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Who said: “Dangerous... to take shelter under a tree, during a thunder-gust. It has been fatal to many, both men and beasts.”
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Often Quotes (106 quotes)

...there is no prescribed route to follow to arrive at a new idea. You have to make the intuitive leap. But the difference is that once you’ve made the intuitive leap you have to justify it by filling in the intermediate steps. In my case, it often happens that I have an idea, but then I try to fill in the intermediate steps and find that they don’t work, so I have to give it up.
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On rencontre sa destinée souvent par des chemins qu’on prend pour l’éviter.
Man meets his destiny oft by the roads he takes to escape it.
In 'L’Horoscope', Fables, Vol. 8, 16. (1668–1679). Reprinted in Fables Choisies de La Fontaine (1730), 198. As translated in Thomas Benfield Harbottle and Colonel Philip Hugh Dalbiac, Dictionary of Quotations (1901), 172.
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[Question: Do you feel that scientists correct themselves as often as they should?]
More often than politicians, but not as often as they should.
Interview with Deborah Solomon, 'The Science of Second-Guessing', in New York Times Magazine (12 Dec 2004), 37.
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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 short, broad man of tremendous vitality, the physical type of Hereward, the last of the English, and his brother-in-arms, Winter, Sylvester’s capacious head was ever lost in the highest cloud-lands of pure mathematics. Often in the dead of night he would get his favorite pupil, that he might communicate the very last product of his creative thought. Everything he saw suggested to him something new in the higher algebra. This transmutation of everything into new mathematics was a revelation to those who knew him intimately. They began to do it themselves. His ease and fertility of invention proved a constant encouragement, while his contempt for provincial stupidities, such as the American hieroglyphics for π and e, which have even found their way into Webster’s Dictionary, made each young worker apply to himself the strictest tests.
In Florian Cajori, Teaching and History of Mathematics in the United States (1890), 265.
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A smart mother makes often a better diagnosis than a poor doctor.
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Abstruse mathematical researches … are … often abused for having no obvious physical application. The fact is that the most useful parts of science have been investigated for the sake of truth, and not for their usefulness. A new branch of mathematics, which has sprung up in the last twenty years, was denounced by the Astronomer Royal before the University of Cambridge as doomed to be forgotten, on account of its uselessness. Now it turns out that the reason why we cannot go further in our investigations of molecular action is that we do not know enough of this branch of mathematics.
In 'Conditions of Mental Development', Lectures and Essays (1901), Vol. 1, 115.
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Algebra is generous; she often gives more than is asked of her.
Quoted in Bulletin American Mathematical Society (1905), 2, 285. As cited in Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 275.
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All versions written for nonscientists speak of fused males as the curious tale of the anglerfish–just as we so often hear about the monkey swinging through the trees, or the worm burrowing through soil. But if nature teaches us any lesson, it loudly proclaims life’s diversity. There ain’t no such abstraction as the clam, the fly, or the anglerfish. Ceratioid anglerfishes come in nearly 100 species, and each has its own peculiarity.
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Although I must say that research problems I worked on were frequently the result of serendipity and often grew out of my interest in some species or some environment which I found to be particularly appealing—marine birds and tropical islands for example.
Bartholomew, April 1993, unpublished remarks when receiving the Miller Award from the Cooper Ornithological Society.
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Anthropologists are a connecting link between poets and scientists though their fieldwork among primitive peoples has often made them forget the language of science.
From Arthur D. Little Lecture (6 Dec 1963) at the London School of Economics, in Saturday Review (1963), 46, No. 4, 83.
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Being perpetually charmed by his familiar siren, that is, by his geometry, he [Archimedes] neglected to eat and drink and took no care of his person; that he was often carried by force to the baths, and when there he would trace geometrical figures in the ashes of the fire, and with his finger draws lines upon his body when it was anointed with oil, being in a state of great ecstasy and divinely possessed by his science.
Plutarch
As translated in George Finlay Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, (1992), 39.
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Better far off to leave half the ruins and nine-tenths of the churches unseen and to see well the rest; to see them not once, but again and often again; to watch them, to learn them, to live with them, to love them, till they have become a part of life and life’s recollections.
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Between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.
From the French, “…entre deux vérités du domaine réel, le chemin le plus facile et le plus court passe bien souvent par le domaine complexe,” in Notice sur les Travaux Scientifiques (1900), 2. Widely seen incorrectly attributed to Hadamard, who quoted it himself as from an unnamed source, and paraphrased thus, “It has been written that the shortest path between two truths in the real domain passes through the complex domain,” in Jacques Hadamard, An Essay on the Psychology of Invention in the Mathematical Field (1945), 123. Now often seen misattributed to Hadamard in a shorter paraphrase, for example as, “The shortest path between two truths in the real domain passes through the complex domain,” in Craig Smorynski, 'The Need for Abstraction', The College Mathematics Journal (Jan 1985), 16, No. 1, 11. For a longer discussion of the Painlevé source, see homepage.math.uiowa.edu/~jorgen/hadamardquotesource.html, which is the source for the English translation in the subject quote.
