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Who said: “Nature does nothing in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.”
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Proof Quotes (287 quotes)

δος μοι που στω και κινω την γην — Dos moi pou sto kai kino taen gaen (in epigram form, as given by Pappus, classical Greek).
δος μοι πα στω και τα γαν κινάσω — Dos moi pa sto kai tan gan kinaso (Doric Greek).
Give me a place to stand on and I can move the Earth.
About four centuries before Pappas, but about three centuries after Archimedes lived, Plutarch had written of Archimedes' understanding of the lever:
Archimedes, a kinsman and friend of King Hiero, wrote to him that with a given force, it was possible to move any given weight; and emboldened, as it is said, by the strength of the proof, he asserted that, if there were another world and he could go to it, he would move this one.
A commonly-seen expanded variation of the aphorism is:
Give me a lever long enough and a place to stand, and I can move the earth.
As attributed to Pappus (4th century A.D.) and Plutarch (c. 46-120 A.D.), in Sherman K. Stein, Archimedes: What Did He Do Besides Cry Eureka? (1999), 5, where it is also stated that Archimedes knew that ropes and pulley exploit “the principle of the lever, where distance is traded for force.” Eduard Jan Dijksterhuis, in his book, Archimedes (1956), Vol. 12., 15. writes that Hiero invited Archimedes to demonstrate his claim on a ship from the royal fleet, drawn up onto land and there loaded with a large crew and freight, and Archimedes easily succeeded. Thomas Little Heath in The Works of Archimedes (1897), xix-xx, states according to Athenaeus, the mechanical contrivance used was not pulleys as given by Plutarch, but a helix., Heath provides cites for Pappus Synagoge, Book VIII, 1060; Plutarch, Marcellus, 14; and Athenaeus v. 207 a-b. What all this boils down to, in the opinion of the Webmaster, is the last-stated aphorism would seem to be not the actual words of Archimedes (c. 287 – 212 B.C.), but restatements of the principle attributed to him, formed by other writers centuries after his lifetime.
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Strictly Germ-proof

The Antiseptic Baby and the Prophylactic Pup
Were playing in the garden when the Bunny gamboled up;
They looked upon the Creature with a loathing undisguised;—
It wasn't Disinfected and it wasn't Sterilized.

They said it was a Microbe and a Hotbed of Disease;
They steamed it in a vapor of a thousand-odd degrees;
They froze it in a freezer that was cold as Banished Hope
And washed it in permanganate with carbolated soap.

In sulphurated hydrogen they steeped its wiggly ears;
They trimmed its frisky whiskers with a pair of hard-boiled shears;
They donned their rubber mittens and they took it by the hand
And elected it a member of the Fumigated Band.

There's not a Micrococcus in the garden where they play;
They bathe in pure iodoform a dozen times a day;
And each imbibes his rations from a Hygienic Cup—
The Bunny and the Baby and the Prophylactic Pup.
Printed in various magazines and medical journals, for example, The Christian Register (11 Oct 1906), 1148, citing Women's Home Companion. (Making fun of the contemporary national passion for sanitation.)
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Das ist nicht Mathematik, das ist Theologie!
This is not mathematics; this is theology.
[Remark about David Hilbert's first proof of his finite basis theorem.]
Attributed. It does not seem to appear in Gordan’s written work. According to Colin McClarty, in 'Theology and its Discontents: the Origin of the Myth of Modern Mathematics' (2008), “The quote first appeared a quarter of a century after the event, as an unexplained side comment in a eulogy to Gordan by his long-time colleague Max Noether. Noether was a reliable witness, but he says little about what Gordan meant.” See Noether's obituary of Gordan in Mathematische Annalen (1914), 75, 18. It is still debated if the quote is pejorative, complimentary or merely a joke.
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Demonstratio longe optima est experientia.
By far the best proof is experience.
Novum Organum, I., 70. In Thomas Benfield Harbottle, Dictionary of Quotations (Classical) (1906), 42.
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Deviner avant de démontrer! Ai-je besoin de rappeler que c'est ainsi que se sont faites toutes les découvertes importantes.
Guessing before proving! Need I remind you that it is so that all important discoveries have been made?
La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 218.
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Die Gewohnheit einer Meinung erzeugt oft völlige Ueberzeugung von ihrer Richtigkeit, sie verbirgt die schwächeren Theile davon, und macht uns unfähig, die Beweise dagegen anzunehmen.
The habit of an opinion often leads to the complete conviction of its truth, it hides the weaker parts of it, and makes us incapable of accepting the proofs against it.
(1827). German text in Ira Freund, The Study of Chemical Composition (1904), 31. Translated form in Carl Schorlemmer, The Rise and Development of Organic Chemistry (1894), 49.
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La vérité ne diffère de l'erreur qu'en deux points: elle est un peu plus difficile à prouver et beaucoup plus difficile à faire admettre. (Dec 1880)
Truth is different from error in two respects: it is a little harder to prove and more difficult to admit.
In Recueil d'Œuvres de Léo Errera: Botanique Générale (1908), 193. Google translation by Webmaster.
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Newsreader: A huge asteroid could destroy Earth! And by coincidence, that's the subject of tonight's miniseries.
Dogbert: In science, researchers proved that this simple device can keep idiots off your television screen. [TV remote control] Click.
Dilbert cartoon strip (30 Apr 1993).
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Arthur Stanley Eddington quote: Proof is an idol before whom the pure mathematician tortures himself. In physics we are generall
Proof is an idol before whom the pure mathematician tortures himself. In physics we are generally content to sacrifice before the lesser shrine of Plausibility.
In Gifford Lecture (1927), Edinburgh, as collected in 'Science and Mysticism', The Nature of the Physical World (1928), 337.
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Question: State the relations existing between the pressure, temperature, and density of a given gas. How is it proved that when a gas expands its temperature is diminished?
Answer: Now the answer to the first part of this question is, that the square root of the pressure increases, the square root of the density decreases, and the absolute temperature remains about the same; but as to the last part of the question about a gas expanding when its temperature is diminished, I expect I am intended to say I don't believe a word of it, for a bladder in front of a fire expands, but its temperature is not at all diminished.
Genuine student answer* to an Acoustics, Light and Heat paper (1880), Science and Art Department, South Kensington, London, collected by Prof. Oliver Lodge. Quoted in Henry B. Wheatley, Literary Blunders (1893), 175, Question 1. (*From a collection in which Answers are not given verbatim et literatim, and some instances may combine several students' blunders.)
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Question: State what are the conditions favourable for the formation of dew. Describe an instrument for determining the dew point, and the method of using it.
Answer: This is easily proved from question 1. A body of gas as it ascends expands, cools, and deposits moisture; so if you walk up a hill the body of gas inside you expands, gives its heat to you, and deposits its moisture in the form of dew or common sweat. Hence these are the favourable conditions; and moreover it explains why you get warm by ascending a hill, in opposition to the well-known law of the Conservation of Energy.
Genuine student answer* to an Acoustics, Light and Heat paper (1880), Science and Art Department, South Kensington, London, collected by Prof. Oliver Lodge. Quoted in Henry B. Wheatley, Literary Blunders (1893), 179, Question 12. (*From a collection in which Answers are not given verbatim et literatim, and some instances may combine several students' blunders.)
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A demonstrative and convincing proof that an acid does consist of pointed parts is, that not only all acid salts do Crystallize into edges, but all Dissolutions of different things, caused by acid liquors, do assume this figure in their Crystallization; these Crystalls consist of points differing both in length and bigness from one another, and this diversity must be attributed to the keener or blunter edges of the different sorts of acids
A Course of Chymistry (1675), trans. W. Harris (1686), 24.
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A diagram is worth a thousand proofs.
In Mathematics Made Difficult (1971). As quoted in Michael Stueben and ‎Diane Sandford, Twenty Years Before the Blackboard (1998), 131.
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A discussion between Haldane and a friend began to take a predictable turn. The friend said with a sigh, “It’s no use going on. I know what you will say next, and I know what you will do next.” The distinguished scientist promptly sat down on the floor, turned two back somersaults, and returned to his seat. “There,” he said with a smile. “That’s to prove that you’re not always right.”
As quoted in Clifton Fadiman (ed.), André Bernard (ed.), Bartlett's Book of Anecdotes (2000), 253.
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A final proof of our ideas can only be obtained by detailed studies on the alterations produced in the amino acid sequence of a protein by mutations of the type discussed here.
In Nobel Lecture (11 Dec 1962). Collected in Nobel Lectures, Physiology or Medicine 1942-1962 (1964).
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A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.
In A Mathematician’s Apology (1940, 2012), 113.
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A mathematician’s reputation rests on the number of bad proofs he has given.
As quoted in John E. Littlewood, A Mathematician’s Miscellany (1953), 41. [His meaning was that an important result is built upon earlier clumsy work. —Webmaster]
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A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.
In Rota's 'Introduction' written (1980) to preface Philip J. Davis and Reuben Hersh, The Mathematical Experience (1981, 2012), xxii.
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A Miracle is a Violation of the Laws of Nature; and as a firm and unalterable Experience has established these Laws, the Proof against a Miracle, from the very Nature of the Fact, is as entire as any Argument from Experience can possibly be imagined. Why is it more than probable, that all Men must die; that Lead cannot, of itself, remain suspended in the Air; that Fire consumes Wood, and is extinguished by Water; unless it be, that these Events are found agreeable to the Laws of Nature, and there is required a Violation of these Laws, or in other Words, a Miracle to prevent them? Nothing is esteem'd a Miracle, if it ever happen in the common Course of Nature... There must, therefore, be a uniform Experience against every miraculous Event, otherwise the Event would not merit that Appellation. And as a uniform Experience amounts to a Proof, there is here a direct and full Proof, from the Nature of the Fact, against the Existence of any Miracle; nor can such a Proof be destroy'd, or the Miracle render'd credible, but by an opposite Proof, which is superior.
An Enquiry Concerning Human Understanding (1748), 180-181.
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A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.
Unterrichtsblätter für Mathematik und Naturwissenschaften (1932), 38, 177-188. As translated by Abe Shenitzer, in 'Part I. Topology and Abstract Algebra as Two Roads of Mathematical Comprehension', The American Mathematical Monthly (May 1995), 102, No. 7, 453.
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A new theory is guilty until proven innocent, and the pre-existing theory innocent until proven guilty ... Continental drift was guilty until proven innocent.
The Nemesis Affair: A Story of the Death of the Dinosaurs and the Ways of Science (1986), 195-205.
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A system such as classical mechanics may be ‘scientific’ to any degree you like; but those who uphold it dogmatically — believing, perhaps, that it is their business to defend such a successful system against criticism as long as it is not conclusively disproved — are adopting the very reverse of that critical attitude which in my view is the proper one for the scientist.
In The Logic of Scientific Discovery (1959, reprint 2002), 28.
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A theory can be proved by experiment; but no path leads from experiment to the birth of a theory.
As quoted in Antonina Vallentin, Einstein: A Biography (1954), 105. The author, a close friend of Einstein’s family, cites the quote only as “which he has recently made public.”
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A witty statesman said you might prove anything with figures.
In Chartism (1839, 1840), 9.
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Absence of proof is not proof of absence.
Seen, for example, in John Bromham Hawes, Early Pulmonary Tuberculosis: Diagnosis, Prognosis and Treatment (1913), 26.
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Acceptance without proof is the fundamental characteristic of Western religion, rejection without proof is the fundamental characteristic of Western science.
In The dancing Wu Li Masters: an Overview of the New Physics (1979), 88.
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An applied mathematician loves the theorem. A pure mathematician loves the proof.
Anonymous
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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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An irrefutable proof that such single-celled primaeval animals really existed as the direct ancestors of Man, is furnished according to the fundamental law of biogeny by the fact that the human egg is nothing more than a simple cell.
Natürlche Schöpfungsgeschichte, trans. E. R. Lankester, The History of Creation (1892), Vol. 2, 381.
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Analogy cannot serve as proof.
Qin Patrice Debré, Louis Pasteur, trans. Elborg Forster (1994), 260.
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And no one has the right to say that no water-babies exist, till they have seen no water-babies existing; which is quite a different thing, mind, from not seeing water-babies; and a thing which nobody ever did, or perhaps will ever do. But surely [if one were caught] ... they would have put it into spirits, or into the Illustrated News, or perhaps cut it into two halves, poor dear little thing, and sent one to Professor Owen, and one to Professor Huxley, to see what they would each say about it.
The Water-babies (1886), 79-80.
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And now the announcement of Watson and Crick about DNA. This is for me the real proof of the existence of God.
