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Who said: “I have no satisfaction in formulas unless I feel their arithmetical magnitude.”
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Set Quotes (56 quotes)

A conflict arises when a religious community insists on the absolute truthfulness of all statements recorded in the Bible. This means an intervention on the part of religion into the sphere of science; this is where the struggle of the Church against the doctrines of Galileo and Darwin belongs. On the other hand, representatives of science have often made an attempt to arrive at fundamental judgments with respect to values and ends on the basis of scientific method, and in this way have set themselves in opposition to religion. These conflicts have all sprung from fatal errors.
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A fateful process is set in motion when the individual is released “to the freedom of his own impotence” and left to justify his existence by his own efforts. The autonomous individual, striving to realize himself and prove his worth, has created all that is great in literature, art, music, science and technology. The autonomous individual, also, when he can neither realize himself nor justify his existence by his own efforts, is a breeding call of frustration, and the seed of the convulsions which shake our world to its foundations.
In The Passionate State of Mind (1955), 18.
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A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales.
Cited as from Fractals: Form, Chance, and Dimension (1977), by J.W. Cannon, in review of The Fractal Geometry of Nature (1982) in The American Mathematical Monthly (Nov 1984), 91, No. 9, 594.
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A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.
In Time Enough for Love: The Lives of Lazarus Long (1987), 248.
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A set is a Many that allows itself to be thought of as a One.
Quoted in Carol Schumacher, Chapter Zero: Fundamental Notions of Abstract Mathematics (1996), 23.
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A very interesting set of compounds that were waiting for the right disease.
[Commenting on AZT and similar drugs he had synthesized.]
Quoted in 'Jerome Horwitz, AZT Creator, Dies at 93', New York Times (21 Sep 2012).
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An evolution is a series of events that in itself as series is purely physical, — a set of necessary occurrences in the world of space and time. An egg develops into a chick; … a planet condenses from the fluid state, and develops the life that for millions of years makes it so wondrous a place. Look upon all these things descriptively, and you shall see nothing but matter moving instant after instant, each instant containing in its full description the necessity of passing over into the next. … But look at the whole appreciatively, historically, synthetically, as a musician listens to a symphony, as a spectator watches a drama. Now you shall seem to have seen, in phenomenal form, a story.
In The Spirit of Modern Philosophy: An Essay in the Form of Lectures (1892), 425.
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As a scientist, I am not sure anymore that life can be reduced to a class struggle, to dialectical materialism, or any set of formulas. Life is spontaneous and it is unpredictable, it is magical. I think that we have struggled so hard with the tangible that we have forgotten the intangible.
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Boundaries which mark off one field of science from another are purely artificial, are set up only for temporary convenience. Let chemists and physicists dig deep enough, and they reach common ground.
From chapter 'Jottings from a Note-Book', in Canadian Stories (1918), 183.
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Bradley is one of the few basketball players who have ever been appreciatively cheered by a disinterested away-from-home crowd while warming up. This curious event occurred last March, just before Princeton eliminated the Virginia Military Institute, the year’s Southern Conference champion, from the NCAA championships. The game was played in Philadelphia and was the last of a tripleheader. The people there were worn out, because most of them were emotionally committed to either Villanova or Temple-two local teams that had just been involved in enervating battles with Providence and Connecticut, respectively, scrambling for a chance at the rest of the country. A group of Princeton players shooting basketballs miscellaneously in preparation for still another game hardly promised to be a high point of the evening, but Bradley, whose routine in the warmup time is a gradual crescendo of activity, is more interesting to watch before a game than most players are in play. In Philadelphia that night, what he did was, for him, anything but unusual. As he does before all games, he began by shooting set shots close to the basket, gradually moving back until he was shooting long sets from 20 feet out, and nearly all of them dropped into the net with an almost mechanical rhythm of accuracy. Then he began a series of expandingly difficult jump shots, and one jumper after another went cleanly through the basket with so few exceptions that the crowd began to murmur. Then he started to perform whirling reverse moves before another cadence of almost steadily accurate jump shots, and the murmur increased. Then he began to sweep hook shots into the air. He moved in a semicircle around the court. First with his right hand, then with his left, he tried seven of these long, graceful shots-the most difficult ones in the orthodoxy of basketball-and ambidextrously made them all. The game had not even begun, but the presumably unimpressible Philadelphians were applauding like an audience at an opera.
A Sense of Where You Are: Bill Bradley at Princeton
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But when science, passing beyond its own limits, assumes to take the place of theology, and sets up its own conception of the order of nature as a sufficient account of its cause, it is invading a province of thought to which it has no claim, and not unreasonably provokes the hostility of its best friends.
