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Essays, reviews, sketches, remembrances, counsel, and other writings.

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A Teacher’s Life – Four Stages in the Practice of Physics

J’aime mieux forger mon âme que la meubler.

Michel Eyquem de Montaigne.

 

A willingness to be puzzled is a valuable trait to cultivate, from childhood to advanced inquiry.

Noam Chomsky.

 

A Story

One of the joys of teaching at St Stephen’s College is that our old students return to us (winter is their chosen season), and when they come back they bring to us new ways of looking at the world, in particular the practice of physics. On one such visit, which I particularly remember, I asked the visiting old student, whom I knew to be a very thoughtful person, to say a few words to the students present, requesting him especially to reflect on what he had learnt at various stages of his career as a student. (He was then doing his PhD.)

He said, “At school I learnt how to solve a problem in one way; at St Stephen’s I learnt how to solve the same problem in many ways; at Cambridge I learnt how to work on my own; at Princeton I learnt how to discover a problem.” This has become one of my stories. I tell it – usually in the classroom but also in conversation with colleagues – to illustrate various aspects of the process of learning. I tell it, for example, to students looking satisfied at having solved a problem, to emphasize how important it is for them not to get stuck in a single approach. I tell it to let students know that the spoon-feeding to which they became accustomed at school, and of which they are not freed at College, could become a serious handicap later. But I tell it most often to drive home a point that is easy to miss in the classroom and the lab – that the knowledge of physics is not enough for the practice of physics.

 

The Practice of Physics

If I had to present my understanding, somewhat idealized, of the practice of physics, I would say that it has the following four stages: discovery, formulation, solution, and return. Stage 1, discovery, consists in sensing the contours of a scientific problem in a phenomenon. In stage 2, formulation, the scientific problem is removed from its physical moorings and placed within the fecund matrix of physics. Experiments are designed and performed, to isolate the parameters that describe the physical situation.  It is described by a mathematical model that fits within the framework of theoretical physics (sometimes, at the highest level of practice, the framework is modified or enlarged in this process). In stage 3, solution, a path is opened up from the general mathematical model to particular solutions that can describe physical situations in terms of measurable parameters. This stage often involves the use of a suite of very sophisticated mathematical and computational techniques, which can take several years to learn. In stage 4, physicists return from the abstractions of the mathematical model to the physical situation in which it originated, and at the same time a whole family of other possibilities, including perhaps some not dreamt of in their philosophy. Experiments are performed to corroborate and test what the model predicts, and to explore what it hints at. The four stages do not necessarily unfold in sequence: stages 1 and 2 do usually precede the others, but stages 3 and 4 usually happen together (and are separated here only for the purposes of this essay). Furthermore, the later stages feed back into the earlier ones, and the process keeps correcting itself. Theoreticians often start at the junction of stages 2 and 3, and, after going to the end of the process, try to imagine what the beginning might have looked like.

 

A Thought Experiment

A good example of the practice of physics can be seen in a pedagogic exercise that was carried out by a teacher in classroom full of 10-year-olds. (This is a true example that I heard about in a talk on Youtube.) She handed out a number of weights attached to strings and asked the kids to play with them. Then she asked them to try to figure out what the time period of oscillation depended on. There was wild debate in the class, with various factions taking different positions – the weight! the length of the string! the shape of the weight! the amplitude of oscillation! gravity! the wind! At this point the teacher calmed the children down and asked them to do a series of experiments to test, one at a time, the parameters that the time period really depended on. Very soon the kids arrived at the astonishing result that the period of the pendulum does not depend on the weight or the amplitude of the oscillation (two parameters that seem so obviously connected with the motion), that in fact it depends only on the length of the pendulum, with longer pendulums having longer periods. What we see is the unfolding of stage 1 and a little of stage 2 in the practice of physics. But the 10-year-olds, while being full of the qualities that make discovery possible – curiosity and enthusiasm and freedom from regimentation – lack the technical skills necessary to go further.

Let us imagine that one of the 10-year-olds, say Chirag, has a sister five years older (let’s call her Gargi). They talk about the pendulum experiment over dinner, and Gargi takes the idea back to school with her. (This happens long before the era of the Internet – so she doesn’t automatically google.) Their teacher suggests that they repeat the experiments done by the 10-year-olds. The students are a little reluctant to repeat what’s been done by their juniors, but the teacher points out that they can do a much better job of understanding the phenomenon, because they’re more mature and know certain techniques that the 10-year-olds don’t, in particular something they have just learnt – graphing. They can characterize the dependence of the time period of oscillation (T) on various parameters, such as the length of the pendulum (l). It’s clear that T increases with l, as discovered by the 10-year-olds, but what the 15-year-olds discover is that the T-l curve is not straight. What kind of curve is it? They argue over it for a while. Then Lilavati, who’s been quietly watching the others, especially the boys, shouting at each other, points out that the longest pendulum in their sample, which is about four times as long the shortest, has about double the time period; could it be, she asks, that T  depends on the square root of l? If so, shouldn’t they be plotting the square of T versus l if they want a straight line? There is some initial resistance. Why should we plot T2 when what we’re observing is T? many ask. But after a small intervention by the teacher, the kids decide to try out Lilavati’s suggestion – and, sure enough, T2-l is a straight line. So Lilavati was right – T depends on √l, not on l. They’re ready to move on.

Now they try to understand why T seems to be independent of weight. The teacher briefly explains that weight and mass aren’t the same thing – that weight is mass times the acceleration due to gravity (g). Could it be, she asks, that T depends on g even if it does not depend on mass? If so, what experiments could they perform? Was there any place in school where their weight might suddenly change? She asks them to imagine standing on weighing scales. Under what conditions would the reading change? What if the floor suddenly fell away under the scales? A lift! A lift! shouts a little boy that everyone calls Tiny. The class runs to the physics lab, borrows a weighing machine, and Tiny and a couple of others rush to the school lifts and keep punching all the buttons until one door finally opens. Of course they’ve forgotten to bring a pendulum, but Miss has followed them with a smile and pendulum. Tiny is given the honour of standing on the weighing scales while holding the pendulum. Gargi times the oscillations. They take the lift all the way to the fourth floor, then down to the ground floor, and back up to the second, where the entire class is waiting impatiently for them. What do the find? It turns out to be a little difficult to do the observations, since the reading on the scales changes only as the lift starts and stops, but still, it looks like T increases when Tiny’s weight decreases, i.e. when the lift starts downward, and decreases when his weight increases. By this time, the principal of the school has arrived to deal with the tamasha by the lift. He sends the students and their teacher back to the classroom. So they decide to test other things.

They still can’t quite believe that the time period could be independent of the amplitude; so they embark on a series of carefully-timed experiments in which the time period is measured for a range of amplitudes. They discover that there indeed is a very slight increase with amplitude; they need to repeat their experiments several times, and get better stop-watches to determine this increase, and in the end they’re convinced that it’s there. But, try as they might, they are unable to figure out how exactly T depends on amplitude; characterizing this dependence is not easy; even Lilavati is unable to suggest anything this time.

So they move on to the relation between T and m, the mass of the bob. It just doesn’t seem possible that T should be independent of mass. So they try a large range of weights, and they begin to see some strange and interesting results. So long as the weight is small in size, much smaller than the length of the string, its magnitude makes absolutely no difference to T, but if it happens to be large in size, then it does seem to make a difference. Once again, the dependence of T on the distribution of mass is highly non-obvious. How does one even talk about distribution of mass? – how is a ball different from a rod of the same mass? At this point, Meghnad, another quiet student, says – In that case, let’s just see what happens if we use a metre rod – at least it looks simple. How are you going to hang it from different points? – it doesn’t have holes. Hmm, maybe I can just hold it between my thumb and forefinger and let it oscillate. Of course there’ll be lots of friction, but it should still oscillate for a while. They do the experiment as Meghnad suggests. And this time they get a crazy, utterly incomprehensible result: T is the same at four points on this pendulum! They try a few other solid objects, and the results become even more complicated. They’re lost; they tell Meghnad that his crazy ideas have only confused them – that they’ll never listen to him again. We’re still at stage 2.

