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Who said: “Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things.”
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Result Quotes (376 quotes)

...the life of the planet began the long, slow process of modulating and regulating the physical conditions of the planet. The oxygen in today's atmosphere is almost entirely the result of photosynthetic living, which had its start with the appearance of blue-green algae among the microorganisms.
In Late Night Thoughts on Listening to Mahler's Ninth Symphony(1984), 74.
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Als Physiker, der sein ganzes Leben der nüchternen Wissenschaft, der Erforschung der Materie widmete, bin ich sicher von dem Verdacht frei, für einen Schwarmgeist gehalten zu werden. Und so sage ich nach meinen Erforschungen des Atoms dieses: Es gibt keine Materie an sich. Alle Materie entsteht und besteht nur durch eine Kraft, welche die Atomteilchen in Schwingung bringt und sie zum winzigsten Sonnensystem des Alls zusammenhält. Da es im ganzen Weltall aber weder eine intelligente Kraft noch eine ewige Kraft gibt - es ist der Menschheit nicht gelungen, das heißersehnte Perpetuum mobile zu erfinden - so müssen wir hinter dieser Kraft einen bewußten intelligenten Geist annehmen. Dieser Geist ist der Urgrund aller Materie.
As a man who has devoted his whole life to the most clear headed science, to the study of matter, I can tell you as a result of my research about atoms this much: There is no matter as such. All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together. We must assume behind this force the existence of a conscious and intelligent mind. This mind is the matrix of all matter.
Lecture, 'Das Wesen der Materie' [The Essence/Nature/Character of Matter], Florence, Italy (1944). Archiv zur Geschichte der Max-Planck-Gesellschaft, Abt. Va, Rep. 11 Planck, Nr. 1797. Excerpt in Gregg Braden, The Spontaneous Healing of Belief: Shattering the Paradigm of False Limits (2009), 334-35. Note: a number of books showing this quote cite it as from Planck's Nobel Prize acceptance speech (1918), which the Webmaster has checked, and does not see this quote therein.
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Man is the result of slow growth; that is why he occupies the position he does in animal life. What does a pup amount to that has gained its growth in a few days or weeks, beside a man who only attains it in as many years.
In Orison Swett Marden, 'Bell Telephone Talk: Hints on Success by Alexander G. Bell', How They Succeeded: Life Stories of Successful Men Told by Themselves (1901), 35.
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[About the demand of the Board of Regents of the University of California that professors sign non-Communist loyalty oaths or lose their jobs within 65 days.] No conceivable damage to the university at the hands of hypothetical Communists among us could possibly have equaled the damage resulting from the unrest, ill-will and suspicion engendered by this series of events.
As quoted in 'Professors in West Call Oath “Indignity”', New York Times (26 Feb 1950), 38.
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[In refutation of evolution] There is not enough evidence, consistent evidence to make it as fact, and I say that because for theory to become a fact, it needs to consistently have the same results after it goes through a series of tests. The tests that they put—that they use to support evolution do not have consistent results. Now too many people are blindly accepting evolution as fact. But when you get down to the hard evidence, it’s merely a theory.
[In favor of the teaching of creationism alongside evolution in schools.]
From interview by Miles O'Brien on CNN (30 Mar 1996). Reported from transcript, via Nexis, in New York Magazine (15 Sep 2010).
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[When recording electrical impulses from a frog nerve-muscle preparation seemed to show a tiresomely oscillating electrical artefact—but only when the muscle was hanging unsupported.] The explanation suddenly dawned on me ... a muscle hanging under its own weight ought, if you come to think of it, to be sending sensory impulses up the nerves coming from the muscle spindles ... That particular day’s work, I think, had all the elements that one could wish for. The new apparatus seemed to be misbehaving very badly indeed, and I suddenly found it was behaving so well that it was opening up an entire new range of data ... it didn’t involve any particular hard work, or any particular intelligence on my part. It was just one of those things which sometimes happens in a laboratory if you stick apparatus together and see what results you get.
From 'Memorable experiences in research', Diabetes (1954), 3, 17-18. As cited in Alan McComa, Galvani's Spark: The Story of the Nerve Impulse (2011), 102-103.
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A general course in mathematics should be required of all officers for its practical value, but no less for its educational value in training the mind to logical forms of thought, in developing the sense of absolute truthfulness, together with a confidence in the accomplishment of definite results by definite means.
In 'Mathematics at West Point and Annapolis', United States Bureau of Education, Bulletin 1912, No. 2, 11.
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A philosopher once said, ‘It is necessary for the very existence of science that the same conditions always produce the same results’. Well, they don’t!
…...
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A pound of energy with an ounce of talent will achieve greater results than a pound of talent with an ounce of energy.
In Getting on in the World; Or, Hints on Success in Life (1873), 55.
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A professor … may be to produce a perfect mathematical work of art, having every axiom stated, every conclusion drawn with flawless logic, the whole syllabus covered. This sounds excellent, but in practice the result is often that the class does not have the faintest idea of what is going on. … The framework is lacking; students do not know where the subject fits in, and this has a paralyzing effect on the mind.
In A Concrete Approach to Abstract Algebra (1959), 1-2.
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A rose is the visible result of an infinitude of complicated goings on in the bosom of the earth and in the air above, and similarly a work of art is the product of strange activities in the human mind.
In Since Cezanne (1922), 40.
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A scientific invention consists of six (or some number) ideas, five of which are absurd but which, with the addition of the sixth and enough rearrangement of the combinations, results in something no one has thought of before.
As quoted in Robert Coughlan, 'Dr. Edward Teller’s Magnificent Obsession', Life (6 Sep 1954), 66.
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A student who wishes now-a-days to study geometry by dividing it sharply from analysis, without taking account of the progress which the latter has made and is making, that student no matter how great his genius, will never be a whole geometer. He will not possess those powerful instruments of research which modern analysis puts into the hands of modern geometry. He will remain ignorant of many geometrical results which are to be found, perhaps implicitly, in the writings of the analyst. And not only will he be unable to use them in his own researches, but he will probably toil to discover them himself, and, as happens very often, he will publish them as new, when really he has only rediscovered them.
From 'On Some Recent Tendencies in Geometrical Investigations', Rivista di Matematica (1891), 43. In Bulletin American Mathematical Society (1904), 443.
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Abraham Maslow, felt … [an] instinctive revolt against the “atmosphere” of Freudian psychology, with its emphasis on sickness and neurosis, and decided that he might obtain some equally interesting results if he studied extremely healthy people.
In Introduction to the New Existentialism (1966), 15.
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Accurate and minute measurement seems to the non-scientific imagination, a less lofty and dignified work than looking for something new. But nearly all the grandest discoveries of science have been but the rewards of accurate measurement and patient long-continued labour in the minute sifting of numerical results.
Presidential inaugural address, to the General Meeting of the British Association, Edinburgh (2 Aug 1871). In Report of the Forty-First Meeting of the British Association for the Advancement of Science (1872), xci.
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Act as if you are going to live for ever and cast your plans way ahead. You must feel responsible without time limitations, and the consideration of whether you may or may not be around to see the results should never enter your thoughts.
In Theodore Rockwell, The Rickover Effect: How One Man Made A Difference (2002), 342.
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After an orange cloud—formed as a result of a dust storm over the Sahara and caught up by air currents—reached the Philippines and settled there with rain, I understood that we are all sailing in the same boat.
In Jack Hassard and Julie Weisberg , Environmental Science on the Net: The Global Thinking Project (1999), 40.
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All palaetiological sciences, all speculations which attempt to ascend from the present to the remote past, by the chain of causation, do also, by an inevitable consequence, urge us to look for the beginning of the state of things which we thus contemplate; but in none of these cases have men been able, by the aid of science, to arrive at a beginning which is homogeneous with the known course of events. The first origin of language, of civilization, of law and government, cannot be clearly made out by reasoning and research; and just as little, we may expect, will a knowledge of the origin of the existing and extinct species of plants and animals, be the result of physiological and geological investigation.
In History of the Inductive Sciences (1837), Vol. 3, 581.
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All that we can hope from these inspirations, which are the fruits of unconscious work, is to obtain points of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work which follows the inspiration, and in which the results of the inspiration are verified and the consequences deduced.
Science and Method (1914, 2003), 62.
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All things on the earth are the result of chemical combination. The operation by which the commingling of molecules and the interchange of atoms take place we can imitate in our laboratories; but in nature they proceed by slow degrees, and, in general, in our hands they are distinguished by suddenness of action. In nature chemical power is distributed over a long period of time, and the process of change is scarcely to be observed. By acts we concentrate chemical force, and expend it in producing a change which occupies but a few hours at most.
In chapter 'Chemical Forces', The Poetry of Science: Or, Studies of the Physical Phenomena of Nature (1848), 235-236. Charles Dicken used this quote, with his own sub-head of 'Relative Importance Of Time To Man And Nature', to conclude his review of the book, published in The Examiner (1848).
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Almost all the greatest discoveries in astronomy have resulted from what we have elsewhere termed Residual Phenomena, of a qualitative or numerical kind, of such portions of the numerical or quantitative results of observation as remain outstanding and unaccounted for, after subducting and allowing for all that would result from the strict application of known principles.
Outlines of Astronomy (1876), 626.
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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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Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility.
In Charles S. Peirce, ‎Charles Hartshorne (ed.), ‎Paul Weiss (ed.), Collected Papers of Charles Sanders Peirce (1931), Vol. 4, 197.
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An experiment is never a failure solely because it fails to achieve predicted results. An experiment is a failure only when it also fails adequately to test the hypothesis in question, when the data it produces don’t prove anything one way or another.
In Zen and the Art of Motorcycle Maintenance: An Inquiry Into Values (1974), 102.
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At last such field studies have been put on a sound basis which should result in the hunting of information rather than specimens.
Concluding line of review (of Hiram Bingham’s 1932 book, Gorillas in a Native Habitat), in journal, Ecology (1933), 14, No. 3, 320. Note that the quote is authored by the reviewer, W.C. Allee, and any source attributing it directly to Bingham himself is incorrect.
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At no period of [Michael Faraday’s] unmatched career was he interested in utility. He was absorbed in disentangling the riddles of the universe, at first chemical riddles, in later periods, physical riddles. As far as he cared, the question of utility was never raised. Any suspicion of utility would have restricted his restless curiosity. In the end, utility resulted, but it was never a criterion to which his ceaseless experimentation could be subjected.
'The Usefulness of Useless Knowledge', Harper's Magazine (Jun/Nov 1939), No. 179, 546. In Hispania (Feb 1944), 27, No. 1, 77.
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Bad science contributes to the steady dumbing down of our nation. Crude beliefs get transmitted to political leaders and the result is considerable damage to society. We see this happening now in the rapid rise of the religious right and how it has taken over large segments of the Republican Party.
As quoted in Kendrick Frazier, 'A Mind at Play: An Interview with Martin Gardner', Skeptical Inquirer (Mar/Apr 1998), 22, No. 2, 37.
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Before an experiment can be performed, it must be planned—the question to nature must be formulated before being posed. Before the result of a measurement can be used, it must be interpreted—nature's answer must be understood properly. These two tasks are those of the theorist, who finds himself always more and more dependent on the tools of abstract mathematics. Of course, this does not mean that the experimenter does not also engage in theoretical deliberations. The foremost classical example of a major achievement produced by such a division of labor is the creation of spectrum analysis by the joint efforts of Robert Bunsen, the experimenter, and Gustav Kirchoff, the theorist. Since then, spectrum analysis has been continually developing and bearing ever richer fruit.
'The Meaning and Limits of Exact Science', Science (30 Sep 1949), 110, No. 2857, 325. Advance reprinting of chapter from book Max Planck, Scientific Autobiography (1949), 110.
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Besides accustoming the student to demand, complete proof, and to know when he has not obtained it, mathematical studies are of immense benefit to his education by habituating him to precision. It is one of the peculiar excellencies of mathematical discipline, that the mathematician is never satisfied with à peu près. He requires the exact truth. Hardly any of the non-mathematical sciences, except chemistry, has this advantage. One of the commonest modes of loose thought, and sources of error both in opinion and in practice, is to overlook the importance of quantities. Mathematicians and chemists are taught by the whole course of their studies, that the most fundamental difference of quality depends on some very slight difference in proportional quantity; and that from the qualities of the influencing elements, without careful attention to their quantities, false expectation would constantly be formed as to the very nature and essential character of the result produced.
In An Examination of Sir William Hamilton’s Philosophy (1878), 611. [The French phrase, à peu près means “approximately”. —Webmaster]
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But it is precisely mathematics, and the pure science generally, from which the general educated public and independent students have been debarred, and into which they have only rarely attained more than a very meagre insight. The reason of this is twofold. In the first place, the ascendant and consecutive character of mathematical knowledge renders its results absolutely insusceptible of presentation to persons who are unacquainted with what has gone before, and so necessitates on the part of its devotees a thorough and patient exploration of the field from the very beginning, as distinguished from those sciences which may, so to speak, be begun at the end, and which are consequently cultivated with the greatest zeal. The second reason is that, partly through the exigencies of academic instruction, but mainly through the martinet traditions of antiquity and the influence of mediaeval logic-mongers, the great bulk of the elementary text-books of mathematics have unconsciously assumed a very repellant form,—something similar to what is termed in the theory of protective mimicry in biology “the terrifying form.” And it is mainly to this formidableness and touch-me-not character of exterior, concealing withal a harmless body, that the undue neglect of typical mathematical studies is to be attributed.
In Editor’s Preface to Augustus De Morgan and Thomas J. McCormack (ed.), Elementary Illustrations of the Differential and Integral Calculus (1899), v.
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Cavendish gave me once some bits of platinum for my experiments, and came to see my results on the decomposition of the alkalis, and seemed to take an interest in them; but he encouraged no intimacy with any one, and received nobody at his own house. … He was acute, sagacious, and profound, and, I think, the most accomplished British philosopher of his time.
As quoted in Victor Robinson, Pathfinders in Medicine (1912), 143.
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Cayley was singularly learned in the work of other men, and catholic in his range of knowledge. Yet he did not read a memoir completely through: his custom was to read only so much as would enable him to grasp the meaning of the symbols and understand its scope. The main result would then become to him a subject of investigation: he would establish it (or test it) by algebraic analysis and, not infrequently, develop it so to obtain other results. This faculty of grasping and testing rapidly the work of others, together with his great knowledge, made him an invaluable referee; his services in this capacity were used through a long series of years by a number of societies to which he was almost in the position of standing mathematical advisor.
In Proceedings of London Royal Society (1895), 58, 11-12.
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Characteristically skeptical of the idea that living things would faithfully follow mathematical formulas, [Robert Harper] seized upon factors in corn which seemed to blend in the hybrid—rather than be represented by plus or minus signs, and put several seasons into throwing doubt upon the concept of immutable hypothetical units of inheritance concocted to account for selected results.
