Mathematicians from the United States

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"I am not shooting at immortality through my books, no. Nor do I think Chopin was shooting at immortality through his music. That strikes me as a very selfish goal, and I don't think Chopin was particularly selfish. I would also say that I think that music comes much closer to capturing the essence of a composer's soul than do a writer's ideas capture the writer's soul. Perhaps some very emotional ideas that I express in my books can get across a bit of the essence of my soul to some readers, but I think that Chopin's music probably does a lot better job (and the same holds, of course, for many composers). I personally don't have any thoughts about "shooting for immortality" when I write. I try to write simply in order to get ideas out there that I believe in and find fascinating, because I'd like to let other people be able share those ideas. But intellectual ideas alone, no matter how fascinating they are, are not enough to transmit a soul across brains. Perhaps, as I say, my autobiographical passages — at least some of them — get tiny shards of my soul across to some people. But such autobiographical story-telling is not nearly as effective a means of soul-transmission as is living with someone you love for many years of your lives, and sharing profound life goals with them — that's for sure!"

- Douglas Hofstadter

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"Of the fifty or hundred systems of philosophy that have been advanced at different times of the world's history, perhaps the larger number have been, not so much results of historical evolution, as happy thoughts which have accidently occurred to their authors. An idea which has been found interesting and fruitful has been adopted, developed, and forced to yield explanations of all sorts of phenomena. … The remaining systems of philosophy have been of the nature of reforms, sometimes amounting to radical revolutions, suggested by certain difficulties which have been found to beset systems previouslv in vogue; and such ought certainly to be in large part the motive of any new theory. … When a man is about to build a house, what a power of thinking he has to do, before he can safely break ground! With what pains he has to excogitate the precise wants that are to be supplied. What a study to ascertain the most available and suitable materials, to determine the mode of construction to which those materials are best adapted, and to answer a hundred such questions! Now without riding the metaphor too far, I think we may safely say that the studies preliminary to the construction of a great theory should be at least as deliberate and thorough as those that are preliminary to the building of a dwelling-house."

- Charles Sanders Peirce

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"Be it understood, then, that what we have to do, as students of phenomenology, is simply to open our mental eyes and look well at the phenomenon and say what are the characteristics that are never wanting in it, whether that phenomenon be something that outward experience forces upon our attention, or whether it be the wildest of dreams, or whether it be the most abstract and general of the conclusions of science. The faculties which we must endeavor to gather for this work are three. The first and foremost is that rare faculty, the faculty of seeing what stares one in the face, just as it presents itself, unreplaced by any interpretation, unsophisticated by any allowance for this or for that supposed modifying circumstance. This is the faculty of the artist who sees for example the apparent colors of nature as they appear. When the ground is covered by snow on which the sun shines brightly except where shadows fall, if you ask any ordinary man what its color appears to be, he will tell you white, pure white, whiter in the sunlight, a little greyish in the shadow. But that is not what is before his eyes that he is describing; it is his theory of what ought to be seen. The artist will tell him that the shadows are not grey but a dull blue and that the snow in the sunshine is of a rich yellow. That artist's observational power is what is most wanted in the study of phenomenology. The second faculty we must strive to arm ourselves with is a resolute discrimination which fastens itself like a bulldog upon the particular feature that we are studying, follows it wherever it may lurk, and detects it beneath all its disguises. The third faculty we shall need is the generalizing power of the mathematician who produces the abstract formula that comprehends the very essence of the feature under examination purified from all admixture of extraneous and irrelevant accompaniments."

- Charles Sanders Peirce

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"There never was a sounder logical maxim of scientific procedure than Ockham's razor: Entia non sunt multiplicanda praeter necessitatem. That is to say; before you try a complicated hypothesis, you should make quite sure that no simplification of it will explain the facts equally well. No matter if it takes fifty generations of arduous experimentation to explode the simpler hypothesis, and no matter how incredible it may seem that that simpler hypothesis should suffice, still fifty generations are nothing in the life of science, which has all time before it; and in the long run, say in some thousands of generations, time will be economized by proceeding in an orderly manner, and by making it an invariable rule to try the simpler hypothesis first. Indeed, one can never be sure that the simpler hypothesis is not the true one, after all, until its cause has been fought out to the bitter end. But you will mark the limitation of my approval of Ockham's razor. It is a sound maxim of scientific procedure. If the question be what one ought to believe, the logic of the situation must take other factors into account. Speaking strictly, belief is out of place in pure theoretical science, which has nothing nearer to it than the establishment of doctrines, and only the provisional establishment of them, at that. Compared with living belief it is nothing but a ghost. If the captain of a vessel on a lee shore in a terrific storm finds himself in a critical position in which he must instantly either put his wheel to port acting on one hypothesis, or put his wheel to starboard acting on the contrary hypothesis, and his vessel will infallibly be dashed to pieces if he decides the question wrongly, Ockham's razor is not worth the stout belief of any common seaman. For stout belief may happen to save the ship, while Entia non sunt multiplicanda praeter necessitatem would be only a stupid way of spelling Shipwreck. Now in matters of real practical concern we are all in something like the situation of that sea-captain."

- Charles Sanders Peirce

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"An "Argument" is any process of thought reasonably tending to produce a definite belief. An "Argumentation" is an Argument proceeding upon definitely formulated premisses. If God Really be, and be benign, then, in view of the generally conceded truth that religion, were it but proved, would be a good outweighing all others, we should naturally expect that there would be some Argument for His Reality that should be obvious to all minds, high and low alike, that should earnestly strive to find the truth of the matter; and further, that this Argument should present its conclusion, not as a proposition of metaphysical theology, but in a form directly applicable to the conduct of life, and full of nutrition for man's highest growth. What I shall refer to as the N.A. — the Neglected Argument — seems to me best to fulfil this condition, and I should not wonder if the majority of those whose own reflections have harvested belief in God must bless the radiance of the N.A. for that wealth. Its persuasiveness is no less than extraordinary; while it is not unknown to anybody. Nevertheless, of all those theologians (within my little range of reading) who, with commendable assiduity, scrape together all the sound reasons they can find or concoct to prove the first proposition of theology, few mention this one, and they most briefly. They probably share those current notions of logic which recognise no other Arguments than Argumentations."

- Charles Sanders Peirce

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"The hypothesis of God is a peculiar one, in that it supposes an infinitely incomprehensible object, although every hypothesis, as such, supposes its object to be truly conceived in the hypothesis. This leaves the hypothesis but one way of understanding itself; namely, as vague yet as true so far as it is definite, and as continually tending to define itself more and more, and without limit. The hypothesis, being thus itself inevitably subject to the law of growth, appears in its vagueness to represent God as so, albeit this is directly contradicted in the hypothesis from its very first phase. But this apparent attribution of growth to God, since it is ineradicable from the hypothesis, cannot, according to the hypothesis, be flatly false. Its implications concerning the Universes will be maintained in the hypothesis, while its implications concerning God will be partly disavowed, and yet held to be less false than their denial would be. Thus the hypothesis will lead to our thinking of features of each Universe as purposed; and this will stand or fall with the hypothesis. Yet a purpose essentially involves growth, and so cannot be attributed to God. Still it will, according to the hypothesis, be less false to speak so than to represent God as purposeless."

- Charles Sanders Peirce

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"My original essay, having been written for a popular monthly, assumes, for no better reason than that real inquiry cannot begin until a state of real doubt arises and ends as soon as Belief is attained, that "a settlement of Belief," or, in other words, a state of satisfaction, is all that Truth, or the aim of inquiry, consists in. The reason I gave for this was so flimsy, while the inference was so nearly the gist of Pragmaticism, that I must confess the argument of that essay might with some justice be said to beg the question. The first part of the essay, however, is occupied with showing that, if Truth consists in satisfaction, it cannot be any actual satisfaction, but must be the satisfaction which would ultimately be found if the inquiry were pushed to its ultimate and indefeasible issue. This, I beg to point out, is a very different position from that of Mr Schiller and the pragmatists of to-day. I trust I shall be believed when I say that it is only a desire to avoid being misunderstood in consequence of my relations with pragmatism, and by no means as arrogating any superior immunity from error which I have too good reason to know that I do not enjoy, that leads me to express my personal sentiments about their tenets. Their avowedly undefinable position, if it be not capable of logical characterisation, seems to me to be characterised by an angry hatred of strict logic, and even some disposition to rate any exact thought which interferes with their doctrines as all humbug. At the same time, it seems to me clear that their approximate acceptance of the Pragmaticist principle, and even that very casting aside of difficult distinctions (although I cannot approve of it), has helped them to a mightily clear discernment of some fundamental truths that other philosophers have seen but through a mist, and most of them not at all. Among such truths — all of them old, of course, yet acknowledged by few — I reckon their denial of necessitarianism; their rejection of any "consciousness" different from a visceral or other external sensation; their acknowledgment that there are, in a Pragmatistical sense, Real habits (which Really would produce effects, under circumstances that may not happen to get actualised, and are thus Real generals); and their insistence upon interpreting all hypostatic abstractions in terms of what they would or might (not actually will) come to in the concrete. It seems to me a pity they should allow a philosophy so instinct with life to become infected with seeds of death in such notions as that of the unreality of all ideas of infinity and that of the mutability of truth, and in such confusions of thought as that of active willing (willing to control thought, to doubt, and to weigh reasons) with willing not to exert the will (willing to believe)."

