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.
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”.
Yet what are all such gaieties to me
Whose thoughts are full of indices and surds?
x² +7x + 53 = 11/3
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.
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.
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.
To say that math has to be useful is like saying that the English language is only good for ordering pizza.
As business reports and scientific research papers, newspapers and other periodicals, databases and electronic mail, textbooks and other books all increase rapidly, the number of interdependencies among them rises exponentially. New ways to link, classify, and order the traffic on the information highways of the future are necessary if we’re to thread our way through the mounds of raw data strewn along them. Some understanding of basic mathematical and statistical ideas is necessary if we’re to avoid the condition described by the computer scientist Jesse Shera’s paraphrase of Coleridge: “Data, data everywhere, but not a thought to think.”
… 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.
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.
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.
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.
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.
Children live in the present. They feel that at any moment something tremendously exciting may happen. Successful teaching makes them feel that something tremendously exciting has happened. Preparation for the business worries of adult life does not meet this specification. Mathematics teaching is practical and purposeful only if it enables children to do better something they desperately want to do here and now.
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.
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.
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.
It is quite natural that if a child of limited intelligence can only do one subject, that subject should be arithmetic. The judgments involved in thinking, “Is this the right block? No, that one’s too long,” and later associating the various blocks with 0, 1, 2, …., and 9 are much simpler than those required to learn the 26 letters of the alphabet and the eccentricities of English and American spelling.
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.
In discovering something for ourselves, we have a sense of freedom and conquest. In memorizing something that another person tells us and that we do not understand, we are slaves.
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.
The best way to learn geometry is to follow the road which the human race originally followed: DO things, MAKE things, NOTICE things, ARRANGE things, and only then reason about things.
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.
… 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.
I ordered a foot-long sandwich from a takeaway restaurant and asked the lady behind the counter to cut it into fourths.
“I’m sorry, I can’t,” she replied. “I already cut it in half.”
A curved line is the loveliest distance between two points.
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.
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
I’m writing a book. I’ve got the page numbers done.
Factorials were someone’s attempt to make math look interesting.
The Doctor: I never make stupid mistakes. Only very, very clever ones.
Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.
Calvin: You can’t just turn on creativity like a faucet. You have to be in the right mood.
Hobbs: What mood is that?
Calvin: Last-minute panic.
If you want to increase your success rate, double your failure rate.
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.
We have admittedly defined the infinite in arithmetic by a loveknot, in this manner ∞; but we possess not therefore the clearer notion of it.
I imagine most of that stuff on the information highway is roadkill anyway.
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.
The man who does not read good books has no advantage over the man who cannot read them.
In the space of one hundred and seventy six years, the Lower Mississippi has shortened itself two hundred and forty two miles. That is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the old oolitic silurian period, just a million years ago next November, the Lower Mississippi river was upwards of one million three hundred miles long, and stuck out over the gulf of Mexico like a fishing rod. And by the same token any person can see that seven hundred forty two years from now the Lower Mississippi will be only a mile and three quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholescale returns of conjecture out of such a trifling investment of fact.
Education … has produced a vast population able to read but unable to distinguish what is worth reading.
What does education often do? It makes a straight-cut ditch of a free, meandering brook.
Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense.
The mistakes are all there, waiting to be made.
Basic research may seem very expensive. I am a well-paid scientist. My hourly wage is equal to that of a plumber, but sometimes my research remains barren of results for weeks, months or years and my conscience begins to bother me for wasting the taxpayer’s money. But in reviewing my life’s work, I have to think that the expense was not wasted. Basic research, to which we owe everything, is relatively very cheap when compared with other outlays of modern society. The other day I made a rough calculation which led me to the conclusion that if one were to add up all the money ever spent by man on basic research, one would find it to be just about equal to the money spent by the Pentagon this past year.
They have likewise discovered two lesser stars, or satellites, which revolve around Mars, whereof the innermost is distant from the centre of the primary planet exactly three of his diameters, and the outermost five; the former revolves in the space of ten hours, and the latter in twenty-one and a half; so that the squares of their periodical times are very near in the same proportion with the cubes of their distances from the centre of Mars, which evidently shows them to be governed by the same law of gravitation, that influences the other heavenly bodies. (This description preceded the discovery of Phobos and Deimos by Asaph Hall in 1877.)
