The Unreasonable Effectiveness of Mathematics

The Unreasonable Effectiveness of Mathematics

R. W. HAMMING

Reprinted From: The American Mathematical Monthly
Volume 87 Number 2 February 1980

Prologue. It is evident from the title that this is a philosophical discussion. I shall not apologize for the philosophy, though I am well aware that most scientists, engineers, and mathematicians have little regard for it; instead, I shall give this short prologue to justify the approach.

Man, so far as we know, has always wondered about himself, the world around him, and what life is all about. We have many myths from the past that tell how and why God, or the gods, made man and the universe. These I shall call theological explanations. They have one principal characteristic in common-there is little point in asking why things are the way they are, since we are given mainly a description of the creation as the gods chose to do it.

Philosophy started when man began to wonder about the world outside of this theological framework. An early example is the description by the philosophers that the world is made of earth, fire, water, and air. No doubt they were told at the time that the gods made things that way and to stop worrying about it.

From these early attempts to explain things slowly came philosophy as well as our present science. Not that science explains "why" things are as they are-gravitation does not explain why things fall-but science gives so many details of "how" that we have the feeling we understand "why." Let us be clear about this point; it is by the sea of interrelated details that science seems to say "why" the universe is as it is.

Our main tool for carrying out the long chains of tight reasoning required by science is mathematics. Indeed, mathematics might be defined as being the mental tool designed for this purpose. Many people through the ages have asked the question I am effectively asking in the title, "Why is mathematics so unreasonably effective?" In asking this we are merely looking more at the logical side and less at the material side of what the universe is and how it works.

Mathematicians working in the foundations of mathematics are concerned mainly with the self-consistency and limitations of the system. They seem not to concern themselves with why the world apparently admits of a logical explanation. In a sense I am in the position of the early Greek philosophers who wondered about the material side, and my answers on the logical side are probably not much better than theirs were in their time. But we must begin somewhere and sometime to explain the phenomenon that the world seems to be organized in a logical pattern that parallels much of mathematics, that mathematics is the language of science and engineering.

Once I had organized the main outline, I had then to consider how best to communicate my ideas and opinions to others. Experience shows that I am not always successful in this matter. It finally occurred to me that the following preliminary remarks would help.

In some respects this discussion is highly theoretical. I have to mention, at least slightly, various theories of the general activity called mathematics, as well as touch on selected parts of it. Furthermore, there are various theories of applications. Thus, to some extent, this leads to a theory of theories. What may surprise you is that I shall take the experimentalist's approach in discussing things. Never mind what the theories are supposed to be, or what you think they should be, or even what the experts in the field assert they are; let us take the scientific attitude and look at what they are. I am well aware that much of what I say, especially about the nature of mathematics, will annoy many mathematicians. My experimental approach is quite foreign to their mentality and preconceived beliefs. So be it!

The inspiration for this article came from the similarly entitled article, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" [1], by E. P. Wigner. It will be noticed that I have left out part of the title, and by those who have already read it that I do not duplicate much of his material (I do not feel I can improve on his presentation). On the other hand, I shall spend relatively more time trying to explain the implied question of the title. But when all my explanations are over, the residue is still so large as to leave the question essentially unanswered.

The Effectiveness of Mathematics. In his paper, Wigner gives a large number of examples of the effectiveness of mathematics in the physical sciences. Let me, therefore, draw on my own experiences that are closer to engineering. My first real experience in the use of mathematics to predict things in the real world was in connection with the design of atomic bombs during the Second World War. How was it that the numbers we so patiently computed on the primitive relay computers agreed so well with what happened on the first test shot at Almagordo? There were, and could be, no small-scale experiments to check the computations directly. Later experience with guided missiles showed me that this was not an isolated phenomenon - constantly what we predict from the manipulation of mathematical symbols is realized in the real world. Naturally, working as I did for the Bell System, I did many telephone computations and other mathematical work on such varied things as traveling wave tubes, the equalization of television lines, the stability of complex communication systems, the blocking of calls through a telephone central office, to name but a few. For glamour, I can cite transistor research, space flight, and computer design, but almost all of science and engineering has used extensive mathematical manipulations with remarkable successes.

Many of you know the story of Maxwell's equations, how to some extent for reasons of symmetry he put in a certain term, and in time the radio waves that the theory predicted were found by Hertz. Many other examples of successfully predicting unknown physical effects from a mathematical formulation are well known and need not be repeated here.

The fundamental role of invariance is stressed by Wigner. It is basic to much of mathematics as well as to science. It was the lack of invariance of Newton's equations (the need for an absolute frame of reference for velocities) that drove Lorentz, Fitzgerald, Poincare, and Einstein to the special theory of relativity.

Wigner also observes that the same mathematical concepts turn up in entirely unexpected connections. For example, the trigonometric functions which occur in Ptolemy's astronomy turn out to be the functions which are invariant with respect to translation (time invariance). They are also the appropriate functions for linear systems. The enormous usefulness of the same pieces of mathematics in widely different situations has no rational explanation (as yet).

Furthermore, the simplicity of mathematics has long been held to be the key to applications in physics. Einstein is the most famous exponent of this belief. But even in mathematics itself the simplicity is remarkable, at least to me; the simplest algebraic equations, linear and quadratic, correspond to the simplest geometric entities, straight lines, circles, and conics. This makes analytic geometry possible in a practical way. How can it be that simple mathematics, being after all a product of the human mind, can be so remarkably useful in so many widely different situations?

Because of these successes of mathematics there is at present a strong trend toward making each of the sciences mathematical. It is usually regarded as a goal to be achieved, if not today, then tomorrow. For this audience I will stick to physics and astronomy for further examples.

Pythagoras is the first man to be recorded who clearly stated that "Mathematics is the way to understand the universe." He said it both loudly and clearly, "Number is the measure of all things."

Kepler is another famous example of this attitude. He passionately believed that God's handiwork could be understood only through mathematics. After twenty years of tedious computations, he found his famous three laws of planetary motion-three comparatively simple mathematical expressions that described the apparently complex motions of the planets.

It was Galileo who said, "The laws of Nature are written in the language of mathematics." Newton used the results of both Kepler and Galileo to deduce the famous Newtonian laws of motion, which together with the law of gravitation are perhaps the most famous example of the unreasonable effectiveness of mathematics in science. They not only predicted where the known planets would be but successfully predicted the positions of unknown planets, the motions of distant stars, tides, and so forth.

Science is composed of laws which were originally based on a small, carefully selected set of observations, often not very accurately measured originally; but the laws have later been found to apply over much wider ranges of observations and much more accurately than the original data justified. Not always, to be sure, but often enough to require explanation.

During my thirty years of practicing mathematics in industry, I often worried about the predictions I made. From the mathematics that I did in my office I confidently (at least to others) predicted some future events-if you do so and so, you will see such and such-and it usually turned out that I was right. How could the phenomena know what I had predicted (based on human-made mathematics) so that it could support my predictions? It is ridiculous to think that is the way things go. No, it is that mathematics provides, somehow, a reliable model for much of what happens in the universe. And since I am able to do only comparatively simple mathematics, how can it be that simple mathematics suffices to predict so much?

I could go on citing more examples illustrating the unreasonable effectiveness of mathematics, but it would only be boring. Indeed, I suspect that many of you know examples that I do not. Let me, therefore, assume that you grant me a very long list of successes, many of them as spectacular as the prediction of a new planet, of a new physical phenomenon, of a new artifact. With limited time, I want to spend it attempting to do what I think Wigner evaded-to give at least some partial answers to the implied question of the title.

What is Mathematics? Having looked at the effectiveness of mathematics, we need to look at the question,"What is Mathematics?" This is the title of a famous book by Courant and Robbins [2]. In it they do not attempt to give a formal definition, rather they are content to show what mathematics is by giving many examples. Similarly, I shall not give a comprehensive definition. But I will come closer than they did to discussing certain salient features of mathematics as I see them.

Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics.

The earliest history of mathematics must, of course, be all speculation, since there is not now, nor does there ever seem likely to be, any actual, convincing evidence. It seems, however, that in the very foundations of primitive life there was built in, for survival purposes if for nothing else, an understanding of cause and effect. Once this trait is built up beyond a single observation to a sequence of, "If this, then that, and then it follows still further that . . . ," we are on the path of the first feature of mathematics I mentioned, long chains of close reasoning. But it is hard for me to see how simple Darwinian survival of the fittest would select for the ability to do the long chains that mathematics and science seem to require.

Geometry seems to have arisen from the problems of decorating the human body for various purposes, such as religious rites, social affairs, and attracting the opposite sex, as well as from the problems of decorating the surfaces of walls, pots, utensils and clothing. This also implies the fourth aspect I mentioned, aesthetic taste, and this is one of the deep foundations of mathematics. Most textbooks repeat the Greeks and say that geometry arose from the needs of the Egyptians to survey the land after each flooding by the Nile River, but I attribute much more to aesthetics than do most historians of mathematics and correspondingly less to immediately utility.

The third aspect of mathematics, numbers, arose from counting. So basic are numbers that a famous mathematician once said, "God made the integers, man did the rest" [3]. The integers seem to us to be so fundamental that we expect to find them wherever we find intelligent life in the universe. I have tried, with little success, to get some of my friends to understand my amazement that the abstraction of integers for counting is both possible and useful. Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, l find it both strange and unexplainable.

In the development of numbers, we next come to the fact that these counting numbers, the integers, were used successfully in measuring how many times a standard length can be used to exhaust the desired length that is being measured. But it must have soon happened, comparatively speaking, that a whole number of units did not exactly fit the length being measured, and the measurers were driven to the fractions-the extra piece that was left over was used to measure the standard length. Fractions are not counting numbers; they are measuring numbers. Because of their common use in measuring, the fractions were, by a suitable extension of ideas, soon found to obey the same rules for manipulations as did the integers, with the added benefit that they made division possible in all cases (I have not yet come to the number zero). Some acquaintance with the fractions soon reveals that between any two fractions you can put as many more as you please and that in some sense they are homogeneously dense everywhere. But when we extend the concept of number to include the fractions, we have to give up the idea of the next number,

This brings us again to Pythagoras, who is reputed to be the first man to prove that the diagonal of a square and the side of the square have no common measure-that they are irrationally related. This observation apparently produced a profound upheaval in Greek: mathematics. Up to that time the discrete number system and the continuous geometry flourished side by side with little conflict. The crisis of incommensurability tripped off the Euclidean approach to mathematics. It is a curious fact that the early Greeks attempted to make mathematics rigorous by replacing the uncertainties of numbers by what they felt was the more certain geometry (due to Eudoxus). It was a major event to Euclid, and as a result you find in The Elements [4] a lot of what we now consider number theory and algebra cast in the form of geometry. Opposed to the early Greeks, who doubted the existence of the real number system, we have decided that there should be a number that measures the length of the diagonal of a unit square (though we need not do so), and that is more or less how we extended the rational number system to include the algebraic numbers. It was the simple desire to measure lengths that did it. How can anyone deny that there is a number to measure the length of any straight line segment?