In 'Robert Almer Harper', National Academy Biographical Memoirs (1948), 25, 233-234.
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Charles Babbage proposed to make an automaton chess-player which should register mechanically the number of games lost and gained in consequence of every sort of move. Thus, the longer the automaton went on playing game, the more experienced it would become by the accumulation of experimental results. Such a machine precisely represents the acquirement of experience by our nervous organization.
In ‘Experimental Legislation’, Popular Science (Apr 1880), 16, 754-5.
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Chemistry works with an enormous number of substances, but cares only for some few of their properties; it is an extensive science. Physics on the other hand works with rather few substances, such as mercury, water, alcohol, glass, air, but analyses the experimental results very thoroughly; it is an intensive science. Physical chemistry is the child of these two sciences; it has inherited the extensive character from chemistry. Upon this depends its all-embracing feature, which has attracted so great admiration. But on the other hand it has its profound quantitative character from the science of physics.
In Theories of Solutions (1912), xix.
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Collective unity is not the result of the brotherly love of the faithful for each other. The loyalty of the true believer is to the whole—the church, party, nation—and not to his fellow true believer. True loyalty between individuals is possible only in a loose and relatively free society.
In The True Believer (1951), 122
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Common sense is in spite of, not as the result of, education.
Anonymous
“Anonymous” because although commonly seen attributed to Victor Hugo, the quote cannot be found in his works.
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Common sense is only the application of theories which have grown and been formulated unconsciously as result of experience.
From 'For Mans Use of God's Gifts', collected in Robert C. Goodpasture (ed.), Engineers and Ivory Towers (1952), 107.
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Consider the very roots of our ability to discern truth. Above all (or perhaps I should say “underneath all”), common sense is what we depend on—that crazily elusive, ubiquitous faculty we all have to some degree or other. … If we apply common sense to itself over and over again, we wind up building a skyscraper. The ground floor of the structure is the ordinary common sense we all have, and the rules for building news floors are implicit in the ground floor itself. However, working it all out is a gigantic task, and the result is a structure that transcends mere common sense.
In Metamagical Themas: Questing for the Essence of Mind and Pattern (1985), 93–94.
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Descriptive geometry has two objects: the first is to establish methods to represent on drawing paper which has only two dimensions,—namely, length and width,—all solids of nature which have three dimensions,—length, width, and depth,—provided, however, that these solids are capable of rigorous definition.
The second object is to furnish means to recognize accordingly an exact description of the forms of solids and to derive thereby all truths which result from their forms and their respective positions.
From On the Purpose of Descriptive Geometry as translated by Arnold Emch in David Eugene Smith, A Source Book in Mathematics (1929), 426.
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Each part of the project had a specific task. These tasks were carefully allocated and supervised so that the sum of their parts would result in the accomplishment of our over-all mission.
In And Now It Can Be Told: The Story Of The Manhattan Project (1962), 414.
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Einstein’s results again turned the tables and now very few philosophers or scientists still think that scientific knowledge is, or can be, proven knowledge.
In 'Falsification and the Methodology of Scientific Research Programmes', in I. Lakatos and A. Musgrave (eds.), Criticism and the Growth of Knowledge: Proceedings of the International Colloquium in the Philosophy of Science, London 1965 (1970), Vol. 4, 92.
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England was nothing, compared to continental nations until she had become commercial … until about the middle of the last century, when a number of ingenious and inventive men, without apparent relation to each other, arose in various parts of the kingdom, succeeded in giving an immense impulse to all the branches of the national industry; the result of which has been a harvest of wealth and prosperity, perhaps without a parallel in the history of the world.
In Lives of the Engineers (1862, 1874), xvii.
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Erasmus Darwin held that every so often you should try a damn-fool experiment. He played the trombone to his tulips. This particular result was, in fact, negative.
In 'The Mathematician’s Art of Work' (1967), collected in Béla Bollobás (ed.), Littlewood’s Miscellany (1986), 194. Webmaster has looked for a primary source to verify this statement and so far has found none. Can you help?
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Every good thought you think is contributing its share to the ultimate result of your life.
…...
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Every improvement that is put upon the real estate is the result of an idea in somebody's head. The skyscraper is another idea; the railroad is another; the telephone and all those things are merely symbols which represent ideas. An andiron, a wash-tub, is the result of an idea that did not exist before.
Speaking to a committee considering a new Copyright Bill (6 Dec 1906). In Mark Twain and William Dean Howells (ed.), Mark Twain’s Speeches? (1910), 320. An andiron is a metal bar, used in a pair, as a stand for logs in a fireplace. The Copyright Bill proposed to give authors, artists and musicians copyright for the term of his life and for 50 years thereafter. John Philip Sousa spoke for the musicians.
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Every well established truth is an addition to the sum of human power, and though it may not find an immediate application to the economy of every day life, we may safely commit it to the stream of time, in the confident anticipation that the world will not fail to realize its beneficial results.
In 'Report of the Secretary', Annual Report of the Board of Regents of the Smithsonian Institution for 1856 (1857), 20.
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Everything material which is the subject of knowledge has number, order, or position; and these are her first outlines for a sketch of the universe. If our feeble hands cannot follow out the details, still her part has been drawn with an unerring pen, and her work cannot be gainsaid. So wide is the range of mathematical sciences, so indefinitely may it extend beyond our actual powers of manipulation that at some moments we are inclined to fall down with even more than reverence before her majestic presence. But so strictly limited are her promises and powers, about so much that we might wish to know does she offer no information whatever, that at other moments we are fain to call her results but a vain thing, and to reject them as a stone where we had asked for bread. If one aspect of the subject encourages our hopes, so does the other tend to chasten our desires, and he is perhaps the wisest, and in the long run the happiest, among his fellows, who has learned not only this science, but also the larger lesson which it directly teaches, namely, to temper our aspirations to that which is possible, to moderate our desires to that which is attainable, to restrict our hopes to that of which accomplishment, if not immediately practicable, is at least distinctly within the range of conception.
From Presidential Address (Aug 1878) to the British Association, Dublin, published in the Report of the 48th Meeting of the British Association for the Advancement of Science (1878), 31.
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Evolution is a blind giant who rolls a snowball down a hill. The ball is made of flakes—circumstances. They contribute to the mass without knowing it. They adhere without intention, and without foreseeing what is to result. When they see the result they marvel at the monster ball and wonder how the contriving of it came to be originally thought out and planned. Whereas there was no such planning, there was only a law: the ball once started, all the circumstances that happened to lie in its path would help to build it, in spite of themselves.
'The Secret History of Eddypus', in Mark Twain and David Ketterer (ed.), Tales of Wonder (2003), 222-23.
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Examples ... show how difficult it often is for an experimenter to interpret his results without the aid of mathematics.
Quoted in E. T. Bell, Men of Mathematics, xvi.
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Experience is never at fault; it is only your judgment that is in error in promising itself such results from experience as are not caused by our experiments. For having given a beginning, what follows from it must necessarily be a natural development of such a beginning, unless it has been subject to a contrary influence, while, if it is affected by any contrary influence, the result which ought to follow from the aforesaid beginning will be found to partake of this contrary influence in a greater or less degree in proportion as the said influence is more or less powerful than the aforesaid beginning.
'Philosophy', in The Notebooks of Leonardo da Vinci, trans. E. MacCurdy (1938), Vol. 1, 70.
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Experiments may be of two kinds: experiments of simple fact, and experiments of quantity. ...[In the latter] the conditions will ... vary, not in quality, but quantity, and the effect will also vary in quantity, so that the result of quantitative induction is also to arrive at some mathematical expression involving the quantity of each condition, and expressing the quantity of the result. In other words, we wish to know what function the effect is of its conditions. We shall find that it is one thing to obtain the numerical results, and quite another thing to detect the law obeyed by those results, the latter being an operation of an inverse and tentative character.
Principles of Science: A Treatise on Logic and Scientific Method (1874, 1892), 439.
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Forces of nature act in a mysterious manner. We can but solve the mystery by deducing the unknown result from the known results of similar events.
In The Words of Gandhi (2001), 87.
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Fourier’s Theorem … is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along a telegraph wire, and the conduction of heat by the earth’s crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance.
In William Thomson and Peter Guthrie Tait, Treatise on Natural Philosophy (1867), Vol. 1, 28.
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Galileo was no idiot. Only an idiot could believe that science requires martyrdom—that may be necessary in religion, but in time a scientific result will establish itself.
As quoted, without citation, in Harold Eves, Mathematical Circles Squared (1971). Collected in Bill Swainson, The Encarta Book of Quotations (2000), 361.
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Geology is part of that remarkable dynamic process of the human mind which is generally called science and to which man is driven by an inquisitive urge. By noticing relationships in the results of his observations, he attempts to order and to explain the infinite variety of phenomena that at first sight may appear to be chaotic. In the history of civilization this type of progressive scientist has been characterized by Prometheus stealing the heavenly fire, by Adam eating from the tree of knowledge, by the Faustian ache for wisdom.
In 'The Scientific Character of Geology', The Journal of Geology (Jul 1961), 69, No. 4, 454.
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Geometric writings are not rare in which one would seek in vain for an idea at all novel, for a result which sooner or later might be of service, for anything in fact which might be destined to survive in the science; and one finds instead treatises on trivial problems or investigations on special forms which have absolutely no use, no importance, which have their origin not in the science itself but in the caprice of the author; or one finds applications of known methods which have already been made thousands of times; or generalizations from known results which are so easily made that the knowledge of the latter suffices to give at once the former. Now such work is not merely useless; it is actually harmful because it produces a real incumbrance in the science and an embarrassment for the more serious investigators; and because often it crowds out certain lines of thought which might well have deserved to be studied.
From 'On Some Recent Tendencies in Geometric Investigations', Rivista di Matematica (1891), 43. In Bulletin American Mathematical Society (1904), 443.
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Geometry, which should only obey Physics, when united with it sometimes commands it. If it happens that the question which we wish to examine is too complicated for all the elements to be able to enter into the analytical comparison which we wish to make, we separate the more inconvenient [elements], we substitute others for them, less troublesome, but also less real, and we are surprised to arrive, notwithstanding a painful labour, only at a result contradicted by nature; as if after having disguised it, cut it short or altered it, a purely mechanical combination could give it back to us.
From Essai d’une nouvelle théorie de la résistance des fluides (1752), translated as an epigram in Ivor Grattan-Guinness, Convolutions in French Mathematics, 1800-1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics (1990), Vol. 1, 33.
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Goethe said that he who cannot draw on 3,000 years of learning is living hand to mouth. It could just as well be said that individuals who do tap deeply into this rich cultural legacy are wealthy indeed. Yet the paradox is that much of this wisdom is buried in a sea of lesser books or like lost treasure beneath an ocean of online ignorance and trivia. That doesn’t mean that with a little bit of diligence you can’t tap into it. Yet many people, perhaps most, never take advantage of all this human experience. They aren’t obtaining knowledge beyond what they need to know for work or to get by. As a result, their view of our amazing world is diminished and their lives greatly circumscribed.
In An Embarrassment of Riches: Tapping Into the World's Greatest Legacy of Wealth (2013), 65.
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Great spirits have always found violent opposition from mediocrities. The latter cannot understand it when a man does not thoughtlessly submit to hereditary prejudices but honestly and courageously uses his intelligence and fulfills the duty to express the results of his thoughts in clear form.
…...
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Here about the beach I wandered, nourishing a youth sublime
With the fairy tales of science, and the long result of Time.
'Locksley Hall' (1842), collected in Alfred Tennyson and William James Rolfe (ed.) The Poetic and Dramatic Works of Alfred, Lord Tennyson (1898), 90.
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Here I shall present, without using Analysis [mathematics], the principles and general results of the Théorie, applying them to the most important questions of life, which are indeed, for the most part, only problems in probability. One may even say, strictly speaking, that almost all our knowledge is only probable; and in the small number of things that we are able to know with certainty, in the mathematical sciences themselves, the principal means of arriving at the truth—induction and analogy—are based on probabilities, so that the whole system of human knowledge is tied up with the theory set out in this essay.
Philosophical Essay on Probabilities (1814), 5th edition (1825), trans. Andrew I. Dale (1995), 1.
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However much the pits may be apparent, yet none, as far as can be comprehended by the senses, passes through the septum of the heart from the right ventricle into the left. I have not seen even the most obscure passages by which the septum of the ventricles is pervious, although they are mentioned by professors of anatomy since they are convinced that blood is carried from the right ventricle into the left. As a result—as I shall declare more openly elsewhere—I am in no little doubt regarding the function of the heart in this part.
In De Humani Corporis Fabrica Libri Septem [Seven Books on the Structure of the Human Body] (revised ed. 1555), 734. Quoted and trans. in Charles Donald O'Malley, Andreas Vesalius of Brussels, 1514-1564 (1964), 281.
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I am afraid I am not in the flight for “aerial navigation”. I was greatly interested in your work with kites; but I have not the smallest molecule of faith in aerial navigation other than ballooning or of expectation of good results from any of the trials we hear of. So you will understand that I would not care to be a member of the aëronautical Society.
Letter (8 Dec 1896) to Baden Powell. This is the full text of the letter. An image of the handwritten original is on the zapatopi.net website
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I am particularly concerned to determine the probability of causes and results, as exhibited in events that occur in large numbers, and to investigate the laws according to which that probability approaches a limit in proportion to the repetition of events. That investigation deserves the attention of mathematicians because of the analysis required. It is primarily there that the approximation of formulas that are functions of large numbers has its most important applications. The investigation will benefit observers in identifying the mean to be chosen among the results of their observations and the probability of the errors still to be apprehended. Lastly, the investigation is one that deserves the attention of philosophers in showing how in the final analysis there is a regularity underlying the very things that seem to us to pertain entirely to chance, and in unveiling the hidden but constant causes on which that regularity depends. It is on the regularity of the main outcomes of events taken in large numbers that various institutions depend, such as annuities, tontines, and insurance policies. Questions about those subjects, as well as about inoculation with vaccine and decisions of electoral assemblies, present no further difficulty in the light of my theory. I limit myself here to resolving the most general of them, but the importance of these concerns in civil life, the moral considerations that complicate them, and the voluminous data that they presuppose require a separate work.
Philosophical Essay on Probabilities (1825), trans. Andrew I. Dale (1995), Introduction.
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I can see him now at the blackboard, chalk in one hand and rubber in the other, writing rapidly and erasing recklessly, pausing every few minutes to face the class and comment earnestly, perhaps on the results of an elaborate calculation, perhaps on the greatness of the Creator, perhaps on the beauty and grandeur of Mathematics, always with a capital M. To him mathematics was not the handmaid of philosophy. It was not a humanly devised instrument of investigation, it was Philosophy itself, the divine revealer of TRUTH.