To move up to stage 3, let us enlarge the conversation at the dining table – let us add an 18-year-old cousin, say Tathagata, who recently joined St Stephen’s College to study physics. He and many of his classmates dream of becoming great physicists. They enjoy the idea of physics. They know about Newton’s laws of motion, about the inverse-square law of gravity, about differential equations, and a lot more. They bring together the law of gravity and the second law of motion. They see that the differential equation is non-linear and therefore outside their ken, but their teacher points out that for small angles sin ϑ = ϑ. On trying this out, they find that the resulting differential equation, which should hold for small-amplitude oscillations, is linear. They know how to solve this equation; they have learnt the technique in their mathematical-physics course. They see that it has sinusoidal solutions with angular frequency √(g/l): thus they show that the time period of a pendulum is 2π √(l/g). What they’d found about the dependence of T on l was right; and now they see how T would have been found to depend on g if they’d been able to do the necessary experiments. They extend their theory to include a damping term. It becomes possible to determine how the time period changes as result of damping. And they see that if damping is sufficiently large, there comes a critical point beyond which the motion of the system is not oscillatory any more – something they hadn’t even imagined. Later their teacher shows them how to handle the case of large-amplitude oscillations as a term-by-term series expansion beginning with 2π √(l/g). It turns out, furthermore, that there is a way to characterize the distribution of mass in pendulums that are not simple – thus the mystery of Meghnad’s pendulum, and of all pendulums whatever their shape or size, is also solved. And oscillating pendulums, they finally realize, are just examples of systems hovering around equilibrium, all of which behave in the same way. Then their teacher shows them how, when two or more pendulums are connected by springs, they can find hidden simple harmonic oscillators behind the now not-so-simple systems; and when he finishes dramatically by repeating the claim in some fancy book, that physics is the subset of all problems reducible to coupled oscillators, they feel very chuffed indeed.

With a single mathematical model of a simple phenomenon – drawing on a universal law of physics that describes all classical motion, another universal law of physics that describes gravity, and an adequate representation of damping – these 18-year-olds have been able build a model that produces, for the behaviour of any pendulum, indeed for any system slightly out of equilibrium, a coherent, convincing, comprehensive, and profound explanation, and one that is, in addition, not just beautiful (in an austere way, no doubt) but simple (if one looks it right), and that leaves them at the threshold of a universe of possibilities. They, and their collaborators – Lilavati, Tiny, Gargi, and Chirag, and all their classmates – have gone through the four stages in the practice of physics: they have returned where they came from, but they have risen above it.

At the end of their first year in college, some of the students – the cousin at the dinner table among them – go to various research institutes to do summer projects. One of them works at NCBS, the National Centre for Biological Sciences, with a physicist turned biologist. The first question she asks him is whether he understands the damped harmonic oscillator. It’s very important in biological systems, she says. Of course! he says, I know it well. Do you understand where all the terms in the equation come from? Yes, I’m pretty sure I do. What about the damping term? Yes, that comes from the velocity-dependent force. Tchch, I mean what’s the physical origin of the velocity-dependent force? Just saying it’s velocity-dependent is not saying much. Well, it’s because of the collision between the air molecules and the object, or because of viscosity if the ambient medium is a liquid. All right, we’ll examine that more closely later; but tell me this, young man: if the oscillator is damped, shouldn’t it come to a halt after enough time has passed? Yes, it must – it does. No, it doesn’t? No? – then what happens? It comes into thermal equilibrium with its surroundings. Thermal equilibrium? Yes, the time averages of its potential energy and its kinetic energy approach kBT/2 each. That’s not obvious in a large pendulum because the corresponding motion is too small to observe, but in a microscopic pendulum you would see it very clearly. Really? – Wow! So the equation we wrote was not right? Well, it was incomplete – you needed to add a noise term: it should be a stochastic differential equation. A stochastic differential equation – what’s that? Come, you’re going to find out, and then you’re going to help me solve this problem I’ve been struggling with….

 

Reality Check

Does it happen like this? No, of course not. Most 10-15-year-old kids in India aren’t lucky enough to have teachers who can guide them in a process of discovery of the kind described above. (The first story here is true, but it didn’t happen here.) Most dinner-table conversations don’t bring together the buoyant, unfettered enthusiasm of a 10-year-old, the somewhat cooler eagerness of a 15-year-old, and the technical skill of an undergraduate. Furthermore, the structure of the narrative above gives us the impression that the enthusiasm and eagerness of the discoverers in this story, the 10- and 15-year-olds, has somehow been communicated to the solvers, the undergraduates. In fact that does not happen most of the time. We see the possibility of that in the summer project experience recounted above, but the college classroom is quite different.

For the vast majority of those studying physics, the burthen of undergraduate education, and indeed also a significant part of graduate school, consists of two parts: first, an introduction to the great paradigms of physics – usually removed from the tortured history of their discovery, and instead recast for greater elegance and ease of comprehension –, and, second, training in the vast panoply of experimental and theoretical methods that stages 2 and 3 in the practice of physics require, again almost always through examples carefully refined to highlight method.

Why is this so? There are good reasons. The repertoire of the practising physicist is so enormous that it would be impossible to teach everything as if it were being discovered for the first time. When it comes to the great paradigms, the way they came into being for the first time is often so fraught with false starts, futile detours, dashed hopes, and sustained misunderstandings, that most students learning physics historically would be filled with despair, and perhaps never really understand the meaning and ramifications of the paradigms, which is what matters most when they are being used. When it comes to methods, removal from context happens almost naturally when one tries bring out the features and power of a method: since many of these methods are intricate and complex, and since a real situation may require the use of several such methods at the same time, it makes sense to remove them from reality and embed them instead in artificial situations that highlight one method at a time with the greatest clarity and force.

However, I don’t think that is the entire truth. It is also true that it is often more satisfying for teachers to teach, and for students to learn, ideas that have been cleaned up and pared down. Teaching and learning become easier in the short term, but the long-term consequences of an excessively hygienic exposure to physics may be quite different. It may produce masters of the entire spectrum of known techniques – who top every exam, are highly regarded by their peers and teachers, gain admission to the most prestigious universities –, masters who, faced with the unforeseen and unknown, lack the conviction and courage to find their way through it. Their beautifully trained systems – like immune systems allowed to grow without adequate exposure to germs – may fail in a crisis or, worse still, leave them ill-equipped to embrace confusion and error, the stuff of real discovery.

The question therefore arises: Should the undergraduate training of a physicist include more than just the imparting of ideas and methods? Is it necessary – would it be useful? – to include a psychological preparation for the uncertainty that appears to be a necessary part of the act of creation and thus a necessary prelude to discovery? I believe it is.

I am sure there are there are young men and women who are naturally endowed with the self-confidence needed to leap into the unknown. But I am equally certain that there are others who can be helped to develop that self-confidence. One tends to regard self-confidence as innate, one of those things – like “intelligence” – that cannot be improved. I myself have a strong tendency to think like that. Yet, twenty years of teaching convince me otherwise – that, given the right conditions many otherwise unwilling to risk the unknown will take the first steps into it, and, once they find their way through, break free. I know that this view is contrary to that of many scientists; having had to fight for their ideas, they believe that those who don’t have the guts don’t deserve to be helped. I too believe that – except that I believe that guts is not innate and immutable: it can be discovered and helped to grow.

I watch a programme on the Internet called Île en Île, in which Francophone writers from all over the world are asked about their writing lives. Most of them are men and women of colour, often belonging to former French colonies, and brought up on French classics written by dead White men. In an interview that I watched recently, the writer in question said, “I didn’t think it – writing, telling our stories – was possible for us: I grew up thinking this was something that they did, not us.” I am sure she speaks not just for coloured, Francophone writers but for the entire world of those dis-advantaged by pre-existing perceptions of their possibilities, which are shaped not just by their innate qualities but equally by an ecosystem of history, culture, and politics over which they have little control.

What this writer discovered in her world I will assume in mine – that there are students out there who, given the right conditions, can develop the courage to be creative.

 

Some Lessons and Suggestions

The search for clarity and order is certainly one of the driving forces of physics, but it is not easy for undergraduates, or even beginning graduate students, to appreciate that this order, if it is to be found by them, must lie on the other side of disorder. A story Steven Weinberg tells in his Four Golden Lessons for aspiring scientists illustrates this point well: “Another lesson to be learned … is that while you are swimming and not sinking you should aim for rough water. When I was teaching at the Massachusetts Institute of Technology in the late 1960s, a student told me that he wanted to go into general relativity rather than the area I was working on, elementary particle physics, because the principles of the former were well known, while the latter seemed like a mess to him. It struck me that he had just given a perfectly good reason for doing the opposite. Particle physics was an area where creative work could still be done. It really was a mess in the 1960s, but since that time the work of many theoretical and experimental physicists has been able to sort it out, and put everything (well, almost everything) together in a beautiful theory known as the standard model. My advice is to go for the messes – that’s where the action is.”

The question for those of us involved with physics education is – How do we inculcate in our students this readiness to dive into the mess? Would it help to present the paradigms of physics always in their own history, thus revealing the difficult paths that led to them? Probably not: it is important not just to know how something was discovered but also to re-imagine it. Furthermore, physics re-imagined is always clearer than physics discovered, and helps faster and clearer learning of the basics – which is important because of the sheer amount of material that an average physics students must learn to reach the stage of research. Nevertheless, a certain exposure to the history of physics – the difficulties, surprises, and serendipities encountered in making discoveries – is, I think, an essential part of physics education. Here is what Weinberg says on this, in the above-mentioned article: “Finally, learn something about the history of science, or at a minimum the history of your own branch of science.” He then adds: “The least important reason for this is that the history may actually be of some use to you in your own scientific work”. While a knowledge of history may not be directly useful, it does, I think, provide a psychological preparedness for the reality of research that is indeed useful.

