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I was more than a little pleased when I got an invitation from the Indian Institute of Science Education and Research, Mohanpur to deliver this year’s commencement address at their convocation on the 11th June. And more than a bit nervous.  After the first set of my reactions had passed, and in the absence of my usual sounding board(s), I started worrying about what to say, what sort of pithy advice to serve up, what note to strike. In the end, here is what resulted, more or less…

[I’m leaving out the introductory salutations, etc. ]

Congratulations foremost, to all those who have been awarded degrees and medals today. This is an important marker, one that most of you would have worked hard to achieve. But then, for all of you, this will just be the beginning of the next chapter of your lives.

We live in interesting times. Some would say too interesting given that the pace of advancement in the sciences appears ever to accelerate, but still.  In the past few months alone we have seen wonderful things – an image of a black hole, news of possible room-temperature superconductivity…  and further back, gravitational waves, the Higgs boson, gene editing through CRISPR-Cas9, stem cells, cancer immunotherapy, treatments for Parkinson’s and Alzheimer’s, finding water on the moon, on Mars, AI, … any number of advances in all the different branches of science. The excitement is real, and for all of you graduating today particularly, these lines from Wordsworth apply so well,

“Bliss it was in that dawn to be alive,

But to be young was very Heaven! “

Bliss indeed to be alive in this age, and to be young! To be sure, there is so much happening in the world of science it is a very good time to be a young scientist.  Many of you will choose a life in science.  Some of you will have already committed to a life in research. Given your initial conditions, though, it is doubtless that science will, in some form or the other, be central to your way of thinking in your future. Even if your passions take you through different paths, what you have imbibed over the past few years will stay with you and shape your approach to life.  There are so many discoveries to be made, so much more to learn… This is just the beginning.

A commitment to a scientific way of life is vital. You would have heard more than once that ours is the only country to have it enshrined in our Constitution that there is a need to have a scientific temper. A scientific temperament is a great asset, regardless of what one does, but doing science is of value. We have seen that our lives have, on average, been immeasurably improved by science, and more to the point, by an overall commitment to science. On occasion some people may profess otherwise, but the average Indian believes very strongly in the transformative power of science and technology.

The widespread support for the IITs and IISERs is part of this belief: we need to have more of you, more that can contribute to the development of science in the country. Your institute is young and very special, and that is a great advantage.  The idea of your institute is also young, so it can rise to new challenges, and can adapt to the changes that these times require, with the agility that only the young possess.

My own academic career has been largely spent in a very different kind of institution – a central University, where the sciences had to learn to coexist with the humanities, language, and a range of other disciplines. There is some advantage in such an environment, not least of which is the respect one learns for very different modes of thought. Speaking from personal experience, the diversity that is inherent in such an institution is very instructive. Much of my own thinking about academia has been moulded by the uniqueness of that experience and the opportunities that it provided.

But to speak of change, and the challenges that changes bring. There is a saying that the more things change, the more they stay the same, but that was said long ago and in another tongue. In the past two decades (a period that spans the life of this institution) alone, the academic workplace has changed very fundamentally, both the laboratory, and the classroom. Part of this has been because of the way in which our country has evolved, but another and more important part has been because of the way in which our world has changed. The role of the teacher has changed drastically in the past few years mainly due to the internet, and the emphasis in educational institutions is slowly shifting, from the passing of examinations to the acquisition of skills. 

How science is done today has slowly but surely changed with the changes in communication and increased mobility. Collaborative science has enabled the pursuit of bigger problems: the solitary scientist, working on an individual problem, is rare outside a few disciplines, and large multidisciplinary groups working on some ambitious projects, are the norm in others. Papers co-authored across national boundaries are commonplace, and those with over a thousand authors are not just confined to particle physics, these days one can even find them in biology!

But in other ways, we are connected in a manner that is global in its essence, and international institutions are fond of articulating grand challenges to highlight the global nature of several problems. Climate change is not restricted to a few nations. Global warming is not only real, it will affect all of us regardless of our individual carbon footprints. The damaged environment – pollution – affects all of us, rich and poor alike. Water scarcity will recognise no national boundaries…. Tackling such issues is going to require our collective efforts and our collective resources, and while we can anticipate that many of these problems will be addressed, the how and when is not clear, and it will need a multiplicity of institutions to show the way forward.  Our graduates must be seen as part of a global cohort and must be prepared as such.