Writing as a Professor Emeritus at Harvard University, a former student of Peirce, in 'Benjamin Peirce: II. Reminiscences', The American Mathematical Monthly (Jan 1925), 32, No. 1, 5.
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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 cannot anyhow be contented to view this wonderful universe, and especially the nature of man, and to conclude that everything is the result of brute force. I am inclined to look at everything as resulting from designed laws, with the details, whether good or bad, left to the working out of what we call chance. Not that this notion at all satisfies me. I feel most deeply that the whole subject is too profound for the human intellect. A dog might as well speculate on the mind of Newton. Let each man hope and believe what he can.
Letter to Asa Gray (22 May 1860). In Charles Darwin and Francis Darwin (ed.), Charles Darwin: His Life Told in an Autobiographical Chapter, and in a Selected Series of His Published Letters (1892), 236.
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I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are permutations and aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then from the top down, the result is always different. Again if you multiply a number by another number before you have had your tea, and then again after, the product will be different. It is also remarkable that the Post-tea product is more likely to agree with other people’s calculations than the Pre-tea result.
Letter to Mrs Arthur Severn (Jul 1878), collected in The Letters of a Noble Woman (Mrs. La Touche of Harristown) (1908), 50. Also in 'Gleanings Far and Near', Mathematical Gazette (May 1924), 12, 95.
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I do not think we can impose limits on research. Through hundreds of thousands of years, man’s intellectual curiosity has been essential to all the gains we have made. Although in recent times we have progressed from chance and hit-or-miss methods to consciously directed research, we still cannot know in advance what the results may be. It would be regressive and dangerous to trammel the free search for new forms of truth.
In Margaret Mead and Rhoda Bubendey Métraux (ed.), Margaret Mead, Some Personal Views (1979), 89.
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I have always supported women’s rights. I moved the admission of women to my college, Gonville and Caius College, Cambridge. The results were wholly good.
From TV show interview with Piers Morgan, on ITV, 'Good Morning Britain'. As quoted in Sarah Knapton, 'Five Most Powerful People in Britain are Women, says Prof Stephen Hawking', Daily Telegraph (20 Mar 2017).
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I have long since come to see that no one deserves either praise or blame for the ideas that come to him, but only for the actions resulting therefrom. Ideas and beliefs are certainly not voluntary acts. They come to us—we hardly know how or whence, and once they have got possession of us we can not reject or change them at will. It is for the common good that the promulgation of ideas should be free—uninfluenced by either praise or blame, reward or punishment. But the actions which result from our ideas may properly be so treated, because it is only by patient thought and work, that new ideas, if good and true, become adopted and utilized; while, if untrue or if not adequately presented to the world, they are rejected or forgotten.
In 'The Origin of the Theory of Natural Selection', Popular Science Monthly (1909), 74, 400.
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Srinivasa Ramanujan quote: I have not trodden through a conventional university course, but I am striking out a new path for mys
I have not trodden through a conventional university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling.”
First letter to G.H. Hardy (16 Jan 1913). In Collected Papers of Srinivasa Ramanujan (1927), xxiii. Hardy notes he did “seem to remember his telling me that his friends had given him some assistance” in writing the letter because Ramanujan's “knowledge of English, at that stage of his life, could scarcely have been sufficient.”
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I hope that in due time the chemists will justify their proceedings by some large generalisations deduced from the infinity of results which they have collected. For me I am left hopelessly behind and I will acknowledge to you that through my bad memory organic chemistry is to me a sealed book. Some of those here, [August] Hoffman for instance, consider all this however as scaffolding, which will disappear when the structure is built. I hope the structure will be worthy of the labour. I should expect a better and a quicker result from the study of the powers of matter, but then I have a predilection that way and am probably prejudiced in judgment.
Letter to Christian Schönbein (9 Dec 1852), The Letters of Faraday and Schoenbein, 1836-1862 (1899), 209-210.
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I like a deep and difficult investigation when I happen to have made it easy to myself, if not to all others; and there is a spirit of gambling in this, whether, as by the cast of a die, a calculation è perte de vue shall bring out a beautiful and perfect result or shall be wholly thrown away. Scientific investigations are a sort of warfare carried on in the closet or on the couch against all one's contemporaries and predecessors; I have often gained a signal victory when I have been half asleep, but more frequently have found, upon being thoroughly awake, that the enemy had still the advantage of me, when I thought I had him fast in a corner, and all this you see keeps me alive.
Letter to Hudson Gurney, quoted in George Peacock, The Life of Thomas Young (1855), 239.
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I like to summarize what I regard as the pedestal-smashing messages of Darwin’s revolution in the following statement, which might be chanted several times a day, like a Hare Krishna mantra, to encourage penetration into the soul: Humans are not the end result of predictable evolutionary progress, but rather a fortuitous cosmic afterthought, a tiny little twig on the enormously arborescent bush of life, which, if replanted from seed, would almost surely not grow this twig again, or perhaps any twig with any property that we would care to call consciousness.
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I noticed affixed to a laboratory door the following words: “Les théories passent. Le Grenouille reste. [The theories pass. The frog remains.] &mdashJean Rostand, Carnets d’un biologiste.” There is a risk that in the less severe discipline of criticism the result may turn out to be different; the theories will remain but the frog may disappear.
In An Appetite for Poetry (1989), 5.
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I pass with relief from the tossing sea of Cause and Theory to the firm ground of Result and Fact.
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I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment.
In Tobias Dantzig and Joseph Mazur (ed.), 'The Two Realities', Number: The Language of Science (1930, ed. by Joseph Mazur 2007), 254.
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I sometimes ask myself how it came about that I was the one to develop the theory of relativity. The reason, I think, is that a normal adult never stops to think about the problem of space and time. These are things which he has thought of as a child. But my intellectual development was retarded, as a result of which I began to wonder about space and time only when I had already grown up.
In Ronald W. Clark, Einstein: The Life and Times (1971), 10.
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I started studying law, but this I could stand just for one semester. I couldn’t stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.
From interview on his 90th birthday. In D J Albers and G L Alexanderson (eds.), Mathematical People: Profiles and Interviews (1985), 245-254.
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I then began to study arithmetical questions without any great apparent result, and without suspecting that they could have the least connexion with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside, and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry.
Science and Method (1908), trans. Francis Maitland (1914), 53-4.
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I think one’s feelings waste themselves in words; they ought all to be distilled into actions, and into actions which bring results.
Letter to Mary Clarke (1844), quoted in Sir Edward Tyas Cook in The Life of Florence Nightingale (1914), 94.
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I undertake my scientific research with the confident assumption that the earth follows the laws of nature which God established at creation. … My studies are performed with the confidence that God will not capriciously confound scientific results by “slipping in” a miracle.
Quoted in Lenny Flank, Deception by Design: The Intelligent Design Movement in America (2007), 81. Also seen as cited from Arthur Newell Strahler, Science and Earth History: the Evolution/Creation Controversy (1987), 40-41.
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I won’t let anyone take us backward, deny our economy the benefits of harnessing a clean energy future, or force our children to endure the catastrophe that would result from unchecked climate change.
In Hillary Clinton, 'Hillary Clinton: America Must Lead at Paris Climate Talks', Time (29 Nov 2015)
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I would picture myself as a virus, or as a cancer cell, for example, and try to sense what it would be like to be either. I would also imagine myself as the immune system, and I would try to reconstruct what I would do as an immune system engaged in combating a virus or cancer cell. When I had played through a series of such scenarios on a particular problem and had acquired new insights, I would design laboratory experiments accordingly… Based upon the results of the experiment, I would then know what question to ask next… When I observed phenomena in the laboratory that I did not understand, I would also ask questions as if interrogating myself: “Why would I do that if I were a virus or a cancer cell, or the immune system?” Before long, this internal dialogue became second nature to me; I found that my mind worked this way all the time.
In Anatomy of Reality: Merging of Intuition and Reason (1983), 7, footnote b, as quoted and cited in Roger Frantz, Two Minds: Intuition and Analysis in the History of Economic Thought (2006), 7.
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If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes.
From final remarks in 'The Semantic Conception of Truth and the Foundations of Semantics' (1944), collected in Leonard Linsky (ed.), Semantics and the Philosophy of Language: A Collection of Readings (1952), 41.
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If gold medals and prizes were awarded to institutions instead of individuals, the Peter Bent Brigham Hospital of 30 years ago would have qualified. The ruling board and administrative structure of that hospital did not falter in their support of the quixotic objective of treating end-stage renal disease despite a long list of tragic failures that resulted from these early efforts.
In Tore Frängsmyr and Jan E. Lindsten (eds.), Nobel Lectures: Physiology Or Medicine: 1981-1990 (1993), 558.
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If materialism is true, it seems to me that we cannot know that it is true. If my opinions are the result of the chemical processes going on in my brain, they are determined by the laws of chemistry, not those of logic.
The Inequality of Man (1932), 162.
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If science is to progress, what we need is the ability to experiment, honesty in reporting results—the results must be reported without somebody saying what they would like the results to have been—and finally—an important thing—the intelligence to interpret the results.
In The Character of Physical Law (1965, 2001), 148.
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If we are to achieve results never before accomplished, we must employ methods never before attempted.
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If we knew all the laws of Nature, we should need only one fact or the description of one actual phenomenon to infer all the particular results at that point. Now we know only a few laws, and our result is vitiated, not, of course, by any confusion or irregularity in Nature, but by our ignorance of essential elements in the calculation. Our notions of law and harmony are commonly confined to those instances which we detect, but the harmony which results from a far greater number of seemingly conflicting, but really concurring, laws which we have not detected, is still more wonderful. The particular laws are as our points of view, as to the traveler, a mountain outline varies with every step, and it has an infinite number of profiles, though absolutely but one form. Even when cleft or bored through, it is not comprehended in its entireness.
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If we work, it is less to obtain those positive results the common people think are our only interest, than to feel that aesthetic emotion and communicate it to those able to experience it.
From the original French, “Si nous travaillons, c’est moins pour obtenir ces résultats auxquels le vulgaire nous croit uniquement attachés, que pour ressentir cette émotion esthétique et la communiquer à ceux qui sont capables de l’éprouver,” quoted in Henri Poincaré,'Notice sur Halphen', Journal de l’École Polytechnique (1890), 60, 143, cited in Oeuvres de G.H. Halphen (1916), Vol. 1, xxiv. As translated in Armand Borel, 'On the Place of Mathematics in Culture', in Armand Borel: Œvres: Collected Papers (1983), Vol. 4, 421.
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If your result needs a statistician then you should design a better experiment.
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If, unwarned by my example, any man shall undertake and shall succeed in really constructing an engine embodying in itself the whole of the executive department of mathematical analysis upon different principles or by simpler mechanical means, I have no fear of leaving my reputation in his charge, for he alone will be fully able to appreciate the nature of my efforts and the value of their results.
In Passages from the Life of a Philosopher (1864), 450.
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In a sense, of course, probability theory in the form of the simple laws of chance is the key to the analysis of warfare;… My own experience of actual operational research work, has however, shown that its is generally possible to avoid using anything more sophisticated. … In fact the wise operational research worker attempts to concentrate his efforts in finding results which are so obvious as not to need elaborate statistical methods to demonstrate their truth. In this sense advanced probability theory is something one has to know about in order to avoid having to use it.
In 'Operations Research', Physics Today (Nov 1951), 19. As cited by Maurice W. Kirby and Jonathan Rosenhead, 'Patrick Blackett (1897)' in Arjang A. Assad (ed.) and Saul I. Gass (ed.),Profiles in Operations Research: Pioneers and Innovators (2011), 25.
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In all cases when a particular agent or cause is to be studied, experiments should be arranged in such a way as to lead if possible to results depending on it alone ; or, if this cannot be done, they should be arranged so as to increase the effects due to the cause to be studied till these so far exceed the unavoidable concomitants, that the latter may be considered as only disturbing, not essentially modifying the effects of the principal agent.
In William Thomson and Peter Guthrie Tait, Treatise on Natural Philosophy (1867), Vol. 1, 305.
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In all likelihood, it is the local conditions of society, which determine the form of the disease, and we can so far think of it as a fairly general result, that the simplest form is the more common, the more paltry and unbalanced the food, and the worse the dwellings are.
From the original German, “Aller Wahrscheinlichkeit nach sind es die lokalen Verhältnisse der Gesellschaft, welche die Form der Krankheit bestimmen, und wir können bis jetzt als ein ziemlich allgemeines Resultathinstellen, daß die einfache Form umso häufiger ist, je armseliger und einseitiger die Nahrungsmittel und je schlechter die Wohnungen sind,” in 'Mittheilungen über die in Oberschlesien herrschende Typhus-Epidemie', R. Virchow and B. Reinhardt, Archiv für pathologische Anatomie und Physiologie und für klinische Medicin (1848), 2, No. 2, 248. English version by Webmaster with Google translate.
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In every case the awakening touch has been the mathematical spirit, the attempt to count, to measure, or to calculate. What to the poet or the seer may appear to be the very death of all his poetry and all his visions—the cold touch of the calculating mind,—this has proved to be the spell by which knowledge has been born, by which new sciences have been created, and hundreds of definite problems put before the minds and into the hands of diligent students. It is the geometrical figure, the dry algebraical formula, which transforms the vague reasoning of the philosopher into a tangible and manageable conception; which represents, though it does not fully describe, which corresponds to, though it does not explain, the things and processes of nature: this clothes the fruitful, but otherwise indefinite, ideas in such a form that the strict logical methods of thought can be applied, that the human mind can in its inner chamber evolve a train of reasoning the result of which corresponds to the phenomena of the outer world.
In A History of European Thought in the Nineteenth Century (1896), Vol. 1, 314.
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In history an additional result is commonly produced by human actions beyond that which they aim at and obtain—that which they immediately recognize and desire. They gratify their own interest; but something further is thereby accomplished, latent in the actions in question, though not present to their consciousness, and not included in their design. … This may be called the cunning of reason.
From Vorlesungen über die philosophie der weltgeschichte (1837), as translated from the Third German Edition by J. Sibree (1857), in The Philosophy of History (1857, 1861), 28-29 & 34. Hegel coined the expression List der Vernunft translated as the cunning of reason. In Hegel’s philosophical view, the expression represents a process by which a certain purpose is realized in the history of mankind that is not conscious to the acting man.
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In Institutions of a lower grade [secondary schools], it [geology] receives far less attention than its merits deserve. Why should not a science, whose facts possess a thrilling interest; whose reasonings are admirably adapted for mental discipline, and often severely tax the strongest powers; and whose results are, many of them, as grand and ennobling as those of Astronomy itself; … why should not such a science be thought as essential in education as the kindred branches of Chemistry and Astronomy?
In 'Preface', Elementary Geology (1840, 1841), vi.
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In many ways, unexpected results are what have most inspired my photography.
