**D D Kosambi** was an Indian mathematician who, in addition to many mathematics and mathematical physics papers, published on genetics, numismatics, Indology, Sanskrit studies, and Indian history. He brought mathematics, statistics, Marxism and critical analysis into his studies of historiography.

- was largely devoted to a single cherished goal: popularising the Buddha's message among his fellow Maharashtrians.
- Dharmanand Kosambi gave up his university position in 1908 and, after lecturing throughout India, he spent a year in Pune, then went to the United States, spending two years at Harvard.
- After returning, he moved to Fergusson College in Pune where he taught for six years.
- During this time in Pune, from 1912 to 1918, Kosambi began his schooling.
- He studied at the New English School where he showed himself to be a brilliant pupil.
- In 1915 he began his secondary schooling, although he was much younger than all the other pupils in his class.
- At this time he was very thin, his heath was poor and he continually suffered from colds and fevers.
- He would rudely talk to anyone, with little regard to the elders without any hesitation.
- Mother was much too indulgent towards him.
- Father was somewhat strict and used to scold him.
- In addition he was quite short-tempered and stubborn.
- He would get angry if anyone dared touch any of his things.
- The sea journey took four months followed by a train journey across the United States to Harvard University in Cambridge, Massachusetts.
- Kosambi began his studies at the Rinz Technical High School but his poor health and weak physique meant to found the physical tasks difficult.
- The principal advised that he transfer to a more academic education, so he was enrolled in the Harvard Grammar School.
- He had his tonsils removed which led to a large improvement in his health.
- After a year at the Harvard Grammar School he entered the Cambridge High and Latin School in 1920.
- He now became interested in all academic subjects but also became physically very fit exercising in the gym, swimming, taking part in athletics and ice skating.
- No other discipline unlocked the door to the atom or to the movement of celestial bodies equally well, as mathematics did.
- Aptitude granted, mathematical research needed the least financial resources of any science.
- Mathematical results possess clarity and give an intellectual satisfaction above any others.
- They have absolute validity in their own domain, due to the rigorous logical process involved, independent of experimental verification upon which applications to the exact sciences must depend.
- They became friends for life.
- Kosambi graduated from the Cambridge High and Latin School in 1924 and was awarded a scholarship to study at Harvard University.
- He postponed his studies, however, and returned to India, sailing first to London then departing on 19 July on the ship Katori Maru for India.
- Kosambi began his studies at Harvard University in January 1926.
- Although his main interest was mathematics, he spread himself very widely across an amazing range of subjects, taking 18 courses in one year.
- He was taught mathematics by G D Birkhoff who worried about the number of courses Kosambi was taking, advising him to concentrate on mathematics.
- In 1929 Kosambi was awarded the degree of Bachelor of Arts 'summa cum laude' but he was not awarded a fellowship to enable him to continue his studies for a doctorate.
- One reason was the start of the Depression which meant fewer fellowships were available, but the main reason seems to have been the fact that Kosambi had not taken G D Birkhoff's advice of concentrating on mathematics and had continued having many other interests.
- Stopping off in France he went to Paris to try to meet Élie Cartan but, failing in this, he sought out Tullio Levi-Civita in Rome but again had no luck.
- Soon, he was appointed to the Banaras Hindu University where he taught mathematics and gave German classes on the side.
- This was not a happy experience for Kosambi who wanted to teach at a higher level and wanted to undertake research.
- In this early work, DDK displays a sophisticated ability to integrate many strands of thought, to generalise observations from one context to another, while retaining sufficient rigour.
- The mention of quasi-periodic motion suggests that some of these topics were possibly covered in the special course on the many-body problem that was given by the mathematician G D Birkhoff.
- Kosambi was not happy with the Banaras Hindu University but the senior staff were not very happy with him since he soon had a reputation of being rude to his seniors.
- He was soon looking for another position and was delighted when André Weil invited him to take up a lectureship at the Aligarh Muslim University.
- André Weil had completed his doctorate in 1928 and after undertaking military service had been appointed as professor of mathematics at Aligarh Muslim University in 1930.
- The marriage to Nalini, who had graduated in mathematics and Sanskrit from Wilson College, was an arranged one.
- Kosambi's next paper, On a generalization of the second theorem of Bourbaki, was published in 1931.
- As described by André Weil, the prank was designed to deflate a senior colleague's ego, presumably by demonstrating the greater familiarity that Kosambi had with modern methods and the then current literature.
