Mathematicians From England

"As the cathode rays carry a charge of negative electricity, are deflected by an electrostatic force as if they were negatively electrified, and are acted on by a magnetic force in just the way in which this force would act on a negatively electrified body moving along the path of these rays, I can see no escape from the conclusion that they are charges of negative electricity carried by particles of matter."

"Thomson's lecture drew from Fitz Gerald the suggestion that "we are dealing with free electrons in these cathode rays"—a remark the point of which will become more evident when we come to consider the direction in which the Maxwellian theory was being developed at this time."

"J. J. Thomson, by a rotating-mirror method, succeeded in measuring the velocity of the cathode rays, finding it to be 1.9 x 107 cm./sec.; a value so much smaller than that of the velocity of light that it was scarcely possible to conceive of the rays as vibrations of the aether."

"Thomson and then his young men demolished a recurrent scientific myth—one that had surfaced again in the 1870's: that there was nothing left to be discovered, nothing new under the sun. Part of the immutable wisdom of the day, endorsed and believed long before the greatest of scientists, Isaac Newton, was a kind of billiard ball theory of the atom, which went back to the ancient Greeks. The word itself is from the Greek atomos, meaning "inidivisible.""

"Notes on Recent Researches in Electricity and Magnetism, published in 1883, had won him enough acclaim at the age of twenty-seven that he was named director of the [Cavendish] laboratory the next year."

"Thomson's work suggested an alternative version—the instability of matter—to that of the indivisible atom. It was revolutionary stuff."

"J. J. Thomson was about to make the most significant find of the late nineteenth century... Thomson had been investigating the nature of cathode rays. He was convinced that they were some kind of electrified particles and, to prove his theory, began testing their behavior in electric or magnetic fields. By measuring both the extent to which such fields deflected them and their electric charge, he discovered that cathode rays consisted of very small negatively charged particles whose mass was about eighteen hundred times smaller than the lightest known substance—the hydrogen atom. ...He initially named these tiny carriers of electricity "corpuscles." Later they would become known as "electrons." The corpuscles were, in fact, the first subatomic particles to be found..."

"His reluctance to pay for elaborate or expensive equipment, perhaps the result of an impoverished childhood, had established the legendary "sealing wax-and-string" tradition of the Cavendish, where everyday materials were ingeniously used to make and patch up experimental equipment, with sealing wax proving particularly useful for vacuum seals."

"Cathode Rays... he adheres to the hypothesis that the rays are due to the violent projection of the negatively charged particles from the cathode. In another abstract from presumably the same lecture, he states that in the cathode discharge the matter is in something beyond the ordinary state and that the carriers of the discharge in a cathode ray are not atoms but something very much smaller; his conclusions are that the particles carrying the charge must be in a much more finely divided state than the ordinary molecule and possibly may be the primordial element; the numerical ration of the mass of the particle to the charge carried is about 1,100 times less than that deduced electrolytically for the hydrogen ion, showing that either the charge must be very great or the particle very small, and it is the latter which he thinks is the case."

"The electron: may it never be of any use to anybody!"

"This example illustrates the differences in the effects which may be produced by research in pure or applied science. A research on the lines of applied science would doubtless have led to improvement and development of the older methods—the research in pure science has given us an entirely new and much more powerful method. In fact, research in applied science leads to reforms, research in pure science leads to revolutions, and revolutions, whether political or industrial, are exceedingly profitable things if you are on the winning side."

"I have described at some length the application of Positive Rays to chemical analysis; one of the main reasons for writing this book was the hope that it might induce others, and especially chemists, to try this method of analysis. I feel sure that there are many problems in chemistry, which could be solved with far greater ease by this than any other method. The method is surprisingly sensitive — more so than even that of spectrum analysis, requires an infinitesimal amount of material, and does not require this to be specially purified; the technique is not difficult if appliances for producing high vacua are available."