As quoted, without source, in Gus Kayafas and Estelle Jussim, Stopping Time: The Photographs of Harold Edgerton (1987), 9.
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In order that the facts obtained by observation and experiment may be capable of being used in furtherance of our exact and solid knowledge, they must be apprehended and analysed according to some Conceptions which, applied for this purpose, give distinct and definite results, such as can be steadily taken hold of and reasoned from.
Philosophy of the Inductive Sciences (1840), Vol. 2, 205.
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In point of fact, no conclusive disproof of a theory can ever be produced; for it is always possible to say that the experimental results are not reliable or that the discrepancies which are asserted to exist between the experimental results and the theory are only apparent and that they will disappear with the advance of our understanding. If you insist on strict proof (or strict disproof) in the empirical sciences, you will never benefit from experience, and never learn from it how wrong you are.
The Logic of Scientific Discovery: Logik Der Forschung (1959, 2002), 28.
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In scientific investigations it is grievously wrong to pander to the public’s impatience for results, or to let them think that for discovery it is necessary only to set up a great manufactory and a system of mass production. If in treatment team work is effective, in research it is the individual who counts first and above all. No great thought has ever sprung from anything but a single mind, suddenly conceiving. Throughout the whole world there has been too violent a forcing of the growth of ideas; too feverish a rush to perform experiments and publish conclusions. A year of vacation for calm detachment with all the individual workers thinking it all over in a desert should be proclaimed.
In Viewless Winds: Being the Recollections and Digressions of an Australian Surgeon (1939), 286.
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In teaching man, experimental science results in lessening his pride more and more by proving to him every day that primary causes, like the objective reality of things, will be hidden from him forever and that he can only know relations.
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In that pure enjoyment experienced on approaching to the ideal, in that eagerness to draw aside the veil from the hidden truth, and even in that discord which exists between the various workers, we ought to see the surest pledges of further scientific success. Science thus advances, discovering new truths, and at the same time obtaining practical results.
In The Principles of Chemistry (1891), Vol. 1, preface, footnote, ix, as translated from the Russian 5th edition by George Kamensky, edited by A. J. Greenaway.
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In the discovery of lemmas the best aid is a mental aptitude for it. For we may see many who are quick at solutions and yet do not work by method ; thus Cratistus in our time was able to obtain the required result from first principles, and those the fewest possible, but it was his natural gift which helped him to the discovery.
Proclus
As given in Euclid, The Thirteen Books of Euclid's Elements, translated from the text of Johan Ludvig Heiberg by Sir Thomas Little Heath, Vol. 1, Introduction and Books 1,2 (1908), 133. The passage also states that Proclus gives the definition of the term lemma as a proposition not proved beforehand. Glenn Raymond Morrow in A Commentary on the First Book of Euclid's Elements (1992), 165, states nothing more seems to be known of Cratistus.
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In the hands of Science and indomitable energy, results the most gigantic and absorbing may be wrought out by skilful combinations of acknowledged data and the simplest means.
From Introduction to Lecture on the Pendulum-Experiments at Harton Pit (1855), iv. The quote comes from the printed Introduction in the published booklet, but was not part of the printed lecture itself. At the request of a “deputation of gentlemen of South Shields,” the printed version of the lecture was written “in extenso, from memory,” after the original talk was given on 24 Oct 1854 at South Shields, County Durham, England.
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In the last four days I have got the spectrum given by Tantalum. Chromium. Manganese. Iron. Nickel. Cobalt. and Copper and part of the Silver spectrum. The chief result is that all the elements give the same kind of spectrum, the result for any metal being quite easy to guess from the results for the others. This shews that the insides of all the atoms are very much alike, and from these results it will be possible to find out something of what the insides are made up of.
Letter to his mother (2 Nov 1913). In J. L. Heilbron (ed.), H. G. J. Moseley: The Life and Letters of an English Physicist 1887-1915 (1974), 209.
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In the medical field [scientific ignorance] could lead to horrendous results. People who don’t understand the difference between a controlled experiment and claims by some quack may die as a result of not taking medical science seriously. One of the most damaging examples of pseudoscience is false memory syndrome. I’m on the board of a foundation exposing this problem.
As quoted by Lawrence Toppman, 'Mastermind', The Charlotte Observer (20 Jun 1993), 6E. As quoted and cited in Dana Richards, 'Martin Gardner: A “Documentary”', collected in Elwyn R. Berlekamp and Tom Rodgers (ed.) The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner (1999), 11.
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In the summer of 1937, … I told Banach about an expression Johnny [von Neumann] had once used in conversation with me in Princeton before stating some non-Jewish mathematician’s result, “Die Goim haben den folgendenSatzbewiesen” (The goys have proved the following theorem). Banach, who was pure goy, thought it was one of the funniest sayings he had ever heard. He was enchanted by its implication that if the goys could do it, Johnny and I ought to be able to do it better. Johnny did not invent this joke, but he liked it and we started using it.
In Adventures of a Mathematician (1976, 1991), 107. Von Neumann, who was raised in Budapest by a Jewish family, knew the Yiddish word “goy” was equivalent to “gentile” or a non-Jew. Stefan Banach, a Polish mathematician, was raised in a Catholic family, hence “pure goy”. Ulam thus gives us the saying so often elsewhere seen attributed to von Neumann without the context: “The goys have proved the following theorem.” It is seen anecdotally as stated by von Neumann to begin a classroom lecture.
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In the twenties the late Dr. Glenn Frank, an eminent social scientist, developed a new statement of the scientific code, which has been referred to as the “Five Fingers of the Scientific Method.” It may be outlined as follows: find the facts; filter the facts; focus the facts; face the facts; follow the facts. The facts or truths are found by experimentation; the motivation is material. The facts are filtered by research into the literature; the motivation is material. The facts are focused by the publication of results; again the motivation is material. Thus the first three-fifths of the scientific method have a material motivation. It is about time scientists acknowledge that there is more to the scientific convention than the material aspect. Returning to the fourth and fifth fingers of Dr. Frank's conception of the scientific method, the facts should be faced by the proper interpretation of them for society. In other words, a scientist must assume social responsibility for his discoveries, which means that he must have a moral motivation. Finally, in the fifth definition of the scientific method, the facts are to be followed by their proper application to everyday life in society, which means moral motivation through responsibility to society.
From 'Scientists and Society', American Scientist (Jul 1954), 42, No. 3, 495.
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In those early days, the Chief Engineer was almost always the Chief Pilot as well. This had the automatic result of eliminating poor engineering very early in aviation.
In The Story of the Winged-S: The Autobiography of Igor I. Sikorsky (2011).
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In working out physical problems there should be, in the first place, no pretence of rigorous formalism. The physics will guide the physicist along somehow to useful and important results, by the constant union of physical and geometrical or analytical ideas. The practice of eliminating the physics by reducing a problem to a purely mathematical exercise should be avoided as much as possible. The physics should be carried on right through, to give life and reality to the problem, and to obtain the great assistance which the physics gives to the mathematics.
In Electromagnetic Theory (1892), Vol. 2, 5.
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Inductive reasoning is, of course, good guessing, not sound reasoning, but the finest results in science have been obtained in this way. Calling the guess a “working hypothesis,” its consequences are tested by experiment in every conceivable way.
In Higher Mathematics for Students of Chemistry and Physics (1902), 2.
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Insanity: doing the same thing over and over again and expecting different results.
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Inspiration plays no less a role in science than it does in the realm of art. It is a childish notion to think that a mathematician attains any scientifically valuable results by sitting at his desk with a ruler, calculating machines or other mechanical means. The mathematical imagination of a Weierstrass is naturally quite differently oriented in meaning and result than is the imagination of an artist, and differs basically in quality. But the psychological processes do not differ. Both are frenzy (in the sense of Plato’s “mania”) and “inspiration.”
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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Intelligence is an extremely subtle concept. It’s a kind of understanding that flourishes if it’s combined with a good memory, but exists anyway even in the absence of good memory. It’s the ability to draw consequences from causes, to make correct inferences, to foresee what might be the result, to work out logical problems, to be reasonable, rational, to have the ability to understand the solution from perhaps insufficient information. You know when a person is intelligent, but you can be easily fooled if you are not yourself intelligent.
In Irv Broughton (ed.), The Writer's Mind: Interviews with American Authors (1990), Vol. 2, 57.
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Inventions and discoveries are of two kinds. The one which we owe to chance, such as those of the mariner’s compass, gunpowder, and in general almost all the discoveries we have made in the arts. The other which we owe to genius: and here we ought to understand by the word discovery, a new combination, or a new relation perceived between certain objects or ideas. A person obtains the title of a man of genius, if the ideas which result from this combination form one grand whole, are fruitful in truths, and are of importance with respect to mankind.
From the original French, “Les inventions ou les découvertes sont de deux espèces. Il en est que nous devons au hazard; telles sont la boussole, la poudre à canon, & généralement presque toutes les découvertes que nous avons faites dans les arts. Il en est d'autres que nous devons au génie: &, par ce mot de découverte, on doit alors entendre une nouvelle combinaison, un rapport nouveau aperçu entre certains objets ou certaines idées. On obtient le titre d'homme de génie, si les idées qui résultent de ce rapport forment un grand ensemble, sont fécondes en vérités & intéressantes pour l'humanité,” in 'Du Génie', L’Esprit (1758), Discourse 4, 476. English version from Claude Adrien Helvétius and William Mudford (trans.), 'Of Genius', De l’Esprit or, Essays on the Mind and its several Faculties (1759), Essay 4, Chap. 1, 241-242.
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Ironically one’s scientific obsolescence is a direct result of the creativity of his peers.
In 'Scientific innovation and creativity: a zoologist’s point of view', American Zoologist (1982), 22, 229.
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It has been said that science is opposed to, and in conflict with revelation. But the history of the former shown that the greater its progress, and the more accurate its investigations and results, the more plainly it is seen not only not to clash with the Latter, but in all things to confirm it. The very sciences from which objections have been brought against religion have, by their own progress, removed those objections, and in the end furnished fall confirmation of the inspired Word of God.
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It has hitherto been a serious impediment to the progress of knowledge, that is in investigating the origin or causes of natural productions, recourse has generally been had to the examination, both by experiment and reasoning, of what might be rather than what is. The laws or processes of nature we have every reason to believe invariable. Their results from time to time vary, according to the combinations of influential circumstances; but the process remains the same. Like the poet or the painter, the chemist may, and no doubt often' does, create combinations which nature never produced; and the possibility of such and such processes giving rise to such and such results, is no proof whatever that they were ever in natural operation.
Considerations on Volcanoes (1825), 243.
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It is admitted by all that a finished or even a competent reasoner is not the work of nature alone; the experience of every day makes it evident that education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history, may be chosen for this purpose. Now of all these, it is desirable to choose the one which admits of the reasoning being verified, that is, in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not. When the guiding property of the loadstone was first ascertained, and it was necessary to learn how to use this new discovery, and to find out how far it might be relied on, it would have been thought advisable to make many passages between ports that were well known before attempting a voyage of discovery. So it is with our reasoning faculties: it is desirable that their powers should be exerted upon objects of such a nature, that we can tell by other means whether the results which we obtain are true or false, and this before it is safe to trust entirely to reason. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:
1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing.
2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general.
3. The demonstration is strictly logical, taking nothing for granted except self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion.
4. When the conclusion is obtained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if, as was said before, reason is not to be the instructor, but the pupil.
5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded. Between the meaning of terms there is no distinction, except a total distinction, and all adjectives and adverbs expressing difference of degrees are avoided.
In On the Study and Difficulties of Mathematics (1898), chap. 1.
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It is better to have a few forms well known than to teach a little about many hundred species. Better a dozen specimens thoroughly studied as the result of the first year’s work, than to have two thousand dollars’ worth of shells and corals bought from a curiosity-shop. The dozen animals would be your own.
Lecture at a teaching laboratory on Penikese Island, Buzzard's Bay. Quoted from the lecture notes by David Starr Jordan, Science Sketches (1911), 147.
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It is easy to obtain confirmations, or verifications, for nearly every theory—if we look for confirmations. Confirmations should count only if they are the result of risky predictions... A theory which is not refutable by any conceivable event is non-scientific. Irrefutability is not a virtue of a theory (as people often think) but a vice. Every genuine test of a theory is an attempt to falsify it, or refute it.
Conjectures and Refutations: The Growth of Scientific Knowledge (1963), 36.
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It is from this absolute indifference and tranquility of the mind, that mathematical speculations derive some of their most considerable advantages; because there is nothing to interest the imagination; because the judgment sits free and unbiased to examine the point. All proportions, every arrangement of quantity, is alike to the understanding, because the same truths result to it from all; from greater from lesser, from equality and inequality.
In On the Sublime and Beautiful, Part 3, sect. 2.
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It is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations [than those given by Archimedes]. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearance, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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It is now necessary to indicate more definitely the reason why mathematics not only carries conviction in itself, but also transmits conviction to the objects to which it is applied. The reason is found, first of all, in the perfect precision with which the elementary mathematical concepts are determined; in this respect each science must look to its own salvation .... But this is not all. As soon as human thought attempts long chains of conclusions, or difficult matters generally, there arises not only the danger of error but also the suspicion of error, because since all details cannot be surveyed with clearness at the same instant one must in the end be satisfied with a belief that nothing has been overlooked from the beginning. Every one knows how much this is the case even in arithmetic, the most elementary use of mathematics. No one would imagine that the higher parts of mathematics fare better in this respect; on the contrary, in more complicated conclusions the uncertainty and suspicion of hidden errors increases in rapid progression. How does mathematics manage to rid itself of this inconvenience which attaches to it in the highest degree? By making proofs more rigorous? By giving new rules according to which the old rules shall be applied? Not in the least. A very great uncertainty continues to attach to the result of each single computation. But there are checks. In the realm of mathematics each point may be reached by a hundred different ways; and if each of a hundred ways leads to the same point, one may be sure that the right point has been reached. A calculation without a check is as good as none. Just so it is with every isolated proof in any speculative science whatever; the proof may be ever so ingenious, and ever so perfectly true and correct, it will still fail to convince permanently. He will therefore be much deceived, who, in metaphysics, or in psychology which depends on metaphysics, hopes to see his greatest care in the precise determination of the concepts and in the logical conclusions rewarded by conviction, much less by success in transmitting conviction to others. Not only must the conclusions support each other, without coercion or suspicion of subreption, but in all matters originating in experience, or judging concerning experience, the results of speculation must be verified by experience, not only superficially, but in countless special cases.
In Werke [Kehrbach] (1890), Bd. 5, 105. As quoted, cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 19.
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It is one of the chief merits of proofs that they instil a certain scepticism as to the result proved.
In The Principles of Mathematics (1903), Vol. 1, 360.
