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Abel, Niels H. (1802  1829)
If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.
In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, p. 188.
[A reply to a question about how he got his expertise:]
By studying the masters and not their pupils.
[About Gauss' mathematical writing style]
He is like the fox, who effaces his tracks in the sand with his tail.
In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, p. 177.
Adams, Douglas (1952  )
Bistromathics itself is simply a revolutionary new way of understanding the behavior of numbers. Just as Einstein observed that space was not an absolute but depended on the observer's movement in space, and that time was not an absolute, but depended on the observer's movement in time, so it is now realized that numbers are not absolute, but depend on the observer's movement in restaurants.
The first nonabsolute number is the number of people for whom the table is reserved. This will vary during the course of the first three telephone calls to the restaurant, and then bear no apparent relation to the number of people who actually turn up, or to the number of people who subsequently join them after the show/match/party/gig, or to the number of people who leave when they see who else has turned up.
The second nonabsolute number is the given time of arrival, which is now known to be one of the most bizarre of mathematical concepts, a recipriversexcluson, a number whose existence can only be defined as being anything other than itself. In other words, the given time of arrival is the one moment of time at which it is impossible that any member of the party will arrive. Recipriversexclusons now play a vital part in many branches of math, including statistics and accountancy and also form the basic equations used to engineer the Somebody Else's Problem field.
The third and most mysterious piece of nonabsoluteness of all lies in the relationship between the number of items on the bill, the cost of each item, the number of people at the table and what they are each prepared to pay for. (The number of people who have actually brought any money is only a subphenomenon of this field.)
Life, the Universe and Everything. New York: Harmony Books, 1982.
Numbers written on restaurant bills within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe.
This single statement took the scientific world by storm. It completely revolutionized it. So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years.
Life, the Universe and Everything. New York: Harmony Books, 1982.
Adams, John (1735  1826)
I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain.
Letter to Abigail Adams, May 12, 1780.
Adler, Alfred
Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing.
"Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.
In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is.
Reflections: mathematics and creativity, New Yorker, 47 (1972), no. 53, 39  45.
The mathematical life of a mathematician is short. Work rarely improves after the age of twentyfive or thirty. If little has been accomplished by then, little will ever be accomplished.
"Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.
Anglin, W.S.
Mathematics is not a careful march down a wellcleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
"Mathematics and History", Mathematical Intelligencer, v. 4, no. 4.
Anonymous
Defendit numerus: There is safety in numbers.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1452.
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]
Referee's report: This paper contains much that is new and much that is true. Unfortunately, that which is true is not new and that which is new is not true.
In H.Eves Return to Mathematical Circles, Boston: Prindle, Weber, and Schmidt, 1988.
Arbuthnot, John
There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning; and when they cannot it's a sign our knowledge of them is very small and confus'd; and when a Mathematical Reasoning can be had it's as great a folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you.
Of the Laws of Chance. (1692)
Aristophanes (ca 444  380 BC)
Meton: With the straight ruler I set to work
To make the circle fourcornered
[First(?) allusion to the problem of squaring the circle]
Aristotle (ca 330 BC)
Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends [mathematics] has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research.
Metaphysica, 1981b
The whole is more than the sum of its parts.
Metaphysica 10f1045a
The socalled 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 15
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.
To Thales the primary question was not what do we know, but how do we know it.
Mathematical Intelligencer v. 6, no. 3, 1984.
The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.
Metaphysica, 31078b.
Aubrey, John (16261697)
[About Thomas Hobbes:]
He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition ... [S]ayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that trueth. This made him in love with geometry.
In O. L. Dick (ed.) Brief Lives, Oxford: Oxford University Press, 1960, p. 604.
Babbage, Charles (17921871)
Errors using inadequate data are much less than those using no data at all.
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 kindof confusion of ideas that could provoke such a question.
Bacon, Sir Francis (15611626)
And as for Mixed Mathematics, I may only make this prediction, that there cannot fail to be more kinds of them, as nature grows further disclosed.
Advancement of Learning book 2; De Augmentis book 3.
Bacon, Roger
For the things of this world cannot be made known without a knowledge of mathematics.
Opus Majus part 4 Distinctia Prima cap 1, 1267.
In the mathematics I can report no deficience, except that it be that men do not sufficiently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures; so in the mathematics, that use which is collateral and intervenient is no less worthy than that which is principal and intended.
John Fauvel and Jeremy Gray (eds.) A History of Mathematics: A Reader, Sheridan House, 1987.
Baker, H. F.
[On the concept of group:]
... what a wealth, what a grandeur of thought may spring from what slight beginnings.
Florian Cajori, A History of Mathematics, New York, 1919, p 283.
Bagehot, Walter
Life is a school of probability.
Quoted in J. R. Newman (ed.) The World of Mathematics, Simon and Schuster, New York, 1956, p. 1360.
Balzac, Honore de (1799  1850)
Numbers are intellectual witnesses that belong only to mankind.
Bell, Eric Temple (18831960)
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.
The Handmaiden of the Sciences.
[Book by that title.]
Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.
"Obvious" is the most dangerous word in mathematics.
The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
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.
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.
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 hogtie, and it is what Euclid did to geometry.
In R CrayshawWilliams The Search For Truth, p. 191.
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.
I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Belloc, Hillaire (18701953)
Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death.
The Silence of the Sea
Bentham, Jeremy (17481832)
O Logic: born gatekeeper to the Temple of Science, victim of capricious destiny: doomed hitherto to be the drudge of pedants: come to the aid of thy master, Legislation.
In J. Browning (ed.) Works.
Bernoulli, Daniel
...it would be better for the true physics if there were no mathematicians on earth.
In The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
Bernoulli, Jacques (Jakob?) (16541705)
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.
Blake
God forbid that Truth should be confined to Mathematical Demonstration!
Notes on Reynold's Discourses, c. 1808.
Bohr, Niels Henrik David (18851962)
An expert is a man who has made all the mistakes, which can be made, in a very narrow field.
The Bible
I returned and saw under the sun that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all.
Ecclesiastes.
Bolyai, János (1802  1860)
Out of nothing I have created a strange new universe.
[A reference to the creation of a noneuclidean geometry.]
Bolyai, Wolfgang (17751856)
[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 give up work on nonEuclidian geometry.]
In P. Davis and R. Hersh The Mathematical Experience , Boston: Houghton Mifflin Co., 1981, p. 220.
Bourbaki
Structures are the weapons of the mathematician.
Bridgman, P. W.
It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention.
The Logic of Modern Physics, New York, 1972.
Brown, George Spencer (1923  )
To arrive at the simplest truth, as Newton knew and practiced, requires years of contemplation. Not activity. Not reasoning. Not calculating. Not busy behaviour of any kind. Not reading. Not talking. Not making an effort. Not thinking. Simply bearing in mind what it is one needs to know. And yet those with the courage to tread this path to real discovery are not only offered practically no guidance on how to do so, they are actively discouraged and have to set about it in secret, pretending meanwhile to be diligently engaged in the frantic diversions and to conform with the deadening personal opinions which are continually being thrust upon them.
The Laws of Form. 1969.
Browne, Sir Thomas (16051682)
God is like a skilful Geometrician.
Religio Medici I, 16.
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, Urnburial and the Garden of Cyrus, 1896.
...indeed what reason may not go to Schoole to the wisdome of Bees, Aunts, and Spiders? what wise hand teacheth them to doe what reason cannot teach us? ruder heads stand amazed at those prodigious pieces of nature, Whales, Elephants, Dromidaries and Camels; these I confesse, are the Colossus and Majestick pieces of her hand; but in these narrow Engines there is more curious Mathematicks, and the civilitie of these little Citizens more neatly sets forth the wisedome of their Maker.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1001.
Buck, Pearl S. (1892  1973)
No one really understood music unless he was a scientist, her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. "And relationships," he had told her, "contained the essential meaning of life."
The Goddess Abides, Pt. I, 1972.
Burke, Edmund
The age of chivalry is gone. That of sophisters, economists and calculators has succeeded.
Reflections on the Revolution in France.