The algebraic numbers, which are roots of polynomials with integer, fractional, and, as was later proved, even algebraic numbers as coefficients, were soon under control by simply extending the same operations that were used on the simpler system of numbers.

However, the measurement of the circumference of a circle with respect to its diameter soon forced us to consider the ratio called pi. This is not an algebraic number, since no linear combination of the power of pi with integer coefficients will exactly vanish. One length, the circumference, being a curved line, and the other length, the diameter, being a straight line, make the existence of the ratio less certain than is the ratio of the diagonal of a square to its side; but since it seems that there ought to be such a number, the transcendental numbers gradually got into the number system. Thus by a further suitable extension of the earlier ideas of numbers, the transcendental numbers were admitted consistently into the number system, though few students are at all comfortable with the technical apparatus we conventionally use to show the consistency.

Further tinkering with the number system brought both the number zero and the negative numbers. This time the extension required that we abandon the division for the single number zero. This seems to round out the real number system for us (as long as we confine ourselves to the process of taking limits of sequences of numbers and do not admit still further operations) -not that we have to this day a firm, logical, simple, foundation for them; but they say that familiarity breeds contempt, and we are all more or less familiar with the real number system. Very few of us in our saner moments believe that the particular postulates that some logicians have dreamed up create the numbers - no, most of us believe that the real numbers are simply there and that it has been an interesting, amusing, and important game to try to find a nice set of postulates to account for them. But let us not confuse ourselves-Zeno's paradoxes are still, even after 2,000 years, too fresh in our minds to delude ourselves that we understand all that we wish we did about the relationship between the discrete number system and the continuous line we want to model. We know, from nonstandard analysis if from no other place, that logicians can make postulates that put still further entities on the real line, but so far few of us have wanted to go down that path. It is only fair to mention that there are some mathematicians who doubt the existence of the conventional real number system. A few computer theoreticians admit the existense of only "the computable numbers."

The next step in the discussion is the complex number system. As I read history, it was Cardan who was the first to understand them in any real sense. In his The Great Art or Rules of Algebra [5] he says, "Putting aside the mental tortures involved multiply (5 + sqrt 15) by (5 - sqrt -15) making 25-(-15) ...." Thus he clearly recognized that the same formal operations on the symbols for complex numbers would give meaningful results. In this way the real number system was gradually extended to the complex number system, except that this time the extension required giving up the property of ordering the numbers-the complex numbers cannot be ordered in the usual sense.

Cauchy was apparently led to the theory of complex variables by the problem of integrating real functions along the real line. He found that by bending the path of integration into the complex plane he could solve real integration problems.

A few years ago I had the pleasure of teaching a course in complex variables. As always happens when I become involved in the topic, I again came away with the feeling that "God made the universe out of complex numbers." Clearly, they play a central role in quantum mechanics. They are a natural tool in many other areas of application, such as electric circuits, fields, and so on.

To summarize, from simple counting using the God-given integers, we made various extensions of the ideas of numbers to include more things. Sometimes the extensions were made for what amounted to aesthetic reasons, and often we gave up some property of the earlier number system. Thus we came to a number system that is unreasonably effective even in mathematics itself; witness the way we have solved many number theory problems of the original highly discrete counting system by using a complex variable.

From the above we see that one of the main strands of mathematics is the extension, the generalization, the abstraction - they are all more or less the same thing-of well-known concepts to new situations. But note that in the very process the definitions themselves are subtly altered. Therefore, what is not so widely recognized, old proofs of theorems may become false proofs. The old proofs no longer cover the newly defined things. The miracle is that almost always the theorems are still true; it is merely a matter of fixing up the proofs. The classic example of this fixing up is Euclid's The Elements [4]. We have found it necessary to add quite a few new postulates (or axioms, if you wish, since we no longer care to distinguish between them) in order to meet current standards of proof. Yet how does it happen that no theorem in all the thirteen books is now false? Not one theorem has been found to be false, though often the proofs given by Euclid seem now to be false. And this phenomenon is not confined to the past. It is claimed that an ex-editor of Mathematical Reviews once said that over half of the new theorems published these days are essentially true though the published proofs are false. How can this be if mathematics is the rigorous deduction of theorems from assumed postulates and earlier results? Well, it is obvious to anyone who is not blinded by authority that mathematics is not what the elementary teachers said it was. It is clearly something else.

What is this "else"? Once you start to look you find that if you were confined to the axioms and postulates then you could deduce very little. The first major step is to introduce new concepts derived from the assumptions, concepts such as triangles. The search for proper concepts and definitions is one of the main features of doing great mathematics.

While on the topic of proofs, classical geometry begins with the theorem and tries to find a proof. Apparently it was only in the 1850's or so that it was clearly recognized that the opposite approach is also valid (it must have been occasionally used before then). Often it is the proof that generates the theorem. We see what we can prove and then examine the proof to see what we have proved! These are often called "proof generated theorems" [6]. A classic example is the concept of uniform convergence. Cauchy had proved that a convergent series of terms, each of which is continuous, converges to a continuous function. At the same time there were known to be Fourier series of continuous functions that converged to a discontinuous limit. By a careful examination of Cauchy's proof, the error was found and fixed up by changing the hypothesis of the theorem to read, "a uniformly convergent series."

More recently, we have had an intense study of what is called the foundations of mathematics-which in my opinion should be regarded as the top battlements of mathematics and not the foundations. It is an interesting field, but the main results of mathematics are impervious to what is found there-we simply will not abandon much of mathematics no matter how illogical it is made to appear by research in the foundations.

I hope that I have shown that mathematics is not the thing it is often assumed to be, that mathematics is constantly changing and hence even if I did succeed in defining it today the definition would not be appropriate tomorrow. Similarly with the idea of rigor-we have a changing standard. The dominant attitude in science is that we are not the center of the universe, that we are not uniquely placed, etc., and similarly it is difficult for me to believe that we have now reached the ultimate of rigor. Thus we cannot be sure of the current proofs of our theorems. Indeed it seems to me:

The Postulates of Mathematics Were Not on the Stone Tablets that Moses Brought Down from Mt. Sinai.

It is necessary to emphasize this. We begin with a vague concept in our minds, then we create various sets of postulates, and gradually we settle down to one particular set. In the rigorous postulational approach the original concept is now replaced by what the postulates define. This makes further evolution of the concept rather difficult and as a result tends to slow down the evolution of mathematics. It is not that the postulation approach is wrong, only that its arbitrariness should be clearly recognized, and we should be prepared to change postulates when the need becomes apparent.

Mathematics has been made by man and therefore is apt to be altered rather continuously by him. Perhaps the original sources of mathematics were forced on us, but as in the example I have used we see that in the development of so simple a concept as number we have made choices for the extensions that were only partly controlled by necessity and often, it seems to me, more by aesthetics. We have tried to make mathematics a consistent, beautiful thing, and by so doing we have had an amazing number of successful applications to the real world.

The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way. For over thirty years I have been making the remark that if you came into my office and showed me a proof that Cauchy's theorem was false I would be very interested, but I believe that in the final analysis we would alter the assumptions until the theorem was true. Thus there are many results in mathematics that are independent of the assumptions and the proof.

How do we decide in a "crisis" what parts of mathematics to keep and what parts to abandon? Usefulness is one main criterion, but often it is usefulness in creating more mathematics rather than in the applications to the real world! So much for my discussion of mathematics.

Some Partial Explanations. I will arrange my explanations of the unreasonable effectiveness of mathematics under four headings.

1. We see what we look for. No one is surprised if after putting on blue tinted glasses the world appears bluish. I propose to show some examples of how much this is true in current science. To do this I am again going to violate a lot of widely, passionately held beliefs. But hear me out.

I picked the example of scientists in the earlier part for a good reason. Pythagoras is to my mind the first great physicist. It was he who found that we live in what the mathematicians call L2-the sum of the squares of the two sides of a right triangle gives the square of the hypotenuse. As I said before, this is not a result of the postulates of geometry-this is one of the results that shaped the postulates.

Let us next consider Galileo. Not too long ago I was trying to put myself in Galileo's shoes, as it were, so that I might feel how he came to discover the law of falling bodies. I try to do this kind of thing so that I can learn to think like the masters did-I deliberately try to think as they might have done.

Well, Galileo was a well-educated man and a master of scholastic arguments. He well knew how to argue the number of angels on the head of a pin, how to argue both sides of any question. He was trained in these arts far better than any of us these days. I picture him sitting one day with a light and a heavy ball, one in each hand, and tossing them gently. He says, hefting them, "It is obvious to anyone that heavy objects fall faster than light ones-and, anyway, Aristotle says so." "But suppose," he says to himself, having that kind of a mind, "that in falling the body broke into two pieces. Of course the two pieces would immediately slow down to their appropriate speeds. But suppose further that one piece happened to touch the other one. Would they now be one piece and both speed up? Suppose I tied the two pieces together. How tightly must I do it to make them one piece? A light string? A rope? Glue? When are two pieces one?"

The more he thought about it-and the more you think about it-the more unreasonable becomes the question of when two bodies are one. There is simply no reasonable answer to the question of how a body knows how heavy it is-if it is one piece, or two, or many. Since falling bodies do something, the only possible thing is that they all fall at the same speed-unless interfered with by other forces. There's nothing else they can do. He may have later made some experiments, but I strongly suspect that something like what I imagined actually happened. I later found a similar story in a book by Polya [7]. Galileo found his law not by experimenting but by simple, plain thinking, by scholastic reasoning.

I know that the textbooks often present the falling body law as an experimental observation; I am claiming that it is a logical law, a consequence of how we tend to think.

Newton, as you read in books, deduced the inverse square law from Kepler's laws, though they often present it the other way; from the inverse square law the textbooks deduce Kepler's laws. But if you believe in anything like the conservation of energy and think that we live in a three-dimensional Euclidean space, then how else could a symmetric central-force field fall off? Measurements of the exponent by doing experiments are to a great extent attempts to find out if we live in a Euclidean space, and not a test of the inverse square law at all.

But if you do not like these two examples, let me turn to the most highly touted law of recent times, the uncertainty principle. It happens that recently I became involved in writing a book on Digital Filters [8] when I knew very little about the topic. As a result I early asked the question, "Why should I do all the analysis in terms of Fourier integrals? Why are they the natural tools for the problem?" I soon found out, as many of you already know, that the eigenfunctions of translation are the complex exponentials. If you want time invariance, and certainly physicists and engineers do (so that an experiment done today or tomorrow will give the same results), then you are led to these functions. Similarly, if you believe in linearity then they are again the eigenfunctions. In quantum mechanics the quantum states are absolutely additive; they are not just a convenient linear approximation. Thus the trigonometric functions are the eigenfunctions one needs in both digital filter theory and quantum mechanics, to name but two places.