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Biological disciplines tend to guide research into certain channels. One consequence is that disciplines are apt to become parochial, or at least to develop blind spots, for example, to treat some questions as “interesting” and to dismiss others as “uninteresting.” As a consequence, readily accessible but unworked areas of genuine biological interest often lie in plain sight but untouched within one discipline while being heavily worked in another. For example, historically insect physiologists have paid relatively little attention to the behavioral and physiological control of body temperature and its energetic and ecological consequences, whereas many students of the comparative physiology of terrestrial vertebrates have been virtually fixated on that topic. For the past 10 years, several of my students and I have exploited this situation by taking the standard questions and techniques from comparative vertebrate physiology and applying them to insects. It is surprising that this pattern of innovation is not more deliberately employed.
In 'Scientific innovation and creativity: a zoologist’s point of view', American Zoologist (1982), 22, 233.
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But, as we consider the totality of similarly broad and fundamental aspects of life, we cannot defend division by two as a natural principle of objective order. Indeed, the ‘stuff’ of the universe often strikes our senses as complex and shaded continua, admittedly with faster and slower moments, and bigger and smaller steps, along the way. Nature does not dictate dualities, trinities, quarterings, or any ‘objective’ basis for human taxonomies; most of our chosen schemes, and our designated numbers of categories, record human choices from a cornucopia of possibilities offered by natural variation from place to place, and permitted by the flexibility of our mental capacities. How many seasons (if we wish to divide by seasons at all) does a year contain? How many stages shall we recognize in a human life?
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Developmental Biology, in capitals, is the wave of the future. The creeping reductionism of biochemistry and molecular biology has taken over the cell and heredity, and looks covetously toward the heights of development and evolution. Recent literature is last year. Ancient literature is a decade ago. The rest is history, doubtfully alive. There is no time and often no opportunity to find and study the work of experimental biologists of 50 or 100 years ago, yet that was a time when the world was fresh.
Developmental biology was a lowercase phrase that graduated about 1950 and had previously lived under the cloak of Experimental Zoology
In obituary by Charles R. Scriver, Biographical Memoirs of Fellows of the Royal Society (Nov 1999), 45, 33.
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Every mathematical book that is worth reading must be read “backwards and forwards”, if I may use the expression. I would modify Lagrange’s advice a little and say, “Go on, but often return to strengthen your faith.” When you come on a hard or dreary passage, pass it over; and come back to it after you have seen its importance or found the need for it further on.
In Algebra, Part 2 (1889), Preface, viii.
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Everybody using C is a dangerous thing. We have other languages that don’t have buffer overflows. But what is the longer-term cost to us as an enterprise in increased vulnerability, increased need for add-on security services or whatever else is involved? Those kinds of questions don’t get asked often enough.
As quoted in magazine article, an interview by John McCormick, 'Computer Security as a Business Enabler', Baseline (7 Jul 2007).
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Evolution is a theory of organic change, but it does not imply, as many people assume, that ceaseless flux is the irreducible state of nature and that structure is but a temporary incarnation of the moment. Change is more often a rapid transition between stable states than a continuous transformation at slow and steady rates. We live in a world of structure and legitimate distinction. Species are the units of nature’s morphology.
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Experience is a jewel, and it need be so, for it is often purchased at an infinite rate.
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False interpretation of the Bible has often clashed with science, as false science has with true interpretation; but true science is the natural ally of religion, for both are from God.
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Geology got into the hands of the theoreticians who were conditioned by the social and political history of their day more than by observations in the field. … We have allowed ourselves to be brainwashed into avoiding any interpretation of the past that involves extreme and what might be termed “catastrophic” processes. However, it seems to me that the stratigraphical record is full of examples of processes that are far from “normal” in the usual sense of the word. In particular we must conclude that sedimentation in the past has often been very rapid indeed and very spasmodic. This may be called the “Phenomenon of the Catastrophic Nature of the Stratigraphic Record.”
In The Nature of the Stratigraphical Record (3rd ed., 1993), 70.
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Good scholars struggle to understand the world in an integral way (pedants bite off tiny bits and worry them to death). These visions of reality ... demand our respect, for they are an intellectual’s only birthright. They are often entirely wrong and always flawed in serious ways, but they must be understood honorably and not subjected to mayhem by the excision of patches.
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Have you ever watched an eagle held captive in a zoo, fat and plump and full of food and safe from danger too?
Then have you seen another wheeling high up in the sky, thin and hard and battle-scarred, but free to soar and fly?
Well, which have you pitied the caged one or his brother? Though safe and warm from foe or storm, the captive, not the other!
There’s something of the eagle in climbers, don’t you see; a secret thing, perhaps the soul, that clamors to be free.
It’s a different sort of freedom from the kind we often mean, not free to work and eat and sleep and live in peace serene.
But freedom like a wild thing to leap and soar and strive, to struggle with the icy blast, to really be alive.
That’s why we climb the mountain’s peak from which the cloud-veils flow, to stand and watch the eagle fly, and soar, and wheel... below...
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I am very astonished that the scientific picture of the real world around me is deficient. It gives a lot of factual information, puts all our experience in a magnificently consistent order, but it is ghastly silent about all and sundry that is really near to our heart, that really matters to us. It cannot tell us a word about red and blue, bitter and sweet, physical pain and physical delight; it knows nothing of beautiful and ugly, good or bad, God and eternity. Science sometimes pretends to answer questions in these domains, but the answers are very often so silly that we are not inclined to take them seriously.
