The value of mathematics is that it is fun and amazing and brings us great joy.  To say that math is important because it is useful is like saying that children are important because we can train them to do spirtually meaningless labor in order to increase corporate profits.  Or is that, in fact what we are saying?

Paul Lockhart in A Mathematician’s Lament, Bellevue Literary Press, New York, 2009, page 121

If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

Paul Lockhart in A Mathematician’s Lament, Bellevue Literary Press, New York, 2009, pages 20-21

There was an old man who said, “Do

Tell me how I’m to add two and two.

I’m not very sure

That it doesn’t make four —

But I fear that it’s almost too few.

Mathematics is music for the mind; music is mathematics for the soul.

— Anonymous

Teacher: You say you got (a + b)/(a – b) to reduce to 1 by cancelling. But if you just cancelled the ‘a’s and the ‘b’s, wouldn’t it be more logical for you to get just “+” divided by “-” ? Student: Yes sir. Then the “-” would cancel out in both places, leaving the “1”.

— Alan Wayne (1909-1996, mathematics author, puzzler, member of the American Mathematical Society)

You boil it in sawdust, you salt it in glue

You condense it with locusts and tape

Still keeping one principal object in view —

To preserve its symmetrical shape.

— Charles Lutwidge Dodgson aka Lewis Carroll (English author, mathematician, logician, Anglican deacon and photographer, 1832 – 1898)

Studying maths and science gives you the tools to think rigorously and analytically, to be able to reason and solve problems — from the simple to the complex — in an innovative way … I enjoy studying the relationships that people have with objects and products around them, and as an engineer, I would be able to enhance these relationships.

— Bill Voss (Year 12 student at the Australian Science and Mathematics School, SA, 2015)

Mathematical language is difficult but imperishable. I do not believe that any Greek scholar of today can understand the idiomatic undertones of Plato’s dialogues, or the jokes of Aristophanes, as thoroughly as mathematicians can understand every shade of meaning of Archimedes’ works.

— Max(well) H A Newman (1897-1984, British mathematician, WWII codebreaker, and designer of early computers)

To say that math has to be useful is like saying that the English language is only good for ordering pizza.

— Alexander Volokh (Associate Professor of Law, Emory Law, Atlanta, Georgia, USA)

… because of the mind-numbing way in which mathematics is generally taught, many people have serious misconceptions about the subject and fail to appreciate its wide applicability. It’s time to let the secret out: Mathematics is not primarily a matter of plugging numbers into formulas and performing rote computations. It is a way of thinking and questioning that may be unfamiliar to many of us, but is available to almost all of us.

— John Allen Paulos (A Mathematician Reads the Newspaper, Anchor Books, New York, 1995, page 03)

Should I admit I had a favourite student? He was a brown-haired, freckled boy, eight years old but small for his age, and the first time I came to the house he fairly shivered with shyness. We started out with the textbooks that the agency loaned me, but they were old and smelly and the leprous covers soon came apart in our hands. A brightly coloured book replaced them, one of the boy’s Christmas gifts, but its jargon was poison to his mind. So we abandoned the books and found some better way to pass the hour together. We talked a lot. It turned out that he had a fondness for collecting football stickers and could recite the names of the players depicted on them by heart. With pride he showed me the accompanying album. ‘Can you tell me how many stickers you have in there?’ I asked him. He admitted to having never totalled them up. The album contained many pages. ‘If we count each sticker one by one it will take quite a long time to reach the last page,’ I said. ‘What if we were to count the stickers two by two instead?’ The boy agreed that would be quicker. Twice as quick, I pointed out, ‘And what if we were to count the stickers in threes? Would we get to the end of the album even faster?’ He nodded. Yes, we would: three times as fast. The boy interrupted. ‘If we counted the stickers five at a time, we would finish five times faster.’ He smiled at my smile. Then we opened the album and counted the stickers on the first page, I placing my larger palm over every five. There were three palms of stickers: fifteen. The second page had slightly fewer stickers (two palms and three fingers, thirteen) – so we carried the difference over to the next. By the seventh page we had reached twenty palms: one hundred stickers. We continued turning the pages, putting my palm on to each, and counting along. In all, the number of stickers rose to over eighty palms (four hundred). After making light work of the album, we considered the case of a giant counting its pages. The giant’s palm, we agreed, would easily count the stickers by the score. What if the same giant wanted to count up to a million? The boy thought for a moment. ‘Perhaps he counts in hundreds: one hundred, two hundred, three hundred . . .’ Did the boy know how many hundreds it would take to reach a million? He shook his head. Ten thousand, I told him. His eyebrows leapt. Finally, he said, ‘He would count in ten thousands then, wouldn’t he?’ I confirmed that he would: it would be like us counting from one to hundred. ‘And if he were really big, he might count in hundred thousands,’ I continued. The giant would reach a million as quickly as we counted from one to ten. Once, during a lesson solving additions, the boy hit upon a small but clever insight. He was copying down his homework for us to go through together. The sum was 12 + 9, but it came out as 19 + 2. The answer, he realised, did not change. Twelve plus nine, and nineteen plus two, both equal twenty-one. The fortuitous error pleased him; it made him pause and think. I paused as well, not wishing to talk for fear of treading on his thoughts. Later I asked him for the answer to a much larger sum, something like 83 + 8. He closed his eyes and said, ‘Eighty-nine, ninety, ninety-one.’ I knew then that he had understood.

— Daniel Tammet (Thinking in Numbers, Hodder& Stoughton, 2013, pages 41-43)

Twice a week I would sit in Mr Baxter’s class and do my best to keep my head down. I was thirteen, going on fourteen. With his predecessors I had excelled at the subject: number theory, statistics, probability, none of them had given me any trouble. Now I found myself an algebraic zero. Things were changing; I was changing. All swelling limbs and sweating brain, suddenly I had more body than I knew what to do with. Arms and legs became the prey of low desktops and narrow corridors, were ambushed by sharp corners. Mr Baxter ignored my plight. Bodies were inimical to mathematics, or so we were led to believe. Bad hair, acrid breath, lumpy skin, all vanished for an hour every Tuesday and Thursday. Young minds in the buff soared into the sphere of pure reason. Pages turned to parallelograms; cities, circumferences; recipes, ratios. Shorn of our bearings, we groped our way around in this rarefied air.

— Daniel Tammet (Thinking in Numbers, Hodder& Stoughton, 2013, pages 38-39)

Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view. Numbers, properly considered, make us better people.

— Daniel Tammet (Thinking in Numbers, Hodder& Stoughton, 2013, page 10)

Imagine. Close your eyes and imagine a space without limits, or the infinitesimal events that can stir up a country’s revolution. Imagine how the perfect game of chess might start and end: a win for white, or black or a draw? Imagine numbers so vast that they exceed every atom in the universe, counting with eleven or twelve fingers instead of ten, reading a single book in an infinite number of ways. Such imagination belongs to everyone. It even possesses its own science: mathematics. Ricardo Nemirovsky and Francesca Ferrara, who specialise in the study of mathematical cognition, write that, ‘Like literary fiction, mathematical imagination entertains pure possibilities.’ This is the distillation of what I take to be interesting and important about the way in which mathematics informs our imaginative life. Often we are barely aware of it, but the play between numerical concepts saturates the way we experience the world.

— Daniel Tammet (Thinking in Numbers, Hodder& Stoughton, 2013, page 2)

The topics and treatment of the mathematics syllabus should be determined by the following principles:

a. The course must be enjoyable and generate steadily increasing enthusiasm in the pupils,

b. It should develop independence and activity of mind, curiosity, observation, and confidence,

c. It should make pupils familiar with the basic ideas and processes of mathematics.

— W W Sawyer (Mathematics Author Extraordinaire, 1911-2008)

The practical value of mathematics lies in the fact that a single mathematical truth has a multitude of applications. If children can handle numbers with confidence and enthusiasm, they will be able to apply arithmetic to any situation that later life may bring.

— W W Sawyer (Mathematics Author Extraordinaire, 1911-2008)

We are all of us imprisoned in our habits. The essential of education is to foster correct habits. It is easier for a mentally defective child to develop the rhythm of research than it is for a normally intelligent adult who has been subjected to fifteen years of parrot learning.

— W W Sawyer (Mathematics Author Extraordinaire, 1911-2008)

Education is essentially the direction of mental energy. Children have abundant energy looking for an outlet. If adult society provides a satisfactory outlet, hobbies develop into professions and adults find life in their work. If adult society fails in providing an outlet, a double disaster occurs. The child has no energy or enthusiasm for work; and the child’s energies are left to find an outlet at random. Society has then abdicated its duty to educate.

— W W Sawyer (Mathematics Author Extraordinaire, 1911-2008)

In the rhythm of rote learning all the emphasis is on the answer. In the rhythm of research, the emphasis is on the two items: understand what the problem is and solve the problem.

— W W Sawyer (Mathematics Author Extraordinaire, 1911-2008)

We accuse statisticians of being overly reductive and turning the world into numbers, but statisticians know well enough how approximate and fallible their numbers are. It is the rest of us who perform the worst reductionism whenever we pretend the numbers give us excessive certainty. Any journalist who acts as if the range of uncertainty does not matter, and reports only one number in place of a spread of doubt, conspires in a foolish delusion for which no self-respecting statistician would ever fall. Statistics is an exercise in coping with, and trying to make sense of, uncertainty, not in producing certainty. It is usually frank in admission of its doubt, and we should be more willing to do the same. If ever you find yourself asking, as you contemplate a number, “How can they be so precise?” the answer is that they probably can’t, and probably weren’t, but the reporting swept doubt under the carpet in the interests of brevity. If, somewhere along the line, the uncertainty has dropped out of a report, it probably will pay to find out what it had to say.

— Michael Blastard and Andrew Dilnot (The Numbers Game, pages 109-110)

… measurement is not passive; it often changes the very thing that we are measuring. And many of the measurements we hear every day, if strained too far, may have both caricatured the world and so changed it in ways we never intended. Numbers are pure and true; counting never is. That limitation does not ruin counting by any means, but if you forget it, the world you think you know through numbers will be a neat and tidy illusion.

— Michael Blastard and Andrew Dilnot (The Numbers Game, page 95)

The normal law of error stands out in the experience of mankind as one of the broadest generalisations of natural philosophy. It serves as the guiding instrument in researches in the physical and social sciences and in medicine, agriculture and engineering. It is an indispensable tool for the analysis and the interpretation of the basic data obtained by observation and experiment.

— William John Youden (Australian born statistician who formulated new statistical techniques in statistical analysis and in designs for experimenters, 1900 – 1971)

The primary purpose of the DATA statement is to give names to constants; instead of referring to pi as 3.141592653589793 at every appearance, the variable PI can be given that value with a DATA statement and used instead of the longer form of the constant. This also simplifies modifying the program, should the value of pi change.
— FORTRAN Manual for Xerox Computers

Factorials were someone’s attempt to make math look interesting.

— Steve Wright (American comedian, actor and writer, 1955 -)

Now let me explain why this makes intuitive sense.

— Larry Wasserman (Canadian statistician)

There is an astonishing imagination, even in the science of mathematics. … We repeat, there was far more imagination in the head of Archimedes than in that of Homer.