As quoted in 'Playboy Interview: Salvador Dalí, a candid conversation with the flamboyantly eccentric grand vizier of surrealism', Playboy Magazine (Jul 1964). Also in Alan Lindsay Mackay, A Dictionary of Scientific Quotations (1991), 66.
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Another roof, another proof.
[His motto, as an itinerant between mathematical friends’ houses at which he collaborated.]
Quoted by Béla Bollobás, 'The Life and Work of Paul Erdos', in Shiing-Shen Chern and Friedrich Hirzebruch, Wolf Prize in Mathematics (2000), 293.
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Any work of science, no matter what its point of departure, cannot become fully convincing until it crosses the boundary between the theoretical and the experimental: Experimentation must give way to argument, and argument must have recourse to experimentation.
The New Scientific Spirit (1934), trans. A. Goldhammer (1984), 3-4.
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Archimedes possessed so high a spirit, so profound a soul, and such treasures of highly scientific knowledge, that though these inventions [used to defend Syracuse against the Romans] had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of the methods and means of proof, most deserve our admiration.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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As arithmetic and algebra are sciences of great clearness, certainty, and extent, which are immediately conversant about signs, upon the skilful use whereof they entirely depend, so a little attention to them may possibly help us to judge of the progress of the mind in other sciences, which, though differing in nature, design, and object, may yet agree in the general methods of proof and inquiry.
In Alciphron: or the Minute Philosopher, Dialogue 7, collected in The Works of George Berkeley D.D. (1784), Vol. 1, 621.
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As history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.
In A Mathematician’s Apology (1940, 1967), 80.
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As our own species is in the process of proving, one cannot have superior science and inferior morals. The combination is unstable and self-destroying.
Voices From the Sky: Previews of the Coming Space Age (1967), 156.
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As we survey all the evidence, the thought insistently arises that some supernatural agency—or, rather, Agency—must be involved. Is it possible that suddenly, without intending to, we have stumbled upon scientific proof of the existence of a Supreme Being? Was it God who stepped in and so providentially crafted the cosmos for our benefit?
In The Symbiotic Universe: Life and Mind in the Cosmos (1988), 27.
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Attainment is a poor measure of capacity, and ignorance no proof of defect.
From 'The Binet-Simon Scale: Practical Use of the Method', Mental and Scholastic Tests (1921), 1.
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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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Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.
'Mathematical Problems', Bulletin of the American Mathematical Society (Jul 1902), 8, 441.
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Biot, who assisted Laplace in revising it [The Mécanique Céleste] for the press, says that Laplace himself was frequently unable to recover the details in the chain of reasoning, and if satisfied that the conclusions were correct, he was content to insert the constantly recurring formula, “Il est àisé a voir” [it is easy to see].
In History of Mathematics (3rd Ed., 1901), 427.
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But I think that in the repeated and almost entire changes of organic types in the successive formations of the earth—in the absence of mammalia in the older, and their very rare appearance (and then in forms entirely. unknown to us) in the newer secondary groups—in the diffusion of warm-blooded quadrupeds (frequently of unknown genera) through the older tertiary systems—in their great abundance (and frequently of known genera) in the upper portions of the same series—and, lastly, in the recent appearance of man on the surface of the earth (now universally admitted—in one word, from all these facts combined, we have a series of proofs the most emphatic and convincing,—that the existing order of nature is not the last of an uninterrupted succession of mere physical events derived from laws now in daily operation: but on the contrary, that the approach to the present system of things has been gradual, and that there has been a progressive development of organic structure subservient to the purposes of life.
'Address to the Geological Society, delivered on the Evening of the 18th of February 1831', Proceedings of the Geological Society (1834), 1, 305-6.
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But neither thirty years, nor thirty centuries, affect the clearness, or the charm, of Geometrical truths. Such a theorem as “the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the sides” is as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it, and celebrated its advent, it is said, by sacrificing a hecatomb of oxen—a method of doing honour to Science that has always seemed to me slightly exaggerated and uncalled-for. One can imagine oneself, even in these degenerate days, marking the epoch of some brilliant scientific discovery by inviting a convivial friend or two, to join one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It would produce a quite inconvenient supply of beef.
Written without pseudonym as Charles L. Dodgson, in Introduction to A New Theory of Parallels (1888, 1890), xvi. Note: a hecatomb is a great public sacrifice, originally of a hundred oxen.
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But no pursuit at Cambridge was followed with nearly so much eagerness or gave me so much pleasure as collecting beetles. It was the mere passion for collecting, for I did not dissect them, and rarely compared their external characters with published descriptions, but got them named anyhow. I will give a proof of my zeal: one day, on tearing off some old bark, I saw two rare beetles, and seized one in each hand; then I saw a third and new kind, which I could not bear to lose, so that I popped the one which I held in my right hand into my mouth. Alas! it ejected some intensely acrid fluid, which burnt my tongue so that I was forced to spit the beetle out, which was lost, as was the third one.
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), 20.
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Chemistry affords two general methods of determining the constituent principles of bodies, the method of analysis, and that of synthesis. When, for instance, by combining water with alkohol, we form the species of liquor called, in commercial language, brandy or spirit of wine, we certainly have a right to conclude, that brandy, or spirit of wine, is composed of alkohol combined with water. We can produce the same result by the analytical method; and in general it ought to be considered as a principle in chemical science, never to rest satisfied without both these species of proofs. We have this advantage in the analysis of atmospherical air, being able both to decompound it, and to form it a new in the most satisfactory manner.
Elements of Chemistry (1790), trans. R. Kerr, 33.
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Considering that, among all those who up to this time made discoveries in the sciences, it was the mathematicians alone who had been able to arrive at demonstrations—that is to say, at proofs certain and evident—I did not doubt that I should begin with the same truths that they have investigated, although I had looked for no other advantage from them than to accustom my mind to nourish itself upon truths and not to be satisfied with false reasons.
In Discourse upon Method, Part 2, in Henry A. Torrey (ed., trans. )Philosophy of Descartes in Extracts from His Writings , (1892), 47-48.
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Deaths, births, and marriages, considering how much they are separately dependent on the freedom of the human will, should seem to be subject to no law according to which any calculation could be made beforehand of their amount; and yet the yearly registers of these events in great countries prove that they go on with as much conformity to the laws of nature as the oscillations of the weather.
'Idea of a Universal history on a Cosmo-Political Plan' (1784). As translated by Thomas De Quinsey in The London Magazine (Oct 1824), 10, 385. Reprinted in 1859 by De Quincey in Vol. 8 of his Collective Edition of his writings.
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Direct observation of the testimony of the earth … is a matter of the laboratory, of the field naturalist, of indefatigable digging among the ancient archives of the earth’s history. If Mr. Bryan, with an open heart and mind, would drop all his books and all the disputations among the doctors and study first hand the simple archives of Nature, all his doubts would disappear; he would not lose his religion; he would become an evolutionist.
'Evolution and Religion', New York Times (5 Mar 1922), 91. Written in response to an article a few days earlier in which William Jennings Bryan challenged the theory of evolution as lacking proof.
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Don’t despise empiric truth. Lots of things work in practice for which the laboratory has never found proof.
Martin H. Fischer, Howard Fabing (ed.) and Ray Marr (ed.), Fischerisms (1944).
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Einstein, twenty-six years old, only three years away from crude privation, still a patent examiner, published in the Annalen der Physik in 1905 five papers on entirely different subjects. Three of them were among the greatest in the history of physics. One, very simple, gave the quantum explanation of the photoelectric effect—it was this work for which, sixteen years later, he was awarded the Nobel prize. Another dealt with the phenomenon of Brownian motion, the apparently erratic movement of tiny particles suspended in a liquid: Einstein showed that these movements satisfied a clear statistical law. This was like a conjuring trick, easy when explained: before it, decent scientists could still doubt the concrete existence of atoms and molecules: this paper was as near to a direct proof of their concreteness as a theoretician could give. The third paper was the special eory of relativity, which quietly amalgamated space, time, and matter into one fundamental unity. This last paper contains no references and quotes no authority. All of them are written in a style unlike any other theoretical physicist's. They contain very little mathematics. There is a good deal of verbal commentary. The conclusions, the bizarre conclusions, emerge as though with the greatest of ease: the reasoning is unbreakable. It looks as though he had reached the conclusions by pure thought, unaided, without listening to the opinions of others. To a surprisingly large extent, that is precisely what he had done.
Variety of Men (1966), 100-1.
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Euclid and Archimedes are allowed to be knowing, and to have demonstrated what they say: and yet whosoever shall read over their writings without perceiving the connection of their proofs, and seeing what they show, though he may understand all their words, yet he is not the more knowing. He may believe, indeed, but does not know what they say, and so is not advanced one jot in mathematical knowledge by all his reading of those approved mathematicians.
In Conduct of the Understanding, sect. 24.
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Evolution ... is really two theories, the vague theory and the precise theory. The vague theory has been abundantly proved.... The precise theory has never been proved at all. However, like relativity, it is accepted on faith.... On getting down to actual details, difficulties begin.
In Science is a Sacred Cow (1950), 101 & 103.
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Evolution is the conviction that organisms developed their current forms by an extended history of continual transformation, and that ties of genealogy bind all living things into one nexus. Panselectionism is a denial of history, for perfection covers the tracks of time. A perfect wing may have evolved to its current state, but it may have been created just as we find it. We simply cannot tell if perfection be our only evidence. As Darwin himself understood so well, the primary proofs of evolution are oddities and imperfections that must record pathways of historical descent–the panda’s thumb and the flamingo’s smile of my book titles (chosen to illustrate this paramount principle of history).
…...
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Evolution is the law of policies: Darwin said it, Socrates endorsed it, Cuvier proved it and established it for all time in his paper on 'The Survival of the Fittest.' These are illustrious names, this is a mighty doctrine: nothing can ever remove it from its firm base, nothing dissolve it, but evolution.
'Three Thousand Years Among the Microbes', Which Was the Dream? (1967), Chap. 8. In Mark Twain and Brian Collins (ed.), When in Doubt, Tell the Truth: and Other Quotations from Mark Twain (1996), 47.
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Faced with the choice between changing one's mind and proving that there is no need to do so, almost everyone gets busy with the proof.
A Contemporary Guide to Economics, Peace, and Laughter (1971), 50.
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Facts are meaningless. You could use facts to prove anything that's even remotely true! Facts, shmacts.
Spoken by character Homer Simpson in TV episode written by David X. Cohen, 'Lisa the Skeptic', of the animated comedy, The Simpsons (23 Nov 1997). Quoted in Kee Malesky, All Facts Considered: The Essential Library of Inessential Knowledge (2010), 201.
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Faith is a knowledge within the heart, beyond the reach of proof.
In The Kahlil Gibran Reader: Inspirational Writings (2006), 38.
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Faith is different from proof; the latter is human, the former is a Gift from God.
In Pensées (1670), Section 10, 11. As translated in Blaise Pascal and W.F. Trotter (trans.), 'Thoughts', No. 248, collected in Charles W. Eliot (ed.), The Harvard Classics (1910), Vol. 48, 92. From the original French, “La foi est différente de la preuve: l’une est humaine, l’autre est un don de Dieu,” in Ernest Havet (ed.), Pensées de Pascal (1892), 276.
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Few intellectual tyrannies can be more recalcitrant than the truths that everybody knows and nearly no one can defend with any decent data (for who needs proof of anything so obvious). And few intellectual activities can be more salutary than attempts to find out whether these rocks of ages might crumble at the slightest tap of an informational hammer.
…...
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Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case,—which would only indicate some defect in the plan or treatment of the whole,—the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method.
In 'Stücke aus dem Lehrbuche der Arithmetik', Werke, Bd. 2 (1904), 296.
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For those who want some proof that physicists are human, the proof is in the idiocy of all the different units which they use for measuring energy.
In The Character of Physical Law (1967, 2001), 75.
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For we may remark generally of our mathematical researches, that these auxiliary quantities, these long and difficult calculations into which we are often drawn, are almost always proofs that we have not in the beginning considered the objects themselves so thoroughly and directly as their nature requires, since all is abridged and simplified, as soon as we place ourselves in a right point of view.
In Théorie Nouvelle de la Rotation des Corps (1834). As translated by Charles Thomas Whitley in Outlines of a New Theory of Rotatory Motion (1834), 4.
