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, " "Abel, Niels H. (1802 - 1829)" "
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.
Abel, Niels H. (1802 - 1829)
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 " "Adams, Douglas (1952 - 2001)" "
Adams, Douglas (1952 - 2001)
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," "Adams, Douglas (1952 - 2001)" "
Adams, Douglas (1952 - 2001)
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 s "Adams, Douglas (1952 - 2001)" "
Adams, Douglas (1952 - 2001)
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 th" "Adams, John (1735 - 1826)" "
Adams, John (1735 - 1826)
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 nothin" "Adler, Alfred" "
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 Mag" "Adler, Alfred" "
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 th" "Adler, Alfred" "
Adler, Alfred
[At a musical concert:]
...the music's pure algebra of enchantment.
Aiken, Conrad
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 conc "Allen, Woody" "
Allen, Woody
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 else" "Anglin, W.S." "
Anglin, W.S.
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 Tho" Anonymous
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" Anonymous
Anonymous
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 Mathematic" "Arbuthnot, John" "
Arbuthnot, John
Meton: With the straight ruler I set to work
To make the circle four-cornered
[First(?) allusion to the problem of squaring the circle]
Aristophanes (ca 444 - 380 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 necessa" Aristotle (ca 330 BC)
Aristotle (ca 330 BC)
The whole is more than the sum of its parts.
Metaphysica 10f-1045a
Aristotle (ca 330 BC)
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.
Aristotle
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 EnglandAscham, Roger (1515-1568)
[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 propositio" "Aubrey, John (1626-1697)" "
Aubrey, John (1626-1697)
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, W. H. (1907-1973)" "
Auden, W. H. (1907-1973)
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.
Augarten, Stan
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 e" St. Augustine (354-430)
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.
DeGe" St. Augustine (354-430)
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.
St. Augustine (354-430)
Errors using inadequate data are much less than those using no data at all.
"Babbage, Charles (1792-1871)" " 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 " "Babbage, Charles (1792-1871)" "
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.
Babbage, Charles (1792-1871)
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, Sir Francis (1561-1626)
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 to" "Bacon, Roger" "
Bacon, Roger
[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.
Baker, H. F.
Life is a school of probability.
Quoted in J. R. Newman (ed.) The World of Mathematics, Simon and Schuster, New York,1956, p. 1360.
Bagehot, Walter
Numbers are intellectual witnesses that belong only to mankind.
"Balzac, Honore de (1799 - 1850)" " Balzac, Honore de (1799 - 1850)
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 app" "Banville, John" "
Banville, John
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)" " 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)" "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 useful" "Bell, Eric Temple (1883-1960)" "
Bell, Eric Temple (1883-1960)
"Obvious" is the most dangerous word in mathematics.
"Bell, Eric Temple (1883-1960)" "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 " "Bell, Eric Temple (1883-1960)" "
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, Ne" "Bell, Eric Temple (1883-1960)" "
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)" "
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.
Bell, Eric Temple (1883-1960)
Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death.
The Silence of the Sea
Belloc, Hillaire (1870-1953)
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.
Bentham, Jeremy (1748-1832)
...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, Daniel
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, Jacques (Jakob?) (1654-1705)
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" "Bernoulli, Johann" "
Bernoulli, Johann
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.
Besicovitch, A.S.
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.
Blake
An expert is a man who has made all the mistakes, which can be made, in a very narrow field.
Bohr, Niels Henrik David (1885-1962)
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.
E" The Bible
The Bible
Out of nothing I have created a strange new universe.
[A reference to the creation of a non-euclidean geometry.]
Bolyai, János (1802 - 1860)
[To son János:]
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 g" "Bolyai, Wolfgang (1775-1856)" "
Bolyai, Wolfgang (1775-1856)
Structures are the weapons of the mathematician.
BourbakiBourbaki
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.
Bridgman, P. W.
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 i" "Brown, George Spencer (1923 - )" "
Brown, George Spencer (1923 - )
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 C" "Browne, Sir Thomas (1605-1682)" "
Browne, Sir Thomas (1605-1682)
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" "Buck, Pearl S. (1892 - 1973)" "
Buck, Pearl S. (1892 - 1973)
The age of chivalry is gone. That of sophisters, economists and calculators has succeeded.
Reflections on the Revolution in France.
Burke, Edmund
To us probability is the very guide of life.
Preface to Analogy.
Butler, Bishop
... 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 requ" "Butler, Samuel (1612 - 1680)" "
Butler, Samuel (1612 - 1680)
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.
Byron
Caballero, James
" "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 repea" "Cardano, Girolamo (1501 - 1576)" "
Cardano, Girolamo (1501 - 1576)
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.
Carlyle, Thomas (1795-1881)
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" "Carroll, Lewis" "
Carroll, Lewis
"Then you should say what you mean," the March Hare went on.
"I do, " Alice hastily replied; "at least I mean what I say, that's the same thing, you know."
"Not the same thing a bit!" said the Hatter. &qu" "Carroll, Lewis" "
Carroll, Lewis
"It's very good jam," said the Queen.
"Well, I don't want any to-day, at any rate."
"You couldn't have it if you did want it," the Queen said. "The rule is jam tomorrow and jam yesterday but never jam to-day.&" "Carroll, Lewis" "
Carroll, Lewis
"When I use a word," Humpty Dumpty said, in a rather scornful tone, "it means just what I choose it to mean - neither more nor less."
"The question is," said Alice, "whether you can make words mean so many differe" "Carroll, Lewis" "
Carroll, Lewis
Entre le pénis et les mathématiques... il n'existe rien. Rien! C'est le vide.
Voyage au bout de la nuit. Paris: Gallimard.
Céline, Louis-Ferdinand (1894 - 1961)
A thing is obvious mathematically after you see it.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Carmichael, R. D.
Men pass away, but their deeds abide.
[His last words (?)]
In H. Eves Mathematical Circles Revisted, Boston: Prindle, Weber and Schmidt, 1971.
Cauchy, Augustin-Louis (1789 - 1857)
As for everything else, so for a mathematical theory: beauty can be perceived but not explained.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Cayley, Arthur
Projective geometry is all geometry.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Cayley, Arthur
...treat Nature by the sphere, the cylinder and the cone...
" "Cézanne, Paul (1839 - 1906)" " Cézanne, Paul (1839 - 1906)
To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls.
In G. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, page 198.
Chebyshev
There is no national science just as there is no national multiplication table; what is national is no longer science.
In V. P. Ponomarev Mysli o nauke Kishinev, 1973.
Chekov, Anton (1860 - 1904)
Poets do not go mad; but chess-players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination.
Orthodo" "Chesterton, G. K. (1874 - 1936)" "
Chesterton, G. K. (1874 - 1936)
You can only find truth with logic if you have already found truth without it.
The Man who was Orthodox. 1963.
Chesterton, G. K. (1874 - 1936)
It isn't that they can't see the solution. It is that they can't see the problem.
The Point of a Pin in The Scandal of Father Brown.
Chesterton, G. K. (1874 - 1936)
"I think you're begging the question," said Haydock, "and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how lik" "Christie, Agatha" "
Christie, Agatha
I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours I found it quite enthralling.
An Aut" "Christie, Agatha" "
Christie, Agatha
It is a good thing from an uneducated man to read books of quotations.
Roving Commission in My Early Life. 1930.
Churchill, [Sir] Winston Spencer (1874-1965)
I had a feeling once about Mathematics - that I saw it all. Depth beyond depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing its s "Churchill, Sir Winston Spencer (1874-1965)" "
Churchill, Sir Winston Spencer (1874-1965)
The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems.
In J.E. Littlewood A Mathematician's Miscellany. Methuen and Co., Ltd. 1953.
Churchman, C. W.
The composer opens the cage door for arithmetic, the draftsman gives geometry its freedom.
" Cocteau Cocteau
...from the time of Kepler to that of Newton, and from Newton to Hartley, not only all things in external nature, but the subtlest mysteries of life and organization, and even of the intellect and moral being, were conjured within the magic circle of m" "Coleridge, Samuel Taylor (1772-1834)" "
Coleridge, Samuel Taylor (1772-1834)
C'este donc par l'étude des mathématiques, et seulement par elle, que l'on peut se faire une idée juste et approfondie de ce que c'est qu'une science.
Quoted by T. H. Huxley in Fortnightly Review, Vol. II, N.S. 5." "Comte, Auguste (1798-1857)" "
Comte, Auguste (1798-1857)
Don't talk to me of your Archimedes' lever. He was an absentminded person with a mathematical imagination. Mathematics commands all my respect, but I have no use for engines. Give me the right word and the right accent and I will move the world.
Pr" "Conrad, Joseph" "
Conrad, Joseph
[Upon proving that the best betting strategy for "Gambler's Ruin" was to bet all on the first trial.]
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 H. Ev "Coolidge, Julian Lowell (1873 - 1954)" "
Coolidge, Julian Lowell (1873 - 1954)
Mathematics is written for mathematicians.
De Revolutionibus.
Copernicus, Nicholaus (1473-1543)
In my experience most mathematicians are intellectually lazy and especially dislike reading experimental papers. He (René Thom) seemed to have very strong biological intuitions but unfortunately of negative sign.
What Mad Pursuit. Lon "Crick, Francis Harry Compton (1916 - )" "
Crick, Francis Harry Compton (1916 - )
Revolutions never occur in mathematics.
Historia Mathematica. 1975.
Crowe, Michael
Just go on..and faith will soon return.
[To a friend hesitant with respect to infinitesimals.]
In P. J. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
D'Alembert, Jean Le Rond (1717-1783)
Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this .... The imagination in a mat" "D'Alembert, Jean Le Rond (1717-1783)" "
D'Alembert, Jean Le Rond (1717-17830
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape wi" Dantzig
Dantzig
Neither in the subjective nor in the objective world can we find a criterion for the reality of the number concept, because the first contains no such concept, and the second contains nothing that is free from the concept. How then can we arrive at a c" Dantzig
Dantzig
Every new body of discovery is mathematical in form, because there is no other guidance we can have.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Darwin, Charles
Mathematics seems to endow one with something like a new sense.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Darwin, Charles
The numbers are a catalyst that can help turn raving madmen into polite humans.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Davis, Philip J.
, Scientific American, 211, (Sept. 1964), 51 - 59.
" "Davis, Philip J." " Davis, Philip J.
One began to hear it said that World War I was the chemists' war, World War II was the physicists' war, World War III (may it never come) will be the mathematicians' war.
The Mathematical Experience, Boston: Birkhäuser, 1981.
Davis, Philip J. and Hersh, Reuben
Mathematics is the only instructional material that can be presented in an entirely undogmatic way.
In The Mathematical Intelligencer, v. 5, no. 2, 1983.
Dehn, Max
[When asked about his age.] I was x years old in the year x^2.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
De Morgan, Augustus (1806-1871)
It is easier to square the circle than to get round a mathematician.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
De Morgan, Augustus (1806-1871)
Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.
Transactions Cambridge Philosophical So" "De Morgan, Augustus (1806-1871)" "
De Morgan, Augustus (1806-1871)
Of all things, good sense is the most fairly distributed: everyone thinks he is so well supplied with it that even those who are the hardest to satisfy in every other respect never desire more of it than they already have.
Discours de la M&eacut" "Descartes, René (1596-1650)" "
Descartes, René (1596-1650)
Each problem that I solved became a rule which served afterwards to solve other problems.
Discours de la Méthode. 1637.
Descartes, René (1596-1650)
If I found any new truths in the sciences, I can say that they follow from, or depend on, five or six principal problems which I succeeded in solving and which I regard as so many battles where the fortunes of war were on my side.
