"It is an important and popular fact that things are not always as what they seem. For instance, on the planet earth, man has always assumed that he was more intelligent than dolphins because he had achieved so much: the wheel, New York, wars. [...] But conversely, the dolphins had always believed that they were far more intelligent than men; for precisely the same reasons."
"Mathematics is less related to accounting than it is to philosophy."
"If you don't like your analyst, see your local algebraist!"
"Combinatorics is not a science, it's an attitude."
"You can only be truly accomplished at something you love. Don't make money
the goal. Instead, pursue the things you love doing, and do them so well that
people can't take their eyes off you. All the other tangible rewards will come
as a result."
"Probable impossibilities are to be preferred to improbable possibilities."
"It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out."
"Combinatorial analysis, in the trivial sense of manipulating binomial and multinomial coefficients, and formally expanding powers of infinite series by applications ad libitum and ad nauseamque of the multinomial theorem, represented the best that academic mathematics could do in the Germany of the late 18th century."
"In the broad light of day mathematicians check their equations and their proofs, leaving no stone
unturned in their search for rigour. But, at night, under the full moon, they dream, they float
among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there
is no art, no mathematics, no life."
"Spending time with math people is a lot of fun. As a result of the play, I've had semi-drunken dinners with mathematicians all over the country. I recommend the experience."
"Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium, maxime dicentes vera, cavendi sunt,
ne consortio daemoniorum animam deceptam, pacto quodam societatis irretiant."
"Z is complicated."
"The irreducible price of learning is realizing that you do not know."
"If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."
"It takes a village to count integer points."
"One and one and one is three."
"Obvious is the most dangerous word in mathematics."
"Mathematics is a collection of cheap tricks and dirty jokes."
"The origins of graph theory are humble, even frivolous."
"No WLAN [wi-fi]. Please just talk to each other and get drunk."
"Life is the twofold internal movement of composition and decomposition at
once general and continuous."
"There are two versions of math in the lives of many Americans: the strange and boring subject that they encountered in classrooms and an interesting set of ideas that is the math of the world, and is curiously different and surprisingly engaging. Our task is to introduce this second version to today's students, get them excited about math, and prepare them for the future."
"5 out of 4 people have trouble with fractions."
"Suppose that you want to teach the 'cat' concept to a very young child. Do you explain that a cat is a relatively
small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I'll bet not. You
probably show the kid a lot of different cats, saying 'kitty' each time, until it gets the idea. To put it more
generally, generalizations are best made by abstraction from experience."
"We all agree that your theory is crazy, but is it crazy enough?"
"My special pleasure in mathematics rested particularly on its purely speculative part."
"Mathematics as we know it and as it has come to shape modern science could never have come into being without some disregard for the dangers of the infinite."
"There are no problems, just pauses between ideas."
"There are three kinds of people: those who can count and those who can't."
"Abstraction consists essentially in the creation and utilization of ambiguity."
"Too many people are thinking of security instead of opportunity. They seem more afraid of life than death."
"The essence of mathematics resides in its freedom."
"Alice laughed: 'There's no use trying,' she said; 'one can't believe impossible things.'
'I daresay you haven't had much practice,' said the Queen. 'When I was younger, I always did it for
half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast.' "
"The peculiar interest of magic squares and all lusus numerorum in general lies in
the fact that they possess the charm of mystery. They appear to betray some hidden
intelligence which by a preconceived plan produces the impression of intentional design,
a phenomenon which finds its close analogue in nature."
"First, it is necessary to study the facts, to multiply the number of observations, and then later to search for
formulas that connect them so as thus to discern the particular laws governing a certain class of phenomena.
In general, it is not until after these particular laws have been established that one can expect to discover
and articulate the more general laws that complete theories by bringing a multitude of apparently very diverse
phenomena together under a single governing principle."
"To make a great dream come true, the first requirement is a great capacity to dream; the second is persistence."
"I'm a mathematical optimist: I deal only with positive integers."
"Everyone engaged in research must have had the experience of working with
feverish and prolonged intensity to write a paper which no one else will
read or to solve a problem which no one else thinks important and which
will bring no conceivable reward -- which may only confirm a general
opinion that the researcher is wasting his time on irrelevancies."
"A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty."
"11:15 Restate my assumptions:
"To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples..."
"I hate algebra."
"I have photographed many people: artists, writers, and scientists, among others. In speaking about their work, mathematicians use the words 'elegance', 'truth', and 'beauty' more than everyone else combined."