            An institution such as IISER-K can play a leadership role in this context.  Many of the problems can be addressed mainly through basic research that integrates both the natural sciences as well as the social sciences and the humanities, with each other and within themselves. And this needs to be done proactively; tomorrow cannot wait.

            Speaking of which, are we still hidebound to a classic ideal, that of narrow categories of disciplines within which teaching happens? This needs to be debated since there really are no easy answers.  What are the exciting areas of the future and what are the skills that students will need in order to tackle the important questions therein? Do we teach too much? And do we permit a student the option to explore what she or he may want to learn as opposed to what they are required to learn?  The changing boundaries of subjects are kind only to those who are prepared and willing to take a plunge. A spectacular cross-disciplinarian of recent times, a Nobel laureate in Chemistry, had his degrees in physics and his work was in biology. More by accident than by design, perhaps, but we need more such happy accidents.

There are other changes that we need to recognise at the same time. The reality of the India of tomorrow and the changing demography needs to be addressed. India has the largest proportion of young people, those that need and want skills and learning.  Increasing numbers of those seeking higher education will necessarily be first generation learners, and the manner in which our classrooms adapt to this change will determine how well we prepare ourselves for the future.

But today is mainly about those that are graduating, those on the brink of a new tomorrow. It is traditional in commencement addresses to give some meaningful advice, some inspiration…. let me do that by drawing from Cavafy’s great poem, Ithaka that captures so well the essence of the many journeys we undertake.

“When you set out for Ithaka

ask that your way be long,

full of adventure, full of instruction.

The Laistrygonians and the Cyclops,

angry Poseidon – do not fear them:

such as these you will never find

as long as your thought is lofty, as long as a rare

emotion touch your spirit and your body.

[…]

Ask that your way be long.

At many a Summer dawn to enter

with what gratitude, what joy –

ports seen for the first time

[…]

Have Ithaka always in your mind.

Your arrival there is what you are destined for.

But don’t in the least hurry the journey.

Better it last for years,

so that when you reach the island you are old,

rich with all you have gained on the way.

[…]

So do choose goals that are ambitious- ask questions for which the answers are worth knowing, and spend the time needed in order to resolve them. There will be distractions aplenty, and distractions everywhere. Disregard them.

Take it slow. The goal is the thing.

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.

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

Attempts 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.

But 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.

‘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, http://www.e-cell.org) 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

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?

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?

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.

Our 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.

When I was in the last years of high school in Mussoorie, a brand new magazine appeared- JS, or Junior Statesman. The first issue came out in January 1967, and it quickly became staple reading for me and my friends, especially during long study periods when it was also forbidden… Although published from Calcutta (I remember going to Desmond Doig’s office once in 1970 or ’71) there was a lot of Kathmandu in it, drawings of hippies, articles by Jug Suraiya, Sashi Tharoor, Zeenat Aman, all exotic names in our boarding school. It was our one great weekly escape that we shared. When I went to college later, I went on to send contributions to JS (and earned some pocket money in the bargain), but regrettably I never kept copies, even though one of the articles I wrote even made it to a cover. And The Statesman has either not digitized this classic (or has chosen to not make the files public) and as a result, the only illustrations I was able to find on the net are very unsatisfactory. See above.

But Kathmandu…

The gap between passing out of school and entering college was a long one: The Senior Cambridge exam was held in November 1968, and college admissions were not until June the following year. And for the not very professionally inclined – I was not going to do either medicine or engineering – there was a lot of time on my hands. As it happened, there was a to-do about the exams my year, and we all had to repeat the school final in February or March 1968, but still, there was a lot of waiting time, and I decided I wanted to travel “abroad”.

I had already traveled outside India and found it quite underwhelming… One afternoon, driving beyond Tuensang in Nagaland, we stopped the jeep at a post along the road that said simply “Burma”. That was about it, but being a little over 14, I jumped out, ran into Burma, expecting I’m not sure what. Of course I could not have expected that it would be very different, but still, there was an unreasonable disappointment… And a year or so later, when living in Ukhrul in Manipur, we drove down to the border town of Moreh and crossed over into Tamu in Burma. Being heavily populated by Tamils at that time, it was actually possible to get by in Tamil, and the only sense of the foreignness of Tamu that I got was from the visibility of a lot of “Made in China” goods and Burmese parasols, neither of which were of interest to me then. But still, seeing the pagoda and having to change money into Kyat was a positive…