- André Weil left the Aligarh Muslim University in 1932 and Kosambi decided he would leave too.
- In 1933 he was appointed to Ferguson College in Pune at a considerably lower salary than he had been receiving but, nevertheless, he was much happier there.
- He planned a long stay in Pune, purchased land and had a bungalow built.
- he first built up a reputation as a serious mathematician, serious enough that he was elected to the Indian Academy of Sciences by C V Raman in 1935 who also probably nominated him for the Ramanujan Medal of the Madras University in 1934.
- He soon had an excellent publication record with eight papers published in 1932-33 in three languages: Modern differential geometries (1932), On differential equations with the group property (1932), Geometrie differentielle et calcul des variations Ⓣ(Differential geometry and calculus of variations) (1932), On the existence of a metric and the inverse variational problem (1932), Affin-geometrische Grundlagen der Einheitlichen Feld-theorie Ⓣ(Affine-geometric foundations of unified field theory) (1932), Parallelism and path-spaces (1933), The problem of differential invariants (1933), and The classification of integers (1933).
- Despite this concentration on mathematics, Kosambi continued to have broad research interests with his non-mathematical publications beginning in 1939 with A note on the trial of Socrates followed by The emergence of national characteristics among three Indo-European people (1940).
- By this time he was also undertaking research on the weights of ancient coins with two papers A statistical study of the weights of the old Indian punch-marked coins and On the weights of old Indian punch-marked coins both being published in 1940.
- He also continued his work on "path-geometry", a term he invented, with Path-equations admitting the Lorentz group (1940) and The concept of isotropy in generalized path-spaces (1940).
- In 1943 Kosambi published Statistics in function space which introduced a method for decomposing a signal into its components, a technique widely used today in image processing and in data analysis.
- It is named today as the Kosambi-Karhunen-Loève theorem or the Karhunen-Loève theorem.
- Kari Karhunen (1915-1992) and Michel Loève (1907-1979) gave this result, independently of Kosambi, in 1947 and 1948 respectively.
- Another important paper by Kosambi around this time was The estimation of map distance from recombination values (1944), a work on genetics before the structure of DNA had been discovered.
- He was not an amicable person and had many disagreeable facets to his personality.
- He was short tempered.
- He would not stand any nonsense and would not hesitate to cut anyone to his size irrespective of his age, seniority or prestige.
- Many times he would lose his temper for small negligible mistakes and offend others.
- He took great pride in his intelligence.
- His intellect was not matched with humility and because of this he tended to underestimate others.
- He was also given to using shock treatment to stir people out of their habitual thinking by taking an extreme stand.
- He would never attempt to interact or mingle with others and if anyone dared to communicate with him he seemed to disappoint them as far as possible.
- Kosambi resigned from Ferguson College in 1945.
- The high academic reputation he had gained, however, led to the nuclear physicist Homi Jehangir Bhabha (1909-1966), who was the founding director of the Tata Institute of Fundamental Research, inviting him to join the Tata Institute.
- On 1 June 1945 Kosambi took up the post of Chair of Mathematics at the Institute.
- He continued to live in Pune, travelling by train to the Tata Institute in Bombay.
- What was now becoming typical, the sixteen years he spent at the Tata Institute started well but deteriorated.
- Homi Bhabha arranged for Kosambi to become president of the Mathematics Section of the 34th Indian Science Congress that was held in Delhi 1-8 January 1947.
- By this time he had tried his hand at fabricating at least one, his "universal" calculating machine, the Kosmagraph ...
- DDK did not proofread this paper, and thus it is not in the form that he would have wished.
- Dhannananda must have had great influence on Damodar's growth.
- However their minds had progressed in two different directions.
- There he talked with, among others, Albert Einstein, J Robert Oppenheimer, Hermann Weyl, John von Neumann, Marston Morse, Oswald Veblen and Carl Ludwig Siegel.
- He left the United States in May of that year, flying from New York to London on 17 May 1949.
- During a short stay in London, he met the historian and Indologist, Arthur Llewellyn Basham (1914-1986), at the School of Oriental and African Studies.
- The two became friends as Kosambi's interest in ancient Indian history increased.
- Back at the Tata Institute of Fundamental Research, Kosambi's research interests broadened.
- Although he continued to undertake research in mathematics and publish research papers in that topic, he was undertaking research into ancient Indian history, on social issues from his Marxist viewpoint and was active in the Peace Movement.
