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Prove Quotes (109 quotes)
Proved Quotes, Proving Quotes

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

Goldsmith: If you put a tub full of blood into a stable, the horses are like to go mad.
Johnson: I doubt that.
Goldsmith: Nay, sir, it is a fact well authenticated.
Thrale: You had better prove it before you put it into your book on natural history. You may do it in my stable if you will.
Johnson: Nay, sir, I would not have him prove it. If he is content to take his information from others, he may get through his book with little trouble, and without much endangering his reputation. But if he makes experiments for so comprehensive a book as his, there would be no end to them; his erroneous assertions would then fall upon himself: and he might be blamed for not having made experiments as to every particular.
In James Boswell, The Life of Samuel Johnson, LL.D.: Comprehending an Account of His Studies and Numerous Works (1785, 1830), 229-230.
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A fateful process is set in motion when the individual is released “to the freedom of his own impotence” and left to justify his existence by his own efforts. The autonomous individual, striving to realize himself and prove his worth, has created all that is great in literature, art, music, science and technology. The autonomous individual, also, when he can neither realize himself nor justify his existence by his own efforts, is a breeding call of frustration, and the seed of the convulsions which shake our world to its foundations.
In The Passionate State of Mind (1955), 18.
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A Frenchman who arrives in London, will find Philosophy, like every Thing else, very much chang’d there. He had left the World a plenum, and he now finds it a vacuum. At Paris the Universe is seen, compos’d of Vortices of subtile Matter; but nothing like it is seen in London. In France, ‘tis the Pressure of the Moon that causes the Tides; but in England ‘tis the Sea that gravitates towards the Moon; so what when you think that the Moon should make it flood with us, those Gentlemen fancy it should be Ebb, which, very unluckily, cannot be prov’d. For to be able to do this, ‘tis necessary the Moon and the Tides should have been enquir’d into, at the very instant of the Creation.
Letter XIV. 'On DesCartes and Sir Isaac Newton', in Letters Concerning the English Nation (1733), 109-110.
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All the experiments which have been hitherto carried out, and those that are still being daily performed, concur in proving that between different bodies, whether principles or compounds, there is an agreement, relation, affinity or attraction (if you will have it so), which disposes certain bodies to unite with one another, while with others they are unable to contract any union: it is this effect, whatever be its cause, which will help us to give a reason for all the phenomena furnished by chemistry, and to tie them together.
From Elemens de Chymie Theorique (1749). As quoted, in Trevor Harvey Levere, Affinity and Matter: Elements of Chemical Philosophy, 1800-1865 (1971), 17.
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Any priest or shaman must be presumed guilty until proved innocent.
In 'From the Notebooks of Lazarus Long', Time Enough for Love: The Lives of Lazarus Long (1973), 256.
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Concepts that have proven useful in ordering thi ngs easily achieve such authority over us that we forget their earthly origins and accept them as unalterable givens.
…...
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Einstein’s results again turned the tables and now very few philosophers or scientists still think that scientific knowledge is, or can be, proven knowledge.
In 'Falsification and the Methodology of Scientific Research Programmes', in I. Lakatos and A. Musgrave (eds.), Criticism and the Growth of Knowledge: Proceedings of the International Colloquium in the Philosophy of Science, London 1965 (1970), Vol. 4, 92.
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Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have ‘proved’ that it involves no contradiction either in its terms or with the truths previously admitted.
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Far must thy researches go
Wouldst thou learn the world to know;
Thou must tempt the dark abyss
Wouldst thou prove what Being is;
Naught but firmness gains the prize,—
Naught but fullness makes us wise,—
Buried deep truth ever lies!
In Edgar A. Bowring (trans.), The Poems of Schiller (1875), 260.
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For it is too bad that there are so few who seek the truth and so few who do not follow a mistaken method in philosophy. This is not, however, the place to lament the misery of our century, but to rejoice with you over such beautiful ideas for proving the truth. So I add only, and I promise, that I shall read your book at leisure; for I am certain that I shall find the noblest things in it. And this I shall do the more gladly, because I accepted the view of Copernicus many years ago, and from this standpoint I have discovered from their origins many natural phenomena, which doubtless cannot be explained on the basis of the more commonly accepted hypothesis.
Letter (4 Aug 1597) to Kepler, expressing thanks and interest in the book Kepler sent him. As quoted in translation in Jackson J. Spielvogel, Western Civilization: Alternate Volume: Since 1300 (2010), Vol. 2, 494.
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For the saving the long progression of the thoughts to remote and first principles in every case, the mind should provide itself several stages; that is to say, intermediate principles, which it might have recourse to in the examining those positions that come in its way. These, though they are not self-evident principles, yet, if they have been made out from them by a wary and unquestionable deduction, may be depended on as certain and infallible truths, and serve as unquestionable truths to prove other points depending upon them, by a nearer and shorter view than remote and general maxims. … And thus mathematicians do, who do not in every new problem run it back to the first axioms through all the whole train of intermediate propositions. Certain theorems that they have settled to themselves upon sure demonstration, serve to resolve to them multitudes of propositions which depend on them, and are as firmly made out from thence as if the mind went afresh over every link of the whole chain that tie them to first self-evident principles.
In The Conduct of the Understanding, Sect. 21.