- Charles Sanders Peirce

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"Intensive research in recent years into the sources of economic growth among both developing and developed nations generally point to a number of important factors: the state of knowledge and skill of a population; the degree of control over indigenous natural resources; the quality of a country's legal system, particularly a strong commitment to a rule of law and protection of property rights; and yes, the extent of a country's openness to trade with the rest of the world. For the United States, arguably the most important factor is the type of rule of law under which economic activity takes place. When asked abroad why the United States has become the most prosperous large economy in the world, I respond, with only mild exaggeration, that our forefathers wrote a constitution and set in motion a system of laws that protects individual rights, especially the right to own property. Nonetheless, the degree of state protection is sometimes in dispute. But by and large, secure property rights are almost universally accepted by Americans as a critical pillar of our economy. While the right of property in the abstract is generally uncontested in all societies embracing democratic market capitalism, different degrees of property protection do apparently foster different economic incentives and outcomes."

- Alan Greenspan

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"I remember a talk that Von Neumann gave at Princeton around 1950, describing the glorious future which he then saw for his computers. Most of the people that he hired for his computer project in the early days were meteorologists. Meteorology was the big thing on his horizon. He said, as soon as we have good computers, we shall be able to divide the phenomena of meteorology cleanly into two categories, the stable and the unstable. The unstable phenomena are those which are upset by small disturbances, the stable phenomena are those which are resilient to small disturbances. He said, as soon as we have some large computers working, the problems of meteorology will be solved. All processes that are stable we shall predict. All processes that are unstable we shall control. He imagined that we needed only to identify the points in space and time at which unstable processes originated, and then a few airplanes carrying smoke generators could fly to those points and introduce the appropriate small disturbances to make the unstable processes flip into the desired directions. A central committee of computer experts and meteorologists would tell the airplanes where to go in order to make sure that no rain would fall on the Fourth of July picnic. This was John von Neumann's dream. This, and the hydrogen bomb, were the main practical benefits which he saw arising from the development of computers."

- John von Neumann

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"Von Neumann was a true genius, the only one I’ve ever known. I’ve met Einstein and Oppenheimer and Teller and—who’s the mad genius from MIT? I don’t mean McCulloch, but a mathematician. Any-way, a whole bunch of those other guys. Von Neumann was the only genius I ever met. The others were supersmart .... And great prima donnas. But von Neumann’s mind was all-encompassing. He could solve problems in any domain. . . . And his mind was always working, always restless. He walked into my living room one night and a half dozen people were already having cocktails, and he disappeared into a corner and stood with his back to us, hands behind him, and after about two minutes turned to me and said, “About two thirds of a liter a week, Leon.” And I had to think about it for three or four minutes, and finally I said, “Yeah, Johnny, that’s just about right.” He’d walked up to the nine-gallon tropical fish aquarium that stood on a table in the corner, had noted the temperature of the water, had made an estimate of the surface area, had seen the gap that existed between the overhead light and the glass to keep the fish from jumping out, made an estimate of the particular escape velocity of the water molecules, integrated and found out how much added water was needed each week for that aquarium. And he was right within a few percent. That’s the kind of thing he did all the time. Another thing that he isn’t known well for was his sense of humor. He really enjoyed dirty limericks. And though we never said anything to each other deliberately, it sort of evolved that whenever we came together, whether it was an hour or a month later, the name of the game was to see who could rush up the fastest and unload the largest number of new limericks. It turned out to be a delightful game. He had oodles of them; I was hard put to keep up with him. His memory was just beyond conception, a photograph for everything he ever learned or saw. Lightning calculator and head screwed on to boot—he put all of those together with a huge creative talent."

- John von Neumann

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""Johnny" von Neumann, as he was always known among scientists, achieved fame first of all as a pure mathematician. I am not qualified to describe his contributions to pure mathematics, which usually related to the most recent, and most abstruse, branches of the subject at the time, but they certainly placed him among the leaders of modern mathematics. In the 1920s, he was interested in the development of quantum mechanics, then in rapid growth, which caused difficulty to many because of the bold use of new mathematical techniques. Von Neumann contributed greatly to making this new subject "respectable"; he pointed out the precise mathematical significance of the new developments and, at the same time, helped greatly to clarify the physical content of the new ideas. He was, in fact, quicker than many physicists in grasping the changes that were then taking place in physics Later he was a frequent visitor to the Atomic Weapons Project at Los Alamos. Here his particular quality of combining powerful mathematical insight with a very practical interest in the problems became familiar to all those associated with the project. He was never satisfied with showing that a problem could be solved on paper, but he took a personal interest in its quantitative application and in its practical realization. His many contributions, particularly to the hydrodynamics of shock waves and detonation waves, which are important both in the design of atomic weapons and in an understanding of their effects, were vital to the success of the project. For a man to whom complicated mathematics presented no difficulty, he could explain his conclusions to the uninitiated with amazing lucidity. After a talk with him one always came away with a feeling that the problem was really simple and transparent. About the same time, he became interested in the application of computing techniques to mathematical problems, and this led him to design the computer now in operation at Princeton and to planning out its applications both to practical problems and to abstract problems in nonlinear equations. He was the antithesis of the conventional image of the "long-haired" mathematics don. Always well-groomed, he had as lively views on international politics and practical affairs as on mathematical problems. His book on the Theory of Games, "including the theory of bluffing at poker," which has proved fruitful for many applications going beyond the field of games of chance and skill, is another example of the happy combination of his command of mathematics with an interest in practical matters. For the last few years, he was a member of the Atomic Energy Commission, and it is worth recording that in a field beset with much controversy, he retained the universal respect and confidence of those who did not agree with his views on policy as well as those who did."

- John von Neumann

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"Many people have wondered how Johnny von Neumann could think so fast and so effectively. How he could find so many original solutions, in areas where most people did not even notice the problems. I think I know a part of the answer, perhaps an important part, Johnny von Neumann enjoyed thinking. I have come to suspect that to most people, thinking is painful. Some of us are addicted to thinking. Some of us find it a necessity. Johnny enjoyed it. I even have a suspicion that he enjoyed practically nothing else. This explains a lot, because what you like, you do well. And he liked thinking, not just in mathematics. He liked thinking in the clear and complete manner of mathematicians, in every field; in mathematics, in physics, in the business world - his father was a banker - and in many other fields. He could and did talk to my 3-year-old son on his own terms, and I sometimes wondered whether his relation to the rest of us were a little bit similar. This also explains his effectiveness in connection with computing machines, because computing machines apply logical processes to fields: not only mathematics, but to others as yet untouched by the logical process. And it is very significant that this revolution, the revolution of the electronic brains, was practically initiated by Johnny von Neumann. I cannot think of Johnny now without remembering a very tragic circumstance when he was dying of cancer. His brain was affected. I visited him frequently and he was trying to do what he always tried to do. And he was trying to argue with me as he used to and it wasn't functioning anymore. And I think that he suffered from this loss more than I have seen any human to suffer in any other circumstance."

- John von Neumann

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"Throughout much of his career, he led a double life: as an intellectual leader in the ivory tower of pure mathematics and as a man of action, in constant demand as an advisor, consultant and decision-maker to what is sometimes called the military-industrial complex of the United States. My own belief is that these two aspects of his double life, his wide-ranging activities as well as his strictly intellectual pursuits, were motivated by two profound convictions. The first was the overriding responsibility that each of us has to make full use of whatever intellectual capabilities we were endowed with. He had the scientist's passion for learning and discovery for its own sake and the genius's ego-driven concern for the significance and durability of his own contributions. The second was the critical importance of an environment of political freedom for the pursuit of the first, and for the welfare of mankind in general. I'm convinced, in fact, that all his involvements with the halls of power were driven by his sense of the fragility of that freedom. By the beginning of the 1930s, if not even earlier, he became convinced that the lights of civilization would be snuffed out all over Europe by the spread of totalitarianism from the right: Nazism and Fascism. So he made an unequivocal commitment to his home in the new world and to fight to preserve and reestablish freedom from that new beachhead. In the 1940s and 1950s, he was equally convinced that the threat to civilization now came from totalitarianism on the left, that is, Soviet Communism, and his commitment was just as unequivocal to fighting it with whatever weapons lay at hand, scientific and economic as well as military. It was a matter of utter indifference to him, I believe, whether the threat came from the right or from the left. What motivated both his intense involvement in the issues of the day and his uncompromisingly hardline attitude was his belief in the overriding importance of political freedom, his strong sense of its continuing fragility, and his conviction that it was in the United States, and the passionate defense of the United States, that its best hope lay."

