Mathematicians From England

"Classical mechanics has been developed continuously from the time of Newton and applied to an ever-widening range of dynamical systems, including the electromagnetic field in interaction with matter. The underlying ideas and the laws governing their application form a simple and elegant scheme, which one would be inclined to think could not be seriously modified without having all its attractive features spout. Nevertheless it has been found possible to set up a new scheme, called quantum mechanics, which is more suitable for the description of phenomena on the atomic scale and which is in some respects more elegant and satisfying than the classical scheme. This possibility is due to the changes which the new scheme involves being of a very profound character and not clashing with the features of the classical theory that make it so attractive, as a result of which all these features can be incorporated in the new scheme."

"I have the best of reasons for being an admirer of Werner Heisenberg. He and I were young research students at the same time, about the same age, working on the same problem. Heisenberg succeeded where I failed. There was a large mass of spectroscopic data accumulated at that time and Heisenberg found out the proper way of handling it. In doing so he started the golden age in theoretical physics, and for a few years after that it was easy for any second rate student to do first rate work."

"If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics led me by the hand. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress."

"I was taught at school never to start a sentence without knowing the end of it."

"The aim of science is to make difficult things understandable in a simpler way; the aim of poetry is to state simple things in an incomprehensible way. The two are incompatible."

"The interpretation of quantum mechanics has been dealt with by many authors, and I do not want to discuss it here. I want to deal with more fundamental things."

"A good deal of my research work in physics has consisted in not setting out to solve some particular problems, but simply examining mathematical quantities of a kind that physicists use and trying to get them together in an interesting way regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later that the work does have an application. Then one has had good luck."

"God used beautiful mathematics in creating the world."

"It seems clear that the present quantum mechanics is not in its final form. Some further changes will be needed, just about as drastic as the changes made in passing from Bohr's orbit theory to quantum mechanics. Some day a new quantum mechanics, a relativistic one, will be discovered, in which we will not have these infinities occurring at all. It might very well be that the new quantum mechanics will have determinism in the way that Einstein wanted."

"My research work was based in pictures. I needed to visualise things and projective geometry was often most useful e.g. in figuring out how a particular quantity transforms under Lorentz transf[ormation]. When I came to publish the results I suppressed the projective geometry as the results could be expressed more concisely in analytic form."

"I went back to Cambridge at the beginning of October, 1925, and resumed my previous style of life, intense thinking about these problems during the week and relaxing on Sunday, going for a long walk in the country alone... It was during one of the Sunday walks in October, 1925, when I was thinking very much about this uv - vu, in spite of my intention to relax, that I thought about Poisson brackets. I remembered something which I had read up previously in advanced books of dynamics about these strange quantities, Poisson brackets, and from what I could remember, there seemed to be a close similarity between a Poisson bracket of two quantities, u and v, and the commutator uv - vu. The idea first came in a flash, I suppose, and provided of course some excitement, and then of course came the reaction "No, this is probably wrong." I did not remember very well the precise formula for a Poisson bracket, and only had some vague recollections. But there were exciting possibilities there, and I thought that I might be getting to some big new idea... it was a Sunday evening then and the libraries were all closed. I just had to wait impatiently through that night without knowing whether this idea was really any good or not, but still I think that my confidence gradually grew during the course of the night. The next morning I hurried along to one of the libraries as soon as it was open, and then I looked up Poisson brackets in Whittaker's Analytical Dynamics, and I found that they were just what I needed."

"I want to emphasize the necessity for a sound mathematical basis for any fundamental physical theory. Any philosophical ideas that one may have play only a subordinate role. Unless such ideas have a mathematical basis they will be ineffective."

"The measure of greatness in a scientific idea is the extent to which it stimulates thought and opens up new lines of research."

"If there is no complete agreement between the results of one's work and the experiment, one should not allow oneself to be too discouraged."

"I don't suppose that applies so much to other physicists; I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do."

"It was a good description to say that it was a game, a very interesting game one could play. Whenever one solved one of the little problems, one could write a paper about it. It was very easy in those days for any second-rate physicist to do first-rate work. There has not been such a glorious time since then."

"With my assumption... life need never end. There is no decisive argument for deciding between [certain] assumptions. I prefer the one that allows the possibility of endless life. One may hope that some day the question will be decided by direct observation."

"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it's the exact opposite!"

"One possibility in this direction is to regard, classically, an electron as the end of a single Faraday line of force. The electric field in this picture from discrete Faraday lines of force, which are to be treated as physical things, like strings. One has then to develop a dynamics for such a string like structure, and quantize it.... In such a theory a bare electron would be inconceivable, since one cannot imagine the end of a piece of string without having the string."

"At the beginning of time the laws of Nature were probably very different from what they are now. Thus we should consider the laws of Nature as continually changing with the epoch, instead of as holding uniformly throughout space-time. This idea was first put forward by Milne, who worked it out on... assumptions... not very satisfying... we should expect them also to depend on position in space, in order to preserve the beautiful idea of the theory of relativity [that] there is fundamental similarity between space and time."