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Chromosomes … [contain] some kind of code-script the entire pattern of the individual’s future development and of its functioning in the mature state. Every complete set of chromosomes contains the full code.
In What is Life? : The Physical Aspect of the Living Cell (1944), 20.
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Contingency is rich and fascinating; it embodies an exquisite tension between the power of individuals to modify history and the intelligible limits set by laws of nature. The details of individual and species’s lives are not mere frills, without power to shape the large-scale course of events, but particulars that can alter entire futures, profoundly and forever.
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Dissection ... teaches us that the body of man is made up of certain kinds of material, so differing from each other in optical and other physical characters and so built up together as to give the body certain structural features. Chemical examination further teaches us that these kinds of material are composed of various chemical substances, a large number of which have this characteristic that they possess a considerable amount of potential energy capable of being set free, rendered actual, by oxidation or some other chemical change. Thus the body as a whole may, from a chemical point of view, be considered as a mass of various chemical substances, representing altogether a considerable capital of potential energy.
From Introduction to A Text Book of Physiology (1876, 1891), Book 1, 1.
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FORTRAN —’the infantile disorder’—, by now nearly 20 years old, is hopelessly inadequate for whatever computer application you have in mind today: it is now too clumsy, too risky, and too expensive to use. PL/I —’the fatal disease’— belongs more to the problem set than to the solution set. It is practically impossible to teach good programming to students that have had a prior exposure to BASIC: as potential programmers they are mentally mutilated beyond hope of regeneration. The use of COBOL cripples the mind; its teaching should, therefore, be regarded as a criminal offence. APL is a mistake, carried through to perfection. It is the language of the future for the programming techniques of the past: it creates a new generation of coding bums.
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Fractals are patterns which occur on many levels. This concept can be applied to any musical parameter. I make melodic fractals, where the pitches of a theme I dream up are used to determine a melodic shape on several levels, in space and time. I make rhythmic fractals, where a set of durations associated with a motive get stretched and compressed and maybe layered on top of each other. I make loudness fractals, where the characteristic loudness of a sound, its envelope shape, is found on several time scales. I even make fractals with the form of a piece, its instrumentation, density, range, and so on. Here I’ve separated the parameters of music, but in a real piece, all of these things are combined, so you might call it a fractal of fractals.
Interview (1999) on The Discovery Channel. As quoted by Benoit B. Manelbrot and Richard Hudson in The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward (2010), 133.
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God is love… . We wouldn’t recognize that love. It might even look like hate. It would be enough to scare us—God’s love. It set fire to a bush in the desert, didn’t it, and smashed open graves and set the dead walking in the dark.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 143
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going to have an industrial society you must have places that will look terrible. Other places you set aside—to say, ‘This is the way it was.’
Assembling California
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Gödel proved that the world of pure mathematics is inexhaustible; no finite set of axioms and rules of inference can ever encompass the whole of mathematics; given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. I hope that an analogous Situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we go into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.
From Lecture 1, 'Philosophy', in a series of four James Arthur Lectures, 'Lectures on Time and its Mysteries' at New York University (Autumn 1978). Printed in 'Time Without End: Physics and Biology in an Open Universe', Reviews of Modern Physics (Jul 1979), 51, 449.
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I am willing to believe that my unobtainable sixty seconds within a sponge or a flatworm might not reveal any mental acuity that I would care to ca ll consciousness. But I am also confident ... that vultures and sloths, as close evolutionary relatives with the same basic set of organs, lie on our side of any meaningful (and necessarily fuzzy) border–and that we are therefore not mistaken when we look them in the eye and see a glimmer of emotional and conceptual affinity.
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I view the major features of my own odyssey as a set of mostly fortunate contingencies. I was not destined by inherited mentality or family tradition to become a paleontologist. I can locate no tradition for scientific or intellectual careers anywhere on either side of my eastern European Jewish background ... I view my serious and lifelong commitment to baseball in entirely the same manner: purely as a contingent circumstance of numerous, albeit not entirely capricious, accidents.
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I would teach the world that science is the best way to understand the world, and that for any set of observations, there is only one correct explanation. Also, science is value-free, as it explains the world as it is. Ethical issues arise only when science is applied to technology – from medicine to industry.
Response to question “What is the one thing everyone should learn about science?” in 'Life Lessons' The Guardian (7 Apr 2005).