Discours de la" "Descartes, René (1596-1650)" "
Descartes, René (1596-1650)
I concluded that I might take as a general rule the principle that all things which we very clearly and obviously conceive are true: only observing, however, that there is some difficulty in rightly determining the objects which we distinctly conceive." "Descartes, René (1596-1650)" "
Descartes, René (1596-1650)
I thought the following four [rules] would be enough, provided that I made a firm and constant resolution not to fail even once in the observance of them. The first was never to accept anything as true if I had not evident knowledge of its being so; th" "Descartes, René (1596-1650)" "
Descartes, René (1596-1650)
These long chains of perfectly simple and easy reasonings by means of which geometers are accustomed to carry out their most difficult demonstrations had led me to fancy that everything that can fall under human knowledge forms a similar sequence; and "Descartes, René (1596-1650)" "
Descartes, René (1596-1650)
When writing about transcendental issues, be transcendentally clear.
In G. Simmons Calculus Gems. New York: McGraw Hill Inc., 1992.
Descartes, René (1596-1650)
If we possessed a thorough knowledge of all the parts of the seed of any animal (e.g. man), we could from that alone, be reasons entirely mathematical and certain, deduce the whole conformation and figure of each of its members, and, conversely if we k" "Descartes, René (1596-1650)" "
Descartes, René (1596-1650)
Cogito Ergo Sum. "I think, therefore I am."
Discours de la Méthode. 1637.
Descartes, René (1596-1650)
I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.
La Geometrie.
Descartes, René (1596-1650)
Perfect numbers like perfect men are very rare.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Descartes, René (1596-1650)
omnia apud me mathematica fiunt.
With me everything turns into mathematics.
Descartes, René (1596-1650)
It is not enough to have a good mind. The main thing is to use it well.
Discours de la Méthode. 1637.
Descartes, René (1596-1650)
If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things.
Discours de la Méthode. 1637.
Descartes, René (1596-1650)
Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. Quantum mechanics for example would be a religion under this definition. But mathematics w" "De Sua, F. (1956)" "
De Sua, F. (1956)
[His epitaph.]
This tomb hold Diophantus Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He " Diophantus
Diophantus
I think that there is a moral to this story, namely that it is more important to have beauty in one's equations that to have them fit experiment. If Schroedinger had been more confident of his work, he could have published it some months earlier, and " "Dirac, Paul Adrien Maurice (1902- )" "
Dirac, Paul Adrien Maurice (1902- )
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
In P. J. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
Dirac, Paul Adrien Maurice (1902- )
In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 19" "Dirac, Paul Adrien Maurice (1902- )" "
Dirac, Paul Adrien Maurice (1902- )
There are three kinds of lies: lies, damned lies, and statistics.
Mark Twain. Autobiography.
Disraeli, Benjamin
Pereant qui ante nos nostra dixerunt.
"To the devil with those who published before us."
[Quoted by St. Jerome, his pupil]
Donatus, Aelius (4th Century)
Detection is, or ought to be, an exact sciences and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces much the same effect as if you worked a love story or an elopement into the f" "Doyle, Sir Arthur Conan (1859-1930)" "
Doyle, Sir Arthur Conan (1859-1930)
When you have eliminated the impossible, what ever remains, however improbable must be the truth.
The Sign of Four.
Doyle, Sir Arthur Conan (1859-1930)
From a drop of water a logician could predict an Atlantic or a Niagara.
A study in Scarlet 1929.
Doyle, Sir Arthur Conan (1859-1930)
It is a capital mistake to theorize before one has data.
Scandal in Bohemia.
Doyle, Sir Arthur Conan (1859-1930)
Mere poets are sottish as mere drunkards are, who live in a continual mist, without seeing or judging anything clearly. A man should be learned in several sciences, and should have a reasonable, philosophical and in some measure a mathematical head, to" "Dryden, John (1631-1700)" "
Dryden, John (1631-1700)
Gauss replied, when asked how soon he expected to reach certain mathematical conclusions, that he had them long ago, all he was worrying about was how to reach them!
In Mechanisms of Discovery in I. S. Gordon and S. Sorkin (eds.) The Armc" "Dubos, René J." "
Dubos, René J.
Logic, like whiskey, loses its beneficial effect when taken in too large quantities.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Dunsany, Lord
But when great and ingenious artists behold their so inept performances, not undeservedly do they ridicule the blindness of such men; since sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge, although with plenty " "Dürer, Albrecht (1471-1528)" "
Dürer, Albrecht (1471-1528)
Whoever ... proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured.
J Heidrich (ed.) Albrecht Dürer's schriftlicher Nachlass Berlin, 1920.
Dürer, Albrecht (1471-1528)
And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art...
Course in the Art of Measurement
Dürer, Albrecht (1471-1528)
I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.
Missed Opportunities, 1972. (Gibbs Lecture?)
Dyson, Freeman
For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created.
Mathematics in the Physical Sciences.
Dyson, Freeman
The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then th" "Dyson, Freeman" "
Dyson, Freeman
Proof is the idol before whom the pure mathematician tortures himself.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Eddington, Sir Arthur (1882-1944)
We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about `and'.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Eddington, Sir Arthur (1882-1944)
We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own." "Eddington, Sir Arthur (1882-1944)" "
Eddington, Sir Arthur (1882-1944)
It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Eddington, Sir Arthur (1882-1944)
I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe and the same number of electrons.
The Philosophy of Physical Science. Cambridge, 1939.
Eddington, Sir Arthur (1882-1944)
To the pure geometer the radius of curvature is an incidental characteristic - like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidenta "Eddington, Sir Arthur (1882-1944)" "
Eddington, Sir Arthur (1882-1944)
Human life is proverbially uncertain; few things are more certain than the solvency of a life-insurance company.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Eddington, Sir Arthur (1882-1944)
When I am violently beset with temptations, or cannot rid myself of evil thoughts, [I resolve] to do some Arithmetic, or Geometry, or some other study, which necessarily engages all my thoughts, and unavoidably keeps them from wandering.
In T. Mall" "Edwards, Jonathon" "
Edwards, Jonathon
" "If you ask mathematicians what they do, yo always get the same answer. They think. They think about difficult and unusual problems. They do not think about ordinary problems: they just write down the answers.
Mathematics Magazine, v. 65 no." "Egrafov, M." "
Egrafov, M.
A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant.
Jagdish Mehra (ed.) The Physicist's Conception of Nature, 1973.
Eigen, Manfred (1927 - )
[During a lecture:]This has been done elegantly by Minkowski; but chalk is cheaper than grey matter, and we will do it as it comes.
[Attributed by Pólya.]
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co. Ltd., 1953.<" "Einstein, Albert (1879-1955)" "
Einstein, Albert (1879-1955)
Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.
Einstein, Albert (1879-1955)
I don't believe in mathematics.
Quoted by Carl Seelig. Albert Einstein.
Einstein, Albert (1879-1955)
Imagination is more important than knowledge.
On Science.
Einstein, Albert (1879-1955)
The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.
What I Believe.
Einstein, Albert (1879-1955)
The bitter and the sweet come from the outside, the hard from within, from one's own efforts.
Out of My Later Years.
Einstein, Albert (1879-1955)
Gott würfelt nicht.
"Einstein, Albert (1879-1955)" " Einstein, Albert (1879-1955)
Common sense is the collection of prejudices acquired by age eighteen.
In E. T. Bell Mathematics, Queen and Servant of the Sciences. 1952.
Einstein, Albert (1879-1955)
God does not care about our mathematical difficulties. He integrates empirically.
L. Infeld Quest, 1942.
Einstein, Albert (1879-1955)
How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?
" "Einstein, Albert (1879-1955)" " Einstein, Albert (1879-1955)
[About Newton]
Nature to him was an open book, whose letters he could read without effort.
In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.
Einstein, Albert (1879-1955)
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Einstein, Albert (1879-1955)
What is this frog and mouse battle among the mathematicians?
[i.e. Brouwer vs. Hilbert]
In H. Eves Mathematical Circles Squared Boston: Prindle, Weber and Schmidt, 1972.
Einstein, Albert (1879-1955)
Raffiniert ist der Herr Gott, aber boshaft ist er nicht. God is subtle, but he is not malicious.
Inscribed in Fine Hall, Princeton University.
Einstein, Albert (1879-1955)
Nature hides her secrets because of her essential loftiness, but not by means of ruse.
" "Einstein, Albert (1879-1955)" " Einstein, Albert (1879-1955)
The human mind has first to construct forms, independently, before we can find them in things.
" "Einstein, Albert (1879-1955)" " Einstein, Albert (1879-1955)
Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
In A. Sommerfelt "To Albert Einstein's Seventieth Birthday" in Paul A. Schilpp (ed.) Albert Einstein, Philosopher-Scientist, Eva" "Einstein, Albert (1879-1955)" "
Einstein, Albert (1879-1955)
Do not worry about your difficulties in mathematics, I assure you that mine are greater.
" "Einstein, Albert (1879-1955)" " Einstein, Albert (1879-1955)
The truth of a theory is in your mind, not in your eyes.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Einstein, Albert (1879-1955)
These thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words afterward.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Einstein, Albert (1879-1955)
A human being is a part of the whole, called by us "Universe," a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the resta kind of optical delusion of his consciousness. This delus" "Einstein, Albert (1879-1955)" "
Einstein, Albert (1879-1955)
The world needs heroes and it's better they be harmless men like me than villains like Hitler.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Einstein, Albert (1879-1955)
It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiousity of inquiry.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Einstein, Albert (1879-1955)
Everything that is really great and inspiring is created by the individual who can labor in freedom.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Einstein, Albert (1879-1955)
The search for truth is more precious than its possession.
The American Mathematical Monthly v. 100 no. 3.
Einstein, Albert (1879-1955)
If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.
We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think nothing. Already the distance-concept is logically arbitrary; there n" "Einstein, Albert (1879-1955)" "
Einstein, Albert (1879-1955)
Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.
The Evolution of Physics.
Einstein, Albert (1879-1955)
Science without religion is lame; religion without science is blind.
Reader's Digest, Nov. 1973.
Einstein, Albert (1879-1955)
The mathematician has reached the highest rung on the ladder of human thought.
The Dance of Life.
Ellis, Havelock
It is here [in mathematics] that the artist has the fullest scope of his imagination.
The Dance of Life.
Ellis, Havelock
God is a child; and when he began to play, he cultivated mathematics. It is the most godly of man's games.
Das blinde Spiel. 1954.
Erath, V.
Mathematics is not yet ready for such problems.
[Attributed by Paul Halmos.]
The American Mathematical Monthly, Nov. 1992
Erdös, Paul
A Mathematician is a machine for turning coffee into theorems.
"Erdös, Paul" "Erdös, Paul
If a nonnegative quantity was so small that it is smaller than any given one, then it certainly could not be anything but zero. To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are n" "Euler, Leonhard (1707 - 1783)" "
Euler, Leonhard (1707 - 1783)
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
In G. Simmons Calculus Gems, New York: " "Euler, Leonhard (1707-1783)" "
Euler, Leonhard (1707-1783)
[upon losing the use of his right eye]
Now I will have less distraction.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Euler, Leonhard (1707-1783)
In the pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven.
Quoted by E.T." "Everett, Edward (1794-1865)" "
Everett, Edward (1794-1865)
A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Eves, Howard W.
An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Eves, Howard W.
Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematici "Eves, Howard W." "
Eves, Howard W.
One is hard pressed to think of universal customs that man has successfully established on earth. There is one, however, of which he can boast the universal adoption of the Hindu-Arabic numerals to record numbers. In this we perhaps have man's unique w" "Eves, Howard W." "
Eves, Howard W.
If the entire Mandelbrot set were placed on an ordinary sheet of paper, the tiny sections of boundary we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only mathematicians have microscopes fine eno" "Ewing, John" "
Ewing, John
Sample recommendation letter:
Dear Search Committee Chair,
I am writing this letter for Mr. John Smith who has applied for a position in your department. I should start by saying that I cannot recommend him too highly.