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It is still false to conclude that man is nothing but the highest animal, or the most progressive product of organic evolution. He is also a fundamentally new sort of animal and one in which, although organic evolution continues on its way, a fundamentally new sort of evolution has also appeared. The basis of this new sort of evolution is a new sort of heredity, the inheritance of learning. This sort of heredity appears modestly in other mammals and even lower in the animal kingdom, but in man it has incomparably fuller development and it combines with man's other characteristics unique in degree with a result that cannot be considered unique only in degree but must also be considered unique in kind.
In The Meaning of Evolution: A Study of the History of Life and of its Significance for Man (1949), 286.
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It is the care we bestow on apparently trifling, unattractive detail and very troublesome minutiae which determines the result.
As quoted in William Bulloch, Obituary, 'Theobald Smith', Journal of Pathology and Bacteriology (May 1935), 40, No. 3, 621-625.
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It is therefore easy to see why the churches have always fought science and persecuted its devotees. On the other hand, I maintain that the cosmic religious feeling is the strongest and noblest motive for scientific research. Only those who realize the immense efforts and, above all, the devotion without which pioneer work in theoretical science cannot be achieved are able to grasp the strength of the emotion out of which alone such work, remote as it is from the immediate realities of life, can issue. What a deep conviction of the rationality of the universe and what a yearning to understand, were it but a feeble reflection of the mind revealed in this world, Kepler and Newton must have had to enable them to spend years of solitary labor in disentangling the principles of celestial mechanics! Those whose acquaintance with scientific research is derived chiefly from its practical results easily develop a completely false notion of the mentality of the men who, surrounded by a skeptical world, have shown the way to kindred spirits scattered wide through the world and through the centuries. Only one who has devoted his life to similar ends can have a vivid realization of what has inspired these men and given them the strength to remain true to their purpose in spite of countless failures. It is cosmic religious feeling that gives a man such strength. A contemporary has said, not unjustly, that in this materialistic age of ours the serious scientific workers are the only profoundly religious people.
…...
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It may be unpopular and out-of-date to say—but I do not think that a scientific result which gives us a better understanding of the world and makes it more harmonious in our eyes should be held in lower esteem than, say, an invention which reduces the cost of paving roads, or improves household plumbing.
From final remarks in 'The Semantic Conception of Truth and the Foundations of Semantics' (1944), collected in Leonard Linsky (ed.), Semantics and the Philosophy of Language: A Collection of Readings (1952), 41.
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It may well be doubted whether, in all the range of Science, there is any field so fascinating to the explorer—so rich in hidden treasures—so fruitful in delightful surprises—as that of Pure Mathematics. The charm lies chiefly, I think, in the absolute certainty of its results: for that is what, beyond all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light … “And if our fate be death, give light and let us die” This is the cry that, through all the ages, is going up from perplexed Humanity, and Science has little else to offer, that will really meet the demands of its votaries, than the conclusions of Pure Mathematics.
Opening of 'Introduction', A New Theory of Parallels (1890), xv. As a non-fiction work, the author’s name on the title page of this book was Charles Lutwidge Dodgson. Being better known for his works of fiction as Lewis Carroll, all quotes relating to this one person, published under either name, are gathered on this single web page under his pen name.
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It may well be doubted whether, in all the range of science, there is any field so fascinating to the explorer—so rich in hidden treasures—so fruitful in delightful surprises—as that of Pure Mathematics. The charm lies chiefly, I think, in the absolute certainty of its results; for that is what, beyond all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light!
Written without pseudonym as Charles L. Dodgson. Opening remarks in Introduction to A New Theory of Parallels (1888, 1890), xv.
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It was found after many troublesome experiments that when the vacuum within the lamp globe was good, and the contact between the carbon and the conductor which supported it sufficient, there was no blackening of the globes, and no appreciable wasting away of the carbons. Thus was swept away a pernicious error, which, like a misleading finger post proclaiming “No road this way,” tended to bar progress along a good thoroughfare. It only remained to perfect the details of the lamp, to find the best material from which to form the carbon, and to fix this material in the lamp in the best manner. These points, I think, I have now satisfactorily settled, and you see the result in the lamp before me on the table.
In Lecture (20 Oct 1880) at Newcastle-Upon-Tyne, as quoted in United States Courts of Appeals Reports: Cases Adjudged in the United States Circuit Court of Appeals (1894), Vol. 11, 419-420.
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It would seem at first sight as if the rapid expansion of the region of mathematics must be a source of danger to its future progress. Not only does the area widen but the subjects of study increase rapidly in number, and the work of the mathematician tends to become more and more specialized. It is, of course, merely a brilliant exaggeration to say that no mathematician is able to understand the work of any other mathematician, but it is certainly true that it is daily becoming more and more difficult for a mathematician to keep himself acquainted, even in a general way, with the progress of any of the branches of mathematics except those which form the field of his own labours. I believe, however, that the increasing extent of the territory of mathematics will always be counteracted by increased facilities in the means of communication. Additional knowledge opens to us new principles and methods which may conduct us with the greatest ease to results which previously were most difficult of access; and improvements in notation may exercise the most powerful effects both in the simplification and accessibility of a subject. It rests with the worker in mathematics not only to explore new truths, but to devise the language by which they may be discovered and expressed; and the genius of a great mathematician displays itself no less in the notation he invents for deciphering his subject than in the results attained. … I have great faith in the power of well-chosen notation to simplify complicated theories and to bring remote ones near and I think it is safe to predict that the increased knowledge of principles and the resulting improvements in the symbolic language of mathematics will always enable us to grapple satisfactorily with the difficulties arising from the mere extent of the subject.
In Presidential Address British Association for the Advancement of Science, Section A., (1890), Nature, 42, 466.
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Keep in mind that new ideas are commonplace, and almost always wrong. Most flashes of insight lead nowhere; statistically, they have a half-life of hours or maybe days. Most experiments to follow up the surviving insights are tedious and consume large amounts of time, only to yield negative or (worse!) ambiguous results.
In Consilience: The Unity of Knowledge (1998, 1999), 60
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Leave the beaten track occasionally and dive into the woods. Every time you do so you will be certain to find something that you have never seen before. Of course, it will be a little thing, but do not ignore it. Follow it up, explore all around it: one discovery will lead to another, and before you know it, you will have something worth thinking about to occupy your mind. All really big discoveries are the results of thought.
Address (22 May 1914) to the graduating class of the Friends’ School, Washington, D.C. Printed in 'Discovery and Invention', The National Geographic Magazine (1914), 25, 650.
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Let U = the University, G = Greek, and P = Professor, Then GP = Greek Professor; let this be reduced to its lowest terms and call the result J.
From an essay concerning the Regius Professorship of Greek, The New Method of Evaluating as Applied to π (1865), as quoted and cited in Stuart Dodgson Collingwood, The Life and Letters of Lewis Carroll (1898), 159. Collingwood explains parenthetically, after "result J", “[i.e., Jowett]”, which was not in the original publication of “The New Method…”.
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Man is but a perambulating tool-box and workshop or office, fashioned for itself by a piece of very clever slime, as the result of long experience. ... Hence we speak of man's body as his “trunk.”
Samuel Butler, Henry Festing Jones (ed.), The Note-Books of Samuel Butler (1917), 18.
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Man is the result of a purposeless and natural process that did not have him in mind.
In The Meaning of Evolution (Rev.Ed. 1967), 345.
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Mathematicians assume the right to choose, within the limits of logical contradiction, what path they please in reaching their results.
In A Letter to American Teachers of History (1910), Introduction, v.
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Mathematicians attach great importance to the elegance of their methods and their results. This is not pure dilettantism. What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details. But this is exactly what yields great results, in fact the more we see this aggregate clearly and at a single glance, the better we perceive its analogies with other neighboring objects, consequently the more chances we have of divining the possible generalizations. Elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together; there again it is fruitful, since it thus unveils for us kinships before unrecognized. It is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set; it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason; that it is to be found in some unexpected law. In a word, the feeling of mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind, and it is because of this very adaptation that this solution can be for us an instrument. Consequently this esthetic satisfaction is bound up with the economy of thought.
In 'The Future of Mathematics', Monist, 20, 80. Translated from the French by George Bruce Halsted.
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Mathematics gives the young man a clear idea of demonstration and habituates him to form long trains of thought and reasoning methodically connected and sustained by the final certainty of the result; and it has the further advantage, from a purely moral point of view, of inspiring an absolute and fanatical respect for truth. In addition to all this, mathematics, and chiefly algebra and infinitesimal calculus, excite to a high degree the conception of the signs and symbols—necessary instruments to extend the power and reach of the human mind by summarizing an aggregate of relations in a condensed form and in a kind of mechanical way. These auxiliaries are of special value in mathematics because they are there adequate to their definitions, a characteristic which they do not possess to the same degree in the physical and mathematical [natural?] sciences.
There are, in fact, a mass of mental and moral faculties that can be put in full play only by instruction in mathematics; and they would be made still more available if the teaching was directed so as to leave free play to the personal work of the student.
In 'Science as an Instrument of Education', Popular Science Monthly (1897), 253.
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Mathematics has beauties of its own—a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found only in the works of the greatest beauty. … When this subject is properly and concretely presented, the mental emotion should be that of enjoyment of beauty, not that of repulsion from the ugly and the unpleasant.
In The Teaching of Mathematics in the Elementary and the Secondary School (1906), 44-45.
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Measurement has too often been the leitmotif of many investigations rather than the experimental examination of hypotheses. Mounds of data are collected, which are statistically decorous and methodologically unimpeachable, but conclusions are often trivial and rarely useful in decision making. This results from an overly rigorous control of an insignificant variable and a widespread deficiency in the framing of pertinent questions. Investigators seem to have settled for what is measurable instead of measuring what they would really like to know.
'Patient Care—Mystical Research or Researchable Mystique/', Clinical Research (1964), 12, no. 4, 422.
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Men who have excessive faith in their theories or ideas are not only ill prepared for making discoveries; they also make very poor observations. Of necessity, they observe with a preconceived idea, and when they devise an experiment, they can see, in its results,only a confirmation of their theory. In this way they distort observation and often neglect very important facts because they do not further their aim.
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 38.
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Men who have excessive faith in their theories … make poor observations, because they choose among the results of their experiments only what suits their object, neglecting whatever is unrelated to it and carefully setting aside everything which might tend toward the idea they wish to combat
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 38.
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Misuse of reason might yet return the world to pre-technological night; plenty of religious zealots hunger for just such a result, and are happy to use the latest technology to effect it.
The Heart of Things: Applying Philosophy to the 21st Century (2006).
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Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature.
In A Budget of Paradoxes (1872), 55.
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Most of the work performed by a development engineer results in failure. The occasional visit of success provides just the excitement an engineer needs to face work the following day.
In Tore Frängsmyr (ed.), Les Prix Nobel. The Nobel Prizes 2002 (2003), 193.
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Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence”.
In 'A Plea for the Mathematician', Nature, 1, 238 in Collected Mathematical Papers, Vol. 2, 655-56.
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My belief (is) that one should take a minimum of care and preparation over first experiments. If they are unsuccessful one is not then discouraged since many possible reasons for failure can be thought of, and improvements can be made. Much can often be learned by the repetition under different conditions, even if the desired result is not obtained. If every conceivable precaution is taken at first, one is often too discouraged to proceed at all.
Nobel Lectures in Chemistry (1999), Vol. 3, 364.
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My colleagues in elementary particle theory in many lands [and I] are driven by the usual insatiable curiosity of the scientist, and our work is a delightful game. I am frequently astonished that it so often results in correct predictions of experimental results. How can it be that writing down a few simple and elegant formulae, like short poems governed by strict rules such as those of the sonnet or the waka, can predict universal regularities of Nature?
Nobel Banquet Speech (10 Dec 1969), in Wilhelm Odelberg (ed.),Les Prix Nobel en 1969 (1970).
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My definition of science is … somewhat as follows: Science is an interconnected series of concepts and conceptual schemes that have developed as a result of experimentation and observation and are fruitful of further experimentation and observations. In this definition the emphasis is on the word “fruitful.”
In Science and Common Sense (1951), 25.
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My eureka moment was in the dead of night, the early hours of the morning, on a cold, cold night, and my feet were so cold, they were aching. But when the result poured out of the charts, you just forget all that. You realize instantly how significant this is—what it is you’ve really landed on—and it’s great!
[About her discovery of the first pulsar radio signals.]
From BBC TV program, Journeys in Time and Space: Invisible Universe (28 Feb 2001).
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My father’s collection of fossils was practically unnamed, but the appearance of Phillips’ book [Geology of the Yorkshire Coast], in which most of our specimens were figured, enabled us to remedy this defect. Every evening was devoted by us to accomplishing the work. This was my first introduction to true scientific study. ... Phillips’ accurate volume initiated an entirely new order of things. Many a time did I mourn over the publication of this book, and the consequences immediately resulting from it. Instead of indulging in the games and idleness to which most lads are prone, my evenings throughout a long winter were devoted to the detested labour of naming these miserable stones. Such is the short-sightedness of boyhood. Pursuing this uncongenial work gave me in my thirteenth year a thorough practical familiarity with the palaeontological treasures of Eastern Yorkshire. This early acquisition happily moulded the entire course of my future life.
In Reminiscences of a Yorkshire naturalist (1896), 12.
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My interest in the biology of tissue and organ transplantation arose from my [WW II] military experience at Valley Forge General Hospital in Pennsylvania … a major plastic surgical center. While there, I spent all my available spare time on the plastic surgical wards which were jammed with hundreds of battle casualties. I enjoyed talking to the patients, helping with dressings, and observing the results of the imaginative reconstructive surgical operations.
As a First Lieutenant with only a nine-month surgical internship, randomly assigned to VFGH to await overseas duty. In Tore Frängsmyr and Jan E. Lindsten (eds.), Nobel Lectures: Physiology Or Medicine: 1981-1990 (1993), 556.
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My original decision to devote myself to science was a direct result of the discovery which has never ceased to fill me with enthusiasm since my early youth—the comprehension of the far from obvious fact that the laws of human reasoning coincide with the laws governing the sequences of the impressions we receive from the world about us; that, therefore, pure reasoning can enable man to gain an insight into the mechanism of the latter. In this connection, it is of paramount importance that the outside world is something independent from man, something absolute, and the quest for the laws which apply to this absolute appeared to me as the most sublime scientific pursuit in life.
'A Scientific Autobiography' (1948), in Scientific Autobiography and Other Papers, trans. Frank Gaynor (1950), 13.
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My visceral perception of brotherhood harmonizes with our best modern biological knowledge ... Many people think (or fear) that equality of human races represents a hope of liberal sentimentality probably squashed by the hard realities of history. They are wrong. This essay can be summarized in a single phrase, a motto if you will: Human equality is a contingent fact of history. Equality is not true by definition; it is neither an ethical principle (though equal treatment may be) nor a statement about norms of social action. It just worked out that way. A hundred different and plausible scenarios for human history would have yielded other results (and moral dilemmas of enormous magnitude). They didn’t happen.
