Abel, Niels H. (1802 - 1829) If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words,the most important parts of mathematics stand without a foundation. In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, p. 188.
Abel, Niels H. (1802 - 1829) [A reply to a question about how he got his expertise:] By studying the masters and not their pupils.
Abel, Niels H. (1802 - 1829) [About Gauss' mathematical writing style] He is like the fox, who effaces his tracks in the sand with his tail. In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, p. 177.
Adams, Douglas (1952 - ) Bistromathics itself is simply a revolutionary new way of understanding the behavior of numbers. Just as Einstein observed that space was not an absolute but depended on the observer's movement in space, and that time was not an absolute, but depended on the observer's movement in time, so it is now realized that numbers are not absolute, but depend on the observer's movement in restaurants. Life, the Universe and Everything. New York: Harmony Books, 1982.
Adams, Douglas (1952 - ) The first nonabsolute number is the number of people for whom the table is reserved. This will vary during the course of the first three telephone calls to the restaurant, and then bear no apparent relation to the number of people who actually turn up, or to the number of people who subsequently join them after the show/match/party/gig, or to the number of people who leave when they see who else has turned up. The second nonabsolute number is the given time of arrival, which is now known to be one of the most bizarre of mathematical concepts, a recipriversexcluson, a number whose existence can only be defined as being anything other than itself. In other words, the given time of arrival is the one moment of time at which it is impossible that any member of the party will arrive. Recipriversexclusons now play a vital part in many branches of math, including statistics and accountancy and also form the basic equations used to engineer the Somebody Else's Problem field. The third and most mysterious piece of nonabsoluteness of all lies in the relationship between the number of items on the bill, the cost of each item, the number of people at the table and what they are each prepared to pay for. (The number of people who have actually brought any money is only a subphenomenon of this field.) Life, the Universe and Everything. New York: Harmony Books, 1982.
Adams, Douglas (1952 - ) Numbers written on restaurant bills within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe. This single statement took the scientific world by storm. It completely revolutionized it. So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years. Life, the Universe and Everything. New York: Harmony Books, 1982.
Adams, John (1735 - 1826) I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain. Letter to Abigail Adams, May 12, 1780.
Adler, Alfred Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing. "Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.
Adler, Alfred The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then, little will ever be accomplished. "Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.
Adler, Alfred In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is. Reflections: mathematics and creativity, New Yorker, 47(1972), no. 53, 39 - 45.
Aiken, Conrad [At a musical concert:] ...the music's pure algebra of enchantment.
Allen, Woody Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals. In Howard Eves' Return to Mathematical Circles, Boston: Prindle, Weber, and Schmidt, 1988.
Anglin, W.S. Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere. "Mathematics and History", Mathematical Intelligencer, v. 4, no. 4.
Anonymous If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom. In Ivor Thomas "Greek Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Anonymous Defendit numerus: There is safety in numbers. In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1452.
Anonymous Like the crest of a peacock so is mathematics at the head of all knowledge. [An old Indian saying. Also, "Like the Crest of a Peacock" is the title of a book by G.G. Joseph]
Anonymous Referee's report: This paper contains much that is new and much that is true. Unfortunately, that which is true is not new and that which is new is not true. In H.Eves Return to Mathematical Circles, Boston: Prindle, Weber, and Schmidt, 1988.
Arbuthnot, John The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning; and when they cannot it's a sign our knowledge of them is very small and confus'd; and when a Mathematical Reasoning can be had it's as great a folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you. Of the Laws of Chance. (1692)
Aristophanes (ca 444 - 380 BC) Meton: With the straight ruler I set to work To make the circle four-cornered [First(?) allusion to the problem of squaring the circle]
Aristotle (ca 330 BC) Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends [mathematics] has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research. Metaphysica, 1-981b
Aristotle (ca 330 BC) The whole is more than the sum of its parts. Metaphysica 10f-1045a
Aristotle The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things. Metaphysica 1-5
Aristotle It is not once nor twice but times without number that the same ideas make their appearance in the world. "On The Heavens", in T. L. Heath Manual of Greek Mathematics, Oxford: Oxford University Press, 1931.
Aristotle To Thales the primary question was not what do we know, but how do we know it. Mathematical Intelligencer v. 6, no. 3, 1984.
Aristotle The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. Metaphysica, 3-1078b.
Ascham, Roger (1515-1568) Mark all mathematical heads which be wholly and only bent on these sciences, how solitary they be themselves, how unfit to live with others, how unapt to serve the world. In E G R Taylor, Mathematical Practitioners of Tudor and Stuart England, Cambridge: Cambridge University Press, 1954.
Aubrey, John (1626-1697) [About Thomas Hobbes:] He was 40 years 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 I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "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, that at last he was demonstratively convinced of that trueth. This made him in love with geometry. In O. L. Dick (ed.) Brief Lives, Oxford: Oxford University Press, 1960, p. 604.