At some point in early 1969, I decided to go to Nepal. Some cajoling of parents was needed, but they gave in soon enough- there was little enough to do in Ukhrul, and those were also days in which, in retrospect, our lives seemed surreally secure. I was a little over 15, and all I had as I set off for Kathmandu was the address of some people at the Indian Embassy, friends of friends. My travel plan was vague. I would fly from Imphal to Calcutta (via Silchar and Agartala) by the Indian Airlines Dakota, take the train (third class, no reservation) to Patna, and fly into Kathmandu from there…

Amazingly enough, it happened pretty much like that. I had an old sleeping bag and a duffel, some money- I’m not even sure how much, except that it was probably a few hundred rupees, a princely sum in those days- and an idea that Nepal was doable, and both exciting and inexpensive. In addition there were student concessions that made every trip half-price or less, and the Indian Airlines flights were adventures in of themselves. To this day I regret not taking the Agartala to Khowai flight when I could have- it was the shortest flight in the world at that time, and at Rs 7 ($1.40 then) surely the cheapest.

I took some train to Patna and landed up early in the morning, and eventually made my way to the Indian Airlines city office by a rickshaw. I managed to get myself the very last seat on the morning flight to Kathmandu, and it is another testimony to the times that none of the IA staff (and from talking to some of the old-timers, I know that they all regret the merger with Air India) found it odd that I would be traveling on my own on an international flight. I can’t swear to it, but from the little scouting I have done on the web and the few old timetables I could find, I think it must have been IC 245, the thrice-weekly Vickers Viscount flight at 9:05 in the morning. That day we were a bit delayed by a massive dust storm – I have a very clear memory of the sky turning a vivid brown – but soon enough, I was in my window seat and on my way…

The window seat. Its funny what one chooses to remember – my eyes must have been glued to the outside for all the 50 minutes of the flight, but I can still see the roseate Himalayan peaks as we came into the Kathmandu valley in the early morning. Time has added hues and tinges to all memories, but I can still sense the excitement that I felt landing at Tribhuvan airport, more than a bit nervous since all I had was a school ID and an address in the Embassy compound that I needed to get to.

And the arrival did not disappoint. In those days, all taxis in Kathmandu were whimsically painted in tiger stripes- and many were Volkswagen Beetles. The Tiger Taxis were a world away from the black and yellow Ambassadors of Calcutta and Delhi- but strangely I was only able to find a few images on the net. Anyhow, I was soon at the Embassy, and spent the subsequent few days doing much the usual things one did before there was Thamel.

Getting back was another trip. I had run out of money, so flying back was not an option, and people told me about taking a bus to Birgunj, on the border between Nepal and India, from where one took a rickshaw to Raxaul on the Indian side. My memory of the ride is a dimmed one, the only striking image I have of that ride is when I looked over my shoulder for that last view of the Himalayas…

And then Birgunj. By the time the bus deposited me in Birgunj, it was dark, all the better for me to make the crossing I thought. I had bought some “foreign goods” – some inexpensive perfumes and gifts for my parents, mainly – and was firmly convinced that this would going to land me in trouble with the border police. Although I had hidden these artfully in my luggage in case of a search, I was quite nervous as we made the cross into Raxaul, and to the train station from where one would take the train back to Calcutta.

I can’t decide whether Google and Google Maps are a good or poor way to recreate memories… There is so much that gets thrown up on a search that I’m not sure if these are my recollections or trancreations thereof. Anyhow, it has been a bonus to discover the old IC timetables and to realize that already in 1970, Indian Airlines flew between 62 cities in India. (The number today is not that much more within the country, while it has greatly increased the number of flights outside the country.) The train ride I took from Raxaul to Patna was via Muzaffarpur, and I do have a vivid memory of crossing the Ganga at Patna by ferry. These were the days before the Mahatma Gandhi Setu was built (as I discovered via G) or the Digha-Sonpur bridge… Blame it on the age, but the excitement of taking a boat trip- it was less than an hour long – to top off everything was a big bonus! And I can add in hindsight, it was one of the nicest ways in which I have arrived in Patna.

There really is not much more to tell. I got back to Calcutta the next day, took a plane the next to  Imphal (via Agartala and Silchar) and reached Ukhrul, feeling very “foreign-returned” and worldly wise. I had had my share of sightseeing, the living goddess, Hanuman Dhoka, Pashupatinath, Swayambhunath, renting a bicycle and riding out to Patan, eating momos for the first time… the works, but as the wise TSE had said, it really was all about the journey.