If you could pick a real number at random, you would be virtually certain to pick one that was transcendental.
The true mathematician is not a juggler of numbers, but a juggler of concepts.
What is the difference between unethical and ethical advertising? Unethical advertising uses falsehoods to deceive the public; ethical advertising uses truth to deceive the public.
One death is a tragedy, a million deaths is a statistic.
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.
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!
Marge: Bart, how many hours a day do you watch TV?
Bart: Six. Seven if there’s something good on.
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.
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.
As a statistician, I want to use mathematics to help deal with practical uncertainty. The natural mathematical way to handle uncertainty is via probability. About the simplest practical probability statement I can think of is “The probability that a fair coin, tossed at random, will come down ‘heads’ is 1/2”. Now try to define “fair coin”, “at random” and “probability 1/2” without using subjective probability or circular definitions. Summary: if a practical probability statement is not subjective, then it must be tautologous, ill-defined, or useless. Of course, for balance, some of the time I teach subjective methods, and some of the time I teach useless methods :-).
For every person wishing to teach there are thirty not wanting to be taught.
They couldn’t hit an elephant at this distance.
The only place where success comes before work is in a dictionary.
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.
Although this may seem a paradox, all exact science is dominated by the idea of approximation.
Geometry has been, throughout, of supreme importance in the history of knowledge.
The lust for academic respectability is the major cause of intellectual whoredom.
— Rousas John Rushdoony (Calvinist philosopher, historian, and theologian, 1916 – 2010)
You know everybody is ignorant, only on different subjects.
Exercise is the beste intrument in learnyng.
My mother made me a scientist without ever intending to.
Every Jewish mother in Brooklyn would ask her child after school: ‘So? Did you learn anything today?’
But not my mother. She always asked me a different question.
‘Izzy,’ she would say, ‘did you ask a good question today?’
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.
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.
Life is good for only two things: discovering mathematics and teaching mathematics.
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.
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.
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.
The man who makes no mistakes does not usually make anything.
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.
79.48% of all statistics are made up on the spot.
Mathematics is no more computation than typing is literature.
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.
Take rest; a field that has rested gives a beautiful crop.
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.
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.
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.
You can tell whether a man is clever by his answers. You can tell whether a man is wise by his questions.
The moving power of mathematical invention is not reasoning but imagination.
The knowledge of numbers is one of the chief distinctions between us and the brutes.
“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.
Euclid alone has looked on Beauty bare.
While attending a recent baseball game with my 7-year old daughter, she asked me how many people I thought were there. I replied “About 27,000”. She looked around for a moment, then turned to me and asked, “Are you counting yourself?”
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.
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.
I must say that I find television very educational. The minute somebody turns it on, I go to the library and read a book.
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.
You cannot feed the hungry on statistics.
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.
If I had eight hours to chop down a tree, I’d spend six sharpening my axe.
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.
“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?”
Statistics are like a bikini. What they reveal is suggestive, but what they conceal is vital.
That is what learning is. You suddenly understand something you’ve understood all your life, but in a new way.
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.
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.)
How come you never see a headline like ‘Psychic Wins Lottery’?
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.
I can’t give you brains, but I can give you a diploma.
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.
I evidently knew more about economics than my examiners.
[On his poor mark in the Civil Service examinations]
For a list of all the ways technology has failed to improve the quality of life, please press three.
Children need models rather than critics.
To teach is to learn twice.
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?
Some of the most important results (e.g. Cauchy’s theorem) are so surprising at first sight that nothing short of a proof can make them credible.
I am a great believer in luck, and I find that the harder I work, the more I have of it.
A great many people think they are thinking when they are merely rearranging their prejudices.
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.
Higher education is not necessarily a guarantee of higher virtue.
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.)
This will never be a civilized country until we spend more money for books than we do on chewing gum.
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.
The human mind once stretched to a new idea never goes back to its original dimensions.
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.
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.
Estimated amount of glucose used by an adult human brain each day, expressed in M&Ms: 250.
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.
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.
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.
To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess.
Mathematics is not a deductive science — that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.
Perplexity is the beginning of knowledge.
I have had my results for a long time: but I do not yet know how I am to arrive at them.
