The Natural Effectiveness of Mathematics in the Biological Sciences

Wigner, E. P., The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pure Appl. Math., 1960, 13. 

Some years ago, too many for my reckoning, I was invited to contribute an article for a special issue of the journal Current Science (Bangalore), on the use of mathematics in  different scientific disciplines. I had occasion to read the article again after some 15 years, mainly to cannibalize it for a talk I had to give yesterday, I should confess. Some embarrassment is inevitable on reading something one has written some time ago (I have almost never looked at my Ph D thesis, for example) but I thought that some of it could be shared, so here is an abbreviated essay where I have not removed all the dated bits… The title, of course, acknowledges a great thinker and physicist, Eugene Wigner.

An increasingly quantitative approach within the biological sciences has been accompanied by a greater degree of mathematical sophistication. However, there is a need for new paradigms within which to treat an array of biological phenomena such as life, development, evolution or cognition. Topics such as game theory, chaos theory and complexity studies are now commonly used in biology, if not yet as analytic tools, as frameworks within which some biological processes can be understood. In addition, there have been great advances in unravelling the mechanism of biological processes from the fundamental cellular level upwards that have also required the input of very advanced methods of mathematical analysis. These range from the combinatorics needed in genome sequencing, to the complex transforms needed for image reconstruction in tomography. In this essay, I discuss some of these applications, and also whether there is any framework other than mathematics within which the human mind can comprehend natural phenomena.

It is a commonplace that in recent years the biological sciences have gradually become more quantitative. Far from being the last refuge of the nonmathematical but scientifically inclined, the modern biological sciences require familiarity with a barrage of sophisticated mathematical and statistical techniques.

By now the role of statistics in biology is traditional, and has been historically derived from the need to systematize a large body of variable data. The relation has been two- sided: biological systems have provided a wealth of information for statisticians and have driven the development of many measures, particularly for determining significance, as in the χ2 or Student’s-t tests. Indeed, Galton’s biometrical laboratory was instrumental in collecting and tabulating a plethora of biological measurements, and these and similar data formed the testing ground for a number of statistical theories.

The role of mathematics in biology is more recent. The phenomenal developments in experimental techniques that have helped to make biology more quantitative have necessitated the applications of a number of different mathematical tools. There have been unexpected and frequently serendipitous applications of techniques developed earlier and in a different context. The widespread use of dynamic programming techniques in computational biology, of stochastic context-free grammars in RNA folding, hidden Markov models for biological feature recognition in DNA sequence analysis, or the theory of games for evolutionary studies are some instances of existing methods finding new arenas for their application. There have also been the mathematics and the mathematical techniques that derive inspiration from biology. The logistic mapping, the discrete dynamical system that is so central to chaos theory, arose first in a model of population dynamics. Attempts to model the human mind have led to the burgeoning field of artificial neural networks, while the theory of evolution finds a direct application in the genetic algorithm for optimisation.

Mathematics is about identifying patterns and learning from them. Much of biology is still most easily described as phenomena. The underlying patterns that appear are nebulous, so extracting a set of rules or laws from the huge body of observations has not always been easy. Or always possible since some experiments (like evolution) are unrepeatable, and separating the essential from the inessential can be very difficult. Detail is somewhat more important in the life sciences: often it has been said that the only law in biology is that to every ‘law’, there is an exception. This makes generalizations difficult: biological systems are more like unhappy families. With the exception of natural selection, there are no clearly established universal laws in biology.

This is, of course, in sharp contrast to the more quantitative physical sciences where the unreasonable effectiveness of mathematics has often been commented upon. It might be held that these observations, coming as they do in the twentieth century, comment on a science that has already had about three centuries of development. The earlier stages of the fields that we now call physics or chemistry were also very poorly described by mathematics—there was no general picture beyond a set of apparently unrelated observations, and it required the genius of a Mendeleev, of a Faraday or Maxwell or Einstein to identify the underlying patterns and expose the mathematical structure that lay under some aspects of these fields. This structure made much of the modern physical sciences possible, and led to some of the most accurate verifications of the laws of physics. As predictive theories, relativity and quantum electrodynamics are unparalleled and have achieved astonishing accuracy. In a more complex setting, the seemingly infinite possibilities of organic chemical reactions have found organizational structure in the Woodward–Hoffman rules that combine an elementary quantum mechanics with notions of graph theory to make precise, semiquantitative predictions of the outcome of a large class of chemical reactions. What will it take to similarly systematize biology? Or to rephrase the question, what will the analogous grand theories in biology be?