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I can see him [Sylvester] now, with his white beard and few locks of gray hair, his forehead wrinkled o’er with thoughts, writing rapidly his figures and formulae on the board, sometimes explaining as he wrote, while we, his listeners, caught the reflected sounds from the board. But stop, something is not right, he pauses, his hand goes to his forehead to help his thought, he goes over the work again, emphasizes the leading points, and finally discovers his difficulty. Perhaps it is some error in his figures, perhaps an oversight in the reasoning. Sometimes, however, the difficulty is not elucidated, and then there is not much to the rest of the lecture. But at the next lecture we would hear of some new discovery that was the outcome of that difficulty, and of some article for the Journal, which he had begun. If a text-book had been taken up at the beginning, with the intention of following it, that text-book was most likely doomed to oblivion for the rest of the term, or until the class had been made listeners to every new thought and principle that had sprung from the laboratory of his mind, in consequence of that first difficulty. Other difficulties would soon appear, so that no text-book could last more than half of the term. In this way his class listened to almost all of the work that subsequently appeared in the Journal. It seemed to be the quality of his mind that he must adhere to one subject. He would think about it, talk about it to his class, and finally write about it for the Journal. The merest accident might start him, but once started, every moment, every thought was given to it, and, as much as possible, he read what others had done in the same direction; but this last seemed to be his real point; he could not read without finding difficulties in the way of understanding the author. Thus, often his own work reproduced what had been done by others, and he did not find it out until too late.
A notable example of this is in his theory of cyclotomic functions, which he had reproduced in several foreign journals, only to find that he had been greatly anticipated by foreign authors. It was manifest, one of the critics said, that the learned professor had not read Rummer’s elementary results in the theory of ideal primes. Yet Professor Smith’s report on the theory of numbers, which contained a full synopsis of Kummer’s theory, was Professor Sylvester’s constant companion.
This weakness of Professor Sylvester, in not being able to read what others had done, is perhaps a concomitant of his peculiar genius. Other minds could pass over little difficulties and not be troubled by them, and so go on to a final understanding of the results of the author. But not so with him. A difficulty, however small, worried him, and he was sure to have difficulties until the subject had been worked over in his own way, to correspond with his own mode of thought. To read the work of others, meant therefore to him an almost independent development of it. Like the man whose pleasure in life is to pioneer the way for society into the forests, his rugged mind could derive satisfaction only in hewing out its own paths; and only when his efforts brought him into the uncleared fields of mathematics did he find his place in the Universe.
In Florian Cajori, Teaching and History of Mathematics in the United States (1890), 266-267.
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I confess, that after I began … to discern how useful mathematicks may be made to physicks, I have often wished that I had employed about the speculative part of geometry, and the cultivation of the specious Algebra I had been taught very young, a good part of that time and industry, that I had spent about surveying and fortification (of which I remember I once wrote an entire treatise) and other parts of practick mathematicks.
In 'The Usefulness of Mathematiks to Natural Philosophy', Works (1772), Vol. 3, 426.
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I do not intend to go deeply into the question how far mathematical studies, as the representatives of conscious logical reasoning, should take a more important place in school education. But it is, in reality, one of the questions of the day. In proportion as the range of science extends, its system and organization must be improved, and it must inevitably come about that individual students will find themselves compelled to go through a stricter course of training than grammar is in a position to supply. What strikes me in my own experience with students who pass from our classical schools to scientific and medical studies, is first, a certain laxity in the application of strictly universal laws. The grammatical rules, in which they have been exercised, are for the most part followed by long lists of exceptions; accordingly they are not in the habit of relying implicitly on the certainty of a legitimate deduction from a strictly universal law. Secondly, I find them for the most part too much inclined to trust to authority, even in cases where they might form an independent judgment. In fact, in philological studies, inasmuch as it is seldom possible to take in the whole of the premises at a glance, and inasmuch as the decision of disputed questions often depends on an aesthetic feeling for beauty of expression, or for the genius of the language, attainable only by long training, it must often happen that the student is referred to authorities even by the best teachers. Both faults are traceable to certain indolence and vagueness of thought, the sad effects of which are not confined to subsequent scientific studies. But certainly the best remedy for both is to be found in mathematics, where there is absolute certainty in the reasoning, and no authority is recognized but that of one’s own intelligence.
In 'On the Relation of Natural Science to Science in general', Popular Lectures on Scientific Subjects, translated by E. Atkinson (1900), 25-26.
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I have often admired the mystical way of Pythagoras, and the secret magick of numbers.
In Religio Medici (1642, 1754), pt. 1, sec. 12, 28.
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I have often been amused by our vulgar tendency to take complex issues, with solutions at neither extreme of a continuum of possibilities, and break them into dichotomies, assigning one group to one pole and the other to an opposite end, with no acknowledgment of subtleties and intermediate positions–and nearly always with moral opprobrium attached to opponents.
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I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, “Oh, I see. That’s how it must be.” This is a sign that they explain it to themselves from within their own system.
Lichtenberg: A Doctrine of Scattered Occasions: Reconstructed From: Reconstructed From His Aphorisms and Reflections (1959), 291.
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I respect Kirkpatrick both for his sponges and for his numinous nummulosphere. It is easy to dismiss a crazy theory with laughter that debars any attempt to understand a man’s motivation–and the nummulosphere is a crazy theory. I find that few men of imagination are not worth my attention. Their ideas may be wrong, even foolish, but their methods often repay a close study ... The different drummer often beats a fruitful tempo.
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I strongly reject any conceptual scheme that places our options on a line, and holds that the only alternative to a pair of extreme positions lies somewhere between them. More fruitful perspectives often require that we step off the line to a site outside the dichotomy.