…...
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Natural powers, principally those of steam and falling water, are subsidized and taken into human employment Spinning-machines, power-looms, and all the mechanical devices, acting, among other operatives, in the factories and work-shops, are but so many laborers. They are usually denominated labor-saving machines, but it would be more just to call them labor-doing machines. They are made to be active agents; to have motion, and to produce effect; and though without intelligence, they are guided by laws of science, which are exact and perfect, and they produce results, therefore, in general, more accurate than the human hand is capable of producing.
Speech in Senate (12 Mar 1838). In The Writings and Speeches of Daniel Webster (1903), Vol. 8, 177.
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Natural selection produces systems that function no better than necessary. It results in ad hoc adaptive solutions to immediate problems. Whatever enhances fitness is selected. The product of natural selection is not perfection but adequacy, not final answers but limited, short-term solutions.
In 'The role of natural history in contemporary biology', BioScience (1986), 36, 325.
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Nature is a source of truth. Experience does not ever err, it is only your judgment that errs in promising itself results which are not caused by your experiments.
The Notebook. As cited in Edward Schwartz, One Step Forward, Two Steps Backward (2003), 38, with caption “examining objects in all their diversity.” Also quoted in Daniel J. Boorstin, The Discoverers (1983), 350.
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Nine-tenths of diseases are medicinal diseases.
[Lamenting the diseases resulting from the continued prescribing of harmful medicinal remedies.]
Quoted by William M. Scribner, 'Treatment of Pneumonia and Croup, Once More, Etc,' in The Medical World (1885), 3, 187.
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No creature is too bulky or formidable for man's destructive energies—none too minute and insignificant for his keen detection and skill of capture. It was ordained from the beginning that we should be the masters and subduers of all inferior animals. Let us remember, however, that we ourselves, like the creatures we slay, subjugate, and modify, are the results of the same Almighty creative will—temporary sojourners here, and co-tenants with the worm and the whale of one small planet. In the exercise, therefore, of those superior powers that have been intrusted to us, let us ever bear in mind that our responsibilities are heightened in proportion.
Lecture to the London Society of Arts, 'The Raw Materials of the Animal Kingdom', collected in Lectures on the Results of the Great Exhibition of 1851' (1852), 131.
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No experimental result can ever kill a theory: any theory can be saved from counterinstances either by some auxiliary hypothesis or by a suitable reinterpretation of its terms.
In 'Falsification and the Methodology of Scientific Research Programmes', in I. Lakatos and A. Musgrave (eds.), Criticism and the Growth of Knowledge: Proceedings of the International Colloquium in the Philosophy of Science, London 1965 (1970), Vol. 4, 116.
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No one believes the results of the computational modeler except the modeler, for only he understands the premises. No one doubts the experimenter’s results except the experimenter, for only he knows his mistakes.
Anonymous
See a similar idea expressed by W.I.B. Beveredge, beginning “No one believes an hypothesis…” on the W.I.B. Beveredge Quotes web page on this site.
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No process of sound reasoning can establish a result not contained in the premises.
In Higher Mathematics for Students of Chemistry and Physics (1902), 2.
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Not long ago the head of what should be a strictly scientific department in one of the major universities commented on the odd (and ominous) phenomenon that persons who can claim to be scientists on the basis of the technical training that won them the degree of Ph.D. are now found certifying the authenticity of the painted rag that is called the “Turin Shroud” or adducing “scientific” arguments to support hoaxes about the “paranormal” or an antiquated religiosity. “You can hire a scientist [sic],” he said, “to prove anything.” He did not adduce himself as proof of his generalization, but he did boast of his cleverness in confining his own research to areas in which the results would not perturb the Establishment or any vociferous gang of shyster-led fanatics. If such is indeed the status of science and scholarship in our darkling age, Send not to ask for whom the bell tolls.
…...
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Nothing that you do in science is guaranteed to result in benefits for mankind. Any discovery, I believe, is morally neutral and it can be turned either to constructive ends or destructive ends. That’s not the fault of science.
Quoted by Jeremy Pearce in 'Arthur Galston, Agent Orange Researcher, Is Dead at 88', New York Times (23 Jun 2008), B6.
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Objections … inspired Kronecker and others to attack Weierstrass’ “sequential” definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth—whatever that may be, if anything—precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals.
In Men of Mathematics (1937), 431-432.
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Often the great scientists, by turning the problem around a bit, changed a defect to an asset. For example, many scientists when they found they couldn't do a problem finally began to study why not. They then turned it around the other way and said, “But of course, this is what it is” and got an important result.
'You and Your Research', Bell Communications Research Colloquium Seminar, 7 Mar 1986.
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On all questions where his passions are strongly engaged, man prizes certitude and fears knowledge. Dispassionate inquiry is welcomed only when the result is indifferent.
In Aspects of Science (1925), 9.
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On the whole, I cannot help saying that it appears to me not a little extraordinary, that a theory so new, and of such importance, overturning every thing that was thought to be the best established in chemistry, should rest on so very narrow and precarious a foundation, the experiments adduced in support of it being not only ambiguous or explicable on either hypothesis, but exceedingly few. I think I have recited them all, and that on which the greatest stress is laid, viz. That of the formation of water from the decomposition of the two kinds of air, has not been sufficiently repeated. Indeed it required so difficult and expensive an apparatus, and so many precautions in the use of it, that the frequent repetition of the experiment cannot be expected; and in these circumstances the practised experimenter cannot help suspecting the accuracy of the result and consequently the certainty of the conclusion.
Considerations on the Doctrine of Phlogiston (1796), 57-8.
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Once a mathematical result is proven to the satisfaction of the discipline, it doesn’t need to be re-evaluated in the light of new evidence or refuted, unless it contains a mistake. If it was true for Archimedes, then it is true today.
In 'The Unplanned Impact of Mathematics', Nature (14 Jul 2011), 475, No. 7355, 166.
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One cannot ignore half of life for the purposes of science, and then claim that the results of science give a full and adequate picture of the meaning of life. All discussions of “life” which begin with a description of man's place on a speck of matter in space, in an endless evolutionary scale, are bound to be half-measures, because they leave out most of the experiences which are important to use as human beings.
In Religion and the Rebel (1957), 309.
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One feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the deliberation with which Archimedes approaches the solution of any one of his main problems. Yet this very characteristic, with its incidental effects, is calculated to excite the more admiration because the method suggests the tactics of some great strategist who foresees everything, eliminates everything not immediately conducive to the execution of his plan, masters every position in its order, and then suddenly (when the very elaboration of the scheme has almost obscured, in the mind of the spectator, its ultimate object) strikes the final blow. Thus we read in Archimedes proposition after proposition the bearing of which is not immediately obvious but which we find infallibly used later on; and we are led by such easy stages that the difficulties of the original problem, as presented at the outset, are scarcely appreciated. As Plutarch says: “It is not possible to find in geometry more difficult and troublesome questions, or more simple and lucid explanations.” But it is decidedly a rhetorical exaggeration when Plutarch goes on to say that we are deceived by the easiness of the successive steps into the belief that anyone could have discovered them for himself. On the contrary, the studied simplicity and the perfect finish of the treatises involve at the same time an element of mystery. Though each step depends on the preceding ones, we are left in the dark as to how they were suggested to Archimedes. There is, in fact, much truth in a remark by Wallis to the effect that he seems “as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results.” Wallis adds with equal reason that not only Archimedes but nearly all the ancients so hid away from posterity their method of Analysis (though it is certain that they had one) that more modern mathematicians found it easier to invent a new Analysis than to seek out the old.
In The Works of Archimedes (1897), Preface, vi.
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One of the most important choices any researcher makes is picking a significant topic to study. If you choose the right problem, you get important results that transform our perception of the underlying structure of the universe. If you don’t choose the right problem, you may work very hard but only get an interesting result.
Unverified - source citation needed. Can you help?
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One of the most striking results of modern investigation has been the way in which several different and quite independent lines of evidence indicate that a very great event occurred about two thousand million years ago. The radio-active evidence for the age of meteorites; and the estimated time for the tidal evolution of the Moon's orbit (though this is much rougher), all agree in their testimony, and, what is far more important, the red-shift in the nebulae indicates that this date is fundamental, not merely in the history of our system, but in that of the material universe as a whole.
The Solar System and its Origin (1935), 137.
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One of the principal results of civilization is to reduce more and more the limits within which the different elements of society fluctuate. The more intelligence increases the more these limits are reduced, and the nearer we approach the beautiful and the good. The perfectibility of the human species results as a necessary consequence of all our researches. Physical defects and monstrosities are gradually disappearing; the frequency and severity of diseases are resisted more successfully by the progress of modern science; the moral qualities of man are proving themselves not less capable of improvement; and the more we advance, the less we shall have need to fear those great political convulsions and wars and their attendant results, which are the scourges of mankind.
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One should guard against inculcating a young man with the idea that success is the aim of life, for a successful man normally receives from his peers an incomparably greater portion than the services he has been able to render them deserve. The value of a man resides in what he gives and not in what he is capable of receiving. The most important motive for study at school, at the university, and in life is the pleasure of working and thereby obtaining results which will serve the community. The most important task for our educators is to awaken and encourage these psychological forces in a young man {or woman}. Such a basis alone can lead to the joy of possessing one of the most precious assets in the world - knowledge or artistic skill.
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One should guard against preaching to the young man success in the customary sense as the aim in life. ... The most important motive for work in school and in life is pleasure in work, pleasure in its result, and the knowledge of the value of the result to the community.
'On Education', address at the State University of New York, Albany (15 Oct 1936) in celebration of the Tercentenary of Higher Education in America, translation prepared by Lina Arronet. In Albert Einstein, The Einstein Reader (2006), 30.
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Only when Genius is married to Science can the highest results be produced.
Education: Intellectual, Moral, and Physical (1889), 81.
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Organization is simply the means by which the acts of ordinary men can be made to add up to extraordinary results. To this idea of progress that does not wait on some lucky break, some chance discovery, or some rare stroke of genius, but instead is achieved through systematic, cumulative effort, the engineer has contributed brilliantly.
In A Professional Guide for Young Engineers (1949, 1967), 36.
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Our children will enjoy in their homes electrical energy too cheap to meter. ... Transmutation of the elements, unlimited power, ability to investigate the working of living cells by tracer atoms, the secret of photosynthesis about to be uncovered, these and a host of other results, all in about fifteen short years. It is not too much to expect that our children will know of great periodic famines in the world only as matters of history, will travel effortlessly over the seas and under the and through the air with a minimum of danger and at great speeds, and will experience a life span far longer than ours, as disease yields and man comes to understand what causes him to age.
Speech at the 20th anniversary of the National Association of Science Writers, New York City (16 Sep 1954), asquoted in 'Abundant Power From Atom Seen', New York Times (17 Sep 1954) 5.
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Our emphasis on science has resulted in an alarming rise in world populations, the demand and ever-increasing emphasis of science to improve their standards and maintain their vigor. I have been forced to the conclusion that an over-emphasis of science weakens character and upsets life's essential balance.
In 'The Wisdom of Wilderness', Life (22 Dec 1967), 63, No. 25, 9.
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Our immediate interests are after all of but small moment. It is what we do for the future, what we add to the sum of man's knowledge, that counts most. As someone has said, 'The individual withers and the world is more and more.' Man dies at 70, 80, or 90, or at some earlier age, but through his power of physical reproduction, and with the means that he has to transmit the results of effort to those who come after him, he may be said to be immortal.
'Willis Rodney Whitney', National Academy of Sciences, Biographical Memoirs (1960), 360.
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Our mind, by virtue of a certain finite, limited capability, is by no means capable of putting a question to Nature that permits a continuous series of answers. The observations, the individual results of measurements, are the answers of Nature to our discontinuous questioning.
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Our school education ignores, in a thousand ways, the rules of healthy development; and the results … are gained very generally at the cost of physical and mental health.
Lecture (2 Dec 1959) delivered in Clinton Hall, New York City. Published in 'Medicine as a Profession for Women', The English Woman’s Journal (1 May 1860), 5, No. 27, 148. (Prepared together with Emily Blackwell.) The Blackwells recognized the connection between health and learning. They also wanted that teachers (of whom 90% were women) should “diffuse among women the physiological and sanitary knowledge which they will need.”
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Part of the charm in solving a differential equation is in the feeling that we are getting something for nothing. So little information appears to go into the solution that there is a sense of surprise over the extensive results that are derived.
Co-author with Jules Alphonse Larrivee, Mathematics and Computers (1957), 40.
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Philosophically, I liked the steady-state cosmology. So I thought that we should report our results as a simple measurement; the measurement might be true after the cosmology was no longer true!
Remarking on the measurement he made with Arno Penzias of the 3 K cosmic background radiation. From Proceedings of workshop, National Radio Astronomy Observatory, Green Bank, West Virginia (4-6 May 1983), 'Discovery of the Cosmic Microwave Background', Serendipitous Discoveries in Radio Astronomy (1983), 195. Also collected in B. Bertotti (ed.) Modern Cosmology in Retrospect (1990), 303.
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Physical science enjoys the distinction of being the most fundamental of the experimental sciences, and its laws are obeyed universally, so far as is known, not merely by inanimate things, but also by living organisms, in their minutest parts, as single individuals, and also as whole communities. It results from this that, however complicated a series of phenomena may be and however many other sciences may enter into its complete presentation, the purely physical aspect, or the application of the known laws of matter and energy, can always be legitimately separated from the other aspects.
In Matter and Energy (1912), 9-10.
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Politicians, real-estate agents, used-car salesmen, and advertising copy-writers are expected to stretch facts in self-serving directions, but scientists who falsify their results are regarded by their peers as committing an inexcusable crime. Yet the sad fact is that the history of science swarms with cases of outright fakery and instances of scientists who unconsciously distorted their work by seeing it through lenses of passionately held beliefs.
Science Good, Bad and Bogus (1981), 123.
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Progress is made by trial and failure; the failures are generally a hundred times more numerous than the successes; yet they are usually left unchronicled. The reason is that the investigator feels that even though he has failed in achieving an expected result, some other more fortunate experimenter may succeed, and it is unwise to discourage his attempts.
From 'Radium and its Products', Harper’s Magazine (Dec 1904), 110, No. 655, 52.
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Questioning the status quo can result in banishment, imprisonment, ridicule or being burned at the stake, depending on your era, your locale, and the sacred cows you wish to butcher.
From post 're:The Pursuit of Knowledge, from Genesis to Google' to the 'Interesting People' List (6 Jan 2005) maintained by David J. Farber, now archived at interesting-people.org website.