Now when you use these eigenfunctions you are naturally led to representing various functions, first as a countable number and then as a non-countable number of them-namely, the Fourier series and the Fourier integral. Well, it is a theorem in the theory of Fourier integrals that the variability of the function multiplied by the variability of its transform exceeds a fixed constant, in one notation l/2pi. This says to me that in any linear, time invariant system you must find an uncertainty principle. The size of Planck's constant is a matter of the detailed identification of the variables with integrals, but the inequality must occur.

As another example of what has often been thought to be a physical discovery but which turns out to have been put in there by ourselves, I turn to the well-known fact that the distribution of physical constants is not uniform; rather the probability of a random physical constant having a leading digit of 1. 2, or 3 is approximately 60%, and of course the leading digits of 5, 6, 7, 8, and 9 occur in total only about 40% of the time. This distribution applies to many types of numbers, including the distribution of the coefficients of a power series having only one singularity on the circle of convergence. A close examination of this phenomenon shows that it is mainly an artifact of the way we use numbers.

Having given four widely different examples of nontrivial situations where it turns out that the original phenomenon arises from the mathematical tools we use and not from the real world, I am ready to strongly suggest that a lot of what we see comes from the glasses we put on. Of course this goes against much of what you have been taught, but consider the arguments carefully. You can say that it was the experiment that forced the model on us, but I suggest that the more you think about the four examples the more uncomfortable you are apt to become. They are not arbitrary theories that I have selected, but ones which are central to physics,

In recent years it was Einstein who most loudly proclaimed the simplicity of the laws of physics, who used mathematics so exclusively as to be popularly known as a mathematician. When examining his special theory of relativity paper [9] one has the feeling that one is dealing with a scholastic philosopher's approach. He knew in advance what the theory should look like. and he explored the theories with mathematical tools, not actual experiments. He was so confident of the rightness of the relativity theories that, when experiments were done to check them, he was not much interested in the outcomes, saying that they had to come out that way or else the experiments were wrong. And many people believe that the two relativity theories rest more on philosophical grounds than on actual experiments.

Thus my first answer to the implied question about the unreasonable effectiveness of mathematics is that we approach the situations with an intellectual apparatus so that we can only find what we do in many cases. It is both that simple, and that awful. What we were taught about the basis of science being experiments in the real world is only partially true. Eddington went further than this; he claimed that a sufficiently wise mind could deduce all of physics. I am only suggesting that a surprising amount can be so deduced. Eddington gave a lovely parable to illustrate this point. He said, "Some men went fishing in the sea with a net, and upon examining what they caught they concluded that there was a minimum size to the fish in the sea."

2. We select the kind of mathematics to use. Mathematics does not always work. When we found that scalars did not work for forces, we invented a new mathematics, vectors. And going further we have invented tensors. In a book I have recently written [10] conventional integers are used for labels, and real numbers are used for probabilities; but otherwise all the arithmetic and algebra that occurs in the book, and there is a lot of both, has the rule that

1+1=0.

Thus my second explanation is that we select the mathematics to fit the situation, and it is simply not true that the same mathematics works every place.

3. Science in fact answers comparatively few problems. We have the illusion that science has answers to most of our questions, but this is not so. From the earliest of times man must have pondered over what Truth, Beauty, and Justice are. But so far as I can see science has contributed nothing to the answers, nor does it seem to me that science will do much in the near future. So long as we use a mathematics in which the whole is the sum of the parts we are not likely to have mathematics as a major tool in examining these famous three questions.

Indeed, to generalize, almost all of our experiences in this world do not fall under the domain of science or mathematics. Furthermore, we know (at least we think we do) that from Godel's theorem there are definite limits to what pure logical manipulation of symbols can do, there are limits to the domain of mathematics. It has been an act of faith on the part of scientists that the world can be explained in the simple terms that mathematics handles. When you consider how much science has not answered then you see that our successes are not so impressive as they might otherwise appear.

4. The evolution of man provided the model. I have already touched on the matter of the evolution of man. I remarked that in the earliest forms of life there must have been the seeds of our current ability to create and follow long chains of close reasoning. Some people [11] have further claimed that Darwinian evolution would naturally select for survival those competing forms of life which had the best models of reality in their minds-"best" meaning best for surviving and propagating. There is no doubt that there is some truth in this. We find, for example, that we can cope with thinking about the world when it is of comparable size to ourselves and our raw unaided senses, but that when we go to the very small or the very large then our thinking has great trouble. We seem not to be able to think appropriately about the extremes beyond normal size.

Just as there are odors that dogs can smell and we cannot, as well as sounds that dogs can hear and we cannot, so too there are wavelengths of light we cannot see and flavors we cannot taste. Why then, given our brains wired the way they are, does the remark "Perhaps there are thoughts we cannot think," surprise you? Evolution, so far, may possibly have blocked us from being able to think in some directions; there could be unthinkable thoughts.

If you recall that modern science is only about 400 years old, and that there have been from 3 to 5 generations per century, then there have been at most 20 generations since Newton and Galileo. If you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics.

Conclusion. From all of this I am forced to conclude both that mathematics is unreasonably effective and that all of the explanations I have given when added together simply are not enough to explain what I set out to account for. I think that we-meaning you, mainly-must continue to try to explain why the logical side of science-meaning mathematics, mainly-is the proper tool for exploring the universe as we perceive it at present. I suspect that my explanations are hardly as good as those of the early Greeks, who said for the material side of the question that the nature of the universe is earth, fire, water, and air. The logical side of the nature of the universe requires further exploration.



I (Larry Frazier, who (with R. Hamming's permission) scanned this and put it online) was pleased to note that 58 people visited this essay in a recent 2-month period. I assume most of you are finding this from a pointer in the Gutenberg Project hierarchy.

On the other hand, I feel like thousands of people should be reading this. It is the most profound essay I have seen regarding philosophy of science; important, significant, in fact, for our whole understanding of thought, of knowing, or reality.

Drop me a note if you have any comments. Larry Frazier



1. E. P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Comm. Pure Appl. Math., 13 (Feb. 1960).

2. R. Courant and H. Robbins, What Is Mathematics? Oxford University Press, 1941.

3. L. Kronecker, Item 1634. in On Mathematics and Mathematicians, by R E Moritz.

4. Euclid, Euclid's Elements, T. E. Heath, Dover Publications, New York, 1956.

5. G. Cardano, The Great Art or Rules of Algebra, transl. by T. R. Witmer, MIT Press, 1968, pp. 219-220

6. Imre Lakatos, Proofs and Refutations; Cambridge University Press, 1976, p. 33.

7. G. Polya, Mathematical Methods in Science, MAA, 1963, pp. 83-85.

8. R. W. Hamming, Digital Filters, Prentice-Hall, Englewood Cliffs, NJ., 1977.

9. G. Holton Thematic Origins of Scientific Thought, Kepler to Einstein, Harvard University Press, 1973.

10. R. W. Hamming, Coding and Information Theory, Prentice-Hall, Englewood Cliffs, NJ., 1980.

11. H. Mohr, Structure and Significance of Science, Springer- Verlag, 1977.

On 2001 May 24 Larry Frazier gave me permission to post this.

Tom Schneider

2003 April 10. I noticed that one paragraph ends incorrectly with "idea of the next number," To determine if there is a corrected copy somewhere I did a search (see below). The Dartmouth version has the same error!

* Google search for The Unreasonable Effectiveness of Mathematics
* The Unreasonable Effectiveness of Mathematics by R. W. HAMMING (at Dartmouth) Another copy of Hamming's article.
* The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner (at Dartmouth) This was cited in Hamming's article. It's also well worth reading.




Schneider Lab
origin: 1998 or 1999 sometime?
updated: 2001 May 24
updated: 2003 Apr 10

THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS IN THE NATURAL SCIENCES

This HTML page was prepared based on the original PDF file found here: http://www.physik.uni-wuerzburg.de/fileadmin/tp3/QM/wigner.pdf

Reprinted from Communications in Pure and Applied Mathematics, Vol. 13, No. I (February 1960).
New York: John Wiley & Sons, Inc.
Copyright © 1960 by John Wiley & Sons, Inc.

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THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS IN THE NATURAL SCIENCES
by Eugene Wigner

Mathematics, rightly viewed, possesses not only truth, but supreme beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

— BERTRAND RUSSELL, Study of Mathematics

There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense. I was even more confused when, not many days later, someone came to me and expressed his bewilderment [ The remark to be quoted was made by F. Werner when he was a student in Princeton.] with the fact that we make a rather narrow selection when choosing the data on which we test our theories. "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?" It has to be admitted that we have no definite evidence that there is no such theory.

The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

Page 2 Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, "What is mathematics?", then, "What is physics?", then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.
WHAT IS MATHEMATICS?

Somebody once said that philosophy is the misuse of a terminology which was invented just for this purpose.[This statement is quoted here from W. Dubislav's Die Philosophie der Mathematik in der Gegenwart (Berlin: Junker and Dunnhaupt Verlag, 1932), p. 1.] In the same vein, I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms. Furthermore, whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics. Thus, the rules for operations with pairs of numbers are obviously designed to give the same results as the operations with fractions which we first learned without reference to "pairs of numbers." The rules for the operations with sequences, that is, with irrational numbers, still belong to the category of rules which were determined so as to reproduce rules for the operations with quantities which were already known to us. Most more advanced mathematical concepts, such as complex numbers, algebras, linear operators, Borel sets - and this list could be continued almost indefinitely - were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty. In fact, the definition of these concepts, with a realization that interesting and ingenious considerations could be applied to them, is the first demonstration of the ingeniousness of the mathematician who defines them. The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used. The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess. However, this is not our present subject. The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity. [ M. Polanyi, in his Personal Knowledge (Chicago: University of Chicago Press, 1958), says: "All these difficulties are but consequences of our refusal to see that mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting" (p. 188)].

The complex numbers provide a particularly striking example for the foregoing. Certainly, nothing in our experience suggests the introduction of these quantities. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many Page 3 beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius. [ The reader may be interested, in this connection, in Hilbert's rather testy remarks about intuitionism which "seeks to break up and to disfigure mathematics," Abh. Math. Sem., Univ. Hamburg, 157 (1922), or Gesammelte Werke (Berlin: Springer, 1935), p. 188.]
WHAT IS PHYSICS?

The physicist is interested in discovering the laws of inanimate nature. In order to understand this statement, it is necessary to analyze the concept, "law of nature."

The world around us is of baffling complexity and the most obvious fact about it is that we cannot predict the future. Although the joke attributes only to the optimist the view that the future is uncertain, the optimist is right in this case: the future is unpredictable. It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. The laws of nature are concerned with such regularities. Galileo's regularity is a prototype of a large class of regularities. It is a surprising regularity for three reasons.

