THE UNREASONABLE W. HAMMING is evident from the title that this is a a - PDF document

THE UNREASONABLE W. HAMMING is evident from the title that this is a apologize for the philosophy, am well aware that most scientists, engineers, and little regard for it; instead, shall give this short prologue to justify the so far 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 made man and shall call theological explanations. They have one principal characteristic little point asking why things are the way they we are of the creation as the when man of this is the description by that the world is earth, fire, water, air. No doubt were told at way and to stop worrying explain things as well as our explains "why" things are as they are-gravitation why things fall-but science gives details of that we have the be clear it is the sea of interrelated details seems to is as it is. Our main tool for out the chains of tight reasoning required mathematics. Indeed, mental tool for this purpose. Many people asked the question title, "Why this we are merely looking more at the logical side and material side of what is and in the foundations of self-consistency and system. They seem not to concern the world apparently admits logical explanation. the position of the early the material answers on the logical side not much better their time. But we must somewhere and that the world seems to be logical pattern that parallels much of that mathematics is the the main then to consider ideas and opinions to others. Experience shows that am not finally occurred to me that the following preliminary.remarks some respects have to theories of the well as there are various theories of to some leads to a theory of may surprise you shall take the discussing things. mind what the theories what the field assert they are; let us the scientific and look at what am well aware of what say, especially received his Ph.D. from the of Illinois in 1942 and has computing, mainly at the Telephone Laboratories. He has Since 1976 has been of the Naval School, Monterey, California, in the He is a past president of the Association Computing Machinery and received Turing prize. His research interests are in numerical software, simulation, system performance and measurement. This article text of a talk at the Section of MAA, February 24, of mathematics, My experimental is quite their mentality be it! for this of Mathematics [1], by E. P. be noticed that I have of the by those read it I do of his (I do not feel I can on his the other of the when all my explanations are over, is still as to leave of Mathematics. In his a large of examples the effectiveness in the on my to engineering. My first use of in the was in the design of atomic Second World was it the numbers so patiently on the relay computers what happened the first at Almagordo? were, and to check the computations was not an isolated we predict is realized as I did for the Bell I did many telephone other mathematical on such as traveling of complex name but a few. For glamour, cite transistor research, flight, and all of mathematical manipulations how to some extent for reasons in time the radio waves that successfully predicting well known and need not The fundamental of invariance to much to science. It was the lack of invariance of Newton's that the same mathematical occur in the functions to translation for linear The enormous of the no rational (as yet). held to be the of this belief. But even at least linear and to the conics. This in a can it be that all a useful in is at trend toward of the It is to be if not then tomorrow. this audience I will stick for further is the first man to be recorded stated that "Mathematics is the the universe." He said it both is the measure of this attitude. After twenty he found his of planetary motion-three comparatively the planets. was Galileo of Nature in the the law in science. not only the positions distant stars, so forth. Science is of laws on a but the wider ranges of observations Not always, to be often enough my thirty of practicing I made. in my office (at least future events-if do so and How could the I had that it It is ridiculous that is it is that mathematics of what in the And since to do mathematics suffices but it would list of successes, of them of a of a new artifact. I want to to do answers to the implied of the at the effectiveness to look at "What is is the title of a famous book Courant and and In it formal definition, to show is by giving I will of what mathematics is to start the far we must for the of mathematics. was the to this in modem in mathematics. it seems to be in abstract in the of aesthetics. of mathematics there is nor does ever seem life there was if for of cause this trait is and then follows still . . we are on first feature of mathematics of close But it is hard me to for the to do the and science R. W. HAMMING HAMMING Geometry seems to have arisen from the the human for various purposes, such rites, social affairs, as well as from problems of decorating the surfaces walls, pots, utensils, the fourth aspect I mentioned, aesthetic taste, and this is one of the deep foundations of mathematics. Most textbooks repeat the Greeks arose from the needs Egyptians to survey the land after each flooding by the Nile but I more to aesthetics than do most historians mathematics and correspondingly less to im- mediately utility. The third aspect of mathematics, numbers, arose from So basic are numbers that a famous mathematician once said, integers, man did the rest" rest" The integers us to find them wherever we find the universe. I have little success, to get some of my friends to understand my that the abstraction of and useful. Is it remarkable that 6 sheep sheep make sheep; that stones plus 7 stones make 13 stones? not a miracle that the universe is so constructed that such a abstraction as a number possible? To me this is of the strongest examples the unreasonable effectiveness of mathematics. Indeed, I find the development of we next come the fact counting numbers, integers, were used times a standard can be used to that is measured. But it must have happened, compara- tively speaking, that a whole of units did fit the length being measured, measurers were driven to was left over was