"The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation."

"If we are honest — and scientists have to be — we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can't for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards — in heaven if not on earth — all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins."

"In the fight between you and the world, back the world."

"Dirac’s relationship with quantum electrodynamics was not an easy one. On the one hand, the theory owes to him much more than to anybody else, especially if one considers the years crucial for its emergence, the late 1920s and early 1930s, when practically all its main concepts, except for that of renormalization, were developed. After this period Dirac also wrote a number of important papers, specifically, on indefinite metrics and quantum dynamics with constraints. On the other hand, since the early 1930s he was an active critic of the theory and tried to develop alternative schemes. He did not become satisfied with the later method of renormalization and regarded it as a mathematical trick rather than a fundamental solution, and died unreconciled with what, to a large extent, was his own brainchild. …"

"Dirac has done more than anyone this century, with the exception of Einstein, to advance physics and change our picture of the universe. He is surely worthy of the memorial in Westminster Abbey. It is just a scandal that it has taken so long."

"Dirac, in his first paper, in contrast to what his “hole”-theory implied, had identified the positively charged particle corresponding to the electron also with the proton. However, after Weyl had pointed out that Dirac’s hole theory led to equal masses, he changed his mind and gave the new particle the same mass as the electron."

"Let me backtrack... to the 1920s when quantum mechanics was created. ...[O]ne of the things that the originators of quantum mechanics... were very unhappy about was that quantum mechanics was not compatible with the special theory of relativity. ...Attempts were made to create a theory of the electron that is consistent with relativity, and one person made such a theory, but it was inconsistent with the electron spin. Another person made a theory that was consistent with electron spin, but not consistent with relativity. Until finally... a then young man... Paul Dirac created the theory of an equation that correctly describes the electron, and is relativistic, and also includes the description of spin... [T]hat was the beginning of the marriage that would take place between quantum mechanics and relativity. ...He originally noticed that the equation had too many solutions... He thought the negatively charged solutions would be electrons and the positive charge solutions would be s... He realized that this made no sense and the two things had to have the same , and finally... realized that he was... predicting [the positively charged electron,] the , just shortly before the positron was... independently and serendipitously discovered."

"Here we find a man with an almost miraculous apprehension of the structure of the physical world, coupled with gentle incomprehension of that less logical, messier world, the world of other people."

"Before World War II there had been considerable theoretical effort directed towards the question of the self-energy of the electron. However, because of the war, interest had remained dormant. Now, the stimulus of results of Lamb and Retherford the latent interest developed into a major attack by theoretical physicists, and within a few years the problem was solved to the satisfaction of nearly everyone. (To the end of his life, however, Dirac maintained that any theory involving the subtraction of infinities was ugly, unsatisfactory and surely incomplete.)"

"When I was a young man, Dirac was my hero. He made a breakthrough, a new method of doing physics. He had the courage to simply guess at the form of an equation, the equation we now call the Dirac equation, and to try to interpret it afterwards. Maxwell in his day got his equations, but only in an enormous mass of 'gear wheels' and so forth."

"The latest and most successful creation of theoretical physics, namely Quantum Mechanics, is fundamentally different in its principles from the two programmes which we will briefly call Newton's and Maxwell's. For the quantities that appear in its laws make no claim to describe Physical Reality itself, but only the probabilities for the appearances of a particular physical reality on which our attention is fixed. Dirac, to whom, in my opinion, we owe the most logically perfect presentation of this theory, rightly points out that it appears, for example, to be by no means easy to give a theoretical description of a photon that shall contain within it the reasons that determine whether or not the photon will pass a polarizator set obliquely in its path."

"As for life within a Legislature,— who can tell how warped and bent and twisted, and accommodated to the exigencies of party struggle become the faculties of belief? Strong and courageous natures know it, and remain strong and courageous in spite of knowledge and practice; but the pliancy of man is beyond admiration, and is nowhere better seen than under the schooling of Parliament."

"What an education follows! It is really a fine comedy, though the players rarely know it. I am but a clumsy performer myself, and have to confess to incurable defects of training, so that I sometimes wonder I have not been hissed off the stage; still I have seen the performance through more than once or twice, and know something about it. Such tender and delicate adjustments and readjustments of convictions to keep the party balance sure! Such abundance of spoonmeat on the one hand, and such careful economy on the other of truths that may prove too strong for weak digestions! Such avowals of readiness to consider seriously any opinion, however obviously absurd, broached by a possible supporter! Such prompt denunciations of all the devices of an irreconcilable opponent!"

"The young man who is moved in any way to contemplate an entry into public life, whose creed is not in absolute inheritance from his fathers, learns first of all to understand that there are two great political organizations, with one of which he must associate himself, learning and echoing its catch-words, accepting its leadership, and steeping himself in the belief that in it are wisdom and truth while the other party is void of both. It is not everyone whose ductile mind takes him through this training, and a goodly number of up-growing men of not the worst promise for the future have to step aside."