- John von Neumann

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"Omni: Do you find it depressing that chess computers are getting so strong? Shannon: I am not depressed by it. I am rooting for the machines! I have always been on the machines' side. Ha-ha! Betty Shannon: Some people get livid when he says that. Shannon: I am not depressed by machines getting better. Whether people will be replaced by machines that have gotten smarter in all things, I can't say. Within a century or so, machines will be doing almost everything better than we can. They already do factory work better than we can, but the highly intellectual stuff is going to come later. It gets harder and harder as you get higher and higher in this game. Omni: Do you agree with Norbert Wiener's denial of any basic distinction between life and nonlife, man and machine? Shannon: That's a heavily loaded question there! I'm an atheist to begin with. I believe in evolutionary theory and that we are basically machines but of a very complex type, far more so than any machine that man has made yet. So that's both a yes and a no. Mechanical doesn't just mean that metal and gears are involved, of course. We are the extreme case: a natural mechanical device. I see no God involved. Omni: Will robots be complex enough to be friends of people? Shannon: I think so. I myself could very easily imagine that happening. I see no limit to the capabilities of machines. As microchips get smaller and faster, I can see them getting better than we are. I can visualize a time in the future when we will be to robots as dogs are to humans. Omni: Can you imagine a robot president of the United States? Shannon: Could be, but I think by then you wouldn't speak of the United States anymore. The world will have a totally different organization. Omni: Is it a big leap from the pedestrian routines of today's chess computers to machines that could grapple, seemingly in a creative, intuitive fashion, with the problems of higher mathematics? Shannon: I see computers proving theorems that have been sitting around that nobody's proved. I don't yet see them creating theories, that is, discovering a new branch of mathematics, as many great mathematicians have in the past. That's a broader, wider thing—more like writing a play—and will be a lot longer in coming. Omni: Is your famous proof that a reliable circuit can be built from unreliable components relevant to the brain's operations? Shannon: The brain can suffer all kinds of damage and yet can still handle things pretty well. It must use some redundancy to take care of faulty operations, such as the death of certain neurons. The modern desk computer generally has no redundancy, so if one part gets into trouble, that will show up in later operations. That we manage to live in spite of all kinds of internal troubles suggests that the brain's design involves a great deal of redundancy or parallelism of multiple units. Omni: Your paper shows that if the relays closed only sixty percent of the time when triggered, you could have highly effective circuitry. Could the brain be using such an approach? Shannon: That the brain has ten billion neurons probably means it was cheaper for biology to make more components than to work out sophisticated circuits. Yet I am totally astounded at how clever and sophisticated some of the things we see in human or animal bodies are. Such long-term, sophisticated changes could be what happened in the brain, but an easier way would be to use paralleling and multiplication to reduce errors of individual neuron operation. And when it all gets going, we have these clever people like Einstein. ... Omni: Has your ambition waned at all? Shannon: I was never motivated by the notion of winning prizes or the desire for financial gain. My motivation in science has always been curiosity about something: How is it put together? What laws or rules govern this situation? Are there any theorems one can prove about what one can or can’t do? After I had found answers, it was always painful to publish, which is where you get the acclaim. Many things I have done and never written up at all. Too lazy, I guess. I have got a file upstairs of unfinished papers! Ha-ha-ha! But that’s true of most of the good scientists I know. Just knowing for ourselves is probably our main motivation."

- Claude Elwood Shannon

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"Since Leibniz there has perhaps been no man who has had a full command of all the intellectual activity of his day. Since that time, science has been increasingly the task of specialists, in fields which show a tendency to grow progressively narrower... Today there are few scholars who can call themselves mathematicians or physicists or biologists without restriction. A man may be a topologist or a coleopterist. He will be filled with the jargon of his field, and will know all its literature and all its ramifications, but, more frequently than not, he will regard the next subject as something belonging to his colleague three doors down the corridor, and will consider any interest in it on his own part as an unwarrantable breach of privacy... There are fields of scientific work, as we shall see in the body of this book, which have been explored from the different sides of pure mathematics, statistics, electrical engineering, and neurophysiology; in which every single notion receives a separate name from each group, and in which important work has been triplicated or quadruplicated, while still other important work is delayed by the unavailability in one field of results that may have already become classical in the next field. It is these boundary regions which offer the richest opportunities to the qualified investigator. They are at the same time the most refractory to the accepted techniques of mass attack and the division of labor. If the difficulty of a physiological problem is mathematical in essence, then physiologists ignorant of mathematics will get precisely as far as one physiologists ignorant of mathematics, and no further. If a physiologist who knows no mathematics works together with a mathematician who knows no physiology, the one will be unable to state his problem in terms that the other can manipulate, and the second will be unable to put the answers in any form that the first can understand... A proper exploration of these blank spaces on the map of science could only be made by a team of scientists, each a specialist in his own field but each possessing a thoroughly sound and trained acquaintance with the fields of his neighbors; all in the habit of working together, of knowing one another's intellectual customs, and of recognizing the significance of a colleague's new suggestion before it has taken on a full formal expression. The mathematician need not have the skill to conduct a physiological experiment, but he must have the skill to understand one, to criticize one, and to suggest one. The physiologist need not be able to prove a certain mathematical theorem, but he must be able to grasp its physiological significance and to tell the mathematician for what he should look. We had dreamed for years of an institution of independent scientists, working together in one of these backwoods of science, not as subordinates of some great executive officer, but joined by the desire, indeed by the spiritual necessity, to understand the region as a whole, and to lend one another the strength of that understanding."

- Norbert Wiener

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"The odors perceived by the ant seem to lead to a highly standardized course of conduct; but the value of a simple stimulus, such as an odor, for conveying information depends not only on the information conveyed by the stimulus itself but on the whole nervous constitution of the sender and receiver of the stimulus as well. Suppose I find myself in the woods with an intelligent savage who cannot speak my language and whose language I cannot speak. Even without any code of sign language common to the two of us, I can learn a great deal from him. All I need to do is to be alert to those moments when he shows the signs of emotion or interest. I then cast my eyes around, perhaps paying special attention to the direction of his glance, and fix in my memory what I see or hear. It will not be long before I discover the things which seem important to him, not because he has communicated them to me by language, but because I myself have observed them. In other words, a signal without an intrinsic content may acquire meaning in his mind by what he observes at the time, and may acquire meaning in my mind by what I observed at the time. The ability that he has to pick out the moments of my special, active attention is in itself a language as varied in possibilities as the range of impressions that the two of us are able to encompass. Thus social animals may have an active, intelligent, flexible means of communication long before the development of language."

- Norbert Wiener

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"It is my thesis that the physical functioning of the living individual and the operation of some of the newer communication machines are precisely parallel in their analogous attempts to control entropy through feedback. Both of them have sensory receptors as one stage in their cycle of operation: that is, in both of them there exists a special apparatus for collecting information from the outer world at low energy levels, and for making it available in the operation of the individual or of the machine. In both cases these external messages are not taken neat, but through the internal transforming powers of the apparatus, whether it be alive or dead. The information is then turned into a new form available for the further stages of performance. In both the animal and the machine this performance is made to be effective on the outer world. In both of them, their performed action on the outer world, and not merely their intended action, is reported back to the central regulatory apparatus. This complex of behavior is ignored by the average man, and in particular does not play the role that it should in our habitual analysis of society; for just as individual physical responses may be seen from this point of view, so may the organic responses of society itself. I do not mean that the sociologist is unaware of the existence and complex nature of communications in society, but until recently he has tended to overlook the extent to which they are the cement which binds its fabric together."

- Norbert Wiener

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"As I near the end of my personal recollections of life at M.I.T., it is impossible to refrain from relating my eye-witness stories about a brilliant man, Norbert Wiener, and his lovable eccentricities. I took two semester courses under Professor Wiener: one was Fourier Series and Fourier Integrals, and the other was, I believe, Operational Calculus. It is vivid in my memory that Professor Wiener would always come to class without any lecture notes. He would first take out his big handkerchief and blow his nose very vigorously and noisily. He would pay very little attention to his class and would seldom announce the subject of his lecture. He would face the blackboard, standing very close to it because he was extremely near-sighted. Although I usually sat in the front row, I had difficulty seeing what he wrote. Most of the other students could not see anything at all. It was most amusing to the class to hear Professor Wiener saying to himself, "This was very wrong, definitely." He would quickly erase all he had written down. He would then start all over again, and sometimes murmur to himself, "This looks all right so far." Minutes later, "This cannot be right either," and he would rub it all out again. This on- again, off-again process continued until the bell signaled the end of the hour. Then Professor Wiener would leave the room without even looking at his audience."