Butler, Bishop
To us probability is the very guide of life.
Preface to Analogy.
Butler, Samuel (1835  1902)
... There can be no doubt about faith and not reason being the ultima ratio. Even Euclid, who has laid himself as little open to the charge of credulity as any writer who ever lived, cannot get beyond this. He has no demonstrable first premise. He requires postulates and axioms which transcend demonstration, and without which he can do nothing. His superstructure indeed is demonstration, but his ground his faith. Nor again can he get further than telling a man he is a fool if he persists in differing from him. He says "which is absurd," and declines to discuss the matter further. Faith and authority, therefore, prove to be as necessary for him as for anyone else.
The Way of All Flesh.
Byron
When Newton saw an apple fall, he found ...
A mode of proving that the earth turnd round
In a most natural whirl, called gravitation;
And thus is the sole mortal who could grapple
Since Adam, with a fall or with an apple.
Caballero, James
I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.
Everybody a mathematician?,CAIP Quarterly 2 (Fall, 1989).
Carlyle, Thomas (1795  1881)
It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity of the universe.
Sartor Resartus III.
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.
A witty statesman said, you might prove anything by figures.
Chartism.
Carroll, Lewis
What I tell you three times is true.
The Hunting of the Snark.
The different branches of Arithmetic  Ambition, Distraction, Uglification, and Derision.
Alice in Wonderland.
"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.
"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
Alice in Wonderland.
"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. "Why, you might just as well say that "I see what I eat" is the same thing as "I eat what I see!"
Alice in Wonderland.
"It's very good jam," said the Queen.
"Well, I don't want any today, 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 today."
"It must come sometimes to "jam today,""Alice objected.
"No it can't," said the Queen. "It's jam every other day; today isn't any other day, you know."
"I don't understand you," said Alice. "It's dreadfully confusing."
Through the Looking Glass.
"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 different things."
"The question is," said Humpty Dumpty, "which is to be master  that's all."
Through the Looking Glass.
Carmichael, R. D.
A thing is obvious mathematically after you see it.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Chebyshev
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.
Chekov, Anton (1860  1904)
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.
Chesterton, G. K. (1874  1936)
Poets do not go mad; but chessplayers 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.
Orthodoxy ch. 2.
You can only find truth with logic if you have already found truth without it.
The Man who was Orthodox. 1963.
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.
Christie, Agatha
"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 likely it is that the hats will get mixed up and in what proportion. If you start thinking about things like that, you would go round the bend. Let me assure you of that!"
The Mirror Crack'd. Toronto: Bantam Books, 1962.
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 Autobiography.
Churchill, [Sir] Winston Spencer (18741965)
It is a good thing for an uneducated man to read books of quotations.
Roving Commission in My Early Life. 1930.
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 sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Churchman, C. W.
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.
Cocteau
The composer opens the cage door for arithmetic, the draftsman gives geometry its freedom.
Coleridge, Samuel Taylor (17721834)
...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 mathematical formulae.
The Theory of Life.
Conrad, Joseph
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.
Preface to A Personal Record.
Coolidge, Julian Lowell (1873  1954)
[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. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Copernicus, Nicholaus (14731543)
Mathematics is written for mathematicians.
De Revolutionibus.
Crick, Francis Harry Compton (1916  )
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. London: Weidenfeld and Nicolson, 1988.
Crowe, Michael
Revolutions never occur in mathematics.
Historia Mathematica. 1975.
D'Alembert, Jean Le Rond (17171783)
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.
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 mathematician who creates makes no less difference than in a poet who invents.... Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer.
Discours Preliminaire de L'Encyclopedie, Tome 1, 1967. pp 47  48.
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 criterion? Not by evidence, for the dice of evidence are loaded. Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiplied necessity that we call mathematics.
How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.
Darwin, Charles
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.
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.
Davis, Philip J.
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.
One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.
Number, Scientific American, 211, (Sept. 1964), 51  59.
Davis, Philip J. and Hersh, Reuben
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.
Dehn, Max
Mathematics is the only instructional material that can be presented in an entirely undogmatic way.
In The Mathematical Intelligencer, v. 5, no. 2, 1983.
De Morgan, Augustus (18061871)
[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.
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.
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 Society, vol. X, 1864, p. 184.
Descartes, René (15961650)
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 Méthode. 1637.
Each problem that I solved became a rule which served afterwards to solve other problems.
Discours de la Méthode. 1637.
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 Méthode. 1637.
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.
Discours de la Méthode. 1637.
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; that is, carefully to avoid precipitancy and prejudice, and to embrace in my judgment only what presented itself to my mind so clearly and distinctly that I had no occasion to doubt it. The second, to divide each problem I examined into as many parts as was feasible, and as was requisite for its better solution. The third, to direct my thoughts in an orderly way; beginning with the simplest objects, those most apt to be known, and ascending little by little, in steps as it were, to the knowledge of the most complex; and establishing an order in thought even when the objects had no natural priority one to another. And the last, to make throughout such complete enumerations and such general surveys that I might be sure of leaving nothing out. 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 that so long as we avoid accepting as true what is not so, and always preserve the right order of deduction of one thing from another, there can be nothing too remote to be reached in the end, or to well hidden to be discovered.
Discours de la Méthode. 1637.
When writing about transcendental issues, be transcendentally clear.
In G. Simmons Calculus Gems. New York: McGraw Hill Inc., 1992.
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 knew several peculiarities of this conformation, we would from those deduce the nature of its seed.
Cogito Ergo Sum. "I think, therefore I am."
Discours de la Méthode. 1637.
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.
Perfect numbers like perfect men are very rare.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
omnia apud me mathematica fiunt.
With me everything turns into mathematics.
It is not enough to have a good mind. The main thing is to use it well.
Discours de la Méthode. 1637.
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.
De Sua, F. (1956)
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 would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Diophantus
[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 kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! latebegotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.
In Ivor Thomas Greek Mathematics, in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Dirac, Paul Adrien Maurice (1902 )
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 Schrodinger had been more confident of his work, he could have published it some months earlier, and he could have published a more accurate equation. It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory.
Scientific American, May 1963.
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.
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, 1977.
Disraeli, Benjamin
There are three kinds of lies: lies, damned lies, and statistics.
Mark Twain. Autobiography.
Donatus, Aelius (4th Century)
Pereant qui ante nos nostra dixerunt.
"To the devil with those who published before us."
[Quoted by St. Jerome, his pupil]
Doyle, Sir Arthur Conan (18591930)
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 fifth proposition of Euclid.
The Sign of Four.
When you have eliminated the impossible, what ever remains, however improbable must be the truth.
The Sign of Four.
From a drop of water a logician could predict an Atlantic or a Niagara.
A study in Scarlet 1929.
It is a capital mistake to theorize before one has data.
Scandal in Bohemia.
Dryden, John (16311700)
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 be a complete and excellent poet.
Notes and Observations on The Empress of Morocco. 1674.
Dubos, René J.
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 Armchair Science Reader, New York: Simon and Schuster, 1959.
Dunsany, Lord
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.
Dürer, Albrecht (14711528)
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 of care and diligence. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who are themselves ignorant of this art.
The Art of Measurement. 1525.
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.
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
Dyson, Freeman
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?)
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.
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 the details miraculously fit. The problem is the overall design.
"Freeman Dyson: Mathematician, Physicist, and Writer". Interview with Donald J. Albers, The College Mathematics Journal, vol 25, no. 1, January 1994.
Eddington, Sir Arthur (18821944)
Proof is the idol before whom the pure mathematician tortures himself.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
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.
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.
Space, Time and Gravitation. 1920.
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.
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.
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 incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the "cat without a grin" and the "grin without a cat" are equally set aside as purely mathematical phantasies.
The Expanding Universe..
Human life is proverbially uncertain; few things are more certain than the solvency of a lifeinsurance company.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Edwards, Jonathon
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. Mallon A Book of One's Own. Ticknor & Fields, New York, 1984, p. 106107.
Egrafov, M.
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. 5, December 1992.
Eigen, Manfred (1927  )
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.