"We may blunder on in spite of repeated miscalculations of the popular will. More penetrating and pernicious is the influence our ill-devised machinery has upon the character of our national life. It eats in and into it. It degrades candidates and electors alike. It does its worst to reduce to sterility of influence many of the best of the component elements of the people. The individuals survive, but with their political activity dead or dying, no opportunities of life and growth being afforded them. Finally it presents as an embodiment of the nation an assembly or assemblies into which none can enter who have not been clipped, and pared, and trimmed, and stretched out of natural shape and likeness to slip along the grooves of supply. A free press, free pulpits, and a free people outside help to correct what would otherwise become intolerable but press, pulpits and people, free as they are, work and live in strict limits of relation to the machinery established among them. The world revolves on its axis subject to the Constitution of the United States, and the most Radical newspaper man in London, if such there be, never lets his imagination range out of hearing of the Clock Tower."

"What a jolly awakening there will be some few years hence, when the inevitable argument of experience will show us a nation contradicting itself through the voices of its chosen representatives! The stupidest politician will sit up, rubbing his eyes. After all, facts are facts, and although we may quote one to another with a chuckle the words of the Wise Statesman, "Lies — damned lies — and statistics," still there are some easy figures the simplest must understand, and the astutest cannot wriggle out of. So we may be led to the serious consideration of change by the evolution of materials of conviction which those who run may read, though some who read may wish to run away from them."

": These have often been attributed to Thomas Jefferson, but also to Thomas Paine, Abraham Lincoln, and many others; Lord Denning in The Road to Justice (1988) states that the phrase originated in a statement of Irish orator John Philpot Curran in 1790: "It is the common fate of the indolent to see their rights become a prey to the active. The condition upon which God hath given liberty to man is eternal vigilance.""

"There is an imperialism that deserves all honor and respect — an imperialism of service in the discharge of great duties. But with too many it is the sense of domination and aggrandisement, the glorification of power. The price of peace is eternal vigilance."

"It is true— it has been already admitted— that the picture will not be universally recognized; but it has been suggested that the failure of recognition lies rather in the degeneracy of the faculty of seeing than in the misrepresentation of the vision to be seen. It may be also confessed that life often survives all the perversities of training. We cannot absolutely nullify the prodigality of nature, try as hard as we may. In spite of most careful management, untractable growths survive in the most provoking way, and intrude themselves into fields believed to be kept free from their presence. And sometimes it happens that the poor party managers have to accommodate themselves to the genius they curse."

"For the theory-practice iteration to work, the scientist must be, as it were, mentally ambidextrous; fascinated equally on the one hand by possible meanings, theories, and tentative models to be induced from data and the practical reality of the real world, and on the other with the factual implications deducible from tentative theories, models and hypotheses."

"The researcher hoping to break new ground in the theory of experimental design should involve himself in the design of actual experiments. The investigator who hopes to revolutionize decision theory should observe and take part in the making of important decisions."

"Since all models are wrong the scientist cannot obtain a "correct" one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity."

"One important idea is that science is a means whereby learning is achieved, not by mere theoretical speculation on the one hand, nor by the undirected accumulation of practical facts on the other, but rather by a motivated iteration between theory and practice."

"All models are wrong; some models are useful."

"We have a large reservoir of engineers (and scientists) with a vast background of engineering know how. They need to learn statistical methods that can tap into the knowledge. Statistics used as a catalyst to engineering creation will, I believe, always result in the fastest and most economical progress…"

"Statistical criteria should (1) be sensitive to change in the specific factors tested, (2) be insensitive to changes, of a magnitude likely to occur in practice, in extraneous factors."

"[Box's 1960 paper Fitting empirical data is] a mature exposition of an important branch of statistics, to which the author has made great contributions. One feature of particular interest is practical discussion of genuinely nonlinear fitting problems and their solution with the help of tact and a special, publicly available, IBM-704 program. Another is insightful comments on the role of prior distributions in statistics."

"George Box is, in the field of the quality sciences, the consummate ‘Renaissance man’ who has made significant and enduring contributions to the profession of quality control and the allied arts and sciences... [His] contributions encompass considerable scope and have already had lasting effect."

"I want to tell you how I got to be a statistician. I was, of course, born in England and in 1939... when war broke out in September of that year, although I was close to getting a degree in Chemistry, I abandoned that and joined the Army. They put me in the Engineers (and when I see a bridge I still catch myself calculating where I would put the charges to blow it up). Before I could actually do any of that I was moved to a highly secret experimental station in the south of England. At the time they were bombing London every night and our job was to help to find out what to do if, one night, they used poisonous gas. Some of England's best scientists were there. There were a lot of experiments with small animals, I was a lab assistant making biochemical determinations, my boss was a professor of physiology dressed up as a colonel, and I was dressed up as a staff sergeant. The results I was getting were very variable and I told my colonel that what we really needed was a statistician. He said "we can't get one, what do you know about it?" I said "Nothing, I once tried to read a book about it by someone called R. A. Fisher but I didn't understand it". He said "You've read the book so you better do it", so I said, "Yes sir""

"Essentially, all models are wrong, but some are useful."