"The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday
life, out of numbers, those weapons of the gods: the gods are there, behind the wall, at play
"A mathematician is a blind man in a dark room looking for a black cat which isn't there."
"Je serais reconnaissant a toute personne ayant compris cette demonstration de me l'expliquer."
"Mathematicians care no more for logic than logicians for mathematics."
"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."
"The grand thing is to be able to reason backwards."
"A lot of mathematicians are a little bit strange in one way or another. It goes with creativity."
"Suppose that we think of the integers lined up like dominoes. The inductive step tells us that they are close enough for each domino to knock over the next one, the base case tells us that the first domino falls over, the conclusion is that they all fall over. The fault in this analogy is that it takes time for each domino to fall and so a domino which is a long way along the line won't fall over for a long time. Mathematical implication is outside time."
"I have not failed. I've just found 10,000 ways that won't work."
"The most beautiful experience is the mysterious. It is the source of true art and science."
"The secret to education is respecting the pupil."
"Analysis is the art of taming infinity."
"Education's purpose is to replace an empty mind with an open one."
"Spoon feeding, in the long run teaches us nothing but the shape of the spoon."
"Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them."
"In 1906, it was estimated that one in eight Americans bought a postcard every day."
"When I am working on a problem I never think about beauty. I only think about how to solve
the problem. But when I have finished, if the solution is not beautiful, I know it is wrong."
"When people tell you something's wrong or doesn't work for them, they are almost always right.
When they tell you exactly what they think is wrong and how to fix it, they are almost always
"Philosophy is written in this grand book--I mean the universe--which stands continually open to our gaze,
but it cannot be understood unless one first learns to comprehend the language and interpret the characters
in which it is written. It is written in the language of mathematics, and its characters are triangles,
circles, and other geometric figures, without which it is humanly impossible to understand a single word of it."
"When I give this talk to a physics audience, I remove the quotes from my 'Theorem'."
"The fact that we're humans is really pretty limiting. I mean, we didn't evolve to
understand the structure of the space-time continuum or do things in 100,000 dimensions. We
know how to stay out of the rain and keep [from] getting eaten by animals."
"Bart: Hey, Houdini! Why don't you saw Martin in half?
"Apu: In fact I can recite pi to 40000 places. The last digit is one!
"Homer: This time tomorrow, you'll be wearing high heels!
"Internet Guy: Your stock is at zero.
"(Homer has disappeared into a wall in the living room.)
"The shortest route between two truths in the real domain passes through the complex domain."
"The only way to learn mathematics is to do mathematics."
"Have you ever observed that we pay much more attention to a wise passage when it is quoted than when we read it in the original author?"
"The purpose of computation is insight, not numbers."
"Inspiration is constantly in the air. It's up to us to develop the sensitivity to pick up on it."
"One good thing about teaching calculus is that you develop a hardened attitude towards repeating yourself."
"I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations,' are simply the notes of our observations."
"The student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight."
"The whole purpose of education is to turn mirrors into windows."
"Es ist eine Tatsache, dass die genauere Kenntnis des Verhaltens einer analytischen Funktion in der Nähe ihrer singulären Stellen eine Quelle von arithmetischen Sätzen ist.
"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."
"I believe that numbers and functions of Analysis are not the arbitrary result of our minds; I think that they exist outside of us, with the same character of necessity as the things of objective reality, and we meet them or discover them, and study them, as do the physicists, the chemists and the zoologists."
"I wrote a few papers on Koszul algebras, but I really don't understand the definition of Koszul algebras."
"Mathematics is for lazy people."
"Die mathematische Forschung besteht aus 10% Intuition und 90% Arbeit."
"Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth."
"No education is politically neutral."
"Good judgement comes from experience. Experience comes from bad judgement."
"Here we will outline what is involved in the computations and list our answers. The details are available from the author on request. The author does hope someone will check the calculations, because he does not have a great deal of faith in his ability to carry through the details in a fault-free manner. He hopes however that the answers are qualitatively correct as stated."
"I do not believe in [writer's block]. My job is to write, not to like what I'm writing. That's why we edit, and nobody gets editor's block."
"For a list of all the ways technology has failed to improve the quality of life, please press three."
"Counting pairs is the oldest trick in combinatorics... Every time we count pairs, we learn something from it."
"Mathematics is an activity governed by the same rules imposed upon the symphonies of Beethoven, the paintings of DaVinci, and the poetry of Homer."
"Ubi materia, ibi geometria."