Einstein, Albert (18791955)
[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.
Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.
I don't believe in mathematics.
Quoted by Carl Seelig. Albert Einstein.
Imagination is more important than knowledge.
On Science.
The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.
What I Believe.
The bitter and the sweet come from the outside, the hard from within, from one's own efforts.
Out of My Later Years.
Gott würfelt nicht.
Common sense is the collection of prejudices acquired by age eighteen.
In E. T. Bell Mathematics, Queen and Servant of the Sciences. 1952.
God does not care about our mathematical difficulties. He integrates empirically.
L. Infeld Quest, 1942.
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?
[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.
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.
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.
Raffiniert ist der Herr Gott, aber boshaft ist er nicht. God is subtle, but he is not malicious.
Inscribed in Fine Hall, Princeton University.
Nature hides her secrets because of her essential loftiness, but not by means of ruse.
The human mind has first to construct forms, independently, before we can find them in things.
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, PhilosopherScientist, Evanston, 1949.
Do not worry about your difficulties in mathematics, I assure you that mine are greater.
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.
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.
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 delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
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.
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.
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.
The search for truth is more precious than its possession.
The American Mathematical Monthly v. 100 no. 3.
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.
Address at the Sorbonne, Paris.
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 distanceconcept is logically arbitrary; there need be no things that correspond to it, even approximately.
"SpaceTime." Encyclopaedia Britannica, 14th ed.
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.
Science without religion is lame; religion without science is blind.
Reader's Digest, Nov. 1973.
Ellis, Havelock
The mathematician has reached the highest rung on the ladder of human thought.
The Dance of Life.
It is here [in mathematics] that the artist has the fullest scope of his imagination.
The Dance of Life.
Erath, V.
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.
Euler, Leonhard (1707  1783)
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 not so many mysteries hidden in this concept as they are usually believed to be. These supposed mysteries have rendered the calculus of the infinitely small quite suspect to many people. Those doubts that remain we shall thoroughly remove in the following pages, where we shall explain this calculus.
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: McGraw Hill Inc., 1992.
[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.
Everett, Edward (17941865
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. Bell in The Queen of the Sciences, Baltimore, 1931.
Eves, Howard W.
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.
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.
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 mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well placed internal charges.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
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 HinduArabic numerals to record numbers. In this we perhaps have man's unique worldwide victory of an idea.
Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Ewing, John
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 enough to actually observe them.
"Can We See the Mandelbrot Set?", The College Mathematics Journal, v. 26, no. 2, March 1995.
de Fermat, Pierre (1601?1665)
[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 assuredly found an admirable proof of this, but the margin is too narrow to contain it.
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.
Feynman, Richard Philips (1918  1988)
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 place to publish, in a dignified manner, what you actually did in order to get to do the work.
Nobel Lecture, 1966.
Finkel, Benjamin Franklin
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 minds have been aroused into activity through the mastery of a single problem.
The American Mathematical Monthly, no. 1.
Flaubert, Gustave (18211880)
Poetry is as exact a science as geometry.
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 passengers aboard, the wind is blowing EastNorthEast, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?
Frankland, W.B.
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, The Story of Euclid. 1901.
Galbraith, John Kenneth
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.
Galilei, Galileo (1564  1642)
[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 means it is humanly impossible to comprehend a single word.
Opere Il Saggiatore p. 171.
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.
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 whatsoever competence are made judges over experts and are granted authority to treat them as they please? These are the novelties which are apt to bring about the ruin of commonwealths and the subversion of the state.
[On the margin of his own copy of Dialogue on the Great World Systems].
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 733.
Galois, Evariste
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.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Galton, [Sir] Francis (18221911)
Whenever you can, count.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
[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.
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error." The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete selfeffacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 1482.
Gardner, Martin
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 who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.
In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.
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 what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure.
Gardner on Gardner: JPBM Communications Award Presentation. FocusThe Newsletter of the Mathematical Association of America v. 14, no. 6, December 1994.
Gauss, Karl Friedrich (17771855)
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 entice him to work on Fermat's Theorem.] In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 312.
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.
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 and completely outside the province of science.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 314.
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 Hill inc., 1992.
God does arithmetic.
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.
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 Calculus Gems, New York: McGraw Hill inc., 1992.
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.
[His motto:]
Few, but ripe.
[His second motto:]
Thou, nature, art my goddess; to thy laws my services are bound...
W. Shakespeare King Lear.
[attributed to him by H.B Lübsen]
Theory attracts practice as the magnet attracts iron.
Foreword of H.B Lübsen's geometry textbook.
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 neversatisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.
Letter to Bolyai, 1808.
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 what made my success possible.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
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 we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
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.
Quoted in J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990.
Gay, John
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.
Gibbs, Josiah Willard (1839  1903)
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.
Mathematics is a language.
Gilbert, W. S. (1836  1911)
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 MajorGeneral.
The Pirates of Penzance. Act 1.
Glaisher, J.W.
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, Boston: Prindle, Weber and Schmidt, 1972.
Goethe
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.
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 all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1754.
Goodman, Nicholas P.
There are no deep theorems  only theorems that we have not understood very well.
The Mathematical Intelligencer, vol. 5, no. 3, 1983.
Gordon, 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.
Graham, Ronald
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 (October 1993) 92103.
Grünbaum, Branko (1926  ), and Shephard, G. C. (?)
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 Applicable Mathematics.
Hadamard, Jacques
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.
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.
Halmos, Paul R.
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, Washington: MAA Spectrum, 1985.
... 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.
...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 be a Mathematician, Washington: MAA Spectrum, 1985.
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 tension.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
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 proof use the hypothesis?
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
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.
Hamilton, [Sir] William Rowan (18051865)
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.
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.
On earth there is nothing great but man; in man there is nothing great but mind.
Lectures on Metaphysics.
Hamming, Richard W.
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 in that plane.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
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.
Hardy, Godfrey H. (1877  1947)
[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 replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
Ramanujan, London: Cambridge Univesity Press, 1940.
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 Mathematician's Apology, London, Cambridge University Press, 1941.
I am interested in mathematics only as a creative art.
A Mathematician's Apology, London, Cambridge University Press, 1941.
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.
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.
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 secondrate minds.
A Mathematician's Apology, London, Cambridge University Press, 1941.
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.
A science is said to be useful if 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 Press, 1941.
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.
A Mathematician's Apology, London, Cambridge University Press, 1941.
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.
A Mathematician's Apology, London, Cambridge University Press, 1941.
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's Apology, London, Cambridge University Press, 1941.
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 than in music. Appearances may suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.
A Mathematician's Apology, London, Cambridge University Press, 1941.
Hardy, Thomas
...he seemed to approach the grave as an hyperbolic curve approaches a lineless directly as he got nearer, till it was doubtful if he would ever reach it at all.
Far from the Madding Crowd.
HarishChandra
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 advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naivete, unburdened by conventional wisdom, can sometimes be a positive asset.
R. Langlands, "HarishChandra," Biographical Memoirs of Fellows of the Royal Society 31 (1985) 197  225.
Harris, Sydney J.
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.
Hawking, Stephen Williams (1942 )
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.
Heath, Sir Thomas
[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, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
A History of Greek Mathematics. 1921.
Heaviside, Oliver (18501925)
[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?
Heinlein, Robert A.
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.
Heisenberg, Werner (19011976)
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.
Hempel, Carl G.
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 that certainty: The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter.
"On the Nature of Mathematical Truth" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.
"Geometry and Empirical Science" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
...to characterize the import of pure geometry, we might use the standard form of a moviedisclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental.
"Geometry and Empirical Science" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Henkin, Leon
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 answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That's so unlike the true nature of mathematics.
L.A. Steen and D.J. Albers (eds.), Teaching Teachers, Teaching Students, Boston: Birkhäuser, 1981, p89.
Hermite, Charles (1822  1901)
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 creation.
In The Mathematical Intelligencer, v. 5, no. 4.
Abel has left mathematicians enough to keep them busy for 500 years.
