Abel, Niels H. (1802 - 1829)
If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words,the most important parts of mathematics stand without a foundation.
In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, p. 188.

Abel, Niels H. (1802 - 1829)
[A reply to a question about how he got his expertise:]
By studying the masters and not their pupils.

Abel, Niels H. (1802 - 1829)
[About Gauss' mathematical writing style]
He is like the fox, who effaces his tracks in the sand with his tail.
In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, p. 177.

Adams, Douglas (1952 - 2001)
Bistromathics itself is simply a revolutionary new way of understanding the behavior of numbers. Just as Einstein observed that space was not an absolute but depended on the observer's movement in space, and that time was not an absolute, but depended on the observer's movement in time, so it is now realized that numbers are not absolute, but depend on the observer's movement in restaurants.
Life, the Universe and Everything. New York: Harmony Books, 1982.

Adams, Douglas (1952 - 2001)
The first nonabsolute number is the number of people for whom the table is reserved. This will vary during the course of the first three telephone calls to the restaurant, and then bear no apparent relation to the number of people who actually turn up, or to the number of people who subsequently join them after the show/match/party/gig, or to the number of people who leave when they see who else has turned up.
The second nonabsolute number is the given time of arrival, which is now known to be one of the most bizarre of mathematical concepts, a recipriversexcluson, a number whose existence can only be defined as being anything other than itself. In other words, the given time of arrival is the one moment of time at which it is impossible that any member of the party will arrive. Recipriversexclusons now play a vital part in many branches of math, including statistics and accountancy and also form the basic equations used to engineer the Somebody Else's Problem field.
The third and most mysterious piece of nonabsoluteness of all lies in the relationship between the number of items on the bill, the cost of each item, the number of people at the table and what they are each prepared to pay for. (The number of people who have actually brought any money is only a subphenomenon of this field.)
Life, the Universe and Everything. New York: Harmony Books, 1982.

Adams, Douglas (1952 - 2001)
Numbers written on restaurant bills within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe.
This single statement took the scientific world by storm. It completely revolutionized it. So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years.
Life, the Universe and Everything. New York: Harmony Books, 1982.

Adams, John (1735 - 1826)
I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain.
Letter to Abigail Adams, May 12, 1780.

Adler, Alfred
Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing.
"Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.

Adler, Alfred
The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then, little will ever be accomplished.
"Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.

Adler, Alfred
In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is.
Reflections: mathematics and creativity, New Yorker, 47(1972), no. 53, 39 - 45.

Aiken, Conrad
[At a musical concert:]
...the music's pure algebra of enchantment.

Allen, Woody
Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
In Howard Eves' Return to Mathematical Circles, Boston: Prindle, Weber, and Schmidt, 1988.

Anglin, W.S.
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
"Mathematics and History", Mathematical Intelligencer, v. 4, no. 4.

Anonymous
If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.
In Ivor Thomas "Greek Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Anonymous
Defendit numerus: There is safety in numbers.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1452.

Anonymous
Like the crest of a peacock so is mathematics at the head of all knowledge.
[An old Indian saying. Also, "Like the Crest of a Peacock" is the title of a book by G.G. Joseph]

Anonymous
Referee's report: This paper contains much that is new and much that is true. Unfortunately, that which is true is not new and that which is new is not true.
In H.Eves Return to Mathematical Circles, Boston: Prindle, Weber, and Schmidt, 1988.

Arbuthnot, John

The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning; and when they cannot it's a sign our knowledge of them is very small and confus'd; and when a Mathematical Reasoning can be had it's as great a folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you.
Of the Laws of Chance. (1692)

Aristophanes (ca 444 - 380 BC)
Meton: With the straight ruler I set to work
To make the circle four-cornered
[First(?) allusion to the problem of squaring the circle]

Aristotle (ca 330 BC)
Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends [mathematics] has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research.
Metaphysica, 1-981b

Aristotle (ca 330 BC)
The whole is more than the sum of its parts.
Metaphysica 10f-1045a

Aristotle
The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.
Metaphysica 1-5

Aristotle
It is not once nor twice but times without number that the same ideas make their appearance in the world.
"On The Heavens", in T. L. Heath Manual of Greek Mathematics, Oxford: Oxford University Press, 1931.

Aristotle
To Thales the primary question was not what do we know, but how do we know it.
Mathematical Intelligencer v. 6, no. 3, 1984.

Aristotle
The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.
Metaphysica, 3-1078b.

Ascham, Roger (1515-1568)
Mark all mathematical heads which be wholly and only bent on these sciences, how solitary they be themselves, how unfit to live with others, how unapt to serve the world.
In E G R Taylor, Mathematical Practitioners of Tudor and Stuart England, Cambridge: Cambridge University Press, 1954.

Aubrey, John (1626-1697)
[About Thomas Hobbes:]
He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that trueth. This made him in love with geometry.
In O. L. Dick (ed.) Brief Lives, Oxford: Oxford University Press, 1960, p. 604.

Auden, W. H. (1907-1973)
How happy the lot of the mathematician. He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve.
The Dyer's Hand, London: Faber & Faber, 1948.

Auden, W. H. (1907-1973)
Thou shalt not answer questionnaires
Or quizzes upon world affairs,
Nor with compliance
Take any test. Thou shalt not sit
with statisticians nor commit
A social science.
"Under which lyre" in Collected Poems of W H Auden, London: Faber and Faber.

Augarten, Stan
Computers are composed of nothing more than logic gates stretched out to the horizon in a vast numerical irrigation system.
State of the Art: A Photographic History of the Integrated Circuit. New York: Ticknor and Fields.

St. Augustine (354-430)
Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist.
The City of God.

St. Augustine (354-430)
The good Christian should beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
DeGenesi ad Litteram, Book II, xviii, 37 [Note: mathematician = astrologer]

St. Augustine (354-430)
If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?
De Magistro ch X, 23.

Babbage, Charles (1792-1871)
Errors using inadequate data are much less than those using no data at all.

Babbage, Charles (1792-1871)
On two occasions I have been asked [by members of Parliament], 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Babbage, Charles (1792-1871)
I wish to God these calculations had been executed by steam.
In H. Eves In Mathematical Circles,, Boston: Prindle, Weber and Schmidt, 1969.