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If basketball was going to enable Bradley to make friends, to prove that a banker’s son is as good as the next fellow, to prove that he could do without being the greatest-end-ever at Missouri, to prove that he was not chicken, and to live up to his mother’s championship standards, and if he was going to have some moments left over to savor his delight in the game, he obviously needed considerable practice, so he borrowed keys to the gym and set a schedule for himself that he adhereded to for four full years—in the school year, three and a half hours every day after school, nine to five on Saturday, one-thirty to five on Sunday, and, in the summer, about three hours a day.
A Sense of Where You Are: Bill Bradley at Princeton
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If texts are unified by a central logic of argument, then their pictorial illustrations are integral to the ensemble, not pretty little trifles included only for aesthetic or commercial value. Primates are visual animals, and (particularly in science) illustration has a language and set of conventions all its own.
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If we long to believe that the stars rise and set for us, that we are the reason there is a Universe, does science do us a disservice in deflating our conceits
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In the social production of their life, men enter into definite relations that are indispensable and independent of their will; these relations of production correspond to a definite stage of development of their material forces of production. The sum total of these relations of production constitutes the economic structure of society - the real foundation, on which rises a legal and political superstructure and to which correspond definite forms of social consciousness. The mode of production of material life determines the social, political and intellectual life process in general. It is not the consciousness of men that determines their being, but, on the contrary, their social being that determines their consciousness. At a certain stage of their development, the material productive forces in society come in conflict with the existing relations of production, or - what is but a legal expression for the same thing - with the property relations within which they have been at work before. From forms of development of the productive forces these relations turn into fetters. Then begins an epoch of social revolution. With the change of the economic foundation the entire immense superstructure is more or less rapidly transformed. In considering such transformations a distinction should always be made between the material transformation of the economic conditions of production, which can be determined with the precision of natural science, and the legal, political, religious, aesthetic or philosophic - in short, ideological forms in which men become conscious of this conflict and fight it out. Just as our opinion of an individual is not based on what he thinks of himself, so we can not judge of such a period of transformation by its own consciousness; on the contrary, this consciousness must be explained rather from the contradictions of material life, from the existing conflict between the social productive forces and the relations of production. No social order ever disappears before all the productive forces for which there is room in it have been developed; and new, higher relations of production never appear before the material conditions of their existence have matured in the womb of the old society itself. Therefore, mankind always sets itself only such tasks as it can solve; since, looking at the matter more closely, we will always find that the task itself arises only when the material conditions necessary for its solution already exist or are at least in the process of formation. In broad outlines we can designate the Asiatic, the ancient, the feudal, and the modern bourgeois modes of production as so many progressive epochs in the economic formation of society. The bourgeois relations of production are the last antagonistic form of the social process of production - antagonistic not in the sense of individual antagonism, but of one arising from the social conditions of life of the individuals; at the same time the productive forces developing in the womb of bourgeois society create the material conditions for the solution of that antagonism. This social formation constitutes, therefore, the closing chapter of the prehistoric stage of human society.
Karl Marx
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Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply. The latter are considered to be not amenable to theoretical treatment and to be determinable only from observation.
From Lecture delivered on presentation of the James Scott prize, (6 Feb 1939), 'The Relation Between Mathematics And Physics', printed in Proceedings of the Royal Society of Edinburgh (1938-1939), 59, Part 2, 125.
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Mathematics… is the set of all possible self-consistent structures, and there are vastly more logical structures than physical principles.
In 'Conclusion', Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension (1995), 328.
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No man ever looks at the world with pristine eyes. He sees it edited by a definite set of customs and institutions and ways of thinking.
In 'The Science of Custom', Patterns of Culture (1934, 2005), 2.
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Nobody knows more than a tiny fragment of science well enough to judge its validity and value at first hand. For the rest he has to rely on views accepted at second hand on the authority of a community of people accredited as scientists. But this accrediting depends in its turn on a complex organization. For each member of the community can judge at first hand only a small number of his fellow members, and yet eventually each is accredited by all. What happens is that each recognizes as scientists a number of others by whom he is recognized as such in return, and these relations form chains which transmit these mutual recognitions at second hand through the whole community. This is how each member becomes directly or indirectly accredited by all. The system extends into the past. Its members recognize the same set of persons as their masters and derive from this allegiance a common tradition, of which each carries on a particular strand.
Personal Knowledge (1958), 163.
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One ought to be ashamed to make use of the wonders of science embodied in a radio set, while appreciating them as little as a cow appreciates the botanical marvels in the plant she munches.