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.
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.
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.
Whenever you can, count.
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.
The only reason some people get lost in thought is because it’s unfamiliar territory.
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.
Mighty is geometry; joined with art, irresistible.
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.
It is here [in mathematics] that the artist has the fullest scope of his imagination.
If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.
We don’t know a millionth of one percent about anything.
The Romans didn’t find algebra very challenging, because X was always 10.
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.
The generation of random numbers is too important to be left to chance.
A man who has committed a mistake and doesn’t correct it is committing another mistake.
If you torture the data long enough, it will confess.
There are two kinds of people: those who finish what they start and so on …
The advancement and perfection of mathematics are intimately connected with the prosperity of the State.
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.”
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.
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).
I can’t say I’ve ever been lost, but I was bewildered once for three days.
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.
If you think education is expensive, try ignorance.
… 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.
Yes, the lectures are optional. Graduation is also optional.
An error doesn’t become a mistake until you refuse to correct it.
‘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.
It is the mark of an educated mind to be able to entertain a thought without accepting it.
That which we must learn to do, we learn by doing.
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.”
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.
Teachers open the door. You enter by yourself.
Zenophobia: the irrational fear of convergent sequences. (Google ‘Zeno’ and you will understand.)
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.
An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn’t care.
‘I checked it very thoroughly,’ said the computer, ‘and that quite definitely is the answer. I think the problem, to be quite honest with you, is that you’ve never actually known what the question is.’
Virtually all of the advantage that wealthy students have over poor students is the result of differences in the way privileged kids learn while they are not in school.
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.
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.
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.
… the sense of entitlement … is an attitude perfectly suited to succeeding in the modern world.
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.”
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.
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.
People don’t rise from nothing. We do owe something to parentage and patronage. The people who stand before kings may look like they did it all by themselves. But in fact they are invariably the beneficiaries of hidden advantages and extraordinary opportunities and cultural legacies that allow them to learn and work hard and make sense of the world in ways others cannot. It makes a difference where and when we grew up. The culture we belong to and the legacies passed down by our forebears shape the patterns of our achievement in ways we cannot begin to imagine. It’s not enough to ask what successful people are like, in other words. It is only by asking where they are from that we can unravel the logic behind who succeeds and who doesn’t.
Algebra was not taught as an academic discipline in European universities until the middle of the sixteenth century.
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.
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.
I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
There is no national science just as there is no national multiplication table; what is national is no longer science.
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!”
If there is a problem you can’t solve, then there is an easier problem you can solve: find it.
I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.
A math student’s best friend is BOB (Back Of the Book), but remember that BOB doesn’t come to school on test days.
Natural numbers are better for your health.
If you play around with your fingers for a while, you’ll see that it’s true.
This [mathematics] does have physical implications. In fact, it’s all tied up with strings!
You mustn’t be too rigid when doing fluid mechanics!
Equations are just the boring parts of mathematics. I attempt to see things in terms of geometry.
Q. What sound does a drowning mathematician make?
A. Log, log, log, log, log, log, log …
Calvin: You know, I don’t think math is a science, I think it’s a religion.
Hobbes: A religion?
Calvin:Yeah. All these equations are like miracles. You take two numbers and when you add them, they magically become one NEW number! No one can say how it happens. You either believe it or you don’t. [Pointing at his math book] This whole book is full of things that have to be accepted on faith! It’s a religion!
Hobbes: And in the public schools no less. Call a lawyer.
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?”
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?”
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.
With me, everything turns into mathematics.
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.
There is a story that John Napier had a pet black rooster that his staff believed was a familiar spirit. One day, a theft occurred on his property and everyone denied any knowledge of it. Apparently, Napier gathered his staff, placed his rooster in a darkened room, and announced that each person was to enter the room alone and place his/her hands on the rooster and come out. The rooster would then tell him who had been involved in the theft. All entered the room and returned, but only one servant returned with clean hands. Napier had covered his pet rooster with lamp black.
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.
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.
An equation for me has no meaning unless it expresses a thought of God.
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.
There is a much quoted story about David Hilbert, who one day noticed that a certain student had stopped attending class. When told that the student had decided to drop mathematics to become a poet, Hilbert replied, “Good — he did not have enough imagination to become a mathematician.”