Knowledge of history, however, is by itself far from enough. The fact that discovery involves falling into confusion before emerging from it needs to be communicated more effectively and through methods that allow students to experience some of that confusion themselves: the textbook, the classroom, the lab must all become arenas for discovery rather than just the learning of paradigms and methods.

The vast majority of problems in undergraduate physics textbooks, even very good ones, are toy problems, which may very effectively teach certain skills but do a poor job of communicating what real physics is like. I am not suggesting that they should all be replaced by discovery-oriented problems: the vast majority of the toy problems are indeed very useful, and necessary, but I think it might be useful for students, after they have done twenty such problems, to try one or two in which they need to find a problem in the fog of a physical situation, formulate it, solve it, and then return to reality. This can happen very naturally if opportunities for serious research by undergraduates exist in an institution. (We have been seeing this for example at St Stephen’s in the last couple of years.) Unfortunately, such opportunities are usually restricted to a few students. For the others, i.e. the majority, it may be useful to assign small projects that are not excessively well-defined and thus give them some taste of the various stages in the practice of physics, and that take their minds away from the regimentation of the classroom.

The centre of interaction between teacher and student, however, remains the classroom. This space is undergoing a revolution in some parts of the world: peer instruction, the inverted classroom, MOOCs, all of these are removing the teacher from his position of authority and making learning more self-driven and democratic. The inverted classroom in particular – where students watch their lectures at home, and in the classroom solve problems, work through difficulties, pose questions, etc with the help of one another and the teacher – promises to be a long-overdue transformation, one that will finally make students the prime movers in education. This will undoubtedly give them the confidence necessary for discovery. (It is not clear that the inverted classroom will come to India any time soon, except perhaps in institutions like the IITs.) None of these changes are by themselves necessarily discovery-oriented, but some of them, like peer instruction and the inverted classroom, are more likely to build the willingness to take risks that is necessary for discovery. Even the traditional classroom leaves plenty of space for independent thinking and exploration if the teacher refuses to play the role of an omniscient and unimpeachable guru who must answer every question posed and clear every doubt that arises.

In physics we have not just the classroom but also the laboratory. The lab can easily be turned into a space for discovery, so long as the only concern of teachers and students is not simply to get the experiments done. The lab offers – far more naturally than the classroom – opportunities for making mistakes and suffering the ensuing confusion, and thus for eventually finding one’s way out of it. Unless lab work is completely vitiated by “perfect” set-ups requiring no more of the students than the pushing of buttons, it automatically helps them develop the very valuable skill of trouble-shooting. The lab, if effectively used, can become a matrix not just for experimental skills but also, interestingly, for theoretical skills. In India, for example, physics labs generally don’t have enough equipment for all the students to do the same experiment at the same time. This makes it difficult to ensure that every student already has the theoretical background to do the experiment allotted to him or her. This is usually seen as a disadvantage, and indeed for most students it is; however, it offers an advantage which our best students have often profited from: being unfamiliar with the theoretical background to the experiment allotted to them, and finding the “theory” provided in most Indian manuals incoherent and woefully inadequate, they are forced to figure out a lot of things by themselves. Just recently, in fact, a very theoretically-oriented student told me how he and a few others had had to work out the theory behind a particular experiment almost entirely on their own, and how they’d learnt greatly from this process.

Another forum in which the idea of discovery can be very effectively brought out is something like a physics society, a student-driven body that organizes talks and encourages discussions. Former students doing research and current students who’ve done summer research projects can be encouraged to talk about their experiences at a forum like the Feynman Club of the Physics Society of St Stephen’s, bringing back into the predictable world of undergraduate physics a whiff of the spontaneity that is a part of all discovery.

Of course, for all these things to happen – for labs, classrooms, physics societies, textbooks and other spaces in a typical physics department to turn into zones of discovery – there has to an acceptance by those in authority – teachers, administrators, scientists guiding summer projects – of the idea that confusion is as important to pedagogy as clarity. It is not an easy to get this point across. Several years ago, I asked a number of students who had just finished their first year in college what their experience so far had been. One of them said something that I’ve never forgotten: “Sir, when I got into St Stephen’s I thought this place would be great: I thought the teachers would be wonderful – I thought that I would just sit back in class, and understand everything.” I was stupefied – not by his thinking of us teachers as less than wonderful, but by this excessively clear statement of what he thought learning meant. Unfortunately, his understanding of learning as a completely passive, utterly confusion-free process is very widespread, and not just among students. There is very little in the Indian schooling system that allows students to imagine that they can make discoveries, and that to do so they must be willing to enter into confusion before emerging from it. I am not sure how school teaching in India can be changed to inculcate the qualities that that discovery requires – curiosity and willingness to take risks, above all. But even college is not, I think, too late. It thus becomes incumbent upon those of us who teach in undergraduate colleges to allow this to become a possibility, instead of imagining that our task is restricted to imparting knowledge and clearing doubts. If students are to develop the independence, resolution, and patience to walk by themselves through darkness into light, teachers must develop the wisdom, maturity, and patience to allow the process to unfold – not always, certainly, but some of the time. And it is equally incumbent upon research scientists who take on undergraduate students for summer projects to allow them to struggle through some of the uncertainty that they know precedes discovery, instead of taking the easy way out and leading them through everything, or, worse still, just helping them learn another technique or method that they haven’t learnt in college.

 

The Heart of the Matter

All of us have a strong tendency to run away from confusion, to stay close to what is clear and certain. We associate confusion with error, and see error as evidence of foolishness, which we are taught from early childhood to avoid, especially in public. It is difficult for our brains to accept confusion and error as the necessary and inevitable foreshadowing of insight and creativity. As a result, we imagine that creative geniuses are creatures who never make mistakes. But that is very far from the truth. Error is not just inevitable, it is essential. One of the clearest statements on this that I have come across is the following passage from Récoltes et Sémailles, by the revolutionary mathematician Alexandre Grothendieck:

La découverte de l’erreur est un des moments cruciaux, un moment créateur entre tous, dans tout travail de découverte, …. Craindre l’erreur et craindre la vérité est une seule et même chose. Celui qui craint de se tromper est impuissant à découvrir. C’est quand nous craignons de nous tromper que l’erreur qui est en nous se fait immuable comme un roc. Car dans notre peur, nous nous accrochons à ce que nous avons décrété “vrai” un jour, ou à ce qui depuis toujours nous a été présenté comme tel. Quand nous sommes mûs, non par la peur de voir s’évanouir une illusoire sécurité, mais par une soif de connaître, alors l’erreur, comme la souffrance ou la tristesse, nous traverse sans se figer jamais, et la trace de son passage est une connaissance renouvelée.

 

(The discovery of error is one of the crucial moments, a moment uniquely creative, in every work of discovery, …. To fear error and to fear the truth are one and the same thing. He who is afraid of fooling himself is powerless to discover. It is when we fear fooling ourselves that the error that is in us becomes immovable as a rock. For, in our fear, we attach ourselves to that which we have one day decided is “true”, or that which was always presented to us as such. When we are moved not by the fear of seeing an illusory security collapse but by a thirst to know, then error, like suffering and sadness, goes through us without ever hardening, and the trace of its passage is a renewed knowing.”)

The most important word in that passage is fear. We fear making mistakes: we fear doing things that make us look unsure, confused, ridiculous, that make us feel as if the ground beneath our feet were crumbling. Do creative people, those like Grothendieck, not know – do they not feel – this fear? Here is what the great American artist Georgia O’Keefe had to say about this: “I have been absolutely terrified every moment of my life and I have never let it keep me from doing a single thing that I wanted to do.”

 

  • Bikram Phookun

Delhi

25th August, 2016.

Summer Project Dos and Don’ts

The first question about summer projects is of course “To do or not to do?” “To do, to do”, I have answered for most of my time as a teacher. I sometimes wonder whether the summer-project experience is as universally useful as I declare it to be – but this article is not about my doubts. It is a compilation of advice, drawing on my own experiences, both as student and as teacher, and those of my students – experiences had as well as experiences not-had.

Be open to possibilities

Most application forms ask you for your “field of interest” and even, sometimes, “sub-field of interest”. I think this is a deeply misguided question for research institutes to ask of undergraduate students, and I don’t understand why they keep doing it. Most of you don’t really know what you are interested in, and that is as it should be. At this time of your life, you should be open to possibilities. In earlier years bewildered applicants would often ask me how to answer this question, and I would usually advise them to (i) leave it blank, (ii) provide a very general answer like “experimental physics”, or (iii) do some research on the kinds of things that they could imagine themselves doing and were qualified to do, and try to find a possible (but not unique) answer.