The inevitable applications of mathematics are those that are a carry-over from the more quantitative physical sciences. As in the other natural sciences, more refined experiments have spearheaded some of these changes. The ability to probe phenomena at finer and finer scales reduces some aspects of biology to chemistry and physics, which makes it necessary to borrow the mathematics that applies there, often without modification. For instance, tomographic techniques rely on a complicated set of mathematical transforms for image reconstruction. These may be largely unknown to the working biologist who uses NMR imaging, but are a crucial component of the methodology, nonetheless. Similarly, the genome revolution was catalysed by the shotgun sequencing strategy which itself relied on sophisticated mathematics and probability theory to ensure that it would work. Several of the problems in computational biology arose (or at least were made more immediate, and their resolution more pressing) by the very rapid increase in experimental power.

The other sort of application of mathematics is, for want of a better descriptor, a systems approach, namely that which is not predicated by the reductionist approach to biology but instead by a need to describe the behaviour of a biological entity in toto.

Even the simplest living organisms appear to be complex, in way that is currently poorly described and poorly understood, and much as one would like, it is not possible to describe in all totality the behaviour of a living organism in the same way as one can the behaviour of, say, a complex material. The promise that there could be mathematical models that capture the essence of this complexity has been held out in the past few decades by several developments, including that of inexpensive computational power which has made possible the study of more realistic models of biological systems. Theoretical developments—cellular automata, chaos theory, neural networks, self-organization— have provided simple mathematical models that seem to capture one or the other aspect of what we understand as ‘complexity’, which itself is an imprecise term. There is one class of applications of mathematical or physical models to biology which attempt to adapt an existing technique to a problem, while another aims to develop the methods that a given problem needs. Each of these approaches have their own value and appeal. In the next sections of this article, I discuss some of the ways in which they have found application in the study of biological systems.

Hamming, R. W., The unreasonable effectiveness of mathematics, Am. Math. Monthly, 1980, 87.

The resonance of the title with those of the well-known essays by Wigner and Hamming is deliberate, as is the dissonance. There are applications in the physical sciences where knowledge of the underlying mathematics can provide very accurate predictions. Comparable situations in the biological sciences may not arise, in part because it may be unnecessary, and in part because biological systems are inherently unpredictable since they are so fundamentally complex. The demands, as it were, that are made of mathematics in the life and physical sciences are very distinct, and therefore, it is very reasonable that the mathematics that finds application in the two areas can also be very different.

Is there any framework other than mathematics within which we can systematize any knowledge? Recent advances in cognitive studies, as well as information that is now coming from the analysis of genomes and genes, suggest that several aspects of human behaviour is instinctual (or ‘hardwired’). That mathematical reasoning is an instinct that we are endowed with is a distinct possibility, and therefore, it may not be given to us (as a species) to comprehend our world in any other manner. This point of view, that it is very natural that we should use mathematics to understand any science, is explored below.

In the last few years there has been a veritable explosion in the study of complex systems. The concept of complexity is itself poorly defined (‘the more complex something is, the more you can talk about it’ ), and as has been pointed out by others, ‘If a concept is not well-defined, it can be abused.’ Nevertheless, there is some unity in what studies of complexity aim to uncover.

A common feature of many complex systems is that they are composed of many interconnected and interacting subunits. Many systems, natural as well as constructed, are, in this sense complex. Examples that are frequently cited apart from those involving living organisms such as ecologies or societies, are the human brain, turbulent flows, market economies or the traffic. A second feature of complex systems is that they are capable of adaptation and organization, and these properties are a consequence of the interconnection and interactions of the subunits. The mathematics of complex systems would thus appear a natural candidate for application to biology. The drawback is that there is, at present, no unifying framework for the study of complex systems although there are some promising leads offered by studies of dynamical systems, cellular automata and random networks.