Bacon, Sir Francis (1561-1626)
And as for Mixed Mathematics, I may only make this prediction, that there cannot fail to be more kinds of them, as nature grows further disclosed.
Advancement of Learning book 2; De Augmentis book 3.

Bacon, Roger
For the things of this world cannot be made known without a knowledge of mathematics.
Opus Majus part 4 Distinctia Prima cap 1, 1267.

Bacon, Roger
In the mathematics I can report no deficience, except that it be that men do not sufficiently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures; so in the mathematics, that use which is collateral and intervenient is no less worthy than that which is principal and intended.
John Fauvel and Jeremy Gray (eds.) A History of Mathematics: A Reader, Sheridan House, 1987.

Baker, H. F.
[On the concept of group:]
... what a wealth, what a grandeur of thought may spring from what slightbeginnings.
Florian Cajori, A History of Mathematics, New York, 1919, p 283.

Bagehot, Walter
Life is a school of probability.
Quoted in J. R. Newman (ed.) The World of Mathematics, Simon and Schuster, New York,1956, p. 1360.

Balzac, Honore de (1799 - 1850)
Numbers are intellectual witnesses that belong only to mankind.

Banville, John
Throughout the 1960s and 1970s devoted Beckett readers greeted each successively shorter volume from the master with a mixture of awe and apprehensiveness; it was like watching a great mathematician wielding an infinitesimal calculus, his equations approaching nearer and still nearer to the null point.
Quoted in a review of Samuel Beckett's Nohow On: I11 Seen I11 Said, Worstward Ho, in The New York Review of Books, August 13, 1992.

Bell, Eric Temple (1883-1960)
Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.
In H. Eves Return to Mathematical Circles., Boston: Prindle, Weber and Schmidt, 1988.

Bell, Eric Temple (1883-1960)
Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.
Mathematics, Queen and Servant of Science, New York, 1951, p 164.

Bell, Eric Temple (1883-1960)
It is the perennial youthfulness of mathematics itself which marks it off with a disconcerting immortality from the other sciences.

Bell, Eric Temple (1883-1960)
The Handmaiden of the Sciences.
[Book by that title.]

Bell, Eric Temple (1883-1960)
Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.

Bell, Eric Temple (1883-1960)
Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.

Bell, Eric Temple (1883-1960)
"Obvious" is the most dangerous word in mathematics.

Bell, Eric Temple (1883-1960)
The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

Bell, Eric Temple (1883-1960)
If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 308.

Bell, Eric Temple (1883-1960)
The longer mathematics lives the more abstract -- and therefore, possibly also the more practical -- it becomes.
In The Mathematical Intelligencer, vol. 13, no. 1, Winter 1991.

Bell, Eric Temple (1883-1960)
The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry.
In R Crayshaw-Williams The Search For Truth, p. 191.

Bell, Eric Temple (1883-1960)
If "Number rules the universe" as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.

Bell, Eric Temple (1883-1960)
I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

Belloc, Hillaire (1870-1953)
Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death.
The Silence of the Sea

Bentham, Jeremy (1748-1832)
O Logic: born gatekeeper to the Temple of Science, victim of capricious destiny: doomed hitherto to be the drudge of pedants: come to the aid of thy master, Legislation.
In J. Browning (ed.) Works.

Bernoulli, Daniel
...it would be better for the true physics if there were no mathematicians on earth.
In The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.

Bernoulli, Jacques (Jakob?) (1654-1705)
I recognize the lion by his paw.
[After reading an anonymous solution to a problem that he realized was Newton's solution.]
In G. Simmons, Calculus Gems, New York: McGraw Hill, 1992, p. 136.

Bernoulli, Johann
But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not.

Besicovitch, A.S.
A mathematician's reputation rests on the number of bad proofs he has given.
In J. E. Littlewood A Mathematician's Miscellany, Methuen & Co. Ltd., 1953.

Blake
God forbid that Truth should be confined to Mathematical Demonstration!
Notes on Reynold's Discourses, c. 1808.

Blake
What is now proved was once only imagin'd.
The Marriage of Heaven and Hell, 1790-3.

Bohr, Niels Henrik David (1885-1962)
An expert is a man who has made all the mistakes, which can be made, in a very narrow field.

The Bible
I returned and saw under the sun that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all.
Ecclesiastes.

Bolyai, János (1802 - 1860)
Out of nothing I have created a strange new universe.
[A reference to the creation of a non-euclidean geometry.]

Bolyai, Wolfgang (1775-1856)
[To son 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 non-Euclidian geometry.]
In P. Davis and R. Hersh The Mathematical Experience , Boston: Houghton Mifflin Co., 1981, p. 220.

Bourbaki
Structures are the weapons of the mathematician.

Bridgman, P. W.
It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention.
The Logic of Modern Physics, New York, 1972.

Brown, George Spencer (1923 - )
To arrive at the simplest truth, as Newton knew and practiced, requires years of contemplation. Not activity Not reasoning. Not calculating. Not busy behaviour of any kind. Not reading. Not talking. Not making an effort. Not thinking. Simply bearing in mind what it is one needs to know. And yet those with the courage to tread this path to real discovery are not only offered practically no guidance on how to do so, they are actively discouraged and have to set abut it in secret, pretending meanwhile to be diligently engaged in the frantic diversions and to conform with the deadening personal opinions which are continually being thrust upon them.
The Laws of Form. 1969.

Browne, Sir Thomas (1605-1682)
God is like a skilful Geometrician.
Religio Medici I, 16.

Browne, Sir Thomas (1605-1682)
All things began in Order, so shall they end, and so shall they begin again, according to the Ordainer of Order, and the mystical mathematicks of the City of Heaven.
Hydriotaphia, Urn-burial and the Garden of Cyrus, 1896.

Browne, Sir Thomas (1605-1682)
...indeed what reason may not go to Schoole to the wisdome of Bees, Aunts, and Spiders? what wise hand teacheth them to doe what reason cannot teach us? ruder heads stand amazed at those prodigious pieces of nature, Whales, Elephants, Dromidaries and Camels; these I confesse, are the Colossus and Majestick pieces of her hand; but in these narrow Engines there is more curious Mathematicks, and the civilitie of these little Citizens more neatly sets forth the wisedome of their Maker.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1001.