The first reason that it is surprising is that it is true not only in Pisa, and in Galileo's time, it is true everywhere on the Earth, was always true, and will always be true. This property of the regularity is a recognized invariance property and, as I had occasion to point out some time ago, without invariance principles similar to those implied in the preceding generalization of Galileo's observation, physics would not be possible. The second surprising feature is that the regularity which we are discussing is independent of so many conditions which could have an effect on it. It is valid no matter whether it rains or not, whether the experiment is carried out in a room or from the Leaning Tower, no matter whether the person who drops the rocks is a man or a woman. It is valid even if the two rocks are dropped, simultaneously and from the same height, by two different people. There are, obviously, innumerable other conditions which are all immaterial from the point of view of the validity of Galileo's regularity. The irrelevancy of so many circumstances which could play a role in the phenomenon observed has also been called an invariance. However, this invariance is of a different character from the preceding one since it cannot be formulated as a general principle. The exploration of the conditions which do, and which do not, influence a phenomenon is part of the early experimental exploration of a field. It is the skill and ingenuity of the experimenter which show him phenomena which depend on a relatively narrow set of relatively easily realizable and reproducible conditions. [ see, in this connection, the graphic essay of M. Deutsch, Daedalus 87, 86 (1958). A. Shimony has called my attention to a similar passage in C. S. Peirce's Essays in the Philosophy of Science (New York: The Liberal Arts Press, 1957), p. 237.] In the present case, Galileo's restriction of his observations to relatively heavy bodies was the most important step in this regard. Again, it is true that if there were no phenomena which are independent of all but a manageably small set of conditions, physics would be impossible.

The preceding two points, though highly significant from the point of view of the philosopher, are not the ones which surprised Galileo most, nor do they contain a specific law of nature. The law of nature is contained in the statement that the length of time which it takes for a heavy object to fall from a given height is independent of the size, material, and shape of the body which drops. In the framework of Newton's second "law," this amounts to the statement that the gravitational force which acts on the falling body is proportional to its mass but independent of the size, material, and shape of the body which falls.

Page 4

The preceding discussion is intended to remind us, first, that it is not at all natural that "laws of nature" exist, much less that man is able to discover them. [ E. Schrodinger, in his What Is Life? (Cambridge: Cambridge University Press, 1945), p. 31, says that this second miracle may well be beyond human understanding.] The present writer had occasion, some time ago, to call attention to the succession of layers of "laws of nature," each layer containing more general and more encompassing laws than the previous one and its discovery constituting a deeper penetration into the structure of the universe than the layers recognized before. However, the point which is most significant in the present context is that all these laws of nature contain, in even their remotest consequences, only a small part of our knowledge of the inanimate world. All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present, except that some aspects of the present state of the world, in practice the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction. The irrelevancy is meant in the sense of the second point in the discussion of Galileo's theorem. [ The writer feels sure that it is unnecessary to mention that Galileo's theorem, as given in the text, does not exhaust the content of Galileo's observations in connection with the laws of freely falling bodies.]

As regards the present state of the world, such as the existence of the earth on which we live and on which Galileo's experiments were performed, the existence of the sun and of all our surroundings, the laws of nature are entirely silent. It is in consonance with this, first, that the laws of nature can be used to predict future events only under exceptional circumstances - when all the relevant determinants of the present state of the world are known. It is also in consonance with this that the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted. Radars and nuclear reactors are examples of such machines.

The principal purpose of the preceding discussion is to point out that the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world. Thus, classical mechanics, which is the best known prototype of a physical theory, gives the second derivatives of the positional coordinates of all bodies, on the basis of the knowledge of the positions, etc., of these bodies. It gives no information on the existence, the present positions, or velocities of these bodies. It should be mentioned, for the sake of accuracy, that we discovered about thirty years ago that even the conditional statements cannot be entirely precise: that the conditional statements are probability laws which enable us only to place intelligent bets on future properties of the inanimate world, based on the knowledge of the present state. They do not allow us to make categorical statements, not even categorical statements conditional on the present state of the world. The probabilistic nature of the "laws of nature" manifests itself in the case of machines also, and can be verified, at least in the case of nuclear reactors, if one runs them at very low power. However, the additional limitation of the scope of the laws of nature which follows from their probabilistic nature will play no role in the rest of the discussion.
THE ROLE OF MATHEMATICS IN PHYSICAL THEORIES

Having refreshed our minds as to the essence of mathematics and physics, we should be in a better position to review the role of mathematics in physical theories. Naturally, we do use mathematics in everyday physics to evaluate the results of the laws of nature, to apply the conditional statements to the particular conditions which happen to prevail or happen to interest us. In order that this be possible, the laws of nature must already be formulated in mathematical language. However, the role of evaluating the consequences of already established Page 5 theories is not the most important role of mathematics in physics. Mathematics, or, rather, applied mathematics, is not so much the master of the situation in this function: it is merely serving as a tool.

Mathematics does play, however, also a more sovereign role in physics. This was already implied in the statement, made when discussing the role of applied mathematics, that the laws of nature must have been formulated in the language of mathematics to be an object for the use of applied mathematics. The statement that the laws of nature are written in the language of mathematics was properly made three hundred years ago; [ It is attributed to Galileo.] it is now more true than ever before. In order to show the importance which mathematical concepts possess in the formulation of the laws of physics, let us recall, as an example, the axioms of quantum mechanics as formulated, explicitly, by the great physicist, Dirac. There are two basic concepts in quantum mechanics: states and observables. The states are vectors in Hilbert space, the observables self-adjoint operators on these vectors. The possible values of the observations are the characteristic values of the operators - but we had better stop here lest we engage in a listing of the mathematical concepts developed in the theory of linear operators.

It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. It is true also that the concepts which were chosen were not selected arbitrarily from a listing of mathematical terms but were developed, in many if not most cases, independently by the physicist and recognized then as having been conceived before by the mathematician. It is not true, however, as is so often stated, that this had to happen because mathematics uses the simplest possible concepts and these were bound to occur in any formalism. As we saw before, the concepts of mathematics are not chosen for their conceptual simplicity - even sequences of pairs of numbers are far from being the simplest concepts - but for their amenability to clever manipulations and to striking, brilliant arguments. Let us not forget that the Hilbert space of quantum mechanics is the complex Hilbert space, with a Hermitean scalar product. Surely to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of the laws of quantum mechanics. Finally, it now begins to appear that not only complex numbers but so-called analytic functions are destined to play a decisive role in the formulation of quantum theory. I am referring to the rapidly developing theory of dispersion relations.

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind's capacity to divine them. The observation which comes closest to an explanation for the mathematical concepts' cropping up in physics which I know is Einstein's statement that the only physical theories which we are willing to accept are the beautiful ones. It stands to argue that the concepts of mathematics, which invite the exercise of so much wit, have the quality of beauty. However, Einstein's observation can at best explain properties of theories which we are willing to believe and has no reference to the intrinsic accuracy of the theory. We shall, therefore, turn to this latter question.
IS THE SUCCESS OF PHYSICAL THEORIES TRULY SURPRISING?

A possible explanation of the physicist's use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the Page 6 conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language. Let us consider a few examples.

The first example is the oft-quoted one of planetary motion. The laws of falling bodies became rather well established as a result of experiments carried out principally in Italy. These experiments could not be very accurate in the sense in which we understand accuracy today partly because of the effect of air resistance and partly because of the impossibility, at that time, to measure short time intervals. Nevertheless, it is not surprising that, as a result of their studies, the Italian natural scientists acquired a familiarity with the ways in which objects travel through the atmosphere. It was Newton who then brought the law of freely falling objects into relation with the motion of the moon, noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence. Philosophically, the law of gravitation as formulated by Newton was repugnant to his time and to himself. Empirically, it was based on very scanty observations. The mathematical language in which it was formulated contained the concept of a second derivative and those of us who have tried to draw an osculating circle to a curve know that the second derivative is not a very immediate concept. The law of gravity which Newton reluctantly established and which he could verify with an accuracy of about 4% has proved to be accurate to less than a ten thousandth of a per cent and became so closely associated with the idea of absolute accuracy that only recently did physicists become again bold enough to inquire into the limitations of its accuracy. [ see, for instance, R. H. Dicke, Am. Sci., 25 (1959).] Certainly, the example of Newton's law, quoted over and over again, must be mentioned first as a monumental example of a law, formulated in terms which appear simple to the mathematician, which has proved accurate beyond all reasonable expectations. Let us just recapitulate our thesis on this example: first, the law, particularly since a second derivative appears in it, is simple only to the mathematician, not to common sense or to non-mathematically-minded freshmen; second, it is a conditional law of very limited scope. It explains nothing about the earth which attracts Galileo's rocks, or about the circular form of the moon's orbit, or about the planets of the sun. The explanation of these initial conditions is left to the geologist and the astronomer, and they have a hard time with them.

The second example is that of ordinary, elementary quantum mechanics. This originated when Max Born noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions. Indeed, they say "if the mechanics as here proposed should already be correct in its essential traits." As a matter of fact, the first application of their mechanics to a realistic problem, that of the hydrogen atom, was given several months later, by Pauli. This application gave results in agreement with experience. This was satisfactory but still understandable because Heisenberg's rules of calculation were abstracted from problems which included the old theory of the hydrogen atom. The miracle occurred only when matrix mechanics, or a mathematically equivalent theory, was applied to problems for which Heisenberg's calculating rules were meaningless. Heisenberg's rules presupposed that the classical equations of motion had solutions with certain periodicity properties; Page 7 and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg's rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we "got something out" of the equations that we did not put in.

The same is true of the qualitative characteristics of the "complex spectra," that is, the spectra of heavier atoms. I wish to recall a conversation with Jordan, who told me, when the qualitative features of the spectra were derived, that a disagreement of the rules derived from quantum mechanical theory and the rules established by empirical research would have provided the last opportunity to make a change in the framework of matrix mechanics. In other words, Jordan felt that we would have been, at least temporarily, helpless had an unexpected disagreement occurred in the theory of the helium atom. This was, at that time, developed by Kellner and by Hilleraas. The mathematical formalism was too dear and unchangeable so that, had the miracle of helium which was mentioned before not occurred, a true crisis would have arisen. Surely, physics would have overcome that crisis in one way or another. It is true, on the other hand, that physics as we know it today would not be possible without a constant recurrence of miracles similar to the one of the helium atom, which is perhaps the most striking miracle that has occurred in the course of the development of elementary quantum mechanics, but by far not the only one. In fact, the number of analogous miracles is limited, in our view, only by our willingness to go after more similar ones. Quantum mechanics had, nevertheless, many almost equally striking successes which gave us the firm conviction that it is, what we call, correct.

The last example is that of quantum electrodynamics, or the theory of the Lamb shift. Whereas Newton's theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg's prescriptions. The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand.