- Norbert Wiener

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"Lysenkoism is held up by bourgeois commentators as the supreme demonstration that conscious ideology cannot inform scientific practice and that "ideology has no place in science." On the other hand, some writers are even now maintaining a Lysenkoist position because they believe that the principles of dialectical materialism contradict the claims of genetics. Both of these claims stem from a vulgarisation of Marxist philosophy through deliberate hostility, in the first case, or ignorance, in the second. Nothing in Marx, Lenin or Mao contradicts the particular physical facts and processes of a particular set of natural phenomena in the objective world, because what they wrote about nature was at a high level of abstraction. The error of the Lysenkoist claim arises from attempting to apply a dialectical analysis of physical problems from the wrong end. Dialectical materialism is not, and has never been, a programmatic method for solving particular physical problems. Rather, dialectical analysis provides an overview and a set of warning signs against particular forms of dogmatism and narrowness of thought. It tells us, "Remember that history may leave an important trace. Remember that being and becoming are dual aspects of nature. Remember that conditions change and that the conditions necessary to the initiation of some process may be destroyed by the process itself. Remember to pay attention to real objects in space and time and not lose them utterly in idealized abstractions. Remember that qualitative effects of context and interaction may be lost when phenomena are isolated." And above all else, "Remember that all the other caveats are only reminders and warning signs whose application to different circumstances of the real world is contingent.""

- Richard Lewontin

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"It is time for students of the evolutionary process, especially those who have been misquoted and used by the creationists, to state clearly that evolution is a fact, not theory, and that what is at issue within biology are questions of details of the process and the relative importance of different mechanisms of evolution. It is a fact that the earth, with liquid water, is more than 3.6 billion years old. It is a fact that cellular life has been around for at least half of that period and that organized multicellular life is at least 800 million years old. It is a fact that major life forms now on earth were not at all represented in the past. There were no birds or mammals 250 million years ago. It is a fact that major life forms of the past are no longer living. There used to be dinosaurs and Pithecanthropus, and there are none now. It is a fact that all living forms come from previous living forms. Therefore, all present forms of life arose from ancestral forms that were different. Birds arose from nonbirds and humans from nonhumans. No person who pretends to any understanding of the natural world can deny these facts any more than she or he can deny that the earth is round, rotates on its axis, and revolves around the sun. The controversies about evolution lie in the realm of the relative importance of various forces in molding evolution."

- Richard Lewontin

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"The great attraction of cultural anthropology in the past was precisely that it seemed to offer such a richness of independent natural experiments; but unfortunately it is now clear that there has been a great deal of historical continuity and exchange among those "independent" experiments, most of which have felt the strong effect of contact with societies organized as modern states. More important, there has never been a human society with unlimited resources, of three sexes, or the power to read other people's minds, or to be transported great distances at the speed of light. How then are we to know the effect on human social organization and history of the need to scrabble for a living, or of the existence of males and females, or of the power to make our tongues drop manna and so to make the worse appear the better reason? A solution to the epistemological impotence of social theory has been to create a literature of imagination and logic in which the consequences of radical alterations in the conditions of human existence are deduced. It is the literature of science fiction. … [S]cience fiction is the laboratory in which extraordinary social conditions, never possible in actuality, are used to illumine the social and historical norm. … Science fiction stories are the Gedanken experiments of social science."

- Richard Lewontin

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"With great perception, Sagan sees that there is an impediment to the popular credibility of scientific claims about the world, an impediment that is almost invisible to most scientists. Many of the most fundamental claims of science are against common sense and seem absurd on their face. Do physicists really expect me to accept without serious qualms that the pungent cheese that I had for lunch is really made up of tiny, tasteless, odorless, colorless packets of energy with nothing but empty space between them? Astronomers tell us without apparent embarrassment that they can see stellar events that occurred millions of years ago, whereas we all know that we see things as they happen. (…) Our willingness to accept scientific claims that are against common sense is the key to an understanding of the real struggle between science and the supernatural. We take the side of science in spite of the patent absurdity of some of its constructs, in spite of its failure to fulfill many of its extravagant promises of health and life, in spite of the tolerance of the scientific community for unsubstantiated just-so stories, because we have a prior commitment, a commitment to materialism. It is not that the methods and institutions of science somehow compel us to accept a material explanation of the phenomenal world, but, on the contrary, that we are forced by our a priori adherence to material causes to create an apparatus of investigation and a set of concepts that produce material explanations, no matter how counter-intuitive, no matter how mystifying to the uninitiated. Moreover, that materialism is absolute, for we cannot allow a Divine Foot in the door. The eminent Kant scholar Lewis Beck used to say that anyone who could believe in God could believe in anything. To appeal to an omnipotent deity is to allow that at any moment the regularities of nature may be ruptured, that miracles may happen."

- Richard Lewontin

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"To say that genetic differences are relevant to hetero- and homosexuality is not, however, to say that there are “genes for homosexuality” or even that there is a “genetic tendency to homosexuality.” This critical point can be illustrated by an example I owe to the philosopher of science, Elliott Sober. If we look at the chromosomes of people who knit and those who do not, we will find that with few exceptions, knitters have two X chromosomes, while people with one X and one Y chromosome almost never knit. Yet it would be absurd to say that we had discovered genes for knitting. The possession of two X chromosomes causes an embryo, with rare exceptions, to develop into an anatomical and physiological female, while the possession of a Y chromosome leads almost always to male development, and in our culture women are taught to knit while men are not. The beauty of this example is its historical (and geographical) contingency. Had we made the observations before the end of the eighteenth century (or even now in a few Irish, Scottish and Newfoundland communities), the result would have been reversed. Hand knitting was men’s work before the introduction of knitting machines around 1790, and was turned into a female domestic occupation only when mechanization made it economically marginal."

- Richard Lewontin

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"When, where, and how were you, Joseph Smith, first called? How old were you? and what were you qualifications? I was between fourteen and fifteen years of age. Had you been to college? No. Had you studied in any seminary of learning? No. Did you know how to read? Yes. How to write? Yes. Did you understand much about arithmetic? No. About grammar? No. Did you understand all the branches of education which are generally taught in our common schools? No. But yet you say the Lord called you when you were but fourteen or fifteen years of age? How did he call you? I will give you a brief history as it came from his own mouth. I have often heard him relate it. He was wrought upon by the Spirit of God, and felt the necessity of repenting of his sins and serving God. He retired from his father's house a little way, and bowed himself down in the wilderness, and called upon the name of the Lord. He was inexperienced, and in great anxiety and trouble of mind in regard to what church he should join. He had been solicited by many churches to join with them, and he was in great anxiety to know which was right. He pleaded with the Lord to give him wisdom on the subject; and while he was thus praying, he beheld a vision, and saw a light approaching him from the heavens; and as it came down and rested on the tops of the trees, it became more glorious; and as it surrounded him, his mind was immediately caught away from beholding surrounding objects. In this cloud of light he saw two glorious personages; and one, pointing to the other, said, "Behold my beloved son! hear ye him.""

- Orson Pratt

0 likesPeople from New York (state)Leaders in The Church of Jesus Christ of Latter-day SaintsMathematicians from the United StatesEditors from the United StatesPioneers from the United States
"By and by an obscure individual, a young man, rose up, and, in the midst of all Christendom, proclaimed the startling news that God had sent an angel to him; that through his faith, prayers, and sincere repentance he had beheld a supernatural vision, that he had seen a pillar of fire descend from Heaven, and saw two glorious personages clothed upon with this pillar of fire, whose countenance shone like the sun at noonday; that he heard one of these personages say, pointing to the other, 'This is my beloved Son, hear ye him.' This occurred before this young man was fifteen years of age; and it was a startling announcement to make in the midst of a generation so completely given up to the traditions of their fathers; and when this was proclaimed by this young, unlettered boy to the priests and the religious societies in the State of New York, they laughed him to scorn. 'What!' said they, "visions and revelations in our day! God speaking to men in our day!" They looked upon him as deluded; they pointed the finger of scorn at him and warned their congregations against him. 'The canon of Scripture is closed up; no more communications are to be expected from Heaven. The ancients saw heavenly visions and personages; they heard the voice of the Lord; they were inspired by the Holy Ghost to receive revelations, but behold no such thing is to be given to man in our day, neither has there been for many generations past.' This was the style of the remarks made by religionists forty years ago. This young man, some four years afterwards, was visited again by a holy angel."

- Orson Pratt

0 likesPeople from New York (state)Leaders in The Church of Jesus Christ of Latter-day SaintsMathematicians from the United StatesEditors from the United StatesPioneers from the United States
"But by and by the time came when the Christian Church apostatized and turned away, and began to follow after their own wisdom, and the Prophets and Apostles ceased, so far as the affairs of the Christian Church on the earth were concerned. Revelations, and visions, and the various gifts of the spirit were also taken away, according to their unbelief and apostacy; but in the latter days God intends to again raise up a Christian Church upon the earth. Do not be startled, you who think that God will no more have a Church on the earth, for he has promised that he would again have one, and that he would set up his kingdom, and when he does you may look out for a great many Prophets and inspired men; and if you ever see a Church arise, calling itself a Christian Church, and it has not inspired Apostles like those in ancient times, you may know that it is a spurious church, and that it makes pretensions to something that it does not enjoy. If you ever find a church called a Christian Church that has no men to foretell future events, you may know, at once, that it is not a Christian Church. If you find a Christian Church that has not the ancient gifts, for instance the gift of healing, opening the eyes of the blind, unstopping the ears of the deaf, causing the tongue of the dumb to speak and the lame to walk; if you ever find a people calling themselves a Christian Church and they have not these gifts among them, you may know with a perfect knowledge that they do not agree with the pattern given in the New Testament. The Christian Church is always characterized with inspired men, whose revelations are just as sacred as any contained in the Bible; and, if written and published, just as binding upon the human family. The Christian Church will always lay hands upon the sick in the name of Jesus, in order that the sick may be healed. The Christian Church will always have those among its members who have heavenly visions, the ministration of angels, and the various gifts that are promised according to the Gospel."