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Gauss was not the son of a mathematician; Handel’s father was a surgeon, of whose musical powers nothing is known; Titian was the son and also the nephew of a lawyer, while he and his brother, Francesco Vecellio, were the first painters in a family which produced a succession of seven other artists with diminishing talents. These facts do not, however, prove that the condition of the nerve-tracts and centres of the brain, which determine the specific talent, appeared for the first time in these men: the appropriate condition surely existed previously in their parents, although it did not achieve expression. They prove, as it seems to me, that a high degree of endowment in a special direction, which we call talent, cannot have arisen from the experience of previous generations, that is, by the exercise of the brain in the same specific direction.
In 'On Heredity', Essays upon Heredity and Kindred Biological Problems (1889), Vol. 1, 96.
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Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it is the truth.
In a lecture, after establishing the relation eπ/2 = ii in a lecture, “which evidently had a strong hold on his imagination. He dropped his chalk and rubber, put his hands in his pockets, and after contemplating the formula a few minutes turned to his class and said [this quote] very slowly and impressively.” As quoted in W. E. Byerly (writing as a Professor Emeritus at Harvard University, but a former student of Peirce), 'Benjamin Peirce: II. Reminiscences', The American Mathematical Monthly (Jan 1925), 32, No. 1, 6.
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God exists since mathematics is consistent, and the Devil exists since we cannot prove it.
Given as “A. Weil has said…”, in Paul C. Rosenbloom, The Elements of Mathematical Logic (1950), 72. Note that Rosenbloom gives the quote in narrative form, not within quotation marks, which suggests these words may not be verbatim. Later texts have added quotation marks [which may not be justified. —Webmaster] As yet, Webmaster has not found an earlier source to validate whether the quotations marks can be used. (Can you help?)
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Gödel proved that the world of pure mathematics is inexhaustible; no finite set of axioms and rules of inference can ever encompass the whole of mathematics; given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. I hope that an analogous Situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we go into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.
From Lecture 1, 'Philosophy', in a series of four James Arthur Lectures, 'Lectures on Time and its Mysteries' at New York University (Autumn 1978). Printed in 'Time Without End: Physics and Biology in an Open Universe', Reviews of Modern Physics (Jul 1979), 51, 449.
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He made an instrument to know If the moon shine at full or no;
That would, as soon as e’er she shone straight,
Whether ‘twere day or night demonstrate;
Tell what her d’ameter to an inch is,
And prove that she’s not made of green cheese.
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He thought he saw an Argument
That proved he was the Pope:
He looked again and found it was
A Bar of Mottled Soap.
“A fact so dread.” he faintly said,
“Extinguishes all hope!”
In Sylvie and Bruno Concluded (1893), 319.
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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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However, if we consider that all the characteristics which have been cited are only differences in degree of structure, may we not suppose that this special condition of organization of man has been gradually acquired at the close of a long period of time, with the aid of circumstances which have proved favorable? What a subject for reflection for those who have the courage to enter into it!
In Recherches sur l'Organization des corps vivans (1802), as translated in Alpheus Spring Packard, Lamarck, the Founder of Evolution: His Life and Work (1901), 363. Packard's italics.
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I am coming more and more to the conviction that the necessity of our geometry cannot be proved, at least neither by, nor for, the human intelligence … One would have to rank geometry not with arithmetic, which stands a priori, but approximately with mechanics.
From Letter (28 Apr 1817) to Olbers, as quoted in Guy Waldo Dunnington, Carl Friedrich Gauss, Titan of Science: A Study of His Life and Work (1955), 180.
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I am more fond of achieving than striving. My theories must prove to be facts or be discarded as worthless. My efforts must soon be crowned with success, or discontinued.
The Rest of My Life, ch. 4 (1937).Wells wrote, in addition to other works, the popular 'Fleming Stone' detective series.
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I believe in an immortal soul. science has proved that nothing disintegrates into nothingness. Life and soul, therefore, cannot disintegrate into nothingness, and so are immortal.
…...
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I believe that the science of chemistry alone almost proves the existence of an intelligent creator.
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I can’t recall a single problem in my life, of any sort, that I ever started on that I didn't solve, or prove that I couldn’t solve it. I never let up, until I had done everything that I could think of, no matter how absurd it might seem as a means to the end I was after.
As quoted in French Strother, 'The Modern Profession of Inventing', World's Work and Play (Jul 1905), 6, No. 32, 186.
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I confess that Fermat’s Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.
From Letter (Mar 1816), replying to Olbers’ attempt to entice him to work on Fermat’s Theorem. In James R. Newman (ed.) The World of Mathematics (1956), Vol. 1, 312-313.
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I have been arranging certain experiments in reference to the notion that Gravity itself may be practically and directly related by experiment to the other powers of matter and this morning proceeded to make them. It was almost with a feeling of awe that I went to work, for if the hope should prove well founded, how great and mighty and sublime in its hitherto unchangeable character is the force I am trying to deal with, and how large may be the new domain of knowledge that may be opened up to the mind of man.
In ‎Thomas Martin (ed.) Faraday’s Diary: Sept. 6, 1847 - Oct. 17, 1851 (1934), 156.
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I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the “Law of Frequency of Error.” The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
In Natural Inheritance (1894), 66.