In fact, there is no " Focus Newsletter (MAA)
Focus Newsletter (MAA)
[In the margin of his copy of Diophantus' Arithmetica, Fermat wrote]
To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have a" "de Fermat, Pierre (1601?-1665)" "
de Fermat, Pierre (1601?-1665)
And perhaps, posterity will thank me for having shown it that the ancients did not know everything.
In D. M. Burton, Elementary Number Theory, Boston: Allyn and Bacon, Inc., 1976.
de Fermat, Pierre (1601?-1665)
We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn't any p" "Feynman, Richard Philips (1918 - 1988)" "
Feynman, Richard Philips (1918 - 1988)
The solution of problems is one of the lowest forms of mathematical research, ... yet its educational value cannot be overestimated. It is the ladder by which the mind ascends into higher fields of original research and investigation. Many dormant mind" "Finkel, Benjamin Franklin" "
Finkel, Benjamin Franklin
The effort of the economist is to "see," to picture the interplay of economic elements. The more clearly cut these elements appear in his vision, the better; the more elements he can grasp and hold in his mind at once, the better. The economi" "Fisher, Irving" "
Fisher, Irving
Natural selection is a mechanism for generating an exceedingly high degree of improbability.
"Fisher, Ronald Aylmer (1890 - 1962)" "Fisher, Ronald Aylmer (1890 - 1962)
To call in the statistician after the experiment is done may be no more than asking hm to perform a postmortem examination: he may be able to say what the experiment died of.
Indian Statistical Congress, Sankhya, ca 1938.
Fisher, Ronald Aylmer (1890-1962)
Poetry is as exact a science as geometry.
"Flaubert, Gustave (1821-1880)" " Flaubert, Gustave (1821-1880)
Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12" "Flaubert, Gustave (1821-1880)" "
Flaubert, Gustave (1821-1880)
Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another.
Quoted in V. H. Larney Abstract Algebra: A First Course, Bost" "Fontenelle, Bernard Le Bovier (1657-1757)" "
Fontenelle, Bernard Le Bovier (1657-1757)
A work of morality, politics, criticism will be more elegant, other things being equal, if it is shaped by the hand of geometry.
Preface sur l'Utilité des Mathématiques et de la Physique, 1729.
Fontenelle, Bernard Le Bovier (1657-1757)
Leibniz never married; he had considered it at the age of fifty; but the person he had in mind asked for time to reflect. This gave Leibniz time to reflect, too, and so he never married.
Eloge de le Leibniz.
Fontenelle, Bernard Le Bovier (1657-1757)
Whereas at the outset geometry is reported to have concerned herself with the measurement of muddy land, she now handles celestial as well as terrestrial problems: she has extended her domain to the furthest bounds of space.
Hodder and Stoughton, <" "Frankland, W.B." "
Frankland, W.B.
For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.
" "Frayn, Michael" "
Frayn, Michael
To your care and recommendation am I indebted for having replaced a half-blind mathematician with a mathematician with both eyes, which will especially please the anatomical members of my Academy.
[To D'Alembert about Lagrange. Euler had vacated th" Frederick the Great (1712-1786)
Frederick the Great (1712-1786)
A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
In Scient "Frege, Gottlob (1848 - 1925)" "
Frege, Gottlob (1848 - 1925)
There can be no question, however, that prolonged commitment to mathematical exercises in economics can be damaging. It leads to the atrophy of judgement and intuition...
Economics, Peace, and Laughter.
Galbraith, John Kenneth
[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which " "Galilei, Galileo (1564 - 1642)" "
Galilei, Galileo (1564 - 1642)
Measure what is measurable, and make measurable what is not so.
Quoted in H. Weyl "Mathematics and the Laws of Nature" in I Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.
Galilei, Galileo (1564 - 1642)
And who can doubt that it will lead to the worst disorders when minds created free by God are compelled to submit slavishly to an outside will? When we are told to deny our senses and subject them to the whim of others? When people devoid of whatsoev "Galilei, Galileo (1564 - 1642)" "
Galilei, Galileo (1564 - 1642)
Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties.
In N. Rose (ed.) Mathema "Galois, Evariste" "
Galois, Evariste
Whenever you can, count.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Galton, [Sir] Francis (1822-1911)
[Statistics are] the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man.
Pearson, The Life and Labours of Francis Galton, 1914.
Galton, Sir Francis (1822-1911)
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 reign" "Galton, Sir Francis (1822-1911)" "
Galton, Sir Francis (1822-1911)
Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals -- the flotsam and jetsam of historical currents. The men wh" "Gardner, Martin" "
Gardner, Martin
Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now" "Gardner, Martin" "
Gardner, Martin
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.
[A reply to Olbers' attempt in 1816 to enti" "Gauss, Karl Friedrich (1777-1855)" "
Gauss, Karl Friedrich (1777-1855)
" "If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 326.
Gauss, Karl Friedrich (1777-1855)
There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us a" "Gauss, Karl Friedrich (1777-1855)" "
Gauss, Karl Friedrich (1777-1855)
You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
In G. Simmons Calculus Gems, New York: McGraw Hi" "Gauss, Karl Friedrich (1777-1855)" "
Gauss, Karl Friedrich (1777-1855)
God does arithmetic.
"Gauss, Karl Friedrich (1777-1855)" " Gauss, Karl Friedrich (1777-1855)
We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.
Letter to Bessel, 1830.
Gauss, Karl Friedrich (1777-1855)
I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.
In G. Simmons Calcu" "Gauss, Karl Friedrich (1777-1855)" "
Gauss, Karl Friedrich (1777-1855)
I have had my results for a long time: but I do not yet know how I am to arrive at them.
In A. Arber The Mind and the Eye 1954.
Gauss, Karl Friedrich (1777-1855)
[His motto:]
Few, but ripe.
Gauss, Karl Friedrich (1777-1855)
[His second motto:]
Thou, nature, art my goddess; to thy laws my services are bound...
W. Shakespeare King Lear.
Gauss, Karl Friedrich (1777-1855)
[attributed to him by H.B Lübsen]
Theory attracts practice as the magnet attracts iron.
Foreword of H.B Lübsen's geometry textbook.
Gauss, Karl Friedrich (1777-1855)
It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satis" "Gauss, Karl Friedrich (1777-1855)" "
Gauss, Karl Friedrich (1777-1855)
Finally, two days ago, I succeeded - not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with " "Gauss, Karl Friedrich (1777-1855)" "
Gauss, Karl Friedrich (1777-1855)
A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that" "Gauss, Karl Friedrich (1777-1855)" "
Gauss, Karl Friedrich (1777-1855)
I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect...geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.<" "Gauss, Karl Friedrich (1777-1855)" "
Gauss, Karl Friedrich (1777-1855)
Lest men suspect your tale untrue,
Keep probability in view.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 1334.
Gay, John
One of the principal objects of theoretical research in my department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.
"Gibbs, Josiah Willard (1839 - 1903)" " Gibbs, Josiah Willard (1839 - 1903)
Mathematics is a language.
"Gibbs, Josiah Willard (1839-1903)" " Gibbs, Josiah Willard (1839-1903)
I'm very good at integral and differential calculus, I know the scientific names of beings animalculous; In short, in matters vegetable, animal, and mineral, I am the very model of a modern Major-General.
The Pirates of Penzance. Act 1.
Gilbert, W. S. (1836 - 1911)
The mathematician requires tact and good taste at every step of his work, and he has to learn to trust to his own instinct to distinguish between what is really worthy of his efforts and what is not.
In H. Eves Mathematical Circles Squared Glaisher, J.W.
And for mathematical science, he that doubts their certainty hath need of a dose of hellebore.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 548.
Glanvill, Joseph
" "
I don't believe in natural science.
[Said to physicist John Bahcall.]
Ed Regis, Who Got Einstein's Office? Addison Wesley, 1987.
Goedel, Kurt
" "It has been said that figures rule the world. Maybe. But I am sure that figures show us whether it is being ruled well or badly.
In J. P. Eckermann, Conversations with Goethe.
Goethe
Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at " Goethe
Goethe
There are no deep theorems -- only theorems that we have not understood very well.
The Mathematical Intelligencer, vol. 5, no. 3, 1983.
Goodman, Nicholas P.
This is not mathematics, it is theology.
[On being exposed to Hilbert's work in invariant theory.]
Quoted in P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
Gordon, P
It wouild be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, `Yes, it is true, but you won't be able to understand the proof.'
John Horgan. Scientific American 269:4 (Oc" "Graham, Ronald" "
Graham, Ronald
Mathematicians have long since regarded it as demeaning to work on problems related to elementary geometry in two or three dimensions, in spite of the fact that it it precisely this sort of mathematics which is of practical value.
Handbook of Ap" "Grünbaum, Branko (1926 - ), and Shephard, G. C. (?)" "
Grünbaum, Branko (1926 - ), and Shephard, G. C. (?)
The shortest path between two truths in the real domain passes through the complex domain.
Quoted in The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
Hadamard, Jacques
Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Hadmard, Jacques
In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind. The catch in this criterion lies in the world "simplest." It is really an aesthetic c "Haldane, John Burdon Sanderson (1892-1964)" "
Haldane, John Burdon Sanderson (1892-1964)
A time will however come (as I believe) when physiology will invade and destroy mathematical physics, as the latter has destroyed geometry.
Daedalus, or Science and the Future, London: Kegan Paul, 1923.
Haldane, John Burdon Sanderson (1892-1964)
Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
I Want to be a Mathematician Halmos, Paul R.
... the student skit at Christmas contained a plaintive line: "Give us Master's exams that our faculty can pass, or give us a faculty that can pass our Master's exams."
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.<" "Halmos, Paul R." "
Halmos, Paul R.
I remember one occasion when I tried to add a little seasoning to a review, but I wasn't allowed to. The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory. The domains of the underlying measures were not" "Halmos, Paul R." "
Halmos, Paul R.
...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
I Want to" "Halmos, Paul R." "
Halmos, Paul R.
The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me -- both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tensio" "Halmos, Paul R." "
Halmos, Paul R.
Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the p" "Halmos, Paul R." "
Halmos, Paul R.
To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
Halmos, Paul R.
Who would not rather have the fame of Archimedes than that of his conqueror Marcellus?
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.
Hamilton, [Sir] William Rowan (1805-1865)
I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc..
In a letter from Tait to Cayley.
Hamilton, Sir William Rowan (1805-1865)
On earth there is nothing great but man; in man there is nothing great but mind.
Lectures on Metaphysics.
Hamilton, Sir William Rowan (1805-1865)
Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly " "Hamming, Richard W." "
Hamming, Richard W.
Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hamming, Richard W.
[On Ramanujan]
I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he r" "Hardy, Godfrey H. (1877 - 1947)" "
Hardy, Godfrey H. (1877 - 1947)
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
A " "Hardy, Godfrey H. (1877 - 1947)" "
Hardy, Godfrey H. (1877 - 1947)
I am interested in mathematics only as a creative art.
A Mathematician's Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 - 1947)
Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
" "Hardy, Godfrey H. (1877 - 1947)" " Hardy, Godfrey H. (1877 - 1947)
In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.
A Mathematician's Apology, London, Cambridge University Press, 1941.
Hardy, Godfrey H. (1877 - 1947)
There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
A Mathematician's Apology, London, Cambridge Univer" "Hardy, Godfrey H. (1877 - 1947)" "
Hardy, Godfrey H. (1877 - 1947)
Young Men should prove theorems, old men should write books.
Quoted by Freeman Dyson in Freeman Dyson: Mathematician, Physicist, and Writer. Interview with Donald J. Albers, The College Mathematics Journal, vol. 25, No. 1, January 1994.
Hardy, Godfrey H. (1877 - 1947)
A science is said to be useful of its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life.
A Mathematician's Apology, London, Cambridge University P" "Hardy, Godfrey H. (1877 - 1947)" "
Hardy, Godfrey H. (1877 - 1947)
The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations.