Buck, Pearl S. (1892 - 1973)
No one really understood music unless he was a scientist, her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. "And relationships," he had told her, "contained the essential meaning of life."
The Goddess Abides, Pt. I, 1972.

Burke, Edmund
The age of chivalry is gone. That of sophisters, economists and calculators has succeeded.
Reflections on the Revolution in France.

Butler, Bishop
To us probability is the very guide of life.
Preface to Analogy.

Butler, Samuel (1612 - 1680)
... There can be no doubt about faith and not reason being the ultima ratio. Even Euclid, who has laid himself as little open to the charge of credulity as any writer who ever lived, cannot get beyond this. He has no demonstrable first premise. He requires postulates and axioms which transcend demonstration, and without which he can do nothing. His superstructure indeed is demonstration, but his ground his faith. Nor again can he get further than telling a man he is a fool if he persists in differing from him. He says "which is absurd," and declines to discuss the matter further. Faith and authority, therefore, prove to be as necessary for him as for anyone else.
The Way of All Flesh.

Byron
When Newton saw an apple fall, he found ...
A mode of proving that the earth turnd round
In a most natural whirl, called gravitation;
And thus is the sole mortal who could grapple
Since Adam, with a fall or with an apple.

Caballero, James

Cardano, Girolamo (1501 - 1576)
To throw in a fair game at Hazards only three-spots, when something great is at stake, or some business is the hazard, is a natural occurrence and deserves to be so deemed; and even when they come up the same way for a second time if the throw be repeated. If the third and fourth plays are the same, surely there is occasion for suspicion on the part of a prudent man.
De Vita Propria Liber.

Carlyle, Thomas (1795 - 1881)
It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity of the universe.
Sartor Resartus III.

Carlyle, Thomas (1795-1881)
Teaching school is but another word for sure and not very slow destruction.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Carlyle, Thomas (1795-1881)
A witty statesman said, you might prove anything by figures.
Chartism.

Carroll, Lewis
What I tell you three times is true.
The Hunting of the Snark.

Carroll, Lewis
The different branches of Arithmetic -- Ambition, Distraction, Uglification, and Derision.
Alice in Wonderland.

Carroll, Lewis
"Can you do addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?" "I don't know," said Alice. "I lost count."
Through the Looking Glass.

Carroll, Lewis
"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
Alice in Wonderland.

Carroll, Lewis
"Then you should say what you mean," the March Hare went on.
"I do, " Alice hastily replied; "at least I mean what I say, that's the same thing, you know."
"Not the same thing a bit!" said the Hatter. "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.

Carroll, Lewis
"It's very good jam," said the Queen.
"Well, I don't want any to-day, at any rate."
"You couldn't have it if you did want it," the Queen said. "The rule is jam tomorrow and jam yesterday but never jam to-day."
"It must come sometimes to "jam to-day,""Alice objected.
"No it can't," said the Queen. "It's jam every other day; to-day isn't any other day, you know."
"I don't understand you," said Alice. "It's dreadfully confusing."
Through the Looking Glass.

Carroll, Lewis
"When I use a word," Humpty Dumpty said, in a rather scornful tone, "it means just what I choose it to mean - neither more nor less."
"The question is," said Alice, "whether you can make words mean so many different things."
"The question is," said Humpty Dumpty, "which is to be master - that's all."
Through the Looking Glass.

Céline, Louis-Ferdinand (1894 - 1961)
Entre le pénis et les mathématiques... il n'existe rien. Rien! C'est le vide.
Voyage au bout de la nuit. Paris: Gallimard.

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.

Cauchy, Augustin-Louis (1789 - 1857)
Men pass away, but their deeds abide.
[His last words (?)]
In H. Eves Mathematical Circles Revisted, Boston: Prindle, Weber and Schmidt, 1971.

Cayley, Arthur
As for everything else, so for a mathematical theory: beauty can be perceived but not explained.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Cayley, Arthur
Projective geometry is all geometry.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Cézanne, Paul (1839 - 1906)
...treat Nature by the sphere, the cylinder and the cone...

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 chess-players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination.
Orthodoxy ch. 2.

Chesterton, G. K. (1874 - 1936)
You can only find truth with logic if you have already found truth without it.
The Man who was Orthodox. 1963.

Chesterton, G. K. (1874 - 1936)
It isn't that they can't see the solution. It is that they can't see the problem.
The Point of a Pin in The Scandal of Father Brown.

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.

Christie, Agatha
I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours I found it quite enthralling.
An Autobiography.

Churchill, [Sir] Winston Spencer (1874-1965)
It is a good thing from an uneducated man to read books of quotations.
Roving Commission in My Early Life. 1930.

Churchill, Sir Winston Spencer (1874-1965)
I had a feeling once about Mathematics - that I saw it all. Depth beyond depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing its 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 (1772-1834)
...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.

Comte, Auguste (1798-1857)
C'este donc par l'étude des mathématiques, et seulement par elle, que l'on peut se faire une idée juste et approfondie de ce que c'est qu'une science.
Quoted by T. H. Huxley in Fortnightly Review, Vol. II, N.S. 5.

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 (1473-1543)
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 (1717-1783)
Just go on..and faith will soon return.
[To a friend hesitant with respect to infinitesimals.]
In P. J. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.

D'Alembert, Jean Le Rond (1717-17830
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
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight.

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.

Darwin, Charles
Mathematics seems to endow one with something like a new sense.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

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.

Davis, Philip J.
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 (1806-1871)
[When asked about his age.] I was x years old in the year x^2.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

De Morgan, Augustus (1806-1871)
It is easier to square the circle than to get round a mathematician.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

De Morgan, Augustus (1806-1871)
Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.
Transactions Cambridge Philosophical Society, vol. X, 1864, p. 184.

Descartes, René (1596-1650)
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.

Descartes, René (1596-1650)
Each problem that I solved became a rule which served afterwards to solve other problems.
Discours de la Méthode. 1637.

