Mathematicians From The United States

"Ascend with me above the dust, above the cloud, to the realms of the higher geometry, where the heavens are never clouded; where there is no impure vapour, and no delusive or imperfect observation, where the new truths are already arisen, while they are yet dimly dawning on the world below; where the earth is a little planet; where the sun has dwindled to a star; where all the stars are lost in the Milky Way to which they belong; where the Milky Way is seen floating through space like any other nebula; where the whole great girdle of nebulae has diminished to an atom and has become as readily and completely submissive to the pen of the geometer, and the slave of his formula, as the single drop, which falls from the clouds, instinct with all the forces of the material world."

"The Key! it is of wonderful construction, with its infinity of combination, and its unlimited capacity to fit every lock. … it is the great master-key which unlocks every door of knowledge and without which no discovery which deserves the name — which is law, and not isolated fact — has been or ever can be made. Fascinated by its symmetry the geometer may at times have been too exclusively engrossed with his science, forgetful of its applications; he may have exalted it into his idol and worshipped it; he may have degraded it into his toy . . . when he should have been hard at work with it, using it for the benefit of mankind and the glory of his Creator."

"Geometry, to which I have devoted my life, is honoured with the title of the Key of Sciences; but it is the Key of an ever open door which refuses to be shut, and through which the whole world is crowding, to make free, in unrestrained license, with the precious treasures within, thoughtless both of lock and key, of the door itself, and even of Science, to which it owes such boundless possessions, the New World included. The door is wide open and all may enter, but all do not enter with equal thoughtlessness. There are a few who wonder, as they approach, at the exhaustless wealth, as the sacred shepherd wondered at the burning bush of Horeb, which was ever burning and never consumed. Casting their shoes from off their feet and the world's iron-shod doubts from their understanding, these children of the faithful take their first step upon the holy ground with reverential awe, and advance almost with timidity, fearful, as the signs of Deity break upon them, lest they be brought face to face with the Almighty."

"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."

"Ideality is preëminently the foundation of Mathematics."

"What is man? … What a strange union of matter and mind! A machine for converting material into spiritual force."

"He was one of the most stimulating men I have ever known."

"I always had the feeling that his attitude toward his loved science was that of a devoted worshipper, rather than a clear expounder. Although we could rarely follow him, we certainly sat up and took notice. … To him mathematics was not a humanly devised instrument of investigation, it was Philosophy itself, the divine revealer of TRUTH."

"There are many cases of these algebras which may obviously be combined into natural classes, but the consideration of this portion of the subject will be reserved to subsequent researches."

"All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations."

"See the sage Rittenhonse, with ardent eye, Lift the long tube and pierce the starry sky; Clear in his view the circling systems roll, And broader splendours gild the central pole. He marks what laws th' eccentric wand'rers bind, Copies Creation in his forming mind, And bids, beneath his hand, in semblance rise, With mimic orbs, the labours of the skies. There wond'ring crowds with raptur'd eye behold The spangled heav'ns their mystic maze unfold; While each glad sage his splendid hall shall grace, With all the spheres that cleave th' ethereal space."

"The direct tendency of (Astronomy) is to dilate the heart with universal benevolence, and to enlarge its views."

"I do not design a machine which will give the ignorant in astronomy a just view of the solar system, but would rather astonish the skilful and curious observer by a most accurate correspondence between the situations and motions of our little representatives of our heavenly bodies and the situations and motions of those bodies themselves. I would have my orrery really useful by making it capable of informing us truly of the astronomical phenomena for any particular point of time, which I do not find that any orrery yet made can do."

"I have no health for a soldier, and as I have no expectation of serving my country in that way, I am spending my time in the old trifling manner, and am so taken with optics, that I do not know whether, if the enemy should invade this part of the country, as Archimedes was slain while making geometrical figures on the sand, so I should die making a telescope."

