Huxley, Thomas Henry (1825-1895) This seems to be one of the many cases in which the admitted accuracy of mathematical processes is allowed to throw a wholly inadmissible appearance of authority over the results obtained by them. Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff of any degree of fineness; but, nevertheless, what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulae will not get a definite result out of loose data. Quarterly Journal of the Geological Society, 25,1869.
Huxley, Thomas Henry (1825-1895) The mathematician starts with a few propositions, the proof of which is so obvious that they are called selfevident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature authority and tradition furnish the data, and the mental operations are deductive. "Scientific Education -Notes of an After-dinner Speech." Macmillan's Magazine Vol XX, 1869.
Huxley, Thomas Henry (1825-1895) It is the first duty of a hypothesis to be intelligible.
Ibn Khaldun (1332-1406) Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence. The Muqaddimah. An Introduction to History.
Isidore of Seville (ca 600 ad) Take from all things their number and all shall perish.
Jacobi, Carl It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world. In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Jacobi, Carl God ever arithmetizes. In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.
Jacobi, Carl One should always generalize. (Man muss immer generalisieren) In P. Davis and R. Hersh The Mathematical Experience, Boston: BirkhSuser, 1981.
Jacobi, Carl The real end of science is the honor of the human mind. In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Jacobi, Carl It is often more convenient to possess the ashes of great men than to possess the men themselves during their lifetime. [Commenting on the return of Descartes' remains to France] In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Jacobi, Carl Mathematics is the science of what is clear by itself. In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
James, William (1842 - 1910) The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal. Collected Essays.
Jeans, Sir James The essential fact is that all the pictures which science now draws of nature, and which alone seem capable of according with observational facts, are mathematical pictures. In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Jeans, Sir James From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician. Mysterious Universe.
Jefferson, Thomas ...the science of calculation also is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not be in indulged in by one who is to have a profession to follow for his subsistence. In J. Robert Oppenheimer "The Encouragement of Science" in I. Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.
Jevons, William Stanley It is clear that Economics, if it is to be a science at all, must be a mathematical science. Theory of Political Economy.
Johnson, Samuel (1709-1784) Sir, I have found you an argument. I am not obliged to find you an understanding. J. Boswell The Life of Samuel Johnson, 1784.
Jowett, Benjamin (1817 - 1893) Logic is neither a science or an art, but a dodge. In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Kant, Emmanual (1724 - 1804) The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience. The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
Kant, Emmanual (1724 - 1804) All human knowledge thus begins with intuitions, proceeds thence to concepts, and ends with ideas. Quoted in Hilbert's Foundations of Geometry.
Kaplan, Abraham Mathematics is not yet capable of coping with the naivete of the mathematician himself. Sociology Learns the Language of Mathematics.
Kaplansky, Irving We [he and Halmos] share a philosophy about linear algebra: we think basis-free, we write basis-free , but when the chips are down we close the office door and compute with matrices like fury. Paul Halmos: Celebrating 50 Years of Mathematics.
Karlin, Samuel (1923 - ) The purpose of models is not to fit the data but to sharpen the questions. 11th R A Fisher Memorial Lecture, Royal Society 20, April 1983.
Kasner, E. and Newman, J. Mathematics is man's own handiwork, subject only to the limitations imposed by the laws of thought. Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Kasner, E. and Newman, J. ...we have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed. Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Kasner, E. and Newman, J. Mathematics is the science which uses easy words for hard ideas. Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Kasner, E. and Newman, J. Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense. Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Kasner, E. and Newman, J. Perhaps the greatest paradox of all is that there are paradoxes in mathematics. Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Kasner, E. and Newman, J. When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home. Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Kasner, E. and Newman, J. R. The testament of science is so continually in a flux that the heresy of yesterday is the gospel of today and the fundamentalism of tomorrow. E. Kasner and J. R. Newman, Mathematics and the Imagination, Simon and Schuster, 1940.
Keller, Helen (1880 - 1968) Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important... The knowledge doesn't make life any sweeter or happier, does it? The Story of My Life. 1903.
Kelley, John A topologist is one who doesn't know the difference between a doughnut and a coffee cup. In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Kepler, Johannes (1571-1630) A mind is accustomed to mathematical deduction, when confronted with the faulty foundations of astrology, resists a long, long time, like an obstinate mule, until compelled by beating and curses to put its foot into that dirty puddle. In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Kepler, Johannes (1571-1630) Where there is matter, there is geometry. (Ubi materia, ibi geometria.) J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990.
Kepler, Johannes (1571-1630) The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.
Kepler, Johannes (1571-1630) Nature uses as little as possible of anything.
Keynes, John Maynard It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability. The Application of Probability to Conduct.
Kleinhenz, Robert J. 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 countrysideand then noting the automobile road up the other side!