Descartes, René (1596-1650)
If I found any new truths in the sciences, I can say that they follow from, or depend on, five or six principal problems which I succeeded in solving and which I regard as so many battles where the fortunes of war were on my side.
Discours de la Méthode. 1637.

Descartes, René (1596-1650)
I concluded that I might take as a general rule the principle that all things which we very clearly and obviously conceive are true: only observing, however, that there is some difficulty in rightly determining the objects which we distinctly conceive.
Discours de la Méthode. 1637.

Descartes, René (1596-1650)
I thought the following four [rules] would be enough, provided that I made a firm and constant resolution not to fail even once in the observance of them. The first was never to accept anything as true if I had not evident knowledge of its being so; 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.
Discours de la Méthode. 1637.

Descartes, René (1596-1650)
These long chains of perfectly simple and easy reasonings by means of which geometers are accustomed to carry out their most difficult demonstrations had led me to fancy that everything that can fall under human knowledge forms a similar sequence; and 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.

Descartes, René (1596-1650)
When writing about transcendental issues, be transcendentally clear.
In G. Simmons Calculus Gems. New York: McGraw Hill Inc., 1992.

Descartes, René (1596-1650)
If we possessed a thorough knowledge of all the parts of the seed of any animal (e.g. man), we could from that alone, be reasons entirely mathematical and certain, deduce the whole conformation and figure of each of its members, and, conversely if we knew several peculiarities of this conformation, we would from those deduce the nature of its seed.

Descartes, René (1596-1650)
Cogito Ergo Sum. "I think, therefore I am."
Discours de la Méthode. 1637.

Descartes, René (1596-1650)
I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.
La Geometrie.

Descartes, René (1596-1650)
Perfect numbers like perfect men are very rare.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

Descartes, René (1596-1650)
omnia apud me mathematica fiunt.
With me everything turns into mathematics.

Descartes, René (1596-1650)
It is not enough to have a good mind. The main thing is to use it well.
Discours de la Méthode. 1637.

Descartes, René (1596-1650)
If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things.
Discours de la Méthode. 1637.

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! late-begotten 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 Schroedinger 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.

Dirac, Paul Adrien Maurice (1902- )
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
In P. J. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.

Dirac, Paul Adrien Maurice (1902- )
In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 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 (1859-1930)
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.

Doyle, Sir Arthur Conan (1859-1930)
When you have eliminated the impossible, what ever remains, however improbable must be the truth.
The Sign of Four.

Doyle, Sir Arthur Conan (1859-1930)
From a drop of water a logician could predict an Atlantic or a Niagara.
A study in Scarlet 1929.

Doyle, Sir Arthur Conan (1859-1930)
It is a capital mistake to theorize before one has data.
Scandal in Bohemia.

Dryden, John (1631-1700)
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 (1471-1528)
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.

Dürer, Albrecht (1471-1528)
Whoever ... proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured.
J Heidrich (ed.) Albrecht Dürer's schriftlicher Nachlass Berlin, 1920.

Dürer, Albrecht (1471-1528)
And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art...
Course in the Art of Measurement

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?)

Dyson, Freeman
For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created.
Mathematics in the Physical Sciences.

Dyson, Freeman
The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then 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 (1882-1944)
Proof is the idol before whom the pure mathematician tortures himself.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Eddington, Sir Arthur (1882-1944)
We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about `and'.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Eddington, Sir Arthur (1882-1944)
We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own.
Space, Time and Gravitation. 1920.

Eddington, Sir Arthur (1882-1944)
It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Eddington, Sir Arthur (1882-1944)
I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe and the same number of electrons.
The Philosophy of Physical Science. Cambridge, 1939.

Eddington, Sir Arthur (1882-1944)
To the pure geometer the radius of curvature is an incidental characteristic - like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely 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..

Eddington, Sir Arthur (1882-1944)
Human life is proverbially uncertain; few things are more certain than the solvency of a life-insurance company.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

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. 106-107.

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 (1879-1955)
[During a lecture:]This has been done elegantly by Minkowski; but chalk is cheaper than grey matter, and we will do it as it comes.
[Attributed by Pólya.]
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co. Ltd., 1953.

Einstein, Albert (1879-1955)
Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.

Einstein, Albert (1879-1955)
I don't believe in mathematics.
Quoted by Carl Seelig. Albert Einstein.

Einstein, Albert (1879-1955)
Imagination is more important than knowledge.
On Science.

Einstein, Albert (1879-1955)
The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.
What I Believe.

Einstein, Albert (1879-1955)
The bitter and the sweet come from the outside, the hard from within, from one's own efforts.
Out of My Later Years.

Einstein, Albert (1879-1955)
Gott würfelt nicht.

Einstein, Albert (1879-1955)
Common sense is the collection of prejudices acquired by age eighteen.
In E. T. Bell Mathematics, Queen and Servant of the Sciences. 1952.

Einstein, Albert (1879-1955)
God does not care about our mathematical difficulties. He integrates empirically.
L. Infeld Quest, 1942.

Einstein, Albert (1879-1955)
How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?

Einstein, Albert (1879-1955)
[About Newton]
Nature to him was an open book, whose letters he could read without effort.
In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.

Einstein, Albert (1879-1955)
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Einstein, Albert (1879-1955)
What is this frog and mouse battle among the mathematicians?
[i.e. Brouwer vs. Hilbert]
In H. Eves Mathematical Circles Squared Boston: Prindle, Weber and Schmidt, 1972.

Einstein, Albert (1879-1955)
Raffiniert ist der Herr Gott, aber boshaft ist er nicht. God is subtle, but he is not malicious.
Inscribed in Fine Hall, Princeton University.

Einstein, Albert (1879-1955)
Nature hides her secrets because of her essential loftiness, but not by means of ruse.

Einstein, Albert (1879-1955)
The human mind has first to construct forms, independently, before we can find them in things.

Einstein, Albert (1879-1955)
Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
In A. Sommerfelt "To Albert Einstein's Seventieth Birthday" in Paul A. Schilpp (ed.) Albert Einstein, Philosopher-Scientist, Evanston, 1949.