That the description of phenomena at one level may be inadequate or irrelevant at another has been noted for a long time. Thus the electronic structure of atoms can be understood quite adequately without reference to quarks, and is itself irrelevant, for the most part, when dealing with the thermodynamics of the material of which the atoms are constituents. Schrödinger, in a chapter of his very influential book (Schrödinger, E., What is Life?, Cambridge University Press, Cambridge, 1967) entitled ‘Is Life Based on the Laws of Physics’, observed that with regard to ‘the structure of living matter, that we must be prepared to finding it working in a manner that cannot be reduced to the ordinary laws of physics’. He further contrasted the laws of physics and chemistry, most of which apply in a statistical sense, to biological phenomena, which, even though they involve large numbers of atoms and molecules, nevertheless have nothing of the uncertainty associated with individual properties of the constituent atoms. Indeed, given a radioactive atom, he says, ‘it’s probable lifetime is much less certain than that of a healthy sparrow’.

But even at a given level, it frequently happens that the properties of a system cannot be simply inferred from those of its constituents. The feature of emergence, namely the existence of properties that are characteristic of the entire system but which are not those of the units, is a common feature of systems that are termed complex.

Distinction should be drawn between the complex and the complicated, though this boundary is itself poorly defined. For instance, it is not clear whether or not in order to be deemed complex, a system requires an involved algorithm (or set of instructions). The algorithmic complexity, defined in terms of the length of the (abstract) program that is required is of limited utility in characterizing most systems

starlingAttempts to decode the principles that govern the manner in which new properties emerge—for example the creation of a thought or an idea, from the firing of millions of neurons in the brain, or the cause of a crash in the stock market from the exit poll predictions in distant electoral constituencies—require new approaches. The principles themselves need not necessarily be profound. A simple example of this is provided by a study of flocking behaviour in bird flight. A purely ‘local’ rule: each bird adopts the average direction and speed of all its neighbours within distance R, say, is enough to ensure that an entire group adopts a common velocity and moves in unison. This behaviour depends on the density of birds as well as the size of R relative to the size of a bird in flight. If R is the size of a bird, then each bird flies on its own path, regardless of its neighbours: there is no flock. However, as R increases to a few times that of the bird, depending on the density, there can be a phase transition, an abrupt change from a random state to one of ordered, coherent, flight. And such a system can adapt rapidly: we have all seen flocks navigate effortlessly through cities, avoiding tall buildings, and weaving their way through the urban landscape at high speed.

boulezBut there are other aspects of complexity. A (western) orchestra, for instance, consisting as it does of several musicians, requires an elaborate set of rules so that the output is the music that the composer intended: a set of music sheets with the detailed score, a proper setting wherein the orchestra can perform, a specific placement of the different musical instruments, and above all, strict obedience to the conductor who controls what is played and when. To term this a complex system would not surprise anyone, but there is a sense in which such a system is not: it cannot adapt. Should the audience demand another piece of music, or music of another genre, an orchestra which has not prepared for it would be helpless and could not perform. Although the procedure for creating the orchestra is undoubtedly complicated, the result is tuned to a single output (or limited set of outputs). There is, of course, emergence: a single tuba could hardly carry a tune, but in concert, the entire orchestra creates the symphony.

Models like this illustrate some of the features that complex systems studies aim to capture: adaptability, emergence and self-organization, all from a set of elementary rules. The emphasis on elementary is deliberate. Most phenomena we see as complex have no obvious underlying conductor, no watchmaker, blind or not who has implemented this as part of a grand design (Dawkins, R., The Blind Watchmaker, Norton, New York, 1996). Therefore, in the past few decades, considerable effort has gone into understanding ‘simple’ systems that give rise to complex behaviour.

logistic‘Simple mathematical models with very complicated dynamics’, a review article published in 1976 (May, R. M., Simple mathematical models with very complicated dynamics. Nature, 1976, 261, 459) was responsible in great measure for the phenomenal growth in the study of chaotic dynamics. In this article—which remains one of the most accessible introductions to chaos theory— May showed that the simplest nonlinear iterative dynamical systems could have orbits that were as unpredictable as a coin-toss experiment. The thrust of much work in the past few decades has been to establish that complex temporal behaviour can result from simple nonlinear dynamical models. Likewise, complex spatial organization can result from relatively simple sets of local rules. Taken together, this would suggest that it might be possible to obtain relatively simple mathematical models that can capture the complex spatiotemporal behaviour of biological systems. 