In recent years fewer students have asked me this question. I now sometimes find first-year undergraduates declaring quite confidently what their area of interest is. Furthermore, many of them seem to know already what area they will eventually do their PhD in, and begin planning a succession of small research experiences that will accelerate them in that direction. I think this is a very foolish tendency, which limits the directions in which they can grow.

If you are passionate about something that is accessible at your level, e.g. astronomy or electronics, then there may be something to be said for following your passion through summer projects – but even then you’re probably better off allowing yourself to sample other possibilities as well.

If on the other hand you are taken with something that is at the moment way beyond your ken – e.g. string theory or black-hole physics –, then you will need a few more years of training before you can do anything significant in these areas. And, in the meantime, you must allow yourself to explore areas of physics that seem more mundane. Remember that one of the great things about physics is the deep connection that exists between different fields. Who would have imagined that theories of gravity would be used to solved problems in condensed-matter physics, or that string theory would have parallels in fluid dynamics, or that thermodynamics would appear in descriptions of gravity?

Conversely, don’t assume, on the basis of the insufficient evidence available, that you can’t ever become proficient at the physics now beyond reach. Give yourself time, try out things, don’t decide on your limitations before you’ve given your strengths a chance to grow.

Do something, don’t just read something

If your declared field of interest is high-energy physics or gravity, you may be offered the opportunity to spend all your time reading up on some beautiful and profound idea from a beautiful and profound book (written by the kind of scientist you would one day like to be). Resist such an “opportunity” – a temptation both to you and to your adviser – at all costs.

One major reason to do a summer project is to develop your independence – something at which classroom education, especially in India, fails you. Without independence you will not be able to do any kind of worthwhile creative work, no matter how well-trained you are. Here is an opportunity to find yourself totally at sea, to almost drown, and then somehow make it to the shore. Don’t lose this opportunity by playing it safe. You will be amazed at how difficult even a simple problem is when you have to recognize it, formulate it, and then find the means to solve it – and how rewarding.

Retracing a path taken by someone else can be useful, if the steps you have to take are long enough to stretch you to your limit, e.g. if you have to work through and make sense of a research paper. It can also be useful sometimes to learn a new technique that would not be taught in a standard undergraduate curriculum. But, by my reckoning, even such learning cannot compare with doing something independently. You are young, and there’s lots of time for you to increase your knowledge, but you’ll be surprised at how little time there is for you to develop your independence.

And don’t imagine that you are too young to make a significant contribution. Turing discovered the Turing Machine when he was an undergraduate, and Heisenberg was in his early twenties when he discovered Matrix Mechanics. So be of good courage. Who knows what you can discover?

Listen to your adviser with respect, but not with reverence

The working relationship that you should aim to develop with your adviser is somewhat different from the one you have with your teachers. If college teachers give you more space than your school teachers did, research advisers have to go one step further. When doing a research project, you should really look only for general guidelines, occasional nudges, and some insight. When you have doubts, struggle with them yourself first. How else will you develop your abilities as a scientific explorer?

When you find yourself bewildered – as we hope you will – your reaction may be either to run to your adviser constantly or to avoid contact with him altogether. Avoid both these extremes. Listen carefully to what your adviser has to say, but then struggle with the problem yourself; see if you can come up with some ideas of your own.You may say to yourself – what contribution can I possibly make? You may be awed by adviser’s knowledge, and dejected at your own. But your very ignorance may allow you to stumble on openings that a more knowledgeable person would not see.

Talk to people from other disciplines

One of the most creative spaces in modern history was Building 20 at MIT. It housed all kind of scientists, engineers, and social scientists, who constantly ran into each other in its long corridors. This happenstance interaction made for a kind of creative hubbub that has rarely been duplicated anywhere else.

So if you go to IISc or some such place for a summer project, and you meet an environmentalist or a chemist over coffee, do try to explain to them what you are doing, and learn in return what they are doing. Having to translate your technical work into everyday language may help you understand it better. Besides, even casual conversations may carry stray insights and ideas that can seed your mind.

But don’t be afraid to work on your own

Sometimes a first-year student will come up to me and ask: “Sir, you tell us to solve problems. But how do I solve problems? When I see one that I have never seen before, my mind goes blank.” That’s just the kind of feeling you’re likely to get when you work on a research problem – except that it will be deeper and last longer. Don’t be afraid of the darkness. When you are in that state, ideas will eventually start bubbling up in your mind. Some will seem completely crazy, some half-crazy, some obviously wrong, and you’ll find yourself puncturing them before they have the chance to form fully. But unless you let at least some of these ideas grow and lift off, you may not be able to find a way out of your impasse.

Borrow from others, by all means, but allow these borrowings to mature and grow in your own mind – or you’ll end up replicating the thinking patterns of the dominant mind in the conversation, which is not likely to be your own, unless you are very confident. Your mind must be open enough to allow ideas to enter from without, but closed enough to create a unique environment within. The crossing of boundaries – from contact into isolation, from authority into innocence, from madness into method, from one discipline into another – is the beginning of the creative process.

And your tenacity – your ability to hold and stay with a problem you’ve made your own – may be the best indicator of your future success as a scientist, far more so than being “smart.”

Be confident, but don’t be cocky

A belief in your ability to do things you haven’t done before can take you very far. But an “attitude” is not a very useful trait in someone training to be a scientist. Whom will you fool, anyway?

Be honest

Don’t pretend to know or understand things that you don’t. Don’t be reflexively defensive when your adviser criticizes your work – which doesn’t mean that you have to accept all his criticism. Don’t pretend to be working any harder than you are.

Work hard

There is no substitute for this.

Give a talk on the work you’ve done

One communcation skill worth learning is how to give a good talk. Most research institutes running summer programmes will give participants the opportunity to give talks on their summer projects. Take this seriously, and do a good job. Prepare, analyze, practise, and time your talk. (Try your own patience, not others’.)

Write a report

Another communcation skill worth learning is how to write a research paper. A project report is your first stab at this. Don’t think of it as just a requirement to be fulfilled. Ask your adviser for suggestions on how to write it. Discuss it with research students who’re stuggling to write papers. Read some of the advice given on the internet on how to write a research paper or report. Read a classic paper or two, e.g. some early paper by Einstein or Feynman, and see if you can communicate your ideas as effectively, as simply, and as powerfully as they do. After you have written your report, comb out all its logical tangles – edit it and rewrite it until it flows and glistens.

Don’t keep thinking about recommendations and testimonials

In the matter of giving letters of recommendation to summer-project students, I have known two extremes among advisers. One adviser I know lets his summer-project students know in advance that he will not entertain requests for recommendations from them. Another claims he prefers those who want such letters, because they’re the ones who will work. I suspect most advisers lie between these two extremes: they will be willing to write you a letter of recommendation if you work hard and well; but if they think your focus is the recommendation you intend to extract from them, they will not appreciate it.

As for an open testimonial from your adviser, if you get one, good, and if not, don’t worry about it. The experience is the main thing. (And, if you ask me for an important letter of recommendation, I will probably write to your adviser and find out what kind of work you did.)

Don’t take this opportunity for granted

The opportunity to do a summer project in a research institute, all expenses paid, is a privilege. Don’t imagine that you are entitled to such a privilege. It is true that research institutes must invest in potential future members of the scientific community if they’re to survive and grow – but in a country teeming with aspirants those members need not have included you. In this, as in almost everything else, you as a Stephanian have an enormous, possibly unfair, advantage over others. So thank your stars, if nothing else. And make good use of the privilege you’ve been given.

If you are offered more than one summer fellowship, try not to hold on to both until it’s too late for the offer you turn down to be made to anyone else.

Remember to thank and acknowledge your adviser

If your adviser is skilful, he may be able to help you to discover a problem, formulate it, solve it, talk about it, write it up, and perhaps even publish a paper on it, “without ever doing anything himself.” Don’t be fooled into thinking that you did all the work on your own. The force field created by an effective and generous adviser can help you to discover things, about yourself and world, that you could not have imagined. Think about this a little after the work is done, and express your gratitude adequately. If you want to publish the work that resulted from your collaboration, or present it at a conference, make sure you do so with the full knowledge and consent of your adviser.