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Formerly one sought the feeling of the grandeur of man by pointing to his divine origin: this has now become a forbidden way, for at its portal stands the ape, together with other gruesome beasts, grinning knowingly as if to say: no further in this direction! One therefore now tries the opposite direction: the way mankind is going shall serve as proof of his grandeur and kinship with God. Alas this, too, is vain! At the end of this way stands the funeral urn of the last man and gravedigger (with the inscription “nihil humani a me alienum puto”). However high mankind may have evolved—and perhaps at the end it will stand even lower than at the beginning!— it cannot pass over into a higher order, as little as the ant and the earwig can at the end of its “earthly course” rise up to kinship with God and eternal life. The becoming drags the has-been along behind it: why should an exception to this eternal spectacle be made on behalf of some little star or for any little species upon it! Away with such sentimentalities!
Daybreak: Thoughts on the Prejudices of Morality (1881), trans. R. J. Hollingdale (1982), 32.
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Geometry enlightens the intellect and sets one’s mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence.
In Ibn Khaldûn, Franz Rosenthal (trans.) and N.J. Dawood (ed.), The Muqaddimah: An Introduction to History (1967, 1969), Vol. 1, 378.
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GRAVITATION, n. The tendency of all bodies to approach one another with a strength proportioned to the quantity of matter they contain—the quantity of matter they contain being ascertained by the strength of their tendency to approach one another. This is a lovely and edifying illustration of how science, having made A the proof of B, makes B the proof of A.
The Collected Works of Ambrose Bierce (1911), Vol. 7, The Devil's Dictionary,  123.
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He was 40 yeares old before he looked on Geometry; which happened accidentally. Being in a Gentleman's Library, Euclid's Elements lay open, and 'twas the 47 El. Libri 1 [Pythagoras' Theorem]. He read the proposition. By G-, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps [and so on] that at last he was demonstratively convinced of that trueth. This made him in love with Geometry .
Of Thomas Hobbes, in 1629.
Brief Lives (1680), edited by Oliver Lawson Dick (1949), 150.
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He [Sylvester] had one remarkable peculiarity. He seldom remembered theorems, propositions, etc., but had always to deduce them when he wished to use them. In this he was the very antithesis of Cayley, who was thoroughly conversant with everything that had been done in every branch of mathematics.
I remember once submitting to Sylvester some investigations that I had been engaged on, and he immediately denied my first statement, saying that such a proposition had never been heard of, let alone proved. To his astonishment, I showed him a paper of his own in which he had proved the proposition; in fact, I believe the object of his paper had been the very proof which was so strange to him.
As quoted by Florian Cajori, in Teaching and History of Mathematics in the United States (1890), 268.
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His [Faraday's] soul was above all littleness and proof to all egotism.
Faraday as a Discoverer (1868), 104.
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I am an organic chemist, albeit one who adheres to the definition of organic chemistry given by the great Swedish chemist Berzelius, namely, the chemistry of substances found in living matter, and my science is one of the more abstruse insofar as it rests on concepts and employs a jargon neither of which is a part of everyday experience. Nevertheless, organic chemistry deals with matters of truly vital Importance and in some of its aspects with which I myself have been particularly concerned it may prove to hold the keys to Life itself.
In 'Synthesis in the Study of Nucleotides', Nobel Lecture, 11 December 1957. In Nobel Lectures: Chemistry 1942-1962 (1964), 522.
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I am now convinced that we have recently become possessed of experimental evidence of the discrete or grained nature of matter, which the atomic hypothesis sought in vain for hundreds and thousands of years. The isolation and counting of gaseous ions, on the one hand, which have crowned with success the long and brilliant researches of J.J. Thomson, and, on the other, agreement of the Brownian movement with the requirements of the kinetic hypothesis, established by many investigators and most conclusively by J. Perrin, justify the most cautious scientist in now speaking of the experimental proof of the atomic nature of matter, The atomic hypothesis is thus raised to the position of a scientifically well-founded theory, and can claim a place in a text-book intended for use as an introduction to the present state of our knowledge of General Chemistry.
In Grundriss der allgemeinen Chemie (4th ed., 1909), Preface, as cited by Erwin N. Hiebert and Hans-Gunther Korber in article on Ostwald in Charles Coulston Gillespie (ed.), Dictionary of Scientific Biography Supplement 1, Vol 15-16, 464.
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I am the most hesitating of men, the most fearful of committing myself when I lack evidence. But on the contrary, no consideration can keep me from defending what I hold as true when I can rely on solid scientific proof.
As quoted in René J. Dubos, Louis Pasteur, Free Lance of Science (1960, 1986), 76.
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I approached the bulk of my schoolwork as a chore rather than an intellectual adventure. The tedium was relieved by a few courses that seem to be qualitatively different. Geometry was the first exciting course I remember. Instead of memorizing facts, we were asked to think in clear, logical steps. Beginning from a few intuitive postulates, far reaching consequences could be derived, and I took immediately to the sport of proving theorems.
Autobiography in Gösta Ekspong (ed.), Nobel Lectures: Physics 1996-2000 (2002), 115.
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I can assure you, reader, that in a very few hours, even during the first day, you will learn more natural philosophy about things contained in this book, than you could learn in fifty years by reading the theories and opinions of the ancient philosophers. Enemies of science will scoff at the astrologers: saying, where is the ladder on which they have climbed to heaven, to know the foundation of the stars? But in this respect I am exempt from such scoffing; for in proving my written reason, I satisfy sight, hearing, and touch: for this reason, defamers will have no power over me: as you will see when you come to see me in my little Academy.
The Admirable Discourses (1580), trans. Aurele La Rocque (1957), 27.
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I can’t prove it, but I’m pretty sure that people gain a selective advantage from believing in things they can’t prove.
In David Stokes, Nicholas Wilson and Martha Mador, Entrepreneurship (2009), 190.
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I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.
Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.
Quoted in interview for PBS TV program Nova. In William Byers, How Mathematicians Think (2007), 1.
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I have always assumed, and I now assume, that he [Robert Oppenheimer] is loyal to the United States. I believe this, and I shall believe it until I see very conclusive proof to the opposite. … [But] I thoroughly disagreed with him in numerous issues and his actions frankly appeared to me confused and complicated. To this extent I feel that I would like to see the vital interests of this country in hands which I understand better, and therefore trust more.
After Teller paid tribute to Oppenheimer’s talents, especially his “very outstanding achievement” as the wartime organizer and director of Los Alamos, Teller continued his testimony to the Gray board hearings (28 Apr 1954) in the Atomic Energy Commission building, “In the Matter of J. Robert Oppenheimer.” The subject quotes were excerpted from Teller’s answers to their questions. As given in Robert Coughlan, 'Dr. Edward Teller’s Magnificent Obsession', Life (6 Sep 1954), 72-74.
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I have had my results for a long time: but I do not yet know how I am to arrive at them.
Quoted in A. Arber, The Mind and the Eye (1954).
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I have tried to avoid long numerical computations, thereby following Riemann’s postulate that proofs should be given through ideas and not voluminous computations.
In Report on Number Theory (1897). As given in epigraph, without citation, in Eberhard Zeidler and Juergen Quandt (trans.), Nonlinear Functional Analysis and its Applications: IV: Applications to Mathematical Physics (2013), 448.
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I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.
Quoted in G. Simmons, Calculus Gems (1992).
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I propose to provide proof... that just as always an alcoholic ferment, the yeast of beer, is found where sugar is converted into alcohol and carbonic acid, so always a special ferment, a lactic yeast, is found where sugar is transformed into lactic acid. And, furthermore, when any plastic nitrogenated substance is able to transform sugar into that acid, the reason is that it is a suitable nutrient for the growth of the [lactic] ferment.
Comptes Rendus (1857), 45, 913.
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I read in the proof sheets of Hardy on Ramanujan: “As someone said, each of the positive integers was one of his personal friends.” My reaction was, “I wonder who said that; I wish I had.” In the next proof-sheets I read (what now stands), “It was Littlewood who said…”. What had happened was that Hardy had received the remark in silence and with poker face, and I wrote it off as a dud.
In Béla Bollobás (ed.), Littlewood’s Miscellany, (1986), 61.
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I stand almost with the others. They believe the world was made for man, I believe it likely that it was made for man; they think there is proof, astronomical mainly, that it was made for man, I think there is evidence only, not proof, that it was made for him. It is too early, yet, to arrange the verdict, the returns are not all in. When they are all in, I think that they will show that the world was made for man; but we must not hurry, we must patiently wait till they are all in.
Attributed.
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I think it’s time we recognized the Dark Ages are over. Galileo and Copernicus have been proven right. The world is in fact round; the Earth does revolve around the sun. I believe God gave us intellect to differentiate between imprisoning dogma and sound ethical science, which is what we must do here today.
Debating federal funding for stem cell research as Republican Representative (CT).
In Eve Herold, George Daley, Stem Cell Wars (2007), 57.
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I was inspired by the remarks in those books; not by the parts in which everything was proved and demonstrated [but by] the remarks about the fact that this doesn’t make any sense. … So I had this as a challenge and an inspiration.
In his Nobel Prize Lecture (11 Dec 1965), 'The Development of the Space-Time View of Quantum Electrodynamics'. Collected in Stig Lundqvist, Nobel Lectures: Physics, 1963-1970 (1998), 156.
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I was pretty good in science. But again, because of the small budget, in science class we couldn’t do experiments in order to prove theories. We just believed everything. Actually I think that class was call Religion. Religion was always an easy class. All you had to do was suspend the logic and reasoning you were taught in all the other classes.
In autobiography, Brain Droppings (1998), 227.
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I've come loaded with statistics, for I've noticed that a man can't prove anything without statistics. No man can.
Speech at the Republican Rally, Hartford Opera House (26 Oct 1880). In Mark Twain and Paul Fatout (ed.,) Mark Twain Speaking (2006), 140.
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If all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime.
In When You Were a Tadpole and I Was a Fish: And Other Speculations About This and That (), 124.
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If in a given community unchecked popular rule means unlimited waste and destruction of the natural resources—soil, fertility, waterpower, forests, game, wild-life generally—which by right belong as much to subsequent generations as to the present generation, then it is sure proof that the present generation is not yet really fit for self-control, that it is not yet really fit to exercise the high and responsible privilege of a rule which shall be both by the people and for the people. The term “for the people” must always include the people unborn as well as the people now alive, or the democratic ideal is not realized.
In A Book-Lover's Holidays in the Open (1916), 319.
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If men of science owe anything to us, we may learn much from them that is essential. For they can show how to test proof, how to secure fulness and soundness in induction, how to restrain and to employ with safety hypothesis and analogy.
Lecture, 'The Study of History' (11 Jun 1895) delivered at Cambridge, published as A Lecture on The Study of History (1895), 54.
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If one proves the equality of two numbers a and b by showing first that “a is less than or equal to b” and then “a is greater than or equal to b”, it is unfair, one should instead show that they are really equal by disclosing the inner ground for their equality.
As quoted, without citation, in biography by Hermann Wehl, Emmy Noether (1935), 18.
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If the finding of Coines, Medals, Urnes, and other Monuments of famous Persons, or Towns, or Utensils, be admitted for unquestionable Proofs, that such Persons or things have, in former Times, had a being, certainly those Petrifactions may be allowed to be of equal Validity and Evidence, that there have been formerly such Vegetables or Animals. These are truly Authentick Antiquity not to be counterfeited, the Stamps, and Impressions, and Characters of Nature that are beyond the Reach and Power of Humane Wit and Invention, and are true universal Characters legible to all rational Men.
Lectures and Discourses of Earthquakes (1668). In The Posthumous Works of Robert Hooke, containing his Cutlerian Lectures and other Discourses read at the Meetings of the Illustrious Royal Society (1705), 449.
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If the love of surgery is a proof of a person’s being adapted for it, then certainly I am fitted to he a surgeon; for thou can’st hardly conceive what a high degree of enjoyment I am from day to day experiencing in this bloody and butchering department of the healing art. I am more and more delighted with my profession.
Letter to his father (1853). In John Vaughan, 'Lord Lister', The Living Age (1918), 297, 361. Reprinted from The Fortnightly Review (1918), 109, 417- .
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If the matter is one that can be settled by observation, make the observation yourself. Aristotle could have avoided the mistake of thinking that women have fewer teeth than men, by the simple device of asking Mrs. Aristotle to keep her mouth open while he counted.
In An Outline of Intellectual Rubbish (1943), 22.
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If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book … it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.
Mathematical Discovery: on Understanding, Learning, and Teaching Problem Solving (1981), 129.