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Quite distinct from the theoretical question of the manner in which mathematics will rescue itself from the perils to which it is exposed by its own prolific nature is the practical problem of finding means of rendering available for the student the results which have been already accumulated, and making it possible for the learner to obtain some idea of the present state of the various departments of mathematics. … The great mass of mathematical literature will be always contained in Journals and Transactions, but there is no reason why it should not be rendered far more useful and accessible than at present by means of treatises or higher text-books. The whole science suffers from want of avenues of approach, and many beautiful branches of mathematics are regarded as difficult and technical merely because they are not easily accessible. … I feel very strongly that any introduction to a new subject written by a competent person confers a real benefit on the whole science. The number of excellent text-books of an elementary kind that are published in this country makes it all the more to be regretted that we have so few that are intended for the advanced student. As an example of the higher kind of text-book, the want of which is so badly felt in many subjects, I may mention the second part of Prof. Chrystal’s Algebra published last year, which in a small compass gives a great mass of valuable and fundamental knowledge that has hitherto been beyond the reach of an ordinary student, though in reality lying so close at hand. I may add that in any treatise or higher text-book it is always desirable that references to the original memoirs should be given, and, if possible, short historic notices also. I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.
In Presidential Address British Association for the Advancement of Science, Section A (1890), Nature, 42, 466.
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Recurrences of like cases in which A is always connected with B, that is, like results under like circumstances, that is again, the essence of the connection of cause and effect, exist but in the abstraction which we perform for the purpose of mentally reproducing the facts. Let a fact become familiar, and we no longer require this putting into relief of its connecting marks, our attention is no longer attracted to the new and surprising, and we cease to speak of cause and effect.
In The Science of Mechanics (1893), 483.
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Research! A mere excuse for idleness; it has never achieved, and will never achieve any results of the slightest value.
As quoted from author’s conversation with Jowett, in Logan Pearsall Smith, Unforgotten Years (1938, 1939), 186-187.
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Results rarely specify their causes unambiguously. If we have no direct evidence of fossils or human chronicles, if we are forced to infer a process only from its modern results, then we are usually stymied or reduced to speculation about probabilities. For many roads lead to almost any Rome.
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Results! Why, man, I have gotten a lot of results. I know several thousand things that won’t work.
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Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”
In 'Ten Lessons I Wish I Had Been Taught', Indiscrete Thoughts (2008), 202.
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Samuel Pierpoint Langley, at that time regarded as one of the most distinguished scientists in the United States … evidently believed that a full sized airplane could be built and flown largely from theory alone. This resulted in two successive disastrous plunges into the Potomac River, the second of which almost drowned his pilot. This experience contrasts with that of two bicycle mechanics Orville and Wilbur Wright who designed, built and flew the first successful airplane. But they did this after hundreds of experiments extending over a number of years.
In article Total Quality: Its Origins and its Future (1995), published at the Center for Quality and Productivity Improvement.
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Scarcely any attempt is entirely a failure; scarcely any theory, the result of steady thought, is altogether false; no tempting form of Error is without some latent charm derived from Truth.
In 'Lecture 7', Lectures on the History of Moral Philosophy in England (1852), 101.
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Science does not present itself to man until mind conquers matter in striving to subject the result of experimental investigation to rational combinations.
In Alexander Humboldt and E.C. Otté (trans.), 'Introduction', Cosmos: Sketch of a Physical Description of the Universe (1852), Vol. 1, 76. The translator’s preface is dated 1844.
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Science is a human activity, and the best way to understand it is to understand the individual human beings who practise it. Science is an art form and not a philosophical method. The great advances in science usually result from new tools rather than from new doctrines. ... Every time we introduce a new tool, it always leads to new and unexpected discoveries, because Nature's imagination is richer than ours.
Concluding remark from 'The Scientist As Rebel' American Mathemtical Monthly (1996), 103, 805. Reprinted in The Scientist as Rebel (2006), 17-18, identified as originally written for a lecture (1992), then published as an essay in the New York Review.
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Science progresses by a series of combinations in which chance plays not the least role. Its life is rough and resembles that of minerals which grow by juxtaposition [accretion]. This applies not only to science such as it emerges [results] from the work of a series of scientists, but also to the particular research of each one of them. In vain would analysts dissimulate: (however abstract it may be, analysis is no more our power than that of others); they do not deduce, they combine, they compare: (it must be sought out, sounded out, solicited.) When they arrive at the truth it is by cannoning from one side to another that they come across it.
English translation from manuscript, in Évariste Galois and Peter M. Neumann, 'Dossier 12: On the progress of pure analysis', The Mathematical Writings of Évariste Galois (2011), 263. A transcription of the original French is on page 262. In the following quote from that page, indicated deletions are omitted, and Webmaster uses parentheses to enclose indications of insertions above the original written line. “La science progresse par une série de combinaisons où le hazard ne joue pas le moindre rôle; sa vie est brute et ressemble à celle des minéraux qui croissent par juxtà position. Cela s’applique non seulement à la science telle qu’elle résulte des travaux d’une série de savants, mais aussi aux recherches particulières à chacun d’eux. En vain les analystes voudraient-ils se le dissimuler: (toute immatérielle qu’elle wst analyse n’est pas pas plus en notre pouvoir que des autres); ils ne déduisent pas, ils combinent, ils comparent: (il faut l’epier, la sonder, la solliciter) quand ils arrivent à la vérité, c’est en heurtant de côté et d’autre qu’il y sont tombés.” Webmaster corrected from typo “put” to “but” in the English text.
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Science was born as a result and consequence of philosophy; it cannot survive without a philosophical base. If philosophy perishes, science will be next to go.
Ayn Rand
For the New Intellectual: The Philosophy of Ayn Rand (1963), 44.
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Scientific inquiry would thus he conceived of as analogous to terrestrial exploration, whose product—geography—yields results of continually smaller significance which fill in ever more minute gaps in our information. In such a view, later investigations yield findings of ever smaller importance, with each successive accretion making a relatively smaller contribution to what has already come to hand. The advance of science leads, step by diminished step, toward a fixed and final view of things.
In The Limits Of Science (1984, Rev. 1999), 67.
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Secondly, the study of mathematics would show them the necessity there is in reasoning, to separate all the distinct ideas, and to see the habitudes that all those concerned in the present inquiry have to one another, and to lay by those which relate not to the proposition in hand, and wholly to leave them out of the reckoning. This is that which, in other respects besides quantity is absolutely requisite to just reasoning, though in them it is not so easily observed and so carefully practised. In those parts of knowledge where it is thought demonstration has nothing to do, men reason as it were in a lump; and if upon a summary and confused view, or upon a partial consideration, they can raise the appearance of a probability, they usually rest content; especially if it be in a dispute where every little straw is laid hold on, and everything that can but be drawn in any way to give color to the argument is advanced with ostentation. But that mind is not in a posture to find truth that does not distinctly take all the parts asunder, and, omitting what is not at all to the point, draws a conclusion from the result of all the particulars which in any way influence it.
In Conduct of the Understanding, Sect. 7.
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So I want to admit the assumption which the astronomer—and indeed any scientist—makes about the Universe he investigates. It is this: that the same physical causes give rise to the same physical results anywhere in the Universe, and at any time, past, present, and future. The fuller examination of this basic assumption, and much else besides, belongs to philosophy. The scientist, for his part, makes the assumption I have mentioned as an act of faith; and he feels confirmed in that faith by his increasing ability to build up a consistent and satisfying picture of the universe and its behavior.
From Science and the Nation (1957), 49. Also quoted in Ronald Keast, Dancing in the Dark: The Waltz in Wonder of Quantum Metaphysics (2009), 106.
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Sociology is the science with the greatest number of methods and the least results.
…...
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Some of the most important results (e.g. Cauchy’s theorem) are so surprising at first sight that nothing short of a proof can make them credible.
As co-author with Bertha Swirls Jeffreys, in Methods of Mathematical Physics (1946, 1999), v.
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Success is a science; if you have the conditions, you get the result.
In Epigrams of Oscar Wilde (2007), 68.
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Suppose the results of a line of study are negative. It might save a lot of otherwise wasted money to know a thing won’t work. But how do you accurately evaluate negative results? ... The power plant in [the recently developed streamline trains] is a Diesel engine of a type which was tried out many [around 25] years ago and found to be a failure. … We didn’t know how to build them. The principle upon which it operated was sound. [Since then much has been] learned in metallurgy [and] the accuracy with which parts can be manufactured
When this type of engine was given another chance it was an immediate success [because now] an accuracy of a quarter of a tenth of a thousandth of an inch [prevents high-pressure oil leaks]. … If we had taken the results of past experience without questioning the reason for the first failure, we would never have had the present light-weight, high-speed Diesel engine which appears to be the spark that will revitalize the railroad business.
'Industrial Prospecting', an address to the Founder Societies of Engineers (20 May 1935). In National Research Council, Reprint and Circular Series of the National Research Council (1933), No. 107, 2-3.
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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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That form of popular science which merely recites the results of investigations, which merely communicates useful knowledge, is from this standpoint bad science, or no science at all. … Apply this test to every work professing to give a popular account of any branch of science. If any such work gives a description of phenomena that appeals to his imagination rather than to his reason, then it is bad science.
From The Grammar of Science (1892), 12.
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That mathematics “do not cultivate the power of generalization,”; … will be admitted by no person of competent knowledge, except in a very qualified sense. The generalizations of mathematics, are, no doubt, a different thing from the generalizations of physical science; but in the difficulty of seizing them, and the mental tension they require, they are no contemptible preparation for the most arduous efforts of the scientific mind. Even the fundamental notions of the higher mathematics, from those of the differential calculus upwards are products of a very high abstraction. … To perceive the mathematical laws common to the results of many mathematical operations, even in so simple a case as that of the binomial theorem, involves a vigorous exercise of the same faculty which gave us Kepler’s laws, and rose through those laws to the theory of universal gravitation. Every process of what has been called Universal Geometry—the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them—is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics.
In An Examination of Sir William Hamilton’s Philosophy (1878), 612-13.
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That the main results of the astronomer’s work are not so immediately practical does not detract from their value. They are, I venture to think, the more to be prized on that account. Astronomy has profoundly influenced the thought of the race. In fact, it has been the keystone in the arch of the sciences under which we have marched out from the darkness of the fifteenth and preceding centuries to the comparative light of to-day.
In 'The Nature of the Astronomer’s Work', North American Review (Jun 1908), 187, No. 631, 915.
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That which is perfect in science, is most commonly the elaborate result of successive improvements, and of various judgments exercised in the rejection of what was wrong, no less than in the adoption of what was right.
Reflection 490, in Lacon: Or Many Things in Few Words, Addressed to Those who Think (1832), 202.
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The “British Association for the Promotion of Science,” … is almost necessary for the purposes of science. The periodical assemblage of persons, pursuing the same or différent branches of knowledge, always produces an excitement which is favourable to the development of new ideas; whilst the long period of repose which succeeds, is advantageous for the prosecution of the reasonings or the experiments then suggested; and the récurrence of the meeting in the succeeding year, will stimulate the activity of the inquirer, by the hope of being then enabled to produce the successful result of his labours.
In 'Future Prospects', On the Economy of Machinery and Manufactures (1st ed., 1832), chap. 32, 274. Note: The British Association for the Advancement of Science held its first meeting at York in 1831, the year before the first publication of this book in 1832.
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The actual evolution of mathematical theories proceeds by a process of induction strictly analogous to the method of induction employed in building up the physical sciences; observation, comparison, classification, trial, and generalisation are essential in both cases. Not only are special results, obtained independently of one another, frequently seen to be really included in some generalisation, but branches of the subject which have been developed quite independently of one another are sometimes found to have connections which enable them to be synthesised in one single body of doctrine. The essential nature of mathematical thought manifests itself in the discernment of fundamental identity in the mathematical aspects of what are superficially very different domains. A striking example of this species of immanent identity of mathematical form was exhibited by the discovery of that distinguished mathematician … Major MacMahon, that all possible Latin squares are capable of enumeration by the consideration of certain differential operators. Here we have a case in which an enumeration, which appears to be not amenable to direct treatment, can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators, the calculus of which belongs superficially to a wholly different region of thought from that relating to Latin squares.
In Presidential Address to British Association for the Advancement of Science (1910), Nature, 84, 290.
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The advancement of science is slow; it is effected only by virtue of hard work and perseverance. And when a result is attained, should we not in recognition connect it with the efforts of those who have preceded us, who have struggled and suffered in advance? Is it not truly a duty to recall the difficulties which they vanquished, the thoughts which guided them; and how men of different nations, ideas, positions, and characters, moved solely by the love of science, have bequeathed to us the unsolved problem? Should not the last comer recall the researches of his predecessors while adding in his turn his contribution of intelligence and of labor? Here is an intellectual collaboration consecrated entirely to the search for truth, and which continues from century to century.
[Respecting how the work of prior researchers had enabled his isolation of fluorine.]
Proceedings of the Royal Institution (1897). In Annual Report of the Board of Regents of the Smithsonian Institution to July 1897 (1898), 262.
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The aid which we feel impelled to give to the helpless is mainly an incidental result of the instinct of sympathy, which was originally acquired as part of the social instincts, but subsequently rendered, in the manner previously indicated, more tender and more widely diffused. Nor could we check our sympathy, even at the urging of hard reason, without deterioration in the noblest part of our nature.
…...
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The Big Idea that had been developed in the seventeenth century ... is now known as the scientific method. It says that the way to proceed when investigating how the world works is to first carry out experiments and/or make observations of the natural world. Then, develop hypotheses to explain these observations, and (crucially) use the hypothesis to make predictions about the future outcome of future experiments and/or observations. After comparing the results of those new observations with the predictions of the hypotheses, discard those hypotheses which make false predictions, and retain (at least, for the time being) any hypothesis that makes accurate predictions, elevating it to the status of a theory. Note that a theory can never be proved right. The best that can be said is that it has passed all the tests applied so far.
In The Fellowship: the Story of a Revolution (2005), 275.
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The books of the great scientists are gathering dust on the shelves of learned libraries. ... While the artist's communication is linked forever with its original form, that of the scientist is modified, amplified, fused with the ideas and results of others and melts into the stream of knowledge and ideas which forms our culture. The scientist has in common with the artist only this: that he can find no better retreat from the world than his work and also no stronger link with the world than his work.
From Nobel Lecture (10 Dec 1969), 'A Physicist's Renewed Look at Biology – Twenty Years Later.' in Nobel Lectures, Physiology or Medicine 1963-1970 (1972), 409.
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The carbon output that melts the ice in the Arctic also causes ocean acidification, which results from the ocean absorbing excess carbon dioxide from the atmosphere (the same carbon dioxide that is the primary cause of global warming, hence the nickname “the other carbon problem”).
In 'What do the Arctic, a Thermostat and COP15 Have in Common?', Huffington Post (18 Mar 2010).
