What are the makings of a truly remarkable scientific discovery ? This question is probably as old as the pursuit of science itself. In fact, a similar question is applicable in equal measure to every form of human creativity. Most of us have paused, at some time or the other, to admire in awe and wonder some truly remarkable piece of art, or music, or poetry, or other manifestation of creativity by a fellow human being, while scarcely believing that human hands or a human mind could have been responsible for such a marvel. In that moment of wonder all else is forgotten, an unalloyed feeling of admiration elevates one’s own spirit, and one appears to get a fleeting glance into the highest reaches of the human mind. What lies behind such instances of supreme creativity? Many philosophers and psychologists, as well as outstanding scientists and mathematicians themselves, have made thorough analyses of the possible answers to the question with which I have begun this essay.1 Much light has been shed on the matter, although crucial aspects of the process remain intangible. What is clear is that there is no unique prescription for success, no well-defined (or definable) route to the peak. It goes without saying that the soil must be fertile–the mind must be a prepared mind, the innate talent must be an exceptional one. These are necessary conditions, but by no means sufficient ones. The willingness to think big, the boldness to grapple with the big problem, the audacity to question authority, the ability to ask the right question, the originality ‘to think out of the box’, the determination to convert a vision into reality, and the perseverance to see the enterprise through–all these qualities must come together for a successful outcome. Over and above all these factors, it appears that luck or chance, too, plays a role. But this is chance in the sense of the discoverer being in the right place at the right time, rather than someone stumbling upon a treasure.
In this essay I shall attempt to describe this process, not by means of a scholarly and abstract analysis (which I am not equipped to carry out), but by the far more informal narration of a specific example of a scientific breakthrough. The instance I have chosen is a very famous one in science, and has all the ingredients listed above for a great success story. It concerns an important unsolved problem in the physical science of the day, and it involves a brilliant (but as-yet-unknown) young scientist at the beginning of his career. As it happens, it is part of one of the most dramatic sagas in the history of human thought. The scientist is none other than Albert Einstein himself, and the saga is the unparalleled scientific revolution wrought by him in 1905, his annus mirabilis. As I have already mentioned, we shall be concerned here with only a particular part of this story.
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The nature of the things around us has excited the curiosity of human beings from the earliest times. Thinking about this question, ancient civilizations seem to have arrived independently at some strikingly similar conclusions. While the actual details differ, the classification of all things into a few ‘elements’ is a common theme. Air, earth, fire and water; or earth, fire, sky and water; or earth, fire, metal, water and wood; or air, earth, fire, sky and water–these are some of the best known lists of elements that, according to ancient thinkers, made up everything on the earth, both living and non-living, as well the rest of the universe as they imagined it to be. It is obvious that this sort of classification is based directly upon unaided human observation, followed by a degree of thought, or extrapolation, or ‘pure reason’.
Equally fascinating is the question of the divisibility of material objects. What happens if one takes, say, a piece of wood, and goes on halving it? Once again, ancient thinkers could only suggest answers based on thought (more accurately, imagination), as the actual splitting process soon becomes impossible to continue in practice. Once again, the deepest thinkers in more than one ancient civilization came up with a possibility that was commendably close to actuality, at least in principle: namely, that all matter was ultimately composed of very tiny constituents or atoms, that were themselves indivisible. This last property, purely a postulate (or rather, a fiat) at this stage, precluded further descent down the ladder of reductionism.2
It is important to note that the conclusions (or theories) described above were based entirely upon observation followed by thought–that is, by ‘reasoning’ or ‘ratiotination’. They were not based on actual experimentation. In particular, they were not based on the currently accepted, time-tested scientific methodology that proceeds according to the following chain: observation –> hypothesis –> experiment –> analysis –> theory –> prediction –> further observation and experimentation –> . . . . We would therefore label these ancient theories as speculations.3 We know now that all matter on earth is composed of a rather small number of elements, listed in the periodic table. We know what makes fire, earth, water and air. We know what atoms are, and what they themselves are made up of–or rather, we have peeled a couple of layers deeper into the ‘cosmic onion’. These insights have been gained painstakingly. The pieces of the jigsaw puzzle of nature have to be fitted together with care and deliberation. And at each level of completion, it becomes evident that we have merely erected the scaffolding needed to begin work on a deeper and more subtle puzzle at the next level.