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If there is any kind of animal which is female and has no male separate from it, it is possible that this may generate a young one from itself. No instance of this worthy of any credit has been observed up to the present at any rate, but one case in the class of fishes makes us hesitate. No male of the so-called erythrinus has ever yet been seen, but females, and specimens full of roe, have been seen. Of this, however, we have as yet no proof worthy of credit.
Aristotle
Generation of Animals, 741a, 32-8. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. I, 1150.
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If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.
In How to Solve It: A New Aspect of Mathematical Method (2004), 15.
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In describing a protein it is now common to distinguish the primary, secondary and tertiary structures. The primary structure is simply the order, or sequence, of the amino-acid residues along the polypeptide chains. This was first determined by Sanger using chemical techniques for the protein insulin, and has since been elucidated for a number of peptides and, in part, for one or two other small proteins. The secondary structure is the type of folding, coiling or puckering adopted by the polypeptide chain: the a-helix structure and the pleated sheet are examples. Secondary structure has been assigned in broad outline to a number of librous proteins such as silk, keratin and collagen; but we are ignorant of the nature of the secondary structure of any globular protein. True, there is suggestive evidence, though as yet no proof, that a-helices occur in globular proteins, to an extent which is difficult to gauge quantitatively in any particular case. The tertiary structure is the way in which the folded or coiled polypeptide chains are disposed to form the protein molecule as a three-dimensional object, in space. The chemical and physical properties of a protein cannot be fully interpreted until all three levels of structure are understood, for these properties depend on the spatial relationships between the amino-acids, and these in turn depend on the tertiary and secondary structures as much as on the primary. Only X-ray diffraction methods seem capable, even in principle, of unravelling the tertiary and secondary structures.
Co-author with G. Bodo, H. M. Dintzis, R. G. Parrish, H. Wyckoff, and D. C. Phillips
'A Three-Dimensional Model of the Myoglobin Molecule Obtained by X-ray Analysis', Nature (1958) 181, 662.
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In Euclid each proposition stands by itself; its connection with others is never indicated; the leading ideas contained in its proof are not stated; general principles do not exist. In modern methods, on the other hand, the greatest importance is attached to the leading thoughts which pervade the whole; and general principles, which bring whole groups of theorems under one aspect, are given rather than separate propositions. The whole tendency is toward generalization. A straight line is considered as given in its entirety, extending both ways to infinity, while Euclid is very careful never to admit anything but finite quantities. The treatment of the infinite is in fact another fundamental difference between the two methods. Euclid avoids it, in modern mathematics it is systematically introduced, for only thus is generality obtained.
In 'Geometry', Encyclopedia Britannica (9th edition).
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In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.
In 'Mathematical Games', Scientific American (Oct 1973), 229, 114.
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In my opinion, there is absolutely no trustworthy proof that talents have been improved by their exercise through the course of a long series of generations. The Bach family shows that musical talent, and the Bernoulli family that mathematical power, can be transmitted from generation to generation, but this teaches us nothing as to the origin of such talents. In both families the high-watermark of talent lies, not at the end of the series of generations, as it should do if the results of practice are transmitted, but in the middle. Again, talents frequently appear in some member of a family which has not been previously distinguished.
In 'On Heredity', Essays upon Heredity and Kindred Biological Problems (1889), Vol. 1, 95-96.
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In order that an inventory of plants may be begun and a classification of them correctly established, we must try to discover criteria of some sort for distinguishing what are called “species”. After a long and considerable investigation, no surer criterion for determining species had occurred to me than distinguishing features that perpetuate themselves in propagation from seed. Thus, no matter what variations occur in the individuals or the species, if they spring from the seed of one and the same plant, they are accidental variations and not such as to distinguish a species. For these variations do not perpetuate themselves in subsequent seeding. Thus, for example, we do not regard caryophylli with full or multiple blossoms as a species distinct from caryophylli with single blossoms, because the former owe their origin to the seed of the latter and if the former are sown from their own seed, they once more produce single-blossom caryophylli. But variations that never have as their source seed from one and the same species may finally be regarded as distinct species. Or, if you make a comparison between any two plants, plants which never spring from each other's seed and never, when their seed is sown, are transmuted one into the other, these plants finally are distinct species. For it is just as in animals: a difference in sex is not enough to prove a difference of species, because each sex is derived from the same seed as far as species is concerned and not infrequently from the same parents; no matter how many and how striking may be the accidental differences between them; no other proof that bull and cow, man and woman belong to the same species is required than the fact that both very frequently spring from the same parents or the same mother. Likewise in the case of plants, there is no surer index of identity of species than that of origin from the seed of one and the same plant, whether it is a matter of individuals or species. For animals that differ in species preserve their distinct species permanently; one species never springs from the seed of another nor vice versa.
John Ray
Historia Plantarum (1686), Vol. 1, 40. Trans. Edmund Silk. Quoted in Barbara G. Beddall, 'Historical Notes on Avian Classification', Systematic Zoology (1957), 6, 133-4.
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In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics,—provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary.
In 'Die Theorie der algebraischen Zahlkorper', Vorwort, Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 4.
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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 the expressions we adopt to prescribe physical phenomena we necessarily hover between two extremes. We either have to choose a word which implies more than we can prove, or we have to use vague and general terms which hide the essential point, instead of bringing it out. The history of electrical theories furnishes a good example.
Opening Address to the Annual Meeting of the British Association by Prof. Arthur Schuster, in Nature (4 Aug 1892), 46, 325.
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In the vestibule of the Manchester Town Hall are placed two life-sized marble statues facing each other. One of these is that of John Dalton … the other that of James Prescott Joule. … Thus honour is done to Manchester’s two greatest sons—to Dalton, the founder of modern Chemistry and of the Atomic Theory, and the laws of chemical-combining proportions; to Joule, the founder of modern Physics and the discoverer of the Law of Conservation of Energy. The one gave to the world the final and satisfactory proof … that in every kind of chemical change no loss of matter occurs; the other proved that in all the varied modes of physical change, no loss of energy takes place.
In John Dalton and the Rise of Modern Chemistry (1895), 7.
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In want of other proofs, the thumb would convince me of the existence of a God.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 42
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In war, science has proven itself an evil genius; it has made war more terrible than it ever was before. Man used to be content to slaughter his fellowmen on a single plane—the earth’s surface. Science has taught him to go down into the water and shoot up from below and to go up into the clouds and shoot down from above, thus making the battlefield three times as bloody as it was before; but science does not teach brotherly love. Science has made war so hellish that civilization was about to commit suicide; and now we are told that newly discovered instruments of destruction will make the cruelties of the late war seem trivial in comparison with the cruelties of wars that may come in the future.
Proposed summation written for the Scopes Monkey Trial (1925), in Genevieve Forbes Herrick and John Origen Herrick, The Life of William Jennings Bryan (1925), 405. This speech was prepared for delivery at the trial, but was never heard there, as both sides mutually agreed to forego arguments to the jury.
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It appears, then, to be a condition of a genuinely scientific hypothesis, that it be not destined always to remain an hypothesis, but be certain to be either proved or disproved by.. .comparison with observed facts.
A System of Logic, Ratiocinative and Inductive (1858), 293.
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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 has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness.
In An Examination of Sir William Hamilton’s Philosophy (1878), 611.
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It is by logic that we prove, but by intuition that we discover.
In Science and Method (1908) translated by Francis Maitland (1914, 2007), 129.
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It is characteristic of science that the full explanations are often seized in their essence by the percipient scientist long in advance of any possible proof.
The Origin of Life, 1967
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It is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician demonstrative proofs.
Aristotle
Nicomachean Ethics, 1094b, 25-7. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. 2, 1730.
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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 the facts that matter, not the proofs. Physics can progress without the proofs, but we can’t go on without the facts … if the facts are right, then the proofs are a matter of playing around with the algebra correctly.
Feynman Lectures on Gravitation, edited by Brian Hatfield (2002), 137.
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It is wrong always, everywhere, and for anyone to believe anything on insufficient evidence.
In 'The Ethics of Belief', Contemporary Review (Jan 1877), collected in Leslie Stephen and Frederick Pollock (eds.), Lectures and Essays: By the Late William Kingdon Clifford, F.R.S. (1886), 346.
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It must be conceded that a theory has an important advantage if its basic concepts and fundamental hypotheses are 'close to experience,' and greater confidence in such a theory is certainly justified. There is less danger of going completely astray, particularly since it takes so much less time and effort to disprove such theories by experience. Yet more and more, as the depth of our knowledge increases, we must give up this advantage in our quest for logical simplicity in the foundations of physical theory...
'On the Generalized Theory of Gravitation', Scientific American (Apr 1950), 13. In David H. Levy (Ed.), The Scientific American Book of the Cosmos (2000), 19.
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It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.
Grundgesetz der Arithmetik(1893), Vol. 2, Section 60, In P. Greach and M. Black (eds., Translations from the Philosophical Writings of Gottlob Frege (1952), 144.
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It reveals to me the causes of many natural phenomena that are entirely incomprehensible in the light of the generally accepted hypotheses. To refute the latter I collected many proofs, but I do not publish them ... I would dare to publish my speculations if there were people men like you.
[Declaring his belief in the heliocentric theory of Copernicus.]
Letter to Kepler (1596). Quoted in Will Durant, Ariel Duran, The Age of Reason Begins (1961), 603. From Hermann Kesten, Copernicus and His World, translated by E.B. Ashton (pseud.) and Norbert Guterman (1945), 348-349.
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It was a dark and stormy night, so R. H. Bing volunteered to drive some stranded mathematicians from the fogged-in Madison airport to Chicago. Freezing rain pelted the windscreen and iced the roadway as Bing drove on—concentrating deeply on the mathematical theorem he was explaining. Soon the windshield was fogged from the energetic explanation. The passengers too had beaded brows, but their sweat arose from fear. As the mathematical description got brighter, the visibility got dimmer. Finally, the conferees felt a trace of hope for their survival when Bing reached forward—apparently to wipe off the moisture from the windshield. Their hope turned to horror when, instead, Bing drew a figure with his finger on the foggy pane and continued his proof—embellishing the illustration with arrows and helpful labels as needed for the demonstration.
In 'R. H. Bing', Biographical Memoirs: National Academy of Sciences (2002), 49. Anecdote based on the recollections of Bing's colleagues, Steve Armentrout and C. E. Burgess. The narrative was given in a memorial tribute at the University of Texas at Austin.
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It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, “Yes, it is true, but you won’t be able to understand the proof.”
As quoted in John Horgan, 'The Death of Proof', Scientific American (Oct 1993), 269, No. 4, 103.
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Just as, in civil History, one consults title-deeds, one studies coins, one deciphers ancient inscriptions, in order to determine the epochs of human revolutions and to fix the dates of moral [i.e. human] events; so, in Natural History, one must excavate the archives of the world, recover ancient monuments from the depths of the earth, collect their remains, and assemble in one body of proofs all the evidence of physical changes that enable us to reach back to the different ages of Nature. This, then, is the order of the times indicated by facts and monuments: these are six epochs in the succession of the first ages of Nature; six spaces of duration, the limits of which although indeterminate are not less real; for these epochs are not like those of civil History ... that we can count and measure exactly; nevertheless we can compare them with each other and estimate their relative duration.
'Des Époques de la Nature', Histoire Naturelle, Générale et Particulière contenant les Époques de la Nature (1778), Supplement Vol. 9, 1-2, 41. Trans. Martin J. Rudwick.
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Let us dismiss the question, “Have you proven that your model is valid?” with a quick NO. Then let us take up the more rewarding and far more challenging question: “Have you proven that your model is useful for learning more… ” [Co-author]
In J.B. Mankin, R.V. O’Neill, H.H. Shugart and B.W. Rust, 'The Importance of Validation in Ecosystem Analysis', in: G.S. Innis (ed.), New Directions in the Analysis of Ecological Systems (1977), Part I, 69.
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Let us now recapitulate all that has been said, and let us conclude that by hermetically sealing the vials, one is not always sure to prevent the birth of the animals in the infusions, boiled or done at room temperature, if the air inside has not felt the ravages of fire. If, on the contrary, this air has been powerfully heated, it will never allow the animals to be born, unless new air penetrates from outside into the vials. This means that it is indispensable for the production of the animals that they be provided with air which has not felt the action of fire. And as it would not be easy to prove that there were no tiny eggs disseminated and floating in the volume of air that the vials contain, it seems to me that suspicion regarding these eggs continues, and that trial by fire has not entirely done away with fears of their existence in the infusions. The partisans of the theory of ovaries will always have these fears and will not easily suffer anyone's undertaking to demolish them.