Reflect on your experience

The assumption I appear to have made in this homily is that you have already decided to pursue a career in research – that you are only waiting to develop your strengths and discover where they intersect with your inclinations. But of course that is not true. Many of you, perhaps most, are not sure whether a career in research is what you want. Some of you are unsure whether you really like physics. Some of you like physics but are unsure whether you have the qualities required of a scientist. Some you have the qualities of a scientist but are unsure whether you are ready to spend ten years on a PhD and post-doctoral positions before finding a stable job. And almost none of you have a clear idea what the life of a scientist is like. Don’t worry – it’s natural to be only partially aware of one’s strengths, weaknesses, and inclinations, and of the kind of life one is prepared for. But there’s no harm in trying to get a preview of one kind of life. So think of a summer project as not just an opportunity to develop your scientific muscles, but also as a foray into a possible future for yourself. Observe your adviser and other scientists; observe PhD students and post-docs, who’re on their way to becoming full-fledged scientists. Ask questions. You may not be able to arrive at any unshakable answers, but that’s all right too – we all step into the future with only a faint idea of what it holds for us.

Good luck!

 

Physics at St Stephen’s in 2035 – a Vision

Later this year it will be twenty-five years since I graduated from college. If I had spent this quarter-century in the wilderness and then returned to St Stephen’s, I would certainly have noticed a few changes – more cars, more women, more Hindi –, but the ovewhelming perception would have been – nothing’s changed! Now let’s look twenty-five years into the future.

You are in your mid-forties, and have not revisited college since you left. In fact you’ve been in the wilderness yourself, cut-off from the world of education, working, let us say, with the tribal people in Bastar. But you’re back in Delhi after two decades, and yesterday when you called up one of your classmates she told you her daughter was hoping to get into St Stephen’s. You’re drawn back to the old days – you have to go back to College. You park your rented car in the underground parking lot where the shooting range used to be, and walk in through the Allnutt gate. Rohtas’s dhaba, you notice immediately, is gone, replaced by a large vending machine. But the cafe is still where it was. You step in; how clean it looks! And no, of course not, that can’t be Bhayyan over there. You decide not to stop for a snack; you never liked the mince cutlet anyway, and you’ve finally outgrown Maggi. Outside, eternal, is the tree and its ring of beautiful people, sitting delicately, legs crossed, like ballet dancers. On the Allnutt Court you see groups of students in the sun. The flowers look beautiful.You walk on, down the lane by Allnutt South. Is it still a girls’ block? you wonder, and then you notice both boys and girls going in and out freely.

You walk into the main cor. The ramp has been re-built, thank god. It has a guard-rail now, and a considerate slope. You look eagerly into Room C. Every student has a small computer, you notice. This doesn’t surprise you – you’re not that disconnected. You rush past rooms C, B, and A, turn quickly into the main foyer and look in through the open door of the hall. The seating is rather plush. You look up. Jesus said, I am the Light of the World … – it’s still there. You go back out into the main corridor, and have a look at Today’s Engagements: the Planning Forum invites you to meet Mr Rahul Gandhi, leader of the Opposition; the Choreo and Music Societies will hold a combined audition this afternoon for their annual musical (wow, you think, they’re really moved on since my time). You walk past the Principal’s office, and turn right. You look into the library. It’s now one vast computer centre. Looks like something out of Star Trek, you think.You can’t wait anymore: you must see what’s happened to the science block. You rush past Rud North, give the old chapel hardly a nod, and then, as you reach Mukh West you begin to run. You’re startled to see a dome-like fibre-glass roof floating over the basketball court, but you don’t stop.

Oh my god – a second storey! And the corridors have been glassed in. You turn left, push open the door, and enter the cool, air-conditioned space outside the NPLT. You look in. The classroom has no ramp anymore, and that huge desk in front is gone. So is Dr Popli’s picture. There are about twenty students sitting in a ring. They’re looking intently into wafer-thin screens. Ah, you think, the teacher must have given them some exercise to do.

You look in through the open door, at the nearest computer screen. Electricty & Magnetism, it says at the top of the page. Old Maxwell’s Equations, you think. Hmm, let’s see: div E equals rho by epsilon naught, …. There’s a video running on the screen. A white man draws a horizontal line, quickly inscribes some positive signs under it, and a vertical arrow – a vector, an electric field –, and then draws a flat little box – a Gaussian pillbox! Yes, of course: the change in epsilon times the the normal component of the electric field is equal to the surface density of free charges. You smile foolishly, aglow with pleasure. You start looking around. Then you notice something funny: there are different videos on different computers. But as you watch you notice that they all seem to be saying the same thing: they’re all about boundary conditions, only some explain them with pictures, some with equations, some talk a lot, some just write enigmatically. No one looks bored. Wow, what a great idea, you think; I could never understand Dr Phookun’s long-winded explanations.

A bell rings. The students take off their headphones. They talk to each other for a while, and then most of them type furiously into what looks like a chat-box. You peer in. I don’t get it, the student is writing, how come only the free charge density appears on the RHS? Ah, questions, they’re posing questions! And you see the answers popping up in the lower panel. Now you can’t stop yourself; you walk into the classroom, and look into a couple of the screens in front of you. At one of them a sardarji is sitting quietly. Then, slowly, with one finger, he types out something. After a pause, you see this message on the lower panel: The data bank bank does not have any answer matching your question, but it will be passed on to Grandmaster Walter Lewin Jr at MIT. If he approves of your question, you will get your answer tommorrow, plus ten bonus points on your Universal Academic Rating.

The class ends. Some of the kids smile at you, and you begin talking to them. You think, let’s see, who would still be around? Dr Garg, Dr Cherian, Dr Phookun, Dr Sanjay, Dr Sangeeta – they must all be gone by now. Maybe Dr Gupta is still here; he looked so young then. You ask after your old teachers. The kids look puzzled. Then someone tells you – but there are no teachers here! No teachers? You mean this isn’t a college any more? Of course it is – it’s a self-learning college. You mean teachers don’t exist any more. They do, but only in, like, really remote areas, and special colleges for the handicapped. And of course there are the Grandmasters at MIT and Caltech; they’re the ones who make our lessons, and our answer banks. But who looks after the college? Well, there’s an administration, but we help too. But why come to college at all then? The discussions really help. And often old students and visitors come and tell us about what’s happening out there. Then there’s ECA. Besides, if we didn’t come to college, how would we meet?

You stagger out.

March 2010.

Isaac Newton (1642/43 – 1727)

Outside the British Library in London there is a bronze statue of Isaac Newton, by Sir Eduardo Paulozzi, based on an engraving by the English mystical poet and artist William Blake. It shows Newton naked and muscular, sitting on a block and bent double, eyes focussed on a divider with which he marks out something – presumably the order of the universe, in a poet’s understanding of the scientist’s vocation. Blake lived in the century after Newton, at the same time as Pierre-Simon de Laplace, the French mathematician whose great work on celestial mechanics started where Newton’s Principia ended and demonstrated, among other things, the long-term stability of the solar system. On being presented this work Napoleon is supposed to have asked Laplace what role God had to play in his scheme of things, to which Laplace replied that God was a hypothesis of which he had no need, since it explained everything but predicted nothing! The supposed ability of the Newtonian mechanics to predict the course of the world was a shock to the religion-infused Middle Ages in Europe, where God’s supervisory role in everyday life was taken for granted. It is the disdain of a mystic for the man ultimately responsible for this worldview that Blake’s engraving of Newton is thought to represent. I like to think, however, that what stirred Blake was an inkling that behind the cold front of Isaac Newton, this god of Godless science, there was a heart that sought out ultimate order with just as great a passion as his own.

Isaac Newton was born on Christmas Day 1642 (which, when the Gregorian calendar was adopted by England in 1752, became 4th January 1643), two months after his father died, to Hannah Ayscough of Woolsthorpe, Lincolnshire, England. When he was two, his mother remarried and left him to the care of her parents. Isaac was unhappy at this abandonment. He was briefly re-united with his mother in 1653, when her second husband died. Shortly thereafter he was sent to nearby Grantham to attend its Free Grammar School. His mother withdrew him from school when reports came in that he was “idle” and “inattentive”, and sent him to manage the considerable properties left to her by Isaac’s father (a well-to-do but illiterate man of farmer stock), a task at which he proved inept. He was sent back to the school about 1659, at the instance of his uncle William Ayscough and the headmaster of the school, and there prepared for admission to university.

In 1661 he went up to Trinity College, Cambridge, to study the Law. He was enrolled as a sizar, i.e. a student who earns an allowance as a servant or helper to another, in this case Humphrey Babington, a distant relative who did not make much of an imposition on Isaac. At any rate, he had time enough to read extensively – not just the canonical Aristotle, but also Descartes, Hobbes, Boyle, Galileo, Kepler, et al. He also learned mathematics seriously – from Euclid and Descartes, and from Wallis, whose work on algebra inspired Isaac to try proving theorems his own way (“Thus Wallis doth it, but it may be done thus…”). By 1665, when Isaac graduated from Cambridge, his mind was ripe for the two-year hiatus from university life that followed when Cambridge had to be closed for the duration of the Great Plague of London (which spread to Cambridge). Isaac returned to Lincolnshire, and in the two following years cultivated in solitude the many ideas that Cambridge had sown in his fertile mind. From this emerged the great work for which he later became famous – on calculus, gravity and mechanics, and optics.