In G. F. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
We are servants rather than masters in mathematics.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Hertz, Heinrich
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 them.
Quoted by ET Bell in Men of Mathematics,, New York, 937.
Hesse, Hermann (18771962)
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 Game, 1943.
Hilbert, David (18621943)
Wir müssen wissen.
Wir werden wissen.
[Engraved on his tombstone in Göttingen.]
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 Science, New York: McGraw Hill Inc., 1951.
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 Schmidt, 1971.
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.
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.
Physics is much too hard for physicists.
C. Reid Hilbert, London: Allen and Unwin, 1970.
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 complicated developments.
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.
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 Minims, Raleigh NC: Rome Press Inc., 1988.
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.
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.
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.
Hirst, Thomas Archer
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 returned, cursing the mode of life I was pursuing; next morning I had already shaken hands, however, with Diff. Calculus, and forgot the ladies....
J. Helen Gardner and Robin J. Wilson, "Thomas Archer Hirst  Mathematician Xtravagant II  Student Days in Germany", The American Mathematical Monthly , v. 6, no. 100.
Hobbes, Thomas
There is more in Mersenne than in all the universities together.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
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.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
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.
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, New York: Simon and Schuster, 1956.
Holmes, Oliver Wendell
Descartes commanded the future from his study more than Napoleon from the throne.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
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.
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 + 2 = 4. Every philosophical proposition has the more general character of the expression a + b = c. We are mere operatives, empirics, and egotists until we learn to think in letters instead of figures.
The Autocrat of the Breakfast Table.
Holt, M. and Marjoram, D. T. E.
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 depends on it.
Mathematics in a Changing World Walker, New York 1973.
Hofstadter, Douglas R. (1945  )
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.
Gödel, Escher, Bach 1979.
Hughes, Richard
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 scientist of today stands counting out loud in the face of the unknown. Numbers come back to him  and he believes in the Great Mathematician.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Hume, David (1711  1776)
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 existence?' No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.
Treatise Concerning Human Understanding.
Huxley, Aldous
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.
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 a monster he was superb.
Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934.
...[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 always itself, would in any case have lost its charm, while the rudiments of algebra continued to grow and blossom in his mind with an unfailing luxuriance. Every day he made the discovery of something which seemed to him exquisitely beautiful; the new toy was inexhaustible in its potentialities.
Young Archimedes.
Huxley, Thomas Henry (18251895)
This seems to be one of the many cases in which the admitted accuracy of mathematical processes is allowed to throw a wholly inadmissible appearance of authority over the results obtained by them. Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff of any degree of fineness; but, nevertheless, what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulae will not get a definite result out of loose data.
Quarterly Journal of the Geological Society, 25,1869.
The mathematician starts with a few propositions, the proof of which is so obvious that they are called selfevident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature authority and tradition furnish the data, and the mental operations are deductive.
"Scientific Education Notes of an Afterdinner Speech." Macmillan's Magazine Vol XX, 1869.
It is the first duty of a hypothesis to be intelligible.
Ibn Khaldun (13321406)
Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence.
The Muqaddimah. An Introduction to History.
Isidore of Seville (ca 600 ad)
Take from all things their number and all shall perish.
Jacobi, Carl
It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
God ever arithmetizes.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.
One should always generalize.
(Man muss immer generalisieren)
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
The real end of science is the honor of the human mind.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
It is often more convenient to possess the ashes of great men than to possess the men themselves during their lifetime.
[Commenting on the return of Descartes' remains to France]
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Mathematics is the science of what is clear by itself.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
James, William (1842  1910)
The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.
Collected Essays.
Jeans, Sir James
The essential fact is that all the pictures which science now draws of nature, and which alone seem capable of according with observational facts, are mathematical pictures.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.
Mysterious Universe.
Jefferson, Thomas
...the science of calculation also is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not be in indulged in by one who is to have a profession to follow for his subsistence.
In J. Robert Oppenheimer "The Encouragement of Science" in I. Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.
Jevons, William Stanley
It is clear that Economics, if it is to be a science at all, must be a mathematical science.
Theory of Political Economy.
Johnson, Samuel (17091784)
Sir, I have found you an argument. I am not obliged to find you an understanding.
J. Boswell The Life of SamuelJohnson, 1784.
Jowett, Benjamin (1817  1893)
Logic is neither a science or an art, but a dodge.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Kant, Emmanual (1724  1804)
The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.
The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
All human knowledge thus begins with intuitions, proceeds thence to concepts, and ends with ideas.
Quoted in Hilbert's Foundations of Geometry.
Kaplan, Abraham
Mathematics is not yet capable of coping with the naivete of the mathematician himself.
Sociology Learns the Language of Mathematics.
Kaplansky, Irving
We (he and Halmos) share a philosophy about linear algebra: we think basisfree, we write basisfree , but when the chips are down we close the office door and compute with matrices like fury.
Paul Halmos: Celebrating 50 Years of Mathematics.
Karlin, Samuel (1923  )
The purpose of models is not to fit the data but to sharpen the questions.
11th R A Fisher Memorial Lecture, Royal Society 20, April 1983.
Kasner, E. and Newman, J.
Mathematics is man's own handiwork, subject only to the limitations imposed by the laws of thought.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
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.
Mathematics and the Imagination, Simon and Schuster, 1940.
...we have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Mathematics is the science which uses easy words for hard ideas.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Perhaps the greatest paradox of all is that there are paradoxes in mathematics.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Keller, Helen (1880  1968)
Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important... The knowledge doesn't make life any sweeter or happier, does it?
The Story of My Life. 1903.
Kepler, Johannes (15711630)
A mind is accustomed to mathematical deduction, when confronted with the faulty foundations of astrology, resists a long, long time, like an obstinate mule, until compelled by beating and curses to put its foot into that dirty puddle.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Where there is matter, there is geometry.
(Ubi materia, ibi geometria.)
J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990
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.
Nature uses as little as possible of anything.
Keynes, John Maynard
It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability.
The Application of Probability to Conduct.
Kleinhenz, Robert J.
When asked what it was like to set about proving something, the mathematician likened proving a theorem to seeing the peak of a mountain and trying to climb to the top. One establishes a base camp and begins scaling the mountain's sheer face, encountering obstacles at every turn, often retracing one's steps and struggling every foot of the journey. Finally when the top is reached, one stands examining the peak, taking in the view of the surrounding countryside and then noting the automobile road up the other side!
Kline, Morris
A proof tells us where to concentrate our doubts.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Statistics: the mathematical theory of ignorance.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Logic is the art of going wrong with confidence.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Universities hire professors the way some men choose wives  they want the ones the others will admire.
Why the Professor Can't Teach. St. Martin's Press, 1977. p 92.
Koestler, Arthur (1905 )
In the index to the six hundred odd pages of Arnold Toynbee's A Study of History, abridged version, the names of Copernicus, Galileo, Descartes and Newton do not occur yet their cosmic quest destroyed the medieval vision of an immutable social order in a walledin universe and transformed the European landscape, society, culture, habits and general outlook, as thoroughly as if a new species had arisen on this planet.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Nobody before the Pythagoreans had thought that mathematical relations held the secret of the universe. Twentyfive centuries later, Europe is still blessed and cursed with their heritage. To nonEuropean civilizations, the idea that numbers are the key to both wisdom and power, seems never to have occurred.
The Sleepwalkers. 1959.
Kraft, Prinz zu HohlenloheIngelfingen (1827  1892)
Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else.
Attributed by Karl Schellbach. In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Kronecker, Leopold (18231891)
Number theorists are like lotuseaters  having once tasted of this food they can never give it up.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
God made the integers, all else is the work of man.
Jahresberichte der Deutschen Mathematiker Vereinigung.
La Touche, Mrs.
I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are permutations and aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then from the top down, the result is always different.
Mathematical Gazette, v. 12.
LaGrange, JosephLouis
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 Mécanique Analytique.
[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.
When we ask advice, we are usually looking for an accomplice.
Lakatos, Imre
That sometimes clear ... and sometimes vague stuff ... which is ... mathematics.