"The minimum I. Q. necessary for one to grasp the concepts of Statistics required for the undergraduate degree is 120.(paraphrased)"

"Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an "intelligence explosion", and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make, provided that the machine is docile enough to tell us how to keep it under control. It is curious that this point is made so seldom outside of science fiction. It is sometimes worthwhile to take science fiction seriously."

"There may be occasions when it is best to behave irrationally, but whether there are should be decided rationally."

"Superficially at least the forces of electricity and magnetism seem to present the same kind of problem as gravitation. Experiment shows that two electrically charged bodies attract one another (or repel if their charges are of the same kind) with a force which conforms to the same mathematical law as the force of gravitation - both forces fall off inversely as the inverse square of the distance. The same is true of the magnetic force also; two magnetic poles attract or repel one another with a force which again follows the law of the inverse square of the distance."

"If we assume that the last breath of, say, Julius Caesar has by now become thoroughly scattered through the atmosphere, then the chances are that each of us inhales one molecule of it with every breath we take."

"The human race, whose intelligence dates back only a single tick of the astronomical clock, could hardly hope to understand so soon what it all means."

"The motion of the stars over our heads is as much an illusion as that of the cows, trees and churches that flash past the windows of our train."

"The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or a picture which helps us understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from reality; it is like making a graven image of a spirit."

"The tendency of modern physics is to resolve the whole material universe into waves, and nothing but waves. These waves are of two kinds: bottled-up waves, which we call matter, and unbottled waves, which we call radiation or light. If annihilation of matter occurs, the process is merely that of unbottling imprisoned wave-energy and setting it free to travel through space. These concepts reduce the whole universe to a world of light, potential or existent, so that the whole story of its creation can be told with perfect accuracy and completeness in the six words: ‘God said, Let there be light’."

"Everything that has been said, and every conclusion that has been tentatively put forward, is quite frankly speculative and uncertain. We have tried to discuss whether present-day science has anything to say on certain difficult questions, which are perhaps set for ever beyond the reach of human understanding. We cannot claim to have discerned more than a very faint glimmer of light at the best; perhaps it was wholly illusory, for certainly we had to strain our eyes very hard to see anything at all. So that our main contention can hardly be that the science of to-day has a pronouncement to make, perhaps it ought rather to be that science should leave off making pronouncements: the river of knowledge has too often turned back on itself."

"the universe can be best pictured, ... as consisting of pure thought, the thought of what, for want of a wider word, we must describe as a mathematical thinker."

"Today there is a wide measure of agreement, which on the physical side of science approaches almost to unanimity, that the stream of knowledge is heading towards a non-mechanical reality; the universe begins to look more like a great thought than like a great machine. Mind no longer appears as an accidental intruder into the realm of matter; we are beginning to suspect that we ought rather to hail it as a creator and governor of the realm of matter..."

"From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician."

"The concepts which now prove to be fundamental to our understanding of nature—a space which is finite; a space which is empty, so that one point [of our 'material' world] differs from another solely in the properties of space itself; four-dimensional, seven- and more dimensional spaces; a space which for ever expands; a sequence of events which follows the laws of probability instead of the law of causation—or alternatively, a sequence of events which can only be fully and consistently described by going outside of space and time—all these concepts seem to my mind to be structures of pure thought, incapable of realisation in any sense which would properly be described as material."

"And the substance out of which this bubble is blown, the soap-film, is empty space welded onto empty time."

"For, for aught we know, or for aught that the new science can say to the contrary, the gods which play the part of fate to the atoms of our brains may be our own minds. Through these atoms our minds may perchance affect the motions of our bodies and so the state of the world around us. To-day science can no longer shut the door on this possibility; she has no longer any unanswerable arguments to bring against our innate conviction of free-will. On the other hand, she gives no hint as to what absence of determinism or causation may mean. If we, and nature in general, do not respond in a unique way to external stimuli, what determines the course of events? If anything at all, we are thrown back on determinism and causation; if nothing at all, how can anything ever occur? As I see it, we are unlikely to reach any definite conclusions on these questions until we have a better understanding of the true nature of time."