"The year 1775 opened with a project intended to bring the abilities of Rittenhouse more effectually into the service of science. The Philosophical Society addressed the colonial legislature of Pennsylvania, praying it to establish a public observatory, and commit it to the care of Rittenhouse. Had the circumstances of the times permitted this project to be carried into effect, it would have enabled him to occupy a great space in the history of astronomy. He had already shown himself the equal, in point of learning and skill as an observer, to any practical astronomer then living; nothing was wanting to make him rank with the Flamsteads, the Halleys, and the Maskelynes, but that he should be permitted to devote his whole mind to this pursuit, and be furnished with those instruments and accommodations, for which no private fortune will suffice. Other men might have been found as well, nay, better qualified for the political pursuits and public offices in which it became his fate to spend the rest of his life; but America has never yet produced any individual who has manifested so great a capacity for extending the domain of practical astronomy. To arrange the details of a disorganized and depreciating currency, to collect and disburse a scanty and ill-paid revenue, were thereafter to be the pursuits of our philosopher; and he was to expend upon the estimates and returns of the tax-gatherer those powers of mind which were capable of grasping, and that mechanical skill which sufficed to imitate, the vast mechanism of the universe."

"Before the nineteenth century, writers on education portayed the "improvement of mind" as an activity mainly suited to gentlemen. ...both Thomas Jefferson and Benjamin Rush had venerated David Rittenhouse as an example of how arduous philosophical investigation might elevate the child of humble parents. But Jefferson and Rush tagged Rittenhouse as a genius, and hence untypical, and each employed Rittenhouse as ammunition in a debate among educated gentlemen. Jefferson invoked Rittenhouse in his Notes on the State of Virginia, a book he wrote to disabuse French philosophes of the notion that all specied degenerated in the New World; Rush used his eulogy of Rittenhouse before the select audience of the American Philosophical Society to ridicule colleges for requiring students to learn the ancient languages."

"'America had not yet produced one good poet.' When we shall have existed as a people as long as the Greeks did before they produced a Homer, the Romans a Virgil, the French a Racine and Voltaire, the English a Shakespeare and Milton, should this reproach be still true, we will inquire from what unfriendly causes it has proceeded, that the other countries of Europe and quarters of the earth shall not have inscribed any name in the roll of poets. But neither has America produced 'one able mathematician, one man of genius in a single art or a single science.' In war we have produced a Washington, whose memory will be adored while liberty shall have votaries, whose name shall triumph over time, and will in future ages assume its just station among the most celebrated worthies of the world, when that wretched philosophy shall be forgotten which would have arranged him among the degeneracies of nature. In physics we have produced a Franklin, than whom no one of the present age has made more important discoveries, nor has enriched philosophy with more, or more ingenious solutions of the phenomena of nature. We have supposed Mr. Rittenhouse second to no astronomer living: that in genius he must be the first, because he is self-taught. As an artist he has exhibited as great a proof of mechanical genius as the world has ever produced. He has not indeed made a world; but he has by imitation approached nearer its Maker than any man who has lived from the creation to this day."

"If our astronomer be judged by the original contributions which, under existing adverse circumstances, he actually did make to astronomy and mathematics, then it must be admitted that he can not be placed in the foremost rank of astronomers then living. Friends will judge him by what he might have done; the world at large will judge him by what he actually accomplished. Our greatest indebtedness to Rittenhouse lies not in the original contributions he made to science, but rather in the interest which he aroused in astronomical pursuits, and in the diffusion of scientific knowledge in the New World which resulted from his efforts."

"Only two transits of Venus had been observed before his time, and of these, the first, in 1639, had been seen by only two persons. These transits happen so seldom that there cannot be more than two in one century, and in some centuries none at all. But the transits of Venus are the best means we have for determining the parallax of the sun. ...The observations were a success and established for Rittenhonse the reputation of an exact and careful astronomer. ...During the transit Rittenhouse saw one phenomenon which escaped the notice of all other astronomers. When the planet had advanced about half of its [Venus'] diameter upon the sun, that part of the edge of the planet which was off the sun's disc appeared illuminated, so that the outline of the entire planet could be seen. But a complete circle of light around Venus would indicate that more than half of Venus is illuminated. This can happen... only when the rays of light are refracted by an atmosphere. Hence... Venus is surrounded, like the earth, by an atmosphere. But this appearance of a ring of light was not confirmed by other astronomers, and the statement of Rittenhouse excited no attention for nearly a century, until his observation was at last confirmed by other astronomers."