Nouvelles Recherches sur les Découvertes Microscopiques, et la Génération des Corps Organisés (1769), 134-5. Quoted in Jacques Roger, The Life Sciences in Eighteenth-Century French Thought, ed. Keith R. Benson and trans. Robert Ellrich (1997), 510-1.
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Liebig was not a teacher in the ordinary sense of the word. Scientifically productive himself in an unusual degree, and rich in chemical ideas, he imparted the latter to his advanced pupils, to be put by them to experimental proof; he thus brought his pupils gradually to think for themselves, besides showing and explaining to them the methods by which chemical problems might be solved experimentally.
As quoted in G. H. Getman, The Life of Ira Remsen (1980), 18-19.
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Littlewood, on Hardy’s own estimate, is the finest mathematician he has ever known. He was the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power.
(1943). In Béla Bollobás, Littlewood's Miscellany (1986), Foreward, 22.
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Logic does not pretend to teach the surgeon what are the symptoms which indicate a violent death. This he must learn from his own experience and observation, or from that of others, his predecessors in his peculiar science. But logic sits in judgment on the sufficiency of that observation and experience to justify his rules, and on the sufficiency of his rules to justify his conduct. It does not give him proofs, but teaches him what makes them proofs, and how he is to judge of them.
In A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation (1843), Vol. 1, 11.
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Long may Louis de Broglie continue to inspire those who suspect that what is proved by impossibility proofs is lack of imagination.
In 'On the Impossible Pilot Wave', Foundations of Physics (Oct 1982), 12, No. 10, 989-999. Reprinted in Speakable and Unspeakable in Quantum Mechanics (1987), 167.
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Lord Kelvin was so satisfied with this triumph of science that he declared himself to be as certain of the existence of the ether as a man can be about anything.... “When you can measure what you are speaking about, and express it in numbers, you know something about it....” Thus did Lord Kelvin lay down the law. And though quite wrong, this time he has the support of official modern Science. It is NOT true that when you can measure what you are speaking about, you know something about it. The fact that you can measure something doesn't even prove that that something exists.... Take the ether, for example: didn't they measure the ratio of its elasticity to its density?
In Science is a Sacred Cow (1950), 69-70; 85.
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Man has been here 32,000 years. That it took a hundred million years to prepare the world for him is proof that that is what it was done for. I suppose it is, I dunno. If The Eiffel Tower were now to represent the world's age, the skin of paint on the pinnacle knob at its summit would represent man’s share of that age; and anybody would perceive that the skin was what the tower was built for. I reckon they would, I dunno.
Declaiming Alfred Russel Wallace's 'anthropocentric' theory, that the universe was created specifically for the evolution of mankind. From 'Was the World Made for Man?' (1903) collected in What is Man?: and Other Philosophical Writings (1973), 106. Twain used the age of the earth accepted in his time; it is now estimated as 4,500 million years. Man’s origin is now estimated as 250,000 years.
For the complete essay, see Was The World Made For Man?.
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Man is the Reasoning Animal. Such is the claim. I think it is open to dispute. Indeed, my experiments have proven to me that he is the Unreasoning Animal. Note his history, as sketched above. It seems plain to me that whatever he is he is not a reasoning animal. His record is the fantastic record of a maniac. I consider that the strongest count against his intelligence is the fact that with that record back of him he blandly sets himself up as the head animal of the lot: whereas by his own standards he is the bottom one.
In truth, man is incurably foolish. Simple things which the other animals easily learn, he is incapable of learning. Among my experiments was this. In an hour I taught a cat and a dog to be friends. I put them in a cage. In another hour I taught them to be friends with a rabbit. In the course of two days I was able to add a fox, a goose, a squirrel and some doves. Finally a monkey. They lived together in peace; even affectionately.
Next, in another cage I confined an Irish Catholic from Tipperary, and as soon as he seemed tame I added a Scotch Presbyterian from Aberdeen. Next a Turk from Constantinople; a Greek Christian from Crete; an Armenian; a Methodist from the wilds of Arkansas; a Buddhist from China; a Brahman from Benares. Finally, a Salvation Army Colonel from Wapping. Then I stayed away two whole days. When I came back to note results, the cage of Higher Animals was all right, but in the other there was but a chaos of gory odds and ends of turbans and fezzes and plaids and bones and flesh—not a specimen left alive. These Reasoning Animals had disagreed on a theological detail and carried the matter to a Higher Court.
In Letters from the Earth: Uncensored Writings (),
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Many 'hard' scientists regard the term 'social science' as an oxymoron. Science means hypotheses you can test, and prove or disprove. Social science is little more than observation putting on airs.
'A Cuba Policy That's Stuck On Plan A', opinion column, The Washington Post (17 Apr 2009)
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Many inventions are not suitable for the people at large because of their carelessness. Before a thing can be marketed to the masses, it must be made practically fool-proof. Its operation must be made extremely simple. That is one reason, I think, why the phonograph has been so universally adopted. Even a child can operate it. … Another reason is that people are far more willing to pay for being amused than for anything else.
As quoted from an interview by B.C. Forbes in The American Magazine (Jan 1921), 86.
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Mathematical proofs are essentially of three different types: pre-formal; formal; post-formal. Roughly the first and third prove something about that sometimes clear and empirical, sometimes vague and ‘quasi-empirical’ stuff, which is the real though rather evasive subject of mathematics.
In Mathematics, Science and Epistemology (1980), Vol. 2, 69.
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Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning.
In 'Mr Locke’s Reply to the Bishop of Worcester’s Answer to his Second Letter', collected in The Works of John Locke (1824), Vol. 3, 428.
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Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science.
In An Introduction to Mathematics (1911), 15.
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Mathematics had never had more than a secondary interest for him [her husband, George Boole]; and even logic he cared for chiefly as a means of clearing the ground of doctrines imagined to be proved, by showing that the evidence on which they were supposed to give rest had no tendency to prove them. But he had been endeavoring to give a more active and positive help than this to the cause of what he deemed pure religion.
In Eleanor Meredith Cobham, Mary Everest Boole: Collected Works (1931), 40.
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Mathematics in gross, it is plain, are a grievance in natural philosophy, and with reason…Mathematical proofs are out of the reach of topical arguments, and are not to be attacked by the equivocal use of words or declamation, that make so great a part of other discourses; nay, even of controversies.
In 'Mr Locke’s Reply to the Bishop of Worcester’s Answer to his Second Letter', collected in The Works of John Locke (1824), Vol. 3, 428.
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Mathematics is not a deductive science—that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experiment and guesswork.
In I Want to be a Mathematician: an Automathography in Three Parts (1985), 321.
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Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional.
(1923). Quoted, without source, in E.T. Bell, Men of Mathematics (1937), Vol. 1, li.
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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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Mssr. Fermat—what have you done?
Your simple conjecture has everyone
Churning out proofs,
Which are nothing but goofs!
Could it be that your statement’s an erudite spoof?
A marginal hoax
That you’ve played on us folks?
But then you’re really not known for your practical jokes.
Or is it then true
That you knew what to do
When n was an integer greater than two?
Oh then why can’t we find
That same proof…are we blind?
You must be reproved, for I’m losing my mind.
In 'Fermat's Last Theorem', Mathematics Magazine (Apr 1986), 59, No. 2, 76.
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My Design in this Book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments: In order to which, I shall premise the following Definitions and Axioms.
Opticks (1704), Book 1, Part 1, Introduction, 1.
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My experiences with science led me to God. They challenge science to prove the existence of God. But must we really light a candle to see the sun?
In letter to California State board of Education (14 Sep 1972).
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My profession often gets bad press for a variety of sins, both actual and imagined: arrogance, venality, insensitivity to moral issues about the use of knowledge, pandering to sources of funding with insufficient worry about attendant degradation of values. As an advocate for science, I plead ‘mildly guilty now and then’ to all these charges. Scientists are human beings subject to all the foibles and temptations of ordinary life. Some of us are moral rocks; others are reeds. I like to think (though I have no proof) that we are better, on average, than members of many other callings on a variety of issues central to the practice of good science: willingness to alter received opinion in the face of uncomfortable data, dedication to discovering and publicizing our best and most honest account of nature’s factuality, judgment of colleagues on the might of their ideas rather than the power of their positions.
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Never leave an unsolved difficulty behind. I mean, don’t go any further in that book till the difficulty is conquered. In this point, Mathematics differs entirely from most other subjects. Suppose you are reading an Italian book, and come to a hopelessly obscure sentence—don’t waste too much time on it, skip it, and go on; you will do very well without it. But if you skip a mathematical difficulty, it is sure to crop up again: you will find some other proof depending on it, and you will only get deeper and deeper into the mud.
From letter to Edith Rix with hints for studying (about Mar 1885), in Stuart Dodgson Collingwood, The Life and Letters of Lewis Carroll (1898), 241.
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No man who has not a decently skeptical mind can claim to be civilized. Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. Then, in the alleged proof, be alert for inexplicit assumptions. Euclid’s notorious oversights drove this lesson home. Thanks to him, I am (I hope!) immune to all propaganda, including that of mathematics itself.
In 'What Mathematics Has Meant to Me', Mathematics Magazine (Jan-Feb 1951), 24, 161.
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No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress. It is also a first step in our attempt to find the principles implicit in familiar observational notions.
Against Method: Outline of an Anarchistic Theory of Knowledge (1975, 1993), 39. This is his summary at the head of Chapter 5.
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Non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be farther from the truth. Rather, there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.
In 'Remark on Non-standard Analysis' (1974), in S. Feferman (ed.), Kurt Gödel Collected Works: Publications 1938-1974 (1990), Vol. 2, 311.
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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 has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, and are still granted in these latter times by the Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity.
Martha Somerville (ed.) Personal Recollections, from Early Life to Old Age, of Mary Somerville (1874), 140-141.
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Of Science generally we can remark, first, that it is the most perfect embodiment of Truth, and of the ways of getting at Truth. More than anything else does it impress the mind with the nature of Evidence, with the labour and precautions necessary to prove a thing. It is the grand corrective of the laxness of the natural man in receiving unaccredited facts and conclusions. It exemplifies the devices for establishing a fact, or a law, under every variety of circumstances; it saps the credit of everything that is affirmed without being properly attested.
In Education as a Science (1879), 147-148.
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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 aim of physical sciences had been to give an exact picture the material world. One achievement of physics in the twentieth century has been to prove that that aim is unattainable.
From The Ascent of Man (1973), 353.
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One never finds fossil bones bearing no resemblance to human bones. Egyptian mummies, which are at least three thousand years old, show that men were the same then. The same applies to other mummified animals such as cats, dogs, crocodiles, falcons, vultures, oxen, ibises, etc. Species, therefore, do not change by degrees, but emerged after the new world was formed. Nor do we find intermediate species between those of the earlier world and those of today's. For example, there is no intermediate bear between our bear and the very different cave bear. To our knowledge, no spontaneous generation occurs in the present-day world. All organized beings owe their life to their fathers. Thus all records corroborate the globe's modernity. Negative proof: the barbaritY of the human species four thousand years ago. Positive proof: the great revolutions and the floods preserved in the traditions of all peoples.
'Note prese al Corso di Cuvier. Corso di Geologia all'Ateneo nel 1805', quoted in Pietro Corsi, The Age of Lamarck, trans. J. Mandelbaum (1988), 183.
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Richard P. Feynman quote: One of the ways of stopping science would be only to do experiments in the region where you know the l
One of the ways of stopping science would be only to do experiments in the region where you know the law. … In other words we are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress.
In The Character of Physical Law (1965, 2001), 158.
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Our duty is to believe that for which we have sufficient evidence, and to suspend our judgment when we have not.
The Use of Life (1895), 225.
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People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs.
In 'On Proof and Progress in Mathematics', For the Learning of Mathematics (Feb 1995), 15, No. 1, 33. Reprinted from Bulletin of the American Mathematical Society (1994), 30, No. 2, 170.
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Physical science comes nearest to that complete system of exact knowledge which all sciences have before them as an ideal. Some fall far short of it. The physicist who inveighs against the lack of coherence and the indefiniteness of theological theories, will probably speak not much less harshly of the theories of biology and psychology. They also fail to come up to his standard of methodology. On the other side of him stands an even superior being—the pure mathematician—who has no high opinion of the methods of deduction used in physics, and does not hide his disapproval of the laxity of what is accepted as proof in physical science. And yet somehow knowledge grows in all these branches. Wherever a way opens we are impelled to seek by the only methods that can be devised for that particular opening, not over-rating the security of our finding, but conscious that in this activity of mind we are obeying the light that is in our nature.
Swarthmore Lecture (1929) at Friends’ House, London, printed in Science and the Unseen World (1929), 77-78.
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Physics is geometric proof on steroids.
SA Sachs
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Physics is NOT a body of indisputable and immutable Truth; it is a body of well-supported probable opinion only .... Physics can never prove things the way things are proved in mathematics, by eliminating ALL of the alternative possibilities. It is not possible to say what the alternative possibilities are.... Write down a number of 20 figures; if you multiply this by a number of, say, 30 figures, you would arrive at some enormous number (of either 49 or 50 figures). If you were to multiply the 30-figure number by the 20-figure number you would arrive at the same enormous 49- or 50-figure number, and you know this to be true without having to do the multiplying. This is the step you can never take in physics.
In Science is a Sacred Cow (1950), 68, 88, 179.
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Proof that a given condition always precedes or accompanies a phenomenon does not warrant concluding with certainty that a given condition is the immediate cause of that phenomenon. It must still be established that when this condition is removed, the phenomen will no longer appear.
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 55.
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PROOF, n. Evidence having a shade more of plausibility than of unlikelihood. The testimony of two credible witnesses as opposed to that of only one.
The Collected Works of Ambrose Bierce (1911), Vol. 7, The Devil's Dictionary,  269-270.
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Proofs are the last thing looked for by a truly religious mind which feels the imaginative fitness of its faith.
Interpretations of Poetry and Religion (1900), 95.
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Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols.
In Die Mathematik die Fackelträgerin einer neuen Zeit (1889), 94. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 11.
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Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astrology the theory of eccentrics and epicycles is considered as established because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this reason were sufficient, since some other theory might explain them.
Summa Theologica [1266-1273], Part I, question 32, article 2 (reply to objection 2), trans. Fathers of the English Dominican Province (i.e. L. Shapeote), revised D. J. Sullivan (1952), Vol. I, 177.
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Religions die when they are proved to be true. Science is the record of dead religions.
In 'Phrases and Philosophies', The Writings of Oscar Wilde:Epigrams: Phrases and Philosophies for the Use of the Young (1907), 141.
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Science as such assuredly has no authority, for she can only say what is, not what is not.
'Is Life Worth Living?' (1895). In The Will to Believe and Other Essays in Popular Philosophy (1897), 56.
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Scientists are supposed to live in ivory towers. Their darkrooms and their vibration-proof benches are supposed to isolate their activities from the disturbances of common life. What they tell us is supposed to be for the ages, not for the next election. But the reality may be otherwise.
…...
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Scientists are supposed to make predictions, probably to prove they are human and can be as mistaken as anyone else. Long-range predictions are better to make because the audience to whom the prediction was made is no longer around to ask questions. The alternative... is to make conflicting predictions, so that one prediction may prove right.
'Fossils—The How and Why of Collecting and Storing', Proceedings of the Biological Society of Washington (1969), 82, 597.
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Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress.
In Lord Kelvin and Peter Guthrie Tait Elements of Natural Philosophy (1873), Preface.
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Sodium thymonucleate fibres give two distinct types of X-ray diagram … [structures A and B]. The X-ray diagram of structure B (see photograph) shows in striking manner the features characteristic of helical structures, first worked out in this laboratory by Stokes (unpublished) and by Crick, Cochran and Vand2. Stokes and Wilkins were the first to propose such structures for nucleic acid as a result of direct studies of nucleic acid fibres, although a helical structure had been previously suggested by Furberg (thesis, London, 1949) on the basis of X-ray studies of nucleosides and nucleotides.
While the X-ray evidence cannot, at present, be taken as direct proof that the structure is helical, other considerations discussed below make the existence of a helical structure highly probable.
From Rosalind Franklin and R. G. Gosling,'Molecular Configuration in Sodium Thymonucleate', Nature (25 Apr 1953), 171, No. 4356, 740.
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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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Some proofs command assent. Others woo and charm the intellect. They evoke delight and an overpowering desire to say, 'Amen, Amen'.
Quoted in H. E. Hunter, The Divine Proportion (1970), 6; but with no footnote identifying primary source.
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Statistics can be made to prove anything—even the truth.
Anonymous
In Evan Esar, 20,000 Quips and Quotes (1995), 765.
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Study actively. Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
In I Want to be a Mathematician: An Automathography (1985), 69.
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Subtlety is not a proof of wisdom. Fools and even madmen are at times extraordinarily subtle. One can add that subtlety rarely combines with genius, which is usually ingenuous, or with greatness of character, which is always frank.
(1827). In Aleksandr Sergeevich Pushkin, John Bayley (ed.), Pushkin on Literature (1986), 211.
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That is the way of the scientist. He will spend thirty years in building up a mountain range of facts with the intent to prove a certain theory; then he is so happy with his achievement that as a rule he overlooks the main chief fact of all—that all his accumulation proves an entirely different thing.
'The Bee'. In What is Man? and Other Essays? (1917), 283.
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That which is provable, ought not to be believed in science without proof.
Was sind und was sollen die Zahlen? (1888), Preface, trans. Ivor Grattan-Guinness.
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The accomplishments of those born blind are a sure proof of how much the spirit can achieve when difficulties are placed in its way.
Aphorism 50 in Notebook D (1773-1775), as translated by R.J. Hollingdale in Aphorisms (1990). Reprinted as The Waste Books (2000), 51.
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The American Cancer Society's position on the question of a possible cause-effect relationship between cigarette smoking and lung cancer is:
1. The evidence to date justifies suspicion that cigarette smoking does, to a degree as yet undetermined, increase the likelihood of developing cancer of the lung.
2. That available evidence does not constitute irrefutable proof that cigarette smoking is wholly or chiefly or partly responsible for lung cancer.
3. That the evidence at hand calls for the extension of statistical and laboratory studies designed to confirm or deny a causual relationship between cigarette smoking and lung cancer.
4. That the society is committed to furthering such intensified investigation as its resources will permit.
Conclusions of statement after a meeting of the ACS board of directors in San Francisco (17 Mar 1954). Quoted in 'Tobacco Industry Denies Cancer Tie'. New York Times (14 Apr 1954), 51.
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The ancestors of the higher animals must be regarded as one-celled beings, similar to the Amœbæ which at the present day occur in our rivers, pools, and lakes. The incontrovertible fact that each human individual develops from an egg, which, in common with those of all animals, is a simple cell, most clearly proves that the most remote ancestors of man were primordial animals of this sort, of a form equivalent to a simple cell. When, therefore, the theory of the animal descent of man is condemned as a “horrible, shocking, and immoral” doctrine, tho unalterable fact, which can be proved at any moment under the microscope, that the human egg is a simple cell, which is in no way different to those of other mammals, must equally be pronounced “horrible, shocking, and immoral.”
Translated from his Ueber die Entstehung und den Stammbaum des Menschengeschlechts, (1873), Vol. 2, as an epigraph to Chap. 6, The Evolution of Man, (1879), Vol 1, 120-121.
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The art of drawing conclusions from experiments and observations consists in evaluating probabilities and in estimating whether they are sufficiently great or numerous enough to constitute proofs. This kind of calculation is more complicated and more diff
…...
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The autopsy of a person who had died from phosphorus poisoning would reveal inflammation and haemorrhage in the stomach and bowel, the liver would show fatty changes and both it, and the kidneys would be enlarged, greasy and of a yellow colour. But the most convincing proof of death due to phosphorus exposure would be to turn off all the lights in the mortuary and see its tell-tale glow...
The 13th Element: The Sordid Tale of Murder, Fire and Phosphorus (U.S., 2000), 191. Also published in Great Britain as The Shocking History of Phosphorus (2000).
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The beautiful has its place in mathematics as elsewhere. The prose of ordinary intercourse and of business correspondence might be held to be the most practical use to which language is put, but we should be poor indeed without the literature of imagination. Mathematics too has its triumphs of the Creative imagination, its beautiful theorems, its proofs and processes whose perfection of form has made them classic. He must be a “practical” man who can see no poetry in mathematics.
In A Scrap-book of Elementary Mathematics: Notes, Recreations, Essays (1908), 208.
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The best men are always first discovered by their enemies: it is the adversary who turns on the searchlight, and the proof of excellence lies in being able to stand the gleam.
Aphorism in The Philistine (Dec 1904), 20, No. 1, 18.
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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 drive behind life has lost none of its power; proof that, impelled by that drive, man can build as well as destroy; that in his nature is more of Vishnu the Creator than of Siva the Destroyer.
From Why We Behave Like Human Beings (1925), 479.
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The fact that man knows right from wrong proves his intellectual superiority to other creatures; but the fact that he can do wrong proves his moral inferiority to any creature that cannot.
Spoken by Old Man in What is Man? In What is Man? and Other Essays (1917), 89.
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The fact that the proof of a theorem consists in the application of certain simple rules of logic does not dispose of the creative element in mathematics, which lies in the choice of the possibilities to be examined.
As co-author with Herbert Robbins, in What Is Mathematics?: An Elementary Approach to Ideas and Methods (1941, 1996), 15.
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The facts obtained in this study may possibly be sufficient proof of the causal relationship, that only the most sceptical can raise the objection that the discovered microorganism is not the cause but only an accompaniment of the disease... It is necessary to obtain a perfect proof to satisfy oneself that the parasite and the disease are ... actually causally related, and that the parasite is the... direct cause of the disease. This can only be done by completely separating the parasite from the diseased organism [and] introducing the isolated parasite into healthy organisms and induce the disease anew with all its characteristic symptoms and properties.
Berliner Klinische Wochenschrift (1882), 393. Quoted in Edward J. Huth and T. Jock Murray (eds.), Medicine in Quotations: Views of Health and Disease Through the Ages (2000), 52.
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The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us.
In John Bartlett, Familiar Quotations: A Collection of Passages, Phrases, and Proverbs Traced (1968), 857.
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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 history of civilization proves beyond doubt just how sterile the repeated attempts of metaphysics to guess at nature’s laws have been. Instead, there is every reason to believe that when the human intellect ignores reality and concentrates within, it can no longer explain the simplest inner workings of life’s machinery or of the world around us.
From Reglas y Consejos sobre Investigacíon Cientifica: Los tónicos de la voluntad. (1897), as translated by Neely and Larry W. Swanson, in Advice for a Young Investigator (1999), 2.
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The History of Evolution is the real source of light in the investigation of organic bodies. It is applicable at every step, and all our ideas of the correlation of organic bodies will be swayed by our knowledge of the history of evolution. To carry the proof of it into all branches of research would be an almost endless task. (1828)
Quoted as an epigraph to Chap. 3, in Ernst Haeckel, The Evolution of Man, (1886), Vol 1, 48.
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The history of science has proved that fundamental research is the lifeblood of individual progress and that the ideas that lead to spectacular advances spring from it.
In J. Edwin Holmström, Records and Research in Engineering and Industrial Science (1956), 7.
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The influence of the leaders is due in very small measure to the arguments they employ, but in a large degree to their prestige. The best proof of this is that, should they by any circumstance lose their prestige, their influence disappears.
From Psychologie des Foules (1895), 175. English text in The Crowd: A Study of the Popular Mind (1897), Book 2, Chap. 5, 197-198. Original French text: “Les meneurs agissent très peu par leurs raisonnements, beaucoup par leur prestige. Et la meilleure preuve, c'est que si une circonstance quelconque les en dépouille, ils n’ont plus d’influence.”
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The mathematician starts with a few propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature authority and tradition furnish the data, and the mental operations are deductive.
In 'Scientific Education: Notes of an After-Dinner Speech' (Delivered to Liverpool Philomathic Society, Apr 1869), published in Macmillan’s Magazine (Jun 1869), 20, No. 116, 177. Collected in Lay Sermons, Addresses, and Reviews (1871), Chap 4, 66.
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The mind can proceed only so far upon what it knows and can prove. There comes a point where the mind takes a higher plane of knowledge, but can never prove how it got there. All great discoveries have involved such a leap
As recollected from a visit some months earlier, and quoted in William Miller, 'Old Man’s Advice to Youth: “Never Lose a Holy Curiosity”', Life (2 May 1955), 64.
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The moment one has offered an original explanation for a phenomenon which seems satisfactory, that moment affection for his intellectual child springs into existence, and as the explanation grows into a definite theory his parental affections cluster about his offspring and it grows more and more dear to him. ... There springs up also unwittingly a pressing of the theory to make it fit the facts and a pressing of the facts to make them fit the theory... To avoid this grave danger, the method of multiple working hypotheses is urged. It differs from the simple working hypothesis in that it distributes the effort and divides the affections... In developing the multiple hypotheses, the effort is to bring up into view every rational exploration of the phenomenon in hand and to develop every tenable hypothesis relative to its nature, cause or origin, and to give to all of these as impartially as possible a working form and a due place in the investigation. The investigator thus becomes the parent of a family of hypotheses; and by his parental relations to all is morally forbidden to fasten his affections unduly upon anyone. ... Each hypothesis suggests its own criteria, its own method of proof, its own method of developing the truth, and if a group of hypotheses encompass the subject on all sides, the total outcome of means and of methods is full and rich.
'Studies for Students. The Method of Multiple Working Hypotheses', Journal of Geology (1897), 5, 840-6.
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The more I study the things of the mind the more mathematical I find them. In them as in mathematics it is a question of quantities; they must be treated with precision. I have never had more satisfaction than in proving this in the realms of art, politics and history.
Notes made after the completion of the third chapter of Vol. 3 of La Rivolution, 22 April 1883. In E. Sparvel-Bayly (trans.), Life and Letters of H. Taine (1902-1908), Vol. 3, 239.
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The most convincing proof of the conversion of heat into living force [vis viva] has been derived from my experiments with the electro-magnetic engine, a machine composed of magnets and bars of iron set in motion by an electrical battery. I have proved by actual experiment that, in exact proportion to the force with which this machine works, heat is abstracted from the electrical battery. You see, therefore, that living force may be converted into heat, and that heat may be converted into living force, or its equivalent attraction through space.
'On Matter, Living Force, and Heat' (1847). In The Scientific Papers of James Prescott Joule (1884), Vol. 1, 270-1.
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The number of hypotheses and theories about climate change are numerous. Quite naturally they have caught the public attention, as any proof of past climactic change points to the possibility of future climate change, which inevitably will have significant implications for global economics.
'Klimaschwankungen seit 1700 nebst Bemerkungen fiber die Klimaschwankungen der Diluvialzeit' Pencks Geographische Abhandlingen (1890) 4,2. In Eduard Brückner, Nico Stehr (Ed.), Hans von Storch (Ed.) and Barbara Stehr, Eduard Brückner (2000), 11.
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The number of travellers by gigs, the outside of coaches, and on horseback, have, since the introduction of railways, been prodigiously diminished; and as, in addition, the members of the medical faculty having lent their aid to run down the use of water-proof (apparently having found it decided enemy against their best friends colds and catarrhs), the use of the article [the Macintosh] in the form of cloaks, etc., has of late become comparatively extinct.
A Biographical Memoir of the late Charles Macintosh Esq FRS (1847), 89.
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The old scientific ideal of episteme — of absolutely certain, demonstrable knowledge — has proved to be an idol. The demand for scientific objectivity makes it inevitable that every scientific statement must remain tentative for ever. (1959)
The Logic of Scientific Discovery: Logik Der Forschung (1959, 2002), 280.
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The only difference between elements and compounds consists in the supposed impossibility of proving the so-called elements to be compounds.
'Faraday Lecture: Elements and Compounds', Journal of the Chemical Society (1904), 85, 520.
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The order of ... successive generations is indeed much more clearly proved than many a legend which has assumed the character of history in the hands of man; for the geological record is the work of God.
Siluria (1872), 476.
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The personal views of the lecturer may seem to be brought forward with undue exclusiveness, but, as it is his business to give a clear exposition of the actual state of the science which he treats, he is obliged to define with precision the principles, the correctness of which he has proved by his own experience.
Cellular Pathology, translated by Frank Chance (1860), xi.
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The picture of scientific method drafted by modern philosophy is very different from traditional conceptions. Gone is the ideal of a universe whose course follows strict rules, a predetermined cosmos that unwinds itself like an unwinding clock. Gone is the ideal of the scientist who knows the absolute truth. The happenings of nature are like rolling dice rather than like revolving stars; they are controlled by probability laws, not by causality, and the scientist resembles a gambler more than a prophet. He can tell you only his best posits—he never knows beforehand whether they will come true. He is a better gambler, though, than the man at the green table, because his statistical methods are superior. And his goal is staked higher—the goal of foretelling the rolling dice of the cosmos. If he is asked why he follows his methods, with what title he makes his predictions, he cannot answer that he has an irrefutable knowledge of the future; he can only lay his best bets. But he can prove that they are best bets, that making them is the best he can do—and if a man does his best, what else can you ask of him?
The Rise of Scientific Philosophy (1951, 1973), 248-9. Collected in James Louis Jarrett and Sterling M. McMurrin (eds.), Contemporary Philosophy: A Book of Readings (1954), 376.
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The proof given by Wright, that non-adaptive differentiation will occur in small populations owing to “drift,” or the chance fixation of some new mutation or recombination, is one of the most important results of mathematical analysis applied to the facts of neo-mendelism. It gives accident as well as adaptation a place in evolution, and at one stroke explains many facts which puzzled earlier selectionists, notably the much greater degree of divergence shown by island than mainland forms, by forms in isolated lakes than in continuous river-systems.
Evolution: The Modern Synthesis (1942), 199-200.
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The publication of the Darwin and Wallace papers in 1858, and still more that of the 'Origin' in 1859, had the effect upon them of the flash of light, which to a man who has lost himself in a dark night, suddenly reveals a road which, whether it takes him straight home or not, certainly goes his way. That which we were looking for, and could not find, was a hypothesis respecting the origin of known organic forms, which assumed the operation of no causes but such as could be proved to be actually at work. We wanted, not to pin our faith to that or any other speculation, but to get hold of clear and definite conceptions which could be brought face to face with facts and have their validity tested. The 'Origin' provided us with the working hypothesis we sought.
'On the Reception of the Origin of Species'. In F. Darwin (ed.), The Life and Letters of Charles Darwin, Including an Autobiographical Chapter (1888), Vol 2, 197.
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The question of the origin of the hypothesis belongs to a domain in which no very general rules can be given; experiment, analogy and constructive intuition play their part here. But once the correct hypothesis is formulated, the principle of mathematical induction is often sufficient to provide the proof.
As co-author with Herbert Robbins, in What Is Mathematics?: An Elementary Approach to Ideas and Methods (1941, 1996), 15.
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The real question is, Did God use evolution as His plan? If it could be shown that man, instead of being made in the image of God, is a development of beasts we would have to accept it, regardless of its effort, for truth is truth and must prevail. But when there is no proof we have a right to consider the effect of the acceptance of an unsupported hypothesis.
'God and Evolution', New York Times (26 Feb 1922), 84. Rebuttals were printed a few days later from Henry Fairfield Osborn and Edwin Grant Conklin.
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The regularity with which we conclude that further advances in a particular field are impossible seems equaled only by the regularity with which events prove that we are of too limited vision. And it always seems to be those who have the fullest opportunity to know who are the most limited in view. What, then, is the trouble? I think that one answer should be: we do not realize sufficiently that the unknown is absolutely infinite, and that new knowledge is always being produced.
Quoted in Guy Suits, 'Willis Rodney Whitney', National Academy of Sciences, Biographical Memoirs (1960), 357.
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The Requisites of a good Hypothesis are:
That It be Intelligible.
That It neither Assume nor Suppose anything Impossible, unintelligible, or demonstrably False.
That It be consistent with Itself.
That It be lit and sufficient to Explicate the Phaenomena, especially the chief.
That It be, at least, consistent, with the rest of the Phaenomena It particularly relates to, and do not contradict any other known Phaenomena of nature, or manifest Physical Truth.
The Qualities and Conditions of an Excellent Hypothesis are:
That It be not Precarious, but have sufficient Grounds In the nature of the Thing Itself or at least be well recommended by some Auxiliary Proofs.
That It be the Simplest of all the good ones we are able to frame, at least containing nothing that is superfluous or Impertinent.
That It be the only Hypothesis that can Explicate the Phaenomena; or at least, that do’s Explicate them so well.
That it enable a skilful Naturailst to foretell future Phaenomena by the Congruity or Incongruity to it; and especially the event of such Experlm’ts as are aptly devis’d to examine It, as Things that ought, or ought not, to be consequent to It.
Boyle Papers, 37. Quoted In Barbara Kaplan (ed.), Divulging of Useful Truths in Physick:The Medical Agenda of Robert Boyle (1993), 50.
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The scientist believes in proof without certainty, the bigot in certainty without proof.
Ashley Montagu (ed.), Science and Creationism (1984), Introduction, 9. Commonly seem derivative quotes include: “Science has proof without any certainty. Creationists have certainty without any proof,” and “Religion gives us certainty without proof; science gives us proof without certainty.”
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The scientist has to take 95 per cent of his subject on trust. He has to because he can't possibly do all the experiments, therefore he has to take on trust the experiments all his colleagues and predecessors have done. Whereas a mathematician doesn't have to take anything on trust. Any theorem that's proved, he doesn't believe it, really, until he goes through the proof himself, and therefore he knows his whole subject from scratch. He's absolutely 100 per cent certain of it. And that gives him an extraordinary conviction of certainty, and an arrogance that scientists don't have.
Essay, 'Private Games', in Lewis Wolpert and Alison Richards, (eds.), A Passion for Science (1988), 61.
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The search for extraterrestrial intelligence (SETI, to us insiders) has so far only proved that no matter what you beam up—the Pythagorean theorem, pictures of attractive nude people, etc.—the big 800 number in the sky does not return calls.
From essay 'First Person Secular: Blocking the Gates to Heaven', Mother Jones Magazine (Jun 1986), 48. Collected in The Worst Years of our Lives: Irreverent Notes from a Decade of Greed (1995), 267.
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The second [argument about motion] is the so-called Achilles, and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
Statement of the Achilles and the Tortoise paradox in the relation of the discrete to the continuous.; perhaps the earliest example of the reductio ad absurdum method of proof.
Zeno
Aristotle, Physics, 239b, 14-6. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. 1, 404.
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The self-fulfilling prophecy is, in the beginning, a false definition of the situation evoking a new behavior which makes the originally false conception come true. The specious validity of the self-fulfilling prophecy perpetuates a reign of error. For the prophet will cite the actual course of events as proof that he was right from the very beginning. … Such are the perversities of social logic.
In article, 'The Self-Fulfilling Prophecy', The Antioch Review (Summer 1948), 8, No. 2, 195-196. Included as Chap. 7 of Social Theory and Social Structure (1949), 181-195. Note: Merton coined the expression “self-fulfilling prophecy.”
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The standard of proof is not very high for an investigation that announces that a plume is responsible for a bit of magma or a bit of chemistry found in, or near, or away from a volcano. The standard is being lowered all the time. Plumes were invented to explain small-scale features such as volcanoes. They were 100 kilometers wide. Then they were used to provide magmas 600 km away from a volcano, or to interact with distant ridges. Then the whole North Atlantic, from Canada to England needed to be serviced by a single plume. Then all of Africa. Then a bit of basalt on the East Pacific Rise was found to be similar to a Hawaiian basalt, so the plume influence was stretched to 5000 kilometers! No reviewer or editor has been found to complain yet. Superplumes are now routinely used to affect geology all around the Pacific. This is called creeping incredulity. It can also be called a Just-So Story.
Please contact Webmaster if you know the primary source for this quote.
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The strangest thing of all is that our ulama these days have divided science into two parts. One they call Muslim science, and one European science. Because of this they forbid others to teach some of the useful sciences. They have not understood that science is that noble thing that has no connection with any nation, and is not distinguished by anything but itself. Rather, everything that is known is known by science, and every nation that becomes renowned becomes renowned through science. Men must be related to science, not science to men. How very strange it is that the Muslims study those sciences that are ascribed to Aristotle with the greatest delight, as if Aristotle were one of the pillars of the Muslims. However, if the discussion relates to Galileo, Newton, and Kepler, they consider them infidels. The father and mother of science is proof, and proof is neither Aristotle nor Galileo. The truth is where there is proof, and those who forbid science and knowledge in the belief that they are safeguarding the Islamic religion are really the enemies of that religion. Lecture on Teaching and Learning (1882).
In Nikki R. Keddie, An Islamic Response to Imperialism (1983), 107.
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The strongest arguments prove nothing so long as the conclusions are not verified by experience. Experimental science is the queen of sciences and the goal of all speculation.
Opus Tertium. Translation as stated in Popular Science (Aug 1901), 337.
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The supposed astronomical proofs of the theory [of relativity], as cited and claimed by Einstein, do not exist. He is a confusionist. The Einstein theory is a fallacy. The theory that ether does not exist, and that gravity is not a force but a property of space can only be described as a crazy vagary, a disgrace to our age.
Quoted in Elizabeth Dilling, A "Who's Who" and Handbook of Radicalism for Patriots (1934), 49.
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The test of a theory is its ability to cope with all the relevant phenomena, not its a priori 'reasonableness'. The latter would have proved a poor guide in the development of science, which often makes progress by its encounter with the totally unexpected and initially extremely puzzling.
'From DAMTP [Department of Applied Mathematics and Theoretical Physics] to Westcott House', Cambridge Review (1981), 103, 61.
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The use of thesis-writing is to train the mind, or to prove that the mind has been trained; the former purpose is, I trust, promoted, the evidences of the latter are scanty and occasional.
From Preface to First Edition to Notes on the Composition of Scientific Papers (1904), v.
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The virtue of a logical proof is not that it compels belief but that it suggests doubts.
In Foundations of Euclidean Geometry (1927), viii. This quote is often seen mis-attributed to Morris Kline, who merely quoted it without citation in his books. The idea was expressed earlier by Bertrand Russell as, “It is one of the chief merits of proofs that they instill a certain skepticism about the result proved.” See the Bertrand Russell Quotes page on this website.
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The vortex theory [of the atom] is only a dream. Itself unproven, it can prove nothing, and any speculations founded upon it are mere dreams about dreams.
Quoted in Henry Smith Williams, 'Some Unsolved Scientific Problems', Harper's New Monthly Magazine (1899-1900), Vol. 100, 779.
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Theorem proving is seductive—and its Lorelei voices can put us on the rocks.
In 'Sunset Salvo', The American Statistician (Feb 1986), 40, No. 1, 74.
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There are many examples of old, incorrect theories that stubbornly persisted, sustained only by the prestige of foolish but well-connected scientists. ... Many of these theories have been killed off only when some decisive experiment exposed their incorrectness.
In Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and The Tenth Dimension (1994), 314.
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There have, however, always been men of high and disciplined spirituality who have insisted on their direct experience of something greater than themselves. Their conviction of the reality of a spiritual life apart from and transcending the life of the body may not lend itself to scientific proof or disproof; nevertheless the remarkable transformation in personality seen in those who rightfully lay claim to such experience is as objective as tomorrow's sunrise. Millions of lesser men draw strength from the contacts they can make through prayer and meditation with this aspect of the inner life.
at a convention of scientists in 1967 at the University of Notre Dame
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There is a reward structure in science that is very interesting: Our highest honors go to those who disprove the findings of the most revered among us. So Einstein is revered not just because he made so many fundamental contributions to science, but because he found an imperfection in the fundamental contribution of Isaac Newton.
In 'Wonder and Skepticism', Skeptical Enquirer (Jan-Feb 1995), 19, No. 1.
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There is a theory that creativity arises when individuals are out of sync with their environment. To put it simply, people who fit in with their communities have insufficient motivation to risk their psyches in creating something truly new, while those who are out of sync are driven by the constant need to prove their worth.
In 'Beyond the Soapsuds Universe', Discover Magazine (1997). The author explains (in a blog found online) that this “idea comes from a theory by Mihaly Csikszentmihalyi, who wrote a book called Flow: the Psychology of Optimal Experience. You do need someone to come in from outside. One of the theories about why mathematicians do their best work when they’re young is because they’re not yet educated enough to know what seems obviously wrong. So they try new things.”
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There is great exhilaration in breaking one of these things. … Ramanujan gives no hints, no proof of his formulas, so everything you do you feel is your own.
[About verifying Ramanujan’s equations in a newly found manuscript.]
Quoted in John Noble Wilford, 'Mathematician's Final Equations Praised', New York Times (9 Jun 1981), C1.
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There is more evidence to prove that saltness [of the sea] is due to the admixture of some substance, besides that which we have adduced. Make a vessel of wax and put it in the sea, fastening its mouth in such a way as to prevent any water getting in. Then the water that percolates through the wax sides of the vessel is sweet, the earthy stuff, the admixture of which makes the water salt, being separated off as it were by a filter.
[This is an example of Aristotle giving proof by experiment, in this case, of desalination by osmosis.]
Aristotle
Meteorology (350 B.C.), Book II, translated by E. W. Webster. Internet Classics Archive, (classics.mit.edu).
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There is no art so difficult as the art of observation: it requires a skillful, sober spirit and a well-trained experience, which can only be acquired by practice; for he is not an observer who only sees the thing before him with his eyes, but he who sees of what parts the thing consists, and in what connexion the parts stand to the whole. One person overlooks half from inattention; another relates more than he sees while he confounds it with that which he figures to himself; another sees the parts of the whole, but he throws things together that ought to be separated. ... When the observer has ascertained the foundation of a phenomenon, and he is able to associate its conditions, he then proves while he endeavours to produce the phenomena at his will, the correctness of his observations by experiment. To make a series of experiments is often to decompose an opinion into its individual parts, and to prove it by a sensible phenomenon. The naturalist makes experiments in order to exhibit a phenomenon in all its different parts. When he is able to show of a series of phenomena, that they are all operations of the same cause, he arrives at a simple expression of their significance, which, in this case, is called a Law of Nature. We speak of a simple property as a Law of Nature when it serves for the explanation of one or more natural phenomena.
'The Study of the Natural Sciences: An Introductory Lecture to the Course of Experimental Chemistry in the University of Munich, for the Winter Session of 1852-53,' as translated and republished in The Medical Times and Gazette (22 Jan 1853), N.S. Vol. 6, 82.
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There is no more convincing proof of the truth of a comprehensive theory than its power of absorbing and finding a place for new facts, and its capability of interpreting phenomena which had been previously looked upon as unaccountable anomalies. It is thus that the law of universal gravitation and the undulatory theory of light have become established and universally accepted by men of science. Fact after fact has been brought forward as being apparently inconsistent with them, and one alter another these very facts have been shown to be the consequences of the laws they were at first supposed to disprove. A false theory will never stand this test. Advancing knowledge brings to light whole groups of facts which it cannot deal with, and its advocates steadily decrease in numbers, notwithstanding the ability and scientific skill with which it may have been supported.
From a review of four books on the subject 'Mimicry, and Other Protective Resemblances Among Animals', in The Westminster Review (Jul 1867), 88, 1. Wallace is identified as the author in the article as reprinted in William Beebe, The Book of Naturalists: An Anthology of the Best Natural History (1988), 108.
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There is one great difficulty with a good hypothesis. When it is completed and rounded, the corners smooth and the content cohesive and coherent, it is likely to become a thing in itself, a work of art. It is then like a finished sonnet or a painting completed. One hates to disturb it. Even if subsequent information should shoot a hole in it, one hates to tear it down because it once was beautiful and whole. One of our leading scientists, having reasoned a reef in the Pacific, was unable for a long time to reconcile the lack of a reef, indicated by soundings, with the reef his mind told him was there.
In John Steinbeck and Edward Flanders Ricketts Sea of Cortez: a Leisurely Journal of Travel and Research (1941), 179-80.
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There is, however, no genius so gifted as not to need control and verification. ... [T]he brightest flashes in the world of thought are incomplete until they have been proved to have their counterparts in the world of fact. Thus the vocation of the true experimentalist may be defined as the continued exercise of spiritual insight, and its incessant correction and realisation. His experiments constitute a body, of which his purified intuitions are, as it were, the soul.
In 'Vitality', Scientific Use of the Imagination and Other Essays (1872), 43.
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There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.
In George Pólya and Gerald L. Alexanderson (ed.) The Pólya Picture Album: Encounters of a Mathematician (1987), 154. Also footnoted in Matti Tedre, The Development of Computer Science: a Sociocultural Perspective (2006), 198, cited as from How to Solve It: A New Aspect of Mathematical Method (1957), xv.
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There was, I think, a feeling that the best science was that done in the simplest way. In experimental work, as in mathematics, there was “style” and a result obtained with simple equipment was more elegant than one obtained with complicated apparatus, just as a mathematical proof derived neatly was better than one involving laborious calculations. Rutherford's first disintegration experiment, and Chadwick's discovery of the neutron had a “style” that is different from that of experiments made with giant accelerators.
From 'Physics in a University Laboratory Before and After World War II', Proceedings of the Royal Society of London, Series A, (1975), 342, 463. As cited in Alan McComas, Galvani's Spark: The Story of the Nerve Impulse (2011), 107.
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There’s no value in digging shallow wells in a hundred places. Decide on one place and dig deep ... If you leave that to dig another well, all the first effort is wasted and there is no proof you won’t hit rock again.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 258
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These machines [used in the defense of the Syracusans against the Romans under Marcellus] he [Archimedes] had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with king Hiero’s desire and request, some time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly-prized art of mechanics, which they employed as an elegant illustration of geometrical truths, and as means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by words and diagrams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extremes, to find the two mean lines of a proportion, both these mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines. But what with Plato’s indignation at it, and his invectives against it as the mere corruption and annihilation of the one good of geometry,—which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help (not to be obtained without base supervisions and depravation) from matter; so it was that mechanics came to be separated from geometry, and, repudiated and neglected by philosophers, took its place as a military art.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects.
In The Philosophy of the Inductive Sciences (1858), Part 1, Bk. 2, chap. 1, sect. 2.
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They [mathematicians] only take those things into consideration, of which they have clear and distinct ideas, designating them by proper, adequate, and invariable names, and premising only a few axioms which are most noted and certain to investigate their affections and draw conclusions from them, and agreeably laying down a very few hypotheses, such as are in the highest degree consonant with reason and not to be denied by anyone in his right mind. In like manner they assign generations or causes easy to be understood and readily admitted by all, they preserve a most accurate order, every proposition immediately following from what is supposed and proved before, and reject all things howsoever specious and probable which can not be inferred and deduced after the same manner.
In Mathematical Lectures (1734), 65-66.
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Think of Adam and Eve like an imaginary number, like the square root of minus one: you can never see any concrete proof that it exists, but if you include it in your equations, you can calculate all manner of things that couldn't be imagined without it.
In The Golden Compass (1995, 2001), 372-373.
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This leads us to ask for the reasons which call for this new theory of transmutation. The beginning of things must needs lie in obscurity, beyond the bounds of proof, though within those of conjecture or of analogical inference. Why not hold fast to the customary view, that all species were directly, instead of indirectly, created after their respective kinds, as we now behold them,--and that in a manner which, passing our comprehension, we intuitively refer to the supernatural? Why this continual striving after “the unattained and dim,”—these anxious endeavors, especially of late years, by naturalists and philosophers of various schools and different tendencies, to penetrate what one of them calls “the mystery of mysteries,” the origin of species? To this, in general, sufficient answer may be found in the activity of the human intellect, “the delirious yet divine desire to know,” stimulated as it has been by its own success in unveiling the laws and processes of inorganic Nature,—in the fact that the principal triumphs of our age in physical science have consisted in tracing connections where none were known before, in reducing heterogeneous phenomena to a common cause or origin, in a manner quite analogous to that of the reduction of supposed independently originated species to a common ultimate origin,—thus, and in various other ways, largely and legitimately extending the domain of secondary causes. Surely the scientific mind of an age which contemplates the solar system as evolved from a common, revolving, fluid mass,— which, through experimental research, has come to regard light, heat, electricity, magnetism, chemical affinity, and mechanical power as varieties or derivative and convertible forms of one force, instead of independent species,—which has brought the so-called elementary kinds of matter, such as the metals, into kindred groups, and raised the question, whether the members of each group may not be mere varieties of one species,—and which speculates steadily in the direction of the ultimate unity of matter, of a sort of prototype or simple element which may be to the ordinary species of matter what the protozoa or component cells of an organism are to the higher sorts of animals and plants,—the mind of such an age cannot be expected to let the old belief about species pass unquestioned.
Asa Gray
'Darwin on the Origin of Species', The Atlantic Monthly (Jul 1860), 112-3. Also in 'Natural Selection Not Inconsistent With Natural Theology', Darwiniana: Essays and Reviews Pertaining to Darwinism (1876), 94-95.
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This marvellous experimental method eliminates certain facts, brings forth others, interrogates nature, compels it to reply and stops only when the mind is fully satisfied. The charm of our studies, the enchantment of science, is that, everywhere and always, we can give the justification of our principles and the proof of our discoveries.
As quoted in René Dubos, Louis Pasteur, Free Lance of Science (1960, 1986), 377.
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This skipping is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal to the reader. In most cases he will be able to go on and later he may return to the parts which he skipped.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 19.
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Those intervening ideas, which serve to show the agreement of any two others, are called proofs; and where the agreement or disagreement is by this means plainly and clearly perceived, it is called demons