آموزش مجازی ریاضیات مدارس و دانشگاه های کشور

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Collected by Mark R. Woodard

**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,

**Abel, Niels H. (1802 - 1829)**

**Abel, Niels H. (1802 - 1829)**

In G. F. Simmons,

**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.

**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.

...

**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,

**Abel, Niels H. (1802 - 1829)**

**Abel, Niels H. (1802 - 1829)**

In G. F. Simmons,

**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.

**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.)

**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.

**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."

**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."

**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,

**Aiken, Conrad**

**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'

**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",

**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.)

**Anonymous**Defendit numerus: There is safety in numbers.

In J. R. Newman (ed.)

**Anonymous**Like the crest of a peacock so is mathematics at the head of all knowledge.

[An old Indian saying. Also, "

**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

**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.

**Aristotle (ca 330 BC)**The whole is more than the sum of its parts.

**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.

**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

**Aristotle**To Thales the primary question was not what do we know, but how do we know it.

**Aristotle**The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.

**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,

**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.

In O. L. Dick (ed.)

**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.

**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

**Augarten, Stan**Computers are composed of nothing more than logic gates stretched out to the horizon in a vast numerical irrigation system.

**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.

**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.

D

**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?

**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

**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.

**Bacon, Roger**For the things of this world cannot be made known without a knowledge of mathematics.

**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.)

**Baker, H. F.**[On the concept of group:]

... what a wealth, what a grandeur of thought may spring from what slightbeginnings.

Florian Cajori,

**Bagehot, Walter**Life is a school of probability.

Quoted in J. R. Newman (ed.)

**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

**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

**Bell, Eric Temple (1883-1960)**Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.

**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

**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.)

**Bell, Eric Temple (1883-1960)**The longer mathematics lives the more abstract -- and therefore, possibly also the more practical -- it becomes.

In

**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

**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

**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

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

**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.)

**Bernoulli, Daniel**...it would be better for the true physics if there were no mathematicians on earth.

In

**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,

**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

**Blake**God forbid that Truth should be confined to Mathematical Demonstration!

**Blake**What is now proved was once only imagin'd.

**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.

**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

**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.

**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.

**Browne, Sir Thomas (1605-1682)**God is like a skilful Geometrician.

**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.

**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.)

**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."

**Burke, Edmund**The age of chivalry is gone. That of sophisters, economists and calculators has succeeded.

**Butler, Bishop**To us probability is the very guide of life.

Preface to

**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.

**Byron**When Newton saw an apple fall, he found ...

A mode of proving that the earth turnd round

In a most natural whirl, called gravitation;

And thus is the sole mortal who could grapple

Since Adam, with a fall or with an apple.

**Caballero, James**

I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.

Everybody a mathematician?,*CAIP Quarterly* 2 (Fall, 1989).

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

**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.

**Carlyle, Thomas (1795-1881)**Teaching school is but another word for sure and not very slow destruction.

In H. Eves

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

**Carroll, Lewis**What I tell you three times is true.

**Carroll, Lewis**The different branches of Arithmetic -- Ambition, Distraction, Uglification, and Derision.

**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."

**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."

**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!"

**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."

**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."

**Céline, Louis-Ferdinand (1894 - 1961)**

Voyage au bout de la nuit.

**Carmichael, R. D.**A thing is obvious mathematically after you see it.

In N. Rose (ed.)

**Cauchy, Augustin-Louis (1789 - 1857)**Men pass away, but their deeds abide.

[His last words (?)]

In H. Eves

**Cayley, Arthur**As for everything else, so for a mathematical theory: beauty can be perceived but not explained.

In J. R. Newman (ed.)

**Cayley, Arthur**Projective geometry is all geometry.

In J. R. Newman (ed.)