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I ought to call myself an agnostic; but, for all practical purposes, I am an atheist. I do not think the existence of the Christian God any more probable than the existence of the Gods of Olympus or Valhalla. To take another illustration: nobody can prove that there is not between the Earth and Mars a china teapot revolving in an elliptical orbit, but nobody thinks this sufficiently likely to be taken into account in practice. I think the Christian God just as unlikely.
Letter (1958) to Mr Major. Collected in Dear Bertrand Russell: A Selection of his Correspondence with the General Public, 1950 - 1968 (1969), 41-42.
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If anyone could prove to me that Christ is outside the truth, and if the truth really did exclude Christ, I should prefer to stay with Christ and not with truth.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 155
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If basketball was going to enable Bradley to make friends, to prove that a banker’s son is as good as the next fellow, to prove that he could do without being the greatest-end-ever at Missouri, to prove that he was not chicken, and to live up to his mother’s championship standards, and if he was going to have some moments left over to savor his delight in the game, he obviously needed considerable practice, so he borrowed keys to the gym and set a schedule for himself that he adhereded to for four full years—in the school year, three and a half hours every day after school, nine to five on Saturday, one-thirty to five on Sunday, and, in the summer, about three hours a day.
A Sense of Where You Are: Bill Bradley at Princeton
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If Einstein’s theory [of relativity] should prove to be correct, as I expect it will, he will be considered the Copernicus of the twentieth century.
As quoted in Philipp Frank and Shuichi Kusaka, Einstein, His Life and Times (1947), 101.
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If finally, the science should prove that society at a certain time revert to the church and recover its old foundation of absolute faith in a personal providence and a revealed religion, it commits suicide.
In The Degradation of the Democratic Dogma (1919), 131.
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If I set out to prove something, I am no real scientist—I have to learn to follow where the facts lead me—I have to learn to whip my prejudices.
Attributed, as cited in Peter McDonald (ed.), Oxford Dictionary of Medical Quotations (2004), 95. Quoted earlier without citation as “If I want to find truth, I must have an open mind. I have to learn to follow where the facts lead me—I have to learn to whip my prejudices,” in World Order (1948), 14, 76.
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If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?
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If you were going to risk all that, not just risk the hardship and the pain but risk your life. Put everything on line for a dream, for something that’s worth nothing, that can’t be proved to anybody. You just have the transient moment on a summit and when you come back down to the valley it goes. It is actually a completely illogical thing to do. It is not justifiable by any rational terms. That’s probably why you do it.
The Beckoning Silence
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If [in a rain forest] the traveler notices a particular species and wishes to find more like it, he must often turn his eyes in vain in every direction. Trees of varied forms, dimensions, and colors are around him, but he rarely sees any of them repeated. Time after time he goes towards a tree which looks like the one he seeks, but a closer examination proves it to be distinct.
In 'Equitorial Vegetation', Natural Selection and Tropical Nature Essays on Descriptive and Theoretical Biology (1891), 267.
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In every case the awakening touch has been the mathematical spirit, the attempt to count, to measure, or to calculate. What to the poet or the seer may appear to be the very death of all his poetry and all his visions—the cold touch of the calculating mind,—this has proved to be the spell by which knowledge has been born, by which new sciences have been created, and hundreds of definite problems put before the minds and into the hands of diligent students. It is the geometrical figure, the dry algebraical formula, which transforms the vague reasoning of the philosopher into a tangible and manageable conception; which represents, though it does not fully describe, which corresponds to, though it does not explain, the things and processes of nature: this clothes the fruitful, but otherwise indefinite, ideas in such a form that the strict logical methods of thought can be applied, that the human mind can in its inner chamber evolve a train of reasoning the result of which corresponds to the phenomena of the outer world.
In A History of European Thought in the Nineteenth Century (1896), Vol. 1, 314.
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In teaching man, experimental science results in lessening his pride more and more by proving to him every day that primary causes, like the objective reality of things, will be hidden from him forever and that he can only know relations.
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In the same sense that our judicial system presumes us to be innocent until proven guilty, a medical care system may work best if it starts with the presumption that most people are healthy. Left to themselves, computers may try to do it in the opposite way, taking it as given that some sort of direct, continual, professional intervention is required all the time, in order to maintain the health of each citizen, and we will end up spending all our money on nothing but this.
In 'Aspects of Biomedical Science Policy', The New England Journal of Medicine (12 Oct 1972), 4. Also published as Occasional Paper of the Institute of Medicine.
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In the summer of 1937, … I told Banach about an expression Johnny [von Neumann] had once used in conversation with me in Princeton before stating some non-Jewish mathematician’s result, “Die Goim haben den folgendenSatzbewiesen” (The goys have proved the following theorem). Banach, who was pure goy, thought it was one of the funniest sayings he had ever heard. He was enchanted by its implication that if the goys could do it, Johnny and I ought to be able to do it better. Johnny did not invent this joke, but he liked it and we started using it.
In Adventures of a Mathematician (1976, 1991), 107. Von Neumann, who was raised in Budapest by a Jewish family, knew the Yiddish word “goy” was equivalent to “gentile” or a non-Jew. Stefan Banach, a Polish mathematician, was raised in a Catholic family, hence “pure goy”. Ulam thus gives us the saying so often elsewhere seen attributed to von Neumann without the context: “The goys have proved the following theorem.” It is seen anecdotally as stated by von Neumann to begin a classroom lecture.
