D D Kosambi: Selected Works in Mathematics and Statistics by Ram Ramaswamy

Re-posted from: And then...
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After what seems an agonizingly long time since the first ideas of the book took root, I got the following letter from my publishers (how sweet that sounds!) last week,

 “We are very pleased to inform you that your book has been published and it is available on http://tinyurl.com/jgn2djj. Customers can order it […] etc.”

D D Kosambi: Selected Works in Mathematics and Statistics is finally done, and is now available in both e and paper formats. The cover on the right shows DDK at three stages of his life, at Harvard, in Aligarh, and finally, in his TIFR years.

To quote from the blurb: This book fills an important gap in studies on D. D. Kosambi. For the first time, the mathematical work of Kosambi is described, collected and presented in a manner that is accessible to non-mathematicians as well. A number of his papers that are difficult to obtain in these areas are made available here. In addition, there are essays by Kosambi that have not been published earlier as well as some of his lesser known works. Each of the twenty four papers is prefaced by a commentary on the significance of the work, and where possible, extracts from technical reviews by other mathematicians.

My personal contribution to the book, other than to edit is, is fairly minimal. Apart from a preface, I have basically tried to describe the academic milieu in which Kosambi found himself at different points in his life, and have also tried to infer what others thought of him in another prefatory essay, “A Scholar in His Time”.

Kosambi gave his academic manifesto in the essay, “Adventure into the Unknown” which also is one of the places where he wrote that Science is the cognition of necessity. (It is quite another matter that the phrase is not one that can be understood in a straightforward manner. Anyhow, as a quote its famous enough.) Reprinting that essay in its entirety seemed appropriate, as also another note “On Statistics” that gives a flavour of DDK’s interdisciplinarity, mixing statistics, erudition, Marxism, etc. The last of the non-mathematical writings is a project completion report submitted by DDK to the Tata Trust in 1945 and it permits, among other things, an inner view of a vastly gifted and somewhat frugal scholar who, in parallel, and for Rs 1800, carried out  6 research projects on issues as diverse as writing a mathematical monograph on Path Spaces, editing a concordance of Bratrihari‘s epigrams, and constructing an electromechanical computational device (the Kosmagraph),  among others.

The remainder of the book is a set of reprints. Of his 67 or so papers in mathematics and statistics, about a third are presented, starting with some of his first papers, Precessions of an Elliptical Orbit and  On a Generalization of the Second Theorem of Bourbaki, and ending with one of the papers he wrote under the peculiar alias of S. Ducray,  Probability and Prime Numbers.

An attempt was made to include all the important papers, in particular the ones that made his reputation such as Parallelism and Path-Spaces that along with two other notes by Cartan and Chern are the basic of the Kosambi-Cartan-Chern theory,  the various papers that laid the foundations of scientific numismatics, as well as the papers that he should have followed up but didn’t, such as Statistics in Function Space that foreshadowed the K-L decomposition. The Kosambi distance in genetics was elaborated in  The Estimation of Map Distances from Recombination Values, and this is also reprinted.

Kosambi’s obsession with a statistical approach to the proof of the Riemann hypothesis resulted in several papers of which An Application of Stochastic Convergence, Statistical Methods in Number Theory, and The Sampling Distribution of Primes are reprinted here.  These, as is well-known, effectively ruined his reputation as a serious mathematician.

Chinese. Japanese. French. German. English. DDK published papers in all these languages, sometimes exclusively, and twice the same article in translation. Also reprinted in this volume are three of the foreign language papers, the ones in German, French, and Chinese. The last is of particular interest since it was written during an exchange visit to China in the late 1950’s and only later published in English.

A number of people have helped me along the way and it is my pleasure to thank them all here. For the initial suggestion that the book be done, and for sustained and general encouragement, I am very grateful to Romila Thapar. I’ve written about this before.  Meera Kosambi was keen to see her father’s mathematical legacy appreciated and was very enthusiastic about bringing out this collection and helped greatly in more ways than I can describe. She passed away in January 2015, when she knew the project was afoot, but not in any way certain as to how it would all come out. Michael Berry, S. G. Dani, and Andrew Odlyzko discussed and advised on various  points of the mathematics.  Indira Chowdhury and  Oindrila Raychaudhuri helped vis-à-vis archival matters.  Rajaram Nityananda had had many of DDK’s papers digitized, a great boon, and one that made the reproduction of some material much easier! Kapilanjan Krishan,  Rahim Rajan, and Mudit Trivedi  helped me locate some of the more obscure of DDK’s papers. K. Srinivas retyped almost all the papers, and Cicilia Edwin painstakingly proofread most of them.  Toshio Yamazaki and Divyabhanusinh Chavda  told me of their interactions with DDK, helping to flesh out the personality. Finally, Aban Mukherji was gracious with permissions, as were all the journal editors who kindly permitted the several articles to be reprinted.