There’s an old saying in mathematics: Problems worthy of attack, Prove their worth by fighting back.
… 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.”
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.
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 …”
Not everything that can be counted counts; and not everything that counts can be counted.
I certainly do care about measuring educational results. But what is an ‘educational result?’ The twinkling eyes of my students, together with their heartfelt and beautifully expressed mathematical arguments are all the results I need.
That’s all well and good in practice, but how does it work in theory?
Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems.
I’m a mathematical optimist: I deal only with positive integers.
USA Today has come out with a new survey — apparently three out of every four people make up 75% of the population.
Obvious is the most dangerous word in mathematics.
Mathematics is less related to accounting than it is to philosophy.
Inscribing a circle, you’ll heed
These instructions on how to proceed:
Once each angle’s bisected,
Those lines are connected,
Constructing the center you’ll need.
— Mephistopheles [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]
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]
In old Greece, certain shoemakers scrape
All their leather with knives of this shape.
Of three half-circles made,
It’s a beautiful blade —
Plane geometry leaves me agape!
— Sheila B (Limerick referring to a shape called the arbelos)
[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]
The diagram drawn by Argand
In a polar-coordinate hand
Isn’t meant to perplex,
But the plane is complex
So at first you might not understand.
— Mike Scholtes [Used with permission ~ Visit The Omnificent English Dictionary In Limerick Form]
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.
I have in later years taken to Euclid, Whitehead, Bertrand Russell, in an elemental way.
There is geometry in the humming of the strings.
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.
… arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up.
The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic.
Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half.
These … tables (values of trigonometry functions), constructed by means of new techniques based principally on the calculus of differences, are one of the most beautiful monuments ever erected to science.
The invention of logarithms, by shortening the labours, double the life of the astronomer.
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.
God made integers; all else is the work of man.
Number theorists are like lotus-eaters — having tasted this food they can never give it up.
No one will expel us from this paradise Cantor has created for us.
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.
We are servants rather than masters in mathematics.
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.
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.
The total number of Dirichlet’s publications is not large: jewels are not weighed on a grocery scale.
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.”
I would rather discover one scientific fact than become King of Persia.
In mathematics the art of proposing a question must be held of higher value than solving it.
The essence of mathematics is its freedom.
We cannot … prove geometrical truths by arithmetic.
If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.
What would life be without arithmetic, but a scene of horrors?
Film is one of the three universal languages, the other two: mathematics and music.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.
I realise that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.
I know this world is ruled by infinite intelligence. Everything that surrounds us — everything that exists — proves that there are infinite laws behind it. There can be no denying this fact. It is mathematical in its precision.
I chose to deal with the science of cryptography. Cryptography began in mathematics. Codes were developed, even from Caesar’s time, based on number theory and mathematical principles. I decided to use those principles and designed a work that is encoded.
A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.
[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 2p – 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.
Mathematicians are finite, flawed beings who spend their lives trying to understand the infinite and perfect.
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.
[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.
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.
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.
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?”
As we passed the battlefields of the Civil War, Johnny [von Neumann] recounted the smallest details of the battles. His knowledge of history was really encyclopedic, but what he liked and knew best was ancient history. He was a great admirer of the concise and wonderful way the Greek historians wrote. His knowledge of Greek enabled him to read Thucydides, Herodotus, and others in the original; his knowledge of Latin was even better.
… 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.
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.
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.
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.”
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!”
Truth is stranger than fiction; fiction has to make sense.
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.
[…] 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.
Still in a daze the next morning, Robert said to his mother, “Do you know the year I was born? It was 6×1 and 8×10 and 9×100 and 1×1000.”
“I don’t know what’s got into the boy lately,” said Robert’s mother, shaking her head.
“Here,” she added, handing him a cup of hot chocolate, “maybe this will help. You say the oddest things.”
Robert drank his hot chocolate in silence. There are some things you can’t tell your mother, he thought.
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.
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.
It was not until 1913 that France recognized the meridian through Greenwich as the prime (zero) meridian, in exchange for England “recognizing” the metric system.
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.’