The preceding three examples, which could be multiplied almost indefinitely, should illustrate the appropriateness and accuracy of the mathematical formulation of the laws of nature in terms of concepts chosen for their manipulability, the "laws of nature" being of almost fantastic accuracy but of strictly limited scope. I propose to refer to the observation which these examples illustrate as the empirical law of epistemology. Together with the laws of invariance of physical theories, it is an indispensable foundation of these theories. Without the laws of invariance the physical theories could have been given no foundation of fact; if the empirical law of epistemology were not correct, we would lack the encouragement and reassurance which are emotional necessities, without which the "laws of nature" could not have been successfully explored. Dr. R. G. Sachs, with whom I discussed the empirical law of epistemology, called it an article of faith of the theoretical physicist, and it is surely that. However, what he called our article of faith can be well supported by actual examples - many examples in addition to the three which have been mentioned.
THE UNIQUENESS OF THE THEORIES OF PHYSICS

The empirical nature of the preceding observation seems to me to be self-evident. It surely is not a "necessity of thought" and it should not be necessary, in order to prove this, to point to the fact that it applies only to a very small part of our knowledge of the inanimate world. It is absurd to believe that the existence of mathematically simple expressions for the second derivative of the position is Page 8 self-evident, when no similar expressions for the position itself or for the velocity exist. It is therefore surprising how readily the wonderful gift contained in the empirical law of epistemology was taken for granted. The ability of the human mind to form a string of 1000 conclusions and still remain "right," which was mentioned before, is a similar gift.

Every empirical law has the disquietire which will be discovered, will fuse into a single consistent unit, or at least asymptotically approach such a fusion. Alternatively, it is possible that there always will be some laws of nature which have nothing in common with each other. At present, this is true, for instance, of the laws of heredity and of physics. It is even possible that some of the laws of nature will be in conflict with each other in their implications, but each convincing enough in its own domain so that we may not be willing to abandon any of them. We may resign ourselves to such a state of affairs or our interest in clearing up the conflict between the various theories may fade out. We may lose interest in the "ultimate truth," that is, in a picture which is a consistent fusion into a single unit of the little pictures, formed on the various aspects of nature.

It may be useful to illustrate the alternatives by an example. We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity. These two theories have their roots in mutually exclusive groups of phenomena. Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding panicles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical concepts - the four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and of conflict, mentioned before, both of which are conceivable.

In order to obtain an indication as to which alternative to expect ultimately, we can pretend to be a little more ignorant than we are and place ourselves at a lower level of knowledge than we actually possess. If we can find a fusion of our theories on this lower level of intelligence, we can confidently expect that we will find a fusion of our theories also at our real level of intelligence. On the other hand, if we would arrive at mutually contradictory theories at a somewhat lower level of knowledge, the possibility of the permanence of conflicting theories cannot be excluded for ourselves either. The level of knowledge and ingenuity is a continuous variable and it is unlikely that a relatively small variation of this continuous variable changes the attainable picture of the world from inconsistent to consistent. [ This passage was written after a great deal of hesitation. The writer is convinced that it is useful, in epistemological discussions, to abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species. However, the writer also realizes that his thinking along the lines indicated in the text was too brief and not subject to sufficient critical appraisal to be reliable.]

Considered from this point of view, the fact that some of the theories which we know to be false give such amazingly accurate results is an adverse factor. Had we somewhat less knowledge, the group of phenomena which these "false" theories explain would appear to us to be large enough to "prove" these theories. However, these theories are considered to be "false" by us just for the reason that they are, in ultimate analysis, incompatible with more encompassing pictures and, if Page 9 sufficiently many such false theories are discovered, they are bound to prove also to be in conflict with each other. Similarly, it is possible that the theories, which we consider to be "proved" by a number of numerical agreements which appears to be large enough for us, are false because they are in conflict with a possible more encompassing theory which is beyond our means of discovery. If this were true, we would have to expect conflicts between our theories as soon as their number grows beyond a certain point and as soon as they cover a sufficiently large number of groups of phenomena. In contrast to the article of faith of the theoretical physicist mentioned before, this is the nightmare of the theorist.

Let us consider a few examples of "false" theories which give, in view of their falseness, alarmingly accurate descriptions of groups of phenomena. With some goodwill, one can dismiss some of the evidence which these examples provide. The success of Bohr's early and pioneering ideas on the atom was always a rather narrow one and the same applies to Ptolemy's epicycles. Our present vantage point gives an accurate description of all phenomena which these more primitive theories can describe. The same is not true any longer of the so-called free-electron theory, which gives a marvelously accurate picture of many, if not most, properties of metals, semiconductors, and insulators. In particular, it explains the fact, never properly understood on the basis of the "real theory," that insulators show a specific resistance to electricity which may be 10 26 times greater than that of metals. In fact, there is no experimental evidence to show that the resistance is not infinite under the conditions under which the free-electron theory would lead us to expect an infinite resistance. Nevertheless, we are convinced that the free-electron theory is a crude approximation which should be replaced, in the description of all phenomena concerning solids, by a more accurate picture.

If viewed from our real vantage point, the situation presented by the free-electron theory is irritating but is not likely to forebode any inconsistencies which are unsurmountable for us. The free-electron theory raises doubts as to how much we should trust numerical agreement between theory and experiment as evidence for the correctness of the theory. We are used to such doubts.

A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world. Mendel's laws of inheritance and the subsequent work on genes may well form the beginning of such a theory as far as biology is concerned. Furthermore,, it is quite possible that an abstract argument can be found which shows that there is a conflict between such a theory and the accepted principles of physics. The argument could be of such abstract nature that it might not be possible to resolve the conflict, in favor of one or of the other theory, by an experiment. Such a situation would put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called "the ultimate truth." The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. Indeed, it is this writer's belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted.

Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Study: Brain games don't make you smarter

Study: Brain games don't make you smarter
AP
By MARIA CHENG, AP Medical Writer Maria Cheng, Ap Medical Writer – 22 mins ago

LONDON – People playing computer games to train their brains might as well be playing Super Mario, new research suggests.

In a six-week study, experts found people who played online games designed to improve their cognitive skills didn't get any smarter.

Researchers recruited participants from viewers of the BBC's science show "Bang Goes the Theory." More than 8,600 people aged 18 to 60 were asked to play online brain games designed by the researchers to improve their memory, reasoning and other skills for at least 10 minutes a day, three times a week.

They were compared to more than 2,700 people who didn't play any brain games, but spent a similar amount of time surfing the Internet and answering general knowledge questions. All participants were given a sort of I.Q. test before and after the experiment.

Researchers said the people who did the brain training didn't do any better on the test after six weeks than people who had simply been on the Internet. On some sections of the test, the people who surfed the Net scored higher than those playing the games.

The study was paid for by the BBC and published online Tuesday by the journal Nature.

"If you're (playing these games) because they're fun, that's absolutely fine," said Adrian Owen, assistant director of the Cognition and Brain Sciences unit at Britain's Medical Research Council, the study's lead author. "But if you're expecting (these games) to improve your I.Q., our data suggests this isn't the case," he said during a press briefing on Tuesday.

One maker of brain games said the BBC study did not apply to its products. Steve Aldrich, CEO of Posit Science, said the company's games, some of which were funded in part by the U.S. National Institutes of Health, have been proven to boost brain power.

"Their conclusion would be like saying, 'I cannot run a mile in under 4 minutes and therefore it is impossible to do so," Aldrich said.

Posit Science has published research in journals including the Proceedings of the National Academy of Sciences showing their games improved memory in older people.

Computer games available online and marketed by companies like Nintendo that supposedly enhance memory, reasoning and other cognitive skills are played by millions of people worldwide, though few studies have examined if the games work.

"There is precious little evidence to suggest the skills used in these games transfer to the real world," said Art Kramer, a professor of psychology and neuroscience at the University of Illinois. He was not linked to the study and has no ties to any companies that make brain training games.

Kramer had several reservations about the BBC study's methodology and said some brain games had small effects in improving people's cognitive skills. "Learning is very specific," he said. "Unless the component you are trained in actually exists in the real world, any transfer will be pretty minimal."

Instead of playing brain games, Kramer said people would be better off getting some exercise. He said physical activity can spark new connections between neurons and produce new brain cells. "Fitness changes the building blocks of the brain's structure," he said.

Still, Kramer said some brain training games worked better than others. He said some games made by Posit Science had shown modest benefits, including improved memory in older people.

Other experts said brain games might be useful, but only if they weren't fun.

"If you set the level for these games to a very high level where you don't get the answers very often and it really annoys you, then it may be useful," said Philip Adey, an emeritus professor of psychology and neuroscience at King's College in London.

If people are enjoying the brain games, Adey said they probably aren't being challenged and might as well be playing a regular video game.

He said people should consider learning a new language or sport if they really wanted to improve their brain power. "To stimulate the intellect, you need a real challenge," Adey said, adding computer games were not an easy shortcut. "Getting smart is hard work."

The Science of Generosity

The Science of Generosity
By Paul J. Zak
Would you help this child?
Published on November 22, 2009
I just finished reading Theodore Malloch's wonderful new book Being Generous (Templeton Press, 2009) that investigates the reasons for and results of generosity. The book draws on a variety of evidence to show that generosity is not only good for society, but good for the individual. Throughout this inspiring book, pithy and interesting one page biographies appear of well-known givers and their motivations for helping others. These range from Johann Sebastian Bach, John D. Rockefeller, and Mother Theresa to Bill and Melinda Gates.
Giving USA reports that in 2005, individuals in the US gave $199 billion dollars to charity. In the same year, 65 million Americans spent an average of 50 hours volunteering to help others. Using the average US hourly wage, this constitutes an additional charitable donation valued at $60 billion. While this pales in comparison to the federal deficit, $259 billion is a big chunk of change. Could science explain this extraordinary generosity?