- Orson Pratt

0 likesPeople from New York (state)Leaders in The Church of Jesus Christ of Latter-day SaintsMathematicians from the United StatesEditors from the United StatesPioneers from the United States
"We planted our crops in the spring, and they came up, and were looking nicely, and we were cheered with the hopes of having a very abundant harvest. But alas! it very soon appeared as if our crops were going to be swallowed up by a vast horde of crickets, that came down from these mountains-crickets very different to what I used to be acquainted with in the State of New York. They were crickets nearly as large as a man's thumb. They came in immense droves, so that men and women with brush could make no headway against them; but we cried unto the Lord in our afflictions, and the Lord heard us, and sent thousands and tens of thousands of a small white bird. I have not seen any of them lately. Many called them gulls, although they were different from the seagulls that live on the Atlantic coast. And what did they do for us? They went to work, and by thousands and tens of thousands, began to devour them up, and still we thought that even they could not prevail against so large and mighty an army. But we noticed, that when they had apparently filled themselves with these crickets, they would go and vomit them up, and again go to work and fill themselves, and so they continued to do, until the land was cleared of crickets, and our crops were saved. There are those who will say that this was one of the natural courses of events, that there was no miracle in it. Let that be as it may, we esteemed it as a blessing from the hand of God; miracle or no miracle, we believe that God had a hand in it, and it does not matter particularly whether strangers believe or not."

- Orson Pratt

0 likesPeople from New York (state)Leaders in The Church of Jesus Christ of Latter-day SaintsMathematicians from the United StatesEditors from the United StatesPioneers from the United States
"Sir, I am fully sensible of the greatness of that freedom, which I take with you on the present occasion; a liberty which seemed to me scarcely allowable, when I reflected on that distinguished and dignified station in which you stand, and the almost general prejudice and prepossession, which is so prevalent in the world against those of my complexion. I suppose it is a truth too well attested to you, to need a proof here, that we are a race of beings, who have long labored under the abuse and censure of the world; that we have long been looked upon with an eye of contempt; and that we have long been considered rather as brutish than human, and scarcely capable of mental endowments. Sir, I hope I may safely admit, in consequence of that report which hath reached me, that you are a man far less inflexible in sentiments of this nature, than many others; that you are measurably friendly, and well disposed towards us; and that you are willing and ready to lend your aid and assistance to our relief, from those many distresses, and numerous calamities, to which we are reduced. Now Sir, if this is founded in truth, I apprehend you will embrace every opportunity, to eradicate that train of absurd and false ideas and opinions, which so generally prevails with respect to us ; and that your sentiments are concurrent with mine, which are, that one universal Father hath given being to us all ; and that he hath not only made us all of one flesh, but that he hath also, without partiality, afforded us all the same sensations and endowed us all with the same faculties ; and that however variable we may be in society or religion, however diversified in situation or color, we are all of the same family, and stand in the same relation to him. Sir, if these are sentiments of which you are fully persuaded, I hope you cannot but acknowledge, that it is the indispensible duty of those, who maintain for themselves the rights of human nature, and who possess the obligations of Christianity, to extend their power and influence to the relief of every part of the human race, from whatever burden or oppression they may unjustly labor under ; and this, I apprehend, a full conviction of the truth and obligation of these principles should lead all to."

- Benjamin Banneker

0 likesAfrican AmericansMathematicians from the United StatesAstronomers from the United StatesFarmersScientists from Maryland
"'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."

- David Rittenhouse

0 likesAcademics from the United StatesAstronomers from the United StatesPeople from PhiladelphiaInventorsMathematicians from the United States
"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."

- David Rittenhouse

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"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."

- David Rittenhouse

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"Looking back over the space of fifty years since I entered Harvard College, Benjamin Peirce still impresses me as having the most massive intellect with which I have ever come in contact, and as being the most profoundly inspiring teacher I ever had. … As soon as he had finished the problem or filled the blackboard he would rub everything out and begin again. He was impatient of detail, and sometimes the result would not come out right; but instead of going over his work to find the error, he would rub it out, saying that he had made a mistake in a sign somewhere, and that we should find it when we went over our notes. Described in this way it may seem strange that such a method of teaching should be inspiring; yet to us it was so to the highest degree. We were carried along by the rush of his thought, by the ease and grasp of his intellectual movement. The inspiration came, I think, partly from his treating us as highly competent pupils, capable of following his line of thought even through errors in transformations; partly from his rapid and graceful methods of proof, which reached a result with the least number of steps in the process, attaining thereby an artistic or literary character; and partly from the quality of his mind which tended to regard any mathematical theorem as a particular case of some more comprehensive one, so that we were led onward to constantly enlarging truths."

- Benjamin Peirce

0 likesMathematicians from the United StatesAcademics from the United StatesScientists from MassachusettsHarvard University facultyHarvard University alumni
"I have not attempted in these papers to put forward any grand view of the nature of philosophy; nor do I have any such grand view to put forward if I would. It will be obvious that I do not agree with those who see philosophy as the history of 'howlers', and progress in philosophy as the debunking of howlers. It will also be obvious that I do not agree with those who see philosophy as the enterprise of putting forward a priori truths about the real world (since, for one thing, there are no a priori truths, in my view). I see philosophy as a field which has certain central questions, for example, the relation between thought and reality, and, to mention some questions about which I have not written, the relation between freedom and responsibility, and the nature of the good life. It seems obvious that in dealing with these questions philosophers have formulated rival research programs, that they have put forward general hypotheses, and that philosophers within each major research program have modified their hypotheses by trial and error, even if they sometimes refuse to admit that that is what they are doing. To that extent philosophy is a 'science'. To argue about whether philosophy is a science in any more serious sense seems to me to be hardly a useful occupation. The important thing is that in spite of the stereotypes of science and philosophy that have become blinkers inhibiting the view of laymen, scientists, and philosophers, science and philosophy are interdependent activities; philosophers have always found it essential to draw upon the scientific knowledge of the time, and scientists have always found it essential to do a certain amount of philosophy in their very scientific work, even if they denied that that was what they were doing. It does not seem to me important to decide whether science is philosophy or philosophy is science as long as one has a conception of both that makes both essential to a responsible view of the real world and of man's place in it."

- Hilary Putnam

0 likesWomen academics from the United StatesPrinceton University facultyCognitive scientistsComputer scientists from the United StatesMathematicians from the United States
"I do not agree with Quine, that there is no analytic-synthetic distinction to be drawn at all. But I do believe that his emphasis on the monolithic character of our conceptual system and his negative emphasis on the silliness of regarding mathematics as consisting in some sense of 'rules of language', represent exceedingly important theoretical insights in philosophy. I think that what we have to do now is to settle the relatively trivial question concerning analytic statements properly so called ('All bachelors are unmarried'). We have to take a fresh look at the framework principles so much discussed by philosophers, disabusing ourselves of the idea that they are 'rules of language' in any literal or lexicographic sense; and above all, we have to take a fresh look at the nature of logical and mathematical truths. With Quine's contribution, we have to face two choices: We can ignore it and go on talking about the 'logic' of individual words. In that direction lies sterility and more, much more, of what we have already read. The other alternative is to face and explore the insight achieved by Quine, trying to reconcile the fact that Quine is overwhelmingly right in his critique of what other philosophers have done with the analytic-synthetic distinction with the fact that Quine is wrong in his literal thesis, namely, that the distinction itself does not exist at all. In the latter direction lies philosophic progress. For philosophic progress is nothing if it is not the discovery of new areas for dialectical exploration."

- Hilary Putnam

0 likesWomen academics from the United StatesPrinceton University facultyCognitive scientistsComputer scientists from the United StatesMathematicians from the United States
"We've watched the Homo Sapiens interest group since the first appearance of the Blight. Where is this "Earth" the humans claim to be from? "Half way around the galaxy," they say, and deep in the Slow Zone. Even their proximate origin, Nyjora, is conveniently in the Slowness. We see an alternative theory: Sometime, maybe further back than the last consistent archives, there was a battle between Powers. The blueprint for this "human race" was written, complete with communication interfaces. Long after the original contestants and their stories had vanished, this race happened to get in position where it could Transcend. And that Transcending was tailor-made, too, re-establishing the Power that had set the trap to begin with.We're not sure of the details, but a scenario such as this is inevitable. What we must do is also clear. Straumli Realm is at the heart of the Blight, obviously beyond all attack. But there are other human colonies. We ask the Net to help in identifying all of them. We ourselves are not a large civilization, but we would be happy to coordinate the information gathering, and the military action that is required to prevent the Blight's spread in the Middle Beyond. For nearly seventeen weeks, we've been calling for action. Had you listened in the beginning, a concerted strike might have been sufficient to destroy the Straumli Realm. Isn't the Fall of Relay enough to wake you up? Friends, if we act together we still have a chance.Death to vermin."