" "Hardy, Godfrey H. (1877 - 1947)" "
Hardy, Godfrey H. (1877 - 1947)
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
A Mathematician" "Hardy, Godfrey H. (1877 - 1947)" "
Hardy, Godfrey H. (1877 - 1947)
The fact is that there are few more "popular" subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics tha" "Hardy, Godfrey H. (1877 - 1947)" "
Hardy, Godfrey H. (1877 - 1947)
...he seemed to approach the grave as an hyperbolic curve approaches a line, less directly as he got nearer, till it was doubtful if he would ever reach it at all.
Far from the Madding Crowd.
Hardy, Thomas
I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocat" Harish-Chandra
Harish-Chandra
The real danger is not that computers will begin to think like men, but that men will begin to think like computers.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Harris, Sydney J.
God not only plays dice. He also sometimes throws the dice where they cannot be seen.
[See related quotation from Albert Einstein.] Nature 1975 257.
Hawking, Stephen Williams (1942- )
" "[The works of Archimedes] are without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose," "Heath, Sir Thomas" "
Heath, Sir Thomas
[Criticized for using formal mathematical manipulations, without understanding how they worked:]
Should I refuse a good dinner simply because I do not understand the process of digestion?
Heaviside, Oliver (1850-1925)
Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.
Time Enough for Love.
Heinlein, Robert A.
An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them.
Physics and Beyond. 1971.
Heisenberg, Werner (1901-1976)
The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as "All bachelors are unmarried," but they also share the complete lack of empirical content which is associated with tha" "Hempel, Carl G." "
Hempel, Carl G.
The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results.
&quo" "Hempel, Carl G." "
Hempel, Carl G.
...to characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similar" "Hempel, Carl G." "
Hempel, Carl G.
One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right "Henkin, Leon" "
Henkin, Leon
There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine cr" "Hermite, Charles (1822 - 1901)" "
Hermite, Charles (1822 - 1901)
Abel has left mathematicians enough to keep them busy for 500 years.
In G. F. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Hermite, Charles (1822-1901)
We are servants rather than masters in mathematics.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Hermite, Charles (1822-1901)
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser that we are, wiser even than their discoverers, that we get more out of them than was originally put into" "Hertz, Heinrich" "
Hertz, Heinrich
You treat world history as a mathematician does mathematics, in which nothing but laws and formulae exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.
The Glass Bead Ga" "Hesse, Hermann (1877-1962)" "
Hesse, Hermann (1877-1962)
Wir müssen wissen.
Wir werden wissen.
[Engraved on his tombstone in Göttingen.]
Hilbert, David (1862-1943)
Before beginning I should put in three years of intensive study, and I haven't that much time to squander on a probable failure.
[On why he didn't try to solve Fermat's last theorem]
Quoted in E.T. Bell Mathematics, Queen and Servant of Scie" "Hilbert, David (1862-1943)" "
Hilbert, David (1862-1943)
Galileo was no idiot. Only an idiot could believe that science requires martyrdom - that may be necessary in religion, but in time a scientific result will establish itself.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and" "Hilbert, David (1862-1943)" "
Hilbert, David (1862-1943)
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hilbert, David (1862-1943)
Physics is much too hard for physicists.
C. Reid Hilbert, London: Allen and Unwin, 1970.
Hilbert, David (1862-1943)
How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more co" "Hilbert, David (1862-1943)" "
Hilbert, David (1862-1943)
The art of doing mathematics consists in finding that special case which contains all the germs of generality.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Hilbert, David (1862-1943)
The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.
In N. Rose Mathematical Maxims and Min" "Hilbert, David (1862-1943)" "
Hilbert, David (1862-1943)
I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations.
Report on Number Theory, 1897.
Hilbert, David (1862-1943)
" "One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt,1971.
Hilbert, David (1862-1943)
Mathematics knows no races or geographic boundaries; for mathematics,the cultural world is one country.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Hilbert, David (1862-1943)
The infinite! No other question has ever moved so profoundly the spirit of man.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Hilbert, David (1862-1943)
10th August 1851: On Tuesday evening at Museum, at a ball in the gardens. The night was chill, I dropped too suddenly from Differential Calculus into ladies' society, and could not give myself freely to the change. After an hour's attempt so to do, I " "Hirst, Thomas Archer" "
Hirst, Thomas Archer
There is more in Mersenne than in all the universities together.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Hobbes, Thomas
To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad.
["This" is that the volume generated by revolving the region under 1/x from 1 to infinity has finite volume.]
Geometry, which is the only science that it hath pleased God hitherto to bestow on mankind.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Hobbes, Thomas
The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning.
In J. R. Newman (ed.) The World of Mathematics<" "Hobbes, Thomas" "
Hobbes, Thomas
Descartes commanded the future from his study more than Napoleon from the throne.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Holmes, Oliver Wendell
Certitude is not the test of certainty. We have been cocksure of many things that are not so.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Holmes, Oliver Wendell
I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension of the following arithmetical formula: 2 " "Holmes, Oliver Wendell" "
Holmes, Oliver Wendell
The truth of the matter is that, though mathematics truth may be beauty, it can be only glimpsed after much hard thinking. Mathematics is difficult for many human minds to grasp because of its hierarchical structure: one thing builds on another and dep" "Holt, M. and Marjoram, D. T. E." "
Holt, M. and Marjoram, D. T. E.
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.
Gödel, Escher, Bach 1979.
Hofstadter, Douglas R. (1945 - )
Science, being human enquiry, can hear no answer except an answer couched somehow in human tones. Primitive man stood in the mountains and shouted against a cliff; the echo brought back his own voice, and he believed in a disembodied spirit. The scient" "Hughes, Richard" "
Hughes, Richard
If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, `Does it contain any abstract reasoning concerning quantity or number?' No. `Does it contain any experimental reasoning concerning matter of fact and existe" "Hume, David (1711 - 1776)" "
Hume, David (1711 - 1776)
I admit that mathematical science is a good thing. But excessive devotion to it is a bad thing.
Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934.
Huxley, Aldous
If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood, and many other desirable things. As a man he was a failure; as " "Huxley, Aldous" "
Huxley, Aldous
...[he] was as much enchanted by the rudiments of algebra as he would have been if I had given him an engine worked by steam, with a methylated spirit lamp to heat the boiler; more enchanted, perhapsfor the engine would have got broken, and, remaining " "Huxley, Aldous" "
Huxley, Aldous
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 work "Huxley, Thomas Henry (1825-1895)" "
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" "Huxley, Thomas Henry (1825-1895)" "
Huxley, Thomas Henry (1825-1895)
It is the first duty of a hypothesis to be intelligible.
"Huxley, Thomas Henry (1825-1895)" " Huxley, Thomas Henry (1825-1895)
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 ap" Ibn Khaldun (1332-1406)
Ibn Khaldun (1332-1406)
Take from all things their number and all shall perish.
Isidore of Seville (ca 600 ad) Isidore of Seville (ca 600 ad)
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 un" "Jacobi, Carl" "
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: Birkhäuser, 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, We" "Jacobi, Carl" "
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.
Jacobi, Carl
The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.
Collected Essays.
James, William (1842 - 1910)
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" "Jeans, Sir James" "
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.
Jeans, Sir James
...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 "Jefferson, Thomas" "
Jefferson, Thomas
It is clear that Economics, if it is to be a science at all, must be a mathematical science.
Theory of Political Economy.
Jevons, William Stanley
Sir, I have found you an argument. I am not obliged to find you an understanding.
J. Boswell The Life of Samuel Johnson, 1784.
Johnson, Samuel (1709-1784)
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.
Jowett, Benjamin (1817 - 1893)
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.
Kant, Emmanual (1724 - 1804)
Mathematics is not yet capable of coping with the naivete of the mathematician himself.
Sociology Learns the Language of Mathematics.
Kaplan, Abraham
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." "Kaplansky, Irving" "
Kaplansky, Irving
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.
Karlin, Samuel (1923 - )
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 Sch" "Kasner, E. and Newman, J." "
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 " "Kasner, E. and Newman, J." "
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 dangero" "Kasner, E. and Newman, J." "
Kasner, E. and Newman, J.
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.
Kasner, E. and Newman, J. R.
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.
Keller, Helen (1880 - 1968)
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.
Kelley, John
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" "Kepler, Johannes (1571-1630)" "
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)" " Kepler, Johannes (1571-1630)
Nature uses as little as possible of anything.
"Kepler, Johannes (1571-1630)" "Kepler, Johannes (1571-1630)
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 begi" "Keynes, John Maynard" "
Keynes, John Maynard
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, encounter" "Kleinhenz, Robert J." "
Kleinhenz, Robert J.
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.
Kline, Morris
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" "Koestler, Arthur (1905- )" "
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 ke" "Koestler, Arthur (1905- )" "
Koestler, Arthur (1905- )
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."]
Kovalevsky, Sonja
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.
Kraft, Prinz zu Hohlenlohe-Ingelfingen (1827 - 1892)
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.
Kronecker, Leopold (1823-1891)
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 nu "La Touche, Mrs." "
La Touche, Mrs.
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 <" "LaGrange, Joseph-Louis" "
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.
" "LaGrange, Joseph-Louis" " LaGrange, Joseph-Louis
That sometimes clear ... and sometimes vague stuff ... which is ... mathematics.
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
Lakatos, Imre
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 o" "Lanczos, Cornelius" "
Lanczos, Cornelius
[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, Me" "Landau, E." "
Landau, E.
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" "Landau, Susan" "
Landau, Susan
He uses statistics as a drunken man uses lamp posts -- for support rather than illumination.
Treasury of Humorous Quotations.
Lang, Andrew (1844-1912)
[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 Langer, Rudoph E.
A good calculator does not need artificial aids.
Tao Te Ching, ch 27.
Lao Tze (604-531 B.C.)
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, bu" "de Laplace, Pierre-Simon (1749 - 1827)" "
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 " "de Laplace, Pierre-Simon (1749 - 1827)" "
de Laplace, Pierre-Simon (1749 - 1827)
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. "Leach, Edmund Ronald (1910 - 1989)" "
Leach, Edmund Ronald (1910 - 1989)
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" "Leacock, Stephen" "
Leacock, Stephen
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.
Lebesgue, Henri (1875 - 1941)
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.)
Lehrer, Thomas Andrew (1928- )
[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 Orléans]
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)" " 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, N" "Leibniz, Gottfried Whilhem (1646-1716)" "
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 W" "Leibniz, Gottfried Whilhem (1646-1716)" "
Leibniz, Gottfried Whilhem (1646-1716)
The soul is the mirror of an indestructible universe.
The Monadology.
Leibniz, Gottfried Whilhem (1646-1716)
Whoever despises the high wisdom of mathematics nourishes himself on delusion and will never still the sophistic sciences whose only product is an eternal uproar.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988. da Vinci, Leonardo (1452-1519)
Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics.
Notebooks, v. 1, ch. 20.
da Vinci, Leonardo (1452 - 1519)
He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast.
"da Vinci, Leonardo (1452-1519)" " da Vinci, Leonardo (1452-1519)
No human investigation can be called real science if it cannot be demonstrated mathematically.
"da Vinci, Leonardo (1452-1519)" "da Vinci, Leonardo (1452-1519)
Inequality is the cause of all local movements.
"da Vinci, Leonardo (1452-1519)" "da Vinci, Leonardo (1452-1519)
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.
Leybourn, William (1626-1700)
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 departu" "Lichtenberg, Georg Christoph (1742 - 1799)" "
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)" "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 thems" "Lichtenberg, Georg Christoph (1742 - 1799)" "
Lichtenberg, Georg Christoph (1742 - 1799)
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.
le Lionnais, Francois
[On the Gaussian curve, remarked to Poincaré:]
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.
Lippman, Gabriel (1845-1921)
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 th" "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
Littlewood, J. E. (1885 -1977)
I recall once saying that when I had given the same lecture several times I couldn't help feeling that they really ought to know it by now.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
Littlewood, J. E. (1885 -1977)
In passing, I firmly believe that research should be offset by a certain amount of teaching, if only as a change from the agony of research. The trouble, however, I freely admit, is that in practice you get either no teaching, or else far too much.