— Voltaire (François Marie Arouet) (French Enlightenment writer, historian and philosopher, 1694 – 1778)

We have admittedly defined the infinite in arithmetic by a loveknot, in this manner ∞; but we possess not therefore the clearer notion of it.

— Voltaire (François Marie Arouet) (French Enlightenment writer, historian and philosopher, 1694 – 1778), in his Philosophical Dictionary (1764): Infinity

I imagine most of that stuff on the information highway is roadkill anyway.

— John Updike (American novelist, poet, short story writer, art critic, and literary critic, 1932 – 2009), observation in 1994

When I was a boy of fourteen, my father was so ignorant I could hardly stand to have the old man around. But when I got to be twenty-one, I was astonished at how much he had learned in seven years.

— Samuel Langhorne Clemens [Mark Twain] (US author and humorist, 1835 – 1910)

The man who does not read good books has no advantage over the man who cannot read them.

— Samuel Langhorne Clemens [Mark Twain] (US author and humorist, 1835 – 1910)

Education … has produced a vast population able to read but unable to distinguish what is worth reading.

— George Macaulay Trevelyan (British historian, 1876 – 1962)

What does education often do? It makes a straight-cut ditch of a free, meandering brook.

— Henry David Thoreau (American author, poet, philosopher, abolitionist, naturalist, tax resister, development critic, surveyor, historian, and leading transcendentalist, 1817 – 1862), in his Journals, October/November 1850

The true mathematician is not a juggler of numbers, but a juggler of concepts.

— Ian Stewart (professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer, 1945 -), in his Concepts of Modern Mathematics (1975)

One death is a tragedy, a million deaths is a statistic.

— Stalin (Iosef Vissarionovich Dzhugashvili) (second leader of the Soviet Union, 1878 – 1953)

A child who is protected from all controversial ideas is as vulnerable as a child who is protected from every germ. The infection, when it comes — and it will come — may overwhelm the system, be it the immune system or the belief system.

— Jane Smiley (American novelist, 1949 -)

Speaker: First, let me assure you that this is not one of those shady pyramid schemes you’ve been hearing about. No sir. Our model is the trapezoid!

— The Simpsons ~ I Married Marge

Marge: Bart, how many hours a day do you watch TV?

Bart: Six. Seven if there’s something good on.

— The Simpsons ~ When Flanders Failed

Ms M: So y = r^3/3. And if you determine the rate of change in this curve correctly, I think you’ll be pleasantly surprised.

Class: [chuckles]

Ms M: Don’t you get it, Bart? Derivative dy = 3r^2/3, or r^2dr, or rdrr. Har-de-har-har, get it?

Bart: [not amused] Oh, yeah. [forced laugh]

— The Simpsons ~ Bart the Genius

We have all heard that a million monkeys banging on a million typewriters will eventually reproduce the entire works of Shakespeare. Now, thanks to the Internet, we know this is not true.

— Robert Silensky (English mathematician?)

For every person wishing to teach there are thirty not wanting to be taught.

— Walter Carruthers Sellar & Robert Julian Yeatman (Scottish and English humorists, authors of 1066 and All That) in And Now All This (1932)

They couldn’t hit an elephant at this distance.

— General John Sedgewick (teacher, career military officer, and Union Army general in the American Civil War, 1813 – 1864), last words at the Battle of Spotsylvania Court House (1864)

The only place where success comes before work is in a dictionary.

— Vidal Sassoon (British hairdresser, businessman, and philanthropist, 1928 – 2012), quoting one of his teachers

But the fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses.  They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers. But they also laughed at Bozo the Clown.

— Carl Edward Sagan (American astronomer, astrophysicist, cosmologist, author, science popularizer and science communicator, 1934 – 1996)

Although this may seem a paradox, all exact science is dominated by the idea of approximation.

— Bertrand Russell (Welsh philosopher, logician, mathematician, historian, and social critic, 1872 –1970)

The lust for academic respectability is the major cause of intellectual whoredom.
— Rousas John Rushdoony (Calvinist philosopher, historian, and theologian, 1916 – 2010)

Exercise is the beste intrument in learnyng.

— Robert Recorde (Welsh physician and mathematician and inventor of the equals sign, c. 1512 – 1558) in The Whetstone of Witte (1557)

A man should never be ashamed to own he has been wrong, which is but saying in other words that he is wiser today than he was yesterday.

— Alexander Pope (18th-century English poet, 1688 – 1744)

Should we teach mathematical proofs in the high school? In my opinion, the answer is yes. … Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture.

— George Polyá (Hungarian-Jewish mathematician, 1887-1985), in Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (1981)

Life is good for only two things: discovering mathematics and teaching mathematics.

— Siméon Denis Poisson (French mathematician, geometer, and physicist, 1781 – 1840) quoted in Mathematics Magazine 64 Number 1 (1991)

The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Of course, I do not here speak of that beauty which strikes the senses, the beauty of qualities and appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts and which a pure intelligence can grasp. This it is which gives body, a structure so to speak, to the iridescent appearances which flatter our senses, and without this support the beauty of these fugitive dreams would be only imperfect, because it would be vague and always fleeting.

— Jules Henri Poincaré (French mathematician, theoretical physicist and philosopher of science, 1854 –1912)

A scientist worthy of his name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
— Jules Henri Poincaré (French mathematician, theoretical physicist and philosopher of science, 1854 –1912) in N Rose, Mathematical Maxims and Minims (1988)

Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.

— Plato (Greek philosopher and mathematician, c. 424 – c. 347 BC)

He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurate with its side.

— Plato (Greek philosopher and mathematician, c. 424 – c. 347 BC)

The man who makes no mistakes does not usually make anything.

— Edward John Phelps ( American lawyer and diplomat from Vermont, 1822 – 1900), in a speech at Mansion House, 24 January 1899

We trained hard — but it seemed that every time we were beginning to form up into teams we were reorganized.  I was to learn later in life that we tend to meet any new situation by reorganizing, and what a wonderful method it can be for creating the illusion of progress while actually producing confusion, inefficiency, and demoralization.

— Gaius Petronius Arbiter (Roman courtier during the reign of Nero, 27 – 66 AD)

79.48% of all statistics are made up on the spot.

— John Allen Paulos (American professor of mathematics and a writer and speaker on mathematics, 1945 -)

Mathematics is no more computation than typing is literature.

— John Allen Paulos (American professor of mathematics and a writer and speaker on mathematics, 1945 -)

Consider a precise number that is well known to generations of parents and doctors: the normal human body temperature of 98.6 Farenheit. Recent investigations involving millions of measurements reveal that this number is wrong; normal human body temperature is actually 98.2 Farenheit. The fault, however, lies not with Dr. Wunderlich’s original measurements — they were averaged and sensibly rounded to the nearest degree: 37 Celsius. When this temperature was converted to Farenheit, however, the rounding was forgotten and 98.6 was taken to be accurate to the nearest tenth of a degree. Had the  original interval between 36.5 and 37.5 Celsius been translated, the equivalent Farenheit temperatures would have ranged from 97.7 to 99.5. Apparently, discalculia can even cause fevers.

— John Allen Paulos (American professor of mathematics and a writer and speaker on mathematics, 1945 -), in A Mathematician Reads the Newspaper (1995)

Take rest; a field that has rested gives a beautiful crop.

— Ovid (Publius Ovidius Naso) (Roman poet, 43 BC – 17 AD)

He had a style all his own. A multiple choice test he once gave to a class of undergraduate students contained a question which had the following four possible answers: choice A — the answer is 3, choice B — “none of the above”, choice C — “none of the above”, choice D — “none of the above.” What made the problem decidedly Roberts was that the only correct answer was B. Robert Bartoszynski (1933 – 1998) was a Professor in the Department of Statistics at Ohio State University.

— Ohio State University, Department of Statistics News, ‘Remembering Robert Bartoszynski’ (1998)

The true foundation of theology is to ascertain the character of God. It is by the art of Statistics that law in the social sphere can be ascertained and codified, and certain aspects of the character of God thereby revealed. The study of statistics is thus a religious service.

— Florence Nightingale (British social reformer and statistician, and the founder of modern nursing, 1820 – 1910), in Florence Nightingale David’s Games, God and Gambling (1962)

The Theory of Groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing.
— James Roy Newman (US mathematician and mathematical historian, 1907 – 1966), in The World of Mathematics (1956)

My favorite (unit) is the megaparsec-barn, a convenient unit of volume roughly 3.1 millilitres, if somewhat long and thin.

— Christopher Neufeld on alt.folklore.computers, 19 May 1993

You can tell whether a man is clever by his answers. You can tell whether a man is wise by his questions.

— Mahfouz Naguib ( Egyptian writer who won the 1988 Nobel Prize for Literature, 1911 – 2006)

The moving power of mathematical invention is not reasoning but imagination.

— August De Morgan (English mathematician and logician, 1806 – 1871)

The knowledge of numbers is one of the chief distinctions between us and the brutes.

— Lady Mary Wortley Montagu (English aristocrat and writer, 1689 – 1762), in a letter, 1753

“It wasn’t an easy sum to do, But that’s what it is,” said Pooh, said he. “That’s what it is,” said Pooh.

— Alan Alexander Milne (English author, 1882 – 1956), in Now We Are Six (1927) ‘Us Two’

Euclid alone has looked on Beauty bare.

— Edna St Vincent Millay (American lyrical poet and playwright, 1892 – 1950), in a sonnet in ‘The Harp-Weaver’ (1923)

I know that you believe that you understood what you think I said, but I am not sure you realize that what you heard is not what I meant.

— Robert McCloskey (US State Department spokesman, 1922 – 1996)

Shakespeare would have grasped wave functions, Donne would have understood complementarity and relative time. They would have been excited. What richness! They would have plundered this new science for their imagery. And they would have educated their audiences too. But you ‘arts’ people, you’re not only ignorant of these magnificent things, you’re rather proud of knowing nothing.

— Ian McEwan (English novelist and screenwriter, 1948 -) in his novel The Child in Time (1987)

I must say that I find television very educational. The minute somebody turns it on, I go to the library and read a book.

— Groucho Marx (American comedian and film and television star, 1890 – 1977)

If you make people think they’re thinking, they’ll love you; but if you really make them think, they’ll hate you.
— Donald Robert Perry Marquis (American humorist, journalist, and author, 1878 – 1937)

We are not retreating — we are advancing in another direction.
— General Douglas MacArthur (American general and field marshal of the Philippine Army and Chief of Staff of the United States Army, 1880 – 1964)

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

— Nicolai Ivanovich Lobachevsky (Russian mathematician and geometer, 1792 – 1856)

You cannot feed the hungry on statistics.

— David Lloyd George (British Liberal politician and statesman, 1863 – 1945), in a speech (1904)

In a Spectator competition, the following won a prize; subject: what would you most like to read on opening the morning paper?

OUR SECOND COMPETITION The First Prize in the second of this year’s competitions goes to Mr. Arthur Robinson, whose witty entry was easily the best of those we received. His choice of what he would like to see on opening his paper was headed ‘Our Second Competition’, and was as follows: ‘The First Prize in the second of this year’s competitions goes to Mr. Arthur Robinson, whose witty entry was easily the best of those we received. His choice of what he would like to see on opening his paper was headed ‘Our Second Competition’, but owing to paper restrictions we cannot print all of it.