- Vernor Vinge

0 likesMathematicians from the United StatesAcademics from the United StatesComputer scientists from the United StatesNovelists from the United StatesScience fiction authors from the United States
"In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the “ordinary” and the “magicians.’’ An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark. They seldom, if ever, have students because they cannot be emulated and it must be terribly frustrating for a brilliant young mind to cope with the mysterious ways in which the magician’s mind works. Richard Feynman is a magician of the highest caliber. Hans Bethe, whom Dyson considers to be his teacher, is an “‘ordinary genius’’; so much so that one may gain the erroneous impression that he is not a genius at all. But it was Feynman, only slightly older than Dyson, who captured the young man’s imagination. To be a physicist must have meant to him to be like Feynman and this, alas, was impossible. And so Dyson fell back on the source of strength he always had in reserve: the mastery of mathematical technique."

- Mark Kac

0 likesMathematicians from PolandJewsMathematicians from the United StatesAcademics from Poland
"The story begins with a somewhat disgruntled hero, who perceived of the world as populated with stupid people, everywhere committing the environmental fallacy. The fallacy was a case not merely of the “mind’s falling into error,” but rather of the mind leading all of us into incredible dangers as it first builds crisis and then attacks crisis. Like all heroes, this one looked about for resources, for aids that would help in a dangerous battle, and he found plenty of support – in both the past and the present. It won’t hurt to summarize the story thus far. If the intellect is to engage in the heroic adventure of securing improvement in the human condition, it cannot rely on “approaches,” like politics and morality, which attempt to tackle problems head-on, within the narrow scope. Attempts to address problems in such a manner simply lead to other problems, to an amplification of difficulty away from real improvement. Thus the key to success in the hero’s attempt seems to be comprehensiveness. Never allow the temptation to be clear, or to use reliable data, or to “come up to the standards of excellence,” divert you from the relevant, even though the relevant may be elusive, weakly supported by data, and requiring loose methods. Thus the academic world of Western twentieth century society is a fearsome enemy of the systems approach, using as it does a politics to concentrate the scholars’ attention on matters that are scholastically respectable but disreputable from a systems-planning point of view."

- C. West Churchman

0 likesAcademics from the United StatesPhilosophers from the United StatesBusiness theorists from the United StatesMathematicians from the United StatesSystems scientists
"Churchman recognized in his critical systemic thinking that the human mind is not able to know the whole. … Yet the human mind, for Churchman, may appreciate the essential quality of the whole. For Churchman, appreciation of this essential quality begins … when first you see the world through the eyes of another. The systems approach, he says, then goes on to discover that every worldview is terribly restricted. Consequently, with Churchman, a rather different kind of question about practice surfaces. … That is, who is to judge that any one bounded appreciation is most relevant or acceptable? Each judgment is based on a rationality of its own that chooses where a boundary is to be drawn, which issues and dilemmas thus get on the agenda, and who will benefit from this. For each choice it is necessary to ask, What are the consequences to be expected insofar as we can evaluate them and, on reflection, how do we feel about that? As Churchman points out, each judgment of this sort is of an ethical nature since it cannot escape the choice of who is to be the client—the beneficiary—and thus which issues and dilemmas will be central to debate and future action. In this way, the spirit of C. West Churchman becomes our moral conscience. A key principle of systemic thinking, according to Churchman, is to remain ethically alert. Boundary judgments facilitate a debate in which we are sensitized to ethical issues and dilemmas."

- C. West Churchman

0 likesAcademics from the United StatesPhilosophers from the United StatesBusiness theorists from the United StatesMathematicians from the United StatesSystems scientists
"The businessman may be compared with two other types of individuals who are essentially concerned with behavior under uncertainty ― the scientist and the statistician. The scientist must choose, on the basis of limited information, among the innumerable logically conceivable laws of nature, a limited number. He cannot know whether his decisions are right or wrong, and, indeed, it is none too clear what is meant by those terms. There is a long history of attempts to reduce scientific method to system, including many which introduce probability theory, but it cannot be said that any great formal success has attended these efforts. If we were to compare the businessman to the scientist, we would be forced to the melancholy conclusion that little of a systematic nature can be said about the former’s decision-making processes. The statistician typically finds himself in situations more similar to that of the businessman. The problem of statistics can be formulated roughly as follows. It is known that one out of a number of hypotheses about a given situation is true. The statistician has the choice of one of a number of different experiments (a series of experiments can be regarded as a single experiment, so that drawing a sample of any size can be included in this schema), the outcome of any one of which is a random variable with a probability distribution depending on which of the unknown hypotheses is correct. On the basis of that outcome, the statistician must take some action (accept or reject a hypothesis, estimate the mean of a distribution to be some particular value, accept or reject a lot of goods, recommend a change in production methods, and so on), the consequences of which depend on the action taken and on the hypothesis that is actually true."

- Kenneth Arrow

0 likesAcademics from the United StatesEconomists from the United StatesMathematicians from the United StatesNobel laureates in EconomicsPeople from New York City
"The conventional view among economists is that education adds to an individual's productivity and therefore increases the market value of his labor. From the viewpoint of formal theory, it does not matter how the student's productivity is increased, but implicitly it is assumed that the student receives cognitive skills through his education. Educators, on the other hand, have long felt that the activity of education is a process of socialization, with the latent content of the process—the acquisition of skills such as the carrying out of assigned tasks, getting along with others, regularity, punctuality, and the like—being at least as important as the manifest objectives of conveying information. This last doctrine has been revived by radical economists, though with a negative rather than a positive valuation. But from the viewpoint of economic theory, the socialization hypothesis is just as much a human capital theory as the cognitive skill acquisition hypothesis. Both hypotheses imply that education supplies skills that lead to higher productivity. I would like to present a very different view. Higher education, in this model, contributes in no way to superior economic performance; it increases neither cognition nor socialization. Instead, higher education serves as a screening device in that it sorts out individuals of differing abilities, thereby conveying information to the purchasers of labor."

- Kenneth Arrow

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"The comparative economic efficiency of capitalism and socialism remains one of the most controversial areas. The classical socialist argument is that the anarchy of production under capitalism leads to great wastage. An appeal to the virtues of the price system is, in fact, only a partial answer to this critique. The central argument, which implies the efficiency of a competitive economic system, presupposes that all relevant goods are available at prices that are the same for all participants and that supplies and demands of all goods balance. Now virtually all economic decisions have implications for supplies and demands on future markets. The concept of capital, the very root of the term “capitalism,” refers to the setting-aside of resources for use in future production and sale. Hence, goods to be produced in the future are effectively economic commodities today. For efficient resource allocation, the prices of future goods should be known today. But they are not. Markets for current goods exist and enable a certain coherence between supply and demand there. But very few such markets exist for delivery of goods in the future. Hence, plans made by different agents may be based on inconsistent assumptions about the future. Investment plans may be excessive or inadequate to meet future demands or to employ the future labor force."

- Kenneth Arrow

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"Ironically, the current conservative model explaining the supposed association of capitalism and democracy relates to the Marxist as a photographic negative to a positive. It too suggests that the political “superstructure” is determined by the “relations of production.” The conservative model contrasts the dispersion of power under capitalist democracy with its concentration under socialism. Political opposition requires resources. The multiplicity of capitalists implies that any dissenting voice can find some support. Under socialism, the argument goes, the controlling political faction can deny its opponents all resources and dismiss them from their employment. This theoretical argument presupposes a monolithic state. It is something of a chicken-and-egg proposition. If the democratic legal tradition is strong, there are many sources of power in a modern state. Adding economic control functions may only increase the diversity of interests within the state and therefore alternative sources of power. It is notoriously harder for the government to regulate its own agencies than private firms. Socialism may easily offer as much pluralism as capitalism. The overpowering force in all these arguments is the empirical evidence of the Soviet Union and the other Communist countries, and it is strong. But the contrary proposition, that capitalism is a positive safeguard for democracy, is hardly a reasonable inference from experience. The example of Nazi Germany shows that no amount of private enterprise prevents the rise of totalitarianism. Indeed, it is hard to see that capitalism formed a significant impediment. Nor is Nazi Germany unique; Fascist Italy, Franco’s Spain, and the recurrent Latin American dictatorships are illustrative counterexamples to the proposition that capitalism implies democracy."