" "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
It is possible for a mathematician to be "too strong" for a given occasion. He forces through, where another might be driven to a different, and possible more fruitful, approach. (So a rock climber might force a dreadful crack, instead of fin" "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
I constantly meet people who are doubtful, generally without due reason, about their potential capacity [as mathematicians]. The first test is whether you got anything out of geometry. To have disliked or failed to get on with other [mathematical] subj" "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
The infinitely competent can be uncreative.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Littlewood, J. E. (1885 -1977)
In presenting a mathematical argument the great thing is to give the educated reader the chance to catch on at once to the momentary point and take details for granted: his successive mouthfuls should be such as can be swallowed at sight; in case of ac "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that "E is dense in E" does not mean the same thing as "E is dense in itself".
A Mathematician's Miscellany, Me" "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
The surprising thing about this paper is that a man who could write it would.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
Littlewood, J. E. (1885 -1977)
A precisian professor had the habit of saying: "... quartic polynomial ax^4+bx^3+cx^2+dx+e , where e need not be the base of the natural logarithms."
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
Littlewood, J. E. (1885 -1977)
I read in the proof sheets of Hardy on Ramanujan: "As someone said, each of the positive integers was one of his personal friends." My reaction was, "I wonder who said that; I wish I had." In the next proof-sheets I read (what now s" "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
We come finally, however, to the relation of the ideal theory to real world, or "real" probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal s" "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of concentration leads to new ideas or methods of o "Littlewood, J. E. (1885 -1977)" "
Littlewood, J. E. (1885 -1977)
There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Lobatchevsky, Nikolai
...mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning.
D. Burton, Elementary Number Theory, Boston: Allyn and Bacon 1980.
Locke, John
Medicine makes people ill, mathematics make them sad and theology makes them sinful.
" "Luther, Martin (1483-1546)" "Luther, Martin (1483-1546)
Archimedes constructing his circle pays with his life for his defective biological adaptation to immediate circumstances.
"Mach, Ernst (1838 - 1916)" " Mach, Ernst (1838 - 1916)
The mathematician who pursues his studies without clear views of this matter, must often have the uncomfortable feeling that his paper and pencil surpass him in intelligence.
"The Economy of Science" in J. R. Newman (ed.) The World of " "Mach, Ernst (1838-1916)" "
Mach, Ernst (1838-1916)
Like the ski resort full of girls hunting for husbands and husbands hunting for girls, the situation is not as symmetrical as it might seem.
A Dictionary of Scientific Quotations, Bristol: IOP Publishing, 1991.
Mackay, Alan Lindsay (1926- )
Truth ... and if mine eyes
Can bear its blaze, and trace its symmetries,
Measure its distance, and its advent wait,
I am no prophet -- I but calculate.
The Poetical Works of Charles Mackay. 1876.
Mackay, Charles (1814-1889)
The concept of number is the obvious distinction between the beast and man. Thanks to number, the cry becomes a song, noise acquires rhythm, the spring is transformed into a dance, force becomes dynamic, and outlines figures.
" Maistre Joseph Marie de (1753 - 1821) Maistre Joseph Marie de (1753 - 1821)
A great truth is a truth whose opposite is also a great truth.
Essay on Freud. 1937.
Mann, Thomas (1875-1955)
I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh.
The Magic Mountain. 1927.
Mann, Thomas (1875-1955)
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 ws eagerly talking. He had made the assertion that one could draw through a give" "Mann, Thomas (1875-1955)" "
Mann, Thomas (1875-1955)
If your new theorem can be stated with great simplicity, then there will exist a pathological exception.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Mathesis, Adrian
All great theorems were discovered after midnight.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Mathesis, Adrian
The greatest unsolved theorem in mathematics is why some people are better at it than others.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Mathesis, Adrian
If you see a formula in the Physical Review that extends over a quarter of a page, forget it. It's wrong. Nature isn't that complicated.
" "Matthias, Bernd T" " Matthias, Bernd T
... that, in a few years, all great physical constants will have been approximately estimated, and that the only occupation which will be left to men of science will be to carry these measurements to another place of decimals.
Scientific Papers<" "Maxwell, James Clerk (1813-1879)" "
Maxwell, James Clerk (1813-1879)
Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man.
J. Dash, Maria Goeppert-Mayer, A Life of One's Own.
Mayer, Maria Goeppert (1906 -1972)
Gel'fand amazed me by talking of mathematics as though it were poetry. He once said about a long paper bristling with formulas that it contained the vague beginnings of an idea which could only hint at and which he had never managed to bring out more "McDuff, Dusa" "
McDuff, Dusa
There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigoro "McShane, E. J." "
McShane, E. J.
It is now quite lawful for a Catholic woman to avoid pregnancy by a resort to mathematics, though she is still forbidden to resort to physics and chemistry.
Notebooks, "Minority Report"." "Mencken, H. L. (1880 - 1956)" "
Mencken, H. L. (1880 - 1956)
Bridges would not be safer if only people who knew the proper definition of a real number were allowed to design them.
"Topological Theory of Defects" in Review of Modern Physics, v. 51 no. 3, July 1979.
Mermin, N. David (1935 -)
Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shift" "Millay, Edna St. Vincent (1892 - 1950)" "
Millay, Edna St. Vincent (1892 - 1950)
From Man or Angel the great Architect
Did wisely to conceal, and not divulge,
His secrets, to be scanned by them who ought
Rather admire. Or, if they list to try
Conjecture, he his fabric of the Heavens
Hath left to their disputes -" "Milton, John (1608 - 1674)" "
Milton, John (1608 - 1674)
Chaos umpire sits
And by decision more
embroils the fray
by which he reigns: next
him high arbiter
Chance governs all.
Milton, John (1608-1674)
From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Minkowski, Herman
Logic doesn't apply to the real world.
D. R. Hofstadter and D. C. Dennett (eds.) The Mind's I, 1981.
Minsky, Marvin Lee (1927 -)
...She knew only that if she did or said thus-and-so, men would unerringly respond with the complimentary thus-and-so. It was like a mathematical formula and no more difficult, for mathematics was the one subject that had come easy to Scarlett in her s" "Mitchell, Margaret" "
Mitchell, Margaret
The mathematician's best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:" "Mittag-Leffler, Gösta" "
Mittag-Leffler, Gösta
Neither you nor I nor anybody else knows what makes a mathematician tick. It is not a question of cleverness. I know many mathematicians who are far abler than I am, but they have not been so lucky. An illustration may be given by considering two miner" "Mordell, L.J." "
Mordell, L.J.
We lay down a fundamental principle of generalization by abstraction:
"The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them w "Moore, E.H. (1862 - 1932)" "
Moore, E.H. (1862 - 1932)
The words figure and fictitious both derive from the same Latin root, fingere. Beware!
Facts from Figures.
Moroney, M.J.
[about Hypatia:]
In an era in which the domain of intellect and politics were almost exclusively male, Theon [her father] was an unusually liberated person who taught an unusually gifted daughter and encouraged her to achieve things that, as far as" "Mueller, Ian" "
Mueller, Ian
A mathematician of the first rank, Laplace quickly revealed himself as only a mediocre administrator; from his first work we saw that we had been deceived. Laplace saw no question from its true point of view; he sought subtleties everywhere; had only d" Napoleon (1769-1821)
Napoleon (1769-1821)
Teach to the the problems, not to the text.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Nebeuts, E. Kim
To state a theorem and then to show examples of it is literally to teach backwards.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Nebeuts, E. Kim
A good preparation takes longer than the delivery.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Nebeuts, E. Kim
The greatest reward lies in making the discovery; recognition can add little or nothing to that.
"Neumann, Franz Ernst (1798 - 1895)" " Neumann, Franz Ernst (1798 - 1895)
In mathematics you don't understand things. You just get used to them.
In G. Zukav The Dancing Wu Li Masters.
von Neumann, Johann (1903 - 1957)
The most painful thing about mathematics is how far away you are from being able to use it after you have learned it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Newman, James R.
The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left" "Newman, James, R." "
Newman, James, R.
It is hard to know what you are talking about in mathematics, yet no one questions the validity of what you say. There is no other realm of discourse half so queer.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster" "Newman, James R." "
Newman, James R.
Mathematical economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or expository device, can be app" "Newman, James R." "
Newman, James R.
To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse i" "Newman, James R." "
Newman, James R.
The Theory of Groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing.
In J. R. Newman (ed.) <" "Newman, James R." "
Newman, James R.
Games are among the most interesting creations of the human mind, and the analysis of their structure is full of adventure and surprises. Unfortunately there is never a lack of mathematicians for the job of transforming delectable ingredients into a di" "Newman, James R." "
Newman, James R.
...from the same principles, I now demonstrate the frame of the System of the World.
Principia Mathematica.
Newton, Isaac (1642-1727)
Hypotheses non fingo.
I feign no hypotheses.
Principia Mathematica.
Newton, Isaac (1642-1727)
To explain all nature is too difficult a task for any one man or even for any one age. `Tis much better to do a little with certainty, and leave the rest for others hat come after you, than to explain all things.
In G. Simmons Calculus Gems," "Newton, Isaac (1642-1727)" "
Newton, Isaac (1642-1727)
The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.
Principia Mathematica.
Newton, Isaac (1642-1727)
The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of mathematics.
" "Newton, Isaac (1642-1727)" " Newton, Isaac (1642-1727)
[His epitaph:]
Who, by vigor of mind almost divine, the motions and figures of the planets, the paths of comets, and the tides of the seas first demonstrated.
Newton, Isaac (1642-1727)
Usually mathematicians have to shoot somebody to get this much publicity.
[On the attention he received after finding the flaw in Intel's Pentium chip in 1994]
Cincinnati Enquirer, December 18, 1994, Section A, page 19.
Thomas R. Nicely
"[Of her:]
Her statistics were more than a study, they were indeed her religion. For her Quetelet was the hero as scientist, and the presentation copy of his Physique sociale is annotated by her on every page. Florence Nightingale believed -- and in" "Nightingale, Florence (1820-1910)" "
Nightingale, Florence (1820-1910)
The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality.
The American Mathematical Monthly, 56, 1949, p19.
Oakley, C.O.
Mathematicians boast of their exacting achievements, but in reality they are absorbed in mental acrobatics and contribute nothing to society.
Complete Works on Japan's Philosophical Thought. 1956.
Ogyu, Sorai (1666 - 1729)
Today, it is not only that our kings do not know mathematics, but our philosophers do not know mathematics and -- to go a step further -- our mathematicians do not know mathematics.
"The Tree of Knowledge" in Harper's, 217, 1958. Oppenheimer, Julius Robert (1904 - 1967)
The calculus is the greatest aid we have to the application of physical truth in the broadest sense of the word.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Osgood, W. F.
We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others.
Pensees. 1670.
Pascal, Blaise (1623-1662)
It is the heart which perceives God and not the reason.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Our nature consists in movement; absolute rest is death.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Man is full of desires: he loves only those who can satisfy them all. "This man is a good mathematician," someone will say. But I have no concern for mathematics; he would take me for a proposition. "That one is a good soldier." He" "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
We run carelessly to the precipice, after we have put something before us to prevent us from seeing it.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Pascal, Blaise (1623-1662)
We do not worry about being respected in towns through which we pass. But if we are going to remain in one for a certain time, we do worry. How long does this time have to be?
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms<" "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
Few men speak humbly of humility, chastely of chastity, skeptically of skepticism.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Pascal, Blaise (1623-1662)
Those who write against vanity want the glory of having written well, and their readers the glory of reading well, and I who write this have the same desire, as perhaps those who read this have also.