Newton returned to Trinity in 1667, where he was elected Fellow in 1668, giving him the freedom to continue his researches. Though Newton was reluctant to publish, his extraordinary discoveries in mathematics and natural philosophy soon became known to the scientific community at Cambridge and beyond, especially as a result of the interest and encouragement of Isaac Barrow, the first Lucasian Professor of Mathematics, and of Barrow’s correspondence on Newton’s work with the mathematician John Collins, who forwarded it to the President of the Royal Society.

In 1669 Barrow resigned from the Lucasian professorship, recommending Newton as successor. In 1670 Newton delivered his first lectures – on optics – as Lucasian Professor. The work that Newton did on optics at this time – both experimental and theoretical – led to the publication in 1704 of his famous work Opticks (the thirty-year delay in publication being partly the result of Newton’s feuding with Robert Hooke, after whose death the book was published). Newton continued his work on mechanics and astronomy, corresponding with well-known astronomer Edmond Halley, who had been able to show that Kepler’s third law implied an inverse-square law of attraction. Halley discovered that Newton’s researches went much deeper, and strongly encouraged him to publish his work. This led in 1687 to the publication of Philosophiæ Naturalis Principia Mathematica, the preface to the first edition of which begins with the words “Since the ancients …made great account of the science of mechanics in the investigation of natural things; and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phænomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy.” In this – the cultivation of mathematics so far as it regards philosophy – Newton succeeded so completely that his methods became the archetype of the physicist’s practice. The watertight logic of mathematically reasoned natural philosophy also had the effect of making God an unnecessary hypothesis.

Yet Newton was a deeply religious man. In the Middle Ages in England it was taken for granted that everyone would be affiliated to the Church. But Newton’s interest in religion went far beyond affiliation to a church. In the General Scholium, an essay that was included in the 3rd edition of the Principia, he wrote “This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being” and then went on, at some length, to deduce the qualities of an omnipresent God. He wrote extensively on religion, engaging himself not just in interpretations of Christian concepts like the Trinity but also in practices that today we would associate with cranks and charlatans, e.g. numerology based on numbers appearing in the Bible.

In fact there was a strain in Newton that was deeply sympathetic to the supernatural. In 1936 a trunk filled with Newton’s notebooks was auctioned by Sotheby’s and acquired by the economist John Maynard Keynes. These notebooks had been examined by the Royal Society after Newton’s death, and deemed unfit for printing. Keynes was fascinated to discover that they contained, in great detail, accounts of Newton’s secret researches on alchemy, carried on during the very years when he was becoming famous as a mathematical philosopher. Keynes was invited by the Royal Society to speak at Newton’s tercentenary celebrations (which took place in 1946); though he died before the event, the text of his proposed lecture, entitled Newton, the Man, survives and contains the famous lines: “Newton was not the first of the age of reason. He was the last of the magicians …”

The received image of Newton the Sage of Reason was perfected after he left Cambridge in 1696 to become Warden of the Royal Mint. As Master of the Mint from 1699 onward he was a powerful, active, and very successful civil servant. For twenty-four years he was the also reigning monarch of the Royal Society (to which he had been elected Fellow in 1672). He was the first scientist to be knighted, in 1705. In his home in London – of which his brilliant and charming niece Catherine Barton was hostess (Newton never married) – he received the greatest intellectuals of Europe. Jonathan Swift wrote about Catherine. Alexander Pope wrote Newton’s epitaph when he died – “Nature and Nature’s laws lay hid by night: / God said Let Newton be! and all was light.” Newton was buried in Westminster Abbey. The pall bearers to this farmer’s son were three earls, two dukes, and the Lord Chancellor; they were followed by the Fellows of the Royal Society. The rather elaborate monument at his grave bears the inscription “Hic depositum est, quod mortale fuit Isaaci Newtoni.” The immortal remains of Isaac Newton cannot be so neatly contained. Pope’s anodyne epitaph speaks only of Newton the master of self-consistent natural philosophy. Yet this reasonable Newton was only a part of – and perhaps a result of – a complex and contradictory creature whose measure we cannot easily take. Perhaps a more appropriate epitaph to Newton would be the defiant words of the mystical American poet Walt Whitman in Song of Myself “Do I contradict myself? Very well, then, I contradict myself. I am large, I contain multitudes.”

The Fear, and the Use, of Mathematics and Physics

The two areas of human enquiry that inspire the greatest terror in the hearts of students are undoubtedly Mathematics and Physics. You may find history, or chemistry, or economics difficult, but your reaction to these subjects, and most others, is almost certainly not fear. On the other hand, when you encounter an equation, your first reaction is to escape to more amiable company. If you compare subjects to people, you will realize that your reaction to maths or physics is very similar to you reaction to a stern, quiet person who is famous for his wisdom but who makes you very uncomfortable indeed. When he speaks you listen dutifully, because you’ve been told his words contain a lot of meaning, but you understand almost nothing, and you end up feeling foolish and exposed; and what is worse, this person does not need to shout to make you feel this way – he just has to look at you. When you see an equation or a mathematical expression you react in the same way. Let us try to understand what mathematics is and why it is so difficult.

Pure mathematics is a kind of language: its symbols carry meaning and these symbols can be combined into expressions in well-defined ways to carry more complex meaning. The rules for constructing expressions in mathematics are very precise and well-defined. As a result it is possible in mathematics to start from simple ideas and rapidly build up mathematical structures that are unbelievably complex and far-reaching. Even to a professional mathematician, the heights that can be reached by this method are astonishing; in fact a true mathematician never loses his joyful amazement at the power and reach of mathematics.

In applied mathematics, of which theoretical physics is the most outstanding example, the meanings carried by the mathematical structures are closely reflected in the physical world. In a typical piece of reasoning in physics we start from a universal law, e.g. Newton’s second law, which relates the acceleration of an object to its physical interactions with other objects. When we throw a ball, its acceleration is due to its gravitational interaction with the Earth. We express this in mathematical form, and, once we do that, we have all the power and reach of pure mathematics at our disposal – i.e. we can now use the rules of mathematics to travel far away from the apparently simple physical facts expressed by our original equation. But because the physical situation has been captured in some fundamental sense by our original equation, the results we arrive at using mathematics remain representations of actual physical situations – including ones that we may never have guessed. For example, in the case of the ball thrown up, our mathematics tells us that if the velocity of the ball is more than 11.2 km/sec, it will escape the gravitational pull of the earth. So, starting from a mathematical representation of an apparently simple situation – a ball thrown upward – we arrive at the conclusion that rockets and interplanetary travel are possible!

This description of pure and applied mathematics also reveals the various reasons why they are reputed to be – and indeed are – so difficult. First, if you see just the final result of a long process of mathematical reasoning, it is virtually impossible to see how it could have been arrived at from its simple origins. Second, if you decide to go through the process of reasoning yourself, you have to make sure that you walk only along the paths allowed by the rules of mathematics – else you fall! When a mathematical structure is first being built there are often big gaps in it, and seasoned mathematicians must therefore become adept at leaping across these gaps, rather like a monkey jumping from tree to tree. In pure mathematics, as the structure is completed, mathematicians try to ensure that every gap is closed – the tree-tops are linked by a network of sky-bridges – and that it becomes possible for anyone with basic skills and courage to walk from one point to another. In physical mathematics, on the other hand, it is common practice to leave many of the gaps unclosed – we are expected to use our “physical intuition” to leap across the gaps. A mathematician is more sensitive to the beauty and intricacy of the structures that they have built, many of which seem to float in the air. A physicist is more interested in how he can get from one point in the forest to another, and to do that he moves sometimes along the ground – this is called reasoning physically – and sometimes high above the ground – this is called mathematical physics.

From my description above you can deduce that someone wishing to be a pure mathematician must possess two distinct abilities: first, he must be able to guess what kinds of mathematical structures can be built – is this conjecture provable? – and are worth building – is it mathematically important? –; and second, he must have the ability to build them, i.e. he must learn the necessary techniques. To be able to walk along mathematical structures that others have built is no mean ability, and certainly a very useful one, but a creator of mathematics must do much more. On the other hand a physicist must be able to walk boldy along the ground level of physical reality, climb up the elevated network called mathematical physics and walk along it, and always keep in mind the connection between the two.