DDK maintained a charmingly frank notebook diary during his Harvard years. On the 19th of January 1927 he notes: A most restless day. I have forgotten to mention Monday the 17th and an important conference with Birkhoff thereon […] Problems: Fermat’s Last Theorem, the Four color map, the functional equation […] Today was unusually restless with a great deal of time spent, possibly wasted in the Widener. Looked up old issues of Outing, Shakespear’s Hindi Readers, most of Burton’s works [of him more later], Roosevelt on African and Brazilian ‘sporting’ – worthless – Stefansson’s excellent and much remembered ‘Friendly Arctic‘…

All this variety in a single day! To recall Wordsworth,  Bliss indeed it was in that dawn to be alive! Kosambi, just out of his teens, was just bursting with energy, both intellectual and physical (for which one must read the diaries in some detail). The earnestness that only comes at that age shines through on the pages quite unselfconsciously:

Exuberance indeed, but also some simplicity: Deep interest, well sustained, is essential in the acquisition of knowledge upon any subject. And the third realization of the day: Life is good.  Yes indeed, to be young was very heaven.

• D D Kosambi: Selected Works in Mathematics and Statistics is published by Springer. ISBN: 978-81-322-3676-4

Reality and Myth

Reproduced from: And Then, a blog by Ram Ramaswamy

by Ram Ramaswamy

Some years ago, a friend of mine at JNU proudly told me about a book that he had picked up from the library “sale”, a book that had once belonged to D D Kosambi (DDK). Apparently it had not been checked out for years, and was therefore deemed unworthy of staying on in the library, as if finding a place on the library shelf was just some sort of evolutionary game, a survival of the fittest and no more…

The JNU had, at some point in time, acquired Kosambi’s personal collection of books, that was, according to Mr R P Nene (DDK’s friend and assistant, in an interview in June 1985) “sold by his family after his death to the JNU at the cost of Rs. 75, 000.” Details of how this happened are not too clear- Kosambi died in 1966, the JNU was founded in 1969, and the initial seed of the JNU library was that of the “prestigious Indian School of International Studies which was later merged with Jawaharlal Nehru University.” Our website goes on to say that the “JNU Library is a depository of all Govt. publications and publications of some important International Organisations like WHO, European Union, United Nations and its allied agencies etc. The Central Library is knowledge hub of Jawaharlal Nehru University, It provides comprehensive access to books, journals, theses and dissertations, reports, surveys covering diverse disciplines.”

The amount paid suggests that the books were viewed as very valuable: Rs 75K in the late 1960’s was a huge sum of money. And given that, it was quite amazing that the collection had not been kept as one, but the books had apparently been shelved by subject (!) and were then like any other books, and so subject to the periodic culling that most libraries undertake to clear shelves and make space for new books. (In some sense I was not too surprised, having myself bought a book that had been owned by AnandaKentish Coomaraswamy and that had somehow made its way to Princeton. The initials AKC were pencilled in on the first page, but apart from the bookplate, there was little else to show that it had been his. Unfortunately that book is no longer with me, and in hindsight, I think that when libraries acquire collections from scholars of note, they should make some attempt to keep the collections intact. Mercifully the JNU has done that now with more recent acquisitions..).

Nevertheless, a chance conversation shortly thereafter on the vagaries of libraries and the nature of intellectual inheritances started me off on my exploration of Kosambi and his mathematics. The idea was, on the face of, a simple one. What was the extent of Kosambi’s mathematical contributions compared to, say, his contributions to history or numismatics. How would the math stack up?  Having been in TIFR before I came to JNU, I had also heard of how he travelled from Pune to Bombay every day on the Deccan Queen, how he was fired from TIFR, etc. etc. But I also found out that precious little was known of DDK’s other life by the historians. That the mathematics was too different and far too difficult is all too true, but still.