"If you had done something twice, you are likely to do it again."
"Mathematics is, in many ways, the most precious response that the human spirit has made to the call of the infinite."
"A good talk contains no proofs; a great talk contains no theorems."
"The worst thing you can do to a problem is solve it completely."
"Science is what we understand well enough to explain to a computer, Art is all the rest."
"In L_infinity there is a lot of room for shortest vectors."
"The greatest reward for a student is not a good grade. It is the willingness of his teacher to listen to him."
"Nein! Wir Sind Dichter."
"The imaginary number is a fine and wonderful resource of the human spirit, almost an amphibian between being and not being."
"Every dimension is special."
"Nature laughs at the difficulties of integration."
"I don't know where my ideas come from. I will admit, however, that one key ingredient is caffeine. I get a couple cups of coffee into me and weird things just start to happen."
"Music is part of Number Theory. Nowadays when a number-theorist applies for a grant, he says that it is good for security, but in those days, way before America, he would say that it's good for music. I will not comment whether we have progressed..."
"Es ist unglaublich, wie unwissend die studirende Jugend auf Universitäten kommt, wenn ich nur 10 Minuten rechne oder geometrisire, so schläft 1/4 derselben sanfft ein."
"A heavy warning used to be given that pictures are not rigorous; this has never had its bluff called and has permanently frightened its victims into playing for safety."
"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood."
"Education is the most powerful weapon which you can use to change the world."
"Der geistige Mensch hat die Wahl (soweit er die Wahl hat), entweder Ironiker oder Radikalist zu sein; ein Drittes ist anständigerweise nicht möglich."
"Math is the only place where truth and beauty mean the same thing."
"For every problem, there is one solution which is simple, neat, and wrong."
"The [...] thing we evolutionary biologists don't do enough of [...] is make an emotional and moral case for the study of evolution. Last night, I concluded my talk with a quote from Dover, PA creationist school board member William Buckingham, who declared, 'Two thousand years ago someone died on a cross. Can't someone take a stand for him?' My response was, 'In the last two minutes, someone died from a bacterial infection. We take a stand for him.'"
"Mathematical study and research are very suggestive of mountaineering. Whymper made several efforts before he
climbed the Matterhorn in the 1860's and even then it cost the life of four of his party. Now, however, any tourist can be
hauled up for a small cost, and perhaps does not appreciate the difficulty of the original ascent. So in mathematics, it
may be found hard to realise the great initial difficulty of making a little step which now seems so natural and obvious,
and it may not be surprising if such a step has been found and lost again."
"But as Nature is the best guide, teaching must be the development of natural inclinations, for which purpose the teacher must watch his pupil and listen to him, not continually bawl words into his ears as if pouring water into a funnel. Good teaching will come from a mind well made rather than well filled."
"Young bloods can't spell but they can rock you in PlayStation this new math it whippin m&8%!2@*^#ers ass you wanna know how to rhyme you better learn how to add it's mathematics."
"Sometimes a good idea comes to you when you are not looking for it. Through
an improbable combination of coincidence, naivete and lucky mistakes ..."
"The world is continuous, but the mind is discrete."
"In mathematics you don't understand things. You just get used to them."
"Everyone else would climb a peak by looking for a path somewhere in the mountain. Nash would climb another
mountain altogether and from that distant peak would shine a searchlight back onto the first peak."
"If I have been able to see further, it was only because I stood on the shoulders of giants."
"Critique the effectiveness of your lesson, not by what answers students give, but by what questions they ask."
"And those who were seen dancing were thought to be insane by those who could not hear the music."
"Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth."
"This isn't right, this isn't even wrong."
"Mathematicians are a bit like the laconic Vermonter who, when asked if he's lived in the state his whole life, replies, "Not yet."
"... and it is probably that there is some secret here which remains to be
"Leo, do you have a religion? You know, a religion, like Jewish, or Christian, or Mathematics...?"
"Computers are useless. They can only give you answers."
"Let no-one ignorant of geometry enter."
"Mathematics is the art of giving the same name to different things."
"Geometry is the science of correct reasoning on incorrect figures."
"Theorems are fun especially when you are the prover, but then the pleasure fades.
What keeps us going are the unsolved problems."
"Where a calculator like the ENIAC today is equipped with 18,000 vacuum tubes
and weighs 30 tons, computers in the future may have only 1,000 vacuum tubes
and perhaps weigh only 1 1/2 tons."