A number of recent ambitious programs (eCell, A multiple algorithm, multiple timescale simulation software environment, intend to study cellular dynamics, metabolism and pathways in totality, entirely in silico. Since the elementary biochemical processes are, by and large, well-understood from a chemical kinetics viewpoint, and in some cases the details of metabolic pathways have also been explored, entire genomes have been sequenced and the genes are known, at least for simple organisms, the attempt is to integrate all this information to have a working computational model of a cell. By including ideas from network theory and chemical kinetics, the global organization of the metabolic pathway in E. coli has been studied computationally. This required the analysis of 739 chemical reactions involving 537 metabolites and was possible for so well-studied an organism, and the model was also able to make predictions that could be experimentally tested. The sheer size of the dynamical system is indicative of the type of complexity that even the simplest biological organisms possess; that it is even possible for us to contemplate and carry out studies of this magnitude is indicative of the analytic tools that we are in a position to deploy to understand this complexity.  

In recent years, there has been considerable debate, and an emerging viewpoint, that the human species has an instinct for language. Champions of this school of thought are Chomsky, and most notably, Steven Pinker who has written extensively and accessibly on the issue

Pinker, S., The Language Instinct, Morrow, New York, 1994

The argument is elaborate but compelling. It is difficult to summarize the entire line of reasoning that was presented in The Language Instinct, but one of the key features is that language is not a cultural invention of our species (like democracy, say), but is hard-wired into our genome. Like the elephant’s trunk or the giraffe’s neck, language is a biological adaptation to communicate information and is unique to our species.

Humans are born endowed with the ability for language, and this ability enables us to learn any specific language, or indeed to create one if needed. Starting with the work of Chomsky in the 1950s, linguists and cognitive scientists have done much to understand the universal mental grammar that we all possess. (The use of stochastic context-free grammars in addressing the problem of RNA folding is one instance of the remarkable applicability of mathematics in biology.) At the same time, however, our thought processes are not language dependent: we do not think in English or Tamil or Hindi, but in some separate and distinct language of thought termed ‘mentalese’.

Language facilitates (and greatly enriches) communication between humans. Many other species do have sophisticated communication abilities—dolphins use sonar, bees dance to guide their hive mates to nectar sources, all birds and animals call to alarm and to attract, ants use pheromones to keep their nestmates in line, etc.—and all species need to have some communication between individuals, at least for propagation. However, none of these alternate instances matches anything like the communication provided by human language.

It is not easy to separate nature from nurture, as endless debates have confirmed, but one method for determining whether or not some aspect of human behaviour is innate is to study cultures that are widely spaced geographically, and at different stages of social development. Such cross-cultural studies can help to identify those aspects of our behaviour that are a consequence of environment, and those that are a consequence of heredity. The anthropologist Donald Brown (Brown, D., Human Universals, Temple University Press, Philadelphia, 1991) has attempted to identify human ‘universals’, a set of behavioural traits that are common to all tribes on the planet.

All of us share several traits beyond possessing language. As a species we have innumerable taboos relating to sex. Some of these, like incest avoidance, appear as innate genetic wisdom, but there are other common traits that are more surprising. Every culture, from the Inuit to the Jarawa, indulges in baby talk. And everybody dreams. Every tribe however ‘primitive’, has a sense of metaphor, a sense of time, and a world view. Language is only one (although perhaps the most striking) of human universals. Other universals that appear on the extensive list in his book, and which are more germane to the argument I make below, are conjectural reasoning, ordering as cognitive pattern (continua), logical notions, numerals (counting; at least ‘one’, ‘two’ and ‘many’) and interpolation.

The last few mentioned human universals all relate to a set of essentially mathematical abilities. The basic nature of enumeration, of counting, of having a sense of numbers is central to a sense of mathematics and brings to mind Kronecker’s assertion, ‘God made the integers, all else is the work of man’. The ability to interpolate, to have a sense of a continuum (more on this below), also contribute to a sense of mathematics, and lead to the question: Analogous to language, do humans possess a mathematics instinct?

Poincaré, H., Mathematics and Science: Last Essays,
Dover, New York, 1963.

Writing a century ago, Poincaré had an inkling that this might be the case. ‘… we possess the capacity to construct a physical and mathematical continuum; and this capacity exists in us before any experience because, without it, experience properly speaking would be impossible and would be reduced to brute sensations, unsuitable for any organization;…’ The added emphasis is mine; the observations are from the concluding paragraph of his essay, ‘Why space has three dimensions’.