To return to our story: By the middle of the nineteenth century, it was becoming clear that all matter consisted of extremely tiny particles called atoms or molecules.4 The evidence came in from different directions: from the study of gases, chemical reactions and heat engines, to name just a few. These studies also led to the inescapable conclusion that, by everyday standards, these molecules were extremely tiny, as well as extremely numerous in any tangible piece of matter. An individual molecule could not be seen even under the best microscopes of the day.5 Estimates suggested that a typical molecule (of water, for instance) had to be about a ten-billionth of a metre in size, and that there had to be something of the order of a hundred thousand billion billion molecules in a tablespoon of water6 (a billion is a thousand millions). One is thus faced with the problem of simultaneously measuring an incredibly small quantity and measuring an incredibly large quantity, neither of them feasible by any sufficiently satisfactory direct means. No wonder, then, that the atomic constitution of matter remained, throughout the nineteenth century, the atomic hypothesis, although incontrovertible evidence in its favour was piling up. While most of the leading authorities in the physical sciences tended to support the hypothesis, there did remain many eminent sceptics. It was imperative to find a sufficiently direct way of settling the question once and for all.
This was precisely the problem that Einstein tackled and solved, in pioneering work that he published in 1905, and for which he was awarded his doctoral degree by the University of Zurich in early 1906. He had, therefore, set his sights on a truly important scientific problem. His approach to it was characteristically original. The phenomenon to which he turned his attention may seem (at first sight) to be rather unconnected to the problem at hand: he considered the apparently spontaneous and unceasing motion of microscopic particles suspended in a fluid such as water. And his solution of the problem was ingenious. He used clever physical arguments to circumvent difficult technical problems that would have arisen in a head-on, mathematically rigorous mode of attack. The ingredients for a breakthrough were in place. The rest is history.
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Imagine an enormous pumpkin floating in the vacuum of outer space in the middle of a dense swarm of the tiniest of mustard seeds, that ceaselessly bounce randomly off each other and the pumpkin. What kind of motion would the pumpkin exhibit?
This scenario helps us form a mental picture of the physical problem that Einstein confronted. It is an analogue, scaled up appropriately in space and time, of the actual phenomenon he sought to explain. The mustard seeds represent the molecules of water in a glass. As we have seen, each molecule is about a tenbillionth of a metre in size. The pumpkin represents one of the tiny particles that spill out when a pollen grain is crushed and suspended in the water. Such a particle is typically about a millionth of a metre in size, but it is nevertheless a giant compared to a molecule of water. Its mass would be something like a hundred billion billion times that of a molecule. Although individual molecules of water are too small to be seen directly even under a microscope, the ‘large’ particles can be observed and their motion studied. In 1827, this was how the Scottish botanist Robert Brown, examining under a microscope the motion of these particles suspended in water, made the first systematic experimental study of what has come to be known as Brownian motion. The “ceaseless and chaotic dance” of the particles was a truly astonishing phenomenon. Although the phenomenon had been observed even earlier, it was not clear whether any vis vitalis (‘life- force’) or biological effects were involved. Brown’s observations ruled out any such biological origin of the motion. In the years that followed, it became clear that internal motions in the liquid were somehow responsible for the motion. But there was no satisfactory explanation either for the apparently perpetual motion of the Brownian particle, or for the irregularity of its trajectory. The conundrum was unravelled Einstein, then merely twenty-six years old, in the seminal papers he wrote on it in 1905. Like the other two major themes (the light quantum and special relativity) upon which Einstein’s golden touch fell in this miraculous year, this work, too, not only solved a very important problem, but also proved to be the key to a vast empire of knowledge.