— John Edensor Littlewood (English mathematician, 1885 – 1977), in A Mathematician’s Miscellany (1953)

If I had eight hours to chop down a tree, I’d spend six sharpening my axe.

— Abraham Lincoln (16th President of the United States, 1809 – 1965)

He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring.

— Abraham Lincoln (16th President of the United States, 1809 – 1965), in his Short Autobiography (1860)

“Give us a copper, Guv” said the beggar to the Treasury statistician, when he waylaid him in Parliament square. “I haven’t eaten for three days.” “Ah,” said the statistician, “and how does that compare with the same period last year?”

— Russell Lewis

Statistics are like a bikini. What they reveal is suggestive, but what they conceal is vital.

— Aaron Levenstein (Associate Professor of Business Administration at Baruch College NY from 1961 to 1981, 1910 – 1986)

That is what learning is. You suddenly understand something you’ve understood all your life, but in a new way.

— Doris Lessing (British novelist, poet, playwright, librettist, biographer and short story writer, 1919 -)

Iron rusts from disuse; stagnant water loses its purity and in cold weather becomes frozen; even so does inaction sap the vigour of the mind.

— Leonardo da Vinci (Italian polymath, 1452 – 1519)

Nessuna humana investigazione si pio dimandara vera scienzia s’essa non passa per le matematiche dimonstrazione.

(No human investigation can be called real science if it cannot be demonstrated mathematically.)

— Leonardo da Vinci (Italian polymath, 1452 – 1519), in his Treatise on Painting, Chapter 1

How come you never see a headline like ‘Psychic Wins Lottery’?

— Jay (James Douglas Muir) Leno (American stand-up comedian, actor, voice actor, writer, producer and television host, 1950 -)

It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used.

— Gottfried Wilhelm von Leibniz, (German mathematician and philosopher, 1646 – 1716)

I can’t give you brains, but I can give you a diploma.

— The Wizard of Oz to the scarecrow in the movie The Wizard of Oz (1939)

It is now proved beyond doubt that smoking is one of the leading causes of statistics.
— Fletcher Knebel (American author, 1911-1993) in Reader’s Digest, December 1961

Universities hire professors the way some men choose wives — they want the ones the others will admire.

— Morris Kline (Professor of Mathematics and a writer and populariser of the history, philosophy, and teaching of mathematics, 1908-1992), in Why the Professor Can’t Teach (1977)

I evidently knew more about economics than my examiners.

[On his poor mark in the Civil Service examinations]

— John Maynard Keynes (British economist, 1883-1946), in Roy Harrod’s ‘Life of John Maynard Keynes’ (1951)

For a list of all the ways technology has failed to improve the quality of life, please press three.

— Alice Kahn

Children need models rather than critics.

— Joseph Joubert (French moralist and essayist, 1754-1824)

To teach is to learn twice.

— Joseph Joubert (French moralist and essayist, 1754-1824)

I stepped on the gas and got to CalTech in a few minutes, only to behold a hideous sight. Although it was daytime, the lights in the buildings were flashing on and off. I saw physicists grappling on the lawn, locked in tooth and nail struggles for pocket calculators. It was horrible. Suddenly a caravan of police cars escorting a large truck with “Bob’s Slide Rules” painted on the sides pulled in front of the Physics building. Before they could come to a stop, a horde of people rushed out, turned over the truck, and began looting it, screaming and ripping open boxes of slide rules. The few cops who got too close were quickly mauled. Some of the mathematicians didn’t even try to get away; they just sat down and began doing functions, right in front of everyone. I turned away, retching. My God, I thought. The computers must be on the fritz … and they haven’t used the chalkboards in years … they must be out of chalk. If they act this way for slide rules … what will happen when the chalk truck arrives?

— Wayne Johnson, Excerpt from sci.astro post, 1995

I am a great believer in luck, and I find that the harder I work, the more I have of it.

— Thomas Jefferson (American Founding Father, the principal author of the Declaration of Independence and the third President of the United States, 1743-1826)

A great many people think they are thinking when they are merely rearranging their prejudices.

— William James (US psychologist and philosopher, 1842 – 1910)

The moral flabbiness born of the exclusive worship of the bitch-goddess success. That — with the squalid cash interpretation put on the word success — is our national disease.

— William James (US psychologist and philosopher, 1842 – 1910), in a letter to H G (Herbert George) Wells, 11 September 1906

Higher education is not necessarily a guarantee of higher virtue.

— Aldous Leonard Huxley (English writer, 1894-1963), in Brave New World Revisited (1958)

On résiste à l’invasion des armées; on ne résiste pas à l’invasion des idées.

(One can resist invasion by armies; one cannot resist invasion by ideas.)

— Victor Hugo (French poet, novelist, and dramatist, 1802-1885), in Histoire d’un Crime (1877), pt. 5, sect. 10

This will never be a civilized country until we spend more money for books than we do on chewing gum.

— Elbert Hubbard (American writer, publisher, artist, and philosopher, 1856-1915)

It is commonly believed that anyone who tabulates numbers is a statistician. This is like believing that anyone who owns a scalpel is a surgeon.

— Robert Hooke, in How to tell the Liars from the Statisticians (1983)

The human mind once stretched to a new idea never goes back to its original dimensions.

— Oliver Wendell Holmes (US author, poet and physician, 1809 – 1894)

Mathematics is a discipline practised in every university in the world, and it is at least as broad a field as biology, in which one researcher tries to understand the AIDS virus while another studies the socialization of wombats.

— Paul Hoffman ( author, science educator, food entrepreneur, 1956- ), in his Introduction to Archimedes’ Revenge (1988)

Math is hard work and it occupies your mind — and it doesn’t hurt to learn all you can of it, no matter what rank you are; everything of any importance is founded on mathematics.

— Robert Anson Heinlein (American science fiction writer, 1907-1988), in Starship Troopers (1959)

Estimated amount of glucose used by an adult human brain each day, expressed in M&Ms: 250.

— Harper’s Index (October 1989)

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.

— Godfrey Harold Hardy (English mathematician, 1877 – 1947)

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

— Godfrey H Hardy (English mathematician, 1877 – 1947), in A Mathematician’s Apology (1941)

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.

— Paul Halmos (Hungarian-born US mathematician, 1916 – 2006)

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

— Johann Wolfgang von Goethe (German writer and politician, 1749-1832)

Perplexity is the beginning of knowledge.

— Kahlil Gibran (Lebanese-American artist, poet, and writer, 1883-1931)

I have had my results for a long time: but I do not yet know how I am to arrive at them.

— Karl Frederich Gauss (German mathematician, 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.

— Karl Frederich Gauss (German mathematician, 1777-1855), in a letter to Bolyai, 1808

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.

— Martin Gardner (US popular mathematics and science writer, 1914 – 2010) quoted by George F Simmons in ‘Calculus Gems’ (1992)

Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It’s made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They’re not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there’s a sense in which matter has completely dissolved and what is left is just a mathematical structure.

— Martin Gardner (US popular mathematics and science writer, 1914 – 2010) in Focus, 14 December 1994

The Universe is a grand book which cannot be read until one first learns to comprehend the language and become familiar with the characters in which it is composed. It is written in the language of mathematics, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

— Galileo Galilei (Italian physicist, mathematician, astronomer, and philosopher, 1564-1642)

The only reason some people get lost in thought is because it’s unfamiliar territory.

— Paul Fix (American film and television character actor, 1901 – 1983)

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. . . If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

— Richard Phillips Feynman (American theoretical physicist, 1918-1988), in The Character of Physical Law (1967)

Mighty is geometry; joined with art, irresistible.

— Euripides (one of the three great tragedians of classical Athens, c480-406 BC)

By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics. Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists.

— Maurits Cornelis Escher (Dutch graphic artist, 1898-1972)

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

— Henry Havelock Ellis (British physician, writer, and social reformer, 1859-1939) in The Dance of Life (1923)

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.

— Albert Einstein (German theoretical physicist, 1879–1955), from an address at the Sorbonne, Paris (1929)

We don’t know a millionth of one percent about anything.

— Thomas Alva Edison (US inventor and businessman, 1847 – 1931)

The Romans didn’t find algebra very challenging, because X was always 10.

— Aaron Dragushan

I remember a farmer friend of mine saying that weighing a hog doesn’t fatten it up. To fatten up a hog, we need to do more than just weigh it. To increase student achievement we need to do more than just measure it. We need to nourish it, invest in it.

— Terry Craney, in his Presidential address to the 2000 WEAC Convention

The generation of random numbers is too important to be left to chance.

— Robert R Coveyou (American research mathematician, 1915 – 1996) – Title of his article published in 1970

A man who has committed a mistake and doesn’t correct it is committing another mistake.

— Confucius (551–479 BC), Chinese teacher, editor, politician, and philosopher

If you torture the data long enough, it will confess.

— Ronald Coase (1910-2013), British economist and author

There are two kinds of people: those who finish what they start and so on …

— Robert Byrne

The advancement and perfection of mathematics are intimately connected with the prosperity of the State.

— Napoleon Bonaparte (1769-1821), French military and political leader

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 was 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.”

— Pearl (Sydenstricker) Buck (1892-1973), American writer and novelist, in The Goddess Abides (1972)

It has been a fortunate fact in the modern history of physical science that the scientist constructing a new theoretical system has nearly always found that the mathematics … required … had already been worked out by pure mathematicians for their own amusement. … The moral for statesmen would seem to be that, for proper scientific “planning”, pure mathematics should be endowed fifty years ahead of scientists.

— Richard Bevan Braithwaite (1900-1990), English philosopher, in Scientific Explanation (1953)

The efforts of computer engineers have already produced a mechanical Briggs (who spent his lifetime computing logarithms) and a mechanical Barlow (whose famous Tables were a life’s work), but no one has ever conceived of a mechanical Napier (for he invented logarithms).

— (Baron) Bertram Vivian Bowden (1910-1989), English scientist and educationist, in Faster than Thought (1953)

I can’t say I’ve ever been lost, but I was bewildered once for three days.

— Daniel Boone (1734-1820), American pioneer, explorer, and frontiersman

The solid wealth of insurance companies and the success of those who organise gambling are some indication of the profits to be derived from the efficient use of chance.

— Edward de Bono (1933- ), Maltese physician, author, inventor and consultant … and originator of the term ‘lateral thinking’

If you think education is expensive, try ignorance.

— Derek Bok (1930- ), former President of Harvard University

… a phenomenon that everybody who teaches mathematics has observed: the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.) The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in a calculus class either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on throughout the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it. This is not just because each previous teacher did such a rotten job. It is because there is not time for enough practice on each new topic; and even it there were, it would be insufferably dull.

— Ralph Philip Boas (mathematician, teacher, and journal editor, 1912-1992)

Yes, the lectures are optional. Graduation is also optional.

— Bob Bickford

An error doesn’t become a mistake until you refuse to correct it.