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The complexity of contemporary biology has led to an extreme specialization, which has inevitably been followed by a breakdown in communication between disciplines. Partly as a result of this, the members of each specialty tend to feel that their own work is fundamental and that the work of other groups, although sometimes technically ingenious, is trivial or at best only peripheral to an understanding of truly basic problems and issues. There is a familiar resolution to this problem but it is sometimes difficulty to accept emotionally. This is the idea that there are a number of levels of biological integration and that each level offers problems and insights that are unique to it; further, that each level finds its explanations of mechanism in the levels below, and its significances in the levels above it.
From 'Interaction of physiology and behavior under natural conditions', collected in R.I. Bowman (ed.), The Galapagos (1966), 39.
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The difficulty really is psychological and exists in the perpetual torment that results from your saying to yourself, “But how can it be like that?” which is a reflection of uncontrolled but utterly vain desire to see it in terms of something familiar. … If you will simply admit that maybe [Nature] does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possible avoid it, "But how can it be like that?" because you will get 'down the drain', into a blind alley from which nobody has escaped. Nobody knows how it can be like that.
[About wave-particle duality.]
'Probability abd Uncertainty—the Quantum Mechanical View of Nature', the sixth of his Messenger Lectures (1964), Cornell University. Collected in The Character of Physical Law (1967), 129.
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The discovery that these soccer-ball-like molecules can be made in large quantities will have an effect on chemistry like the sowing of a bucket of flower seeds—the results will spring up everywhere from now on. I’d be surprised if we don’t see thousands of new fullerene compounds in the next few years, some of which are almost certain to have important uses.
As quoted in Malcolm W. Browne, 'Bizarre New Class of Molecules Spawns Its Own Branch of Chemistry', New York Times (25 Dec 1990), Late Edition (East Coast), L37.
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The divine tape recorder holds a million scenarios, each perfectly sensible. Little quirks at the outset, occurring for no particular reason, unleash cascades of consequences that make a particular feature seem inevitable in retrospect. But the slightest early nudge contacts a different groove, and history veers into another plausible channel, diverging continually from its original pathway. The end results are so different, the initial perturbation so apparently trivial.
…...
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The Engineer is one who, in the world of physics and applied sciences, begets new things, or adapts old things to new and better uses; above all, one who, in that field, attains new results in the best way and at lowest cost.
From Address on 'Industrial Engineering' at Purdue University (24 Feb 1905). Reprinted by Yale & Towne Mfg Co of New York and Stamford, Conn. for the use of students in its works.
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The experimental verification of a theory concerning any natural phenomenon generally rests on the result of an integration.
Epigraph in Higher Mathematics for Students of Chemistry and Physics: With Special Reference to Practical Work (1902), Chap 4, 150.
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The final result of our researches has widened the class of substances sensitive to light vibrations, until we can propound the fact of such sensitiveness being a general property of all matter.
In Paper (27 Aug 1880) read to the American Association for the Advancement of Science, 'On the Production and Reproduction of Sound by Light'. Printed in The American Journal of Science (Oct 1880), 20, 3rd Ser., No. 118, 305.
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The final results [of work on the theory of relativity] appear almost simple; any intelligent undergraduate can understand them without much trouble. But the years of searching in the dark for a truth that one feels, but cannot express; the intense effort and the alternations of confidence and misgiving, until one breaks through to clarity and understanding, are only known to him who has himself experienced them.
Concluding remark of George Gibson lecture at the University of Glasgow, 'The Origins of the General Theory of Relativity', (20 Jun 1933). Published by Glasgow University as The Origins of the General Theory of Relativity: Being the First Lecture on the George A. Gibson Foundation in the University of Glasgow, Delivered on June 20th, 1933 (1933), 11. Also quoted in 'No Hitching Posts' The Atlantic (1936), 157, 251.
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The first steps in Agriculture, Astronomy, Zoology, (those first steps which the farmer, the hunter, and the sailor take,) teach that nature's dice are always loaded; that in her heaps and rubbish are concealed sure and useful results.
In Nature (1849), 36.
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The fundamental characteristic of the scientific method is honesty. In dealing with any question, science asks no favors. ... I believe that constant use of the scientific method must in the end leave its impress upon him who uses it. ... A life spent in accordance with scientific teachings would be of a high order. It would practically conform to the teachings of the highest types of religion. The motives would be different, but so far as conduct is concerned the results would be practically identical.
Address as its retiring president, to the American Association for the Advancement of Science, St. Louis (28 Dec 1903). 'Scientific Investigation and Progress', Nature 928 Jan 1904), 69:1787, 309.
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The future is not a result of choices among alternative paths offered by the present, but a place that is created-created first in the mind and will, created next in activity. The future is not some place we are going to, but one we are creating. The paths are not to be found, but made, and the activity of making them, changes both the maker and the destination.
…...
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The generalized theory of relativity has furnished still more remarkable results. This considers not only uniform but also accelerated motion. In particular, it is based on the impossibility of distinguishing an acceleration from the gravitation or other force which produces it. Three consequences of the theory may be mentioned of which two have been confirmed while the third is still on trial: (1) It gives a correct explanation of the residual motion of forty-three seconds of arc per century of the perihelion of Mercury. (2) It predicts the deviation which a ray of light from a star should experience on passing near a large gravitating body, the sun, namely, 1".7. On Newton's corpuscular theory this should be only half as great. As a result of the measurements of the photographs of the eclipse of 1921 the number found was much nearer to the prediction of Einstein, and was inversely proportional to the distance from the center of the sun, in further confirmation of the theory. (3) The theory predicts a displacement of the solar spectral lines, and it seems that this prediction is also verified.
Studies in Optics (1927), 160-1.
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The genius of Laplace was a perfect sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results. In truth, in truism if the reader please, Laplace was neither Lagrange nor Euler, as every student is made to feel. The second is power and symmetry, the third power and simplicity; the first is power without either symmetry or simplicity. But, nevertheless, Laplace never attempted investigation of a subject without leaving upon it the marks of difficulties conquered: sometimes clumsily, sometimes indirectly, always without minuteness of design or arrangement of detail; but still, his end is obtained and the difficulty is conquered.
In 'Review of “Théorie Analytique des Probabilites” par M. le Marquis de Laplace, 3eme edition. Paris. 1820', Dublin Review (1837), 2, 348.
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The great art consists in devising décisive experiments, leaving no place to the imagination of the observer. Imagination is needed to give wings to thought at the beginning of experimental investigations on any given subject. When, however, the time has come to conclude, and to interpret the facts derived from observations, imagination must submit to the factual results of the experiments.
Speech (8 Jul 1876), to the French Academy of Medicine. As translated in René J. Dubos, Louis Pasteur, Free Lance of Science (1950, 1986), 376. Date of speech identified in Maurice B. Strauss, Familiar Medical Quotations (1968), 502.
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The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
In History of Mathematics (3rd Ed., 1901), 468.
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The greater part of our daily actions are the result of hidden motives which escape our observation.
From original French, “La plupart de nos actions journalières ne sont que l’effet de mobiles cachés qui nous échappent,” in Psychologie des Foules (1895), 16. English text in The Crowd: A Study of the Popular Mind (1897), Book 1, Chap. 1, 7. [A closer translation could be: “Most of our daily actions are just the effect of hidden motives that we don’t notice.” —Webmaster]
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The habitat of an organism is the place where it lives, or the place where one would go to find it. The ecological niche, on the other hand, is the position or status of an organism within its community and ecosystem resulting from the organism’s structural adaptations, physiological responses and specific behavior (inherited and/or learned). The ecological niche of an organism depends not only on where it lives, but also on what it does. By analogy, it may be said that the habitat is the organism’s ‘address,’ and the niche is its ‘profession,’ biologically speaking.
Fundamentals of Ecology
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The highest mathematical principles may be involved in the production of the simplest mechanical result.
In 'Academical Education', Orations and Speeches on Various Occasions (1870), Vol. 3, 513. This is seen misattributed to Eric Temple Bell, but he only quoted it, attributing it to Everett, in for example, Mathematics: Queen and Servant of Sciences (1938), 20.
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The highest result of education is tolerance.
In Optimism: An Essay (1903), 40.
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The human mechanism is marvelous. But why not—it is the result of three-and-a-half billion years of tinkering.
Epigraph in Isaac Asimov’s Book of Science and Nature Quotations (1988), 12.
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The hybridoma technology was a by-product of basic research. Its success in practical applications is to a large extent the result of unexpected and unpredictable properties of the method. It thus represents another clear-cut example of the enormous practical impact of an investment in research which might not have been considered commercially worthwhile, or of immediate medical relevance. It resulted from esoteric speculations, for curiosity’s sake, only motivated by a desire to understand nature.
From Nobel Lecture (8 Dec 1984), collected in Tore Frängsmyr and Jan Lindsten (eds.), Nobel Lectures in Physiology Or Medicine: 1981-1990 (1993), 267-268.
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The hypothetical character of continual creation has been pointed out, but why is it more of a hypothesis to say that creation is taking place now than that it took place in the past? On the contrary, the hypothesis of continual creation is more fertile in that it answers more questions and yields more results, and results that are, at least in principle, observable. To push the entire question of creation into the past is to restrict science to a discussion of what happened after creation while forbidding it to examine creation itself. This is a counsel of despair to be taken only if everything else fails.
From Cosmology (), 152.
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The idea of a method that contains firm, unchanging, and absolutely binding principles for conducting the business of science meets considerable difficulty when confronted with the results of historical research. We find, then, that there is not a single rule, however plausible, and however firmly grounded in epistemology, that is not violated at some time or another.
Against Method: Outline of an Anarchistic Theory of Knowledge (1975, 1993), 14.
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The idea of making a fault a subject of study and not an object to be merely determined has been the most important step in the course of my methods of observation. If I have obtained some new results it is to this that I owe it.
'Notice sur les Travaux Scientifiques de Marcel Bertrand' (1894). In Geological Society of London, The Quarterly Journal of the Geological Society of London (May 1908), 64, li.
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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 importance of a result is largely relative, is judged differently by different men, and changes with the times and circumstances. It has often happened that great importance has been attached to a problem merely on account of the difficulties which it presented; and indeed if for its solution it has been necessary to invent new methods, noteworthy artifices, etc., the science has gained more perhaps through these than through the final result. In general we may call important all investigations relating to things which in themselves are important; all those which have a large degree of generality, or which unite under a single point of view subjects apparently distinct, simplifying and elucidating them; all those which lead to results that promise to be the source of numerous consequences; etc.
From 'On Some Recent Tendencies in Geometric Investigations', Rivista di Matematica (1891), 44. In Bulletin American Mathematical Society (1904), 444.
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The importance of rice will grow in the coming decades because of potential changes in temperature, precipitation, and sea-level rise, as a result of global warming. Rice grows under a wide range of latitudes and altitudes and can become the anchor of food security in a world confronted with the challenge of climate change.
In 'Science and Shaping the Future of Rice', collected in Pramod K. Aggarwal et al. (eds.), 206 International Rice Congress: Science, Technology, and Trade for Peace and Prosperity (2007), 4.
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The incessant call in this country for practical results and the confounding of mechanical inventions with scientific discoveries has a very prejudicial influence on science. … A single scientific principle may include a thousand applications and is therefore though if not of immediate use of vastly more importance even in a practical view.
Presidential address to the American Association for the Advancement of Science (22 Aug 1850),The Papers of Joseph Henry, Vol. 8, 101-102.
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The inventor and the research man are confused because they both examine results of physical or chemical operations. But they are exact opposites, mirror images of one another. The research man does something and does not care [exactly] what it is that happens, he measures whatever it is. The inventor wants something to happen, but does not care how it happens or what it is that happens if it is not what he wants.
Aphorism listed Frederick Seitz, The Cosmic Inventor: Reginald Aubrey Fessenden (1866-1932) (1999), 54, being Transactions of the American Philosophical Society, Held at Philadelphia For Promoting Useful Knowledge, Vol. 86, Pt. 6.
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The logic of the subject [algebra], which, both educationally and scientifically speaking, is the most important part of it, is wholly neglected. The whole training consists in example grinding. What should have been merely the help to attain the end has become the end itself. The result is that algebra, as we teach it, is neither an art nor a science, but an ill-digested farrago of rules, whose object is the solution of examination problems. … The result, so far as problems worked in examinations go, is, after all, very miserable, as the reiterated complaints of examiners show; the effect on the examinee is a well-known enervation of mind, an almost incurable superficiality, which might be called Problematic Paralysis—a disease which unfits a man to follow an argument extending beyond the length of a printed octavo page.
In Presidential Address British Association for the Advancement of Science (1885), Nature, 32, 447-448.
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The long-range trend toward federal regulation, which found its beginnings in the Interstate Commerce Act of 1887 and the Sherman Act of 1890, which was quickened by a large number of measures in the Progressive era, and which has found its consummation in our time, was thus at first the response of a predominantly individualistic public to the uncontrolled and starkly original collectivism of big business. In America the growth of the national state and its regulative power has never been accepted with complacency by any large part of the middle-class public, which has not relaxed its suspicion of authority, and which even now gives repeated evidence of its intense dislike of statism. In our time this growth has been possible only under the stress of great national emergencies, domestic or military, and even then only in the face of continuous resistance from a substantial part of the public. In the Progressive era it was possible only because of widespread and urgent fear of business consolidation and private business authority. Since it has become common in recent years for ideologists of the extreme right to portray the growth of statism as the result of a sinister conspiracy of collectivists inspired by foreign ideologies, it is perhaps worth emphasizing that the first important steps toward the modern organization of society were taken by arch-individualists—the tycoons of the Gilded Age—and that the primitive beginning of modern statism was largely the work of men who were trying to save what they could of the eminently native Yankee values of individualism and enterprise.
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The love of mathematics is daily on the increase, not only with us but in the army. The result of this was unmistakably apparent in our last campaigns. Bonaparte himself has a mathematical head, and though all who study this science may not become geometricians like Laplace or Lagrange, or heroes like Bonaparte, there is yet left an influence upon the mind which enables them to accomplish more than they could possibly have achieved without this training.
In Letter (26 Jan 1798) to Von Zach. As quoted in translation in Karl Bruhns (ed.), Jane Lassell (trans.) and Caroline Lassell (trans.), Life of Alexander von Humboldt (1872), Vol. 1, 232. [Webmaster assigns this quote to Jérôme Lalande as an informed guess for the following reasons. The cited text gives only the last names, Lalande and von Zach, but it does also give a source footnote to a Allgemeine geographische Ephemeriden, 1, 340. The journal editor, Franz Xaver von Zach, was a Hungarian astronomer. Jérôme Lalande was a French astronomer, living at the same time, who called himself Jérôme Le Français de la Lande. Their names are seen referred to together in the same journal, Vol. 6, 360.]
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The maintenance of security … required every member of the project to attend strictly to his own business. The result was an operation whose efficiency was without precedent.
In And Now It Can Be Told: The Story Of The Manhattan Project (1962), 414.
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