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Only the individual can think, and thereby create new values for society–nay, even set up new moral standards to which the life of the community conforms. Without creative, independently thinking and judging personalities the upward development of society is as unthinkable as the development of the individual personality without the nourishing soil of the community.
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Our world is not an optimal place, fine tuned by omnipotent forces of selection. It is a quirky mass of imperfections, working well enough (often admirably); a jury-rigged set of adaptations built of curious parts made available by past histories in different contexts ... A world optimally adapted to current environments is a world without history, and a world without history might have been created as we find it. History matters; it confounds perfection and proves that current life transformed its own past.
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Science gives us the grounds of premises from which religious truths are to be inferred; but it does not set about inferring them, much less does it reach the inference;Mthat is not its province. It brings before us phenomena, and it leaves us, if we will, to call them works of design, wisdom, or benevolence; and further still, if we will, to proceed to confess an Intelligent Creator. We have to take its facts, and to give them a meaning, and to draw our own conclusions from them. First comes Knowledge, then a view, then reasoning, then belief. This is why Science has so little of a religious tendency; deductions have no power of persuasion. The heart is commonly reached, not through the reason, but through the imagination, by means of direct impressions, by the testimony of facts and events, by history, by description. Persons influence us, voices melt us, looks subdue us, deeds inflame us. Many a man will live and die upon a dogma; no man will be a martyr for a conclusion.
Tamworth Reading Room (1841).
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Science is a capital or fund perpetually reinvested; it accumulates, rolls up, is carried forward by every new man. Every man of science has all the science before him to go upon, to set himself up in business with. What an enormous sum Darwin availed himself of and reinvested! Not so in literature; to every poet, to every artist, it is still the first day of creation, so far as the essentials of his task are concerned. Literature is not so much a fund to be reinvested as it is a crop to be ever new-grown.
Indoor Studies, vol. 12, Collected Works, Houghton (1913).
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Science is a set of rules that keep the scientists from lying to each other.
Quoted in Alexander Kohn, False Prophets (1986, 1997), 2.
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Science, in its ultimate ideal, consists of a set of propositions arranged in a hierarchy, the lowest level of the hierarchy being concerned with particular facts, and the highest with some general law, governing everything in the universe. The various levels in the hierarchy have a two-fold logical connection, travelling one up, one down; the upward connection proceeds by induction, the downward by deduction.
In The Scientific Outlook (1931, 2009), 38.
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Science, in the immediate, produces knowledge and, indirectly, means of action. It leads to methodical action if definite goals are set up in advance. For the function of setting up goals and passing statements of value transcends its domain. While it is true that science, to the extent of its grasp of causative connections, may reach important conclusions as to the compatibility and incompatibility of goals and evaluations, the independent and fundamental definitions regarding goals and values remain beyond science’s reach.
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Setting an example is not the main means of influencing another, it is the only means.
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She has the sort of body you go to see in marble. She has golden hair. Quickly, deftly, she reaches with both hands behind her back and unclasps her top. Setting it on her lap, she swivels ninety degrees to face the towboat square. Shoulders back, cheeks high, she holds her pose without retreat. In her ample presentation there is defiance of gravity. There is no angle of repose. She is a siren and these are her songs.
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Spong and I had also several fine discourses upon the globes this afternoon, particularly why the fixed stars do not rise and set at the same hour all the year long, which he could not demonstrate nor I neither.
Entry for 19 Aug 1666. In Samuel Pepys and Henry B. Wheatley (ed.), The Diary of Samuel Pepys (1895, 1900), Vol. 5, 382. He spent the day with Mr Reeves and Mr Spong.
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The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 94.
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The essential character of a species in biology is, that it is a group of living organisms, separated from all other such groups by a set of distinctive characters, having relations to the environment not identical with those of any other group of organisms, and having the power of continuously reproducing its like. Genera are merely assemblages of a number of these species which have a closer resemblance to each other in certain important and often prominent characters than they have to any other species.
In 'The Method of Organic Evolution', Fortnightly Review (1895), 57, 441.
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The Himalayas are the crowning achievement of the Indo-Australian plate. India in the Oligocene crashed head on into Tibet, hit so hard that it not only folded and buckled the plate boundaries but also plowed into the newly created Tibetan plateau and drove the Himalayas five and a half miles into the sky. The mountains are in some trouble. India has not stopped pushing them, and they are still going up. Their height and volume are already so great they are beginning to melt in their own self-generated radioactive heat. When the climbers in 1953 planted their flags on the highest mountain, they set them in snow over the skeletons of creatures that had lived in a warm clear ocean that India, moving north, blanked out. Possibly as much as 20,000 feet below the sea floor, the skeletal remains had turned into rock. This one fact is a treatise in itself on the movements of the surface of the earth.