If mathematics is an instinct, then it could have evolved like any other trait. Indeed, it could have co-evolved with language, and that is an argument that Keith Devlin has made recently (Devlin, K., The Maths Gene, Wiedenfeld and Nicolson, London, 2000).

At some level, mathematics is about finding patterns and generalizing them and about perceiving structures and extending them. Devlin suggests that the ability for mathematics resides in our ability for language. Similar abstractions are necessary in both contexts. The concept of the number three, for example, is unrelated either to the written or spoken word three, or the symbol 3 or even the more suggestive alternate, III. Mathematical thought proceeds in its version of mentalese.

An innate mathematical sense need not translate into universal mathematical sophistication, just as an innate language sense does not translate into universal poetic ability. But the thesis that we have it in the genes begs the question of whether mathematical ability confers evolutionary advantage, namely, is the human race selected by a sense for mathematics?

Wilson, E. O., Consilience: The Unity of Knowledge, A. A. Knopf,
New York, 1998.

To know the answer to this requires more information and knowledge than we have at present. Our understanding of what constitutes human nature in all its complexity is at the most basic level. The sociobiologist E. O. Wilson has been at the vanguard of a multidisciplinary effort toward consilience, gathering a coherent and holistic view of current knowledge which is not subdivided in subdisciplinary approaches. This may eventually be one of the grand theories in biology, but its resolution is well in the future. We need to learn more about ourselves.

Traditionally, any sense of understanding physical phenomena has been based on having the requisite mathematical substructure, and this tradition traces backward from the present, via Einstein, Maxwell and Newton, to Archimedes and surely beyond.

Such practice has not, in large measure, been the case in biology. The view that I have put forth above ascribes this in part to the stage of development that the discipline finds itself in at this point in time, and in part, to the manner in which biological knowledge integrates mathematical analysis. The complexity of most biological systems, the competing effects that give rise to organization, and the dynamical instabilities that underlie essentially all processes make the system fundamentally unpredictable, all require that the role played by mathematics in the biological sciences is of necessity very different from that in the physical sciences.

Serendipity can only occasionally provide a ready-made solution to an existing problem whereby one or the other already developed mathematical method can find application in biology. Just as, for example, the research of Poincaré in the area of dynamical systems gave birth to topology, the study of complex biological systems may require the creation of new mathematical tools, techniques, and possibly new disciplines.

F1.largeOur instincts for language and mathematics, consequences of our particular evolutionary history, are unique endowments. While they have greatly facilitated human development, it is also worth considering that there are modes of thought that may be denied to us, as Hamming has observed , similar to our inability to perceive some wavelengths of light or to taste certain flavours. ‘Evolution, so far, may possibly have blocked us from being able to think in some directions; there could be unthinkable thoughts.’ In this sense, it is impossible for us to think non-mathematically, and therefore there is no framework other than mathematics that can confer us with a sense of understanding of any area of inquiry.

In biology, as Dobzhansky’s famous statement goes, nothing makes sense except in the light of evolution. To adapt this aphorism, even in biology nothing can really make sense to us except in the light of mathematics. The required mathematics, though, may not all be uncovered yet.


Shameless Self-Promotion

DDKCover.pngAfter 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 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 WordsworthBliss 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.

Carrying on

ccxCommenting on my last post, an old classmate wrote to say “Ram, we are both at an age where we mark the passage of time by composing eulogies for our friends and loved ones. One day someone else will do the same for us….”

True enough. I found that in the past year or so, I’ve done this four times, and each time has been painful in its own way… The passage of the years does indeed makes these occasions more frequent, but every passing is none the easier for that. And every cliché in the book has some ring of truth to it, each day has its own new regrets.

I have been overwhelmed by the several letters that friends from all over the world have written in the past few weeks. And touched by the genuine expressions of grief, by the concern and the affection. I am beginning to respond to these, but each response goes with its own memories, so this note is both to acknowledge how heartwarming it has been to read each message and to say  I will write back, but maybe slowly.  We will meet, and when we do we will speak of other things, without forgetting this connection.