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
Lanczos, Cornelius
Most of the arts, as painting, sculpture, and music, have emotional appeal to the general public. This is because these arts can be experienced by some one or more of our senses. Such is not true of the art of mathematics; this art can be appreciated only by mathematicians, and to become a mathematician requires a long period of intensive training. The community of mathematicians is similar to an imaginary community of musical composers whose only satisfaction is obtained by the interchange among themselves of the musical scores they compose.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Landau, E.
[Asked for a testimony to the effect that Emmy Noether was a great woman mathematician, he said:]
I can testify that she is a great mathematician, but that she is a woman, I cannot swear.
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co ltd., 1953.
Landau, Susan
There's a touch of the priesthood in the academic world, a sense that a scholar should not be distracted by the mundane tasks of daytoday living. I used to have great stretches of time to work. Now I have research thoughts while making peanut butter and jelly sandwiches. Sure it's impossible to write down ideas while reading "curious George" to a twoyearold. On the other hand, as my husband was leaving graduate school for his first job, his thesis advisor told him, "You may wonder how a professor gets any research done when one has to teach, advise students, serve on committees, referee papers, write letters of recommendation, interview prospective faculty. Well, I take long showers."
In Her Own Words: Six Mathematicians Comment on Their Lives and Careers. Notices of the AMS, V. 38, no. 7 (September 1991), p. 704.
Lang, Andrew (18441912)
He uses statistics as a drunken man uses lamp posts  for support rather than illumination.
Treasury of Humorous Quotations.
Langer, Rudoph E.
[about Fourier] It was, no doubt, partially because of his very disregard for rigor that he was able to take conceptual steps which were inherently impossible to men of more critical genius.
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
Lao Tze (604531 B.C.)
A good calculator does not need artificial aids.
Tao Te Ching, ch 27.
de Laplace, PierreSimon (1749  1827)
Read Euler: he is our master in everything.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
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.
Napoleon: You have written this huge book on the system of the world without once mentioning the author of the universe.
Laplace: Sire, I had no need of that hypothesis.
Later when told by Napoleon about the incident, Lagrange commented: Ah, but that is a fine hypothesis. It explains so many things.
DeMorgan's Budget of Paradoxes.
[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.
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Leach, Edmund Ronald (1910  1989)
How can a modern anthropologist embark upon a generalization with any hope of arriving at a satisfactory conclusion? By thinking of the organizational ideas that are present in any society as a mathematical pattern.
Rethinking Anthropology. 1961.
Leacock, Stephen
How can you shorten the subject? That stern struggle with the multiplication table, for many people not yet ended in victory, how can you make it less? Square root, as obdurate as a hardwood stump in a pasturenothing but years of effort can extract it. You can't hurry the process. Or pass from arithmetic to algebra; you can't shoulder your way past quadratic equations or ripple through the binomial theorem. Instead, the other way; your feet are impeded in the tangled growth, your pace slackens, you sink and fall somewhere near the binomial theorem with the calculus in sight on the horizon. So died, for each of us, still bravely fighting, our mathematical training; except for a set of people called "mathematicians"  born so, like crooks.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Lebesgue, Henri (1875  1941)
In my opinion, a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy  an opinion, moreover, which has been expressed by many philosophers.
Scientific American, 211, September 1964, p. 129.
Leibniz, Gottfried Whilhem (16461716)
[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]
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.
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.
The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.
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.
In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
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.
Although the whole of this life were said to be nothing but a dream and the physical world nothing but a phantasm, I should call this dream or phantasm real enough, if, using reason well, we were never deceived by it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
The soul is the mirror of an indestructible universe.
The Monadology.
da Vinci, Leonardo (14521519)
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.
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.
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.
No human investigation can be called real science if it cannot be demonstrated mathematically.
Inequality is the cause of all local movements.
Leybourn, William (16261700)
But leaving those of the Body, I shall proceed to such Recreation as adorn the Mind; of which those of the Mathematicks are inferior to none.
Pleasure with Profit, 1694.
Lichtenberg, Georg Christoph (1742  1799)
All mathematical laws which we find in Nature are always suspect to me, in spite of their beauty. They give me no pleasure. They are merely auxiliaries. At close range it is all not true.
In J P Stern Lichtenberg, 1959.
The great trick of regarding small departures from the truth as the truth itself  on which is founded the entire integral calculus  is also the basis of our witty speculations, where the whole thing would often collapse if we considered the departures with philosophical rigour.
Aphorisms.
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 ax. Hence mathematical language has great advantages over the common language.
I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, "Oh, I see. That's how it must be." This is a sign that they explain it to themselves from within their own system.
le Lionnais, Francois
Who has not be amazed to learn that the function y = e^x , like a phoenix rising again from its own ashes, is its own derivative?
Great Currents of Mathematical Thought, vol. 1, New York: Dover Publications.
Lippman, Gabriel (18451921)
[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.
Littlewood, J. E. (1885 1977)
It is true that I should have been surprised in the past to learn that Professor Hardy had joined the Oxford Group. But one could not say the adverse chance was 1:10. Mathematics is a dangerous profession; an appreciable proportion of us go mad, and then this particular event would be quite likely.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
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.
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.
"The Mathematician's Art of Work" in Béla Bollobás (ed.) Littlewood's Miscellany, Cambridge: Cambridge University Press, 1986.
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 finding a subtle and delicate route.)
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
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] subjects need mean nothing; much drill and drudgery is unavoidable before they can get started, and bad teaching can make them unintelligible even to a born mathematician.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
The infinitely competent can be uncreative.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
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 accidents, or in case he wishes for once to check in detail, he should have only a clearly circumscribed little problem to solve (e.g. to check an identity: two trivialities omitted can add up to an impasse). The unpractised writer, even after the dawn of a conscience, gives him no such chance; before he can spot the point he has to tease his way through a maze of symbols of which not the tiniest suffix can be skipped.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
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, Methuen Co. Ltd., 1953.
The surprising thing about this paper is that a man who could write it would.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
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.
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 proofsheets I read (what now stands), "It was Littlewood who said..."
A Mathematician's Miscellany, Methuen Co. Ltd, 1953.
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 system runs parallel to the usual theory: "If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician". In practice he is apt to say: "try this; if it works that will justify it". But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.
A Mathematician's Miscellany, Methuen Co. Ltd, 1953.
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 obvious interest, or else one just leaves the problem alone. "Perfect numbers" certainly never did any good, but then they never did any particular harm.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
Lobatchevsky, Nikolai
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.
Locke, John
...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.
Luther, Martin (14831546)
Medicine makes people ill, mathematics make them sad and theology makes them sinful.
Mach, Ernst (1838  1916)
Archimedes constructing his circle pays with his life for his defective biological adaptation to immediate circumstances.
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 Mathematics, New York: Simon and Schuster, 1956.
Mackay, Alan Lindsay (1926 )
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, Charles (18141889)
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.
Maistre Joseph Marie de (1753  1821)
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.
Mann, Thomas (18751955)
A great truth is a truth whose opposite is also a great truth.
Essay on Freud. 1937.
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.
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 given point more than one parallel to a straight line; Frau Hagenström had cried out that this was impossible, and he had gone on to prove it so conclusively that his hearers were constrained to behave as though they understood.
Little Herr Friedemann.
Mathesis, Adrian
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.
All great theorems were discovered after midnight.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
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.
Matthias, Bernd T
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.
Maxwell, James Clerk (18131879)
... 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 2, 244, October 1871.
Mayer, Maria Goeppert (1906 1972)
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 GoeppertMayer, A Life of One's Own.
McDuff, Dusa
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 clearly. I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gel'fand found hedgehogs lurking in the rows of his spectral sequences!
Mathematical Notices v. 38, no. 3, March 1991, pp. 1857.
McShane, E. J.
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 rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.
Bulletin of the American Mathematical Society, v. 69, p. 611, 1963.
Mencken, H.L. (1880  1956)
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".
Mermin, N. David (1935 )
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.