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Induction may be defined, the operation of discovering and proving general propositions.
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, 347.
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It is certainly true that all physical phenomena are subject to strictly mathematical conditions, and mathematical processes are unassailable in themselves. The trouble arises from the data employed. Most phenomena are so highly complex that one can never be quite sure that he is dealing with all the factors until the experiment proves it. So that experiment is rather the criterion of mathematical conclusions and must lead the way.
In Matter, Ether, Motion (1894), 89.
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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 through science that we prove, but through intuition that we discover.
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It is very different to make a practical system and to introduce it. A few experiments in the laboratory would prove the practicability of system long before it could be brought into general use. You can take a pipe and put a little coal in it, close it up, heat it and light the gas that comes out of the stem, but that is not introducing gas lighting. I'll bet that if it were discovered to-morrow in New York that gas could be made out of coal it would be at least five years before the system would be in general use.
From the New York Herald (30 Jan 1879), as cited in Leslie Tomory, 'Building the First Gas Network, 1812-1820', Technology and Culture (Jan 2011), 52, No. 1, 75-102.
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It usually develops that after much laborious and frustrating effort the investigator of environmental physiology succeeds in proving that the animal in question can actually exist where it lives. It is always somewhat discouraging for an investigator to realize that his efforts can be made to appear so trite, but this statement does not belittle the ecological physiologist. If his data assist the understanding of the ways in which an animal manages to live where it does, he makes an important contribution to the study of distribution, for the present is necessarily a key to the past.”
From 'The role of physiology in the distribution of terrestrial vertebrates', collected in C.L. Hubbs (ed.), Zoogeography: Publ. 51 (1958), 84.
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It was his [Leibnitz’s] love of method and order, and the conviction that such order and harmony existed in the real world, and that our success in understanding it depended upon the degree and order which we could attain in our own thoughts, that originally was probably nothing more than a habit which by degrees grew into a formal rule. This habit was acquired by early occupation with legal and mathematical questions. We have seen how the theory of combinations and arrangements of elements had a special interest for him. We also saw how mathematical calculations served him as a type and model of clear and orderly reasoning, and how he tried to introduce method and system into logical discussions, by reducing to a small number of terms the multitude of compound notions he had to deal with. This tendency increased in strength, and even in those early years he elaborated the idea of a general arithmetic, with a universal language of symbols, or a characteristic which would be applicable to all reasoning processes, and reduce philosophical investigations to that simplicity and certainty which the use of algebraic symbols had introduced into mathematics.
A mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations. Wherever progress depends upon precision and clearness of thought, and wherever such can be gained by reducing a variety of investigations to a general method, by bringing a multitude of notions under a common term or symbol, it proves inestimable. It necessarily imports the special qualities of number—viz., their continuity, infinity and infinite divisibility—like mathematical quantities—and destroys the notion that irreconcilable contrasts exist in nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud, Leibnitz expresses it as his opinion that geometry, or the philosophy of space, forms a step to the philosophy of motion—i.e., of corporeal things—and the philosophy of motion a step to the philosophy of mind.
In Leibnitz (1884), 44-45. [The first sentence is reworded to better introduce the quotation. —Webmaster]
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It was said round 1912 that it gave him [Edmund Landau] the same pleasure when someone else proved a good theorem as if he had done it himself.
In A Mathematician’s Miscellany (1953), reissued as Béla Bollobás, Littlewood’s Miscellany (1986), 24. Notice that Littlewood states this in the third person—about Landau. The words are found on the web paraphrased as a first person quote (It gives me … as when I do it myself), sometimes citing a later author, Desmond MacHale, Comic Sections (1993). Webmaster has not yet found a primary source to authenticate the first person version, and since the earlier occurrence is hedged with “it was said”, Webmaster suggests the first person version is not an authentic verbatim quote.
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Keep some souvenirs of your past, or how will you ever prove it wasn’t all a dream?
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Logicians have but ill defined
As rational the human mind;
Reason, they say, belongs to man,
But let them prove it if they can.
In 'The Logicians Refuted', The Poems and Plays of Oliver Goldsmith (1818), 58.
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Many orthodox people speak as though it were the business of sceptics to disprove received dogmas rather than of dogmatists to prove them. This is, of course, a mistake.
In unpublished manuscript, 'Is There a God', (5 Mar 1952) written for the magazine, Illustrated. Collected in Bertrand Russell, John G. Slater (ed.) and Peter Köllner (ed.) The Collected Papers of Bertrand Russell: Volume II: Last Philosophical Testament: 1943-68 (1997), 547.
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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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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 is a public activity. It occurs in a social context and has social consequences. Posing a problem, formulating a definition, proving a theorem are none of them private acts. They are all part of that larger social process we call science.
In 'Mathematics as an Objective Science', The American Mathematical Monthly (Aug-Sep 1979), 86, No. 7, 542. Reprinted in The Mathematical Intelligencer (1983), 5, No. 3.
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Metaphysics: An attempt to prove the incredible by an appeal to the unintelligible.
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No experience whatsoever could prove that the heavens rotate daily and not the earth.
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No science doth make known the first principles whereon it buildeth; but they are always taken as plain and manifest in themselves, or as proved and granted already, some former knowledge having made them evident.
In The of Mr. Richard Hooker in Eight Books Of the Laws of Ecclesiastical Polity (1723), Book 3, 69.