Einstein, Albert (1879-1955)
Do not worry about your difficulties in mathematics, I assure you that mine are greater.

Einstein, Albert (1879-1955)
The truth of a theory is in your mind, not in your eyes.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

Einstein, Albert (1879-1955)
These thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words afterward.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

Einstein, Albert (1879-1955)
A human being is a part of the whole, called by us "Universe," a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the resta kind of optical delusion of his consciousness. This 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.

Einstein, Albert (1879-1955)
The world needs heroes and it's better they be harmless men like me than villains like Hitler.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

Einstein, Albert (1879-1955)
It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiousity of inquiry.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

Einstein, Albert (1879-1955)
Everything that is really great and inspiring is created by the individual who can labor in freedom.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

Einstein, Albert (1879-1955)
The search for truth is more precious than its possession.
The American Mathematical Monthly v. 100 no. 3.

Einstein, Albert (1879-1955)
If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.
Address at the Sorbonne, Paris.

Einstein, Albert (1879-1955)
We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think nothing. Already the distance-concept is logically arbitrary; there need be no things that correspond to it, even approximately.
"Space-Time." Encyclopaedia Britannica, 14th ed.

Einstein, Albert (1879-1955)
Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.
The Evolution of Physics.

Einstein, Albert (1879-1955)
Science without religion is lame; religion without science is blind.
Reader's Digest, Nov. 1973.

Ellis, Havelock
The mathematician has reached the highest rung on the ladder of human thought.
The Dance of Life.

Ellis, Havelock
It is here [in mathematics] that the artist has the fullest scope of his imagination.
The Dance of Life.

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.

Erdös, Paul
Mathematics is not yet ready for such problems.
[Attributed by Paul Halmos.]
The American Mathematical Monthly, Nov. 1992

Erdös, Paul
A Mathematician is a machine for turning coffee into theorems.

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.

Euler, Leonhard (1707-1783)
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

Euler, Leonhard (1707-1783)
[upon losing the use of his right eye]
Now I will have less distraction.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Everett, Edward (1794-1865)
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.

Eves, Howard W.
An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Eves, Howard W.
Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later 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.

Eves, Howard W.
One is hard pressed to think of universal customs that man has successfully established on earth. There is one, however, of which he can boast the universal adoption of the Hindu-Arabic numerals to record numbers. In this we perhaps have man's unique 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.

Focus Newsletter (MAA)
Sample recommendation letter:
Dear Search Committee Chair,
I am writing this letter for Mr. John Smith who has applied for a position in your department. I should start by saying that I cannot recommend him too highly.
In fact, there is no other student with whom I can adequately compare him, and I am sure that the amount of mathematics he knows will surprise you.
His dissertation is the sort of work you don't expect to see these days. It definitely demonstrates his complete capabilities.
In closing, let me say that you will be fortunate if you can get him to work for you.
Sincerely,
A. D. Visor (Prof.)

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.

de Fermat, Pierre (1601?-1665)
And perhaps, posterity will thank me for having shown it that the ancients did not know everything.
In D. M. Burton, Elementary Number Theory, Boston: Allyn and Bacon, Inc., 1976.

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.

Fisher, Irving
The effort of the economist is to "see," to picture the interplay of economic elements. The more clearly cut these elements appear in his vision, the better; the more elements he can grasp and hold in his mind at once, the better. The economic world is a misty region. The first explorers used unaided vision. Mathematics is the lantern by which what before was dimly visible now looms up in firm, bold outlines. The old phantasmagoria disappear. We see better. We also see further.
Transactions of Conn. Academy, 1892.

Fisher, Ronald Aylmer (1890 - 1962)
Natural selection is a mechanism for generating an exceedingly high degree of improbability.

Fisher, Ronald Aylmer (1890-1962)
To call in the statistician after the experiment is done may be no more than asking hm to perform a postmortem examination: he may be able to say what the experiment died of.
Indian Statistical Congress, Sankhya, ca 1938.

Flaubert, Gustave (1821-1880)
Poetry is as exact a science as geometry.

Flaubert, Gustave (1821-1880)
Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?

Fontenelle, Bernard Le Bovier (1657-1757)
Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another.
Quoted in V. H. Larney Abstract Algebra: A First Course, Boston: Prindle, Weber and Schmidt, 1975.

Fontenelle, Bernard Le Bovier (1657-1757)
A work of morality, politics, criticism will be more elegant, other things being equal, if it is shaped by the hand of geometry.
Preface sur l'Utilité des Mathématiques et de la Physique, 1729.

Fontenelle, Bernard Le Bovier (1657-1757)
Leibniz never married; he had considered it at the age of fifty; but the person he had in mind asked for time to reflect. This gave Leibniz time to reflect, too, and so he never married.
Eloge de le Leibniz.

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.

Frayn, Michael
For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.
Constructions. 1974.

Frederick the Great (1712-1786)
To your care and recommendation am I indebted for having replaced a half-blind mathematician with a mathematician with both eyes, which will especially please the anatomical members of my Academy.
[To D'Alembert about Lagrange. Euler had vacated the post.]
In D. M. Burton, Elementary Number Theory, Boston: Allyn and Bacon, Inc., 1976.

Frege, Gottlob (1848 - 1925)
A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
In Scientific American, May 1984, p 77.

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.

Galilei, Galileo (1564 - 1642)
Measure what is measurable, and make measurable what is not so.
Quoted in H. Weyl "Mathematics and the Laws of Nature" in I Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.

Galilei, Galileo (1564 - 1642)
And who can doubt that it will lead to the worst disorders when minds created free by God are compelled to submit slavishly to an outside will? When we are told to deny our senses and subject them to the whim of others? When people devoid of 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 (1822-1911)
Whenever you can, count.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Galton, Sir Francis (1822-1911)
[Statistics are] the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man.
Pearson, The Life and Labours of Francis Galton, 1914.

Galton, Sir Francis (1822-1911)
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error." The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
In 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.

Gardner, Martin
Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now 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. Focus-The Newsletter of the Mathematical Association of America v. 14, no. 6, December 1994.

Gauss, Karl Friedrich (1777-1855) 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.

Gauss, Karl Friedrich (1777-1855)
If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 326.