Millay, Edna St. Vincent (1892  1950)
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 shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
Milton, John (1608  1674)
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  perhaps to move
His laughter at their quaint opinions wide
Hereafter, when they come to model Heaven
And calculate the stars: how they will wield
The mighty frame: how build, unbuild, contrive
To save appearances; how gird the Sphere
With Centric and Eccentric scribbled o'er,
Cycle and Epicycle, Orb in Orb.
Paradise Lost.
Chaos umpire sits
And by decision more
embroils the fray
by which he reigns: next
him high arbiter
Chance governs all.
Minkowski, Herman
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.
Minsky, Marvin Lee (1927 )
Logic doesn't apply to the real world.
D. R. Hofstadter and D. C. Dennett (eds.) The Mind's I, 1981.
Mitchell, Margaret
...She knew only that if she did or said thusandso, men would unerringly respond with the complimentary thusandso. It was like a mathematical formula and no more difficult, for mathematics was the one subject that had come easy to Scarlett in her schooldays.
Gone With the Wind.
MittagLeffler, Gösta
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:Rome Press Inc., 1988.
Mordell, L.J. (1888  ?)
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 miners. One may be an expert geologist, but he does not find the golden nuggets that the ignorant miner does.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Moore, E.H. (1862  1932)
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 with respect to those central features...."
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.
Moroney, M.J.
The words figure and fictitious both derive from the same Latin root, fingere. Beware!
Facts from Figures.
Napoleon (17691821)
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 doubtful ideas, and finally carried the spirit of the infinitely small into administration.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Nebeuts, E. Kim
Teach to the the problems, not to the text.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
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.
A good preparation takes longer than the delivery.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Neumann, Franz Ernst (1798  1895)
The greatest reward lies in making the discovery; recognition can add little or nothing to that.
von Neumann, Johann (1903  1957)
In mathematics you don't understand things. You just get used to them.
In G. Zukav The Dancing Wu Li Masters.
Newman, James R.
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.
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 to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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, 1956.
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 applied to economic reasoning. There have even been rumors that mathematics is used in economics (and in other social sciences) either for the deliberate purpose of mystification or to confer dignity upon commonplacesas French was once used in diplomatic communications. .... 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 it is desirable to know what you are talking about.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.) The World of Mathematics, New York: Simon and Schuster, 1956.
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 dish that tastes like a damp blanket.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Newton, Isaac (16421727)
...from the same principles, I now demonstrate the frame of the System of the World.
Principia Mathematica.
Hypotheses non fingo.
I feign no hypotheses.
Principia Mathematica.
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, New York: McGraw Hill Inc., 1992.
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.
The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of mathematics.
[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.
Thomas R. Nicely
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.
Nightingale, Florence (18201910)
[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 all the actions of her life acted upon that belief  that the administrator could only be successful if he were guided by statistical knowledge. The legislator  to say nothing of the politician  too often failed for want of this knowledge. Nay, she went further; she held that the universe  including human communities  was evolving in accordance with a divine plan; that it was man's business to endeavor to understand this plan and guide his actions in sympathy with it. But to understand God's thoughts, she held we must study statistics, for these are the measure of His purpose. Thus the study of statistics was for her a religious duty.
K. Pearson The Life, Letters and Labours for Francis Galton, vol. 2, 1924.
Oakley, C.O.
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.
Ogyu, Sorai (1666  1729)
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.
Oppenheimer, Julius Robert (1904  1967)
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.
Osgood, W. F.
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.
Pascal, Blaise (16231662)
We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others.
Pensees. 1670.
It is the heart which perceives God and not the reason.
Pensees. 1670.
Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.
Pensees. 1670.
Our nature consists in movement; absolute rest is death.
Pensees. 1670.
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 would take me for a besieged town. I need, that is to say, a decent man who can accommodate himself to all my desires in a general sort of way.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
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.
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, New York: Viking Press, 1966.
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.
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 Viking Book of Aphorisms, New York: Viking Press, 1966.
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.
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.
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.
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: Viking Press, 1966.
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.
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.
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.
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 principles do not understand matters of feeling at all, because they look for first principles and are unable to comprehend at a glance.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
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.
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.
Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.
Pensees. 1670.
We arrive at truth, not by reason only, but also by the heart.
Pensees. 1670.
When the passions become masters, they are vices.
Pensees. 1670.
Men despise religion; they hate it, and they fear it is true.
Pensees. 1670.
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.
It is not certain that everything is uncertain.
Pensees. 1670.
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.
Pensees. 1670.
The sole cause of man's unhappiness is that he does not know how to stay quietly in his room.
Pensees. 1670.
Reason's last step is the recognition that there are an infinite number of things which are beyond it.
Pensees. 1670.
Through space the universe grasps me and swallows me up like a speck; through thought I grasp it.
Pensees. 1670.
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.
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.
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.
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.
The last thing one knows when writing a book is what to put first.
Pensees. 1670.
What is man in nature? Nothing in relation to the infinite, all in relation to nothing, a mean between nothing and everything.
Pensees. 1670.
[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.
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.
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: McGraw Hill Inc., 1992.
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.
The more intelligent one is, the more men of originality one finds. Ordinary people find no difference between men.
Pensees. 1670.
However vast a man's spiritual resources, he is capable of but one great passion.
Discours sur les passions de l'amour. 1653.
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 diverse lovable parts of that which it loves.
Discours sur les passions de l'amour. 1653.
Passano, L.M.
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.
Pasteur, Louis
Chance favors only the prepared mind.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988
Pearson, Karl
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, Raleigh NC:Rome Press Inc., 1988.
Peirce, Benjamin (18091880)
Mathematics is the science which draws necessary conclusions.
Memoir read before the National Academy of Sciences in Washington, 1870.
Peirce, Charles Sanders (18391914)
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: Simon and Schuster, 1956.
...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logiclogic, especiallyis in its early stages in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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 universality; their practical infallibility.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
The pragmatist knows that doubt is an art which hs to be acquired with difficulty.
Collected Papers.
Pedersen, Jean
Geometry is a skill of the eyes and the hands as well as of the mind.
Plato (ca 429347 BC)
He who can properly define and divide is to be considered a god.
The ludicrous state of solid geometry made me pass over this branch. Republic, VII, 528.
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.
Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.
The knowledge of which geometry aims is the knowledge of the eternal.
Republic, VII, 52.
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.
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.
Plutarch (ca 46127)
[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 geometrical figures in the ashes of the fire, and with his finger draws lines upon his body when it was anointed with oil, being in a state of great ecstasy and divinely possessed by his science.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Poe, Edgar Allen
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.
Poincaré, Jules Henri (18541912)
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].
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. 1, Winter 1991.
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 see clearly and to comprehend at once both the ensemble and the details.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.
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.
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.
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 strangers to one another.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
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.
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.
The mind uses its faculty for creativity only when experience forces it to do so.
Mathematical discoveries, small or great, are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
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 suppose the earth turns round" have the same meaning; there is nothing more in the one than in the other.
La Science et l'hypothèse.
...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 and Method.
Poisson, Siméon (17811840)
Life is good for only two things, discovering mathematics and teaching mathematics.
Mathematics Magazine, v. 64, no. 1, Feb. 1991.
Polyá, George (1887, 1985)
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.
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 it should be d. Some of his sayings are handed down from generation to generation.
"In order to solve this differential equation you look at it till a solution occurs to you."
"This principle is so perfectly general that no particular application of it is possible."
"Geometry is the science of correct reasoning on incorrect figures."
"My method to overcome a difficulty is to go round it."
"What is the difference between method and device? A method is a device which you used twice."
How to Solve It. Princeton: Princeton University Press. 1945.
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: Birkhäuser, 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.
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.
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 should understand and impress on his students the view that no problem whatever is completely exhausted.
One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
How to Solve It. Princeton: Princeton University Press. 1945.
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 very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
How to Solve It. Princeton: Princeton University Press. 1945.
Pope, Alexander (16881744)
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 the status quo]
[Aaron Hill's version:
O'er Nature's laws God cast the veil of night,
Out blaz'd a Newton's souland all was light.
Order is Heaven's first law.
An Essay on Man IV.