used to the standard Fractions are not counting numbers, Because of their common use the fractions were, by a suitable the same rules for as did the with the they made in all come to the zero). Some with the fractions soon reveals that put as more as you please and that when we extend the of number to include have to give up idea of the next number. to be the first man to that the of a of the no common measure-that irrationally related. apparently produced to that time discrete number the continuous side with little conflict. crisis of incommensurability tripped off the Euclidean approach to mathematics. It is a curious fact that the rigorous by uncertainties of numbers geometry (due a result a lot of what we now consider number the form geometry. Opposed to the early Greeks, doubted the existence the real decided that there should be a number that measures the of a square (though need not do and that is less how we extended the include the was the desire to measure that did it. How can that there a number any straight algebraic numbers, are roots integer, fractional, and, as was soon under control by simply the same were used on simpler system measurement of circumference of a circle with to its to consider the not an since no UNREASONABLE EFFECTIVENESS of the of pi with integer a curved the other the existence of the ratio less is the ratio of the diagonal of a square but since that there to be a number, the number Thus by suitable extension of the earlier were admitted are at all comfortable with the we conventionally the consistency. the number both the and the the extension we abandon the single This seems to out the real number us (as as we confine of taking of sequences of numbers and do not admit -not that we have to this a firm, logical, but they we are all more or less the real Very few of us in our saner create the of us that the real numbers are simply that it has been an interesting, to try to find a nice to account for them. let us not 2,000 years, in our minds to all that we did the relationship and the we want to know, from if from other place, put still the real but so far of us have go down that path. It is only fair that there some mathematicians doubt the of the few computer the existence of only The next is the I read first to real sense. In his The Great Art or Rules Rules he says, "Putting aside the mental making 25-(-15) the same formal on the In this the real to the that this the numbers-the usual sense. led to functions along complex plane he could solve real few years I had a course I become the universe role in mechanics. They natural tool such as electric so on. we made various ideas of numbers to the extensions were to aesthetic the earlier we came to a even in we have of the the above we see that one of mathematics is the all more or less the same to new in the very process subtly altered. of theorems The old proofs no longer the newly The miracle is that always the theorems still true; it is merely a of fixing up the proofs. example of this fixing up is Euclid's Elements [4]. We have it necessary to add quite a few new postulates axioms, if you wish, we no longer care to to meet of proof. Yet how it happen that no in all the thirteen books Not one theorem has been found to be false, often the given by seem now to be false. this phenomenon is not confined the past. It is that an ex-editor of Mathematical Reviews once that over half of the new are essentially the published are false. be if is the of theorems earlier results? Well, it is obvious not blinded by authority what the elementary said it was. It is clearly something is this "else"? Once you start to find that were confined to the and postulates then you first major step is to introduce concepts derived the assumptions, such as search for and definitions is one the main features of doing great While on the with the theorem and tries to find it was in the 1850's or so that it was that the is also Often it is the we can and then examine the to see we have are often "proof generated generated A classic example is the concept of uniform convergence. Cauchy had proved that a a continuous function. At the same time be Fourier of continuous to a discontinuous the error was found and fixed up by my opinion as the battlements of mathematics and not the foundations. It is an interesting field, but the of mathematics are is found much of matter how it is made to appear the foundations. shown that is not is often assumed be, that hence even did succeed would not be have a standard. The not the center that we not uniquely it is difficult for me to we have now reached the ultimate we cannot be sure the current of Mathematics on the that Moses It is this. We various sets we settle to one In the what the makes further evolution rather difficult and as a result tends to slow down the of mathematics. It is not that the wrong, only that its change postulates the need man and continuously by him. Perhaps the original on us, but as in the example have used see that the development of so a concept as number we have for the me, more by aesthetics. tried to so doing we have an amazing to the The idea the postulates not correspond If the Pythagorean to not for a way to alter until it was true. not the I have if you and showed me a was false be very I believe the final the assumptions until the was true. Thus there of the do we decide of mathematics and what but often it ics rather to the world! So much for will arrange my explanations of the We see look for. No one on blue world appears of how much this is true a lot of held beliefs. in the earlier is to mind the he who the mathematicians of the the two As I said before, not a result is one Let us Not too long ago I was in Galileo's as it so that feel how came to the law to do kind of thing so that can learn to think masters did-I deliberately try to think was a well-educated a master of scholastic He well how to argue the number on the both sides He was in these arts of us these in each is obvious fall faster that kind of a into two down to their that one to touch other one. and both the two I do it to make them more he the more think about it-the more of when two bodies of how a it is is that all fall at the same can do. He have later like what a