- Other than mathematics, all these other interests which occupied his time were irrelevant to the Tata Institute which aimed at being a top class research establishment in nuclear science and mathematics.
- Things went from bad to worse when Kosambi published two incorrect proofs of the Riemann hypothesis, using probability.
- The second proof appears to be erroneous, in that it invokes a sieve theorem under inappropriate conditions.
- The author must replace verbal descriptions, qualitative comparisons and intuition by precise definitions, equations and inequalities, and rigorous reasoning, if he is to claim to have proved a theorem of the magnitude of the Riemann hypothesis.
- Perhaps his colleagues were more annoyed that Kosambi published a paper on such a significant mathematical topic in the Journal of the Indian Society of Agricultural Statistics.
- In 1962 his contract with the Tata Institute was not renewed.
- Kosambi spent his last years continuing to undertake research in mathematics and publishing regularly on statistical methods in number theory, some of these papers being published under the pseudonym S Ducray.
- He published further major books on ancient Indian history: Myth and Reality: Studies in the Formation of Indian Culture (1962); The Culture and Civilisation of Ancient India in Historical Outline (1965); and Ancient India: A History of Its Culture and Civilization (1965).
- Free from teaching duties, he filled the years from 1962 to 1966 doing the research he enjoyed.
- In February 1965 he had an article on numismatics, Scientific Numismatics, published in Scientific American.
- On the 28th of June a complete check-up of his heath was done and he was declared absolutely hale and hearty by his doctor.
- But the night of the same June 28 proved to be his last.
- As usual Kosambi was busy reading and writing in his study late in the night.
- In the small hours of the morning of June 29 he went to bed destined not to wake up ever again.
- Since he was still in bed late next morning, his near ones hesitatingly opened the door of his room, only to find him dead.
- He died in his sleep because of heart failure.
- Four decades of indefatigable incessant striving in research abruptly came to an end.
- Let us end with brief quotes about various aspects of Kosambi's scholarship.
- Yet his admirers are not limited to those who are either Marxists or scholars of antiquity.
- Kosambi is quoted approvingly by medievalists writing on Indian feudalism and by historians of modern India on themes that range from forms of popular expression in the 1857 revolt to the advent of colonialism.
- That he was by profession and training a mathematician rather than a historian makes his work and its influence all the more remarkable.
- He simultaneously worked in both areas for much of his adult life, and to understand the body of his work either in the social sciences or in mathematics, an appreciation of the complementarity of his interests is essential.
- An understanding of Kosambi the historian can only be enhanced by an appreciation of Kosambi the mathematician.
- For those not trained in the subject, the mathematics may itself be subtle and difficult to follow in its entirety, but a knowledge of the intellectual preoccupations, the questions that concerned him, and the techniques and tools at his disposal can help in bringing out the very natural manner in which DDK's mathematics informed and refined his approach to history.
- In a fundamental way, Kosambi embodied the multidisciplinary approach, channelling diverse interests - indeed combining them - to create scholarship of high order.
- The first is his search for a method in the editing of Sanskrit literary texts, and the second his search for a theory in the reading of these texts.
- In the former case, if judged by the practices of editing Sanskrit literary texts in India at the time, Kosambi emerges as a remarkable pioneer, his concrete accomplishments hardly in danger of being superseded anytime soon.
- If Kosambi's theory has proven to be flawed, we have only come to know the flaws and sought ways to overcome them because he had the courage to enunciate the theory in the first place.
- A theoretical framework that was new to the study of Indian history supported his reconstruction of the religion of the Indus valley, as well as his explanations for the spectacular rise and fall of Buddhism, and the enduring appeal of the Krishna myths.
- From today's perspective his work betrays a few blind spots, but it remains largely relevant for the intellectual leap it took in exploring the essential relation between faith and socio-economic factors, and its consciously creative use of Marxism.
- He also worked on statistics in infinite dimensions, computing, and probabilistic number theory.
- A rejection of the value of specialisation saw him leave Harvard.
- The high value he placed on research saw his exit from Banaras Hindu University and Aligarh Muslim University.
- His attempt to impart real knowledge of mathematics saw him sacked from Fergusson College, Pune.
- His insistence on ethical and relevant research led to his exit from the Tata Institute of Fundamental Research where, too, the diversity of his interests was portrayed negatively, though he continued his mathematical research till the end of his life.

Born 31 July 1907, Kosben, Goa, now India. Died 29 June 1966, Pune, India.

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Origin India

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive