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Proof Quotes (133 quotes)

"I should have more faith," he said; "I ought to know by this time that when a fact appears opposed to a long train of deductions it invariably proves to be capable of bearing some other interpretation."
Spoken by character, Sherlock Holmes, in A Study in Scarlet (1887), in Works of Arthur Conan Doyle (1902), Vol. 11, 106.
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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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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 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 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 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.
The Sunday Times (18 Jul 1976)
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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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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.
In Playboy Magazine (Jul 1964), as cited 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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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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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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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].
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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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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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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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 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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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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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 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 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).

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 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 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 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 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 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 (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 the 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.
One gave to the world the final 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.
John Dalton and the Rise of Modern Chemistry (1895), 7.
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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 a 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 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 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 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 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.
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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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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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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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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 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.
I Want to be a Mathematician: an Automathography in Three Parts (1985), 321.
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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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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.
'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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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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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.

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, 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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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 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 (1879, 1902), Preface.
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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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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.

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 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 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 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.
History of Mathematics (3rd Ed., 1901), 468.
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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
Ronald W. Clark, Einstein: The Life and Times (1984), 755.
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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 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 (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 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 scientist believes in proof without certainty, the bigot in certainty without proof.
Ashley Montagu (ed.), Science and Creationism (1984), Introduction, 9.
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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.
The Self-Fulfilling Prophecy (1948), 477. Merton is credited with coining the modern use of the expression “self-fulfilling prophecy.”
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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 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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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.
Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and The Tenth Dimension (1994), 314.
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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 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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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.
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 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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To divide a cube into two other cubes, a fourth power, or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.
[Known as Fermat's Last Theorem, the proof of which remained elusive until 1994.]
Theorem, beside the eighth proposition of the second book of Diophantus, in Précis des Oeuvres Mathématiques de P. Fermat et de l'Arithmetique de Diophante (1853), 53-54. As translated by Vera Sandford in David Eugene Smith, A Source Book in Mathematics (1929), 212.
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Unconfusion submits
its confusion to proof; it's
not a Herod's oath that cannot change.
The Mind is an Enchanting Thing'. Complete Poems (1994), 135.

We know that there exist true propositions which we can never formally prove. What about propositions whose proofs require arguments beyond our capabilities? What about propositions whose proofs require millions of pages? Or a million, million pages? Are there proofs that are possible, but beyond us?
Mathematical Mysteries (1999), 295.
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We need only reflect on what has been prov'd at large, that we are never sensible of any connexion betwixt causes and effects, and that 'tis only by our experience of their constant conjunction, we can arrive at any knowledge of this relation.
A Treatise on Human Nature (1739-40), ed. L. A. Selby-Bigge (1888), book 1, part 4, section 165, 247.
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We often hear that mathematics consists mainly of “proving theorems.” Is a writer's job mainly that of “writing sentences?”
In Rota's 'Introduction' written (1980) to preface Philip J. Davis and Reuben Hersh, The Mathematical Experience (1981, 2012), xxii.
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What is possible can never be demonstrated to be false; and 'tis possible the course of nature may change, since we can conceive such a change. Nay, I will go farther, and assert, that he could not so much as prove by any probable arguments, that the future must be conformable to the past. All probable arguments are built on the supposition, that there is this conformity betwixt the future and the past, and therefore can never prove it. This conformity is a matter of fact, and if it must be proved, will admit of no proof but from experience. But our experience in the past can be a proof of nothing for the future, but upon a supposition, that there is a resemblance betwixt them. This therefore is a point, which can admit of no proof at all, and which we take for granted without any proof.
An Abstract of A Treatise on Human Nature (1740), ed. John Maynard Keynes and Piero Sraffa (1938), 15.
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You can be a thorough-going Neo-Darwinian without imagination, metaphysics, poetry, conscience, or decency. For 'Natural Selection' has no moral significance: it deals with that part of evolution which has no purpose, no intelligence, and might more appropriately be called accidental selection, or better still, Unnatural Selection, since nothing is more unnatural than an accident. If it could be proved that the whole universe had been produced by such Selection, only fools and rascals could bear to live.
Back to Methuselah: A Metabiological Pentateuch (1921), Penguin edition (1939), xlii.
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You must not say that this cannot be, or that that is contrary to nature. You do not know what Nature is, or what she can do; and nobody knows; not even Sir Roderick Murchison, or Professor Huxley, or Mr. Darwin, or Professor Faraday, or Mr. Grove, or any other of the great men whom good boys are taught to respect. They are very wise men; and you must listen respectfully to all they say: but even if they should say, which I am sure they never would, 'That cannot exist. That is contrary to nature,' you must wait a little, and see; for perhaps even they may be wrong.
The Water-babies (1886), 81.
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[The] weakness of biological balance studies has aptly been illustrated by comparison with the working of a slot machine. A penny brings forth one package of chewing gum; two pennies bring forth two. Interpreted according to the reasoning of balance physiology, the first observation is an indication of the conversion of copper into gum; the second constitutes proof.
[Co-author with David Rittenberg (1906-70).]
'The Application of Isotopes to the Study of Intermediary Metabolism', Science (1938), 87, 222.
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[T]he 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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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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- 100 -
Sophie Germain
Gertrude Elion
Ernest Rutherford
James Chadwick
Marcel Proust
William Harvey
Johann Goethe
John Keynes
Carl Gauss
Paul Feyerabend
- 90 -
Antoine Lavoisier
Lise Meitner
Charles Babbage
Ibn Khaldun
Euclid
Ralph Emerson
Robert Bunsen
Frederick Banting
Andre Ampere
Winston Churchill
- 80 -
John Locke
Bronislaw Malinowski
Bible
Thomas Huxley
Alessandro Volta
Erwin Schrodinger
Wilhelm Roentgen
Louis Pasteur
Bertrand Russell
Jean Lamarck
- 70 -
Samuel Morse
John Wheeler
Nicolaus Copernicus
Robert Fulton
Pierre Laplace
Humphry Davy
Thomas Edison
Lord Kelvin
Theodore Roosevelt
Carolus Linnaeus
- 60 -
Francis Galton
Linus Pauling
Immanuel Kant
Martin Fischer
Robert Boyle
Karl Popper
Paul Dirac
Avicenna
James Watson
William Shakespeare
- 50 -
Stephen Hawking
Niels Bohr
Nikola Tesla
Rachel Carson
Max Planck
Henry Adams
Richard Dawkins
Werner Heisenberg
Alfred Wegener
John Dalton
- 40 -
Pierre Fermat
Edward Wilson
Johannes Kepler
Gustave Eiffel
Giordano Bruno
JJ Thomson
Thomas Kuhn
Leonardo DaVinci
Archimedes
David Hume
- 30 -
Andreas Vesalius
Rudolf Virchow
Richard Feynman
James Hutton
Alexander Fleming
Emile Durkheim
Benjamin Franklin
Robert Oppenheimer
Robert Hooke
Charles Kettering
- 20 -
Carl Sagan
James Maxwell
Marie Curie
Rene Descartes
Francis Crick
Hippocrates
Michael Faraday
Srinivasa Ramanujan
Francis Bacon
Galileo Galilei
- 10 -
Aristotle
John Watson
Rosalind Franklin
Michio Kaku
Isaac Asimov
Charles Darwin
Sigmund Freud
Albert Einstein
Florence Nightingale
Isaac Newton

New Book


The Simpsons and their Mathematical Secrets,
by Simon Singh

Cleverly embedded in many Simpsons plots are subtle references to mathematics, because the show's brilliant comedy writers with advanced degrees in math or science. Singh offers an entirely new insight into the most successful show in television history.