similar in a book Polya [7]. his law the textbooks law as an R. W. HAMMING HAMMING 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 Eucidean space, then could a 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 Eucidean space, and not a test of the inverse square law at all. But if you do not like these two examples, highly touted law of uncertainty principle. became involved in a book on Digital Filters Filters when I knew very little about the As a do all the in terms of Fourier the natural tools you already that the translation are complex exponentials. want time certainly physicists and engineers or tomorrow will the same led to these functions. believe in states are absolutely additive; they Thus the are the one needs quantum mechanics, to name but two use these eigenfunctions you led to a countable number and then as a non-countable number of Fourier series integral. Well, is a theorem of Fourier of the function multiplied by its transform exceeds a fixed one notation time invariant system you must find an uncertainty principle. The size of Planck's constant matter of the detailed identification of must occur. another example what has been thought to be a physical discovery but which by ourselves, to the fact that the distribution of constants is not rather the a random physical leading digit 1, 2, or 3 is and of course the 5, 6, 7, 8, and 9 40% of the time. This distribution applies to many types numbers, including the distribution the coefficients power series having on the circle of close examination of the we use numbers. Having given of nontrivial situations where it turns out original phenomenon arises from the mathematical tools use and from the real am ready to strongly suggest that a lot of we see the glasses we put on. goes against have been but consider the You can was the that forced the model the more about the the more uncomfortable are not theories that I have but ones which are central Einstein who most loudly proclaimed the simplicity of the laws of physics, who used mathematics so extensively as to be popularly known as a mathematician. special theory relativity paper [9] one has the feeling that is dealing scholastic philosopher's approach. He what the theory should look like, and he the theories confident of the of the were done to much interested outcomes, saying come out that else the many people that the philosophical grounds than on to the implied question about the unreasonable effectiveness of mathematics is that we the situations with an intellectual we can THE UNREASONABLE EFFECTIVENESS OF only find what we do many cases. 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 we invented a new mathematics, vectors. further we have invented tensors. a book recently written [10] conventional integers are used for labels, and real numbers are used for otherwise all the arithmetic and that occurs and there is a lot of has the rule that is that we select the mathematics to situation, and not true that the same mathematics every place. comparatively few problems. We have the illusion that science has answers to most of our this is not so. the earliest of times man must have over what Truth, Beauty, and Justice are. But so far as see science has to the seem to me that science future. So long use a mathematics which the of the are not likely to have mathematics as a major examining these famous three questions. Indeed, to generalize, almost all of our experiences this world do not fall under the domain or mathematics. theorem there are definite limits to what pure logical manipulation do, there are limits to the domain of mathematics. It has been act of faith on the scientists that the world can be explained the simple terms that mathematics handles. When much science has not answered see that our successes are not the model. touched on the matter of evolution of man. remarked that of life there have been the seeds of our current to create and people [11] further claimed that Darwinian evolution select for survival those forms of life which had the best models of is no doubt that there is some truth that we can about the world when it size to ourselves and but that when we or the very large then our not to be able to think Just as there odors that smell and we as well as sounds hear and there are we cannot see and flavors we cannot Why then, given our brains wired the way they are, remark, "Perhaps think," surprise you? Evolution, far, may possibly have blocked us from think in some could be unthinkable recall that modem science years old, and that there have been from 3 generations per century, then there have at most since Newton and you pick 4,000 years for the science, generally, you get bound of 200 generations. Considering the effects evolution we are for via selection of does not seem to that evolution can than a small ROBERT M. M. Conclusion. From all of this I am forced to conclude both that mathematics is unreasonably effective and that all when added together simply are not enough to explain set out to account I think that we-meaning you, explain why side of proper tool for exploring the universe as we perceive it at present. suspect that my are hardly as good as those of the early Greeks, who said the material side of the question nature of the universe is earth, fire, air. The side of the nature of the universe requires further exploration. E. P. Wigner, The unreasonable of mathematics in the natural sciences, Comm. Pure Appl. 13 (Feb. R. Courant and H. Robbins, What Is Mathematics? Oxford University Press, L. Kronecker, R. E. T. L. Dover Publications, New York, The Great Rules of transl. by T. R. Witmer, MIT Press, 1968, pp. Proofs and Press, 1976, p. 7. G. P6lya, Science, MAA, 1963, pp. R. W. Prentice-Hall, Englewood Cliffs, N.J., Scientific Thought, Kepler to Einstein, Harvard University Press, R. W. Hamming, Cliffs, N.J., Science, Springer-Verlag, THE NOTION first few we can the sequence see the be the the or [2] for material on inference of integer sequences). Sometimes we feel is best as two more simpler been intertwined, three sequences 1, 2, The author is a graduate of the in Mathematics Ph.D. at under Richard M. Karp. in combinatorics and probability The reader is invited the unusual to prove is based on the