W. H. Auden and L. Kronenberger (eds.) The Vi" "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
Our notion of symmetry is derived form the human face. Hence, we demand symmetry horizontally and in breadth only, not vertically nor in depth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966" "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
When we encounter a natural style we are always surprised and delighted, for we thought to see an author and found a man.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Pascal, Blaise (1623-1662)
Everything that is written merely to please the author is worthless.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Pascal, Blaise (1623-1662)
I cannot judge my work while I am doing it. I have to do as painters do, stand back and view it from a distance, but not too great a distance. How great? Guess.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: " "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Pascal, Blaise (1623-1662)
All err the more dangerously because each follows a truth. Their mistake lies not in following a falsehood but in not following another truth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966." "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
Perfect clarity would profit the intellect but damage the will.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Pascal, Blaise (1623-1662)
Those who are accustomed to judge by feeling do not understand the process of reasoning, because they want to comprehend at a glance and are not used to seeking for first principles. Those, on the other hand, who are accustomed to reason from first pri" "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
To deny, to believe, and to doubt well are to a man as the race is to a horse.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Pascal, Blaise (1623-1662)
Words differently arranged have a different meaning and meanings differently arranged have a different effect.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Pascal, Blaise (1623-1662)
Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.
Pensees. 1670.
Pascal, Blaise (1623-1662)
We arrive at truth, not by reason only, but also by the heart.
Pensees. 1670.
Pascal, Blaise (1623-1662)
When the passions become masters, they are vices.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Men despise religion; they hate it, and they fear it is true.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Religion is so great a thing that it is right that those who will not take the trouble to seek it if it be obscure, should be deprived of it.
Pensees. 1670.
Pascal, Blaise (1623-1662)
It is not certain that everything is uncertain.
Pensees. 1670.
Pascal, Blaise (1623-1662)
We are so presumptuous that we should like to be known all over the world, even by people who will only come when we are no more. Such is our vanity that the good opinion of half a dozen of the people around us gives us pleasure and satisfaction.
<" "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
The sole cause of man's unhappiness is that he does not know how to stay quietly in his room.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Reason's last step is the recognition that there are an infinite number of things which are beyond it.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Through space the universe grasps me and swallows me up like a speck; through thought I grasp it.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Let no one say that I have said nothing new... the arrangement of the subject is new. When we play tennis, we both play with the same ball, but one of us places it better.
Pensees. 1670.
Pascal, Blaise (1623-1662)
The excitement that a gambler feels when making a bet is equal to the amount he might win times the probability of winning it.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Pascal, Blaise (1623-1662)
Reason is the slow and tortuous method by which these who do not know the truth discover it. The heart has its own reason which reason does not know.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Reverend Fathers, my letters did not usually follow each other at such close intervals, nor were they so long.... This one would not be so long had I but the leisure to make it shorter.
Lettres provinciales.
Pascal, Blaise (1623-1662)
The last thing one knows when writing a book is what to put first.
Pensees. 1670.
Pascal, Blaise (1623-1662)
What is man in nature? Nothing in relation to the infinite, all in relation to nothing, a mean between nothing and everything.
Pensees. 1670.
Pascal, Blaise (1623-1662)
[I feel] engulfed in the infinite immensity of spaces whereof I know nothing, and which know nothing of me, I am terrified The eternal silence of these infinite spaces alarms me.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Let us weigh the gain and the loss in wagering that God is. Let us consider the two possibilities. If you gain, you gain all; if you lose, you lose nothing. Hesitate not, then, to wager that He is.
Pensees. 1670.
Pascal, Blaise (1623-1662)
Look somewhere else for someone who can follow you in your researches about numbers. For my part, I confess that they are far beyond me, and I am competent only to admire them.
[Written to Fermat]
In G. Simmons Calculus Gems, New York: M" "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
The more I see of men, the better I like my dog.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Pascal, Blaise (1623-1662)
The more intelligent one is, the more men of originality one finds. Ordinary people find no difference between men.
Pensees. 1670.
Pascal, Blaise (1623-1662)
However vast a man's spiritual resources, he is capable of but one great passion.
Discours sur les passions de l'amour. 1653.
Pascal, Blaise (1623-1662)
There are two types of mind ... the mathematical, and what might be called the intuitive. The former arrives at its views slowly, but they are firm and rigid; the latter is endowed with greater flexibility and applies itself simultaneously to the diver" "Pascal, Blaise (1623-1662)" "
Pascal, Blaise (1623-1662)
This trend [emphasizing applied mathematics over pure mathematics] will make the queen of the sciences into the quean of the sciences.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Passano, L.M.
Chance favors only the prepared mind.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Pasteur, Louis
" "The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe.
In N. Rose Mathematical Maxims and Minims, Ra" "Pearson, Karl" "
Pearson, Karl
Mathematics is the science which draws necessary conclusions.
Memoir read before the National Academy of Sciences in Washington, 1870.
Peirce, Benjamin (1809-1880)
The one [the logician] studies the science of drawing conclusions, the other [the mathematician] the science which draws necessary conclusions.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: " "Peirce, Charles Sanders (1839-1914)" "
Peirce, Charles Sanders (1839-1914)
...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating into airy nothingness, degenerating, as the Germa" "Peirce, Charles Sanders (1839-1914)" "
Peirce, Charles Sanders (1839-1914)
Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad" "Peirce, Charles Sanders (1839-1914)" "
Peirce, Charles Sanders (1839-1914)
The pragmatist knows that doubt is an art which hs to be acquired with difficulty.
Collected Papers.
Peirce, Charles Sanders (1839-1914)
Geometry is a skill of the eyes and the hands as well as of the mind.
"Pedersen, Jean" " Pedersen, Jean
He who can properly define and divide is to be considered a god.
Plato (ca 429-347 BC) Plato (ca 429-347 BC)
The ludicrous state of solid geometry made me pass over this branch. Republic, VII, 528.
" Plato (ca 429-347 BC)Plato (ca 429-347 BC)
He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side.
Plato (ca 429-347 BC) Plato (ca 429-347 BC)
Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.
" Plato (ca 429-347 BC) Plato (ca 429-347 BC)
The knowledge of which geometry aims is the knowledge of the eternal.
Republic, VII, 52.
Plato (ca 429-347 BC)
I have hardly ever known a mathematician who was capable of reasoning.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Plato (ca 429-347 BC)
There still remain three studies suitable for free man. Arithmetic is one of them.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Plato (ca 429-347 BC)
[about Archimedes:]
... being perpetually charmed by his familiar siren, that is, by his geometry, he neglected to eat and drink and took no care of his person; that he was often carried by force to the baths, and when there he would trace geometric" Plutarch (ca 46-127)
Plutarch (ca 46-127)
To speak algebraically, Mr. M. is execrable, but Mr. G. is (x + 1)- ecrable.
[Discussing fellow writers Cornelius Mathews and William Ellery Channing.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Poe, Edgar Allen
Mathematics is the art of giving the same name to different things.
[As opposed to the quotation: Poetry is the art of giving different names to the same thing].
Poincaré, Jules Henri (1854-1912)
Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered.
[Whether or not he actually said this is a matter of debate amongst historians of mathematics.]
The Mathematical Intelligencer, vol 13, no." "Poincaré, Jules Henri (1854-1912)" "
Poincaré, Jules Henri (1854-1912)
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to " "Poincaré, Jules Henri (1854-1912)" "
Poincaré, Jules Henri (1854-1912)
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 imagi" "Poincaré, Jules Henri (1854-1912)" "
Poincaré, Jules Henri (1854-1912)
Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Poincaré, Jules Henri (1854-1912)
Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.
La Science et l'hypothèse.
Poincaré, Jules Henri (1854-1912)
A scientist worthy of his name, about all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988" "Poincaré, Jules Henri (1854-1912)" "
Poincaré, Jules Henri (1854-1912)
The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strange" "Poincaré, Jules Henri (1854-1912)" "
Poincaré, Jules Henri (1854-1912)
Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.
" "Poincaré, Jules Henri (1854-1912)" " Poincaré, Jules Henri (1854-1912)
Thought is only a flash between two long nights, but this flash is everything.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Poincaré, Jules Henri (1854-1912)
The mind uses its faculty for creativity only when experience forces it to do so.
"Poincaré, Jules Henri (1854-1912)" " Poincaré, Jules Henri (1854-1912)
Mathematical discoveries, small or greatare never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
" "Poincaré, Jules Henri (1854-1912)" " Poincaré, Jules Henri (1854-1912)
Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence.... The two propositions: "The earth turns round" and "it is more convenient to suppo" "Poincaré, Jules Henri (1854-1912)" "
Poincaré, Jules Henri (1854-1912)
...by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.
Science " "Poincaré, Jules Henri (1854-1912)" "
Poincaré, Jules Henri (1854-1912)
Life is good for only two things, discovering mathematics and teaching mathematics.
Mathematics Magazine, v. 64, no. 1, Feb. 1991.
Poisson, Siméon (1781-1840)
Mathematics consists of proving the most obvious thing in the least obvious way.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Polyá, George (1887, 1985)
The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back to the class. He writes a, he says b, he means c; but i" "Polyá, George (1887, 1985)" "
Polyá, George (1887, 1985)
Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
D. J. Albers and G. L. Alexanderson, Mathematical People, Boston: Birkhä" "Polyá, George (1887, 1985)" "
Polyá, George (1887, 1985)
There are many questions which fools can ask that wise men cannot answer.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Polyá, George (1887, 1985)
When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
The American Mathematical Monthly, v. 100, no. 3.
Polyá, George (1887, 1985)
Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher sh" "Polyá, George (1887, 1985)" "
Polyá, George (1887, 1985)
In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is v" "Polyá, George (1887, 1985)" "
Polyá, George (1887, 1985)
Epitaph on Newton:
Nature and Nature's law lay hid in night:
God said, "Let Newton be!," and all was light.
[added by Sir John Collings Squire:
It did not last: the Devil shouting "Ho.
Let Einstein be," restored " "Pope, Alexander (1688-1744)" "
Pope, Alexander (1688-1744)
Order is Heaven's first law.
An Essay on Man IV.
Pope, Alexander (1688-1744)
See skulking Truth to her old cavern fled,
Mountains of Casuistry heap'd o'er her head!
Philosophy, that lean'd on Heav'n before,
Shrinks to her second cause, and is no more.
Physic of Metaphysic begs defence,
And Metaphysic calls f" "Pope, Alexander (1688-1744)" "
Pope, Alexander (1688-1744)
One of the endearing things about mathematicians is the extent to which they will go to avoid doing any real work.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Pordage, Matthew
It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspec" Proclus Diadochus (412 - 485)
Proclus Diadochus (412 - 485)
The Mean Value Theorem is the midwife of calculus -- not very important or glamorous by itself, but often helping to delivery other theorems that are of major significance.
Calculus with Analytic Geomety, fifth edition, Englewood Cliffs, NJ:" "Purcell, E. and Varberg, D." "
Purcell, E. and Varberg, D.
Inspiration is needed in geometry, just as much as in poetry.
Likhtenshtein
Pushkin, Aleksandr Sergeyevich (1799 - 1837)
Just as the introduction of the irrational numbers ... is a convenient myth [which] simplifies the laws of arithmetic ... so physical objects are postulated entities which round out and simplify our account of the flux of existence... The conceptional "Quine, Willard Van Orman" "
Quine, Willard Van Orman
In an examination those who do not wish to know ask questions of those who cannot tell.
Some Thoughts on Examinations.
Raleigh, [Sir] Walter Alexander (1861-1922)
To avoide the tediouse repetition of these woordes: is equalle to: I will settle as I doe often in woorke use, a paire of paralleles, or gemowe [twin] lines of one lengthe: =, bicause noe .2. thynges, can be moare equalle.
In G. Simmons Calculu" "Recorde, Robert (1557)" "
Recorde, Robert (1557)
It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchal" "Reid, Thomas" "
Reid, Thomas
The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.
Souvenirs d'enfance et de jeunesse.
Renan, Ernest
If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.