**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,

**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

**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.

**Chesterton, G. K. (1874 - 1936)**You can only find truth with logic if you have already found truth without it.

**Chesterton, G. K. (1874 - 1936)**It isn't that they can't see the solution. It is that they can't see the problem.

**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!"

**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.

**Churchill, [Sir] Winston Spencer (1874-1965)**It is a good thing from an uneducated man to read books of quotations.

Roving Commission in

**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

**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

**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.

**Comte, Auguste (1798-1857)**C'este donc par l'étude des mathématiques,

Quoted by T. H. Huxley in

**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

**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

**Copernicus, Nicholaus (1473-1543)**Mathematics is written for mathematicians.

**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.

**Crowe, Michael**Revolutions never occur in mathematics.

**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

**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.

**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.)

**Darwin, Charles**Mathematics seems to endow one with something like a new sense.

In N. Rose (ed.)

**Davis, Philip J.**The numbers are a catalyst that can help turn raving madmen into polite humans.

In N. Rose (ed.)

**Davis, Philip J.**One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.

**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.

**Dehn, Max**Mathematics is the only instructional material that can be presented in an entirely undogmatic way.

In

**De Morgan, Augustus (1806-1871)**[When asked about his age.] I was x years old in the year x^2.

In H. Eves

**De Morgan, Augustus (1806-1871)**It is easier to square the circle than to get round a mathematician.

In H. Eves

**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.

**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.

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

**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.

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

**Descartes, René (1596-1650)**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.

**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.

**Descartes, René (1596-1650)**When writing about transcendental issues, be transcendentally clear.

In G. Simmons

**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)**

**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.

**Descartes, René (1596-1650)**Perfect numbers like perfect men are very rare.

In H. Eves

**Descartes, René (1596-1650)**

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.

**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.

**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

**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

**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.

**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

**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

**Disraeli, Benjamin**There are three kinds of lies: lies, damned lies, and statistics.

Mark Twain.

**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.

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

**Doyle, Sir Arthur Conan (1859-1930)**From a drop of water a logician could predict an Atlantic or a Niagara.

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

**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.

**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

**Dunsany, Lord**Logic, like whiskey, loses its beneficial effect when taken in too large quantities.

In J. R. Newman (ed.)

**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.

**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.)

**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...

**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.

**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.

**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,

**Eddington, Sir Arthur (1882-1944)**Proof is the idol before whom the pure mathematician tortures himself.

In N. Rose

**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

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

**Eddington, Sir Arthur (1882-1944)**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.)

**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.

**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.

**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.)

**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.

**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.)

**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,

**Einstein, Albert (1879-1955)**Everything should be made as simple as possible, but not simpler.

**Einstein, Albert (1879-1955)**I don't believe in mathematics.

Quoted by Carl Seelig.

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

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

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

**Einstein, Albert (1879-1955)**

**Einstein, Albert (1879-1955)**Common sense is the collection of prejudices acquired by age eighteen.

In E. T. Bell

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

L. Infeld

**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

**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.)

**Einstein, Albert (1879-1955)**What is this frog and mouse battle among the mathematicians?

[i.e. Brouwer vs. Hilbert]

In H. Eves

**Einstein, Albert (1879-1955)**

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.)

**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

**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

**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

**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

**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

**Einstein, Albert (1879-1955)**Everything that is really great and inspiring is created by the individual who can labor in freedom.

In H. Eves

**Einstein, Albert (1879-1955)**The search for truth is more precious than its possession.

**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."

**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.

**Einstein, Albert (1879-1955)**Science without religion is lame; religion without science is blind.

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

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

**Erath, V.**God is a child; and when he began to play, he cultivated mathematics. It is the most godly of man's games.

**Erdös, Paul**Mathematics is not yet ready for such problems.

[Attributed by Paul Halmos.]

**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

**Euler, Leonhard (1707-1783)**[upon losing the use of his right eye]

Now I will have less distraction.

In H. Eves

**Everett, Edward (1794-1865)**In the pure mathematics we contemplate absolute truths which exi

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