"As a mechanic, Rittenhouse became celebrated for the extreme exactness and finish of his workmanship. Especially celebrated were his chronometer clocks. It was while thus engaged in the manufacture of clocks that he planned and executed an instrument which brought into play both his mechanical and mathematical skill. ...the orrery. It was, indeed, intended to be a sort of a perpetual astronomical almanac, in which the results, instead of being exhibited in tables, were to be actually exhibited to the eye. His orrery greatly exceeded all others in precision. It attracted very general attention among well informed persons... There arose a lively competition between different colleges in this country for the possession of this orrery."

"His invention, whatever it may have been, was not of sufficient importance to deserve the name of an "invention of fluxions." If Rittenhouse actually made an invention of such transcending magnitude before the age of twenty, and at a time when he had hardly begun his scientific studies, how is it that he made not the slightest approach to any similar discovery during the forty-four years of his maturer life? Though always a passionate lover of scientific pursuits, he made no original contributions whatever to the science of pure mathematics. Science is indebteded to him chiefly for his orreries and the observations of the transit of Venus. ...the alleged invention of fluxions was little more than a "rumor set afloat by idle gossip." It serves to show us, however, in what unbounded admiration he was held by his countrymen."

"The most noted mathematician and astronomer of early times [in the U.S.] was not a professor in a college, nor had he been trained within college walls. We have reference to David Rittonhouse."

"Among the books he inherited from his uncle was an English translation of the "Principia" of Newton. Such was the progress which he made in mathematical knowledge, although now destitute of any aid, that he was enabled to accomplish the perusal of this work, for the proper understanding of which so much acquaintance with geometry and algebra is necessary, before he had attained his nineteenth year. Newton, as is well known, from deference to the practice of the ancient philosophers, adopts in this work the synthetic method of demonstration, and gives no clue to the analytic process by which the truth of his propositions was first discovered by him. Unlike the English followers of this distinguished philosopher, who contented themselves, for a time, with following implicitly in the path of geometric demonstration, which he had thus pointed out, Rittenhouse applied himself to search for an instrument, which might be applied to the purpose of similar discoveries, and in his researches attained the principles of the method of fluxions. So ignorant was he of the progress which this calculus had made, and of the discussions in relation to its invention and improvement, that he for a time considered it as a new discovery of his own. In this impression, however, he could not have long continued; as he made, in his nineteenth year, an acquaintance who was well qualified to set him right in this important point."

"It was during the residence of our ingenious philosopher with his father in the country that he made himself master of Sir Isaac Newton's Principia, which he read in the English translation of Mr. Motte. It was here, likewise, he became acquainted with the science of fluxions; of which sublime invention he believed himself, for a while, to be the author, nor did he know for some years afterwards that a contest had been carried on between Sir Isaac Newton and Leibnitz for the honor of the great and useful discovery. What a mind was here! Without literary friends or society, and with but two or three books, he became, before he had reached his four and twentieth year, the rival of two of the greatest mathematicians in Europe."

"Eppie's memory is like sticky flypaper."

"To have a good memory the first thing you have to do is to trust your memory."

"… one can still say that quantum mechanics is the key to understanding magnetism. When one enters the first room with this key there are unexpected rooms beyond, but it is always the master key that unlocks each door."

"The answer to the ancient question "Why is there something rather than nothing?" would then be that ‘nothing’ is unstable."

"Einstein’s great friend and intellectual sparring partner Niels Bohr had a nuanced view of truth. Whereas according to Bohr, the opposite of a simple truth is a falsehood, the opposite of a deep truth is another deep truth. In that spirit, let us introduce the concept of a deep falsehood, whose opposite is likewise a deep falsehood. It seems fitting to conclude this essay with an epigram that, paired with the one we started with, gives a nice example: “Naïveté is doing the same thing over and over, and always expecting the same result.”"

"Ironically, conventional quantum mechanics itself involves a vast expansion of physical reality, which may be enough to avoid Einstein Insanity. The equations of quantum dynamics allow physicists to predict the future values of the wave function, given its present value. According to the Schrödinger equation, the wave function evolves in a completely predictable way. But in practice we never have access to the full wave function, either at present or in the future, so this “predictability” is unattainable. If the wave function provides the ultimate description of reality — a controversial issue! — we must conclude that “God plays a deep yet strictly rule-based game, which looks like dice to us.”"