Bottom of Form
My lab has been investigating the biological basis for generosity, focusing on the neuroactive hormone oxytocin. We were specifically interested in generosity, or "liberality in giving," rather than people simply giving small gifts to others. Many people have an urge to give just a bit, but we wanted to know why someone would ever give more than they had to. We used a task called the Ultimatum Game in which people are randomly and anonymously paired by computer in a large lab. After extensive instruction and without a speck of deception, people are endowed with a sum of money like $40 and then asked to propose a split of this money to the other person in their pair. No communication before or after the proposal is allowed. The receiver then decides if s/he wants to accept the proposal or reject it. If accepted, the money is paid privately to each person and the experiment ends. But, if the proposal is rejected, both individuals earn nothing.
How much would you offer as a split? In Western countries, offers less than 30% of the endowment are nearly always rejected. Why? Easy--it is simply unfair. We turned this question on its head: why would anyone offer more than one needs to have the offer accepted? We did this by having each person make decisions both as proposer and to identify their smallest acceptable offer as responder. Later, we randomized which role they would actually play and this determined their earnings. Generosity is the difference between what one offers and the smallest amount one is willing to accept.
I had a hunch that oxytocin, which I had already shown causes us to trust others as I discussed in a recent article in Scientific American would also make people generous. So, we infused 40IU oxytocin into half the participants using a nasal spray, and similarly administered salt water to the other half, without them knowing which one they had gotten. They then made decisions in the Ultimatum Game. In a 2007 publication, my team reported that oxytocin increased generosity by 80% compared to the placebo group.
This was a huge effect in an experiment where we tormented people by putting two teaspoons of liquid up their noses. The next question was why oxytocin caused generosity.
Giving to others is often prompted by understanding their perspective. How would you feel if you lost your house to a hurricane or fire, or found yourself homeless after looking for work for a year. We can image how awful these situations would be and this motivates us to help others. Shortly after the August 2005 hurricane Katrina disaster I asked my lab who had donated money to the relief efforts. Several students raised their hands and when I asked them why, most related highly emotional stories of suffering they had seen on TV. The stories were often so emotion-laden that their eyes teared up on the telling.
This gave me and my graduate student Jorge Barraza an idea to run an experiment that simulated this effect. We had participants watch one of two 100 second videos. Both videos feature a father with his four year old son. The son is bald from chemotherapy due to his terminal brain cancer. In the emotional video, the father discussion how it feels to know his son is dying. In the neutral video, the father and son are having a day at the zoo and cancer and death are not mentioned. You can see the video in an earlier PT Blog I wrote. I showed the emotional video recently to group of lawyers at a conference and one-third of them cried so much I had to stop my lecture. If it makes lawyers cry, you know that regular humans are really affected by it.
We drew blood before and after people watched one of the two videos and found that doing nothing more watching the emotional video produced a huge 157% spike in oxytocin levels. Oxytocin levels actually fell for those who watched the neutral video. We then asked people how they felt after seeing the videos. For the emotional video, the change in oxytocin was correlated with feelings of empathy (after we controlled for the distress people reported that correlated with the stress hormone cortisol). Oxytocin connects us to others and lets us understand their emotions.
The most amazing part was that after the videos people made decisions in the Ultimatum Game so we could see if empathic engagement would make people more generous towards another person in the lab. It did. Generosity towards another meant that the giver earned less money for his or her participation in this long and unpleasant experiment.
As participants were leaving the experiment, we also gave them a chance to donate some of their earnings to charity. One-third of the participants did so, averaging a six dollar donation (this was about one-quarter of the average earnings). Who donated? Those who were the most generous and most empathically engaged by the video.
It may very well be the case that those profiled in Being Generous release more oxytocin than others and this partially explains their generosity. Oxytocin connects us to others and social connections are a powerful way to increase one's own happiness. If you want to connect to others, being generous is a great start. You can follow Malloch in this--he is donating all book royalties to the charity portal Global Giving. If you would like to choose a project to donate to, go to www.globalgiving. com. You just might feel the joy of generosity.
---Paul J. Zak is a neuroeconomist and director of the Center for Neuroeconomics Studies at Claremont Graduate University in Claremont, CA
(Thanks for Anand Damma of Keltronians yahoo group)

Calcutta if you must exile me wound my lips before I go

Calcutta if you must exile me wound my lips before I go

only words remain and the gentle touch of your finger on my lips Calcutta
burn my eyes before I go into the night

the headless corpse in a Dhakuria bylane the battered youth his brains blown
out and the silent vigil that takes you to Pataldanga Lane where they
will gun you down without vengeance or hate

Calcutta if you must exile me burn my eyes before I go

they will pull you down from the Ochterlony monument and torture each broken
rib beneath your upthrust breasts they will tear the anguish from
your sullen eyes and thrust the bayonet between your thighs

Calcutta they will tear you apart Jarasandha-like
they will tie your hands on either side and hang you from a wordless cross
and when your silence protests they will execute all the words that
you met and synchronised Calcutta they will burn you at the stake

Calcutta flex the vengeance in your thighs and burn silently in the despair
of flesh
if you feel like suicide take a rickshaw to Sonagachhi and share the sullen
pride in the eyes of women who have wilfully died

wait for me outside the Ujjala theatre and I will bring you the blood of that
armless leper who went mad before hunger and death met in his wounds

I will show you the fatigue of that woman who died near Chitpur out of sheer
boredom and the cages of Burrabazar where passion hides in the
wrinkles of virgins who have aged waiting for a sexless war that
never came
only obscene lust remains in their eyes after time has wintered their
exacting thighs and I will show you the hawker who died with Calcutta
in his eyes
Calcutta if you must exile me destroy my sanity before I go

How 1 Autistic Young Man Runs a Business

How 1 Autistic Young Man Runs a Business

U.S. News & World Report

By Nancy Shute Nancy Shute – Fri Apr 3, 12:46 pm ET

Joe Steffy is off to Overland Park, Kan., this week to do a PowerPoint presentation on his business, Poppin' Joe's Kettle Korn . He's a 23-year-old small-business man with a goal of $100,000 in sales by 2012. Joe also has autism and Down syndrome and is nonverbal. When he gives his talk, he will push buttons on an augmentative speech device to deliver the words. His audience will be parents who fervently hope their own special-needs children will be able to work, too.

Joe's parents, Ray and Janet, didn't agree with the school district assessment in their home town of Louisburg, Kan., that said Joe would never be able to work or live independently. "I'm one who can easily get ticked off," says Ray. "That ticked me off. We saw more in Joe than that. We set out to prove to the school that he had capabilities." They came across kettle corn while on a trip to Alaska and realized that all that popping, scooping, and serving suited Joe's love of work.

The path to Joe Steffy's success was not an easy one; Ray Steffy worked closely with Dave Hammis, an advocate for self-employment for people with disabilities in Middletown, Ohio, who trains business owners, government employees, and parents on how to make use of state and federal programs. The Steffys wrote up a business plan and helped Joe secure $25,000 in grants from programs like Social Security Administration's Plan to Achieve Self-Support program (PASS).

In 2005, Poppin' Joe's Kettle Korn was born. Sales have grown from $16,000 in 2005 to $50,000 in 2008, both from selling at festivals and from delivering popcorn to local outlets. Joe has five part-time employees, and his parents help out with driving and other tasks. "Pop and everyone that works with him knows whatever Joe wants to do you let him do, because he's the boss," Ray says. "If he wants to pop, he'll shove Dad out of the way and pop."

If the business stays on track, it should be grossing more than $100,000 in three years, and the Steffys are seeking a business partner who can work with Joe to manage the business. Joe is no longer on Social Security disability payments; instead, he pays state sales tax and state and federal income tax. He rents his own house and is helped by caregivers who are paid by a state program.

"It's been hard work, from the standpoint of physical work," says Ray Steffy, who is 67. "But a parent with a child like Joe has a choice. You can either kick in and do this kind of thing, or you can sit and fret emotionally with the amount of energy, worrying about what's going to happen to them."

The payoff for that effort, as far as the Steffys are concerned, has been priceless. They see their son make a local popcorn delivery, accept payment, fold it, and put it in his pocket. When he walks out, his dad says, Joe looks 3 inches taller than when he walked in.

TEN tips for 2009.

TEN tips for 2009.
1) Do not get into trouble.

2.) Aim for greater heights.

3) Stay focused on your job.

4) Exercise to maintain good health.

5) Practice Team work,

6) Rely on your trusted partner to watch your back.

7) Save for rainy days.

8) Rest and relax.

9) Always smile when your boss is around.

10) Nothing is impossible.

Warren buffets advice for 2009

Warren buffets advice for 2009

We begin this New Year with dampened enthusiasm and dented optimism. Our happiness is diluted and our peace is threatened by the financial illness that has infected our families, organizations and nations. Everyone is desperate to find a remedy that will cure their financial illness and help them recover their financial health. They expect the financial experts to provide them with remedies, forgetting the fact that it is these experts who created this financial mess.

Every new year, I adopt a couple of old maxims as my beacons to guide my future. This self-prescribed therapy has ensured that with each passing year, I grow wiser and not older. This year, I invite you to tap into the financial wisdom of our elders along with me, and become financially wiser.

* Hard work: All hard work bring a profit, but mere talk leads only to poverty.

* Laziness: A sleeping lobster is carried away by the water current.

* Earnings: Never depend on a single source of income. [ At least make your Investments get you second earning ]

* Spending: If you buy things you don't need, you'll soon sell things you need.

* Savings: Don't save what is left after spending; Spend what is left after saving.

* Borrowings: The borrower becomes the lender's slave.

* Accounting: It's no use carrying an umbrella, if your shoes are leaking.

* Auditing: Beware of little expenses; A small leak can sink a large ship.

* Risk-taking: Never test the depth of the river with both feet. [ Have an alternate plan ready ]

* Investment: Don't put all your eggs in one basket.

I'm certain that those who have already been practicing these principles remain financially healthy. I'm equally confident that
those who resolve to start practicing these principles will quickly regain their financial health.

Let us become wiser and lead a happy, healthy, prosperous and peaceful life..

Keep the Spark Alive

(Thanks N.Krishnakumar of Keltron for sending this)

Keep the Spark Alive

Inaugural Speech for the new batch at the Symbiosis BBA program, Pune 23rd June, 2008

By Chetan Bhagat

Good Morning everyone and thank you for giving me this chance to speak to you. This day is about you. You, who have come to this college, leaving the comfort of your homes (or in some cases discomfort), to become something in your life. I am sure you are excited. There are few days in human life when one is truly elated. The first day in college is one of them. When you were getting ready today, you felt a tingling in your stomach. What would the auditorium be like, what would the teachers be like, who are my new classmates - there is so much to be curious about. I call this excitement, the spark within you that makes you feel truly alive today. Today I am going to talk about keeping the spark shining. Or to put it another way, how to be happy most, if not all the time.
Where do these sparks start? I think we are born with them. My 3-year old twin boys have a million sparks. A little Spiderman toy can make them jump on the bed. They get thrills from creaky swings in the park. A story from daddy gets them excited. They do a daily countdown for birthday party – several months in advance – just for the day they will cut their own birthday cake.
I see students like you, and I still see some sparks. But when I see older people, the spark is difficult to find. That means as we age, the spark fades. People whose spark has faded too much are dull, dejected, aimless and bitter. Remember Kareena in the first half of Jab We Met vs the second half? That is what happens when the spark is lost. So how to save the spark?
Imagine the spark to be a lamp's flame. The first aspect is nurturing - to give your spark the fuel, continuously. The second is to guard against storms.
To nurture, always have goals. It is human nature to strive, improve and achieve full potential. In fact, that is success. It is what is possible for you. It isn't any external measure - a certain cost to company pay package, a particular car or house.
Most of us are from middle class families. To us, having material landmarks is success and rightly so. When you have grown up where money constraints force everyday choices, financial freedom is a big achievement. But it isn't the purpose of life. If that was the case, Mr. Ambani would not show up for work. Shah Rukh Khan would stay at home and not dance anymore. Steve Jobs won't be working hard to make a better iPhone, as he sold Pixar for billions of dollars already. Why do they do it? What makes them come to work everyday? They do it because it makes them happy. They do it because it makes them feel alive. Just getting better from current levels feels good. If you study hard, you can improve your rank. If you make an effort to interact with people, you will do better in interviews. If you practice, your cricket will get better. You may also know that you cannot become Tendulkar, yet. But you can get to the next level. Striving for that next level is important.
Nature designed with a random set of genes and circumstances in which we were born. To be happy, we have to accept it and make the most of nature's design. Are you? Goals will help you do that.
I must add, don't just have career or academic goals. Set goals to give you a balanced, successful life. I use the word balanced before successful. Balanced means ensuring your health, relationships, mental peace are all in good order.
There is no point of getting a promotion on the day of your breakup. There is no fun in driving a car if your back hurts. Shopping is not enjoyable if your mind is full of tensions.
You must have read some quotes - Life is a tough race, it is a marathon or whatever. No, from what I have seen so far, life is one of those races in nursery school, where you have to run with a marble in a spoon kept in your mouth. If the marble falls, there is no point coming first. Same with life, where health and relationships are the marble. Your striving is only worth it if there is harmony in your life. Else, you may achieve the success, but this spark, this feeling of being excited and alive, will start to die.