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Noise proves nothing. Often a hen who has merely laid an egg cackles as if she laid an asteroid. — Pudd’nhead Wilson’s New Calendar
Epigraph to Chap. 5, In More Tramps Abroad (1897), Vol. 1, 40. Also published under the title Following the Equator (1897).
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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 can so quickly blur and distort the facts as desire—the wish to use the facts for some purpose of your own—and nothing can so surely destroy the truth. As soon as the witness wants to prove something he is no longer impartial and his evidence is no longer to be trusted.
From 'Getting at the Truth', The Saturday Review (19 Sep 1953), 36, No. 38, 12. Excerpted in Meta Riley Emberger and Marian Ross Hall, Scientific Writing (1955), 400.
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Of possible quadruple algebras the one that had seemed to him by far the most beautiful and remarkable was practically identical with quaternions, and that he thought it most interesting that a calculus which so strongly appealed to the human mind by its intrinsic beauty and symmetry should prove to be especially adapted to the study of natural phenomena. The mind of man and that of Nature’s God must work in the same channels.
As quoted in W. E. Byerly (writing as a Professor Emeritus at Harvard University, but a former student at a Peirce lecture on Hamilton’s new calculus of quaternions), 'Benjamin Peirce: II. Reminiscences', The American Mathematical Monthly (Jan 1925), 32, No. 1, 6.
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One of the principal results of civilization is to reduce more and more the limits within which the different elements of society fluctuate. The more intelligence increases the more these limits are reduced, and the nearer we approach the beautiful and the good. The perfectibility of the human species results as a necessary consequence of all our researches. Physical defects and monstrosities are gradually disappearing; the frequency and severity of diseases are resisted more successfully by the progress of modern science; the moral qualities of man are proving themselves not less capable of improvement; and the more we advance, the less we shall have need to fear those great political convulsions and wars and their attendant results, which are the scourges of mankind.
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Our world is not an optimal place, fine tuned by omnipotent forces of selection. It is a quirky mass of imperfections, working well enough (often admirably); a jury-rigged set of adaptations built of curious parts made available by past histories in different contexts ... A world optimally adapted to current environments is a world without history, and a world without history might have been created as we find it. History matters; it confounds perfection and proves that current life transformed its own past.
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Prove all things; hold fast that which is good.
Bible
In Thessalonians 5:21, King James Version, Bible. For example, in Robert Carroll and Stephen Prickett, eds., The Bible: Authorized King James Version (2008), 255.
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Psychoanalysis is a science conducted by lunatics for lunatics. They are generally concerned with proving that people are irresponsible; and they certainly succeed in proving that some people are.
From Illustrated London News (23 Jun 1928). In Dale Ahlquist (ed.) The Universe According to G.K. Chesterton: A Dictionary of the Mad, Mundane and Metaphysical (2013), 93.
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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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Science is often regarded as the most objective and truth-directed of human enterprises, and since direct observation is supposed to be the favored route to factuality, many people equate respectable science with visual scrutiny–just the facts ma’am, and palpably before my eyes. But science is a battery of observational and inferential methods, all directed to the testing of propositions that can, in principle, be definitely proven false ... At all scales, from smallest to largest, quickest to slowest, many well-documented conclusions of science lie beyond the strictly limited domain of direct observation. No one has ever seen an electron or a black hole, the events of a picosecond or a geological eon.
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Science is really in the business of disproving current models or changing them to conform to new information. In essence, we are constantly proving our latest ideas wrong.
John Mitchinson and John Lloyd, If Ignorance Is Bliss, Why Aren't There More Happy People?: Smart Quotes for Dumb Times (2009), 274.
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Since the examination of consistency is a task that cannot be avoided, it appears necessary to axiomatize logic itself and to prove that number theory and set theory are only parts of logic. This method was prepared long ago (not least by Frege’s profound investigations); it has been most successfully explained by the acute mathematician and logician Russell. One could regard the completion of this magnificent Russellian enterprise of the axiomatization of logic as the crowning achievement of the work of axiomatization as a whole.
Address (11 Sep 1917), 'Axiomatisches Denken' delivered before the Swiss Mathematical Society in Zürich. Translated by Ewald as 'Axiomatic Thought', (1918), in William Bragg Ewald, From Kant to Hilbert (1996), Vol. 2, 1113.
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Some of the men stood talking in this room, and at the right of the door a little knot had formed round a small table, the center of which was the mathematics student, who was eagerly talking. He had made the assertion that one could draw through a given point more than one parallel to a straight line; Frau Hagenström had cried out that this was impossible, and he had gone on to prove it so conclusively that his hearers were constrained to behave as though they understood.
In Little Herr Friedemann (1961), 25.
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The average gambler will say “The player who stakes his whole fortune on a single play is a fool, and the science of mathematics can not prove him to be otherwise.” The reply is obvious: “The science of mathematics never attempts the impossible, it merely shows that other players are greater fools.”
Concluding remarks to his mathematical proof, with certain assumptions, that the best betting strategy for “Gambler’s Ruin” would be to always make his largest stake on his first play. In 'Gambler’s Ruin', Annals of Mathematics (Jul 1909), 2nd Series, 10, No. 4, 189. This is also seen, without primary source, quoted as “It is true that a man who does this is a fool. I have only proved that a man who does anything else is an even bigger fool,” in Harold Eves, Return to Mathematical Circles (1988), 39.