Gauss, Karl Friedrich (1777-1855)
There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 314.

Gauss, Karl Friedrich (1777-1855)
You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.

Gauss, Karl Friedrich (1777-1855)
God does arithmetic.

Gauss, Karl Friedrich (1777-1855)
We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.
Letter to Bessel, 1830.

Gauss, Karl Friedrich (1777-1855)
I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.
In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.

Gauss, Karl Friedrich (1777-1855)
I have had my results for a long time: but I do not yet know how I am to arrive at them.
In A. Arber The Mind and the Eye 1954.

Gauss, Karl Friedrich (1777-1855)
[His motto:]
Few, but ripe.

Gauss, Karl Friedrich (1777-1855)
[His second motto:]
Thou, nature, art my goddess; to thy laws my services are bound...
W. Shakespeare King Lear.

Gauss, Karl Friedrich (1777-1855)
[attributed to him by H.B Lübsen]
Theory attracts practice as the magnet attracts iron.
Foreword of H.B Lübsen's geometry textbook.

Gauss, Karl Friedrich (1777-1855)
It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied 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.

Gauss, Karl Friedrich (1777-1855)
Finally, two days ago, I succeeded - not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

Gauss, Karl Friedrich (1777-1855)
A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that 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.

Gauss, Karl Friedrich (1777-1855)
I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect...geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.
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.

Gibbs, Josiah Willard (1839-1903)
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 Major-General.
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.

Glanvill, Joseph
And for mathematical science, he that doubts their certainty hath need of a dose of hellebore.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 548.

Goedel, Kurt
I don't believe in natural science.
[Said to physicist John Bahcall.]
Ed Regis, Who Got Einstein's Office? Addison Wesley, 1987.

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.

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) 92-103.

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.

Hadmard, Jacques
Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

Haldane, John Burdon Sanderson (1892-1964)
In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind. The catch in this criterion lies in the world "simplest." It is really an aesthetic canon such as we find implicit in our criticisms of poetry or painting. The layman finds such a law as dx/dt = K(d^2x/dy^2) much less simple than "it oozes," of which it is the mathematical statement. The physicist reverses this judgment, and his statement is certainly the more fruitful of the two, so far as prediction is concerned. It is, however, a statement about something very unfamiliar to the plainman, namely, the rate of change of a rate of change.
Possible Worlds, 1927.

Haldane, John Burdon Sanderson (1892-1964)
A time will however come (as I believe) when physiology will invade and destroy mathematical physics, as the latter has destroyed geometry.
Daedalus, or Science and the Future, London: Kegan Paul, 1923.

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.

Halmos, Paul R.
... the student skit at Christmas contained a plaintive line: "Give us Master's exams that our faculty can pass, or give us a faculty that can pass our Master's exams."
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

Halmos, Paul R.
I remember one occasion when I tried to add a little seasoning to a review, but I wasn't allowed to. The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory. The domains of the underlying measures were not sets but elements of more general Boolean algebras, and their range consisted not of positive numbers but of certain abstract equivalence classes. My proposed first sentence was: "The author discusses valueless measures in pointless spaces."
I want to be a Mathematician, Washington: MAA Spectrum, 1985, p. 120.

Halmos, Paul R.
...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

Halmos, Paul R.
The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me -- both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

Halmos, Paul R.
Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

Halmos, Paul R.
To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

Hamilton, [Sir] William Rowan (1805-1865)
Who would not rather have the fame of Archimedes than that of his conqueror Marcellus?
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.

Hamilton, Sir William Rowan (1805-1865)
I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc..
In a letter from Tait to Cayley.

Hamilton, Sir William Rowan (1805-1865)
On earth there is nothing great but man; in man there is nothing great but mind.
Lectures on Metaphysics.

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.

Hamming, Richard W.
Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

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.

Hardy, Godfrey H. (1877 - 1947)
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
A Mathematician's Apology, London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
I am interested in mathematics only as a creative art.
A Mathematician's Apology, London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

Hardy, Godfrey H. (1877 - 1947)
In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.
A Mathematician's Apology, London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
A Mathematician's Apology, London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
Young Men should prove theorems, old men should write books.
Quoted by Freeman Dyson in Freeman Dyson: Mathematician, Physicist, and Writer. Interview with Donald J. Albers, The College Mathematics Journal, vol. 25, No. 1, January 1994.

Hardy, Godfrey H. (1877 - 1947)
A science is said to be useful of its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life.
A Mathematician's Apology, London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.
A Mathematician's Apology, London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
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.

Hardy, Godfrey H. (1877 - 1947)
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
A Mathematician's Apology, London, Cambridge University Press,1941.

Hardy, Godfrey H. (1877 - 1947)
The fact is that there are few more "popular" subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics 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 line, less directly as he got nearer, till it was doubtful if he would ever reach it at all.
Far from the Madding Crowd.

Harish-Chandra
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, "Harish-Chandra," 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 (1850-1925)
[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 (1901-1976)
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.

Hempel, Carl G.
The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results.
"Geometry and Empirical Science" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Hempel, Carl G.
...to characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any 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.

Hermite, Charles (1822-1901)
Abel has left mathematicians enough to keep them busy for 500 years.
In G. F. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

Hermite, Charles (1822-1901)
We are servants rather than masters in mathematics.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

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 (1877-1962)
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 (1862-1943)
Wir müssen wissen.
Wir werden wissen.
[Engraved on his tombstone in Göttingen.]

Hilbert, David (1862-1943)
Before beginning I should put in three years of intensive study, and I haven't that much time to squander on a probable failure.
[On why he didn't try to solve Fermat's last theorem]
Quoted in E.T. Bell Mathematics, Queen and Servant of Science, New York: McGraw Hill Inc., 1951.

Hilbert, David (1862-1943)
Galileo was no idiot. Only an idiot could believe that science requires martyrdom - that may be necessary in religion, but in time a scientific result will establish itself.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1971.

Hilbert, David (1862-1943)
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Hilbert, David (1862-1943)
Physics is much too hard for physicists.
C. Reid Hilbert, London: Allen and Unwin, 1970.

Hilbert, David (1862-1943)
How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.