One last thing about nurturing the spark - don't take life seriously. One of my yoga teachers used to make students laugh during classes. One student asked him if these jokes would take away something from the yoga practice. The teacher said - don't be serious, be sincere. This quote has defined my work ever since. Whether its my writing, my job, my relationships or any of my goals. I get thousands of opinions on my writing everyday. There is heaps of praise, there is intense criticism. If I take it all seriously, how will I write? Or rather, how will I live? Life is not to be taken seriously, as we are really temporary here. We are like a pre-paid card with limited validity. If we are lucky, we may last another 50 years. And 50 years is just 2,500 weekends. Do we really need to get so worked up? It's ok, bunk a few classes, goof up a few interviews, fall in love. We are people, not programmed devices.
I've told you three things - reasonable goals, balance and not taking it too seriously that will nurture the spark. However, there are four storms in life that will threaten to completely put out the flame. These must be guarded against. These are disappointment, frustration, unfairness and loneliness of purpose.
Disappointment will come when your effort does not give you the expected return. If things don't go as planned or if you face failure. Failure is extremely difficult to handle, but those that do come out stronger. What did this failure teach me? is the question you will need to ask. You will feel miserable. You will want to quit, like I wanted to when nine publishers rejected my first book. Some IITians kill themselves over low grades – how silly is that? But that is how much failure can hurt you. But it's life. If challenges could always be overcome, they would cease to be a challenge. And remember - if you are failing at something, that means you are at your limit or potential. And that's where you want to be.
Disappointment's cousin is frustration, the second storm. Have you ever been frustrated? It happens when things are stuck. This is especially relevant in India. From traffic jams to getting that job you deserve, sometimes things take so long that you don't know if you chose the right goal. After books, I set the goal of writing for Bollywood, as I thought they needed writers. I am called extremely lucky, but it took me five years to get close to a release. Frustration saps excitement, and turns your initial energy into something negative, making you a bitter person. How did I deal with it? A realistic assessment of the time involved – movies take a long time to make even though they are watched quickly, seeking a certain enjoyment in the process rather than the end result – at least I was learning how to write scripts, having a side plan – I had my third book to write and even something as simple as pleasurable distractions in your life - friends, food, travel can help you overcome it. Remember, nothing is to be taken seriously. Frustration is a sign somewhere, you took it too seriously.
Unfairness - this is hardest to deal with, but unfortunately that is how our country works. People with connections, rich dads, beautiful faces, pedigree find it easier to make it – not just in Bollywood, but everywhere. And sometimes it is just plain luck. There are so few opportunities in India, so many stars need to be aligned for you to make it happen. Merit and hard work is not always linked to achievement in the short term, but the long term correlation is high, and ultimately things do work out. But realize, there will be some people luckier than you. In fact, to have an opportunity to go to college and understand this speech in English means you are pretty damm lucky by Indian standards. Let's be grateful for what we have and get the strength to accept what we don't. I have so much love from my readers that other writers cannot even imagine it. However, I don't get literary praise. It's ok. I don't look like Aishwarya Rai, but I have two boys who I think are more beautiful than her. It's ok. Don't let unfairness kill your spark.
Finally, the last point that can kill your spark is isolation. As you grow older you will realize you are unique. When you are little, all kids want Ice cream and Spiderman. As you grow older to college, you still are a lot like your friends. But ten years later and you realize you are unique. What you want, what you believe in, what makes you feel, may be different from even the people closest to you. This can create conflict as your goals may not match with others. . And you may drop some of them. Basketball captains in college invariably stop playing basketball by the time they have their second child. They give up something that meant so much to them. They do it for their family. But in doing that, the spark dies. Never, ever make that compromise. Love yourself first, and then others.
There you go. I've told you the four thunderstorms - disappointment, frustration, unfairness and isolation. You cannot avoid them, as like the monsoon they will come into your life at regular intervals. You just need to keep the raincoat handy to not let the spark die.
I welcome you again to the most wonderful years of your life. If someone gave me the choice to go back in time, I will surely choose college. But I also hope that ten years later as well, your eyes will shine the same way as they do today. That you will Keep the Spark alive, not only through college, but through the next 2,500 weekends. And I hope not just you, but my whole country will keep that spark alive, as we really need it now more than any moment in history. And there is something cool about saying - I come from the land of a billion sparks.
Thank You!

Text of President Barack Obama's inaugural address

By The Associated Press The Associated Press Tue Jan 20, 5:04 pm ET

Text of President Barack Obama's inaugural address on Tuesday, as delivered.

OBAMA: My fellow citizens:

I stand here today humbled by the task before us, grateful for the trust you have bestowed, mindful of the sacrifices borne by our ancestors. I thank President Bush for his service to our nation, as well as the generosity and cooperation he has shown throughout this transition.

Forty-four Americans have now taken the presidential oath. The words have been spoken during rising tides of prosperity and the still waters of peace. Yet, every so often the oath is taken amidst gathering clouds and raging storms. At these moments, America has carried on not simply because of the skill or vision of those in high office, but because we the people have remained faithful to the ideals of our forebears, and true to our founding documents.

So it has been. So it must be with this generation of Americans.

That we are in the midst of crisis is now well understood. Our nation is at war, against a far-reaching network of violence and hatred. Our economy is badly weakened, a consequence of greed and irresponsibility on the part of some, but also our collective failure to make hard choices and prepare the nation for a new age. Homes have been lost; jobs shed; businesses shuttered. Our health care is too costly; our schools fail too many; and each day brings further evidence that the ways we use energy strengthen our adversaries and threaten our planet.

These are the indicators of crisis, subject to data and statistics. Less measurable but no less profound is a sapping of confidence across our land — a nagging fear that America's decline is inevitable, and that the next generation must lower its sights.

Today I say to you that the challenges we face are real. They are serious and they are many. They will not be met easily or in a short span of time. But know this, America — they will be met.

On this day, we gather because we have chosen hope over fear, unity of purpose over conflict and discord.

On this day, we come to proclaim an end to the petty grievances and false promises, the recriminations and worn out dogmas, that for far too long have strangled our politics.

We remain a young nation, but in the words of Scripture, the time has come to set aside childish things. The time has come to reaffirm our enduring spirit; to choose our better history; to carry forward that precious gift, that noble idea, passed on from generation to generation: the God-given promise that all are equal, all are free and all deserve a chance to pursue their full measure of happiness.

In reaffirming the greatness of our nation, we understand that greatness is never a given. It must be earned. Our journey has never been one of shortcuts or settling for less. It has not been the path for the faint-hearted — for those who prefer leisure over work, or seek only the pleasures of riches and fame. Rather, it has been the risk-takers, the doers, the makers of things — some celebrated but more often men and women obscure in their labor, who have carried us up the long, rugged path towards prosperity and freedom.

For us, they packed up their few worldly possessions and traveled across oceans in search of a new life.

For us, they toiled in sweatshops and settled the West; endured the lash of the whip and plowed the hard earth.

For us, they fought and died, in places like Concord and Gettysburg; Normandy and Khe Sanh.

Time and again these men and women struggled and sacrificed and worked till their hands were raw so that we might live a better life. They saw America as bigger than the sum of our individual ambitions; greater than all the differences of birth or wealth or faction.

This is the journey we continue today. We remain the most prosperous, powerful nation on Earth. Our workers are no less productive than when this crisis began. Our minds are no less inventive, our goods and services no less needed than they were last week or last month or last year. Our capacity remains undiminished. But our time of standing pat, of protecting narrow interests and putting off unpleasant decisions — that time has surely passed. Starting today, we must pick ourselves up, dust ourselves off, and begin again the work of remaking America.

For everywhere we look, there is work to be done. The state of the economy calls for action, bold and swift, and we will act — not only to create new jobs, but to lay a new foundation for growth. We will build the roads and bridges, the electric grids and digital lines that feed our commerce and bind us together. We will restore science to its rightful place, and wield technology's wonders to raise health care's quality and lower its cost. We will harness the sun and the winds and the soil to fuel our cars and run our factories. And we will transform our schools and colleges and universities to meet the demands of a new age. All this we can do. All this we will do.

Now, there are some who question the scale of our ambitions — who suggest that our system cannot tolerate too many big plans. Their memories are short. For they have forgotten what this country has already done; what free men and women can achieve when imagination is joined to common purpose, and necessity to courage.

What the cynics fail to understand is that the ground has shifted beneath them — that the stale political arguments that have consumed us for so long no longer apply. The question we ask today is not whether our government is too big or too small, but whether it works — whether it helps families find jobs at a decent wage, care they can afford, a retirement that is dignified. Where the answer is yes, we intend to move forward. Where the answer is no, programs will end. Those of us who manage the public's dollars will be held to account — to spend wisely, reform bad habits, and do our business in the light of day — because only then can we restore the vital trust between a people and their government.

Nor is the question before us whether the market is a force for good or ill. Its power to generate wealth and expand freedom is unmatched, but this crisis has reminded us that without a watchful eye, the market can spin out of control — and that a nation cannot prosper long when it favors only the prosperous. The success of our economy has always depended not just on the size of our gross domestic product, but on the reach of our prosperity; on our ability to extend opportunity to every willing heart — not out of charity, but because it is the surest route to our common good.

As for our common defense, we reject as false the choice between our safety and our ideals. Our founding fathers ... our found fathers, faced with perils we can scarcely imagine, drafted a charter to assure the rule of law and the rights of man, a charter expanded by the blood of generations. Those ideals still light the world, and we will not give them up for expedience's sake. And so to all the other peoples and governments who are watching today, from the grandest capitals to the small village where my father was born: know that America is a friend of each nation and every man, woman, and child who seeks a future of peace and dignity, and that we are ready to lead once more.

Recall that earlier generations faced down fascism and communism not just with missiles and tanks, but with sturdy alliances and enduring convictions. They understood that our power alone cannot protect us, nor does it entitle us to do as we please. Instead, they knew that our power grows through its prudent use; our security emanates from the justness of our cause, the force of our example, the tempering qualities of humility and restraint.