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If I were not a physicist, I would probably be a musician. I often think in music. I live my dreams in music. I see my life in terms of music... I get most joy in life out of music.
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If it were always necessary to reduce everything to intuitive knowledge, demonstration would often be insufferably prolix. This is why mathematicians have had the cleverness to divide the difficulties and to demonstrate separately the intervening propositions. And there is art also in this; for as the mediate truths (which are called lemmas, since they appear to be a digression) may be assigned in many ways, it is well, in order to aid the understanding and memory, to choose of them those which greatly shorten the process, and appear memorable and worthy in themselves of being demonstrated. But there is another obstacle, viz.: that it is not easy to demonstrate all the axioms, and to reduce demonstrations wholly to intuitive knowledge. And if we had chosen to wait for that, perhaps we should not yet have the science of geometry.
In Gottfried Wilhelm Leibnitz and Alfred Gideon Langley (trans.), New Essays Concerning Human Understanding (1896), 413-414.
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In all matters of opinion and science ... the difference between men is ... oftener found to lie in generals than in particulars; and to be less in reality than in appearance. An explication of the terms commonly ends the controversy, and the disputants are surprised to find that they had been quarrelling, while at bottom they agreed in their judgement.
Dissertation IV, 'Of the Standard of Taste', Four Dissertations (1757), 204.
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In natural history, great discovery often requires a map to a hidden mine filled with gems then easily gathered by conventional tools, not a shiny new space-age machine for penetrating previously inaccessible worlds.
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In the matter of physics, the first lessons should contain nothing but what is experimental and interesting to see. A pretty experiment is in itself often more valuable than twenty formulae extracted from our minds.
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Inspiration in the field of science by no means plays any greater role, as academic conceit fancies, than it does in the field of mastering problems of practical life by a modern entrepreneur. On the other hand, and this also is often misconstrued, inspiration plays no less a role in science than it does in the realm of art.
Max Weber
From a Speech (1918) presented at Munich University, published in 1919, and collected in 'Wissenschaft als Beruf', Gessammelte Aufsätze zur Wissenschaftslehre (1922), 524-525. As given in H.H. Gerth and C. Wright-Mills (translators and eds.), 'Science as a Vocation', Max Weber: Essays in Sociology (1946), 136.
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It has often been said that power corrupts. But it is perhaps equally important to realize that weakness, too, corrupts. Power corrupts the few, while weakness corrupts the many. Hatred, malice, rudeness, intolerance, and suspicion are the faults of weakness. The resentment of the weak does not spring from any injustice done to them but from the sense of inadequacy and impotence. We cannot win the weak by sharing our wealth with them. They feel our generosity as oppression. St. Vincent De Paul cautioned his disciples to deport themselves so that the poor “will forgive them the bread you give them.”
In 'The Awakening of Asia', The Ordeal of Change (1963), 12.
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It is characteristic of science that the full explanations are often seized in their essence by the percipient scientist long in advance of any possible proof.
The Origin of Life, 1967
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It is easier to love humanity as a whole than to love one’s neighbor. There may even be a certain antagonism between love of humanity and love of neighbor; a low capacity for getting along with those near us often goes hand in hand with a high receptivity to the idea of the brotherhood of men. About a hundred years ago a Russian landowner by the name of Petrashevsky recorded a remarkable conclusion: “Finding nothing worthy of my attachment either among women or among men, I have vowed myself to the service of mankind.” He became a follower of Fourier, and installed a phalanstery on his estate. The end of the experiment was sad, but what one might perhaps have expected: the peasants—Petrashevsky’s neighbors-burned the phalanstery.
In 'Brotherhood', The Ordeal of Change (1963), 91.
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It is not a case we are treating; it is a living, palpitating, alas, too often suffering fellow creature.
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It is often more convenient to possess the ashes of great men than to possess the men themselves during their lifetime.
Commenting on the return of Descartes’ remains to France. Quoted, without citation, in Eric Temple Bell Men of Mathematics (1937), 52.
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It is the individual only who is timeless. Societies, cultures, and civilizations - past and present - are often incomprehensible to outsiders, but the individual’s hunger, anxieties, dreams, and preoccupations have remained unchanged through the millennia. Thus, we are up against the paradox that the individual who is more complex, unpredictable, and mysterious than any communal entity is the one nearest to our understanding; so near that even the interval of millennia cannot weaken our feeling of kinshiIf in some manner the voice of an individual reaches us from the remotest distance of time, it is a timeless voice speaking about ourselves.
In Reflections on the Human Condition (1973), 97.
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It is the object of science to replace, or save, experiences, by the reproduction and anticipation of facts in thought. Memory is handier than experience, and often answers the same purpose. This economical office of science, which fills its whole life, is apparent at first glance; and with its full recognition all mysticism in science disappears.
In 'The Economy of Science', The Science of Mechanics: A Critical and Historical Exposition of Its Principles (1893), 4.
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It seems rather incongruous that in a society of super-sophisticated communications, we often suffer from a shortage of listeners.
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It’s among the intelligentsia … that we often find the glib compulsion to explain everything and to understand nothing.
The Rape of the Mind World 56
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Men of genius are often dull and inert in society; as the blazing meteor, when it descends to earth, is only a stone.
Louis Klopsch, Many Thoughts of Many Minds (1896), 106.