A Physicist and a Gentleman

Dr Deepak Kumar (1988)My friend and colleague, Deepak Kumar, passed away all of a sudden late Monday (25th January) night. I had seen him that day, sharing a cup of tea with another member of the faculty in the afternoon sun on the lawns of the School of Physical Sciences at JNU. The spot where he sat was directly visible from my office window- Deepak often sat there and had his lunch. I hadn’t spoken to him that particular day, but that was not unusual – there were many days like that. But it was not just another day, not like any other.

Deepak was one of the first to join the School as Professor when it was formed, and he brought a decade or more of experience at the University of Roorkee. As it happened that greatly helped the School in its early, formative years, and set the mark for how it developed subsequently, defined what it’s core values were, and the sense of purpose and commitment that it has had since.

Colleague for almost 30 years, Deepak has been a friend for a little over that, and if I were to have to characterize him, the title of this post says it as well as anything. Deepak was a scholar in the true sense of the word, and one for whom the world of physics was all absorbing. Although his professional interests were in condensed matter physics, he was both knowledgeable about, and was interested in a huge range of topics. One could go to him for just about any doubt, count on him to give the right bit of advice, and if the matter happened to be something that he knew well, his intellectual generosity was limitless.

This is not exaggeration. Not for nothing was Deepak the most collaborative colleague that we have at the SPS:  of the 20 or so faculty that we have in physics, Deepak has actually written papers with no less than seven of us. And with something like twice that many students, either as their formal or informal supervisor, as a mentor in the best tradition.  Indeed, he mentored the first Ph. D. that was awarded from the SPS, and both directly and indirectly showed many of us the way in which one could bring out the very best in our students.

There is so much to say about Deepak- his academic contributions in condensed matter and statistical physics, the several awards, the recognition. But this above all: This was too soon and too sudden. There were many many good years of physics one could have had from him, and many years of physics that he would have enjoyed.  Even the last day, on Monday, he gave a lecture, there was another scheduled this week. And last semester he taught a course for the MSc Physics seniors. He was working to the end, and he went with his academic boots on…

I know his ethos will continue to guide us, and I can only hope that we will not forget his calming spirit that often brought hot tempers down, his somewhat other-worldly smile, and his gentle sense of humour that helped us all see that there were many ways of reaching conclusions. We all will miss him deeply, the community that he helped build at JNU, and the larger community of physicists in the country that knew and admired him.

The Mother of All Chemistry Departments

snake1972 was a very good year to join the IIT Kanpur Chemistry Department as an MSc student. Some 15 of us, bright eyed and bushy-tailed for the most part, did. There was an incredible air of modernity about the place, from the architecture, to the teachers, their teaching, the labs, the hostels, the facilities. The passing years have coloured the memories and blurred some of the edges, but nevertheless, I can still remember the freshness of the campus and the feeling that we had arrived somewhere special.

Most of us- barring the Delhi University sophisticates- were from colleges in somewhat provincial universities. And in those days, universities in Madras, Kolkata, Pune and Bombay all were to varying degrees provincial, and we had classmates from Madurai, Kolhapur and Burdwan as well… All plagued by poor and outdated syllabi, bad teaching, the works. Many of us were also scholarship holders of the National Science Talent Scheme, that great initiative of the NCERT, and we had been exposed to some of the more modern ideas, so we knew the good places to go to. And without doubt, IIT-K was the place to go to if you wanted to do chemistry, with the added attraction that if one did reasonably well, it was a direct line thereafter to the US aka “Fatherland”.

The Chemistry Department, to put it mildly, was rocking! Our teachers were (and many still are) legendary. Almost from day one, the classes were in a completely different category from what we had been used to- no notes for one thing, surprise quizzes, open book examinations… It was not unusual to get homework from the latest issue of JACS, the Journal of the American Chemical Society- giving us the feeling that this was what an international education was all about. And it was.

Arguably, the Chemistry Department at IITK in those years was competitive with the best in the US. The faculty line-up was exceptional and the publication standards were better than most. All the big names were there- and let me not name them, the faculty at that time was the who’s who of Indian chemistry. But more than being famous, they were really inspirational. I can still recall- almost verbatim- a course in Group Theory that we all took in the second or third semester (another innovation in 1972!). And the course in Synthetic Organic Chem. or that in Phys. Chem… The geeks amongst us (mostly all) had it good.