An invitation from Bangalore University to speak at a meeting on Pedagogy and Research in the Sciences in Universities (and to listen to other academics) has given me an occasion to think again of what it is that we do when we teach at university, what we do right, and what we do not. There is a certain amorphousness to the enterprise of instruction above a certain level, where intentions and motivations are (in my opinion properly) a bit blurred…

It also brought to mind, oddly enough, The Lesson,  that wonderful play by Ionesco that was such a staple of the amateur drama circuit when I was in college, and which is such a commentary on the nature of the teacher-student relationship and of pedagogy itself.

But I digress. Academic life at an Indian university – and here context is everything – has been undergoing considerable change in the past few years, both because the classroom is changing, and because the student is evolving. Sometimes a little too rapidly for one’s liking. And if truth be told, a little too rapidly for the average instructor’s comprehension, and that includes me.

What is so special about pedagogy and research in the sciences at universities? For one, these are now just one of the several institutions where a science-rich (i.e. a curriculum with a majority of courses in scientific areas) education can be obtained. The IITs, the IISERs and a few other institutions all vie to offer a range of sciences at the undergraduate and Masters’ levels, some in integrated five-year programmes. At the Ph. D. stage, the competition is even wider with all the national laboratories essentially being deemed-to-be-universities. So one difficulty that universities face in re science students is to get them there in the first place.

The difficulty is truly undeserved. There is enough evidence to suggest that students draw significant benefit from having an intellectually diverse academic environment even if their intended area of study, their specialization, is in itself quite narrow. But of course, the range of specializations at a given place is not as important as the quality of the people there, so its a toss-up as to what is preferable. And over the years, as we (as a nation – we are all responsible!) have encouraged the flight of intellect and academic talent out of the universities into the cloisters of research institutes, by policies of preferential funding for the latter over the former, by differential service conditions and numerous other marks of privilege, it is with a heavily biased coin that one calls the toss.

The advantage that universities have, of numbers, is one we should hold on to. A middling sized masters’ class in physics at JNU now has about 30 students, at Bangalore University this number would be 80, and at Delhi about 200. And these numbers offer several opportunities. There are always (and the emphasis is deserved) exceptionally gifted students in such classes, and also always, some really poor students. That’s just the law of large numbers at play. The width of the ability distribution makes teaching such classes a challenge, and (so long as one  is not terminally cynical) there is much to learn from the process.

There are matters we take for granted, such as the purpose of higher education (at the postgraduate level and above) in our country. Academic policy makers will trot out graphs to suggest that there is a positive correlation between GDP and the quantum of original research or the number of scientific papers published. So one purpose of university departments is to “produce” postgraduates, Masters and Ph D’s for somewhat ephemeral ends. But then some of these postgraduates will end up in research careers as well, and perhaps in teaching positions, so that the concerns of the seminar acquire an existential angst that is all too familiar.

One can go on. But to return to the theme of the seminar, namely Pedagogy and Research in the Sciences, my few points are the following.

The first is that the role of the teacher in the university classroom has changed drastically in the past few years. The teacher today is not the central source of information in the way he or she was say, twenty years ago. One still gets respect as a teacher, but not in the simplistic ways of the past – one has to earn it every day… This was always true, of course, but now there is a constant comparison with material all over the web, lectures on any topic easily available on YouTube, MOOCs, Wikis… the list is long.

Next, and this might not be true for all subjects, we tend to over-teach. At any rate, we do tend to over-regulate curricula. While that is grist for another post, it is true that students today are left with few real choices,  the real freedom to design one’s own learning that comes with the semester system being sacrificed for some form of academic uniformity. I’ve seen “elective” courses being made effectively compulsory year after year, the need for completing a set of credit requirements competing with a shortage of teaching staff. Students are integral to the pedagogic process, and not just as the intended targets. They are as responsible in determining outcomes as teachers are, in ways that need a somewhat more individualized approach.