"Life exists in the universe only because the carbon atom possesses certain exceptional properties."

"One must stand stiller than still."

"Into such a universe we have stumbled, if not exactly by mistake, at least as the result of what may properly be described as an accident. The use of such a word need not imply any surprise that our earth exists, for accidents will happen, and if the universe goes on for long enough, enough, every conceivable accident is likely to happen in time. It was, I think, Huxley who said that six monkeys, set to strum unintelligently on typewriters for millions of millions of years, would be bound in time to write all the books in the British Museum. If we examined the last page which a particular monkey had typed, and found that it had chanced, in its blind strumming, to type a Shakespeare sonnet, we should rightly regard the occurrence as a remarkable accident, but if we looked through all the millions of pages the monkeys had turned off in untold millions of years, we might be sure of finding a Shakespeare sonnet somewhere amongst thein, the product of the blind play of chance. In the same way, millions of millions of stars wandering blindly through space for millions of millions of years are bound to meet with every kind of accident; a limited number are bound to meet with that special kind of accident which calls planetary systems into being."

"Therefore the observer may well look to Jeans' theory for the thread of physical significance that shall vitalize a system of classification of non-galactic nebulae. In the scheme presently to be proposed, a conscious attempt was made to ignore the theory and to arrange the data purely from an observational point of view. The analogy however was so suggestive that at several points... there was no hesitation in accepting the one favored by Jeans' theory of spirals."

"Jeans was not a man of many friends, partly because of his temperamental shyness and reticence and partly because of his intolerance of what he deemed to be second-rate. With his own quick perception he lacked the patience which would have enabled him to understand and appreciate a slower-moving mind and consequently he missed those intimacies which he fundamentally desired."

"Two particular cross-sections, he claimed, were of special interest: first, a cross-section near the beginning of time (the creation of the world); secondly, a cross-section only slightly differing from the present. In the latter case, all those parts of the universe not in our immediate vicinity could be disregarded..."

"He then embarked on a criticism of causality, as expressed by Kant or Bertrand Russel. He says that there is no scientific justification for supposing that the happenings of the world can be divided into detached events, and 'strung in pairs, like a row of dominoes, each being the cause of the event which follows and at the same time the effect of that which precedes.' He warned... at the same time against the other extreme... it was not necessary for all previous events in the history of the world to be considered as separate causes. For one thing, the effects of the earlier of them were already taken into account in the later..."

"He pointed out that amongst the hindrances to a joint discussion by philosophers and physicists are differences of idiom, if not language. He stated that, whether one understands the meaning of a sentence in Newton or not, one knows at least the meaning of the words, whereas philosophy has no agreed terminology. He was right in pointing out that various old problems in philosophy owed their existence to imperfections of language... he argued that the philosopher thinks and speaks in the subjective, the scientist in objective, terms."

"What philosophical conclusions should we draw from the abstract style of the superstring theory? We might conclude, as Sir James Jeans concluded long ago, that the Great Architect of the Universe now begins to appear as a Pure Mathematician, and that if we work hard enough at mathematics we shall be able to read his mind. Or we might conclude that our pursuit of abstractions is leading us far away from those parts of the creation which are most interesting from a human point of view. It is too early yet to come to conclusions."

"To be historically accurate, Hubble failed to acknowledge two of his pivotal sources for those ideas which now bear his name: Reynolds and Jeans. As agreed by Allan Sandage, the graphical representation of the Hubble tuning fork [style diagram of the Hubble sequence] must be attributed to Sir James Jeans - a scientist who adored music, and who wrote the famous book Science and Music on that theme. In the Lowell Observatory archives, Hubble revealed to Slipher that he had "been trying to construct a classification of non-galactic nebulae analogous to Jeans' evolution sequence, but from purely observational material.""

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth... No force of gravitation is... needed to impress curvature on world-lines; the curvature is inherent in the space..."