Returning to our analogy: the pumpkin does move under the constant buffeting of the mustard seeds, contrary to what we might have guessed at first sight. But surely it would never really get very far from wherever it started, because the seeds hit it randomly from all directions, and the pumpkin is an enormous object (both in size and mass) compared to the seeds? Astonishingly enough, this is not so. The almost imperceptible effects of the tiny but numerous and incessant collisions would csause it to tremble and jiggle about in a highly irregular fashion. These jiggles would occur on a hierarchy of scales: in principle, the graph of its path would be extremely jagged, and would remain so even if the resolution used in sketching the path were made finer. When averaged over all possible directions of motion, of course, the different displacements of the pumpkin from its starting point would tend to neutralize each other–because a step in any direction is as likely as an equally long step in the opposite direction. There can therefore be no particular direction, relative to its starting point, in which the pumpkin is more likely to find itself than any other direction, at any given time. But, given sufficient time, the pumpkin might find itself at quite a distance from where it started–in precisely the same way as it is possible to lose a substantial fortune over a period of time by continually placing small “can’t-lose-much” bets, as many have found to their chagrin! In fact, it turns out that the average value of the square of the pumpkin’s distance from its starting point increases steadily, being exactly proportional to the time for which we follow its motion. Einstein’s deep insight lay in recognizing that this average or mean squared distance, rather than the velocity of the particle, was the quantity to be studied and measured in such random motion.
Einstein’s interest in the problem arose from a deep-seated conviction that atoms were real, that their sizes and properties could be experimentally determined, and that their populations in ordinary bits of matter, albeit astronomically large, could be estimated reliably. Using ingenious arguments involving kinetic theory and the dynamics of random molecular motion on the one hand, and heat and thermodynamics on the other, Einstein gave the first correct explanation of the phenomenon. In the process, he showed how its observation could be used to determine the typical sizes of molecules (recall that this is about a ten-billionth of a metre). Moreover, it could also be used to determine Avogadro’s number (recall that this is of the order of a hundred thousand billion billion). Einstein’s predictions were tested and thoroughly vindicated within a few years after they were made. This marked, once and for all, the end of all debate on whether atoms really existed, or whether they were merely convenient mental constructs. The atomic hypothesis was no longer a mere hypothesis. The great physicist Richard Feynman once opined famously that if humanity were to face cataclysmic destruction, while being permitted to pass on a single piece of knowledge to survivors to enable them to build a civilization afresh, it would have to be the fact that matter consisted of atoms. In the light of this profoundly insightful remark, the importance of Einstein’s breakthrough is manifest.
What is even more remarkable is that Einstein used his uncanny instinct for the physics of the problem to explain Brownian motion quantitatively, carefully avoiding the pitfalls arising from the subtleties of the random processes representing Brownian motion in the strict mathematical sense, even though he was not aware of the relevant rigorous mathematics itself at the time. With the passage of time, it has become possible to gauge Einstein’s true strengths with something approaching dispassionate objectivity, on the basis of his total contribution to several areas of physics, such as statistical mechanics, quantum physics, relativity and gravitation. There can be no doubt that he had the most exceptionally deep insight into fundamental concepts in physics such as the role of fluctuations, symmetry, invariance and causality, among others.
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How has the analysis of Brownian motion (or a random walk, as a version of it is called) and of its subsequent generalizations contributed to human knowledge? Incredibly enough, in the century that has elapsed since 1905, it has turned out that the mathematics of this random, irregular motion is the fundamental paradigm for a staggeringly large number of phenomena in subjects ranging from astronomy through economics and meteorology to zoology! This commonality is yet another example of how profoundly unifying mathematics can be, in serving as a bridge across apparently unconnected disciplines. Even a minute part of the remarkably diverse list of applications that have emerged over the past one hundred years is quite astounding. The random processes representing Brownian motion and its off-shoots are relevant to the dynamics of objects as varied as polymers in solution, chemically reacting molecules, neutrons in a nuclear reactor, clouds, sand-piles, avalanches, insect swarms, animal herds, and star clusters; to techniques of computation; to fundamental issues in the theory of probability, quantum mechanics and quantum field theory; to the fluctuations of the stock market; and so on, apparently endlessly.