Auden, W. H. (1907-1973) How happy the lot of the mathematician. He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve. The Dyer's Hand, London: Faber & Faber, 1948.
Auden, W. H. (1907-1973) Thou shalt not answer questionnaires Or quizzes upon world affairs, Nor with compliance Take any test. Thou shalt not sit with statisticians nor commit A social science. "Under which lyre" in Collected Poems of W H Auden, London: Faber and Faber.
Augarten, Stan Computers are composed of nothing more than logic gates stretched out to the horizon in a vast numerical irrigation system. State of the Art: A Photographic History of the Integrated Circuit. New York: Ticknor and Fields.
St. Augustine (354-430) Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist. The City of God.
St. Augustine (354-430) The good Christian should beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell. DeGenesi ad Litteram, Book II, xviii, 37 [Note: mathematician = astrologer]
St. Augustine (354-430) If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me? De Magistro ch X, 23.
Babbage, Charles (1792-1871) Errors using inadequate data are much less than those using no data at all.
Babbage, Charles (1792-1871) On two occasions I have been asked [by members of Parliament], 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.
Babbage, Charles (1792-1871) I wish to God these calculations had been executed by steam. In H. Eves In Mathematical Circles,, Boston: Prindle, Weber and Schmidt, 1969.
Bacon, Sir Francis (1561-1626) And as for Mixed Mathematics, I may only make this prediction, that there cannot fail to be more kinds of them, as nature grows further disclosed. Advancement of Learning book 2; De Augmentis book 3.
Bacon, Roger For the things of this world cannot be made known without a knowledge of mathematics. Opus Majus part 4 Distinctia Prima cap 1, 1267.
Bacon, Roger In the mathematics I can report no deficience, except that it be that men do not sufficiently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures; so in the mathematics, that use which is collateral and intervenient is no less worthy than that which is principal and intended. John Fauvel and Jeremy Gray (eds.) A History of Mathematics: A Reader, Sheridan House, 1987.
Baker, H. F. [On the concept of group:] ... what a wealth, what a grandeur of thought may spring from what slightbeginnings. Florian Cajori, A History of Mathematics, New York, 1919, p 283.
Bagehot, Walter Life is a school of probability. Quoted in J. R. Newman (ed.) The World of Mathematics, Simon and Schuster, New York,1956, p. 1360.
Balzac, Honore de (1799 - 1850) Numbers are intellectual witnesses that belong only to mankind.
Banville, John Throughout the 1960s and 1970s devoted Beckett readers greeted each successively shorter volume from the master with a mixture of awe and apprehensiveness; it was like watching a great mathematician wielding an infinitesimal calculus, his equations approaching nearer and still nearer to the null point. Quoted in a review of Samuel Beckett's Nohow On: I11 Seen I11 Said, Worstward Ho, in The New York Review of Books, August 13, 1992.
Bell, Eric Temple (1883-1960) Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. In H. Eves Return to Mathematical Circles., Boston: Prindle, Weber and Schmidt, 1988.
Bell, Eric Temple (1883-1960) Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos. Mathematics, Queen and Servant of Science, New York, 1951, p 164.
Bell, Eric Temple (1883-1960) It is the perennial youthfulness of mathematics itself which marks it off with a disconcerting immortality from the other sciences.
Bell, Eric Temple (1883-1960) The Handmaiden of the Sciences. [Book by that title.]
Bell, Eric Temple (1883-1960) Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.
Bell, Eric Temple (1883-1960) Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.
Bell, Eric Temple (1883-1960) "Obvious" is the most dangerous word in mathematics.
Bell, Eric Temple (1883-1960) The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular. In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Bell, Eric Temple (1883-1960) If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics. In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 308.
Bell, Eric Temple (1883-1960) The longer mathematics lives the more abstract -- and therefore, possibly also the more practical -- it becomes. In The Mathematical Intelligencer, vol. 13, no. 1, Winter 1991.
Bell, Eric Temple (1883-1960) The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry. In R Crayshaw-Williams The Search For Truth, p. 191.
Bell, Eric Temple (1883-1960) If "Number rules the universe" as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number. In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.
Bell, Eric Temple (1883-1960) I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly. In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Belloc, Hillaire (1870-1953) Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death. The Silence of the Sea
Bentham, Jeremy (1748-1832) O Logic: born gatekeeper to the Temple of Science, victim of capricious destiny: doomed hitherto to be the drudge of pedants: come to the aid of thy master, Legislation. In J. Browning (ed.) Works.