My main concern however, relates to the modularization of pedagogic practice. Over the years the educational pattern in the country has evolved from the system of annual examinations to a semester system, much like the rest of the world. The standard model of academic teaching that has been adopted in most of our progressive universities is one of semester-wise teaching of a number of courses, each considered a stand-alone subject.  It is too late in the day to debate whether it is a good or bad thing, especially when there is so much regulatory pressure to conform to some mythical global standard, but one aspect of such instruction is that it poses new challenges to the didactic process and these challenges have not been seriously addressed.

The unity of a discipline – that which a Department of Study strives to convey – is often missed out in a compartmentalized approach to teaching. The modularity of semester based courses, and in some instances, this is broken up into even smaller modules- can make a discipline seem more granular than it necessarily is.

The contrasting styles of Hindustani and Carnatic classical music offer some parallels to these two pedagogic approaches. A night-long concert of Hindustani music may well be devoted to a single raga, developing and evolving a theme gradually. A Carnatic concert of a few hours, on the other hand would have several compositions in different ragas and with different tempi. There are those who swear by one and those that swear by another, but in the end, both have their adherents as well as those that simply don’t get either.

But again, I digress. Modular teaching can lead to modularized learning, with some students feeling that specific topics belong to a specific courses (“… but Hermite polynomials are in quantum mechanics, not mathematical physics … “ or similar plaintive cries that can be heard when discussing the “unfairness” of a question paper…).  And this is a concern that is shared by many. Graduate schools in the US often (if not uniformly) have comprehensives and cumulative examinations to address these types of issues, and quite successfully, but our teaching practices have not yet adopted such measures. Some years ago I offered the graduating batch of students in the MSc programme an optional examination after all the others were over. It happened to be a Friday, 13th May and hence the drama in the poster; the few students who took it were enthusiastic  (if somewhat bewildered) but the answer scripts were more revealing of the outcomes of two years of teaching and learning…

In the end, my point is a simple one.   Our pedagogy can benefit from the explicit recognition of the collective nature of teaching, where the efficacy of what one teacher imparts depends crucially on what is taught by another. In addition, the question of timing is also important. Especially for core courses, a greater level of coordination and planning is needed. This can greatly improve learning outcomes in all disciplines, but it also requires that we rethink our practices.

Let me close by quoting from a blogpost by Keeling and Hersh on higher education in the US that has considerable resonance with some of the points above.  “One challenge […] is that it requires faculty to come together – collectively – and agree on which outcomes, expectations, and standards they share and endorse, and then reinforce them throughout their various courses and programs. It demands a different institutional culture of learning […].” And, I might add,  teaching.

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.”

Today, D D Kosambi’s significance as a historian greatly overshadows his reputation and contributions in mathematics. Kosambi 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. In a fundamental way, Kosambi embodied the multidisciplinary approach, channelling diverse interests – indeed combining them – to create scholarship of high order.

The same article was published in a book, Unsettling the Past (Permanent Black, 2012)  that Meera Kosambi brought out later that year. Perhaps conscious of the limited time that was left to her, she  was keen to see the intellectual legacies of Kosambi Père et Fils firmly in place. (For whatever reason, and in my opinion unfairly, she constantly judged herself by their mythical standards and fell short. As indeed, who would not…) In the aftermath of the the D D Kosambi centenary (2007-8), Meeratai published at a furious pace- Dharmanand Kosambi’s writings in translation, his autobiography Nivedan, Gender, Culture, and Performance: Marathi Theatre and Cinema before Independence, Crossing Thresholds: Feminist Essays in Social History, Mahatma Gandhi and Prema Kantak: Exploring a Relationship, Exploring History, Feminist Vision or ‘Treason Against Men’?: Kashibai Kanitkar and the Engendering of Marathi Literature, Women Writing Gender: Marathi Fiction Before Independence, something like ten books in the five years that I knew her. She always had a book to write, and as those who knew her more closely told me, when the end was near, she was ready to go only after she finished the very last bits on her forthcoming Pandita Ramabai: Life and landmark writings. 

Meera Kosambi also gave the Nehru Memorial Museum and Library most of what she could of DDK’s papers and writings so that they could be properly archived. But fortunately, this was after she let me, and more importantly, Rajaram Nityananda, then Director of the National Centre for Radio Astrophysics in Pune look through them. In the process Rajaram had many of the papers digitised, and there was much in there that was (and still is) waiting to be discovered – unfinished manuscripts, unpublished notes, letters and so forth.