Mathematicians are fat, scruffy and have no friends — in any language. Youngsters from seven countries, asked to come up with a portrait of the typical mathematician, showed a badly dressed, middle-aged nerd with no social life. … Most children drew white men with glasses, often with a beard, bald head or weird hair, and shirt pockets filled with pens, who were working at a blackboard or computer. Finnish children had an even more disturbing view of maths teachers: several portrayed them forcing children to do sums at gunpoint.
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.
It is difficult to give an idea of the vast extent of modern mathematics. The word “extent” is not the right one: I mean extent crowded with beautiful details — not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood and flower …
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.
[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.
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.
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.
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.
How wonderful that we have met with a paradox. Now we have some hope of making progress.
Don’t worry about your difficulties in mathematics; I can assure you that mine are still greater.
[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.
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me — both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension.
The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization.
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.
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 …
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.
The laws of nature are but the mathematical thoughts of God.
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 …
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.
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.
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.
Mathematics is as much an aspect of culture as it is a collection of algorithms.
The human mind has never invented a labor-saving machine equal to algebra.
It is a mathematical fact that fifty percent of all doctors graduate in the bottom half of their class.
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.
Statistics: The only science that enables different experts using the same figures to draw different conclusions.
[A mathematician is a] scientist who can figure out anything except such simple things as squaring the circle and trisecting an angle.
I never did very well in math — I could never seem to persuade the teacher that I hadn’t meant my answers literally.
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.
It’s not that I’m so smart, it’s just that I stay with problems longer.
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.
Wherever there is number, there is beauty.
Mathematics possesses not only truth, but supreme beauty.
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.
A teacher’s day is half bureaucracy, half crisis, half monotony and one-eighth epiphany. Never mind the arithmetic.
The only way to learn mathematics is to do mathematics.
Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
In my free time I do differential and integral calculus.
A man has one hundred dollars and you leave him with two dollars. That’s subtraction.
Calculus is the most powerful weapon of thought yet devised by the wit of man.
It is clear that the chief end of mathematical study must be to make the students think.
Mathematics has beauties of its own — a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found only in the works of the greatest beauty When this subject is properly … presented, the mental emotion should be that of enjoyment of beauty, not that of repulsion from the ugly and the unpleasant.
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.
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.
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.
A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction.
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.
Fourier is a mathematical poem.
Four circles to the kissing come,
The smaller are the benter.
The bend is just the inverse of
The distance from the centre.
Though their intrigue left Euclid dumb
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line
And concave bends have minus sign,
The sum of squares of all four bends
Is half the square of their sum.
One merit of mathematics few will deny: it says more in fewer words than any other science. The formula, eiπ = -1 expressed a world of thought, of truth, of poetry, and of the religious spirit “God eternally geometrizes.”
I cannot do it without comp[u]ters.
[From The Winter’s Tale]
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.
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.
A good notation has a subtlety and suggestiveness which at times make it almost seem like a live teacher.
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.
In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.
If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.
The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.
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.
To avoide the tediouse repetition of these woordes: is equalle to: I will settle as I doe often in woorke use, a paire of paralleles, or gemowe [twin] lines of one lengthe: ===, bicause noe .2. thynges, can be moare equalle.
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.
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.
[Florence Nightingale’s] statistics were more than a study, they were indeed her religion. For her Quetelet was the hero as scientist, and the presentation copy of his Physique Sociale is annotated by her on every page. Florence Nightingale believed — and in all the actions of her life acted upon that belief — that the administrator could only be successful if he were guided by statistical knowledge. The legislator — to say nothing of the politician — too often failed for want of this knowledge. Nay, she went further; she held that the universe — including human communities — was evolving in accordance with a divine plan; that it was man’s business to endeavour to understand this plan and guide his actions in sympathy with it. But to understand God’s thoughts, she held we must study statistics, for these are the measure of His purpose. Thus the study of statistics was for her a religious duty.
To state a theorem and then to show examples of it is literally to teach backwards.
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.
The mediocre teacher tells. The good teacher explains. The superior teacher demonstrates. The great teacher inspires.
Teach to the problems, not to the text.
I constantly meet people who are doubtful, generally without due reason, about their potential capacity [as mathematicians]. The first test is whether you got anything out of geometry. To have disliked or failed to get on with other [mathematical] subjects need mean nothing; much drill and drudgery is unavoidable before they can get started, and bad teaching can make them unintelligible even to a born mathematician.