- Kenneth Arrow

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"Any argument seeking to establish the presence of irrational economic behavior always meets a standard counterargument: if most agents are irrational, then a rational individual can make a lot of money; eventually, therefore, the rational individuals will take over all the wealth. Hence, rational behavior will be the effective norm. There are two rebuttals to the counterargument. (1) Not all arbitrage possibilities exist. For example, corporate profits, even though they may be down, are very distinctly positive in real terms after all necessary adjustments, including taxes. Yet there seems no way by which the average investor in corporate securities can get a positive real rate of return. (2) More important, if everyone else is “irrational,” it by no means follows that one can make money by being rational, at least in the short run. With discounting, even eventual success may not be worthwhile. Consider, for example, a firm that engages in research and development which depresses the current profit and loss statement. Irrational investors look only at this information, and therefore the price of the stock is below the expected value of future dividends based on the profitable outcomes of the research and development. In a perfectly working market with rational individuals, stock prices would gradually rise as the realization date approached, but prices in the actual market would be constant. A rational investor would understand the future value of the stocks, but he or she could not realize any part of this gain during the gestation period. Although the rational investor may get rewarded eventually if the stock is held long enough, he or she is losing liquidity during an intervening period which may be long. Hence, the demand for the stock even by the rational buyers will be depressed. As Keynes argued long ago, the value of a security depends in good measure on other people’s opinions."

- Kenneth Arrow

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"There’s a famous Churchill quote: “democracy is the worst form of government, except for all the others.” That applies to regulation as well. In the late 1800s, we had a natural monopoly: railroads. The government created a regulatory enterprise – the Interstate Commerce Commission – and of course it was captured. But it still made a difference. I think we just have to accept that capture does occur, but it’s limited. The Federal Trade Commission, for example, is a pretty active body. Monopolies have been broken up. The AT&T telephone monopoly was broken up in 1982 – Stigler was still writing about regulatory capture then. AT&T was a classic monopoly, but a pretty benevolent one. It delivered good service – rates were too high, but not by that much. It didn’t necessarily pass on value to the consumers, but Bell Labs were a source of great innovative function. Nevertheless, it was broken up by antitrust measures, brought on by government. So there is regulatory capture, but it is by no means complete. Regulations do play a role. One example of non-regulatory capture fighting against effective regulation was during the run-up to the 2008 crash, when several officials argued for CDOs to be regulated, which means they would have had to meet certain requirements of transparency. This was not accepted. I’m not saying the crash could have been avoided if that happened, but that would have made a big difference."

- Kenneth Arrow

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"I was influenced very early on, my first time at the—I spent a year at the Center at Palo Alto, the Center for Advanced Study there, and I became a good friend of Kenneth Arrow. I think he was not at the Center that year but he lived in Palo Alto. [...] Marvelous man. Both as a person and as a scholar. And I became as I say greatly influenced by the way in which he dealt with phenomena. [...] So I was influenced by that as a model, a way of thinking more abstractly, perhaps, than customary, about democratic theory. Making clear the premises, the epistomological assumptions and matters of that kind, and I think that sort of set the stage. And then once you get in of course, into that field, which was not highly— I don’t know how to put this properly—as a formal field of political science was not highly developed at the time, once you get into it you quickly become aware of how rich the potential subject matter is. One of the enormous changes, perhaps anticipating your question, one of the changes in the world is the extraordinary increase in the number of countries that, by the standards that we use today, can be called democratic—always, I repeat this and repeat this, but, always keeping in mind the difference between the ideal and the threshold at which we now accept a country as democratic, or a polyarchy as I would say. And the enormous increase in the number of those available for study—when I was a graduate student, there were maybe half a dozen countries that you could study: France and Britain and, I’m not quite sure of Canada at that time . . . and then the expansion created out there a field . . . that was both a challenge and an opportunity."

- Kenneth Arrow

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"The interesting thing is that Ken Arrow was a dove and Bob Solow was then a hawk, in 1963-4. We used to have huge debates. Hahn was very hawkish, Meade was hawkish, only Arrow and what's now the Cambridge left were doves, particularly the Asians, for obvious reasons. After all, it was the Asians who were being napalmed. There were terrible fights going on, and the beginning of huge rifts in the faculty over the Vietnam war. Solow switched, later on, and to his everlasting credit came out and said he'd switched. Arrow was always a dove. That's why, fond as I am of Bob Solow, and much thought I admire him, I've always admired Arrow more, because Arrow, I think, has always had the right instincts. He's a self-declared socialist, he was a dove, he was always active on civil rights, he fought for Sam Bowles at Harvard, and so on. He's always gone out on a limb for the right issues - on the left issues, actually. By that time (1967-8), I was in the thick of moratoria and death threats and bombs and all the rest of it. I wrote the survey in about four months. It was refereed by Arrow, Stiglitz, Samuelson and one other - I've never found out who it was. Samuelson recommended publication 'as is'. Arrow wrote to me and said, 'May I use you brilliant survey for my graduate class at Harvard?', which is the letter I prize most of any I've received in all the world. Stiglitz didn't say anything at the time, but he saved it all up for when he wrote that very critical review article of my book (which he kept calling my article!) in the Journal of Political Economy."

- Kenneth Arrow

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"I’m an optimist, so I am hopeful. I don’t think Samuelsonian economics can be the long-run equilibrium of such an important field such as economics. I think that because of the two secret sins of Samuelsonian economics, the qualitative theorems and significance testing in the absence of the loss function, we can’t make scientific progress. We can do a lot of other stuff. We can take a look at tables and see how large schooling is in the national economy; that kind of thing is science. So I’m sure we can make some progress, off the center of the scientific stage of so-called mainstream Samuelsonian economics. An interesting concrete example of this was a presentation Ken Arrow made when he came to Iowa, where he was trying to make the obvious point that we shouldn’t spend all of our time in undergraduate economics preparing people for graduate school. This is a point that I have made over and over again, that we should be preparing students for life, for business and law school. Arrow said, “Look, the main argument for this is not very complicated: as we all know, half of 1 percent of our students, if that, go on to graduate school in economics, end of argument.” The argument is not special to Arrow, but the fact that it came from Arrow was very interesting. He didn’t come and say there is an existence theorem proof that there doesn’t exist some social welfare function such that blah blah blah. He didn’t do that. He said, “Look, here’s the number that shows obviously that the policy of making undergraduate programs into junior graduate programs is a mistake.”"

- Kenneth Arrow

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"After earning the PhD degree and acquiring some relatively extensive experience in digital computers… It was time to leave the University. The result of an extensive search for the right job was a family move to Arlington Heights, Illinois, where it was a short commute to the Research Laboratories of the Pure Oil Company at Crystal Lake. I was given the title of Mathematical and Computer Consultant. The Labs were set in a beautiful campus, the professional personnel were eager to learn what I had to teach and to include me in many interesting projects where my knowledge and skills could be put to good use. I was encouraged to initiate my own program of research. I went to work with enthusiasm. The corporate headquarters of Pure Oil were located in down town Chicago. Pure Oil had been trying to install an IBM 705 computer system for all their accounting needs including calculation of all data necessary for the management of exploration, drilling, refining and distribution of oil products and even royalties to shareholders in oil wells. Typical for those early days, the programming team was in deep difficulties and needed help; they lacked adequate resources and suitable training. The Executive Vice President of Pure Oil, when he heard that there was a computer expert already on the payroll at the Crystal Lake lab, ended our family blissful dream and I was reassigned to the down town office."

- A. Wayne Wymore

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"Our so-called "Arabic" notation owes its excellence to the application of the principle of local value and the use of a symbol for zero. It is now conclusively established that the principle of local value was used by the ns much earlier than by the Hindus and that the Maya of Central America used the principle and symbols for zero in a well-developed numeral system of their own. The notation of Babylonia used the scale of 60, that of the Maya, the scale 20 (except in one step). It follows, therefore, that the present controversy on the origin of our numerals does not involve the question of the first use of local value and symbols for zero; it concerns itself only with the time and place of the first application of local value to the decimal scale and with the origin of the forms or shapes of our ten numerals. ... Hurt by the alleged arrogance of certain Greek scholars, Sebokht praises the science of the Hindus and speaks of "their valuable methods of computation. . . . I wish only to say that this computation is done by means of nine signs." Unfortunately, he leaves it to us to guess whether or not he used the zero. The passage, written about 662 A.D., is the earliest reference that has been found outside of India to our numerals. ...The form of the symbols with the zero, used in India, differed so widely from the old forms without the zero used there, that the former seem to have had an independent origin and to have been imported into India. ...The following are outstanding facts: 1. The earliest reliable record of the use of our numerals with zero is an inscription of 867 A.D. in India. 2. The validity of the testimony of early Arabic writers ascribing to India the numerals with zero is shaken, but not destroyed. 3. There is not a scintilla of evidence in the form of old manuscripts or numeral inscriptions to support the Greek origin of our numerals. 4. At present the hypothesis of the Hindu origin of our numerals stands without serious rival. But this hypothesis is by no means firmly established."