— Orlando Aloysius Battista (Canadian-American chemist and author, 1917-1995)

‘Every minute dies a man, / Every minute one is born’; I need hardly point out to you that this calculation would tend to keep the sum total of the world’s population in a state of perpetual equipoise, whereas it is a well-known fact that the said sum total is constantly on the increase. I would therefore take the liberty of suggesting that in the next edition of your excellent poem the erroneous calculation to which I refer should be corrected as follows: ‘Every moment dies a man / And one and a sixteenth is born.’ I may add that the exact figures are 1.067, but something must, of course, be conceded to the laws of metre.

— Charles Babbage (English polymath, 1791-1871) in an unpublished letter to Lord Alfred Tennyson

It is the mark of an educated mind to be able to entertain a thought without accepting it.

— Aristotle (Greek philosopher and polymath, student of Plato, and teacher of Alexander the Great)

That which we must learn to do, we learn by doing.

— Aristotle (Greek philosopher and polymath, student of Plato, and teacher of Alexander the Great) in Nicomachean Ethics II

During a mathematicians’ dinner in Kingston, Canada, in 1979, the conversation turned to Fermats last theorem, and Enrico Bombieri proposed a problem: to show that the equation xCn + yCn = zCn where n ≥ 3 has no nontrivial solution. (Roger) Apéry left the table and came back at breakfast with the solution n = 3, x = 10, y = 16, z = 17. Bombiery replied stiffly, “I said nontrivial.”

— François Apéry (writing about his father), Mathematical Intelligencer 18(2):54–61 (1996)

I am returning this otherwise good typing paper to you because someone has printed gibberish all over it and put your name at the top.

— Attributed to an anonymous English Professor at Ohio University

Teachers open the door. You enter by yourself.

— Anonymous

Zenophobia: the irrational fear of convergent sequences. (Google ‘Zeno’ and you will understand.)

— Anonymous

The integral z squared dz

from one to the cube root of three,

times the cosine

of three pi over nine

is the log of the cube root of e.

— Anonymous

When students sit down to take the TIMSS exam [a mathematics and science test given to elementary and junior high students around the world], they also have to fill out a questionnaire. … it is so tedious and demanding that many students leave as many as ten or twenty questions blank. Now, here’s the interesting part. … countries whose students are willing to concentrate and sit still long enough and focus on answering every single question in an endless questionnaire are the same countries whose students do the best job of solving math problems.  … [Erling] Boe’s point is that we could predict precisely the order in which every country would finish in the Math Olympics without asking a single math question. All we would have to do is give them some task measuring how hard they were willing to work.

— Malcolm Gladwell, Outliers ~ The Story of Success, pages 247-8

Those three things — autonomy, complexity, and a connection between effort and reward — are, most people agree, the three qualities that work has to have if it is to be satisfying.

— Malcolm Gladwell, Outliers ~ The Story of Success, pages 149

Every experience he had had outside of his own mind had ended in frustration. He knew he needed to do a better job of navigating the world, but he didn’t know how. He couldn’t even talk to his calculus teacher, for goodness’ sake. These were things that others, with lesser minds, could master easily. But that’s because those others had had help along the way, and Chris Langan never had. It wasn’t an excuse. It was a fact. He’d had to make his way alone, and no one — not rock stars, not professional athletes, not software billionaires, and not even geniuses — ever makes it alone.

— Malcolm Gladwell, Outliers ~ The Story of Success, pages 114-5

… the sense of entitlement … is an attitude perfectly suited to succeeding in the modern world.

— Malcolm Gladwell, Outliers ~ The Story of Success, page 108

In a devastating critique, the sociologist Pitirim Sorokin once showed that if Terman had simply put together a randomly selected group of children from the same kinds of family backgrounds as the Termites — and dispensed with IQs altogether — he would have ended up with a group doing almost as many impressive things as his painstakingly selected group of geniuses. “By no stretch of the imagination or of standards of genius,” Sorokin concluded, “is the ‘gifted group’ as a whole ‘gifted.’” By the time Terman came out with his fourth volume of Genetic Studies of Genius, the word “genius” had all but vanished. “We have seen,” Terman concluded, with more than a touch of disappointment, “that intellect and achievement are far from perfectly correlated.”

— Malcolm Gladwell, Outliers ~ The Story of Success, page 90

The striking thing about Ericsson’s study is that he and his colleagues couldn’t find any “naturals,” musicians who floated effortlessly to the top while practicing a fraction of the time their peers did. Nor could they find any “grinds,” people who worked harder than everyone else, yet just didn’t have what it takes to break the top ranks. Their research suggests that once a musician has enough ability to get into a top music school, the thing that distinguishes one performer from another is how hard he or she works. That’s it. And what’s more, the people at the very top don’t work just harder or even much harder than everyone else. They work much, much harder.

— Malcolm Gladwell, Outliers ~ The Story of Success, page 39

The question is this: is there such a thing as innate talent? The obvious answer is yes.  … Achievement is talent plus preparation. The problem with this view is that the closer psychologists look at the careers of the gifted, the smaller the role innate talent seems to play and the bigger the role preparation seems to play.

— Malcolm Gladwell, Outliers ~ The Story of Success, page 38

Algebra was not taught as an academic discipline in European universities until the middle of the sixteenth century.

— Keith Devlin (Man of Numbers, page 107)

A more substantial printed book in the vernacular abbacus tradition was the scholarly work Summa de arithmetica, geometric, proportions et proportionalità, written by the mathematician Luca Pacioli just a few years later, in 1494. A significant difference between Pacioli’s book and Treviso Arithmetic is that Pacioli dealt with negative numbers. The concept of negative numbers was new in Europe, and Pacioli is believed to have provided the first printed explanation.

— Keith Devlin (Man of Numbers, page 106)

Symbolic algebra did not appear until 1591, when the French amateur mathematician and astronomer François Viète published his book In artem analyticam isagoge (Introduction to the analytic art), explaining how to formulate and solve an equation in symbolic form, much as we do today.

— Keith Devlin (Man of Numbers, page 65)

There is no national science just as there is no national multiplication table; what is national is no longer science.

— Anton Chekov (Russian dramatist and author, 1860-1904)

In 1915, Emma Noether arrived in Göttingen but was denied the private-docent status. The argument was that a woman cannot attend the University senate (the faculty meetings). Hilbert’s reaction was: “Gentlemen! We are a university, not a bath-house!”

— Anecdote about Emma Noether (German mathematician, 1882-1935) and David Hilbert (German mathematician, 1862-1943)

Albert Einstein and others have described Emma Noether as the most important woman in the history of mathematics!  Please read about her.

If there is a problem you can’t solve, then there is an easier problem you can solve: find it.

— George Polya (Hungarian-Jewish mathematician, 1887-1985)

A math student’s best friend is BOB (Back Of the Book), but remember that BOB doesn’t come to school on test days.

— Josh Folb

Natural numbers are better for your health.

— Unknown

If you play around with your fingers for a while, you’ll see that it’s true.

— Unknown lecturer (appealing to students to understand a proof)

This [mathematics] does have physical implications.  In fact, it’s all tied up with strings!

— Unknown

You mustn’t be too rigid when doing fluid mechanics!

— Unknown

Equations are just the boring parts of mathematics.  I attempt to see things in terms of geometry.

— Stephen Hawking (English theoretical physicist, cosmologist and author)

Q.  What sound does a drowning mathematician make?

A.  Log, log, log, log, log, log, log …

— Quoted by Ian Stewart in his Hoard of Mathematical Treasures (Profile Books, 2009, page 49)

Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he’s scribbled on, scrunches them all up, and throws them in the trash can.

— John von Neumann’s housekeeper, describing her employer

Last time, I asked: “What does mathematics mean to you?” And some people answered: “The manipulation of numbers, the manipulation of structures.” And if I had asked what music means to you, would you have answered: “The manipulation of notes?”

— Serge Lang (1927 – 2005), French-born American mathematician (mainly number theory).

Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown? The fly actually hits each train an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. The easy way is as follows: Since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly was flying for two hours. Since the fly was flying at a rate of 75 miles per hour, the fly must have flown 150 miles. That’s all there is to it.

When this problem was posed to John von Neumann, he immediately replied, “150 miles.”

“It is very strange,” said the poser, “but nearly everyone tries to sum the infinite series.”

“What do you mean, strange?” asked Von Neumann. “That’s how I did it!”

Q: What is the world’s longest song?

A: “Aleph-nought bottles sitting on the wall.”

Q: How do you tell that you are in the hands of the Mathematical Mafia?

A: They make you an offer that you can’t understand.

If I have seen farther than others, it is because I was standing on the shoulders of giants.

— Isaac Newton

[Corollary: If I have not seen as far as others, it is because giants were standing on my shoulders. — Hal Abelson]

“Do you love your mathematics more than me?”

“Of course not, dear — I love you much more.”

“Then prove it!”

“OK … let R be the set of all lovable objects …”

Yeah, I used to think it was just recreational … then I started doin’ it during the week … you know, simple stuff: differentiation, kinematics. Then I got into integration by parts … I started doin’ it every night: path integrals, holomorphic functions. Now I’m on Diophantine equations and sinking deeper into transfinite analysis. Don’t let them tell you it’s just recreational. Fortunately, I can quit any time I want.

This is a one line proof … if we start sufficiently far to the left.

A mathematician, native Texan, once was asked in his class: “What is mathematics good for?”

He replied: “This question makes me sick. Like when you show somebody the Grand Canyon for the first time, and he asks you ‘What’s is good for?’ What would you do? Why, you would kick the guy off the cliff.”

Maths Teacher: Now suppose the number of sheep is x …

Student: Yes sir, but what happens if the number of sheep is not x?

— John Edensor Littlewood (british Mathematician, 1885 – 1977) in A Mathematician’s Miscellany

We often hear that mathematics consists mainly of “proving theorems.” Is a writer’s job mainly that of “writing sentences?”

— Gian-Carlo (Juan Carlos) Rota (1932 – 1999) was an Italian-born American mathematician and philosopher.

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.

— Leon Albert Henkin (1921 – 2006) was a logician at the University of California, Berkeley.

William Feller was a probability theorist at Princeton University. One day he and his wife wanted to move a large table from one room of their large house to another, but, try as they might, they couldn’t get it though the door. They pushed and pulled and tipped the table on its side and generally tried everything they could, but it just wouldn’t go. Eventually, Feller went back to his desk and worked out a mathematical proof that the table would never be able to pass through the door. While he was doing this, his wife got the table through the door.

— Ian Stewart, from “Professor Stewart’s Hoard of Mathematical Treasures” (Profile Books, 2009), page 83.

You can do mathematics anywhere. I once had flash of insight into a stubborn problem in the middle of a back somersault with a triple twist.

— Ronald Graham, Chief Scientist at Cal-(IT)² who has a remarkable number named after him (Graham’s Number)!

The first sign of senility is that a man forgets his theorems, the second is that he forgets to zip up, the third sign is that he forgets to zip down.

— Stanislaw Ulam (Polish-US mathematician and inventor of the Monte Carlo method of computation, 1909 – 1984)

An equation for me has no meaning unless it expresses a thought of God.

— Srinivasa Ramanujan (Indian mathematician, 1887 – 1920) India’s Mathematics Day is held on his birthday, 22 December!

Under her masculine nom de plume, Germain shared her work on Fermat’s Last Theorem in a letter to University of Göttingen professor Carl Gauss, one of the most famous mathematicians and astronomers of the day. The letter began with an apology: “Unfortunately, the depth of my intellect does not equal the voracity of my appetite, and I feel a kind of temerity in troubling a man of genius.” Gauss wrote back with words of encouragement: “I am delighted that arithmetic has found in you so able a friend.” Gauss did not learn of her true identity until 1807. When Napoleon’s forces moved into Prussia, Germain feared that Gauss might come to the same end as Archimedes, so she asked a commander she knew in the French Army to ensure his safety. He sought out Gauss and told him that his life was safe on account of the intercession of one Sophie Germain. Gauss had no idea who his mysterious benefactor was until Germain admitted her deception in a subsequent letter. Gauss was delighted by the turn of events: “A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare…. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and superior genius.” For all their corresponding and mutual admiration, Gauss and Germain never met. He persuaded the University of Göttingen to grant her an honorary degree, but before she could make the trip from France, she died at the age of fifty-five after a two-year battle with breast cancer.

— Paul Hoffman, The Man Who Loved Only Numbers, Fourth Estate, London, 1999, pages 190-1.

There’s an old saying in mathematics: Problems worthy of attack, Prove their worth by fighting back.

— Probably Piet Hein (Danish scientist, mathematician, inventor, designer, author, and poet … and inventor of the game of Hex)

… mathematics has an aesthetic side. A conjecture can be “obvious” or “unexpected.” A result can be “trivial” or “beautiful.” A proof can be “messy,” “surprising,” or, as Erdős would say, “straight from the Book.” In a good proof, wrote Hardy, “there is a very high degree of unexpectedness, combined with inevitability and economy. The argument takes so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching consequences; but there is no escape from the conclusions.”

— Paul Hoffman, The Man Who Loved Only Numbers, Fourth Estate, London, 1999, page 43.

In 1971, Erdős’s mother died of a bleeding ulcer in Calgary, Canada, where Erdős was giving a lecture. Apparently, she had been misdiagnosed, and otherwise her life might have been saved. Soon afterward Erdős started taking a lot of pills, first antidepressants and then amphetamines. As one of Hungary’s leading scientists, he had no trouble getting sympathetic Hungarian doctors to prescribe drugs.

“I was very depressed,” Erdős said, “and Paul Turin, an old friend, reminded me, ‘A strong fortress is our mathematics.’”

Erdős took the advice to heart and started putting in nineteen-hour days, churning out papers that would change the course of mathematical history.

— Paul Hoffman, The Man Who Loved Only Numbers, Fourth Estate, London, 1999, page 143.

The most exciting phrase to hear in science, the one that heralds the most discoveries, is not “Eureka!” (I found it!) but “That’s funny …”

— Isaac Asimov (1920 – 1992) Biochemist, science fiction author and prolific writer

Not everything that can be counted counts; and not everything that counts can be counted.

— Albert Einstein (German theoretical physicist, 1879 – 1955)

That’s all well and good in practice, but how does it work in theory?

— Shmuel Weinberger (1963 – ) US topologist

USA Today has come out with a new survey — apparently three out of every four people make up 75% of the population.

— David Letterman (1947 – ) US television host and comedian

Mathematics is less related to accounting than it is to philosophy.

— Leonard Adleman (1945 -) Coinventor of RSA encryption and researcher into DNA computing

An arithmetician once tried

To multiply, add, and divide

Sets of numbers he’d found

To describe objects round

But the effort just left him pi-eyed.

— Basingstoke [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

The addend’s the number you add

To the augend, the number you had.

With computers it’s quick,

On an abacus, slick,

Done by hand it might slow you a tad.

— Basingstoke [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

An aliquot evenly splits,

As three into twelve nicely fits.

If, instead, you contrive

To divide twelve by five,

Five’s an aliquant, leaving some bits.

— Basingstoke [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

This geometry textbook provides

A solution that helpfully guides

As I take my exam.

“What’s a chiliagon?”

“Ma’am It’s a figure with one thousand sides.”

— Sheila B [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

See, O kids, I bring knowledge of rounds.

Truly not every limerick confounds!

Summing lettering, see,

To set numerals free,

Yields an answer that π’s sum expounds!

— PGS [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

[Note that the number of letters in each word gives π to 26 decimal places, i.e. 3.14159265358979323846264338]

I’m awash in these scraps, and I’ve learned,

Where a one-sided band is concerned,

That a Möbius strip

With my scissors’ neat snip

Into links can be magically turned.

— Mary [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

A one and a one and a one

And a one and a one and a one

And a one and a one

And a one and a one

Equal ten. That’s how adding is done.

— Dr Alphabet [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

It was maths lessons I used to dread.

You see, averages messed up my head.

Mean, median, mode:

Is that some sort of code?

At the end of it all I just fled.

— Dave Jermy [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

When an affine equation you see,

Such as y = mx + b,

Then its graph is a line.

This equation works fine

Unless vertical (x = d).

— Betty [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

There is a popular idea that the study of mathematics and the application of formal logic will increase one’s ability to think logically in life, and consequently the student trained in logic will be more successful in solving problems. This is absolutely not true. … What will be of help in the lives of our students is self-discipline, responsible attitudes, persistence, love of learning, respect for others, honest self-analysis, and the self-esteem that comes from meeting rigorous challenges. The value of school is not so much in the studies, but through the studies. Contrary to Locke’s opinion, there is nothing wrong with teaching algebra and geome­try to children, if only the algebra and geometry are not the goal, but rather the medium through which character is developed.

— Michael Stueben (Twenty Years Before the Blackboard, page 11)

There is geometry in the humming of the strings.

— Pythagoras (Greek philosopher and mathematician, c. 570 – c. 495 BC)

I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.

— Henri Poincaré (French mathematician, theoretical physicist and philosopher of science, 1854 –1912)

The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic.

— Gottfried Wilhelm Leibniz (German mathematician and philosopher, 1646 – 1716)

The invention of logarithms, by shortening the labours, double the life of the astronomer.

— Pierre-Simon de Laplace, (French mathematician and astronomer, 1749 – 1827)

As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.

— Joseph Louis Lagrange (Italian-born French mathematician and astronomer, 1736 – 1813)

God made integers; all else is the work of man.

— Leopold Kronecker (German mathematician, 1823 – 1891)

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.

— Charles Hermite (French mathematician, 1822 – 1901)

In most sciences one generation tears down what another has built, and what one has established, another undoes. In Mathematics alone each generation adds a new storey to the old structure.

— Hermann Hankel (German mathematician, 1839 – 1873)

To parents who despair because their children are unable to master the first problems in arithmetic I can dedicate my examples. For, in arithmetic, until the seventh grade I was last or nearly last.

— Jacques Salomon Hadamard (French mathematician, 1865 – 1963)

The total number of Dirichlet’s publications is not large: jewels are not weighed on a grocery scale.

— Karl Frederich Gauss (German mathematician, 1777-1855)

A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, “What do I get by learning these things?” So Euclid called a slave and said “Give him threepence, since he must make a gain out of what he learns.”

— Euclid of Alexandria (Greek mathematician, c. 325 – 265 BC in Alexandria, Egypt)

In mathematics the art of proposing a question must be held of higher value than solving it.

— Georg Cantor (German mathematician and inventor of set theory, 1845 – 1918)

We cannot … prove geometrical truths by arithmetic.

— Aristotle (Greek philosopher and polymath, 384 – 322 BC)

Film is one of the three universal languages, the other two: mathematics and music.

— Frank Capra (Sicilian-born US film director, 1897-1991)

The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

— Charles Caleb Colton (English cleric and writer, 1780 – 1832)

[The French priest Marin] Mersenne was intrigued by numbers of the form 2ⁿ – 1 […] he asserted that the only primes between 2 and 257 for which 2^p – 1 is prime are p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257. Unfortunately, Father Mersenne’s assertion contained sins of both commission and omission. For instance, he missed the fact that the number 2⁶¹ – 1 is prime. On the other hand, 2⁶⁷ – 1 turned out not to be prime at all. This latter fact was established in 1876 by Edouard Lucas (1842-1891), who demonstrated that the number was composite using an argument so indirect that it did not explicitly exhibit any of the factors.

— William Dunham (US writer about the history of mathematics)

Mathematicians are finite, flawed beings who spend their lives trying to understand the infinite and perfect.

— Bruce Schechter (US physicist and author)

Mathematics addiction is reasonably common among the young. Like music, mathematics exists in a separate world of form, relationship, and beauty, which is why history is full of mathematical and musical prodigies and is relatively devoid of toddling attorneys and infant stockbrokers. Children who are not old enough to cross the street can explore the infinite spaces of mathematics.

— Bruce Schechter (US physicist and author)

[Gauss’s work on the geometry of space] was the kind of work that would keep mathematicians and physicists busy for a century. Only one thing stood in the way of his revolution. He kept his work secret.  […] It wasn’t the church Gauss feared, it was its remnant, the secular philosophers.

— Leonard Mlodinow (US physicist and author)

Pythagoras could have pushed up the invention of the real number system by many centuries, had he done a simple thing: given the diagonal a name, say, d, or even better, [symbol for the square-root of 2]. Had he done that, he might have pre-empted Descartes’s coordinate revolution, for, absent a numerical representation, the need to describe this new number begged for the invention of the number line. Instead, Pythagoras retreated from his promising practice of associating geometric figures with numbers, and proclaimed that some lengths cannot be expressed as a number. The Pythagoreans called such lengths alogon, “not a ratio”, which we today translate as “irrational”. The word alogon had a double meaning, though: it also meant “not to be spoken”. Pythagoras had solved his dilemma with a doctrine that would have been hard to defend, so, in keeping with his overall doctrine of secrecy, he banned his followers from revealing the embarrassing paradox. Not obeyed. According to legend, one of his followers, Hippasus, did reveal the paradox. Today people are murdered for many reasons — love, politics, money, religion — but not because somebody squealed about the square root of 2. To the Pythagoreans, though, mathematics was a religion, so when Hippasus broke the oath of silence, he was assassinated. Resistance to irrationals continued for thousands of years. In the late nineteenth century, when the gifted German mathematician Georg Cantor did groundbreaking work to put them on firmer footing, his former teacher, a crab named Leopold Kronecker who “opposed” the irrationals, violently disagreed with Cantor and sabotaged his career at every turn. Cantor, unable to tolerate this, had a breakdown and spent his last days in a mental institution.

— Leonard Mlodinow (US physicist and author)

With all its grandiose vistas, appreciation of beauty, and vision of new realities, mathematics has an addictive property which is less obvious or healthy. It is perhaps akin to the action of some chemical drugs. The smallest puzzle, immediately recognizable as trivial or repetitive can exert such an addictive influence. One can get drawn in by starting to solve such puzzles. I remember when the Mathematical Monthly occasionally published problems sent in by a French geometer concerning banal arrangements of circles, lines and triangles on the plane. “Belanglos,” as the Germans say, but nevertheless these figures could draw you in once you started to think about how to solve them, even when realizing all the time that a solution could hardly lead to more exciting or more general topics. This is much in contrast to what I said about the history of Fermat’s theorem, which led to the creation of vast new algebraical concepts. The difference lies perhaps in that little problems can be solved with a moderate effort whereas Fermat’s is still unsolved and a continuing challenge. Nevertheless both types of mathematical curiosities have a strongly addictive quality for the would-be mathematician which exists on all levels from trivia to the most inspiring aspects.