"To put it crudely, theorists can be tempted to think along the lines “If people as clever as us haven’t explained it, that’s because it can’t be explained – it’s just an accident.” I believe there are at least two important regularities among standard model parameters that do have deeper explanations, namely the unification of couplings and the smallness of the QCD θ parameter. There may well be others."

"In most theoretical embodiments of inflationary cosmology, the currently observed universe appears as a small part of a much larger multiverse. In this framework to hold throughout the universe need not hold through all space. They can be accidents of our local geography, so to speak. If that is so, then it is valid – indeed, necessary – to consider selection effects. It may be that some of the “fundamental constants”, in particular, cannot be determined by theoretical reasoning, even in principle, because they really are different elsewhere."

"The phase transition paradigm: The standard model of fundamental physics incorporates, as one of its foundational principles, the idea that “empty space” or “vacuum” can exist in different phases, typically associated with different amounts of symmetry. Moreover, the laws of the standard model itself suggest that phase transitions will occur, as functions of temperature. Extensions of the standard model to build in higher symmetry (gauge unification or especially supersymmetry) can support effective vacua with radically different properties, separated by great distance or by domain walls. That would be a form of failure of universality, in our sense, whose existence is suggested by the standard model."

"The happy coincidences between life’s requirements and nature’s choices of parameter-values might be just a series of flukes, but one could be forgiven for beginning to suspect that something deeper is at work. That suspicion is the first deep root of anthropic reasoning."

"The traditional “cosmological” Multiverse considers that there might be physical realms inaccessible to us due to their separation in space-time. The quantum Multiverse arises from entities that occupy the same space-time, but are distant in Hilbert space – or in the jargon, decoherent."

"A common habit of thought... is the idea that space is [a] simple receptacle in which bodies move around, with no two bodies present at the same point. ...In modern quantum physics generally, and in the standard model of fundamental physics in particular, physical space appears as a far more flexible framework. Many kinds of particles can be present at the same point in space at the same time. Indeed, the primary ingredients of the standard model are not particles at all, but an abundance of quantum fields, each a complex object in itself, and all omnipresent."

"In the past scientists have repeatedly reached “intellectual closure” on inadequate pictures of the universe, and underestimated its scale."

"The “Copernican Principle” or “Cosmic Mediocrity”... states, basically, that Earth does not occupy a privileged place in the universe. Universality asserts more, namely that there are no privileged places or times."

"Unassuming in manner, genial and kindly in his intercourse with his fellow-men, never showing impatience or irritation, devoid of personal ambition of the baser sort or of the slightest desire to exalt himself, he went far toward realizing the ideal of the unselfish, Christian gentleman. In the minds of those who knew him, the greatness of his intellectual achievements will never overshadow the beauty and dignity of his life."

"If I have had any success in mathematical physics, it is, I think, because I have been able to dodge mathematical difficulties."

"A mathematician may say anything he pleases, but a physicist must be at least partially sane."

"I wish to know systems."

"Anyone having these desires will make these researches."

"The whole is simpler than its parts."

"Mathematics is a language."

"We avoid the gravest difficulties when, giving up the attempt to frame hypotheses concerning the constitution of matter, we pursue statistical inquiries as a branch of rational mechanics."

"The laws of thermodynamics, as empirically determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results."

"In all these papers we see a love of honest work, an aversion to shams, a caution in the enunciation of conclusions, a distrust of rash generalizations and speculations based on uncertain premises. He was never anxious to add one more guess on doubtful matters in the hope of hitting the truth, or what might pass as such for a time, but was always ready to take infinite pains in the most careful testing of every theory. With these qualities was united a modesty which forbade the pushing of his own claims and desired no reputation except the unsought tribute of competent judges."

"His true monument lies not on the shelves of libraries, but in the thoughts of men, and in the history of more than one science."

"One of the principal objects of theoretical research is to find the point of view from which the subject appears in the greatest simplicity."