P. Turán, "The Work of Alfréd Rényi", Matematikai Lapok 21, 1970, pp 199 - 210.
Rényi, Alfréd
Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematicians' stock of formulae. Some verbal statements " "Richardson, Lewis Fry (1881 - 1953)" "
Richardson, Lewis Fry (1881 - 1953)
(after Edna St. Vincent Millay)
...Euclid alone
Has looked on Beauty bare.
He turned away at once;
Far too polite to stare.
The Mathematical Intelligencer, V. 16, no. 4 (Fall 1994), p. 20.
Riskin, Adrian
The magic words are squeamish ossifrage
[This sentence is the result when a coded message in Martin Gardner's column about factoring the famous number RSA-129 is decoded. See the article whose title is the above sentence by Barry Cipra, SIAM New" "R. Rivest, A. Shamir, and L. Adleman" "
R. Rivest, A. Shamir, and L. Adleman
It was by just such a hazard, as if a man should let fall a handful of sand upon a table and the particles of it should be so ranged that we could read distinctly on it a whole page of Virgil's Aenead.
Traité de Physique, Paris, 1671." "Rohault, Jacques (17th century)" "
Rohault, Jacques (17th century)
[with Norbert Wiener]
The best material model of a cat is another, or preferably the same, cat.
Philosophy of Science 1945.
Rosenblueth, A
You know we all became mathematicians for the same reason: we were lazy.
"Rosenlicht, Max (1949)" " Rosenlicht, Max (1949)
In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.
Mathematics Is an Edifice, Not" "Rossi, Hugo"
Hugo Rossi
"We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?"
In preface to P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.Rota, Gian-carlo
How dare we speak of the laws of chance? Is not chance the antithesis of all law?
Calcul des probabilités.
Russell, Bertrand (1872-1970)
Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Russell, Bertrand (1872-1970)
Although this may seem a paradox, all exact science is dominated by the idea of approximation.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Russell, Bertrand (1872-1970)
At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mat" "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
A good notation has a subtlety and suggestiveness which at times make it almost seem like a live teacher.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Russell, Bertrand (1872-1970)
If I were a medical man, I should prescribe a holiday to any patient who considered his work important.
The Autobiography of Bertrand Russell .
Russell, Bertrand (1872-1970)
Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.
The Scien" "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway about the flux. A little of this, but n" "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press I" "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Russell, Bertrand (1872-1970)
The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
Principles " "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
A habit of basing convictions upon evidence, and of giving to them only that degree or certainty which the evidence warrants, would, if it became general, cure most of the ills from which the world suffers.
In G. Simmons Calculus Gems, New Y" "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil.
Introduction to Mathematical Philosophy, New York and London, 1919, p 71.
Russell, Bertrand (1872-1970)
Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives' mouths.
The Impact of Science on Society, 1952.
Russell, Bertrand (1872-1970)
[Upon hearing via Littlewood an exposition on the theory of relativity:]
To think I have spent my life on absolute muck.
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
Russell, Bertrand (1872-1970)
"But," you might say, "none of this shakes my belief that 2 and 2 are 4." You are quite right, except in marginal cases -- and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length " "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to acce" "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
Men who are unhappy, like men who sleep badly, are always proud of the fact.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Russell, Bertrand (1872-1970)
Work is of two kinds: first, altering the position of matter at or near the earth's surface relatively to other such matter; second, telling other people to do so. The first kind is unpleasant and ill paid; the second is pleasant and highly paid.
" "Russell, Bertrand (1872-1970)" " Russell, Bertrand (1872-1970)
A sense of duty is useful in work but offensive in personal relations. Certain characteristics of the subject are clear. To begin with, we do not, in this subject, deal with particular things or particular properties: we deal formally with what can b" "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
The desire to understand the world and the desire to reform it are the two great engines of progress.
Marriage and Morals.
Russell, Bertrand (1872-1970)
It can be shown that a mathematical web of some kind can be woven about any universe containing several objects. The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance.
W. H. Auden an "Russell, Bertrand (1872-1970)" "
Russell, Bertrand (1872-1970)
If your experiment needs statistics, you ought to have done a better experiment.
In N. T. J. Bailey the Mathematical Approach to Biology and Medicine, New York: Wiley, 1967.
Rutherford, Ernest (1871-1937)
The modern, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers not because it is abstract, but because it is a representation of actual things.
" "Sanford, T. H." "
Sanford, T. H.
It is a pleasant surprise to him (the pure mathematician) and an added problem if he finds that the arts can use his calculations, or that the senses can verify them, much as if a composer found that sailors could heave better when singing his songs.
The main duty of the historian of mathematics, as well as his fondest privilege, is to explain the humanity of mathematics, to illustrate its greatness, beauty and dignity, and to describe how the incessant efforts and accumulated genius of many genera" "Sarton, G." "
Sarton, G.
The biologist can push it back to the original protist, and the chemist can push it back to the crystal, but none of them touch the real question of why or how the thing began at all. The astronomer goes back untold million of years and ends in gas and" "Sayers, Dorothy L." "
Sayers, Dorothy L.
Of all the intellectual faculties, judgment is the last to mature. A child under the age of fifteen should confine its attention either to subjects like mathematics, in which errors of judgment are impossible, or to subjects in which they are not very" Schopenhauer
Schopenhauer
If you would make a man happy, do not add to his possessions but subtract from the sum of his desires.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Seneca
I cannot do it without comp[u]ters.
The Winter's Tale.
Shakespeare, William (1564 - 1616)
Though this be madness, yet there is method in't.
" "Shakespeare, William (1564-1616)" " Shakespeare, William (1564-1616)
O God! I could be bounded in a nutshell, and count myself king of infinite space, were it not that I have bad dreams.
Hamlet.
Shakespeare, William (1564-1616)
I am ill at these numbers.
Hamlet.
Shakespeare, William (1564-1616)
Tyndall declared that he saw in Matter the promise and potency of all forms of life, and with his Irish graphic lucidity made a picture of a world of magnetic atoms, each atom with a positive and a negative pole, arranging itself by attraction and repu" "Shaw, George Bernard (1856-1950)" "
Shaw, George Bernard (1856-1950)
The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Shaw, J. B.
Mathematical rigor is like clothing; in its style it ought to suit the occasion, and it diminishes comfort and restrains freedom of movement if it is either too loose or too tight.
In The Mathematical Intelligencer, v. 13, no. 1, Winter 1991" "Simmons, G. F." "
Simmons, G. F.
...[E.H.] Moore ws 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 ro "Slaught, H.E." "
Slaught, H.E.
I have no faith in political arithmetic.
"Smith, Adam" " Smith, Adam
One merit of mathematics few will deny: it says more in fewer words than any other science. The formula, e^iπ = -1 expressed a world of thought, of truth, of poetry, and of the religious spirit "God eternally geometrizes."
In N. Ro" "Smith, David Eugene" "
Smith, David Eugene
[His toast:]
Pure mathematics, may it never be of any use to anyone.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Smith, Henry John Stephen (1826 - 1883)
It is the peculiar beauty of this method, gentlemen, and one which endears it to the really scientific mind, that under no circumstance can it be of the smallest possible utility.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Web" "Smith, Henry John Stephen (1826-1883)" "
Smith, Henry John Stephen (1826-1883)
Four circles to the kissing come,
The smaller are the benter.
The bend is just the inverse of
The distance from the centre.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead s" "Soddy, Frederick (1877-1956)" "
Soddy, Frederick (1877-1956)
Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, and are still granted in these latter times by the Diffe" "Somerville, Mary (1780-1872)" "
Somerville, Mary (1780-1872)
The mathematic, then, is an art. As such it has its styles and style periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes form epo" "Spengler, Oswald (1880 -1936)" "
Spengler, Oswald (1880 -1936)
For all their wealth of content, for all the sum of history and social institution invested in them, music, mathematics, and chess are resplendently useless (applied mathematics is a higher plumbing, a kind of music for the police band). They are metap" "Steiner, G." "
Steiner, G.
Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional.
In E. T. Bell Men" "Steinmetz, Charles P." "
Steinmetz, Charles P.
Kepler's principal goal was to explain the relationship between the existence of five planets (and their motions) and the five regular solids. It is customary to sneer at Kepler for this. It is instructive to compare this with the current attempts to "Sternberg, S." "
Sternberg, S.
The successes of the differential equation paradigm were impressive and extensive. Many problems, including basic and important ones, led to equations that could be solved. A process of self-selection set in, whereby equations that could not be solved " "Stewart, Ian" "
Stewart, Ian
The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming " "Sullivan, John William Navin (1886 - 1937)" "
Sullivan, John William Navin (1886 - 1937)
Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it in" "Sullivan, John William Navin (1886-1937)" "
Sullivan, John William Navin (1886-1937)
The control of large numbers is possible, and like unto that of small numbers, if we subdivide them.
Sun Tze Ping Fa.
Sun Tze (5th - 6th century)
If they would, for Example, praise the Beauty of a Woman, or any other Animal, they describe it by Rhombs, Circles, Parallelograms, Ellipses, and other geometrical terms ...
"A Voyage to Laputa" in Gulliver's Travels.
Swift, Jonathan
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, whic" Jonathan Swift
Jonathan Swift
...there is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectua" "Sylvester, J.J. (1814 - 1897)" "
Sylvester, J.J. (1814 - 1897)
So long as a man remains a gregarious and sociable being, he cannot cut himself off from the gratification of the instinct of imparting what he is learning, of propagating through others the ideas and impressions seething in his own brain, without stun" "Sylvester, J.J. (1814 - 1897)" "
Sylvester, J.J. (1814 - 1897)
[on graph theory...]
The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables h" "Sylvester, J.J. (1814 - 1897)" "
Sylvester, J.J. (1814 - 1897)
Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn tog" "Sylvester, J.J. (1814 - 1897)" "
Sylvester, J.J. (1814 - 1897)
The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concer" "Sylvester, J.J. (1814 - 1897)" "
Sylvester, J.J. (1814 - 1897)
I know, indeed, and can conceive of no pursuit so antagonistic to the cultivation of the oratorical faculty ... as the study of Mathematics. An eloquent mathematician must, from the nature of things, ever remain as rare a phenomenon as a talking fish," "Sylvester, J.J. (1814 - 1897)" "
Sylvester, J.J. (1814 - 1897)
I will be sufficiently rewarded if when telling it to others you will not claim the discovery as your own, but will say it was mine.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Thales (CA 600 BC)
Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed. They are no exceptions to the rule that God always geometrizes." "Thompson, D'Arcy Wentworth (1860-1948)" "
Thompson, D'Arcy Wentworth (1860-1948)
Fourier is a mathematical poem.
"Thomson, [Lord Kelvin] William (1824-1907)" " Thomson, [Lord Kelvin] William (1824-1907)
He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The " Thoreau
Thoreau
The story was told that the young Dirichlet had as a constant companion all his travels, like a devout man with his prayer book, an old, worn copy of the Disquisitiones Arithmeticae of Gauss.
In G. Simmons Calculus Gems, New York: McG" Tietze
Tietze
How often might a man, after he had jumbled a set of letters in a bag, fling them out upon the ground before they would fall into an exact poem, yea, or so much as make a good discourse in prose. And may not a little book be as easily made by chance as" "Tillotson, Archbishop" "
Tillotson, Archbishop
Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study," "Titchmarsh, E. C." "
Titchmarsh, E. C.
It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Titchmarsh, E. C.
[Asked whether he would like to see an experimental demonstration of conical refraction]
No. I have been teaching it all my life, and I do not want to have my ideas upset.
Todhunter, Isaac (1820 - 1910)
A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, " "Tolstoy, [Count] Lev Nikolgevich (1828-1920)" "
Tolstoy, [Count] Lev Nikolgevich (1828-1920)
A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 19" "Tolstoy, Count Lev Nikolgevich (1828-1920)" "
Tolstoy, Count Lev Nikolgevich (1828-1920)
This paper gives wrong solutions to trivial problems. The basic error,however, is not new.
