have handled the imports from Meluhha and exported Mesopotamian goods to their homeland. Since the translator worked with Meluhhans and Mesopotamians, he would need to speak multiple languages. This also suggests that there could exist a bi-lingual tablet somewhere in the region where Shu-ilishu lived. If such a tablet is found, it could be the Rosetta stone which would solve a 134 year old mystery forever.

Finally when it was decoded, the Linear B tablets did not reveal an epic like the Iliad or the Odyssey; they found a record of crops, goods, animals and gifts offered to gods. They revealed the working of the society, about the status of various holdings, and about the personnel who worked there. They revealed their food habits, religious habits and how they spent their time. It revealed the pyramidal structure of the Minoan society with elites at the top, craftsmen and herdsmen below them and slaves at the bottom. The tablets were predominantly economic and was concerned with keeping track of  the goods produced and exchanged.

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Did Nehru promote a dynasty?


Myth 1: Nehru promoted a `dynasty’

This myth draws support from the fact that Nehru’s daughter and grandson also served as Prime Minister, that his grand daughter-in-law has sought that post too, and, most recently, that her son, Nehru’s great-grandson, has joined politics as the heir-apparent of the Congress party.

In truth, Nehru had nothing to do with the “dynasty”. He had no idea, nor desire, that his daughter would become Prime Minister of India. It was Mrs. Indira Gandhi who converted the Indian National Congress into a family business. She first brought in her son Sanjay and, after his death, his brother Rajiv.

In each case, it was made clear that the son would succeed Mrs. Gandhi as head of Congress and head of Government. Thus, the “Nehru-Gandhi dynasty” should properly be known as the “(Indira) Gandhi'” dynasty. [Five myths about Nehru]

In their book Six Thousand Days, Amiya Rao and B.G. Rao refute this fairy tale.  As Nehru became the supreme leader of the country and the party becoming a rubber stamp to his whimsical ideas, there were questions as to who his successor would be.  The clues lay in Nehru’s actions.

Indira Gandhi went with him to all political and administrative functions and got introduced to all the state leaders of the Congress party. She addressed election meetings, presided over Congress women’s committees, was the hostess  when Nehru entertained visitors and he took her with him when he traveled abroad.  With her pedigree, she was able to waltz into the Congress President U.N.Dhebar’s Working Committee without holding any junior position.  On June 18, 1957, after checking with Maulana Kalam Azad and Rajendra Prasad,  Durga Das wrote the following line in his weekly column in The Hindustan Times, “if he (Nehru) is consciously trying to build anyone as his successor, he is building up his daughter.”

Nehru’s colleagues have acknowledged another thing as well. If he wanted to get something done and was uncomfortable doing it, he would get others to do it. The Congressmen were only too willing to oblige. Six months after she was nominated to the Working Committee, Nehru resigned from the powerful Central Parliamentary Board and Dhebar on cue, nominated Indira Gandhi. This was the committee which picked the candidates for elections and decided on the political fate of thousands of Congressmen and women. In 1959, while the father was Prime Minister, the daughter became the president of the party. Nehru expressed “surprise” and spun it as the desire for a section of Congressmen to see young leadership.

Following this Indira was everywhere:  on the executive board of UNESCO, traveling East Africa on Nehru’s behalf, as special messenger to President Johnson, as chairman of National Integration Committee and even on the National Defence Committee, which did not have place for Gen. Cariappa. It was obvious that the daughter was being groomed to occupy the Prime Minister’s chair. A 1961 report from Palam airport observed Cabinet ministers, fixers, chelas and about two hundred odd people walking behind her as she skipped the customs shed and walked directly to the aircraft.

According to the Raos, Nehru had molded the party to accept Indira Gandhi as the Prime Minister. He skillfully removed all the strong contenders to the post like Jagjivan Ram and Morarji Desai by making statements that Congress does not encourage people getting addicted to power. He kept people like Gulzarilal Nanda, who would not be a strong contender, but would be a stop gap arrangement. All of Nehru’s actions indicated that he was promoting the dynasty, but it is strange that modern historians have no recollection of all these events.

Reference:

  1. Adapted from Rao, Amiya Rao, B. G. Six thousand days : Jawaharlal Nehru, Prime Minister, Sterling Publishers, 1974.
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