Kline, Morris A proof tells us where to concentrate our doubts. In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Kline, Morris Statistics: the mathematical theory of ignorance. In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Kline, Morris Logic is the art of going wrong with confidence. In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Kline, Morris Universities hire professors the way some men choose wives - they want the ones the others will admire. Why the Professor Can't Teach. St. Martin's Press, 1977, p 92.
Koestler, Arthur (1905- ) In the index to the six hundred odd pages of Arnold Toynbee's A Study of History, abridged version, the names of Copernicus, Galileo, Descartes and Newton do not occur yet their cosmic quest destroyed the medieval vision of an immutable social order in a walled-in universe and transformed the European landscape, society, culture, habits and general outlook, as thoroughly as if a new species had arisen on this planet. In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Koestler, Arthur (1905- ) Nobody before the Pythagoreans had thought that mathematical relations held the secret of the universe. Twenty-five centuries later, Europe is still blessed and cursed with their heritage. To non-European civilizations, the idea that numbers are the key to both wisdom and power, seems never to have occurred. The Sleepwalkers. 1959.
Kovalevsky, Sonja Say what you know, do what you must, come what may. [Motto on her paper "On the Problem of the Rotation of a Solid Body about a Fixed Point."]
Kraft, Prinz zu Hohlenlohe-Ingelfingen (1827 - 1892) Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else. Attributed by Karl Schellbach. In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Kronecker, Leopold (1823 - 1891) God made the integers, all else is the work of man. Jahresberichte der Deutschen Mathematiker Vereinigung.
Kronecker, Leopold (1823-1891) Number theorists are like lotus-eaters -- having once tasted of this food they can never give it up. In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
La Touche, Mrs. I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are permutations and aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then from the top down, the result is always different. Mathematical Gazette, v. 12.
LaGrange, Joseph-Louis The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure. Preface to MŽcanique Analytique.
LaGrange, Joseph-Louis [said about the chemist Lavoisier:] It took the mob only a moment to remove his head; a century will not suffice to reproduce it. H. Eves An Introduction to the History of Mathematics, 5th Ed., Saunders.
LaGrange, Joseph-Louis When we ask advice, we are usually looking for an accomplice.
Lakatos, Imre That sometimes clear ... and sometimes vague stuff ... which is ... mathematics. In P. Davis and R. Hersh The Mathematical Experience, Boston: BirkhSuser, 1981.
Lanczos, Cornelius Most of the arts, as painting, sculpture, and music, have emotional appeal to the general public. This is because these arts can be experienced by some one or more of our senses. Such is not true of the art of mathematics; this art can be appreciated only by mathematicians, and to become a mathematician requires a long period of intensive training. The community of mathematicians is similar to an imaginary community of musical composers whose only satisfaction is obtained by the interchange among themselves of the musical scores they compose. In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Landau, E. [Asked for a testimony to the effect that Emmy Noether was a great woman mathematician, he said:] I can testify that she is a great mathematician, but that she is a woman, I cannot swear. J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co ltd., 1953.
Landau, Susan There's a touch of the priesthood in the academic world, a sense that a scholar should not be distracted by the mundane tasks of day-to-day living. I used to have great stretches of time to work. Now I have research thoughts while making peanut butter and jelly sandwiches. Sure it's impossible to write down ideas while reading "curious George" to a two-year-old. On the other hand, as my husband was leaving graduate school for his first job, his thesis advisor told him, "You may wonder how a professor gets any research done when one has to teach, advise students, serve on committees, referee papers, write letters of recommendation, interview prospective faculty. Well, I take long showers." In Her Own Words: Six Mathematicians Comment on Their Lives and Careers. Notices of the AMS, V. 38, no. 7 (September 1991), p. 704.
Lang, Andrew (1844-1912) He uses statistics as a drunken man uses lamp posts -- for support rather than illumination. Treasury of Humorous Quotations.
Langer, Rudoph E. [about Fourier] It was, no doubt, partially because of his very disregard for rigor that he was able to take conceptual steps which were inherently impossible to men of more critical genius. In P. Davis and R. Hersh The Mathematical Experience, Boston: BirkhSuser, 1981.
Lao Tze (604-531 B.C.) A good calculator does not need artificial aids. Tao Te Ching, ch 27.
de Laplace, Pierre-Simon (1749 - 1827) What we know is not much. What we do not know is immense. (Allegedly his last words.) DeMorgan's Budget of Paradoxes.
de Laplace, Pierre-Simon (1749 - 1827) [His last words, according to De Morgan:] Man follows only phantoms. DeMorgan's Budget of Paradoxes.
de Laplace, Pierre-Simon (1749 - 1827) Nature laughs at the difficulties of integration. In J. W. Krutch "The Colloid and the Crystal", in I. Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.
de Laplace, Pierre-Simon (1749 - 1827) Read Euler: he is our master in everything. In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
de Laplace, Pierre-Simon (1749 - 1827) Such is the advantage of a well constructed language that its simplified notation often becomes the source of profound theories. In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
de Laplace, Pierre-Simon (1749 - 1827) Napoleon: You have written this huge book on the system of the world without once mentioning the author of the universe. Laplace: Sire, I had no need of that hypothesis. Later when told by Napoleon about the incident, Lagrange commented: Ah, but that is a fine hypothesis. It explains so many things. DeMorgan's Budget of Paradoxes.
de Laplace, Pierre-Simon (1749 - 1827) [said about Napier's logarithms:] ...by shortening the labors doubled the life of the astronomer. In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
de Laplace, Pierre-Simon (1749 - 1827) It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Leach, Edmund Ronald (1910 - 1989) How can a modern anthropologist embark upon a generalization with any hope of arriving at a satisfactory conclusion? By thinking of the organizational ideas that are present in any society as a mathematical pattern. Rethinking Anthropology. 1961.