There are of course many areas outside physics in which mathematics is indispensable. All branches of engineering use a combination of physical and mathematical reasoning, much as physicists do. Theoretical computer science is pure mathematics. Even in some areas of science that have traditionally been practised without mathematics, it is beginning to make inroads. For example, biology, once the most non-mathematical science, has recently begun to discover that mathematical reasoning can go a long way in explaining its phenomena. As a result, physicists are moving into biology in droves, leading to the a new area called biophysics. In chemistry too sophisticated mathematical models are used to determine the details of the chemical bond. Economics and its offshoots, e.g. finance, have of late become increasingly mathematical. The stock market and its various goods – options, derivatives, etc – are priced according to mathematical models. The increasing complexity of these models means that many people with PhDs in mathematics and physics and computer science have begun working in finance, especially in the US.

You may ask what role mathematics and physics have to play in the ordinary lives that most of us lead. Is it necessary for an IAS officer or journalist or a doctor to be familiar with the methods of mathematics and physics? One sometimes reads articles condemning the common man for being mathematically and scientifically illiterate in a world where our technology depends so much on advanced mathematics. I feel, however, that this line of argument misses the point in one way. When calculators didn’t exist, most of us knew our multiplication tables. Now that calculators are available to everyone, do we insist that everyone still remember their multiplication tables? Of course not. Music compression using mp3 depends on a beautiful and powerful mathematical idea called the Fourier transform. Does that mean that all of us must learn about Fourier transforms? Of course not.The experts must of course know their mathematics, or our computers and mobile phones will not work, but it is not just futile but pointless to insist that everyone should be familiar with the mathematics that underlies computers and mobile phones and other conveniences. Of course, everyone is welcome to learn about such mathematics, and feel enriched and inspired by it, but it is unlikely to help administer districts better or write more vivid reports or heal patients more effectively.

Does that mean I think mathematics and physics can be dispensed with altogether by the ordinary person? Not at all. If we believe that an educated person should be familiar, at some level, with the world in which he lives, then perhaps we should encourage him to develop an appreciation of some ideas in mathematics and physics – ideas that are reasonably simple yet convey the spirit and the methods of these disciplines. For example, the idea of the proof, so central to mathematics, is conveyed by fairly simple theorems in geometry, of the kind that used to be taught in all schools when I was a child. The subtlety of the concept of number can be appreciated by trying to undertand why mathematicians are so taken up with prime numbers. The idea of symmetry, which is fairly intuitive and physical, leads to the beautiful and abstract mathematics of group theory. Similarly, in physics the average person can try to learn from popular sources some of the central ideas of physics. A ball follows a path, but an electron does not – this is the strange world of quantum mechanics. One can never catch up with light – this is electromagnetic theory, optics, and relativity –, and so on. You can go beyond the basics too, and discover from popular sources what the major areas of research in mathematics and physics are. A large number of books explain these ideas for the non-scientific reader, and of course the internet is an inexhaustible resource for someone who wishes to learn.

That brings us to my final point. Suppose that you do not wish to spend your time appreciating mathematics and physics. What you want to know is whether they can help you in your day-to-day life. If I were asked to choose one area of mathematics – beyond addition and multiplication – that could help virtually everyone, it would be statistics. And one idea from physics that every minister and IAS officer and journalist should learn is how to make an intelligent estimate.

The basic ideas of statistics appear in virtually all areas of life, though we are often unaware of them. Consider for example the result of a blood test. You find that your cholesterol is 190. Unhappy that it is close to the danger mark, you get it tested again. This time you find that it is 180, and you feel reassured. Being an “optimist” you prefer to believe the second result! But suppose instead that you tried to understand this disparity in test results; then you would have to learn about the basic ideas of a statistical distribution and its mean and standard deviation. You would realize that if you got your blood tested a very large number of times, the results would yield a distribution, which would resemble the famous bell curve. Its mean would tend towards your true cholesterol level. More importantly – and this point is usually missed, even by doctors – the width of the distribution, or its standard deviation, would tell you how far you can trust the number produced by any one test, and help you to distinguish a significant difference between two test results from a insignificant difference. Another important statisical idea is that of a correlation: if you have data on two variables in a populations – e.g. height and weight –, how do you test whether they are related? Obviously, taller people are, on the whole, heavier than shorter people (in spite of the existence of short fat people and tall thin people). If you make a plot of height versus weight of a population, this correlation between height and weight will show itself as a band. If you understand this method, and it refinements, you can use correlations between variables not so obviously connected, to understand how much one factor influences another. And by looking at the way correlations change with location, and with time, you can draw important conclusions on how to invest resourses. For example, if you plot an indicator of health, say longevity, versus income, you are likely to find that in a sample of poor countries longevity increases with wealth, whereas in a sample of rich countries, longevity is not as strongly correlated with wealth. Whether you are talking about medical tests, the census, average rainfall, change in prices, variation in nutrition levels – the ideas of statistics will help you to draw useful conclusions from the information you have.

The idea from physics that I choose as the most useful for the common man is intelligent estimation. Before embarking on a calculation or an experiment that may consume a lot of time and resources, it is often important to have some idea where one is headed. For this, physicists must learn to do what they call “order-of-magnitude” calculations. The great physicist Enrico Fermi expressed this in his own way: he said that every physicist should know how to estimate the number of barbers in the city of Chicago! Let us try to estimate the number of barbers in the city of Delhi. – The average man has one haircut a month, i.e. one-thirtieth of a haircut per day. The male population of Delhi is between 60 lakhs and 90 lakhs, which means that there are 2 to 3 lakh haircuts per day. The average barber probably cuts 15 to 20 heads of hair in a day. … Put these together and you will find that the city of Delhi has between 10,000 and 20,000 barbers. To determine exactly how many barbers there are you need a survey, but notice how easily you can deduce that the number can’t be 1000 (or Delhi-ites would have to grow their hair like sadhus) or 100,000 (or the barbers would be mostly unemployed). A problem that when first posed sounds impossible to answer becomes simple when broken appropriately into parts. The same method can be used to estimate the power consumption or the petrol consumption in a city, the appropriate fare for a half-hour autorickshaw ride, the amount of grain needed to feed a population, the number of doctors needed in a district – the questions are endless. Normally one looks to experts for answers to these questions – and indeed experts are needed for precise answers –, but if you learn the physicist’s science of intelligent estimation, you can check whether the authorities know what they are talking about, and you don’t always have to wait for their answers.

A Letter of Recommendation

In the fourteen years since I joined the Physics Department of ________ I have written several hundred letters of recommendation – to Indian and foreign research institutions and to foundations providing scholarships. The most substantial ones are those written in support of applications to graduate programmes. I would like to share in this note the process of writing one.

What an institution seeks from you, in addition to your marks etc, is a statement of purpose, which is your perception, given your past and present, of the direction your life could take. What they seek from me, the recommender, is the perception of an expert who, knowing you and others like you, can discover your potential and place you in the spectrum of possibilities. There are a hundred questions they might have about you. Some are posed explicitly, but there are many others to which I imagine they might want answers. I try to answer at least a few of them.

Are you eager, attentive, interested? – even when the teacher isn’t observing you? Can you see past the exams? Do you always put in your best, or only when your enthusiasm is aroused? Can you sustain your enthusiasm? Will it survive the rigours of undependent physics? Are you a classroom star or a proto-scientist? Are you capable, resourceful, cooperative, flexible? – independent, open-minded, alert, observant, curious, playful? Do you find possibilities in confusion, or are you afraid of it? Do you face difficulty with courage and humour? Do you have an unusual take on things? – make interesting observations, ask probing questions? Do you follow up on the questions you pose? Do you have the confidence and ability to try out things on your own? (“Thus Wallis doth it, but it may be done thus…”, said Newton, as an undergraduate at Trinity.) Do you stay with a problem until it opens itself to you? Do you enjoy solving problems at all? – even when the answers aren’t given? Do you try to make sense of what you did and what you got? Can you catch a hint and take it further? Can you make connections between different areas of physics? – between physics and the world around you? Can you estimate the number of barbers in the city of Delhi? Do you know why you’re doing the experiment you’re doing, or do you just want to finish the damned thing? Do you ever make discoveries when doing experiments or is your work always perfect? Do you like trouble-shooting, or do you run for help when the BG shows no deflection? Are you interested in instruments and electronics? Do you like taking things apart and putting them together again? How about ideas – do you like taking them apart? Are you comfortable with abstraction? Do you think mathematically or physically? Can you describe a physical situation mathematically and then find physics in the mathematics? Are you quick, or slow but interesting? Do you find different solutions to the same problem and the same solution to different problems? Do you know the thousand names of the simple harmonic oscillator? What if it isn’t simple? Can you solve it on the computer? Do you know enough physics and mathematics for the programme you’re applying to? Are you versatile, drawn in different directions by your many talents? What turns you on? What do you do best? Do your inclinations meet your abilities? Do you manage your time well? Are you ambitious? Do you have any idea what it means to do research? Have you done any interesting projects in college? – summer projects elsewhere? Do you work well with others? – and alone? Do I have any feedback, from you and from your guides? Have you given talks on them? Did your talks seem well-prepared, clear, insightful? Do you like discussing physics? Would you make a good teacher? Are you helpful? – humble enough to admit your mistakes? – honest, intellectually and otherwise? Do you add to the classroom? Anything else? – Do you have any special qualities? Are you disconcertingly direct? Are you unusually logical? Do you always spot my mistakes? What about your mistakes – are they enlightening? Do you jump to the right conclusions?