In particular, there are typescripts of many articles and notes. DDK’s notebooks from  Harvard, where he had taken lecture notes as an undergraduate are also among the papers. And not just mathematics notebooks, either- in short, material that is ripe for some amount of scholarly analysis. For instance, there is an article on C V Raman, (the somewhat more palatable) half of which was published in People’s War in 1945 by “Indian Scientist”. The latter half of the article seems to have been (correctly, in the opinion of several who have read it) deemed to be left better unsaid. There are also the edited typescripts of several essays.

And then there are letters, several of them. Like many in the pre-email era, DDK was a chatty correspondent. He made good and lasting friends as a teenager and young adult, those that stood by him pretty much through his life. Letters to the (R. J.) Conklins- a few of which have appeared in Unsettling give an unusual and somewhat American view of India in the 1930’s in the period when the freedom movement gained ground. Norbert Wiener remained a close friend, and professionally, he remained close to André Weil through his life, and for a brief while to Élie Cartan and Tulio Levi-Civita as well. Their correspondence, what remains of it, needs to be seen, as also his letters to Divyabhanusinh Chavda. The pictures that emerge of DDK, warts and all, need to be confronted. Of his personal angularities DDK was all too aware as he was of his many talents and his intellect. He was unapologetic, but also, occasionally, un-self-critical, refusing to see what he might have achieved through compromise and (academic) collaboration. As he says, when speaking of the Riemann debacle, at the end of  his autobiographical essay, Adventure into the Unknown, with ample self-justification and more than a hint of self pity,

“Let me admit at once that I made every sort of mistake in the first presentation. There is no excuse for this, though there were strong reasons: I had to fight for my results over three long years between waves of agony from chronic arthritis, against massive daily doses of aspirin, splitting headaches, fever, lack of assistance and steady disparagement. It was much more difficult to discover good mathematicians who were able to see the main point of the proof than it had been to make the original mathematical discovery. How much of this is due to my own disagreeable personality and what part to the spirit of a tight medieval guild that rules mathematical circles in certain countries with an `affluent society’ need not be considered here. There is a surely a great deal to be said for the notion that the success of science is fundamentally related to the particular form of society”.

Not being au fait with either the historian or the mathematician’s insider view of DDK, I had to discover many things afresh, such as his use of pseudonyms. This is one area where it is clear that some serious work is needed, given the range of his contributions; I have only poor hypotheses as to why he did this. In three articles, DDK signed himself as Ahriman, Vidyarthi, Indian Scientist, but after he invented the persona of S. Ducray  in the early 1960’s  he used it professionally in no less than four papers. Meanwhile, Kosambi as Kosambi was publishing Myth and Reality in 1963, and The Culture and Civilisation of Ancient India in Historical Outline in 1965, as well as the occasional mathematics paper sent off to select journals! In many letters  to friends, DDK signs off as Ducray; in one of the papers in the Journal of the University of Bombay, Ducray says “My debt to Prof. Kosambi is obvious”, while in another, the acknowledgment reads “This paper would not have been possible without the constant labour of Prof. D. D. Kosambi.”

Weird. DDK had, as has now been recounted several times, been responsible for the first mention of the name Bourbaki in the mathematics literature. Although the name Bourbaki is a collective pseudonym for a group of (largely French) mathematicians, there is a somewhat light-hearted frivolity to the naming part of the enterprise while the mathematics itself is of the highest order. Not so with the Ducray business, and it is difficult to see it through a hagiographer’s eye, as a piece of childish chicanery and no more.

Clearly there is much more work that needs to be done on Kosambi himself and this goes beyond putting together his articles and situating them. This blogpost started off as a way of my keeping track and not just a vehicle for self-promotion. Two of the discursive essays that were found among his papers have now been printed for the first time in the book Adventures into the Unknown that I have edited, and which is published by Three Essays Collective, Gurgaon. My “Collected Works” of DDK project is also under way, though it will probably take the better part of this year to complete, and will become a “Selected Works in Mathematics and Statistics” in the process.

Within a relatively short working life – he died at 59 – DDK made many original and fundamental contributions to many aspects of scholarly enquiry. The Promethean epithet applies very aptly to Kosambi, though- when the Gods give such gifts, and they do not give these to many, there is a price to be paid… A verse from Bhartṛhari that DDK surely knew (having three monumental contributions on the works of the 7th c. poet) says it all too well:

Nor do the gods appear in warrior’s armour clad
To strike them down with sword and spear
Those whom they would destroy
They first make mad.