— Stanislaw M Ulam (Polish born US mathematician, 1909 – 1984)

Mathematicians … appreciate a certain type of logical non sequitur or logical puzzle. For instance, the story of the Jewish mother who gives a present of two ties to her son-in-law. The next time she sees him, he is wearing one of them, and she asks, “You don’t like the other one?”

— Stanislaw M Ulam (Polish born US mathematician, 1909 – 1984)

… there are many books that violate the principle of having something to say by trying to say too many things. Teachers of elementary mathematics in the U.S.A. frequently complain that all calculus books are bad. That is a case in point. Calculus books are bad because there is no such subject as calculus; it is not a subject because it is many Subjects. What we call calculus nowadays is the union of a dab of logic and set theory, some axiomatic theory of complete ordered fields, analytic geometry and topology, the latter in both the “general” sense (limits and continuous functions) and the algebraic sense (orientation), real-variable theory properly so called (differentiation), the combinatoric symbol manipulation called formal integration, the first steps of lowdimensional measure theory, some differential geometry, the first steps of the classical analysis of the trigonometric, exponential, and logarithmic functions, and, depending on the space available and the personal inclinations of the author, some cook-book differential equations, elementary mechanics, and a small assortment of applied mathematics. Any one of these is hard to write a good book on; the mixture is impossible.

— Paul Halmos (Hungarian-born US mathematician, 1916 – 2006)

Sonya Kovalevskaya had been an extraordinarily versatile and talented woman. With the greatest of ease she could turn from a lecture on Abel’s functions, to research on Saturn’s rings, to the writing of verse in French or a novel in Russian or a play in Swedish, to sewing a lace collar for her little daughter Fufi. In reply to a friend’s surprise at her involvement in literature as well as mathematics, she wrote, “Many who have never had an opportunity of knowing any more about mathematics confound it with arithmetic and consider it an arid science. In reality however, it is a science which requires a great amount of imagination, and one of the leading mathematicians of our century states the case quite correctly when he says that it is impossible to be a mathematician without being a poet in soul. … one must renounce the ancient prejudice that a poet must invent something that does not exist, that imagination and invention are identical. It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing.” At the time of her death, Sonya Kovalevskaya was at the very height of her fame. By penetrating deeply into the methods of mathematical research, she had made brilliant discoveries. Her contributions are considered equal to those of any mathematician of her day by any of her colleagues who are qualified to judge.

— Teri Perl (US mathematics author)

When I turned two I was really anxious, because I’d doubled my age in a year. I thought, if this keeps up, by the time I’m six I’ll be ninety.

— Steven Wright (US comedian, actor and writer, 1955 – )

As a result of Cantor’s developments, one could divide the mathematical community into three sorts. There were the finitists, typified by the attitudes of Aristotle or Gauss, who would only speak of potential infinities, not of actual infinities. Then there were the intuitionists like Kronecker and Brouwer who denied that there was any meaningful content to the notion of quantities that are anything but finite. Infinities are just potentialities that can never be actually realised. To manipulate them and include them within the realm of mathematics would be like letting wolves into the sheepfold. Then there were the transfinitists like Cantor himself, who ascribe the same degree of reality to actual completed infinities as they did to finite quantities. In between, there existed a breed of manipulative transfinitists, typified by Hilbert, who felt no compunction or need to ascribe any ontological status to infinities but admitted them as useful ingredients of mathematical formalism whose presence was useful in simplifying and unifying other mathematical theories. “No one,” he predicted, “though he speak with the tongue of angels, will keep people from using the principle of the excluded middle.”

— John Barrow (English cosmologist, theoretical physicist and mathematician, 1952 – )

One is reminded of the story about an astronomer who began a public lecture about stars with the words “Stars are pretty simple things …” only to hear a voice calling from the back of the room, “You’d look pretty simple too from a distance of a hundred light years!”

— John Barrow (English cosmologist, theoretical physicist and mathematician, 1952 – )

Truth is stranger than fiction; fiction has to make sense.

— Leo Rosten (Russian-born US teacher and humorist, 1908 – 1997)

In Samoa, when elementary schools were first established, the natives developed an absolute craze for arithmetical calculations. They laid aside their weapons and were to be seen going about armed with slate and pencil, setting sums and problems to one another and to European visitors. The Honourable Frederick Walpole declares that his visit to the beautiful island was positively embittered by ceaseless multiplication and division.

—Robert Briffault (French social anthropologist and novelist, 1876 – 1948)

[…] it has been suggested that if we were to define a religion to be a system of thought which contains unprovable statements, so it contains an element of faith, then Gödel has taught us that not only is mathematics a religion but it is the only religion able to prove itself to be one.

— John Barrow (English cosmologist, theoretical physicist and mathematician, 1952 – )

What is the origin of the urge, the fascination that drives physicists, mathematicians, and presumably other scientists as well? Psychoanalysis suggests that it is sexual curiosity. You start by asking where little babies come from, one thing leads to another, and you find yourself preparing nitroglycerine or solving differential equations. This explanation is somewhat irritating, and therefore probably basically correct.

— David Ruelle, (Belgian-French mathematical physicist, 1935 – )

Those who taught me the infinitesimal calculus did not know the valid proofs of its fundamental theorems and tried to persuade me to accept the official sophistries as an act of faith.

— Bertrand Russell (Welsh philosopher, logician, mathematician, historian, and social critic, 1872 –1970)

The Mock Turtle went on. ‘We had the best of educations … Reeling and Writhing, of course, to begin with, and then the different branches of Arithmetic: Ambition, Distraction, Uglification, and Derision.’

— Charles Lutwidge Dodgson aka Lewis Carroll (English author, mathematician, logician, Anglican deacon and photographer, 1832–1898) [from Alice’s Adventures in Wonderland]

I have no fault with those who teach geometry. That science is the only one which has not produced sects; it is founded on analysis, and on synthesis and on the calculus; it does not occupy itself with probable truths; moreover it has the same method in every country.

— Frederick the Great of Prussia (1712-1786)

Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true. The author tends to belong to this second school of thought.

— Richard Hamming (US mathematician, 1915 – 1998)

[Bhaskara] wrote […] a book he called Lilavati, ‘Charming Girl’ — perhaps because it was full of problems such as this: “Beautiful and dear delightful girl, whose eyes are like a faun’s! If you are skilled in multiplication, tell me, what is 135 times 12?” They don’t write math books like that any more.

— Robert Kaplan (US mathematician)

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

— Henri Poincaré (French mathematician, theoretical physicist and philosopher of science, 1854 –1912)

Mathematicians do not deal in objects, but in relations between objects, thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.

— Henri Poincaré (French mathematician, theoretical physicist and philosopher of science, 1854 –1912)

I must also add that I do mathematics also because it is difficult, and it is a very beautiful challenge for the mind. I do mathematics to prove to myself that I am capable of meeting this challenge, and winning it.

— Serge Lang (French-born American mathematician, 1927 – 2005)

How wonderful that we have met with a paradox. Now we have some hope of making progress.

— Niels Bohr (Danish physicist, 1885 – 1962)

Don’t worry about your difficulties in mathematics; I can assure you that mine are still greater.

— Albert Einstein (German theoretical physicist, 1879–1955)

[Arithmetic] is one of the oldest branches, perhaps the very oldest branch, of human knowledge; and yet some of its most abstruse secrets lie close to its tritest truths.

— Henry John Stephen Smith (Irish-born English mathematician, 1826 – 1883)

The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization.

— John Kemeny (Hungarian-American mathematician, 1926 – 1992)

Pure mathematics is the world’s best game.  It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly.  It’s free.  It can be played anywhere — Archimedes did it in a bathtub.

— Richard J Trudeau (US mathematician)

I’ve dealt with numbers all my life, of course, and after a while you begin to feel that each number has a personality of its own.  A twelve is very different from a thirteen, for example.  Twelve is upright, conscientious, intelligent, whereas thirteen is a loner, a shady character who won’t think twice about breaking the law to get what he wants.  Eleven is tough, an outdoorsman who likes tramping through woods and scaling mountains; ten is rather simpleminded, a bland figure who always does what he’s told; nine is deep and mystical, a Buddha of contemplation …

— Paul Auster (US author, 1947 – )

Mathematics may be defined as the economy of counting.  There is no problem in the whole of mathematics which cannot be solved by direct counting.

— Ernst Mach (Austrian physicist and philosopher, 1838 – 1916)

The laws of nature are but the mathematical thoughts of God.

— Euclid of Alexandria (Greek mathematician, c. 325 – 265 BC in Alexandria, Egypt)

In the binary system we count on our fists instead of on our fingers.

One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers …

— Heinrich Hertz (German physicist, 1857 – 1894)

To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology.

— Hilda Phoebe Hudson (English mathematician, 1881 – 1965)

We could use up two Eternities in learning all that is to be learned about our own world and the thousands of nations that have arisen and flourished and vanished from it.  Mathematics alone would occupy me eight million years.

— Samuel Clemens [Mark Twain] (US author and humorist, 1835 – 1910)

How many times can you subtract 7 from 83, and what is left afterwards?

You can subtract it as many times as you want, and it leaves 76 every time.

The hardest arithmetic to master is that which enables us to count our blessings.

— Eric Hoffer (US social writer, 1902 – 1983)

Mathematics is as much an aspect of culture as it is a collection of algorithms.

— Carl Boyer (US historian of science and mathematics, 1906 – 1976)

The human mind has never invented a labor-saving machine equal to algebra.

There was a blithe certainty that came from first comprehending the full Einstein field equations, arabesques of Greek letters clinging tenuously to the page, a gossamer web.  They seemed insubstantial when you first saw them, a string of squiggles.  Yet to follow the delicate tensors as they contracted, as the superscripts paired with subscripts, collapsing mathematically into concrete classical entities — potential; mass; forces vectoring in a curved geometry — that was a sublime experience.  The iron fist of the real, inside the velvet glove of airy mathematics.

— Gregory Benford (US science fiction author and astrophysicist, 1941 – )

Statistics: The only science that enables different experts using the same figures to draw different conclusions.

— Evan Esar (US humorist, 1899 – 1995)

[A mathematician is a] scientist who can figure out anything except such simple things as squaring the circle and trisecting an angle.

— Evan Esar (US humorist, 1899 – 1995)

I never did very well in math — I could never seem to persuade the teacher that I hadn’t meant my answers literally.

— Calvin Trillin (US journalist and humorist, 1935 – )

Pure mathematics is, in its way, the poetry of logical ideas.

— Albert Einstein (German theoretical physicist, 1879–1955) in a letter to the New York Times, 05 May 1935

The true spirit of delight, the exaltation … which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

— Bertrand Russell (Welsh philosopher, logician, mathematician, historian, and social critic, 1872 –1970)

It’s not that I’m so smart, it’s just that I stay with problems longer.

— Albert Einstein (German theoretical physicist, 1879–1955)

In a completely rational society, the best of us would aspire to be teachers and the rest of us would have to settle for something less, because passing civilization along from one generation to the next ought to be the highest honor and highest responsibility anyone could have.