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The dexterous management of terms and being able to fend and prove with them, I know has and does pass in the world for a great part of learning; but it is learning distinct from knowledge, for knowledge consists only in perceiving the habitudes and relations of ideas one to another, which is done without words; the intervention of sounds helps nothing to it. And hence we see that there is least use of distinction where there is most knowledge: I mean in mathematics, where men have determined ideas with known names to them; and so, there being no room for equivocations, there is no need of distinctions.
In Conduct of the Understanding, Sect. 31.
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The existence of trousers proves that God meant us to be bipeds.
Caption to cartoon showing one 19th century man holding a book, speaking to another holding a book titled “Darwin.” Accompanying article by Ralph Estling, 'The Trouble With Thinking Backwards', New Scientist (2 Jun 1983), 620.
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The fact that man produces a concept ‘I’ besides the totality of his mental and emotional experiences or perceptions does not prove that there must be any specific existence behind such a concept. We are succumbing to illusions produced by our self-created language, without reaching a better understanding of anything. Most of so-called philosophy is due to this kind of fallacy.
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The fact that your patient gets well does not prove that your diagnosis was correct.
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The facts proved by geology are briefly these: that during an immense, but unknown period, the surface of the earth has undergone successive changes; land has sunk beneath the ocean, while fresh land has risen up from it; mountain chains have been elevated; islands have been formed into continents, and continents submerged till they have become islands; and these changes have taken place, not once merely, but perhaps hundreds, perhaps thousands of times.
In 'On the Law which has regulated the Introduction of New Species', The Annals and Magazine of Natural History (1855), 16, No. 93, 184.
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The individual on his own is stable only so long as he is possessed of self-esteem. The maintenance of self-esteem is a continuous task which taxes all of the individual’s powers and inner resources. We have to prove our worth and justify our existence anew each day. When, for whatever reason, self-esteem is unattainable, the autonomous individual becomes a highly explosive entity. He turns away from an unpromising self and plunges into the pursuit of pride—the explosive substitute for self-esteem. All social disturbances and upheavals have their roots in crises of individual self-esteem, and the great endeavor in which the masses most readily unite is basically a search for pride.
In The Passionate State of Mind (1955), 18
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The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.
Opening of Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree. The paper was originally published (1824) in French, as a pamphlet, in Oslo. Collected in Œuvres Complètes (1881), Vol. 1, 28. Translation by W.H. Langdon collected in David Eugene Smith, A Source Book in Mathematics (2012), 261. In this work, he showed why—despite two centuries of efforts by mathematicians—solving equations of the fifth degree would remain futile. The insights from this paper led to the modern theory of equations.
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The motto in the pursuit of knowledge, of whatever kind, has always been, “Hope all things;—Prove all things.”
From Address (Oct 1874) delivered at Guy’s Hospital, 'On The Study of Medicine', printed in British Medical journal (1874), 2, 425. Collected in Sir William Withey Gull and Theodore Dyke Acland (ed.), A Collection of the Published Writings of William Withey Gull (1896), 5.
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The oceans are the planet’s last great living wilderness, man’s only remaining frontier on Earth, and perhaps his last chance to prove himself as a rational species.
The Forests of the Sea
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The one quality that seems to be so universal among eccentrics is … so subjective as to be incapable of being proved or disproved, yet … eccentrics appear to be happier than the rest of us.
From a summary his study of 1,000 people, done at Royal Edinburgh Hospital in Scotland. In David Weeks, David Joseph Weeks and Jamie James, Eccentrics (1995), 38.
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The professor may choose familiar topics as a starting point. The students collect material, work problems, observe regularities, frame hypotheses, discover and prove theorems for themselves. … the student knows what he is doing and where he is going; he is secure in his mastery of the subject, strengthened in confidence of himself. He has had the experience of discovering mathematics. He no longer thinks of mathematics as static dogma learned by rote. He sees mathematics as something growing and developing, mathematical concepts as something continually revised and enriched in the light of new knowledge. The course may have covered a very limited region, but it should leave the student ready to explore further on his own.
In A Concrete Approach to Abstract Algebra (1959), 1-2.
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The proving power of the intellect or the senses was questioned by the skeptics more than two thousand years ago; but they were browbeaten into confusion by the glory of Newtonian physics.
In Radio Lecture (30 Jun 1973) broadcast by the Open University, collected in Imre Lakatos, John Worrall (ed.) and Gregory Currie (ed.), 'Introduction: Science and Pseudoscience', The Methodology of Scientific Research Programmes (1978, 1980), Vol. 1, 8.
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The rainbow, “the bridge of the gods,” proved to be the bridge to our understanding of light—much more important.
Epigraph in Isaac Asimov’s Book of Science and Nature Quotations (1988), 189.
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The reasoning of mathematicians is founded on certain and infallible principles. Every word they use conveys a determinate idea, and by accurate definitions they excite the same ideas in the mind of the reader that were in the mind of the writer. When they have defined the terms they intend to make use of, they premise a few axioms, or self-evident principles, that every one must assent to as soon as proposed. They then take for granted certain postulates, that no one can deny them, such as, that a right line may be drawn from any given point to another, and from these plain, simple principles they have raised most astonishing speculations, and proved the extent of the human mind to be more spacious and capacious than any other science.