You may wonder how on earth I can answer all these questions about you (though at other times you’re furious with me for not acknowledging your subtlest abilities and fathoming your deepest desires). For most of you, I would be able to answer no more than two or three of these questions. For some rare students I might be able answer more than two-thirds of them. With even a few significant answers I may be able to make a recommendation that counts. Where do I find my answers? In tests: If your written answers are uncharacteristically lucid, get straight to the point, show a different way of thinking, explain what you’re doing instead of brandishing formulas; if you always attempt the challenging questions and not just the standard ones; if I say to myself, “She can read my mind!” or “Wow, I never thought of that!” – I will remember. In the classroom: If you are always attentive, participate in discussions, make unusual observations, are willing to take risks, ask thoughtful or startling or deep questions, make a presentation that takes my breath away – I will remember. In the lab: If I ask you about your potential divider, or your error calculations, or whether you know your result’s in the right ball park, and I see you thinking; if your circuits are beautiful; if you get caught up in trying to figure something out; if I see you trouble-shooting or handling equipment joyfully; if you’re unusually regular in submitting your file – I will remember. In the Physics Society: If you show initiative, get others excited, take the lead, keep your commitments, organize something well, give a good talk – I will remember. Elsewhere: If we meet and you tell me that you’re really enjoying the maths, or that electronics is, really, fun, or that you’re writing this intricate programme; if I see you following a trail like a bloodhound; if your summer project made a big impact on you; if another teacher says something that makes me change my mind about you; if I am taken aback by your integrity; if I see you helping your classmates – I will remember. If your summer-project adviser says to me, “______, that guy was amazing: I just outlined what needed to be done, and he did the whole thing and came back. He was better than my PhD students” – I will remember. Hell, if I see you take one step off the straight and narrow path, one chance in the wilderness, I will remember. But if you always hold back, lest you make a fool of yourself and earn my scorn, I won’t know very much about you. And if you approach me for a letter of recommendation, I may not have the heart to refuse, because I don’t want you to feel rejected, but what I write will not win anyone over.

One question you probably ask yourself is – “Does Dr _________ like me?” My affection for you, real or imagined, is not as important as you think. It is true that on very rare occasions I have been swayed into writing a stronger recommendation than, in retrospect, I think the student deserved, but this happens very rarely indeed – and to the extent that it happens my recommendation is compromised. It may be difficult for you to accept this – but my recommendation is credible to the extent that it is not swayed by the ease with which you and I converse. Nor must it be swayed easily by your self-perception. What is asked of me is a kind of doctor’s view of a patient he knows well. I must know what you think and say of yourself, which is why I listen to you and ask you for your statement of purpose, but I must be able to see you not just as the unduplicable person you know yourself to be but also as one of many who have gone through this process – this suffering! – before and will go through it again in future. So it’s more important that I know you well than that I like you well. That said, I must add that I do not appreciate requests for recommendation from students who have made it amply clear that my teaching – in its broadest sense – has meant little or nothing to them, or from former students who, having snapped all links, reappear only to seek a recommendation, and then plunge back into the darkness from which they emerged. I usually agree even in these cases, and write fairly and perhaps knowledgeably, but I do so with a feeling of being used.

I like to write a letter of recommendation with goodwill and enthusiasm, at least the former. I was on one occasion asked by a student whether I would write him a good recommendation, and if not would I please tell him in advance. I explained that a recommender was not bound to write a good recommendation, only an honest one. However, as a personal policy – probably the policy of many other referees as well –, if I feel I cannot write a recommendation that is on the whole positive, I try to let the student know in advance. This I do especially if the student happens to be someone who, on the basis of his academic standing or for other reasons, would automatically expect a strong recommendation. Most of my letters, however, will, explicitly or through silence, convey not just your strengths but also your weaknesses. Here is how I organize a typical letter of recommendation.

In the first paragraph I establish my credentials, by specifying how I know you – what classes and labs I have taught you, whether I have interviewed you, talked to you outside the classroom, talked to others about you, observed you giving talks, etc. Then I write about your strengths as an undergraduate student. If I am writing to an institution that is unfamiliar with St Stephen’s College, I describe what makes a physics degree from Delhi University, and especially from St Stephen’s, stronger than degrees from other places; I also inform them about the weaknesses of our programme – lack of problem solving and of computational physics. If you have done anything to overcome the weaknesses of the curriculum, e.g. if you are an expert at programming or love problem solving, I mention that. If you have gone on to another institution after St Stephen’s, say IIT or Cambridge, I add a line or two about the ways in which your education there is likely to have complemented what you learnt at College. Then I go on to those of your characteristics that are less student-like and more scientist-like, e.g. the way you do projects or other independent work. I give them my perception of your maturity, self-confidence, ability to do a sustained piece of work, ability to work alone or in a group, and so on. If you are applying to a course outside physics, and I know something of what it requires, I say a word about how your training in physics may lead to it. If there are lacunae in your training, e.g. if you know little programming and are not used to problem solving, I may be quite explicit about them. If you’re easily distracted, a last-minute crammer, I may mention that, but I will normally put it in the the context of a system that encourages such qualities, if I think that you’re capable of focussed hard work in the right circumstances. If I don’t see in you certain other traits of character and mind that graduate study requires, I usually convey that through silence or by pointing to the possibility of growth. However, there are certain character traits that I may try to forewarn them about; e.g. if you are someone who needs to be handled delicately, I may let them know. If you are someone who I think is very capable but who has so far been resolutely unadventurous, I may state that as well. On the whole, though, I am far more gentle about your weaknesses than you might imagine. I try to see you as you might be. If it is a letter to a foreign university, I add a line about your fluency in English. If I know someone who is already in the programme to which you are applying, I compare you with him or her. I do not usually write about your achievements outside physics, but if you are unusually gifted at something and I feel that it says something essential about you, I add that a word about that, and how it connects with, adds or takes away from your commitment to physics; if you are applying for the Rhodes Scholarship, then of course I write a little more about your extra-curricular activities. I finish with an overall recommendation – strong, very strong, etc. In addition to information I try to communicate through tone, emphasis and example a picture of you as a whole person; I may tell a story or two to show how I discovered something about you. Sometimes I go a little overboard. (“_______, on the basis of your recommendation, we could get her married.”) But on the whole I think I do a pretty good job. I may be tempted to tell you what I wrote about you, but, though I may discuss your strengths and weaknesses with you in another setting, I desist from revealing the contents of my letter to you.

I believe strongly that a letter or recommendation to be effective must be confidential. I am required in it to express opinions about you that in normal circumstances would not be explicit. There is no reason, for example, for you to know in what respect I think you are superior or inferior to X or Y, but such information is essential to the institutions to which you are applying; in fact openly bandying such knowledge would destroy or seriously endanger the relationship of which this act of recommendation is a part. Asking for a letter of recommendation is ultimately an act of trust.

**

There are certain protocols that go with the act of asking for a letter of recommendation. First, ask yourself whether I have had the chance to get to know you well enough to write a convincing letter. If you think so, approach me in person if you are in College, or write to me or call me if you are elsewhere. Do not provide my name as a referee without my explicit permission, no matter how sure you may be that I will agree. Provide me with the information I seek, usually your statement of purpose, your CV, etc. I will usually want to talk to you about your application, and may want to interview you. If you are handing me a printed form, make sure your details are complete, and that you provide, as a courtesy, an envelope with your name and the name of the programme written clearly on it. Do not wait until the last moment to ask me for the reco: I do not appreciate being asked to provide a letter tomorrow; I recommend that you let me know month or so in advance that you intend to ask me for a letter, and then forewarn me and hand me the required material at least two weeks in advance. (For summer programmes I can fill a form at short notice, but even there I prefer to have some time.) After the letter is complete I will inform you, and if you are in College I expect you to come personally and collect it. If the letter is to be sent online, I will let you know after I have submitted it. Have the courtesy, in either case, to acknowledge my letter with a word of thanks (if you can’t resist giving me a blinding smile, that will quite in order, but nothing more is needed); remember, I don’t like being taken for granted any more than you do. When you hear back from the institutions to which you have applied, once again be so kind as to let me know what happened; I am interested. I do not expect you to write back to me from Cambridge or Cornell, but if you do, I’ll be happy to know how you’re faring in the programme to which I have recommended you. Besides, the information you provide will prove useful to other students who will, after you, come to me for counsel and recommendations.

Good luck.

December, 2010.