— Lee Iacocca (US businessman, 1924 – )

Wherever there is number, there is beauty.

— Proclus (Greek philosopher, 412 – 485 AD)

Mathematics possesses not only truth, but supreme beauty.

— Bertrand Russell (Welsh philosopher, logician, mathematician, historian, and social critic, 1872 –1970)

Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience.

— Keith Devlin (English-born US mathematician and popular science writer)

In my free time I do differential and integral calculus.

— Karl Marx (German philosopher, economist, sociologist and historian, 1818 – 1883)

A man has one hundred dollars and you leave him with two dollars. That’s subtraction.

— Mae West (US actress and playwright, 1893 – 1980)

Calculus is the most powerful weapon of thought yet devised by the wit of man.

— William Benjamin Smith (US mathematician, 1850 – 1934)

I’m sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it.

— Malcolm X (US African-American human rights activist, 1925 – 1965)

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race.

— Alfred North Whitehead (English mathematician and philosopher, 1861 – 1947)

So far as the mere imparting of information is concerned, no university has had any justification for existence since the popularisation of printing in the fifteenth century.

— Alfred North Whitehead (English mathematician and philosopher, 1861 – 1947)

The story was told that the young Dirichlet had as a constant companion all his travels, like a devout man with his prayer book, an old, worn copy of the Disquisitiones Arithmeticae of Gauss.

— Heinrich Tietze (Austrian mathematician, 1880 – 1964)

Fourier is a mathematical poem.

— [Lord Kelvin] William Thomson (Irish physicist, 1824 – 1907)

One merit of mathematics few will deny: it says more in fewer words than any other science. The formula, e^iπ = -1 expressed a world of thought, of truth, of poetry, and of the religious spirit “God eternally geometrizes.”

— David Eugene Smith (US mathematician, 1860 – 1944)

I cannot do it without comp[u]ters.

[From The Winter’s Tale]

— William Shakespeare, (English poet and playwright, 1564 – 1616)

The main duty of the historian of mathematics, as well as his fondest privilege, is to explain the humanity of mathematics, to illustrate its greatness, beauty and dignity, and to describe how the incessant efforts and accumulated genius of many generations have built up that magnificent monument, the object of our most legitimate pride as men, and of our wonder, humility and thankfulness, as individuals. The study of the history of mathematics will not make better mathematicians but gentler ones, it will enrich their minds, mellow their hearts, and bring out their finer qualities.

— George Sarton (Belgian chemist and historian of science, 1884 – 1956)

With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved.

— Bertrand Russell (Welsh philosopher, logician, mathematician, historian, and social critic, 1872 –1970)

At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness.

— Bertrand Russell (Welsh philosopher, logician, mathematician, historian, and social critic, 1872 –1970)

If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.

— Alfréd Rényi (Hungarian mathematician, 1921 – 1970)

The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.

— Ernest Renan (French expert of Middle East ancient languages and civilisations, 1823 – 1892)

It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics.

— Thomas Reid (Scottish philosopher, 1710 – 1796)

The Mean Value Theorem is the midwife of calculus — not very important or glamorous by itself, but often helping to delivery other theorems that are of major significance.

— Edwin Joseph Purcell (1901? – ) and Dale E Varberg (1958-1989), US mathematicians.

The study of mathematics cannot be replaced by any other activity that will train and develop man’s purely logical faculties to the same level of rationality.

— Cletus O Oakley (US mathematician, 1899 – 1990)

To state a theorem and then to show examples of it is literally to teach backwards.

— E Kim Nebeuts (Mathematics author)

The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked.

— James Roy Newman (US mathematician and mathematical historian, 1907 – 1966)

The mediocre teacher tells. The good teacher explains. The superior teacher demonstrates. The great teacher inspires.

— William Arthur Ward (US writer of inspirational maxims, 1921 – 1994)

Who has not be amazed to learn that the function y = e^x, like a phoenix rising again from its own ashes, is its own derivative?

— François le Lionnais (French chemical engineer, mathematician and poet, 1901 – 1984)

He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.

— Gottfried Wilhelm Leibniz, (German mathematician and philosopher, 1646 – 1716)

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.

— Pierre-Simon de Laplace, (French mathematician and astronomer, 1749 – 1827)

Read Euler: he is our master in everything.
— Pierre-Simon de Laplace, (French mathematician and astronomer, 1749 – 1827)

He uses statistics as a drunken man uses lamp posts — for support rather than illumination.
— Andrew Lang (Scottish poet, novelist and contributor to the field of anthropology, 1844 – 1912)

In an era in which the domain of intellect and politics were almost exclusively male, Theon [Hypatia’s father] was an unusually liberated person who taught an unusually gifted daughter and encouraged her to achieve things that, as far as we know, no woman before her did or perhaps even dreamed of doing.

— Ian Mueller (US philosopher of science, 1938 – 2010)

… She knew only that if she did or said thus-and-so, men would unerringly respond with the complementary thus-and-so. It was like a mathematical formula and no more difficult, for mathematics was the one subject that had come easy to Scarlett in her schooldays.

— Margaret Mitchell, (US author, 1900 – 1949) [From Gone With the Wind]

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.

— Johannes Kepler (German mathematician, astronomer and astrologer, 1571 – 1630)

Perhaps the greatest paradox of all is that there are paradoxes in mathematics.

— Edward Kasner (1878 – 1955) and James Roy Newman (1907–1966), US mathematicians.

[Edward Kasner’s 9 year old nephew, Milton, originally coined the term ‘googol’ and the company name ‘Google’ derives from a misspelling of this.]

The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.

— William James (US psychologist and philosopher, 1842 – 1910)

Descartes commanded the future from his study more than Napoleon from the throne.

— Oliver Wendell Holmes (US author, poet and physician, 1809 – 1894)

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.

— David Hilbert (German mathematician, 1862 – 1943)

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours 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.

— Godfrey H Hardy (English mathematician, 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.

— Godfrey H Hardy (English mathematician, 1877 – 1947)

God does arithmetic.

— Karl Frederich Gauss (German mathematician, 1777 – 1855)

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.

— Benjamin Franklin Finkel (US mathematician and educator, 1865 – 1947)

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. [Written in the margin of his copy of Diophantus’ Arithmetica]

— Pierre de Fermat (French lawyer and amateur mathematician, 1601? – 1665)

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?

— Albert Einstein (German theoretical physicist, 1879 – 1955)

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.’

— Sir Arthur Eddington (British astrophysicist, 1882 – 1944)

The composer opens the cage door for arithmetic, the draftsman gives geometry its freedom.

— Jean Cocteau (French poet, playwright, artist and filmmaker, 1889 –1963)

I recognize the lion by his paw.

— Jacob (Jacques) Bernoulli (Swiss mathematician and discoverer of e, 1654 – 1705)

[After reading Newton’s anonymous solution to a problem]

Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death.

— Hillaire Belloc (Anglo-French writer and historian, 1870 – 1953)

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.

— William S Anglin (Canadian author of mathematics books, 1949 – 2004)

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.

— John Adams (second president of the USA under the current constitution, 1735 – 1826)

[In a letter to Abigail Adams, May 12, 1780]

By studying the masters and not their pupils.

— Niels Henrik Abel (Norwegian mathematician, 1802 – 1829)

[In reply to a question about how he got his expertise]

Lottery: A tax on people who are bad at mathematics.

The optimist says the glass is half full, the pessimist says the glass is half empty, and the engineer says the glass is twice as large as it need be.

Moriarty: How are you at Mathematics?

Seagoon: I speak it like a native.

— Spike Milligan (Irish comic actor and author, 1918 – 2002)

All over China, parents tell their children to stop complaining and to finish their quadratic equations and trigonometric functions because there are sixty-five million American kids going to bed with no math at all.

— Michael Cunningham (US writer, 1952 – )

Arithmetic is being able to count up to twenty without taking off your shoes.

Mathematical illiteracy strikes 8 out of 5 people.

A mathematics professor is one who talks in someone else’s sleep.

Constants aren’t, variables won’t.

Operator! Give me the number for 911!

— Homer Simpson (fictional TV character)

Oh, people can come up with statistics to prove anything, Kent. 14% of people know that.

— Homer Simpson (fictional TV character)

The code of the schoolyard, Marge … the rules that teach a boy to be a man!  Let’s see.  Don’t tattle. Always make fun of those different from you. Never say anything, unless you’re sure everyone feels exactly the same way you do.

— Homer Simpson (fictional TV character)

He said, “You’re under arrest. You have the right to remain silent. Do you wish to retain that right?” I thought, “Ooooh! A paradox …”

— Emo Phillips (US entertainer and comedian)

I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use.

— Galileo Galilei (Italian physicist, mathematician, astronomer, and philosopher, 1564 – 1642)

4 nickels = 2 paradigms

A physicist, a mathematician and a logician were travelling through the British Isles together.

Just after they had crossed the border into Scotland, they saw a black sheep in profile on the top of a hill.

The physicist remarked, “Look at that!  All the sheep in Scotland must be black!”

The mathematician corrected him, “Not necessarily … all we can conclude is that at least one sheep in Scotland is black.”

The logician, however, had the last say.  “You are both quite wrong,” he observed, “All we can really conclude is that at least one sheep in Scotland is black on at least one side!”

Interesting Theorem: All positive integers are interesting.

Proof: Assume the contrary. Then there is a lowest non-interesting positive integer. But, hey, that’s pretty interesting! A contradiction.

If there is a 50-50 chance that something can go wrong, then 9 times out of ten it will.

— Attributed to Paul Harvey

The great majority of people have more than the average number of legs.

Statistics means never having to say you’re certain.

— Anonymous

The first mathematician goes off to the washroom, and in his absence the second calls over the waitress. He tells her that in a few minutes, after his friend has returned, he will call her over and ask her a question. All she has to do is answer one third x cubed. She repeats “one thir – dex cue?” He repeats “one third x cubed.” Her: “One thir dex cuebd?” “Yes, that’s right,” he says. So she agrees, and goes off mumbling to herself, “one thir dex cuebd …” The first guy returns and the second proposes a bet to prove his point, that most people do know something about basic math. He says he will ask the blonde waitress an integral, and the first laughingly agrees. The second man calls over the waitress and asks “what is the integral of x squared?” The waitress says “one third x cubed” and while walking away, turns back and says over her shoulder “plus a constant!”

Theorem: Consider the set of all sets that have never been considered.

Hey!  They’re all gone!!  Oh, well, never mind …

The ink of the scholar is more sacred than the blood of the martyr.

— Muhammad (Prophet of Islam, c. 570 – 632)

The only time my education was interrupted was when I was in school.

— George Bernard Shaw

Bodily exercise, when compulsory, does no harm to the body; but knowledge which is acquired under compulsion obtains no hold on the mind .

— Plato (Greek philosopher and mathematician, c. 424 – c. 347 BC)

Numbers are like people; torture them enough and they’ll tell you anything.

There was magic behind what Napier did with equations. He would take one that had some terms equal to some others and by repeated passes of al-jabr bring all the terms to the left, leaving only a zero on the right. These were what he called his ‘equations to nothing’. Why was the trick so important? It depended on what at first seems rather inconsequential: if a product of two or more factors is equal to zero, then at least one of them must be zero as well.