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My passion for social justice has often brought me into conflict with people, as did my aversion to any obligation and dependence I do not regard as absolutely necessary. I always have a high regard for the individual and have an insuperable distaste for violence and clubmanship.
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My position is a naturalistic one; I see philosophy not as an a priori propaedeutic or groundwork for science, but as continuous with science. I see philosophy and science as in the same boat—a boat which, to revert to Neurath’s figure as I so often do, we can rebuild only at sea while staying afloat in it. There is no external vantage point, no first philosophy.
Ontological Relativity and Other Essays, pp. 126-127, Columbia University Press (1969).
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My profession often gets bad press for a variety of sins, both actual and imagined: arrogance, venality, insensitivity to moral issues about the use of knowledge, pandering to sources of funding with insufficient worry about attendant degradation of values. As an advocate for science, I plead ‘mildly guilty now and then’ to all these charges. Scientists are human beings subject to all the foibles and temptations of ordinary life. Some of us are moral rocks; others are reeds. I like to think (though I have no proof) that we are better, on average, than members of many other callings on a variety of issues central to the practice of good science: willingness to alter received opinion in the face of uncomfortable data, dedication to discovering and publicizing our best and most honest account of nature’s factuality, judgment of colleagues on the might of their ideas rather than the power of their positions.
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Natural historians tend to avoid tendentious preaching in this philosophical mode (although I often fall victim to such temptations in these essays). Our favored style of doubting is empirical: if I wish to question your proposed generality, I will search for a counterexample in flesh and blood. Such counterexamples exist in abundance, for the form a staple in a standard genre of writing in natural history–the “wonderment of oddity” or “strange ways of the beaver” tradition.
In 'Reversing Established Orders', Leonardo’s Mountain of Clams and the Diet of Worms (2011), 394.
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Now, at Suiattle Pass, Brower was still talking about butterflies. He said he had raised them from time to time and had often watched them emerge from the chrysalis—first a crack in the case, then a feeler, and in an hour a butterfly. He said he had felt that he wanted to help, to speed them through the long and awkward procedure; and he had once tried. The butterflies came out with extended abdomens, and their wings were balled together like miniature clenched fists. Nothing happened. They sat there until they died. ‘I have never gotten over that,’ he said. ‘That kind of information is all over in the country, but it’s not in town.”
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Often in evolutionary processes a species must adapt to new conditions in order to survive. Today the atomic bomb has altered profoundly the nature of the world as we know it, and the human race consequently finds itself in a new habitat to which it must adapt its thinking.
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Our remedies oft in ourselves do lie, which we ascribe to heaven.
In All's Well That Ends Well, Act 1, 1, 199-200. Collected in The Complete Works of William Shakespeare (1906), Vol. 1, 299.
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Our situation on this earth seems strange. Every one of us appears here involuntarily and uninvited for a short stay, without knowing the whys and the wherefore. In our daily lives we only feel that man is here for the sake of others, for those whom we love and for many other beings whose fate is connected with our own. I am often worried at the thought that my life is based to such a large extent on the work of my fellow human beings and I am aware of my great indebtedness to them.
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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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People who wouldn’t think of talking with their mouths full often speak with their heads empty.
In Wolfgang Mieder, Wolfgang Mieder, Stewart A. Kingsbury (eds.), A Dictionary of American Proverbs (1992), 472.
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Personally, I feel sorry for those who seem to measure their patriotism by how often and how viciously they can criticise our government.
As recorded in United States Congressional Serial Set (2000), 51.
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Relatively few benefits have flowed to the people who live closest to the more than 3,000 protected areas that have been established in tropical countries during the past 50 years. For this reason, the preservation of biodiversity is often thought of as something that poor people are asked to do to fulfill the wishes of rich people living in comfort thousands of miles away.
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Science is not gadgetry. The desirable adjuncts of modern living, although in many instances made possible by science, certainly do not constitute science. Basic scientific knowledge often (but not always) is a prerequisite to such developments, but technology primarily deserves the credit for having the financial courage, the ingenuity, and the driving energy to see to it that so-called ‘pure knowledge’ is in fact brought to the practical service of man. And it should also be recognized that those who have the urge to apply knowledge usefully have themselves often made significant contribution to pure knowledge and have even more often served as a stimulation to the activities of a pure researcher.
Warren Weaver (1894–1978), U.S. mathematician, scientist, educator. Science and Imagination, ch. 1, Basic Books (1967).
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Science is often regarded as the most objective and truth-directed of human enterprises, and since direct observation is supposed to be the favored route to factuality, many people equate respectable science with visual scrutiny–just the facts ma’am, and palpably before my eyes. But science is a battery of observational and inferential methods, all directed to the testing of propositions that can, in principle, be definitely proven false ... At all scales, from smallest to largest, quickest to slowest, many well-documented conclusions of science lie beyond the strictly limited domain of direct observation. No one has ever seen an electron or a black hole, the events of a picosecond or a geological eon.
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Since most callers have until moments before been completely unaware that there are bears in New Jersey, there is often in their voices a component of alarm, up to and including terror. McConnell’s response is calmer than pavement. She speaks in tones that range from ho to hum. “Yes, there are bears in your area,” she says, and goes on to say, with an added hint of congratulation, “You live in beautiful bear habitat.
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Surprises in science often arise from new tools rather than from new concepts.
(1999). In Marc J. Madou, Fundamentals of Microfabrication: the Science of Miniaturization (2nd ed., 2002), 379.