Hilbert, David (1862-1943)
The art of doing mathematics consists in finding that special case which contains all the germs of generality.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Hilbert, David (1862-1943)
The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Hilbert, David (1862-1943)
I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations.
Report on Number Theory, 1897.

Hilbert, David (1862-1943)
One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt,1971.

Hilbert, David (1862-1943)
Mathematics knows no races or geographic boundaries; for mathematics,the cultural world is one country.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

Hilbert, David (1862-1943)
The infinite! No other question has ever moved so profoundly the spirit of man.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

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.

Hobbes, Thomas
To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad.
["This" is that the volume generated by revolving the region under 1/x from 1 to infinity has finite volume.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Hobbes, Thomas
Geometry, which is the only science that it hath pleased God hitherto to bestow on mankind.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Hobbes, Thomas
The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning.
In J. R. Newman (ed.) The World of Mathematics, 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.

Holmes, Oliver Wendell
Certitude is not the test of certainty. We have been cocksure of many things that are not so.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

Holmes, Oliver Wendell
I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension of the following arithmetical formula: 2 + 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.

Huxley, Aldous
If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood, and many other desirable things. As a man he was a failure; as a monster he was superb.
Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934.

Huxley, Aldous
...[he] was as much enchanted by the rudiments of algebra as he would have been if I had given him an engine worked by steam, with a methylated spirit lamp to heat the boiler; more enchanted, perhapsfor the engine would have got broken, and, remaining 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 (1825-1895)
This seems to be one of the many cases in which the admitted accuracy of mathematical processes is allowed to throw a wholly inadmissible appearance of authority over the results obtained by them. Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff of any degree of fineness; but, nevertheless, what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulae will not get a definite result out of loose data.
Quarterly Journal of the Geological Society, 25,1869.

Huxley, Thomas Henry (1825-1895)
The mathematician starts with a few propositions, the proof of which is so obvious that they are called selfevident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature authority and tradition furnish the data, and the mental operations are deductive.
"Scientific Education -Notes of an After-dinner Speech." Macmillan's Magazine Vol XX, 1869.

Huxley, Thomas Henry (1825-1895)
It is the first duty of a hypothesis to be intelligible.

Ibn Khaldun (1332-1406)
Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence.
The Muqaddimah. An Introduction to History.

Isidore of Seville (ca 600 ad)
Take from all things their number and all shall perish.

Jacobi, Carl
It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

Jacobi, Carl
God ever arithmetizes.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.

Jacobi, Carl
One should always generalize.
(Man muss immer generalisieren)
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.

Jacobi, Carl
The real end of science is the honor of the human mind.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Jacobi, Carl
It is often more convenient to possess the ashes of great men than to possess the men themselves during their lifetime.
[Commenting on the return of Descartes' remains to France]
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

Jacobi, Carl
Mathematics is the science of what is clear by itself.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

James, William (1842 - 1910)
The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.
Collected Essays.

Jeans, Sir James
The essential fact is that all the pictures which science now draws of nature, and which alone seem capable of according with observational facts, are mathematical pictures.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Jeans, Sir James
From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.
Mysterious Universe.

Jefferson, Thomas
...the science of calculation also is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not be in indulged in by one who is to have a profession to follow for his subsistence.
In J. Robert Oppenheimer "The Encouragement of Science" in I. Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.

Jevons, William Stanley
It is clear that Economics, if it is to be a science at all, must be a mathematical science.
Theory of Political Economy.

Johnson, Samuel (1709-1784)
Sir, I have found you an argument. I am not obliged to find you an understanding.
J. Boswell The Life of Samuel Johnson, 1784.

Jowett, Benjamin (1817 - 1893)
Logic is neither a science or an art, but a dodge.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Kant, Emmanual (1724 - 1804)
The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.
The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.

Kant, Emmanual (1724 - 1804)

All human knowledge thus begins with intuitions, proceeds thence to concepts, and ends with ideas.
Quoted in Hilbert's Foundations of Geometry.

Kaplan, Abraham
Mathematics is not yet capable of coping with the naivete of the mathematician himself.
Sociology Learns the Language of Mathematics.

Kaplansky, Irving
We [he and Halmos] share a philosophy about linear algebra: we think basis-free, we write basis-free , but when the chips are down we close the office door and compute with matrices like fury.
Paul Halmos: Celebrating 50 Years of Mathematics.

Karlin, Samuel (1923 - )
The purpose of models is not to fit the data but to sharpen the questions.
11th R A Fisher Memorial Lecture, Royal Society 20, April 1983.

Kasner, E. and Newman, J.
Mathematics is man's own handiwork, subject only to the limitations imposed by the laws of thought.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

Kasner, E. and Newman, J.
...we have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

Kasner, E. and Newman, J.
Mathematics is the science which uses easy words for hard ideas.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

Kasner, E. and Newman, J.
Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

Kasner, E. and Newman, J.
Perhaps the greatest paradox of all is that there are paradoxes in mathematics.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

Kasner, E. and Newman, J.
When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

Kasner, E. and Newman, J. R.
The testament of science is so continually in a flux that the heresy of yesterday is the gospel of today and the fundamentalism of tomorrow.
E. Kasner and J. R. Newman, Mathematics and the Imagination, Simon and Schuster, 1940.

Keller, Helen (1880 - 1968)
Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important... The knowledge doesn't make life any sweeter or happier, does it?
The Story of My Life. 1903.

Kelley, John
A topologist is one who doesn't know the difference between a doughnut and a coffee cup.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

Kepler, Johannes (1571-1630)
A mind is accustomed to mathematical deduction, when confronted with the faulty foundations of astrology, resists a long, long time, like an obstinate mule, until compelled by beating and curses to put its foot into that dirty puddle.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

Kepler, Johannes (1571-1630)
Where there is matter, there is geometry.
(Ubi materia, ibi geometria.)
J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990

Kepler, Johannes (1571-1630)
The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.

Kepler, Johannes (1571-1630)
Nature uses as little as possible of anything.

Keynes, John Maynard
It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability.
The Application of Probability to Conduct.