Mathematical Reviews 12, p561.
Truesdell, Clifford
Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: `Great God, grant that twice two be not four'.
" "Turgenev, Ivan Sergeievich (1818 - 1883)" " Turgenev, Ivan Sergeievich (1818 - 1883)
Attaching significance to invariants is an effort to recognize what, because of its form or colour or meaning or otherwise, is important or significant in what is only trivial or ephemeral. A simple instance of failing in this is provided by the poll-m" "Turnbull, H.W." "
Turnbull, H.W.
In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In th" "Ulam, Stanislaw" "
Ulam, Stanislaw
In the physical world, one cannot increase the size or quantity of anything without changing its quality. Similar figures exist only in pure geometry.
" "Valéry, Paul (1871 - 1945)" " Valéry, Paul (1871 - 1945)
This new integral of Lebesque is proving itself a wonderful tool. I might compare it with a modern Krupp gun, so easily does it penetrate barriers which were impregnable.
Bulletin of the American Mathematical Society, vol. 23, 1916.
van Vleck, E. B.
The outcome of any serious research can only be to make two questions grow where only one grew before.
The Place of Science in Modern Civilization and Other Essays.
Veblen, Thorstein (1857-1929)
Invention is the mother of necessity.
J. Gross, The Oxford Book of Aphorisms, Oxford: Oxford University Press, 1983.
Veblen, Thorstein (1857-1929)
Vous avez trouve par de long ennuis
Ce que Newton trouva sans sortir de chez lui.
[Written to La Condamine after his measurement of the equator.]
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Voltaire (1694-1778)
He who has heard the same thing told by 12,000 eye-witnesses has only 12,000 probabilities, which are equal to one strong probability, which is far from certain.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1" Voltaire (1694-1778)
Voltaire (1694-1778)
There are no sects in geometry.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1962.
Voltaire (1694-1778)
Angling may be said to be so like mathematics that it can never be fully learned.
The Compleat Angler, 1653.
Walton, Izaak
For twenty pages perhaps, he read slowly, carefully, dutifully, with pauses for self-examination and working out examples. Then, just as it was working up and the pauses should have been more scrupulous than ever, a kind of swoon and ecstasy would fall" "Warner, Sylvia Townsend" "
Warner, Sylvia Townsend
Theology, Mr. Fortune found, is a more accommodating subject than mathematics; its technique of exposition allows greater latitude. For instance when you are gravelled for matter there is always the moral to fall back upon. Comparisons too may be draw" "Warner, Sylvia Townsend" "
Warner, Sylvia Townsend
He resumed:
"In order to ascertain the height of the tree I must be in such a position that the top of the tree is exactly in a line with the top of a measuring stick or any straight object would do, such as an umbrella which I shall secure in" "Warner, Sylvia Townsend" "
Warner, Sylvia Townsend
What if angry vectors veer
Round your sleeping head, and form.
There's never need to fear
Violence of the poor world's abstract storm.
Lullaby in Encounter, 1957.
Warren, Robert Penn (1905-)
Every mathematician worthy of the name has experienced ... the state of lucid exaltation in which one thought succeeds another as if miraculously... this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager " "Weil, Andre (1906 -1998)" "
Weil, Andre (1906 -1998)
God exists since mathematics is consistent, and the Devil exists since we cannot prove it.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Weil, Andre (1906- 1998)
Algebra and money are essentially levelers; the first intellectually, the second effectively.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Weil, Simone (1909 - 1943)
Prayers for the condemned man will be offered on an adding machine. Numbers constitute the only universal language.
Miss Lonelyhearts.
West, Nathanael
Our federal income tax law defines the tax y to be paid in terms of the income x; it does so in a clumsy enough way by pasting several linear functions together, each valid in another interval or bracket of income. An archeologist who, five thousand y" "Weyl, Hermann (1885 - 1955)" "
Weyl, Hermann (1885 - 1955)
We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim a" "Weyl, Hermann (1885 - 1955)" "
Weyl, Hermann (1885 - 1955)
A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.
Weyl, Hermann (1885 - 1955)
The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in t "Weyl, Hermann (1885-1955)" "
Weyl, Hermann (1885-1955)
My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.
In an obituary by Freeman J. Dyson in Nature, March 10, 1956.
Weyl, Hermann (1885 - 1955)
... numbers have neither substance, nor meaning, nor qualities. They are nothing but marks, and all that is in them we have put into them by the simple rule of straight succession.
"Mathematics and the Laws of Nature" in The Armchair S" "Weyl, Hermann (1885 - 1955)" "
Weyl, Hermann (1885 - 1955)
Without the concepts, methods and results found and developed by previous generations right down to Greek antiquity one cannot understand either the aims or achievements of mathematics in the last 50 years.
[Said in 1950]
The American Mathem" "Weyl, Hermann (1885 - 1955)" "
Weyl, Hermann (1885 - 1955)
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.
The American Mathematical Monthly, November, 1992.
Weyl, Hermann (1885 - 1955)
Nobody since Newton has been able to use geometrical methods to the same extent for the like purposes; and as we read the Principia we feel as when we are in an ancient armoury where the weapons are of gigantic size; and as we look at them we marvel wh Whewell
Whewell
The science of pure mathematics ... may claim to be the most original creation of the human spirit.
Science and the Modern World.
Whitehead, Alfred North (1861 - 1947)
Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about "any" things or about "some" things, without specifications of definite particular things.
" "Whitehead, Alfred North (1861 - 1947)" " Whitehead, Alfred North (1861 - 1947)
So far as the mere imparting of information is concerned, no university has had any justification for existence since the popularization of printing in the fifteenth century.
The Aims of Education.
Whitehead, Alfred North (1861 - 1947)
No Roman ever died in contemplation over a geometrical diagram.
[A reference to the death of Archimedes.]
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Whitehead, Alfred North (1861 - 1947)
Life is an offensive, directed against the repetitious mechanism of the Universe.
Adventures of Ideas, 1933.
Whitehead, Alfred North (1861 - 1947)
There is no nature at an instant.
"Whitehead, Alfred North (1861 - 1947)" " Whitehead, Alfred North (1861 - 1947)
Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Whitehead, Alfred North (1861 - 1947)
There is a tradition of opposition between adherents of induction and of deduction. In my view it would be just as sensible for the two ends of a worm to quarrel.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988. Whitehead, Alfred North (1861 - 1947)
It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extendi" "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
Our minds are finite, and yet even in these circumstances of finitude we are surrounded by possibilities that are infinite, and the purpose of life is to grasp as much as we can out of that infinitude.
In N. Rose Mathematical Maxims and Minims Whitehead, Alfred North (1861 - 1947)
In modern times the belief that the ultimate explanation of all things was to be found in Newtonian mechanics was an adumbration of the truth that all science, as it grows towards perfection, becomes mathematical in its ideas.
In N. Rose Mathema" "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on pap" "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
Be relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race.
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birk" "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
Everything of importance has been said before by somebody who did not discover it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Whitehead, Alfred North (1861 - 1947)
Seek simplicity, and distrust it.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Whitehead, Alfred North (1861 - 1947)
Fundamental progress has to do with the reinterpretation of basic ideas.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Whitehead, Alfred North (1861 - 1947)
We think in generalities, but we live in details.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Whitehead, Alfred North (1861 - 1947)
Apart from blunt truth, our lives sink decadently amid the perfume of hints and suggestions.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Whitehead, Alfred North (1861 - 1947)
"Necessity is the mother of invention" is a silly proverb. "Necessity is the mother of futile dodges" is much nearer the truth.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966. Whitehead, Alfred North (1861 - 1947)
It is more important that a proposition be interesting than that it be true. This statement is almost a tautology. For the energy of operation of a proposition in an occasion of experience is its interest and is its importance. But of course a true pro "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
War can protect; it cannot create.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Whitehead, Alfred North (1861 - 1947)
The progress of Science consists in observing interconnections and in showing with a patient ingenuity that the events of this ever-shifting world are but examples of a few general relations, called laws. To see what is general in what is particular, a" "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
Through and through the world is infested with quantity: To talk sense is to talk quantities. It is not use saying the nation is large .. How large? It is no use saying the radium is scarce ... How scarce? You cannot evade quantity. You may fly to poet "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
"One and one make two" assumes that the changes in the shift of circumstance are unimportant. But it is impossible for us to analyze this notion of unimportant change.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, Ne" "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certa "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
The study of mathematics is apt to commence in disappointment....We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this greatest science eludes the ef" "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
In the study of ideas, it is necessary to remember that insistence on hard-headed clarity issues from sentimental feeling, as it were a mist, cloaking the perplexities of fact. Insistence on clarity at all costs is based on sheer superstition as to the" "Whitehead, Alfred North (1861 - 1947)" "
Whitehead, Alfred North (1861 - 1947)
Familiar things happen, and mankind does not bother about them. It requires a very unusual mind to undertake the analysis of the obvious.
Science and the Modern World.
Whitehead, Alfred North (1861 - 1947)
Do I contradict myself? Very well then I contradict myself. (I am large, I contains multitudes).
Song of Myself, 1939.
Whitman, Walt (1819-1892)
When I heard the learn'd astronomer,
When the proofs, the figure, were ranged in columns before me,
When I was shown the charts and diagrams, to add, divide, and measure them,
When I sitting heard the astronomer where he lectured with much " "Whitman, Walt (1819-1892)" "
Whitman, Walt (1819-1892)
A professor is one who can speak on any subject -- for precisely fifty minutes.
"Wiener, Norbert (1894 - 1964)" " Wiener, Norbert (1894 - 1964)
The modern physicist is a quantum theorist on Monday, Wednesday, and Friday and a student of gravitational relativity theory on Tuesday, Thursday, and Saturday. On Sunday he is neither, but is praying to his God that someone, preferably himself, will f" "Wiener, Norbert (1894-1964)" "
Wiener, Norbert (1894-1964)
Progress imposes not only new possibilities for the future but new restrictions.
The Human Use of Human Beings.
Wiener, Norbert (1894-1964)
The Advantage is that mathematics is a field in which one's blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is" "Wiener, Norbert (1894-1964)" "
Wiener, Norbert (1894-1964)
There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned.
Introduction to the Foundations of Mathematics.
Wilder, R. L.
Mathematics was born and nurtured in a cultural environment. Without the perspective which the cultural background affords, a proper appreciation of the content and state of present-day mathematics is hardly possible.
In The American Mathematica" "Wilder, R. L." "
Wilder, R. L.
[Occam's Razor:]
Entities should not be multiplied unnecessarily.
Quodlibeta.
William of Occam (1300-1439)
A monument to Newton! a monument to Shakespeare! Look up to Heaven look into the Human Heart. Till the planets and the passionsthe affections and the fixed stars are extinguishedtheir names cannot die.
"Wilson, John (1741 - 1793)" " Wilson, John (1741 - 1793)
We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry.
Tractatus Logico Philosophicus, New York, 1922.
Wittgenstein, Ludwig (1889-1951)
Mathematics is a logical method ... Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathem" "Wittgenstein, Ludwig (1889-1951)" "
Wittgenstein, Ludwig (1889-1951)
There can never be surprises in logic.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Wittgenstein, Ludwig (1889-1951)
The riddle does not exist. If a question can be put at all, then it can also be answered.
Tractatus Logico Philosophicus, New York, 1922.
Wittgenstein, Ludwig (1889-1951)
[Mathematics] is an independent world
Created out of pure intelligence.
Wordsworth, William (1770 - 1850)
In things to be seen at once, much variety makes confusion, another vice of beauty. In things that are not seen at once, and have no respect one to another, great variety is commendable, provided this variety transgress not the rules of optics and geom" "Wren, Sir Christopoher" "
Wren, Sir Christopoher
I'm sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it.
Mascot.
X, Malcom
Mathematics has beauties of its own -- a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found only in the works of the greatest beauty When this subject is " "Young, J. W. A." "
Young, J. W. A.