To start with, I thought it would be good to put together his life mathematical, namely all his math and stats papers. Much of that was on the web, except that it was in bits and pieces, and all over the place. No single bibliography was accurate, and no matter where I looked, there were gaps. Many of the Indian journals where he published were not (and still are not) digitized. Some of the names that were given in the existing lists were incomplete or incorrect, many papers were missing. The Rendiconti della Reale Accademia Nazionale dei Lincei or the  Sitzungsberichten der Preussische Akademie der Wissenschaften, Physikalisch-mathematische klassewere both uncommon journals that were impossible to locate anywhere in India, for instance. I went to the Ramanujan Institute in Chennai in late 2009, looking for copies of the Journal of the Indian Mathematical Society and Mathematics Student where DDK had published a lot of his work in the 1930’s and 1940’s. It’s too depressing to recount that visit… Nothing could be located, and I left empty-handed after a wasted morning.

In 2010, though, I was visiting professor at the University of Tokyo for a month, and luckily, the Komaba campus where I was located, housed the mathematics department and more importantly, its library. It took a few hours spread out over several days, but before long, I not only had the bulk of DDK’s papers in photocopy or in digital form, but I also discovered, via MathSciNet, of DDK’s nom de plume S. Ducray, under which name he had written four papers. I also had access to the reviews of many of his mathematical papers by others, and could see many very famous names among the reviewers. As an aside, I should add that the library of Tokyo University is one of the few that have the complete sets of Journal of the Indian Mathematical Society and Mathematics Student, including the volumes published during the World War II years, when Japan and India were on different sides…

A Scholar in his time: The contemporary views of Kosambi the mathematician

Thanks to Arvind Gupta for sending this paper.

A SCHOLAR IN HIS TIME: CONTEMPORARY VIEWS OF KOSAMBI THE MATHEMATICIAN

RAMAKRISHNA RAMASWAMY

University of Hyderabad, Hyderabad, TS 500 034

“Kosambi introduced a new method into historical scholarship, essentially by application of modern mathematics.” J. D. Bernal [1], who shared some of his interests and much of his politics, summarized the unique talents of DDK [2] in an obituary that appeared in the journal Nature, adding, “Indians were not themselves historians: they left few documents and never gave dates. One thing the Indians of all periods did leave behind, however, were hoards of coins. [...] By statistical study of the weights of the coins, Kosambi was able to establish the amount of time that had elapsed while they were in circulation . . . ”

The facts of DDK’s academic life, in brief are as follows. He attended high–school in the US, in Cambridge, MA, and undergraduate college at Harvard, graduating in 1929. Returning to India, he then worked as a mathematician at Banaras Hindu University (1930-31), Aligarh Muslim University (1931-33), Fergusson College, Pune (1933-45), and the Tata Institute of Fundamental Research (1945-62), after which he held an emeritus fellowship of the CSIR until his death at the age of 59, in 1966.

Today the significance of D. D. Kosambi’s mathematical contributions [3–71] tends to be underplayed, given the impact of his scholarship as historian, and Indologist. His work in the latter areas has been collected in several volumes [72] and critical commentaries have appeared over the years [73, 74], but his work in mathematics has not been compiled and reviewed to the same extent [75, 76, 77, 78]. Indeed, a complete bibliography is not available in the public domain so far [79]. This asymmetry is unfortunate since, as commented elsewhere [75], an understanding of Kosambi the historian can only be enhanced by an appreciation of Kosambi the mathematician [80].

DDK is known for several contributions, some of which like the Kosambi-Cartan-Chern (KCC) theory [81], carry his name, and some like the Karhunen–Loève expansion [37, 39, 82], that do not. The Kosambi mapping function in genetics [40] continues to be used to this day [83], but the path geometry that he studied for much of his life [84] has not found further application. DDK’s final years were mired in controversies, both personal and professional. His papers on the Riemann hypothesis (RH) [65, 66] brought him a great deal of criticism and not a little ridicule, while his personal politics put him in direct conflict with Homi Bhabha and the Department of Atomic Energy. This contributed to his eventual and somewhat ignominious ouster from employment at the Tata Institute of Fundamental Research. His early and passionate advocacy of solar energy was practical and based on sound scientific common sense. In some of his arguments, he seems even somewhat Gandhian, and although this was a contrary position to hold in the TIFR at that time, the essential validity of his argument remains to this day [85].