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 for aid on Sense!
See Mystery to Mathematics fly!
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Pordage, Matthew
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.
Proclus Diadochus (412  485)
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 aspect of living things, was taken to the place where he began and there is for ever beaten by the waves.
Scholium to Book X of Euclid V.
Purcell, E. and Varberg, D.
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: Prentice Hall, 1987.
Pushkin, Aleksandr Sergeyevich (1799  1837)
Inspiration is needed in geometry, just as much as in poetry.
Likhtenshtein
Quine, Willard Van Orman
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 scheme of physical objects is [likewise] a convenient myth, simpler than the literal truth and yet containing that literal truth as a scattered part.
In J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990.
Raleigh, [Sir} Walter Alexander (18611922)
In an examination those who do not wish to know ask questions of those who cannot tell.
Some Thoughts on Examinations.
Recorde, Robert (1557)
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 Calculus Gems, New York: McGraw Hill Inc., 1992.
Reid, Thomas
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 unchallenged leadership in the field of mathematics.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Renan, Ernest
The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.
Souvenirs d'enfance et de jeunesse.
Rényi, Alfréd
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.
Richardson, Lewis Fry (1881  1953)
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 have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless.
Mathematics of War and Foreign Politics.
Riskin, Adrian
(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.
Rivest, R., Shamir, A., and Adleman, L.
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 RSA129 is decoded. See the article whose title is the above sentence by Barry Cipra, SIAM News July 1994, 1, 1213.]
Rohault, Jacques (17th century)
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.
Rosenblueth, A
[with Norbert Wiener]
The best material model of a cat is another, or preferably the same, cat.
Philosophy of Science 1945.
Rosenlicht, Max (1949)
You know we all became mathematicians for the same reason: we were lazy.
Hugo Rossi
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 a Toolbox, Notices of the AMS, v. 43, no. 10, October 1996.
Rota, Giancarlo
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.
Russell, Bertrand (18721970)
How dare we speak of the laws of chance? Is not chance the antithesis of all law?
Calcul des probabilités.
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.
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.
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 thirtyeight, mathematics was my chief interest and my chief source of happiness.
The Autobiography of Bertrand Russell .
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.
If I were a medical man, I should prescribe a holiday to any patient who considered his work important.
The Autobiography of Bertrand Russell .
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 Scientific Outlook, 1931.
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 not much, I have achieved.
The Autobiography of Bertrand Russell .
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 Inc., 1988.
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.
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 of Mathematics. 1903.
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 York: McGraw Hill Inc., 1992.
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.
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.
[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.
"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 is less than a meter. Two must be two of something, and the proposition "2 and 2 are 4" is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. "Well, at any rate there are four animals," you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. "Well, then living organisms," you say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: "Two entities and two entities are four entities." When you have told me what you mean by "entity," we will resume the argument.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
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 accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
Portraits from Memory.
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.
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.
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 be said about "any" thing or "any" property. We are prepared to say that one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians or pure mathematicians, we have never heard of Socrates or Plato. A world in which there were no such individuals would still be a world in which one and one are two. It is not open to us, as pure mathematicians or logicians, to mention anything at all, because, if we do so we introduce something irrelevant and not formal.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
The desire to understand the world and the desire to reform it are the two great engines of progress.
Marriage and Morals.
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 and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.
Rutherford, Ernest (18711937)
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.
Sanford, T. H.
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.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Santayana, George
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.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Sarton, G.
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 generations have built up that magnificent monument, the object of our most legitimate pride as men, and of our wonder, humility and thankfulness, as individuals. The study of the history of mathematics will not make better mathematicians but gentler ones, it will enrich their minds, mellow their hearts, and bring out their finer qualities.
Sayers, Dorothy L.
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 emptiness, and then the mathematician sweeps the whole cosmos into unreality and leaves one with mind as the only thing of which we have any immediate apprehension. Cogito ergo sum, ergo omnia esse videntur. All this bother, and we are no further than Descartes. Have you noticed that the astronomers and mathematicians are much the most cheerful people of the lot? I suppose that perpetually contemplating things on so vast a scale makes them feel either that it doesn't matter a hoot anyway, or that anything so large and elaborate must have some sense in it somewhere.
With R. Eustace, The Documents in the Case, New York: Harper and Row, 1930, p 54.
Schopenhauer
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 dangerous, like languages, natural science, history, etc.
Seneca
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.
Shakespeare, William (1564  1616)
I cannot do it without comp[u]ters.
The Winter's Tale.
Though this be madness, yet there is method in't.
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.
I am ill at these numbers.
Hamlet.
Shaw, George Bernard (18561950)
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 repulsion in orderly crystalline structure. Such a picture is dangerously fascinating to thinkers oppressed by the bloody disorders of the living world. Craving for purer subjects of thought, they find in the contemplation of crystals and magnets a happiness more dramatic and less childish than the happiness found by mathematicians in abstract numbers, because they see in the crystals beauty and movement without the corrupting appetites of fleshly vitality.
Preface to Back to Methuselah.
Shaw, J. B.
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.
Simmons, G. F.
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.
Slaught, H.E.
...[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 room observed it). He stopped and reexamined the doubtful step for several minutes and then, convinced of the error, he abruptly dismissed the meeting  to the astonishment of most of the audience. It was an evidence of intellectual courage as well as honesty and doubtless won for him the supreme admiration of every person in the group  an admiration which was in no wise diminished, but rather increased, when at a later meeting he announced that after all he had been able to prove the step to be correct.
In The American Mathematical Monthly, 40 (1933), 191195.
Smith, Adam
I have no faith in political arithmetic.
Smith, David Eugene
One merit of mathematics few will deny: it says more in fewer words than any other science. The formula, e^ip = 1 expressed a world of thought, of truth, of poetry, and of the religious spirit "God eternally geometrizes."
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Smith, Henry John Stephen (1826  1883)
[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.
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, Weber and Schmidt, 1972.
Somerville, Mary (17801872)
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 Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity.
Martha Somerville (ed.) Personal Recollections of Mary Somerville, Boston, 1874.
Spengler, Oswald (1880 1936)
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 epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) sideglance at contemporary mathematics.
The Decline of the West.
Steiner, G.
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 metaphysically trivial, irresponsible. They refuse to relate outward, to take reality for arbiter. This is the source of their witchery.
The American Mathematical Monthly, v. 101, no. 9, November, 1994.
Steinmetz, Charles P.
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 of Mathematics, New York: Simona and Schuster, 1937.
Sternberg, S.
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 "explain" the zoology of elementary particles in terms of irreducible representations of Lie groups.
Stewart, Ian
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 selfselection set in, whereby equations that could not be solved were automatically of less interest than those that could.
Does God Play Dice? The Mathematics of Chaos. Blackwell, Cambridge, MA, 1989, p. 39.
Sullivan, John William Navin (1886  1937)
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 acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of science.
Aspects of Science, 1925.
Mathematics, as much as music or any other art, is one of the means by which we rise to a complete selfconsciousness. 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 informs us of much that depends on our minds.
Aspects of Science, 1925.
Sun Tze (5th  6th century)
The control of large numbers is possible, and like unto that of small numbers, if we subdivide them.
Sun Tze Ping Fa.
Swift, Jonathan
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.
What vexes me most is, that my female friends, who could bear me very well a dozen years ago, have now forsaken me, although I am not so old in proportion to them as I formerly was: which I can prove by arithmetic, for then I was double their age, which now I am not.
Letter to Alexander Pope. 7 Feb. 1736.
Sylvester, J.J. (1814  1897)
...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 intellectual being....
Presidential Address to British Association, 1869.
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 stunting and atrophying his moral nature and drying up the surest sources of his future intellectual replenishment.
[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 have in explaining the laws of procreation.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
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 together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul.
Presidential Address to British Association, 1869.
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 concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.
Presidential Address to British Association, 1869.
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, and it is certain that the more anyone gives himself up to the study of oratorical effect the less will he find himself in a fit state to mathematicize.
Thales (CA 600 BC)
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.