It was a time, the first that I remember, when I was immersed in a group that, by and large, loved a subject. We talked chemistry, did homework together, did projects (some crazier than others). My undergraduate years had been spent largely in goofing off- most of those who came to the BSc course were there to pick up a degree and move on to the rest of their life- IAS, MBA, whatever- and the few who were interested in the life academic were oddities.

173_001Peer group pressure (and there was plenty of it!)  and teachers apart, there was a steadily growing set of seniors that were setting standards. The ones who had gotten into Harvard, or Chicago, or wherever. The ones who had written research papers as MSc students (and in Nature, no less). The ones who were clearly going to be the next big things… This made us, for the most part, academically very ambitious. In the days before rankings had reduced everything to labels like top ten and so on, there was mostly reputation to go by, and when we applied, the bar was always set high. A few in our class decided not to go on with a Ph D in chemistry- IIM Ahmedabad and BARC were the alternate choices, but for the rest, the next step was to Harvard, Princeton, Berkeley, Chicago, Indiana, SUNY, and so on. But in 1973 the level of competition that one sees today was just not there; all it took to make an application was a respectable GRE score and an aerogramme…

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Prodipto Banerjea (note the deadpan expression) and I in a friend’s hostel room in Hall V, IITK. Early 1973, given our attire and the cut-outs on the wall.

This post actually started out as a long answer to a short question by one of my students as to what “were your thoughts then?  On aspects of academic/extra-academic life… I mean, how did you see the world that time?”. Truth be told, the thoughts were all in the short term. Very short term (as the pictures on the hostel room wall might suggest). Competition was strong, so one wanted to be at a local maximum at the very least, but one could also see what the milieu was throwing up. The B. Tech. batches were very gifted and this was before the coaching classes had dulled the sheen of the JEE rankings. The talents were visible and aplenty, with enough 10. someones, as well as the clearly very cool set.  The faculty was very liberal- in some ways more than what we see now: I recall, as an MSc Chemistry student, taking MSc and Ph D courses in the Physics Department, for credit. Not too many questions asked, and it figures on my transcript. (At the two Universities where I have taught recently, I can say with certainty that if this happened at all, it happened with much sturm und drang.)

And our teachers experimented with pedagogy. With a lot of thought, as even a casual look at the course curriculum would tell- it is, even now, a surprisingly modern curriculum. And with an ability to change. Willingly, as some teachers introduced Bio into Chem (it was not that common then) and unwillingly, as when some of us trashed the attendance requirement and told the instructor we would only do tutorials and the final exam, not go to his classes.

There was a downside, of course. We did not share a certain kind of easy friendship that a less competitive atmosphere might have engendered. Of my 14 classmates, I have not met 4 in the last 40 years, and only 4 of them more than once or twice in all that time. Five of our class chose careers outside science, four were in industry, one went to a national laboratory. Academics eventually attracted only five of us, two in India and three in the US, making the connections more and more tenuous with the passage of time. And now most of us are reaching retirement, so in retrospect, and there is only retrospect now, this was a major shortcoming. A sense of community certainly helps beyond the science, and grass being greener apart, I think that other groups of the same times have bonded better. Maybe it was that we were only together for two years- not a long time, admittedly- but still.

17235.iconBut there was more, much more to IIT K than just the classes, and enough attention to these aspects had been given when the institute was set up. Extracurricular activities apart, there was an airstrip, and a TV station as well- that actually broadcast programs on campus, including the 1973 England vs. India test match that was played in Kanpur, Gavaskar and Bedi being the stars then. And as for the airstrip, I’ve forgotten the chap’s name, but his nickname was Pilot because he knew flying, and I- in retrospect foolhardily- went up with him in a glider. Given the level of safety that we all subscribed to, its a miracle that there were no major accidents! (I would do it again gladly, of course.)

But to get back to the title of this post, the IIT-K Chemistry department was, in many ways, the progenitor of many others that were set up in the 70’s both in style and in content. Many of those who taught us were to leave shortly thereafter to take up positions in Hyderabad, Kolkata, Bangalore and elsewhere. In a sense, the research and teaching culture spread, and flourished.

There is a very real Masonry of IIT-K Chem alums: strong ties bind us to where it all began. For all of us- teachers and students alike- this was a great initial condition to have.