It is also interesting to note that, in the case of Brownian motion, too, the development of the relevant underlying mathematics actually preceded its incorporation into the realm of physics. As I have indicated, Einstein himself seems to have been completely unaware of these formal mathematical results, and so these did not anticipate Einstein’s work, or serve as a precursor for it, by any means. By 1900, fully five years ahead of Einstein, Louis Bachelier had worked out substantial parts of it in his doctoral thesis on the theory of speculation, a mathematical model of the fluctuating prices of shares in the stock market. Although Bachelier’s work did not receive the attention it deserved for some time, 9 it was really the progenitor of many deep results in the theory of probability in the hands of brilliant mathematicians such as Norbert Wiener and Andrey Nikolaevich Kolmogorov.7 A hundred years later, we have come full circle. The application of the theory of probability and random processes to the problems in finance that Bachelier studied, and to their extensions, is a major current preoccupation.
We have seen how Einstein’s explanation of Brownian motion is a classic example of a scientific discovery that qualifies as a genuine breakthrough. I would now like to go a step further, and call attention to a wider implication of such singular events. Once again, the most compelling evidence comes from the annus mirabilis itself. A century after 1905, with the prime perspective afforded by hindsight, what can we say about Einstein’s scientific achievements in that year,8 and how do these compare with other stupendous human achievements? In the judgment made by Abraham Pais in his definitive biography9 of Einstein: ‘No one before or since has widened the horizons of physics in so short a time as Einstein did in 1905.’ It is almost impossible (and perhaps ultimately irrelevant), however, to try to make a comparison between the highest peaks of excellence when these are widely separated in time and circumstance. But human interest in records is insatiable, and leads us to ask: Can we identify the most intense and sustained mental effort ever by a single person leading to the most profound results? It is impossible to give a unique or definitive answer to this question. Newton, Darwin and Einstein, each at the peak of his creativity, would certainly be in the ultimately exclusive club that we may accept, as a more meaningful compromise, in the place of any single person. When they scaled their respective heights, they changed something forever. Their discoveries represent watersheds for the human race itself, as they separate distinct eras in humankind’s understanding of the universe in which it lives, and of its place in it. It is a comfortingly salubrious fact that these are watersheds in a far more profound sense than mere political events–however earth-shaking the latter may appear to be when they occur, or even in the long run, for that matter. Ozymandias wasn’t Archimedes, after all.
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The genius underlying a scientific breakthrough is indeed ‘a miracle of rare device’, in the immortal words of Coleridge. It gives us a glimpse of the heights to which the human mind can soar. But the grandeur of nature also demands a certain measure of circumspection and respect–humility would not be an inappropriate word–in our efforts to understand it. Moreover, nature even seems to dictate the very form that this humility must take, if we are to make headway in our comprehension. It is on this apposite note that I would like to end this essay.
If a single unambiguous lesson can be gleaned from the developments in the sciences during the last one hundred years or so, it is surely that nature simply cannot be second-guessed. Sherlock Holmes’ dictum to avoid theorizing before one has all the evidence (because ‘it biases the judgment’) cannot be ignored when it comes to unlocking the secrets of nature. The eminent biologist J. B. S. Haldane came close to capturing the essence of the state of affairs when he said: ‘Now my own suspicion is that the Universe is not only queerer than we suppose, but queerer than we can suppose.’10 Both relativity and quantum mechanics, the two cornerstones of our current understanding of the physical universe, expose in particularly dramatic fashion how naive and misleading our so-called ‘intuition’ can be, with regard to the most fundamental aspects of nature. This is why it is not appropriate to begin scientific enquiry with a pre-conceived set of notions that are postulated to be ‘absolute truths’ simply because they appear to be self-evident or based on ‘common sense’. To quote Einstein’s fitting aphorism: ‘Common sense [in this context] is the collection of prejudices acquired by age eighteen.’