"MacPherson told me that my theorem can be viewed as blah blah blah Grothendick blah blah blah, which makes it much more respectable."
"Quite frankly, teachers are the only profession that teach our children."
"Cambridge gave me various kinds of teaching and
supervision, but possibly the most important influences were Jeffrey
Goldstone and Peter Swinnerton-Dyer, who encouraged me to continue to
think for myself and not to take the technical details too seriously."
"Mathematics is persistent intellectual honesty."
"It is a pain to think about convergence but sometimes you really have to."
"I like to think of mathematicians as forming a nation of our own without distinctions of geographical origin, race, creed, sex, age or even time... all dedicated to the most beautiful of the arts and sciences."
"God created infinity, and man, unable to understand infinity, created finite sets."
"It is because modern education is so seldom inspired by a great hope that it so seldom achieves a great result. The wish to preserve the past rather than the hope of creating the future dominates the minds of those who control the teaching of the youth."
"We haven't the money, so we have to think."
"If you want to build a ship, don't drum up people to collect wood and don't assign them tasks and work, but rather teach them to long for the endless immensity of the sea."
"All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident."
"The problem is that we tend to live among the set of puny integers and generally ignore the vast infinitude of larger ones. How trite and limiting our view!"
"A felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem."
"It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathematicians have a good ear; some not (and some prefer the slangy expressions such as 'iff'). That's life."
"We learned to be happy.
We danced 'round the hall.
And learning to count was the key to it all."
"We've taught you that the earth is round,
That red and white make pink,
And something else that matters more --
We've taught you how to think."
"Though this be madness, yet there is method in it."
"log log log x goes to infinity with great dignity."
"He [Taniyama] was gifted with the special capability of making many mistakes, mostly in the right direction. I envied
him for this and tried to imitate him, but found it quite difficult to make good mistakes."
"Serious numbers will speak to us always."
"In theory, there is no difference between theory and practice. But, in practice, there is."
"The very term 'combinatorial methods' has an oxymoronic character."
"Mathematics is the science of patterns, and nature exploits just about every pattern that there is."
"1*1 = 1, unzweifelhaft. Aber 1*2 ist nicht 1, weil das Quadrat einer gegebenen Zahl grösser sein mss als die Zahl selbst. Die Wurzel aus 1 kann logischerweise nicht 1 sein, weil die Wurzel aus einer Zahl kleiner sein muss als die Zahl selbst. Aber mathematisch oder formal ist sqrt(1)=1. Die Mathematik widerspricht in diesem Falle der Logik oder der reinen Vernunft, und darum ist die Mathematik in diesem Kardinalfalle vernunftwidrig. Auf dieser Sinnlosigkeit, der 1, bauen sich dann alle Werte auf, und in diesen falschen Werten fusst die mathematische Wissenschaft, die 'einzig exakte, unfehlbare'. Aber dies ist Mathematik! Ein artiges Spiel für Leute, die nichts zu tun haben."
"If you think it's simple, then you have misunderstood the problem."
"The only reason that we like complex numbers is that we don't like real numbers."
"We value what we measure because we do not know how to measure what we really value."
"The purely formal language of geometry describes adequately the reality of space. We might say, in this sense, that geometry is successful magic. I should like to state a converse: is not all magic, to the extent that it is successful, geometry?"
"Get your facts first, and then you can distort them as much as you please."
"A beautiful problem is like a funny joke."
"Mathematics is an escape from reality."
"Phyllis explained to him, trying to give of her deeper self, 'Don't you find it so beautiful, math? Like an endless sheet of gold chains, each link locked into the one before it, the theorems and functions, one thing making the next inevitable. It's music, hanging there in the middle of space, meaning nothing but itself, and so moving...'"
"A lack of seriousness has led to all sorts of wonderful insights."
"A mathematician who is not also something of a poet will never be a complete mathematician."
"Every mathematician worthy of the name has experienced, if only rarely, the state of lucid
exaltation in which one thought succeeds another as if miraculously, and in which the unconscious
(however one interprets that word) seems to play a role."
"That's all well and good in practice, but how does it work in theory?"
"My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful."
"We think in generalities, but we live in details."
"Life is too important to be taken seriously."
"I never use a computer."
"A generating function is a clothesline on which we hang up a sequence of numbers for display."
"1/r^2 has a nasty singularity at r=0, but it did not bother Newton--the Moon is far enough."
"All means (even continuous) sanctify the discrete end." (email on Wed, 14 Aug 2002 16:29:35 -0400 (EDT))