Who has not be amazed to learn that the function y = ex, like a phoenix rising again from its own ashes, is its own derivative?
Whoever despises the high wisdom of mathematics nourishes himself on delusion and will never still the sophistic sciences whose only product is an eternal uproar.
He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.
The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.
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.
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.
The words ‘figure’ and ‘fictitious’ both derive from the same Latin root, fingere. Beware!
… 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.
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.
Perhaps the greatest paradox of all is that there are paradoxes in mathematics.
The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.
Descartes commanded the future from his study more than Napoleon from the throne.
To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad. [Observing that the volume generated by revolving the region under y = 1/x from 1 to infinity has finite volume.]
Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.
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.
I am interested in mathematics only as a creative art.
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.
I remember once going to see [Srinivasa Ramanujan] when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
— Godfrey H Hardy (English mathematician, 1877 – 1947)
[Srinivasa Ramanujan was an Indian mathematician, 1887 – 1920]
I’m very good at integral and differential calculus,
I know the scientific names of beings animalculous;
In short, in matters vegetable, animal, and mineral,
I am the very model of a modern Major-General.
[The Pirates of Penzance, Act 1]
God does arithmetic.
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.
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]
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?
When I am violently beset with temptations, or cannot rid myself of evil thoughts, [I resolve] to do some Arithmetic, or Geometry, or some other study, which necessarily engages all my thoughts, and unavoidably keeps them from wandering.
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.’
And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art …
The composer opens the cage door for arithmetic, the draftsman gives geometry its freedom.
I recognize the lion by his paw.
Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death.
[Thomas Hobbes] was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman’s library, Euclid’s Elements lay open, and “twas the 47 El. libri I” [Pythagoras’ Theorem]. He read the proposition . “By God,” sayd he, “this is impossible.” So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that trueth. This made him in love with geometry.
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.
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.
By studying the masters and not their pupils.
If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.
The reason that every major university maintains a department of mathematics is that it’s cheaper than institutionalising all those people.
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.
Gentlemen, eiπ + 1 = 0 is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be truth.
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.
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.
Great fleas have little fleas upon their backs to bit ‘em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on;
While these again have greater still, and greater still, and so on.
Constants aren’t, variables won’t.
Operator! Give me the number for 911!
Oh, people can come up with statistics to prove anything, Kent. 14% of people know that.
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.
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 …”
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.
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.
The great majority of people have more than the average number of legs.
Statistics means never having to say you’re certain.
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.
The only time my education was interrupted was when I was in school.
Bodily exercise, when compulsory, does no harm to the body; but knowledge which is acquired under compulsion obtains no hold on the mind .
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.
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.
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.
If only one lesson might be drawn from this analysis of talent, it would be that high-level mathematics departs radically from its popular portrayal as a dryly rational discipline, dominated by sheer deductive power, on which emotions have no bearing. Quite the contrary; the most potent of human emotions — love, hope, pain, or despair — hold sway over the relationship these mathematicians entertain with their number friends. When there is a passion for mathematics, talent does not lag very far behind. If, conversely, a child develops math anxiety, this phobia can prevent even the simplest of mathematical concepts from falling into place.
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.
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.
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.
“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]
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.
A person who can, within a year, solve x² – 92y² = 1 is a mathematician.
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.
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.
The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry.
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]
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.
The number you have dialled is imaginary. Please rotate your phone 90 degrees and try again.
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.
If a man’s wit be wandering, let him study the mathematics.
A man who is not somewhat of a poet can never be a mathematician.
The essence of all things is numbers.
Let no one ignorant of geometry enter here.
[Inscription at the entrance to his Academy in Athens]
With the help of God, and with His precious assistance, I say that algebra is a scientific art.
Reserve your right to think, for even to think wrongly is better than not to think at all.
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.
Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.
Mathematics is the queen of sciences and arithmetic is the queen of mathematics.
Mathematics is the pen with which God has written the universe.
Highness, there is no royal road to geometry.
[Euclid to Ptolemy I]
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.
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]
An adult gave a child a new coin.
‘What do you say?’ asked the adult.
The child replied, ‘One …’
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.
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.
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.”