Thompson, D'Arcy Wentworth (18601948)
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. Their problems of form are in the first instance mathematical problems, their problems of growth are essentially physical problems, and the morphologist is, ipso facto, a student of physical science.
On Growth and Form, 1917.
Thomson, [Lord Kelvin] William (18241907)
Fourier is a mathematical poem.
Thoreau
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 study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical.
Tietze
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: McGraw Hill Inc., 1992.
Tillotson, Archbishop
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 this great volume of the world.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Titchmarsh, E. C.
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, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
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.
Todhunter, Isaac (1820  1910)
[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.
Tolstoy, [Count] Lev Nikolgevich (18281920)
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, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable human wills, is continuous. To understand the laws of this continuous movement is the aim of history . Only by taking an infinitesimally small unit for observation (the differential of history, that is, the individual tendencies of man) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.
War and Peace.
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, 1989.
Truesdell, Clifford
This paper gives wrong solutions to trivial problems. The basic error, however, is not new.
Mathematical Reviews 12, p561.
Turgenev, Ivan Sergeievich (1818  1883)
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'.
Turnbull, H.W.
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 pollman at Cambridge, who learned perfectly how to factorize a^2  b^2 but was floored because the examiner unkindly asked for the factors of p^2  q^2 .
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Ulam, Stanislaw
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 their unhappiness over the events of this world, some immerse themselves in a kind of selfsufficiency in mathematics. (Some have engaged in it for this reason alone.)
Adventures of a Mathematician, Scribner's, New York, 1976.
Valéry, Paul (1871  1945)
In the physical world, one cannot increase the size or quantity of anything without changing its quality. Similar figures exist only in pure geometry.
van Vleck, E. B.
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.
Veblen, Thorstein (18571929)
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.
Invention is the mother of necessity.
J. Gross, The Oxford Book of Aphorisms, Oxford: Oxford University Press, 1983.
Voltaire (16941778)
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.
He who has heard the same thing told by 12,000 eyewitnesses 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, 1956.
There are no sects in geometry.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1962.
Walton, Izaak
Angling may be said to be so like mathematics that it can never be fully learned.
The Compleat Angler, 1653.
Warner, Sylvia Townsend
For twenty pages perhaps, he read slowly, carefully, dutifully, with pauses for selfexamination 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 on him, and he read ravening on, sitting up till dawn to finish the book, as though it were a novel. After that his passion was stayed; the book went back to the Library and he was done with mathematics till the next bout. Not much remained with him after these orgies, but something remained: a sensation in the mind, a worshiping acknowledgment of something isolated and unassailable, or a remembered mental joy at the rightness of thoughts coming together to a conclusion, accurate thoughts, thoughts in just intonation, coming together like unaccompanied voices coming to a close.
Mr. Fortune's Maggot.
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 drawn, leading cases cited, types and antetypes analysed and anecdotes introduced. Except for Archimedes mathematics is singularly naked of anecdotes.
Mr. Fortune's Maggot.
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 stickor any straight object would do, such as an umbrellawhich I shall secure in an upright position between my feet. Knowing then that the ratio that the height of the tree bears to the length of the measuring stick must equal the ratio that the distance from my eye to the base of the tree bears to my height, and knowing (or being able to find out) my height, the length of the measuring stick and the distance from my eye to the base of the tree, I can, therefore, calculate the height of the tree."
"What is an umbrella?"
Mr. Fortune's Maggot.
Warren, Robert Penn (1905)
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.
Weil, Andre (1906  1998)
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 to repeat it but unable to do it at will, unless perhaps by dogged work...
The Apprenticeship of a Mathematician.
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, Simone (1909  1943)
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.
Weyl, Hermann (1885  1955)
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 years from now, shall unearth some of our income tax returns together with relics of engineering works and mathematical books, will probably date them a couple of centuries earlier, certainly before Galileo and Vieta.
The Mathematical Way of Thinking, an address given at the Bicentennial Conference at the University of Pennsylvania, 1940.
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 and of the road; we want to understand the idea of the proof, the deeper context.
Unterrichtsblätter für Mathematik und Naturwissenschaften, 38, 177188 (1932). Translation by Abe Shenitzer appeared in The American Mathematical Monthly, v. 102, no. 7 (AugustSeptember 1995), p. 646.
 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.
 Unterrichtsblätter für Mathematik und Naturwissenschaften, 38, 177188 (1932). Translation by Abe Shenitzer appeared in The American Mathematical Monthly, v. 102, no. 7 (AugustSeptember 1995), p. 646.
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 the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity.
1951.
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.
... 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 Science Reader, New York: Simon and Schuster, 1959.
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 Mathematical Monthly, v. 100. p. 93.
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.
The American Mathematical Monthly, November, 1992.
Whewell
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 what manner of man he was who could use as a weapon what we can scarcely lift as a burden.
In E. N. Da C. Andrade "Isaac Newton" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Whitehead, Alfred North
The science of pure mathematics ... may claim to be the most original creation of the human spirit.
Science and the Modern World.
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 efforts of our mental weapons to grasp it.
An Introduction to Mathematics
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.
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.
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.
Life is an offensive, directed against the repetitious mechanism of the Universe.
Adventures of Ideas, 1933.
There is no nature at an instant.
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.
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.
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 extending the number of important operations which we can perform without thinking about them.
An Introduction to Mathematics.
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, Raleigh NC:Rome Press Inc., 1988.
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 Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
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 paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghostlike character of the real variable.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
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: Birkhäuser, 1981.
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.
Seek simplicity, and distrust it.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
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.
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.
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.
"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.
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 proposition is more apt to be interesting than a false one.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
War can protect; it cannot create.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
The progress of Science consists in observing interconnections and in showing with a patient ingenuity that the events of this evershifting world are but examples of a few general relations, called laws. To see what is general in what is particular, and what is permanent in what is transitory, is the aim of scientific thought.
An Introduction to Mathematics.
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 poetry and music, and quantity and number will face you in your rhythms and your octaves.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
"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, New York: Viking Press, 1966.
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 certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming ... and a little mad.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
In the study of ideas, it is necessary to remember that insistence on hardheaded 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 mode in which human intelligence functions. Our reasonings grasp at straws for premises and float on gossamers for deductions.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.
Whitman, Walt (18191892)
Do I contradict myself? Very well then I contradict myself. (I am large, I contains multitudes).
Song of Myself, 1939.
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 applause in the lecture room,
How soon unaccountable I became tired and sick,
Till rising and gliding out I wander'd off by myself,
In the mystical moist nightair, and from time to time,
Look'd up in perfect silence at the stars.
Wiener, Norbert (1894  1964)
A professor is one who can speak on any subject  for precisely fifty minutes.
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 find the reconciliation between the two views.
Progress imposes not only new possibilities for the future but new restrictions.
The Human Use of Human Beings.
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 only one's best moments that count and not one's worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician's reputation.
ExProdigy: My Childhood and Youth.
Wilder, R. L.
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.
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 presentday mathematics is hardly possible.
In The American Mathematical Monthly, March 1994.
William of Occam (13001439)
[Occam's Razor:]
Entities should not be multiplied unnecessarily.
Quodlibeta.
Wilson, John (1741  1793)
A monument to Newton! a monument to Shakespeare! Look up to Heaven look into the Human Heart. Till the planets and the passions the affections and the fixed stars are extinguished their names cannot die.
Wittgenstein, Ludwig (18891951)
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.
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 mathematics to others which equally do not belong to mathematics.
Tractatus Logico Philosophicus, New York, 1922, p. 169.
There can never be surprises in logic.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.
Wordsworth, William (1770  1850)
[Mathematics] is an independent world
Created out of pure intelligence.
Wren, Sir Christopoher
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 geometry.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
X, Malcom
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.
Young, J. W. A.
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 properly ... presented, the mental emotion should be that of enjoyment of beauty, not that of repulsion from the ugly and the unpleasant.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Zeeman, E. Christopher (1925  )
Technical skill is mastery of complexity while creativity is mastery of simplicity.
Catastrophe Theory, 1977.