If by some fiat, I had to restrict all this writing to one sentence; this is the one I would choose: the summit of Mount Everest is marine limestone.
Annals of the Former World
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The history of mathematics, as of any science, is to some extent the story of the continual replacement of one set of misconceptions by another. This is of course no cause for despair, for the newly instated assumptions very often possess the merit of being closer approximations to truth than those that they replace.
In 'Consistency and Completeness—A Résumé', The American Mathematical Monthly (May 1956), 63, No.5, 295.
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The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of science.
Aspects of Science: Second Series (1926), 92.
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The more a man is imbued with the ordered regularity of all events the firmer becomes his conviction that there is no room left by the side of this ordered regularity for causes of a different nature. For him neither the rule of human nor the rule of divine will exists as an independent cause of natural events. To be sure, the doctrine of a personal God interfering with natural events could never be refuted, in the real sense, by science, for this doctrine can always take refuge in those domains in which scientific knowledge has not yet been able to set foot.
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The number of fixed stars which observers have been able to see without artificial powers of sight up to this day can be counted. It is therefore decidedly a great feat to add to their number, and to set distinctly before the eyes other stars in myriads, which have never been seen before, and which surpass the old, previously known stars in number more than ten times.
In pamphlet, The Sidereal Messenger (1610), reprinted in The Sidereal Messenger of Galileo Galilei: And a Part of the Preface to the Preface to Kepler's Dioptrics Containing the Original Account of Galileo's Astronomical Discoveries (1880), 7-8.
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The universe is an asymmetrical entity. I am inclined to believe that life as it is manifested to us must be a function of the asymmetry of the universe or of the consequence of this fact. The universe is asymmetrical; for if one placed the entire set of bodies that compose the solar system, each moving in its own way, before a mirror, the image shown would not be superimposable on the reality.
Rene Vallery-Radot, Vie de Pasteur (1900), 79. Quoted in Patrice Debre, Louis Pasteur, trans. Elborg Forster (1994), 78.
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Theory provides the maps that turn an uncoordinated set of experiments or computer simulations into a cumulative exploration.
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There may only be a small number of laws, which are self-consistent and which lead to complicated beings like ourselves. … And even if there is only one unique set of possible laws, it is only a set of equations. What is it that breathes fire into the equations and makes a universe for them to govern? Is the ultimate unified theory so compelling that it brings about its own existence?
Lecture (1987), 'The Origin of the Universe', collected in Black Holes And Baby Universes And Other Essays (1993), 99.
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This method is, to define as the number of a class the class of all classes similar to the given class. Membership of this class of classes (considered as a predicate) is a common property of all the similar classes and of no others; moreover every class of the set of similar classes has to the set of a relation which it has to nothing else, and which every class has to its own set. Thus the conditions are completely fulfilled by this class of classes, and it has the merit of being determinate when a class is given, and of being different for two classes which are not similar. This, then, is an irreproachable definition of the number of a class in purely logical terms.
The Principles of Mathematics (1903), 115.
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We ourselves introduce that order and regularity in the appearance which we entitle ‘nature’. We could never find them in appearances had we not ourselves, by the nature of our own mind, originally set them there.
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When the climbers in 1953 planted their flags on the highest mountain, they set them in snow over the skeletons of creatures that had lived in the warm clear ocean that India, moving north, blanked out. Possibly as much as twenty thousand feet below the seafloor, the skeletal remains had turned into rock. This one fact is a treatise in itself on the movements of the surface of the earth. If by some fiat I had to restrict all this writing to one sentence, this is the one I would choose: The summit of Mt. Everest is marine limestone.
Annals of the Former World
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When you can dump a load of bricks on a corner lot, and let me watch them arrange themselves into a house — when you can empty a handful of springs and wheels and screws on my desk, and let me see them gather themselves together into a watch — it will be easier for me to believe that all these thousands of worlds could have been created, balanced, and set to moving in their separate orbits, all without any directing intelligence at all.
In 'If A Man Die, Shall He Live again?', More Power To You: Fifty Editorials From Every Week (1917), 218-219.
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Whoever undertakes to set himself up as a judge of Truth and knowledge is shipwrecked by the laughter of the gods.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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- 90 -
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- 80 -
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- 70 -
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- 60 -
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- 50 -
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- 40 -
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- 20 -
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