In Diary, Works (1850), Vol. 2, 21.
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The story of a theory’s failure often strikes readers as sad and unsatisfying. Since science thrives on self-correction, we who practice this most challenging of human arts do not share such a feeling. We may be unhappy if a favored hypothesis loses or chagrined if theories that we proposed prove inadequate. But refutation almost always contains positive lessons that overwhelm disappointment, even when no new and comprehensive theory has yet filled the void.
…...
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The Synthesis consists in assuming the Causes discovered and established as Principles, and by them explaining the Phænomena proceeding from them, and proving the Explanations.
From 'Query 31', Opticks (1704, 2nd ed., 1718), 380-381.
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The whole history of physics proves that a new discovery is quite likely lurking at the next decimal place.
In 'The Romance of the Next Decimal Place', Science (1 Jan 1932), 75, No. 1931, 3.
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The work of Planck and Einstein proved that light behaved as particles in some ways and that the ether therefore was not needed for light to travel through a vacuum. When this was done, the ether was no longer useful and it was dropped with a glad cry. The ether has never been required since. It does not exist now; in fact, it never existed.
In Asimov on Physics (1976), 85. Also in Isaac Asimov’s Book of Science and Nature Quotations (1988), 212.
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Things sweet to taste prove in digestion sour.
Louis Klopsch, Many Thoughts of Many Minds (1896), 67.
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This alleged damage which the small radioactivity is causing—supposedly cancer and leukemia—has not been proved, to the best of my knowledge, by decent and clear statistics. It is possible that there is damage. It is even possible, to my mind, that there is no damage; and there is the possibility, further, that very small amounts of radioactivity are helpful.
From debate (20 Feb 1958) between Linus Pauling and Edward Teller on WQED-TV, San Francisco. Transcript published as Fallout and Disarmament: The Pauling-Teller Debate (1958). Reprinted in 'Fallout and Disarmament: A Debate between Linus Pauling and Edward Teller', Daedalus (Spring 1958), 87, No. 2, 155.
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This new integral of Lebesgue is proving itself a wonderful tool. I might compare it with a modern Krupp gun, so easily does it penetrate barriers which were impregnable.
In 'Current Tendencies of Mathematical Research', Bulletin of the American Mathematical Society (Oct 1916), 23, 7. Previously given as an address read at the Quarter Centennial of the University of Chicago, before a conference of the mathematical, physical and astronomical departments.
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Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the logic piano imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
From 'Les Mathématiques et la Logique', Science et Méthode (1908, 1920), Livre 2, Chap. 3, Sec. 2, 157. English as in Henri Poincaré and George Bruce Halsted (trans.), 'Mathematics and Logic', Science and Method collected in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method (1913), 451. From the French, “Ainsi, c’est bien entendu, pour démontrer un théorème, il n’est pas nécessaire ni même utile de savoir ce qu’il veut dire. On pourrait remplacer le géomètre par le piano à raisonner imaginé par Stanley Jevons; ou, si l’on aime mieux, on pourrait imaginer une machine où l’on introduirait les axiomes par un bout pendant qu’on recueillerait les théorèmes à l’autre bout, comme cette machine légendaire de Chicago où les porcs entrent vivants et d’où ils sortent transformés en jambons et en saucisses. Pas plus que ces machines, le mathématicien n’a besoin de comprendre ce qu’il fait”.
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To illustrate the apparent contrast between statistics and truth … may I quote a remark I once overheard: “There are three kinds of lies: white lies, which are justifiable; common lies—these have no justification; and statistics.” Our meaning is similar when we say: “Anything can be proved by figures”; or, modifying a well-known quotation from Goethe, with numbers “all men may contend their charming systems to defend.”
In Probability, Statistics, and Truth (1939), 1.
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We may summarize … the fundamental characteristics and limitations of mathematics as follows: mathematics is ultimately an experimental science, for freedom from contradiction cannot be proved, but only postulated and checked by observation, and similarly existence can only be postulated and checked by observation. Furthermore, mathematics requires the fundamental device of all thought, of analyzing experience into static bits with static meanings.
In The Nature of Physical Theory (1936), 58.
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We must reject the false choice between combating climate change and fostering strong economic growth. If any country can prove that, it’s the United States.
In Hillary Clinton, 'Hillary Clinton: America Must Lead at Paris Climate Talks', Time (29 Nov 2015).
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We must somehow keep the dreams of space exploration alive, for in the long run they will prove to be of far more importance to the human race than the attainment of material benefits. Like Darwin, we have set sail upon an ocean: the cosmic sea of the Universe. There can be no turning back. To do so could well prove to be a guarantee of extinction. When a nation, or a race or a planet turns its back on the future, to concentrate on the present, it cannot see what lies ahead. It can neither plan nor prepare for the future, and thus discards the vital opportunity for determining its evolutionary heritage and perhaps its survival.
…...
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What is now proved was once only imagin’d.
In 'Proverbs', The Poems: With Specimens of the Prose Writings of William Blake (1885), 280.
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What is truth ? A man may prove much that has no other truth but in him, and all be a turnip lantern leading to a precipice over the sea.
In Sir William Withey Gull and Theodore Dyke Acland (ed.), A Collection of the Published Writings of William Withey Gull (1896), li.