— Stanislas Dehaene (French mathematician and author in the area of Cognitive Neuroscience of Numeracy, 1965 – )

If you hang around with nice people you get nice friends, hang around with smart people and you get smart friends, hang around with yo-yos and you get yo-yos for friends. It’s simple mathematics.

— Sylvester Stallone in ‘Rocky’

When a shepherd, alone in his pasture, rediscovers Pythagoras’ theorem, his talent is no less than that of his renowned predecessor to whose work he was never exposed.

— Stanislas Dehaene (French mathematician and author in the area of Cognitive Neuroscience of Numeracy, 1965 – )

Today, society no longer values mental calculation. Great show-business human calculators are hard to come by. Thus, the professionals of centuries gone by appear all the more prodigious. Nowadays, in the West at least, whoever forced a child to calculate several hours a day would expose himself to a lawsuit — though our society condones the dedication of the same amount of time to piano or chess playing.

— Stanislas Dehaene (French mathematician and author in the area of Cognitive Neuroscience of Numeracy, 1965 – )

Could anyone, with sufficient training, turn into a calculating prodigy, or does it take a special, biological “gift”? To tease apart nature from nurture, a few researchers have tried to turn average students into calculating or memory prodigies through intensive training. Their results prove that passion breeds talent.

— Stanislas Dehaene (French mathematician and author in the area of Cognitive Neuroscience of Numeracy, 1965 – )

Do not forget that before the age of six or seven, children do not yet despise mathematics. Everything that looks mysterious and excites their imagination feels like a game to them. They are open and ready to develop a passion for numbers if only one were willing to show them how magical arithmetic can be.

— Stanislas Dehaene (French mathematician and author in the area of Cognitive Neuroscience of Numeracy, 1965 – )

“What’s one and one and 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.”

“She can’t do addition.” said the Red Queen.

[from Through the Looking Glass]

— Charles Lutwidge Dodgson aka Lewis Carroll (English author, mathematician, logician, Anglican deacon and photographer, 1832 – 1898)

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.

— Albert Einstein (German theoretical physicist, 1879 – 1955)

A person who can, within a year, solve x² – 92y² = 1 is a mathematician.

— Brahmagupta (Indian mathematician and astronomer, 598 – 668)

Relations between pure and applied mathematicians are based on trust and understanding. Namely, pure mathematicians do not trust applied mathematicians, and applied mathematicians do not understand pure mathematicians.

A topologist is a person who doesn’t know the difference between a coffee cup and a donut.

— John Kelley (US mathematician, 1916 – 1999)

Gödel can’t prove he was here.

I heard that parallel lines actually do meet, but they are very discrete.

Mathematics is not a spectator sport.

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.

[Fictional character, Lazarus Long]

— Robert A Heinlein, Time Enough for Love

Engineers like to solve problems. If there are no problems handily available, they will create their own problems. Normal people don’t understand this concept; they believe that if it ain’t broke, don’t fix it.  Engineers believe that if it ain’t broke, it doesn’t have enough features yet.

Customer: “How much is a large order of Fibonaccos?”

Cashier:  “It’s the price of a small order plus the price of a medium order.”

Q: Why do Computer Scientists get Halloween and Christmas mixed up?

A: Because Oct. 31 = Dec. 25.

For π, we have in France:

Que j’aime a faire apprendre un nombre utile aux sages

Immortel Archimede, artiste, ingenieur

Qui de ton jugement peut priser la valeur?

Pour moi ton probleme eut de pareils avantages.

[The number of letters in each word gives π to 30 decimal places, i.e. 3.141592653589793238462643383279]

Now I will a rhyme construct

By chosen words the young instruct.

Cunningly devised endeavour,

Con it and remember ever.

Widths of circle here you see.

Sketched out in strange obscurity.

[The number of letters in each word gives π to 30 decimal places, i.e. 3.141592653589793238462643383279]

Old trigonometry teachers never die, they just lose their identities.

Old mathematicians never die, they just go off on a tangent.

Old mathematicians never die, they just disintegrate.

Old mathematicians never die; they just lose some of their functions.

Lobachevski was out of line.

Klein bottle for rent — inquire within.

On average, people are mean.

Complex numbers are unreal.

I like angles … to a degree.

Vectors can be ‘arrowing.

Algebra is x-sighting.

I’m partial to fractions.

I could go on and on about sequences.

I’ll do algebra, I’ll do trig, and I’ll even do statistics, but graphing is where I draw the line!

A biologist, a physicist and a mathematician were sitting in a street cafe watching the crowd.

Across the street they saw a man and a woman entering a building. Ten minutes they reappeared together with a third person.

They have multiplied, said the biologist.

Oh no, an error in measurement, the physicist sighed.

If exactly one person now enters the building, it will be empty again, the mathematician concluded.

One and one make two,

But if one and one should marry,

Isn’t it queer — Within a year

There’s two and one to carry.

Math and Alcohol don’t mix, so … please don’t drink and derive.

Average means something that hens lay eggs on.

What is eighteen plus fifteen? Let’s see.

Eight plus five is thirteen. You agree?

Mark down “three” (we’re half done)

And then carry the one.

That’s three ones for ten each: thirty-three.

— Chris J Strolin [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]

A Dozen, a Gross and a Score,

plus three times the square root of four,

divided by seven,

plus five times eleven,

equals nine times itself, nothing more.

— Jon Saxton (author of mathematics textbooks)

If a man’s wit be wandering, let him study the mathematics.

— Sir Francis Bacon (English philosopher, statesman and scientist, 1561 – 1626)

A man who is not somewhat of a poet can never be a mathematician.

— Karl Weierstrass (German mathematician, 1815 – 1897)

The essence of all things is numbers.

— Pythagoras (Greek philosopher and mathematician, c. 570 – c. 495 BC)

Let no one ignorant of geometry enter here.

[Inscription at the entrance to his Academy in Athens]

— Plato (Greek philosopher and mathematician, c. 424 – c. 347 BC)

With the help of God, and with His precious assistance, I say that algebra is a scientific art.

— Omar Khayyám (Persian philosopher, mathematician, astronomer and poet, 1048 – 1131)

Reserve your right to think, for even to think wrongly is better than not to think at all.

— Theon (Greek mathematician and last Curator of the Library of Alexandria, c. 335 – c. 405) to his daughter, Hypatia (Greek Neoplatonist mathematician, philosopher and astronomer, c. 350 – 415)

Napoleon’s Theorem:  If the centroids of equilateral triangles constructed on the three sides of any triangle are joined, the shape is an equilateral triangle.

— Napoleon Bonaparte (French military leader and emperor, 1769 – 1821)

Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.

— Archimedes (Greek mathematician and inventor, c.  287 – c.  212 BC)

Mathematics is the queen of sciences and arithmetic is the queen of mathematics.

— Karl Frederich Gauss (German mathematician, 1777 – 1855)

Mathematics is the pen with which God has written the universe.

— Galileo Galilei (Italian physicist, mathematician, astronomer, and philosopher, 1564 – 1642)

Highness, there is no royal road to geometry.

[Euclid to Ptolemy I]

— Euclid of Alexandria (Greek mathematician, c. 325 – 265 BC in Alexandria, Egypt)

Each year, there’s a soccer game between a marketing department and the support staff. Each year, the support staff team wins. One year a memo was circulated to all staff: “The marketing department is pleased to announce, that for this year’s soccer season, we came in second place, having lost only one game all year.  The support department, however, has had a dismal season, as they only won a single game.”

I haven’t eaten for 26 days, 3 hours and 20 minutes.

Why not, mum?

Because I didn’t want to have a mouthful of food when you phoned!

Never forget that 2 + 2 = 5 for extremely large values of 2.

We’re going to turn this team around 360 degrees.

Jason Kidd (on joining the Dallas Mavericks)

There are two types of people: those who divide people into two types, and those who don’t.

There are three kinds of people: those who understand numbers and those who don’t.

There are 10 kinds of people: those who understand binary code and those who don’t.

Statisticians know that if you put a man’s head in a sauna and his feet in a deep freeze, he will feel pretty good … on average.

7 out of 5 people do not understand fractions.

A mathematician is a device for turning coffee into theorems.

[Often attributed to Paul Erdös]

— Alfréd Rényi (Hungarian mathematician, 1921 – 1970)

An adult gave a child a new coin.

‘What do you say?’ asked the adult.

The child replied, ‘One …’

Unknown

Some English boys were revising for a forthcoming mathematics test. Several of them seemed quite lost on the topic of ratios, so the teacher asked the class, ‘Does anybody know what a ratio is?’

A voice piped up: ‘Please, miss, I think he was a sailor.’

There was a kid who hung around the local grocery store where the bigger boys always teased him. They said that his belt didn’t go through all the loops. To prove the kid’s stupidity, the bigger boys frequently offered him a choice between a nickel and a dime. ‘He’ll always take the nickel’, they said, ‘because it’s bigger’.

The grocer took the kid aside and said, ‘Those boys are making fun of you. They think you don’t know a dime is worth more than a nickel. Are you grabbing the nickel because it’s bigger? Or what?’

The kid looked at the grocer and whispered, ‘No. But if I took the dime they’d quit doing it.’

A linguistics professor was lecturing to his class one day. ‘In English,’ he said, ‘a double negative forms a positive. In some languages, though, such as Russian, a double negative is still a negative. However, there is no language where a double positive can form a negative.’ A voice from the back of the room piped up: ‘Yeah, right.’

The difference between an introverted and extroverted mathematician is that an introverted mathematician looks at his shoes while talking to you while an extroverted mathematician looks at your shoes.

A mathematician is a blind man in a dark room looking for a black cat that isn’t there.

Unknown (often wrongly attributed to Charles Darwin)

One day a farmer called up an engineer, a physicist and a mathematician and asked them to fence off the largest possible area with the least amount of fence.

The engineer made the fence in a circle and proclaimed that he had the most efficient design.

The physicist made a long, straight line and proclaimed, ‘We can assume the length is infinite.’ He pointed out that fencing off half of the earth was certainly the most efficient way to do it.

The mathematician built a tiny fence around himself and said ‘I declare myself to be on the outside.’

A confused driving student one night

Made a left by mistake at the light

Then she turned left twice more

With intent to be sure

For she knew that three wrongs make a right.

Sign at a swimming pool: ‘Due to a water shortage, only lanes one and four will be open, thank you.’

Arithmetic is where the answer is right and everything is nice and you can look out of the window and see the blue sky – or the answer is wrong and you have to start over and try again and see how it comes out this time.

Carl Sandburg (US writer and poet, 1878 – 1967)

There was a young man from Trinity,

Who solved the square root of infinity.

While counting the digits,

He was seized by the fidgets,

Dropped science, and took up divinity.

Groucho Marx says he once had a nurse who was so conceited that when she took his pulse she subtracted ten points for her personality.

A prospective army enlistee came into a recruiting centre and, after a lengthy interview, was asked by the recruiter if, because of his education, he might be interested in a commission. The applicant thought and then replied, “No, I don’t think so. The way I shoot, I might be better off on a straight salary.”

— Ralph Williams (Reader’s Digest, December 1971)