Kleinhenz, Robert J.
When asked what it was like to set about proving something, the mathematician likened proving a theorem to seeing the peak of a mountain and trying to climb to the top. One establishes a base camp and begins scaling the mountain's sheer face, encountering obstacles at every turn, often retracing one's steps and struggling every foot of the journey. Finally when the top is reached, one stands examining the peak, taking in the view of the surrounding countrysideand then noting the automobile road up the other side!

Kline, Morris
A proof tells us where to concentrate our doubts.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

Kline, Morris
Statistics: the mathematical theory of ignorance.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

Kline, Morris
Logic is the art of going wrong with confidence.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

Kline, Morris
Universities hire professors the way some men choose wives - they want the ones the others will admire.
Why the Professor Can't Teach. St. Martin's Press, 1977. p 92.

Koestler, Arthur (1905- )
In the index to the six hundred odd pages of Arnold Toynbee's A Study of History, abridged version, the names of Copernicus, Galileo, Descartes and Newton do not occur yet their cosmic quest destroyed the medieval vision of an immutable social order in a walled-in universe and transformed the European landscape, society, culture, habits and general outlook, as thoroughly as if a new species had arisen on this planet.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

Koestler, Arthur (1905- )
Nobody before the Pythagoreans had thought that mathematical relations held the secret of the universe. Twenty-five centuries later, Europe is still blessed and cursed with their heritage. To non-European civilizations, the idea that numbers are the key to both wisdom and power, seems never to have occurred.
The Sleepwalkers. 1959.

Kovalevsky, Sonja
Say what you know, do what you must, come what may.
[Motto on her paper "On the Problem of the Rotation of a Solid Body about a Fixed Point."]

Kraft, Prinz zu Hohlenlohe-Ingelfingen (1827 - 1892)
Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else.
Attributed by Karl Schellbach. In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

Kronecker, Leopold (1823 - 1891)
God made the integers, all else is the work of man.
Jahresberichte der Deutschen Mathematiker Vereinigung.

Kronecker, Leopold (1823-1891)
Number theorists are like lotus-eaters -- having once tasted of this food they can never give it up.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

La Touche, Mrs.
I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are permutations and aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then from the top down, the result is always different.
Mathematical Gazette, v. 12.

LaGrange, Joseph-Louis
The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure.
Preface to Mécanique Analytique.

LaGrange, Joseph-Louis
[said about the chemist Lavoisier:]
It took the mob only a moment to remove his head; a century will not suffice to reproduce it.
H. Eves An Introduction to the History of Mathematics, 5th Ed., Saunders.

LaGrange, Joseph-Louis
When we ask advice, we are usually looking for an accomplice.

Lakatos, Imre
That sometimes clear ... and sometimes vague stuff ... which is ... mathematics.
In P. Davis and R. Hersh The Mathematical Experience, Boston: 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 day-to-day living. I used to have great stretches of time to work. Now I have research thoughts while making peanut butter and jelly sandwiches. Sure it's impossible to write down ideas while reading "curious George" to a two-year-old. On the other hand, as my husband was leaving graduate school for his first job, his thesis advisor told him, "You may wonder how a professor gets any research done when one has to teach, advise students, serve on committees, referee papers, write letters of recommendation, interview prospective faculty. Well, I take long showers."
In Her Own Words: Six Mathematicians Comment on Their Lives and Careers. Notices of the AMS, V. 38, no. 7 (September 1991), p. 704.

Lang, Andrew (1844-1912)
He uses statistics as a drunken man uses lamp posts -- for support rather than illumination.
Treasury of Humorous Quotations.

Langer, Rudoph E.
[about Fourier] It was, no doubt, partially because of his very disregard for rigor that he was able to take conceptual steps which were inherently impossible to men of more critical genius.
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.

Lao Tze (604-531 B.C.)
A good calculator does not need artificial aids.
Tao Te Ching, ch 27.

de Laplace, Pierre-Simon (1749 - 1827)
What we know is not much. What we do not know is immense.
(Allegedly his last words.)
DeMorgan's Budget of Paradoxes.

de Laplace, Pierre-Simon (1749 - 1827)
[His last words, according to De Morgan:]
Man follows only phantoms.
DeMorgan's Budget of Paradoxes.

de Laplace, Pierre-Simon (1749 - 1827)
Nature laughs at the difficulties of integration.
In J. W. Krutch "The Colloid and the Crystal", in I. Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.

de Laplace, Pierre-Simon (1749 - 1827)
Read Euler: he is our master in everything.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

de Laplace, Pierre-Simon (1749 - 1827)
Such is the advantage of a well constructed language that its simplified notation often becomes the source of profound theories.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

de Laplace, Pierre-Simon (1749 - 1827)
Napoleon: You have written this huge book on the system of the world without once mentioning the author of the universe.
Laplace: Sire, I had no need of that hypothesis.
Later when told by Napoleon about the incident, Lagrange commented: Ah, but that is a fine hypothesis. It explains so many things.
DeMorgan's Budget of Paradoxes.

de Laplace, Pierre-Simon (1749 - 1827)
[said about Napier's logarithms:]
...by shortening the labors doubled the life of the astronomer.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

de Laplace, Pierre-Simon (1749 - 1827)
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

Leach, Edmund Ronald (1910 - 1989)
How can a modern anthropologist embark upon a generalization with any hope of arriving at a satisfactory conclusion? By thinking of the organizational ideas that are present in any society as a mathematical pattern.
Rethinking Anthropology. 1961.

Leacock, Stephen
How can you shorten the subject? That stern struggle with the multiplication table, for many people not yet ended in victory, how can you make it less? Square root, as obdurate as a hardwood stump in a pasturenothing but years of effort can extract it. You can't hurry the process. Or pass from arithmetic to algebra; you can't shoulder your way past quadratic equations or ripple through the binomial theorem. Instead, the other way; your feet are impeded in the tangled growth, your pace slackens, you sink and fall somewhere near the binomial theorem with the calculus in sight on the horizon. So died, for each of us, still bravely fighting, our mathematical training; except for a set of people called "mathematicians" -- born so, like crooks.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

Lebesgue, Henri (1875 - 1941)
In my opinion, a mathematician, in so far as he is a mathemat