Leacock, Stephen How can you shorten the subject? That stern struggle with the multiplication table, for many people not yet ended in victory, how can you make it less? Square root, as obdurate as a hardwood stump in a pasturenothing but years of effort can extract it. You can't hurry the process. Or pass from arithmetic to algebra; you can't shoulder your way past quadratic equations or ripple through the binomial theorem. Instead, the other way; your feet are impeded in the tangled growth, your pace slackens, you sink and fall somewhere near the binomial theorem with the calculus in sight on the horizon. So died, for each of us, still bravely fighting, our mathematical training; except for a set of people called "mathematicians" -- born so, like crooks. In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Lebesgue, Henri (1875 - 1941) In my opinion, a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy -- an opinion, moreover, which has been expressed by many philosophers. Scientific American, 211, September 1964, p. 129.
Lehrer, Thomas Andrew (1928- ) In one word he told me the secret of success in mathematics: plagiarize only be sure always to call it please research. Lobachevski (A musical recording.)
Leibniz, Gottfried Whilhem (1646-1716) [about him:] It is rare to find learned men who are clean, do not stink and have a sense of humour. [attributed variously to Charles Louis de Secondat Montesquieu and to the Duchess of OrlŽans]
Leibniz, Gottfried Whilhem (1646-1716) Nothing is more important than to see the sources of invention which are, in my opinion more interesting than the inventions themselves. J. Koenderink, Solid Shape, Cambridge Mass.: MIT Press, 1990.
Leibniz, Gottfried Whilhem (1646-1716) Music is the pleasure the human soul experiences from counting without being aware that it is counting. In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Leibniz, Gottfried Whilhem (1646-1716) The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.
Leibniz, Gottfried Whilhem (1646-1716) He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times. In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Leibniz, Gottfried Whilhem (1646-1716) In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished. In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Leibniz, Gottfried Whilhem (1646-1716) The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road. New Essays Concerning Human Understanding, IV, XII.
Leibniz, Gottfried Whilhem (1646-1716) Although the whole of this life were said to be nothing but a dream and the physical world nothing but a phantasm, I should call this dream or phantasm real enough, if, using reason well, we were never deceived by it. In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Leibniz, Gottfried Whilhem (1646-1716) The soul is the mirror of an indestructible universe. The Monadology.
Leybourn, William (1626-1700) But leaving those of the Body, I shall proceed to such Recreation as adorn the Mind; of which those of the Mathematicks are inferior to none. Pleasure with Profit, 1694.
Lichtenberg, Georg Christoph (1742 - 1799) All mathematical laws which we find in Nature are always suspect to me, in spite of their beauty. They give me no pleasure. They are merely auxiliaries. At close range it is all not true. In J P Stern Lichtenberg, 1959.
Lichtenberg, Georg Christoph (1742 - 1799) The great trick of regarding small departures from the truth as the truth itself -- on which is founded the entire integral calculus -- is also the basis of our witty speculations, where the whole thing would often collapse if we considered the departures with philosophical rigour. Aphorisms.
Lichtenberg, Georg Christoph (1742 - 1799) In mathematical analysis we call x the undetermined part of line a: the rest we don't call y, as we do in common life, but a-x. Hence mathematical language has great advantages over the common language.
Lichtenberg, Georg Christoph (1742 - 1799) I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, "Oh, I see. That's how it must be." This is a sign that they explain it to themselves from within their own system.
le Lionnais, Francois Who has not be amazed to learn that the function y = e^x , like a phoenix rising again from its own ashes, is its own derivative? Great Currents of Mathematical Thought, vol. 1, New York: Dover Publications.
Lippman, Gabriel (1845-1921) [On the Gaussian curve, remarked to PoincarŽ:] Experimentalists think that it is a mathematical theorem while the mathematicians believe it to be an experimental fact. In D'Arcy Thompson On Growth and Form, 1917.
Littlewood, J. E. (1885 -1977) It is true that I should have been surprised in the past to learn that Professor Hardy had joined the Oxford Group. But one could not say the adverse chance was 1:10. Mathematics is a dangerous profession; an appreciable proportion of us go mad, and then this particular event would be quite likely. A Mathematician's Miscellany, Methuen and Co. ltd., 1953.