There is a good reason for the gross inadequacy of our perception of the world around us by means of our unaided senses, and of any description of it in terms of everyday language, as far as the proper scientific understanding of the physical universe is concerned. Obviously, the intuition we acquire with regard to the natural world is based on the behaviour of objects and phenomena we encounter in it in everyday life, and our discernment of this behaviour with the help of our senses. Owing to this very same pair of factors, the information thus gathered is restricted to a rather narrow range of variation (or window) of the values of physical parameters such as mass, length, time, velocity, momentum, force, and so on. This is the so-called ‘world of middle dimensions’. It informs our beliefs and our expectations (in short, our intuition!) regarding the nature of the physical world. Broadly speaking, what has been hard-wired into our brains, for reasons based on survival in the evolutionary sense, is the potential for the cognition of the world of middle dimensions at a certain level, and for the skills to manipulate and manoeuvre objects in it. It is highly presumptuous to expect nature itself to be restricted to this narrow range of values of physical quantities. And, indeed, nature spills out on both sides of this imagined straitjacket in a most spectacular manner! It turns out that our own windows of mass, length and time only encompass about seven or eight orders of magnitude, while our present state of knowledge shows that nature and natural phenomena extend over at least ten times that number of orders of magnitude.11 It is hardly surprising, then, that our naive, essentially hard-wired, intuition and common sense are totally inadequate to comprehend the universe unaided. We need the assistance of better, finer and more powerful tools to do so. As of now, these are provided by our instruments that extend our senses as we seek to probe ever deeper and farther into the mysteries of nature, and by our mathematics that enables us to analyze our findings and express the results in a context-free, semantics-independent manner. We know of no other sure way to get there. But it is a firm and well-lit path. And the view along the way is truly magnificent.
This article was originally published in‘Science, Literature and Aesthetics’, pp. 9-16Editor: Amiya DevVol. XV, Part 3 of ‘History of Science, Philosophy and Culture inIndian Civilization’General Editor: D. P. ChattopadhyayaPublished by Centre for Studies in CivilizationNew Delhi, 2009.
- This essay is developed from, and is a considerably expanded version of, an earlier article of mine titled Of random walks and fluctuating fortunes, published in Frontline, March 2005. ↩
- One is reminded of a joke that goes the other way, in an infinite regress: A certain cosmogony holds that the world is supported on the back of a giant turtle that is, in turn, supported on the back of another turtle. A novice wants to know what holds up the second turtle, and the local sage tells him that it is yet another turtle. After several iterations of the same question and answer, the sage ends the discussion with a sad shake of the head, declaring, “It’s no use, son–it’s turtles all the way!” ↩
- As far as science is concerned, we have come of age, or at least emerged from infancy (so we think). We have learnt the hard way that nature cannot be second-guessed (more on this later). ↩
- The distinction between an atom and a molecule did not become fully clear until a little later. A molecule is a cluster of atoms, sometimes as few as just two atoms, bound together by interatomic forces. ↩
- An impossibility that remains true to this day, but the reason for it is well understood. It has to do with the fact that the wave length of visible light is typically thousands of times bigger than the size of an atom. However, we now have other kinds of ‘microscopes’ with which we can ‘see’ individual atoms as directly as it can ever be done. ↩
- This number is close to the number of molecules in a certain standard amount of matter under certain standard conditions of temperature and pressure, called Avogadro’s number. ↩
- In fact, Brownian motion is also called the Wiener process in the mathematical literature. ↩
- Recall that Einstein made three breakthroughs in 1905. Besides the theory of Brownian motion, he also unveiled special relativity and the light quantum. ↩
- A. Pais, Subtle is the Lord: The Science and the Life of Albert Einstein (Oxford: Oxford University Press, 1983). ↩
- J. B. S. Haldane, in the essay On being the Right Size, in Possible Worlds: And Other Essays (London: Chatto and Windus, 1932). ↩
- If two quantities differ by a single order of magnitude, then one of them is about ten times as big as the other. ↩