"Minkowski... supposed that this fourth dimension of time was not detached from and independent of the three dimensions of space. He introduced a new four-dimensional space to which ordinary space contributed three dimensions, and time one; we may call it 'space-time'. ...The succession of positions which a particle occupied in ordinary space at a succession of instants of time would be represented by a line in space-time; this he called the 'world-line' of the particle. ...Newton's absolute space and absolute time fell out of science, and they carried much with them in their fall. First to go was the concept of simultaneity. ...It now became necessary to find a way of treating gravitation which should not involve simultaneity. Einstein found through the medium of his 'Principle of Equivalence'."

"It can hardly be a matter for surprise that our race has not succeeded in solving any large part of its most difficult problems in the first millionth part of its existence. Perhaps life would be a duller affair if it had. For to many it is not knowledge but the quest for knowledge that gives the greater interest to thought - to travel hopefully is better than to arrive."

"Physics and philosophy are at most a few thousand years old, but probably have lives of thousands of millions of years stretching away in front of them. They are only just beginning to get under way..."

"When two hypotheses are possible, we provisionally choose that which our minds adjudge to be simpler, on the supposition that this is the more likely to lead in the direction of truth. It includes as a special case the principle of Occam's razor-entia non multiplicana praeter necessitatem."

"The complete closed world consists of three parts-substratum, phenomenal world, and observer. By our experiments we drag up activities from the substratum into the phenomenal world of space and time, but there is no clear line of demarcation between subject and object, and by performing observations on the world, we alter it, much as a fisherman dragging up fish from the depths of the seas disturbs the waters and also damages the fish."

"...a detailed mathematical discussion shows that whatever kind of wave-packet we select to represent the electron, the product of the two uncertainties of position and momentum can never be less than h, which is precisely what Heisenberg found..."

"We saw that radiation cannot suitably be pictured as particles when it is traveling through space. There is a corresponding property for electrons; these should not be pictured as waves so long as they are traveling through empty space."

"The main result reached by the new theory was that the classical mechanics can be made to account for the whole range of spectral phenomena, provided entirely new meanings are given to such symbols as p and q which had hitherto been taken to describe the position and motion of the electron. ...The most significant of the new properties is that the product pq is no longer the same as the product qp - in other words the order in which the two factors are multiplied together is no longer a matter of indifference. The difference between pq and qp is found to be always the same, being Planck's constant h multiplied by a numerical multiplier."

"In the interior of the atom, Bohr had tried the plan of retaining the particle-electron and modifying the classical mechanics. Heisenberg took the opposite course, his procedure amounting in effect to retaining the classical mechanics, at least in form, and modifying the electron. Actually, the electron dropped out all together, because it exists only as a matter of inference and not of direct observation. For the same reason, the new theory contains no mention of atoms, nuclei, protons, or of electricity in any shape or form. The existences of all these are matters of inference, and Heisenberg's purely mathematical theory could no more make contact with them than with the efficiency of a turbine or with the price of wheat."

"A similar situation occurred in astronomy, where the Newtonian law of gravitation had been found to predict the orbits of the outer planets with great accuracy, but had failed with the orbits of Mercury and Venus. The relativity theory of gravitation had provided the necessary modification of Newton's law, and in working out the details of the new theory, Einstein had utilized the fact that Newtonian law gave the right result at great distances from the sun. Heisenberg, confronted with a similar problem, was able to avail himself of the fact that the classical mechanics gave the right result at great distances from the atomic nucleus. Here, and here alone Heisenberg's theory made contact with the world of the older physics."

"Bohr had... discovered that the frequencies corresponding to very large integers could be calculated accurately from the classical mechanics; they were simply the number of times that an ordinary electron would complete the circuit of its orbit in one second when it was at a very great distance from the nucleus of the atom to which it belonged. This could only mean that when an electron receded to a great distance from the nucleus of its atom, it not only assumed the properties of an ordinary electron, but also behaved as directed by the classical mechanics. Yet the classical mechanics failed completely for the calculation of frequencies corresponding to small orbits."