Bernoulli, Daniel ...it would be better for the true physics if there were no mathematicians on earth. In The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
Bernoulli, Jacques (Jakob?) (1654-1705) I recognize the lion by his paw. [After reading an anonymous solution to a problem that he realized was Newton's solution.] In G. Simmons, Calculus Gems, New York: McGraw Hill, 1992, p. 136.
Bernoulli, Johann But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not.
Besicovitch, A.S. A mathematician's reputation rests on the number of bad proofs he has given. In J. E. Littlewood A Mathematician's Miscellany, Methuen & Co. Ltd., 1953.
Blake God forbid that Truth should be confined to Mathematical Demonstration! Notes on Reynold's Discourses, c. 1808.
Blake What is now proved was once only imagin'd. The Marriage of Heaven and Hell, 1790-3.
Bohr, Niels Henrik David (1885-1962) An expert is a man who has made all the mistakes, which can be made, in a very narrow field.
The Bible I returned and saw under the sun that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all. Ecclesiastes.
Bolyai, J‡nos (1802 - 1860) Out of nothing I have created a strange new universe. [A reference to the creation of a non-euclidean geometry.]
Bolyai, Wolfgang (1775-1856) [To son J‡nos:] For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life. [Bolyai's father urging him to give up work on non-Euclidian geometry.] In P. Davis and R. Hersh The Mathematical Experience , Boston: Houghton Mifflin Co., 1981, p. 220.
Bourbaki Structures are the weapons of the mathematician.
Bridgman, P. W. It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention. The Logic of Modern Physics, New York, 1972.
Brown, George Spencer (1923 - ) To arrive at the simplest truth, as Newton knew and practiced, requires years of contemplation. Not activity Not reasoning. Not calculating. Not busy behaviour of any kind. Not reading. Not talking. Not making an effort. Not thinking. Simply bearing in mind what it is one needs to know. And yet those with the courage to tread this path to real discovery are not only offered practically no guidance on how to do so, they are actively discouraged and have to set about it in secret, pretending meanwhile to be diligently engaged in the frantic diversions and to conform with the deadening personal opinions which are continually being thrust upon them. The Laws of Form. 1969.
Browne, Sir Thomas (1605-1682) God is like a skilful Geometrician. Religio Medici I, 16.
Browne, Sir Thomas (1605-1682) All things began in Order, so shall they end, and so shall they begin again, according to the Ordainer of Order, and the mystical mathematicks of the City of Heaven. Hydriotaphia, Urn-burial and the Garden of Cyrus, 1896.
Browne, Sir Thomas (1605-1682) ...indeed what reason may not go to Schoole to the wisdome of Bees, Aunts, and Spiders? what wise hand teacheth them to doe what reason cannot teach us? ruder heads stand amazed at those prodigious pieces of nature, Whales, Elephants, Dromidaries and Camels; these I confesse, are the Colossus and Majestick pieces of her hand; but in these narrow Engines there is more curious Mathematicks, and the civilitie of these little Citizens more neatly sets forth the wisedome of their Maker. In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1001.
Buck, Pearl S. (1892 - 1973) No one really understood music unless he was a scientist, her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. "And relationships," he had told her, "contained the essential meaning of life." The Goddess Abides, Pt. I, 1972.
Burke, Edmund The age of chivalry is gone. That of sophisters, economists and calculators has succeeded. Reflections on the Revolution in France.
Butler, Bishop To us probability is the very guide of life. Preface to Analogy.
Butler, Samuel (1612 - 1680) ... 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. The Way of All Flesh.
Byron When Newton saw an apple fall, he found ... A mode of proving that the earth turnd round In a most natural whirl, called gravitation; And thus is the sole mortal who could grapple Since Adam, with a fall or with an apple.
Caballero, James I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors. Everybody a mathematician?,CAIP Quarterly 2 (Fall, 1989).
Cardano, Girolamo (1501 - 1576) To throw in a fair game at Hazards only three-spots, when something great is at stake, or some business is the hazard, is a natural occurrence and deserves to be so deemed; and even when they come up the same way for a second time if the throw be repeated. If the third and fourth plays are the same, surely there is occasion for suspicion on the part of a prudent man. De Vita Propria Liber.
Carlyle, Thomas (1795 - 1881) It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity of the universe. Sartor Resartus III.
Carlyle, Thomas (1795-1881) Teaching school is but another word for sure and not very slow destruction. In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Carlyle, Thomas (1795-1881) A witty statesman said, you might prove anything by figures. Chartism.
Carroll, Lewis What I tell you three times is true. The Hunting of the Snark.
Carroll, Lewis The different branches of Arithmetic -- Ambition, Distraction, Uglification, and Derision. Alice in Wonderland.
Carroll, Lewis "Can you do addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?" "I don't know," said Alice. "I lost count." Through the Looking Glass.
Carroll, Lewis "Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." Alice in Wonderland.