- Florian Cajori

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"In ancient classical literature the rainbow sometimes was deified as Iris; at other times it was regarded merely as the route traversed by the messenger of Hera. The conception of the rainbow as a pathway or bridge has been widespread. For some it has been the best of all bridges, built out of three colors; for others the phrase "building on the rainbow" has meant a bootless enterprise. North American Indians were among those who thought of the rainbow as the Pathway of Souls, an interpretation found in many other places. Among the Japanese the rainbow is identified as the "Floating Bridge of Heaven"; and Hawaiian and Polynesian myths allude to the bow as the path to the upper world. In the Austrian Alps the souls of the righteous are said to ascend the bow to heaven; and in New Zealand the dead chieftains are believed to pass along it to reach their new home. In parts of France the rainbow is called the pont du St. Esprit, and in many places it is the bridge of St. Bernard or of St. Martin or of St. Peter. Basque pilgrims knew it as the 'puente de Roma'. Sometimes it is called instead the Croy de St. Denis (or of St. Leonard or of St. Bernard or of St. Martin). In Italy the name arcu de Santa Marina is relatively familiar. Associations of the rainbow and the milky way are frequent. The Arabic name for the milky way is equivalent to Gate of Heaven, and in Russia the analogous role was played by the rainbow. Elsewhere also the bow has been called the Gate of Paradise; and by some the rainbow has been thought to be a ray of light which falls on the earth when Peter opens the heavenly gate. In parts of France the rainbow is known as the porte de St. Jacques, while the milky way is called chemin de St. Jacques. In Swabia and Bavaria saints pass by the rainbow from heaven to earth; while in Polynesia this is the route of the gods themselves. In Eddic literature the bow served as a link between the gods and man — the Bifrost bridge, guarded by Heimdel, over which the gods passed daily. At the time of the Gotterdamerung the sons of Muspell will cross the bridge and then demolish it. Sometimes also in the Eddas the rainbow is interpreted as a necklace worn by Freyja, the "necklace of the Brisings," alluded to in Beowulf; again it is the bow of Thor from which he shoots arrows at evil spirits. Among the Finns it has been an arc which hurls arrows of fire, in Mozambique it is the arm of a conquering god. In the Japanese Ko-Ji-Ki (or Records of Ancient Matters), compiled presumably in 712, the creation of the island of Onogoro is related to the rainbow. Deities, standing upon the "floating bridge of heaven," thrust down a jeweled spear into the brine and stirred with it. When the spear was withdrawn, the brine that dripped down from the end was piled up in the form of the island. In myth and legend the rainbow has been regarded variously as a harbinger of misfortune and as a sign of good luck. Some have held it to be a bad sign if the feet of the bow rest on water, whereas a rainbow arching from dry land to dry land is a good augury. Dreambooks held that when one dreams of seeing a rainbow, he will give or receive a gift according as the bow is seen in the west or the east. The Crown-prince Frederick August took it as a good omen when, upon his receiving the kingdom form Napoleon in 1806, a rainbow appeared; but others interpreted it as boding ill, a view confirmed by the war and destruction of Saxony which ensued. By many, a rainbow appearing at the birth of a child is taken to be a favorable sign; but in Slavonic accounts a glance from the fay who sits at the foot of the rainbow, combing herself, brings death."

- Carl Benjamin Boyer

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"For reasons mentioned at the beginning of this section, we cannot offer here a precise structural definition of semantical category and will content ourselves with the following approximate formulation: two expressions belong to the same semantical category if (I) there is a sentential function which contains one of these expressions, and if (2) no sentential function which contains one of these expressions ceases to be a sentential function if this expression is replaced in it by the other. It follows from this that the relation of belonging to the same category is reflective, symmetrical and transitive. By applying the principle of abstraction, all the expressions of the language which are parts of sentential functions can be divided into mutually exclusive classes, for two expressions are put into one and the same class if and only if they belong to the same semantical category, and each of these classes is called a semantical category. Among the simplest examples of semantical categories it suffices to mention the category of the sentential functions, together with the categories which include respectively the names of individuals, of classes of individuals, of two-termed relations between individuals, and so on. Variables (or expressions with variables) which represent names of the given categories likewise belong to the same category."

- Alfred Tarski

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"It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B.C. In 's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position... and an arrangement of terms in a kind of notation. The rules given by Liu Hui form a kind of rhetorical algebra. The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving s. ...Sun-tzï solved such problems by analysis and was content with a single result... The Chinese certainly knew how to solve quadratics as early as the 1st century B.C., and rules given even as early as the K'iu-ch'ang Suan-shu... involve the solution of such equations. Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should... By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625). The culmination of Chinese is found in the 13th century. ...numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner (1819)."

- David Eugene Smith

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"In June 1936, Einstein and Rosen sent the paper... "Do Gravitational Waves Exist?" to The Physical Review... [which] rejected the paper, provoking Einstein's furious reaction. Einstein told the editor he... saw no reason to address the erroneous comments of his anonymous expert [Howard Percy Robertson] and... preferred to publish the paper elsewhere. ...Leopold Infeld arrived in Princeton to replace Rosen as... assistant. Einstein explained to him his proof of the non-existence of gravity waves. ...Infeld told Robertson [then professor of theoretical physics at Princeton] about Einstein's... paper[.] Robertson... found a trivial mistake [by] Infeld [and] clarified... the mistake in Einstein's explanation... The linearized approximation [led] to plane transverse gravitational waves... introduc[ing]... coordinate singularities... not real singularities. ...Robertson ...suggested ...the so-called Einstein-Rosen metric... be transformed... to cylindrical coordinates. ...the singularity can be regarded as describing a material source. The solution describe[s]...cylindrical... rather than plane gravitational waves. ...with Robertson's help (still not knowing it was Robertson who had [refereed] The Physical Review) ...Einstein ...revis[ed the] ...paper and added a section: "Rigorous Solution for Cylindrical Waves"... The new version of the paper was re-titled "On Gravitational Waves"..."

- Howard P. Robertson

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"Early in his college days, Minsky had had the good fortune to encounter Andrew Gleason. Gleason was only six years older than Minsky, but he was already recognized as one of the world’s premier problem-solvers in mathematics; he seemed able to solve any well-formulated mathematics problem almost instantly... “I couldn’t understand how anyone that age could know so much mathematics,” Minsky told me. “But the most remarkable thing about him was his plan. When we were talking once, I asked him what he was doing. He told me that he was working on Hilbert’s fifth problem.” Gleason said he had a plan that consisted of three steps, each of which he thought would take him three years to work out. Our conversation must have taken place in 1947, when I was a sophomore. Well, the solution took him only about five more years... I couldn’t understand how anyone that age could understand the subject well enough to have such a plan and to have an estimate of the difficulty in filling in each of the steps. Now that I’m older, I still can’t understand it. Anyway, Gleason made me realize for the first time that mathematics was a landscape with discernible canyons and mountain passes, and things like that. In high school, I had seen mathematics simply as a bunch of skills that were fun to master—but I had never thought of it as a journey and a universe to explore. No one else I knew at that time had that vision, either."

- Andrew Gleason

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"Interviewer: You are an Augustinian. How does Augustinian spirituality characterise your ministry? Cardinal Prevost: We could say several things... As my episcopal motto suggests, unity and communion are part of the charism of the Order of Saint Augustine and also of my way of acting and thinking. I think it is very important to promote communion in the Church, and we know well that communion, participation and mission are the three key words of the Synod. So, as an Augustinian, promoting unity and communion is fundamental for me. St Augustine also speaks a lot about unity in the Church and the need to live it, about the fact that there is a certain guarantee of unity in listening to the Bishop of Rome, in being part of the Church of Rome. In this sense too, therefore, I feel that the Pope's new call is a way of living my unity and participation in the Church, in obedience to the Holy Father. This too is very Augustinian. Interviewer: How much does the figure of Augustine inspire your choices, your steps, your service in the Church? Cardinal Prevost: St Augustine is certainly a great figure not only for the order but for everyone. I wish I had more time to study and read him. He has so much to offer the Church, even the Church of today. Then there is what I said before: unity in the Church and fidelity to the Bishop of Rome, always seeking to promote communion. Living unity in the Church, as Augustine recommends, means living united in Christ."

- Pope Leo XIV

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"One should mention right at the start that one still does not understand whether quantum mechanics and special relativity are compatible at a fundamental level in our Minkowski four-space world. One generally assumes that this means finding a complete Yang-Mills gauge theory or the interaction of gauge fields with fermionic matter fields, the simplest form being quantum chromodynamics (QCD). Associated with this picture is the belief that the fundamental vector meson excitations are massive (as opposed to photons, which arise in the limiting case of an abelian gauge symmetry. The proof of the existence of a “mass gap” appears a necessary integral part of solving the entire puzzle. This question remains one of the deepest open issues in theoretical physics, as well as in mathematics. Basically the question remains: can one give a mathematical foundation to the theory of fields in four-dimensions? In other words, can do quantum mechanics and special relativity lie on the same footing as the classical physics of Newton, Maxwell, Einstein, or Schrödinger—all of which fits into a mathematical framework that we describe as the language of physics. This glaring gap in our fundamental knowledge even dwarfs questions of whether there are other more complicated and sophisticated approaches to physics—those that incorporate gravity, strings, or branes—for understanding their fundamental significance lies far in the future. In fact, one believes that stringy proposals, if they can be fully implemented, have limiting cases that appear as relativistic quantum fields, just as relativistic quantum fields describe non-relativistic quantum theory and classical physics in various limiting cases."

- Arthur Jaffe

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