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What vexes me most is, that my female friends, who could bear me very well a dozen years ago, have now forsaken me, although I am not so old in proportion to them as I formerly was: which I can prove by arithmetic, for then I was double their age, which now I am not.
In Letter (7 Feb 1736) to Alexander Pope, The Works of Jonathan Swift (1841), Vol. 2, 764.
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When asked what it was like to set about proving something, the mathematician likened proving a theorem to seeing the peak of a mountain and trying to climb to the top. One establishes a base camp and begins scaling the mountain’s sheer face, encountering obstacles at every turn, often retracing one’s steps and struggling every foot of the journey. Finally when the top is reached, one stands examining the peak, taking in the view of the surrounding countryside and then noting the automobile road up the other side!
Space-filler in The Two-Year College Mathematics Journal (Nov 1980), 11, No. 5, 295.
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Whoever … proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured.
In J. Heidrich (ed.), 'Von Menschlicher Proportion', Albrecht Dürer’s Schriftlicher Nachlass (1920), 270. From the original German: “Welcher aber … durch die Geometria sein Ding beweist und die gründliche Wahrheit anzeigt, dem soll alle Welt glauben. Denn da ist man gefangen.”
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X-rays will prove to be a hoax.
In Marc J. Madou, Fundamentals of Microfabrication: the Science of Miniaturization (2nd ed., 2002), 325.
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[E.H.] Moore was presenting a paper on a highly technical topic to a large gathering of faculty and graduate students from all parts of the country. When half way through he discovered what seemed to be an error (though probably no one else in the room observed it). He stopped and re-examined the doubtful step for several minutes and then, convinced of the error, he abruptly dismissed the meeting—to the astonishment of most of the audience. It was an evidence of intellectual courage as well as honesty and doubtless won for him the supreme admiration of every person in the group—an admiration which was in no wise diminished, but rather increased, when at a later meeting he announced that after all he had been able to prove the step to be correct.
In Obituary, 'Eliakim Hastings Moore', The American Mathematical Monthly (Apr 1933), 40, 191.
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[J.J.] Sylvester’s methods! He had none. “Three lectures will be delivered on a New Universal Algebra,” he would say; then, “The course must be extended to twelve.” It did last all the rest of that year. The following year the course was to be Substitutions-Théorie, by Netto. We all got the text. He lectured about three times, following the text closely and stopping sharp at the end of the hour. Then he began to think about matrices again. “I must give one lecture a week on those,” he said. He could not confine himself to the hour, nor to the one lecture a week. Two weeks were passed, and Netto was forgotten entirely and never mentioned again. Statements like the following were not unfrequent in his lectures: “I haven’t proved this, but I am as sure as I can be of anything that it must be so. From this it will follow, etc.” At the next lecture it turned out that what he was so sure of was false. Never mind, he kept on forever guessing and trying, and presently a wonderful discovery followed, then another and another. Afterward he would go back and work it all over again, and surprise us with all sorts of side lights. He then made another leap in the dark, more treasures were discovered, and so on forever.
As quoted by Florian Cajori, in Teaching and History of Mathematics in the United States (1890), 265-266.
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[Someone] remarked to me once: Physicians should not say, I have cured this man, but, This man didn’t die in my care. In physics too one might say, For such and such a phenomenon I have determined causes whose absurdity cannot finally be proved, instead of saying, I have explained it.
As quoted in Joseph Peter Stern, Lichtenberg: A Doctrine of Scattered Occasions: Reconstructed From His Aphorisms and Reflections (1959), 297.
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~~[Misquoted ?]~~ It gives me the same pleasure when someone else proves a good theorem as when I do it myself.
Webmaster believes this is a first person paraphrase misquoted from “It was said round 1912 that it gave him the same pleasure when someone else proved a good theorem as if he had done it himself,” as given in Littlewood, A Mathematician’s Miscellany (1953), reissued as Béla Bollobás, Littlewood’s Miscellany (1986), 24. Notice that Littlewood states this in the third person—about Landau. As a first person quote, it is sometimes cited to a much later author, Desmond MacHale, Comic Sections (1993). Webmaster has not yet found a primary source to authenticate this first person version, and since the earlier occurrence is hedged with “it was said”, Webmaster suggests this first person version is a misquote, and is not an authentic verbatim quote.
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… There can be no doubt about faith and not reason being the ultima ratio. Even Euclid, who has laid himself as little open to the charge of credulity as any writer who ever lived, cannot get beyond this. He has no demonstrable first premise. He requires postulates and axioms which transcend demonstration, and without which he can do nothing. His superstructure indeed is demonstration, but his ground his faith. Nor again can he get further than telling a man he is a fool if he persists in differing from him. He says “which is absurd,” and declines to discuss the matter further. Faith and authority, therefore, prove to be as necessary for him as for anyone else.
In The Way of All Flesh (1917), 319-320.
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“Divide et impera” is as true in algebra as in statecraft; but no less true and even more fertile is the maxim “auge et impera”.The more to do or to prove, the easier the doing or the proof.
In 'Proof of the Fundamental Theorem of Invariants', Philosophic Magazine (1878), 186. In Collected Mathematical Papers, 3, 126. [The Latin phrases, “Divide/auge et impera” translate as “Divide/increase and rule”.
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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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