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Technocrats are turning us into daredevils. The haphazard gambles they are imposing on us too often jeopardize our safety for goals that do not advance the human cause but undermine it. By staking our lives on their schemes, decision makers are not meeting the mandate of a democratic society; they are betraying it. They are not ennobling us; they are victimizing us. And, in acquiescing to risks that have resulted in irreversible damage to the environment, we ourselves are not only forfeiting our own rights as citizens. We are, in turn, victimizing the ultimate nonvolunteers: the defenseless, voiceless—voteless—children of the future.
In Jacques Cousteau and Susan Schiefelbein, The Human, the Orchid, and the Octopus: Exploring and Conserving Our Natural World (2007), 85.
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The authority of those who profess to teach is often a positive hindrance to those who desire to learn.
In De Natura Deorum: Academica (c. 45 B.C.), as translated to English (1933), 13.
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The average English author [of mathematical texts] leaves one under the impression that he has made a bargain with his reader to put before him the truth, the greater part of the truth, and nothing but the truth; and that if he has put the facts of his subject into his book, however difficult it may be to unearth them, he has fulfilled his contract with his reader. This is a very much mistaken view, because effective teaching requires a great deal more than a bare recitation of facts, even if these are duly set forth in logical order—as in English books they often are not. The probable difficulties which will occur to the student, the objections which the intelligent student will naturally and necessarily raise to some statement of fact or theory—these things our authors seldom or never notice, and yet a recognition and anticipation of them by the author would be often of priceless value to the student. Again, a touch of humour (strange as the contention may seem) in mathematical works is not only possible with perfect propriety, but very helpful; and I could give instances of this even from the pure mathematics of Salmon and the physics of Clerk Maxwell.
In Perry, Teaching of Mathematics (1902), 59-61.
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The basic ideas and simplest facts of set-theoretic topology are needed in the most diverse areas of mathematics; the concepts of topological and metric spaces, of compactness, the properties of continuous functions and the like are often indispensable.
As co-author with H. Hopf, Topologie I (1935), 23. As quoted, translated and cited in Reinhold Remmert, Theory of Complex Functions (1991), 9-10. From the original German, “Die Grundbegriffe und die einfachsten Tatsachen aus der mengentheoretischen Topologie braucht man in sehr verschiedenen Gebieten der Mathematik; die Begriffe des topologischen und des metrischen Raumes, der Kompaktheit, die Eigenschaften stetiger Abbildungen u. dgl. sind oft unentbehrlich.”
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The chances for favorable serendipity are increased if one studies an animal that is not one of the common laboratory species. Atypical animals, or preparations, force one to use non-standard approaches and non-standard techniques, and even to think nonstandard ideas. My own preference is to seek out species which show some extreme of adaptation. Such organisms often force one to abandon standard methods and standard points of view. Almost inevitably they lead one to ask new questions, and most importantly in trying to comprehend their special and often unusual adaptations one often serendipitously stumbles upon new insights.
In 'Scientific innovation and creativity: a zoologist’s point of view', American Zoologist (1982), 22, 234.
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The concepts most familiar to us are often the most mysterious.
John Mitchinson and John Lloyd, If Ignorance Is Bliss, Why Aren't There More Happy People?: Smart Quotes for Dumb Times (2009), 217.
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The creative scientist studies nature with the rapt gaze of the lover, and is guided as often by aesthetics as by rational considerations in guessing how nature works.
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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 facts of nature are what they are, but we can only view them through the spectacles of our mind. Our mind works largely by metaphor and comparison, not always (or often) by relentless logic. When we are caught in conceptual traps, the best exit is often a change in metaphor–not because the new guideline will be truer to nature (for neither the old nor the new metaphor lies ‘out there’ in the woods), but because we need a shift to more fruitful perspectives, and metaphor is often the best agent of conceptual transition.
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The human mind delights in finding pattern–so much so that we often mistake coincidence or forced analogy for profound meaning. No other habit of thought lies so deeply within the soul of a small creature trying to make sense of a complex world not constructed for it.
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The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics.
In The Philosophy of the Inductive Sciences (1868), Part 1, Bk. 2, chap. 1, sect. 4.
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The Mean Value Theorem is the midwife of calculus—not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance.
With co-author D. Varberg in Calculus with Analytic Geometry (1984), 191.
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The most learned are often the most narrow-minded.
In Hialmer Day Gould, New Practical Spelling (1905), 14.
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The mythology of science asserts that with many different scientists all asking their own questions and evaluating the answers independently, whatever personal bias creeps into their individual answers is cancelled out when the large picture is put together. This might conceivably be so if scientists were women and men from all sorts of different cultural and social backgrounds who came to science with very different ideologies and interests. But since, in fact, they have been predominantly university-trained white males from privileged social backgrounds, the bias has been narrow and the product often reveals more about the investigator than about the subject being researched.
'Have Only Men Evolved?' Women Look at Biology Looking At Women, eds. Ruth Hubbard, Mary Sue Henifin, and Barbara Fried (1979).
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The question is not whether “big is ugly,” “small is beautiful,” or technology is “appropriate.” It is whether technologists will be ready for the demanding, often frustrating task of working with critical laypeople to develop what is needed or whether th
Technology Review (Feb 1980).
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The reputation of science which ought to be the most lasting, as synonymous with truth, is often the least so. One discovery supersedes another; and the progress of light throws the past into obscurity. What is become of the Blacks, the Lavoisiers, the Priestleys, in chemistry? … When any set of men think theirs the only science worth studying, and themselves the only infallible persons in it, it is a sign how frail the traces are of past excellence in it.