We are the keepers of this legacy. Guided by these principles once more, we can meet those new threats that demand even greater effort — even greater cooperation and understanding between nations. We will begin to responsibly leave Iraq to its people, and forge a hard-earned peace in Afghanistan. With old friends and former foes, we will work tirelessly to lessen the nuclear threat, and roll back the specter of a warming planet. We will not apologize for our way of life, nor will we waver in its defense, and for those who seek to advance their aims by inducing terror and slaughtering innocents, we say to you now that our spirit is stronger and cannot be broken; you cannot outlast us, and we will defeat you.

For we know that our patchwork heritage is a strength, not a weakness. We are a nation of Christians and Muslims, Jews and Hindus — and non-believers. We are shaped by every language and culture, drawn from every end of this Earth; and because we have tasted the bitter swill of civil war and segregation, and emerged from that dark chapter stronger and more united, we cannot help but believe that the old hatreds shall someday pass; that the lines of tribe shall soon dissolve; that as the world grows smaller, our common humanity shall reveal itself; and that America must play its role in ushering in a new era of peace.

To the Muslim world, we seek a new way forward, based on mutual interest and mutual respect. To those leaders around the globe who seek to sow conflict, or blame their society's ills on the West — know that your people will judge you on what you can build, not what you destroy. To those who cling to power through corruption and deceit and the silencing of dissent, know that you are on the wrong side of history; but that we will extend a hand if you are willing to unclench your fist.

To the people of poor nations, we pledge to work alongside you to make your farms flourish and let clean waters flow; to nourish starved bodies and feed hungry minds. And to those nations like ours that enjoy relative plenty, we say we can no longer afford indifference to the suffering outside our borders; nor can we consume the world's resources without regard to effect. For the world has changed, and we must change with it.

As we consider the road that unfolds before us, we remember with humble gratitude those brave Americans who, at this very hour, patrol far-off deserts and distant mountains. They have something to tell us, just as the fallen heroes who lie in Arlington whisper through the ages. We honor them not only because they are guardians of our liberty, but because they embody the spirit of service; a willingness to find meaning in something greater than themselves. And yet, at this moment — a moment that will define a generation — it is precisely this spirit that must inhabit us all.

For as much as government can do and must do, it is ultimately the faith and determination of the American people upon which this nation relies. It is the kindness to take in a stranger when the levees break, the selflessness of workers who would rather cut their hours than see a friend lose their job which sees us through our darkest hours. It is the firefighter's courage to storm a stairway filled with smoke, but also a parent's willingness to nurture a child, that finally decides our fate.

Our challenges may be new. The instruments with which we meet them may be new. But those values upon which our success depends — hard work and honesty, courage and fair play, tolerance and curiosity, loyalty and patriotism — these things are old. These things are true. They have been the quiet force of progress throughout our history. What is demanded then is a return to these truths. What is required of us now is a new era of responsibility — a recognition, on the part of every American, that we have duties to ourselves, our nation, and the world, duties that we do not grudgingly accept but rather seize gladly, firm in the knowledge that there is nothing so satisfying to the spirit, so defining of our character, than giving our all to a difficult task.

This is the price and the promise of citizenship.

This is the source of our confidence — the knowledge that God calls on us to shape an uncertain destiny.

This is the meaning of our liberty and our creed — why men and women and children of every race and every faith can join in celebration across this magnificent Mall, and why a man whose father less than sixty years ago might not have been served at a local restaurant can now stand before you to take a most sacred oath.

So let us mark this day with remembrance, of who we are and how far we have traveled. In the year of America's birth, in the coldest of months, a small band of patriots huddled by dying campfires on the shores of an icy river. The capital was abandoned. The enemy was advancing. The snow was stained with blood. At a moment when the outcome of our revolution was most in doubt, the father of our nation ordered these words be read to the people:

"Let it be told to the future world ... that in the depth of winter, when nothing but hope and virtue could survive...that the city and the country, alarmed at one common danger, came forth to meet (it)."

America, in the face of our common dangers, in this winter of our hardship, let us remember these timeless words. With hope and virtue, let us brave once more the icy currents, and endure what storms may come. Let it be said by our children's children that when we were tested we refused to let this journey end, that we did not turn back nor did we falter; and with eyes fixed on the horizon and God's grace upon us, we carried forth that great gift of freedom and delivered it safely to future generations.

Thank you. God bless you. And God bless the United States of America.

Clinton vows robust diplomacy as State Dept chief

WASHINGTON – Hillary Rodham Clinton took charge of the State Department on Thursday, proclaiming the start of a new era of robust U.S. diplomacy to tackle the world's crises and improve America's standing abroad.

Before a raucous, cheering crowd of about 1,000 people, the nation's 67th secretary of state pledged to boost the morale and resources of the diplomatic corps and promised them a difficult but exciting road ahead.

"I believe with all of my heart that this is a new era for America," she said to loud applause in the main lobby of the department's headquarters, which President Barack Obama visited later in the day to underscore his administration's commitment to diplomacy.

With Obama at her side in the ornate Ben Franklin Room, Clinton introduced former Senate Majority Leader George J. Mitchell, D-Maine, as a special envoy for the Middle East. Former U.N. ambassador Richard Holbrooke was announced as a special adviser on Afghanistan and Pakistan.

The posts are the first of several new special envoys the administration plans to name to deal with particularly vexing problems abroad.

Clinton began her first day on the job at the State Department one day after her Senate confirmation.

"This is going to be a challenging time and it will require 21st century tools and solutions to meet our problems and seize our opportunities," Clinton said at her welcoming. "I'm going to be asking a lot of you. I want you to think outside the proverbial box. I want you to give me the best advice you can."

"I want you to understand there is nothing that I welcome more than a good debate and the kind of dialogue that will make us better," she said. "We cannot be our best if we don't demand that from ourselves and each other."

In her spirited 10-minute pep talk, she spoke of the importance of defense, diplomacy and development — the "three legs to the stool of American foreign policy" — and noted that the State Department is in charge of two of them.

"We are responsible for two of the three legs," said the former New York senator and first lady. "And we will make clear as we go forward that diplomacy and development are essential tools in achieving the long-term objectives of the United States."

Clinton's mandate from Obama is to step up diplomatic efforts and restore the nation's tattered image overseas. She has vowed to make use of "smart power" to deal with international challenges.

"At the heart of smart power are smart people, and you are those people," she told the assembled throng. "And you are the ones that we will count on and turn to for the advice and counsel, the expertise and experience to make good on the promises of this new administration."

Clinton takes over an agency that was often sidelined during George W. Bush's eight-year presidency, particularly in his first term over the decision to go to war in Iraq. Although former Secretary of State Condoleezza Rice restored some of the department's influence, diplomats still complained of a lack of access to the top, as well as funding.

In introductory remarks, Steve Kashkett, vice president of the union that represents diplomats, noted that Obama and Clinton had both "decried the neglect that the foreign service and the State Department as a whole have suffered in recent years."

Clinton, meanwhile, sought to reassure frustrated diplomats that they will be heard.

"This is a team, and you are the members of that team," she said. "We are not any longer going to tolerate the kind of divisiveness that has paralyzed and undermined our ability to get things done for America."

She predicted her team would experience "a great adventure. We'll have some ups and some downs. We'll face some obstacles along the way. But be of good cheer and be of strong heart, and do not grow weary as we attempt to do good on behalf of our country and the world. ... And now, ladies and gentlemen, let's get to work."

'What I Want for You — and Every Child
in America'

By President-elect Barack Obama

Publication Date: 01/14/2009

Cover Photo By Kwaku Alston/Corbis Barack and Michelle Obama with daughters Sasha, 7, and Malia, 10.

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Next Tuesday, Barack Obama will be sworn in as our 44th President. On this historic occasion, PARADE asked the President-elect, who is also a devoted family man, to get personal and tell us what he wants for his children. Here, he shares his letter to them.


Dear Malia and Sasha,

I know that you've both had a lot of fun these last two years on the campaign trail, going to picnics and parades and state fairs, eating all sorts of junk food your mother and I probably shouldn't have let you have. But I also know that it hasn't always been easy for you and Mom, and that as excited as you both are about that new puppy, it doesn't make up for all the time we've been apart. I know how much I've missed these past two years, and today I want to tell you a little more about why I decided to take our family on this journey.

When I was a young man, I thought life was all about me-about how I'd make my way in the world, become successful, and get the things I want. But then the two of you came into my world with all your curiosity and mischief and those smiles that never fail to fill my heart and light up my day. And suddenly, all my big plans for myself didn't seem so important anymore. I soon found that the greatest joy in my life was the joy I saw in yours. And I realized that my own life wouldn't count for much unless I was able to ensure that you had every opportunity for happiness and fulfillment in yours. In the end, girls, that's why I ran for President: because of what I want for you and for every child in this nation.

I want all our children to go to schools worthy of their potential-schools that challenge them, inspire them, and instill in them a sense of wonder about the world around them. I want them to have the chance to go to college-even if their parents aren't rich. And I want them to get good jobs: jobs that pay well and give them benefits like health care, jobs that let them spend time with their own kids and retire with dignity.

I want us to push the boundaries of discovery so that you'll live to see new technologies and inventions that improve our lives and make our planet cleaner and safer. And I want us to push our own human boundaries to reach beyond the divides of race and region, gender and religion that keep us from seeing the best in each other.

Sometimes we have to send our young men and women into war and other dangerous situations to protect our country-but when we do, I want to make sure that it is only for a very good reason, that we try our best to settle our differences with others peacefully, and that we do everything possible to keep our servicemen and women safe. And I want every child to understand that the blessings these brave Americans fight for are not free-that with the great privilege of being a citizen of this nation comes great responsibility.

Sasha (l) and Malia Obama at play in New Hampshire in 2007.

Bumper cars at the Iowa State Fair in August 2007.

That was the lesson your grandmother tried to teach me when I was your age, reading me the opening lines of the Declaration of Independence and telling me about the men and women who marched for equality because they believed those words put to paper two centuries ago should mean something.

She helped me understand that America is great not because it is perfect but because it can always be made better-and that the unfinished work of perfecting our union falls to each of us. It's a charge we pass on to our children, coming closer with each new generation to what we know America should be.

I hope both of you will take up that work, righting the wrongs that you see and working to give others the chances you've had. Not just because you have an obligation to give something back to this country that has given our family so much-although you do have that obligation. But because you have an obligation to yourself. Because it is only when you hitch your wagon to something larger than yourself that you will realize your true potential.

These are the things I want for you-to grow up in a world with no limits on your dreams and no achievements beyond your reach, and to grow into compassionate, committed women who will help build that world. And I want every child to have the same chances to learn and dream and grow and thrive that you girls have. That's why I've taken our family on this great adventure.

I am so proud of both of you. I love you more than you can ever know. And I am grateful every day for your patience, poise, grace, and humor as we prepare to start our new life together in the White House.

Love, Dad