Characteristics: In the Manner of Rochefoucault's Maxims (1837), 148-149.
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The science of calculation … is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not to be indulged in by one who is to have a profession to follow for his subsistence.
In Letter (18 Jun 1799) to William G. Munford. On founders.archives.gov website.
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The skein of human continuity must often become this tenuous across the centuries (hanging by a thread, in the old cliche’), but the circle remains unbroken if I can touch the ink of Lavoisier’s own name, written by his own hand. A candle of light, nurtured by the oxygen of his greatest discovery, never burns out if we cherish the intellectual heritage of such unfractured filiation across the ages. We may also wish to contemplate the genuine physical thread of nucleic acid that ties each of us to the common bacterial ancestor of all living creatures, born on Lavoisier’s ancienne terre more than 3.5 billion years ago– and never since disrupted, not for one moment, not for one generation. Such a legacy must be worth preserving from all the guillotines of our folly.
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The story of a theory’s failure often strikes readers as sad and unsatisfying. Since science thrives on self-correction, we who practice this most challenging of human arts do not share such a feeling. We may be unhappy if a favored hypothesis loses or chagrined if theories that we proposed prove inadequate. But refutation almost always contains positive lessons that overwhelm disappointment, even when no new and comprehensive theory has yet filled the void.
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The student of mathematics often finds it hard to throw off the uncomfortable feeling that his science, in the person of his pencil, surpasses him in intelligence,—an impression which the great Euler confessed he often could not get rid of. This feeling finds a sort of justification when we reflect that the majority of the ideas we deal with were conceived by others, often centuries ago. In a great measure it is really the intelligence of other people that confronts us in science.
In Popular Scientific Lectures (1910), 196.
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The student should read his author with the most sustained attention, in order to discover the meaning of every sentence. If the book is well written, it will endure and repay his close attention: the text ought to be fairly intelligible, even without illustrative examples. Often, far too often, a reader hurries over the text without any sincere and vigorous effort to understand it; and rushes to some example to clear up what ought not to have been obscure, if it had been adequately considered. The habit of scrupulously investigating the text seems to me important on several grounds. The close scrutiny of language is a very valuable exercise both for studious and practical life. In the higher departments of mathematics the habit is indispensable: in the long investigations which occur there it would be impossible to interpose illustrative examples at every stage, the student must therefore encounter and master, sentence by sentence, an extensive and complicated argument.
In 'Private Study of Mathematics', Conflict of Studies and other Essays (1873), 67.
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The theory of punctuated equilibrium, proposed by Niles Eldredge and myself, is not, as so often misunderstood, a radical claim for truly sudden change, but a recognition that ordinary processes of speciation, properly conceived as glacially slow by the standard of our own life-span, do not resolve into geological time as long sequences of insensibly graded intermediates (the traditional, or gradualistic, view), but as geologically ‘sudden’ origins at single bedding planes.
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The weight of any heavy body of known weight at a particular distance from the center of the world varies according to the variation of its distance therefrom: so that as often as it is removed from the center, it becomes heavier, and when brought near to it, is lighter. On this account, the relation of gravity to gravity is as the relation of distance to distance from the center.
From Book of the Balance of Wisdom. As cited in an epigraph in Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, Gravity (1973), 37.
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Theories rarely arise as patient inferences forced by accumulated facts. Theories are mental constructs potentiated by complex external prods (including, in idealized cases, a commanding push from empirical reality) . But the prods often in clude dreams, quirks, and errors–just as we may obtain crucial bursts of energy from foodstuffs or pharmaceuticals of no objective or enduring value. Great truth can emerge from small error. Evolution is thrilling, liberating, and correct. And Macrauchenia is a litoptern.
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There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.
In 'Integrals Devised for Special Purposes', Bulletin of the American Mathematical Society (1963), 69, 611.
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They who clamor loudest for freedom are often the ones least likely to be happy in a free society. The frustrated, oppressed by their shortcomings, blame their failure on existing restraints. Actually, their innermost desire is for an end to the “free for all.” They want to eliminate free competition and the ruthless testing to which the individual is continually subjected in a free society.
In The True Believer: Thoughts on the Nature of Mass Movements (1951), 32.
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This theme of mutually invisible life at widely differing scales bears an important implication for the ‘culture wars’ that supposedly now envelop our universities and our intellectual discourse in general ... One side of this false dichotomy features the postmodern relativists who argue that all culturally bound modes of perception must be equally valid, and that no factual truth therefore exists. The other side includes the benighted, old-fashioned realists who insist that flies truly have two wings, and that Shakespeare really did mean what he thought he was saying. The principle of scaling provides a resolution for the false parts of this silly dichotomy. Facts are facts and cannot be denied by any rational being. (Often, facts are also not at all easy to determine or specify–but this question raises different issues for another time.) Facts, however, may also be highly scale dependent–and the perceptions of one world may have no validity or expression in the domain of another. The one-page map of Maine cannot recognize the separate boulders of Acadia, but both provide equally valid representations of a factual coastline.
The World as I See It (1999)
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Too often I would hear men boast of the miles covered that day, rarely of what they had seen.
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Until its results have gone through the painful process of publication, preferably in a refereed journal of high standards, scientific research is just play. Publication is an indispensable part of science. “Publish or perish” is not an indictment of the system of academia; it is a partial prescription for creativity and innovation. Sustained and substantial publication favors creativity. Novelty of conception has a large component of unpredictability. ... One is often a poor judge of the relative value of his own creative efforts. An artist’s ranking of his own works is rarely the same as that of critics or of history. Most scientists have had similar experiences. One’s supply of reprints for a pot-boiler is rapidly exhausted, while a major monograph that is one’s pride and joy goes unnoticed. The strategy of choice is to increase the odds favoring creativity by being productive.
In 'Scientific innovation and creativity: a zoologist’s point of view', American Zoologist (1982), 22, 233-234.
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Very few people, including authors willing to commit to paper, ever really read primary sources–certainly not in necessary depth and contemplation, and often not at all ... When writers close themselves off to the documents of scholarship, and then rely only on seeing or asking, they become conduits and sieves rather than thinkers. When, on the other hand, you study the great works of predecessors engaged in the same struggle, you enter a dialogue with human history and the rich variety of our own intellectual traditions. You insert yourself, and your own organizing powers, into this history–and you become an active agent, not merely a ‘reporter.’
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We often think that when we have completed our study of one we know all about two, because “two” is “one and one.” We forget that we still have to make a study of “and.”
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We often think, naïvely, that missing data are the primary impediments to intellectual progress–just find the right facts and all problems will dissipate. But barriers are often deeper and more abstract in thought. We must have access to the right metaphor, not only to the requisite information. Revolutionary thinkers are not, primarily, gatherers of fact s, but weavers of new intellectual structures.
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Were we in danger of stopping [the experiment] when we liked the answer? I realized then, as I have often said since, that nature does not “know” what experiment a scientist is trying to do. “God loves the noise as much as the signal.”
In Confessions of a Technophile (1994), 45, with embedded quote by Branscomb cited in footnote as from 'Physics and the APS in 1979', Physics Today (Apr 1980), 33, No. 4, 49.
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When one door of happiness closes, another opens; but often we look so long at the closed door that we do not see the one which has been opened for us.
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Whenever I get gloomy with the state of the world, I think about the arrivals gate at Heathrow Airport. General opinion’s starting to make out that we live in a world of hatred and greed, but I don’t see that. It seems to me that love is everywhere. Often it’s not particularly dignified or newsworthy, but it’s always there - fathers and sons, mothers and daughters, husbands and wives, boyfriends, girlfriends, old friends. When the planes hit the Twin Towers, as far as I know none of the phone calls from the people on board were messages of hate or revenge - they were all messages of love. If you look for it, I’ve got a sneaky feeling you’ll find that love actually is all around.
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While the method of the natural sciences is... analytic, the method of the social sciences is better described as compositive or synthetic. It is the so-called wholes, the groups of elements which are structurally connected, which we learn to single out from the totality of observed phenomena... Insofar as we analyze individual thought in the social sciences the purpose is not to explain that thought, but merely to distinguish the possible types of elements with which we shall have to reckon in the construction of different patterns of social relationships. It is a mistake... to believe that their aim is to explain conscious action ... The problems which they try to answer arise only insofar as the conscious action of many men produce undesigned results... If social phenomena showed no order except insofar as they were consciously designed, there would indeed be no room for theoretical sciences of society and there would be, as is often argued, only problems of psychology. It is only insofar as some sort of order arises as a result of individual action but without being designed by any individual that a problem is raised which demands a theoretical explanation... people dominated by the scientistic prejudice are often inclined to deny the existence of any such order... it can be shown briefly and without any technical apparatus how the independent actions of individuals will produce an order which is no part of their intentions... The way in which footpaths are formed in a wild broken country is such an instance. At first everyone will seek for himself what seems to him the best path. But the fact that such a path has been used once is likely to make it easier to traverse and therefore more likely to be used again; and thus gradually more and more clearly defined tracks arise and come to be used to the exclusion of other possible ways. Human movements through the region come to conform to a definite pattern which, although the result of deliberate decision of many people, has yet not be consciously designed by anyone.
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Wisdom comes from experience. Experience is often a result of lack of wisdom.
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Wisdom oft lurks beneath a tattered coat.
From the original Latin, “Saepe est etiam sub palliolo sordido sapientia,” in Fabulae Incertae, Fragment XVIII, (II). As translated in Thomas Benfield Harbottle, Dictionary of Quotations (Classical) (1897), 256.
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[Mathematics is] the study of ideal constructions (often applicable to real problems), and the discovery thereby of relations between the parts of these constructions, before unknown.
In 'Mathematics', Century Dictionary.
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“Crawling at your feet,” said the Gnat … “you may observe a Bread-and-Butterfly. …”
“And what does it live on?”
“Weak tea with cream in it.”
A new difficulty came into Alice's head. “Supposing it couldn't find any?” she suggested.
“Then it would die, of course.”
“But that must happen very often,” Alice remarked thoughtfully.
“It always happens,” said the Gnat.
In Through the Looking Glass: And what Alice Found There (1893), 66-67.
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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 -
Antoine Lavoisier
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Euclid
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Andre Ampere
Winston Churchill
- 80 -
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Bible
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- 70 -
Samuel Morse
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- 60 -
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Karl Popper
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Avicenna
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- 50 -
Stephen Hawking
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- 40 -
Pierre Fermat
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JJ Thomson
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- 30 -
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- 20 -
Carl Sagan
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- 10 -
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