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This was supplied by the mean value theorem; and it was Cauchy's great service to have recognized its fundamental importance… Because of this, we adjudge Cauchy as the founder of exact infinitesimal calculus.
F. Klein, quoted in Analysis by Its History by E. Hairer and G. Wanner.	1117

[Mathematical] analysis is the art of taming infinity.
Neil Falkner, from August-September, 2009 American Mathematical Monthly, p. 658.	1261

Inner city education must change. Our responsibility is not merely to provide access to knowledge; we must produce educated people.
James Farmer, in Today's Education, April 1969.	117

Mathematics serves as a handmaiden for the explanation of the quantitative situations in other subjects, such as economics, physics, navigation, finance, biology and even the arts.
H. F. Fehr, quoted in Out of the Mouths of Mathematicians, By R. Schmalz.	118

On the other hand, it is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as a sum of two fourth powers or, in general for any number which is a power greater than the second to be written as a sum of two like powers. For this I have discovered a truly wonderful proof, but the margin is too small to contain it.
P. Fermat, quoted in Excursions in Calculus, by Robert Young.	119

How many great theorems can be discovered?
Jessica Ferris, Westfield State College student	1131

From a long view of the history of mankind - seen from, say, the thousand years from now - there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale in provincial insignificance in comparison with this important scientific event of the same decade.
Richard Feynman, quoted in Calculus with Analytic Geometry by George F. Simmons	859

I learned from her [my mother] that the highest forms of understanding we can achieve are laughter and human compassion.
Richard Feynman, What Do You Care What Other People Think?	925

I was born not knowing and have only had a little time to change that here and there.
Richard Feynman,	851

Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry.
Richard Feynman,	872

By that experience [what determines a person's internal time clock for counting] Tukey and I discovered that what goes on in different people's heads when they think they're doing the same thing - something as simple as counting - is different for different people… I often think about that, especially when I'm teaching some esoteric technique such as integrating Bessel functions. When I see equations, I see the letters in colors - I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-blueish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students.
Richard Feynman, from The Pleasure of Finding Things Out, pp. 222-3.	1599

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.
Richard Feynman,	1602

What I cannot create, I do not understand.
Richard Feynman,	1606

I don't know what's the matter with people: they don't learn by understanding, they learn by some other way - by rote or something. Their knowledge is so fragile!
Richard Feynman,	1607

You can know the name of a bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bird... So let's look at the bird and see what it's doing -- that's what counts. I learned very early the difference between knowing the name of something and knowing something.
Richard Feynman,	850

For a successful technology, reality must take precedence over public relations, for nature cannot be fooled.
Richard Feynman, concluding sentence from his "Report to the Space Shuttle Challenger Inquiry"	1600

You say you are a nameless man. You are not to your wife and to your child. You will not long remain so to your immediate colleagues if you can answer their simple questions when they come into your office. You are not nameless to me. Do not remain nameless to yourself - it is too sad a way to be. Know your place in the world and evaluate yourself fairly, not in terms of the naive ideals of your own youth, nor in terms of what you erroneously imagine your teacher's ideals are.
Richard Feynman,	1604

I took this stuff that I got out of your seal and I put it in ice water, and I discovered that when you put some pressure on it for a while and then undo it, it doesn't stretch back. It stays the same dimension. In other words, for a few seconds at least and more seconds than that, there is no resilience in this particular material when it is at a temperature of 32 degrees. I believe that has some significance to our problem.
Richard Feynman,	1601

If I wish to know something, my first impulse does not lead me to seek the knowledge on my own. It leads me to seek out someone who knows it and ask him to tell it to me. If I am ignorant to begin with, how else can I find the knowledge I seek? … This view of knowledge is promulgated by almost every culture, because it gives the culture, which has the power to designate who the Authorities are, control over what counts as knowledge… But this view of knowledge has a serious flaw in it, which reveals itself as soon as we ask how the Authority got his knowledge. We will be told, of course, that he got it from another Authority, someone he consulted when he was ignorant. But what about that Authority? We have here an infinite regress, a chain that avoids our question by constantly creating new links. There are only two possible answers to the question: Either the first link in the chain is an omniscient being whose knowledge can't be questioned (i.e., a god), or he got his knowledge from an entirely different process than being told… If the first link in the chain acquired the knowledge in question by "some other process," then it is the process of attaining knowledge, and not the person who acquired it, that makes the knowledge legitimate. We call this process 'inquiry.' The chief thrust of the Scientific Revolution was its development of a kind of inquiry that was reliable and produced spectacular results, a process that has come to be called the 'scientific method.' …In doing so scientists, mathematicians, and philosophers of the 17th century stood on the shoulders of a powerful ally from the 5th century B.C., the Socrates who… insisted that Men continue searching, relying on no authority other than his own mind.
Donald L. Finkel, from Teaching with Your Mouth Shut, pp. 34-5.	1735

Dewey distinguishes between intelligence and thinking. We always manifest our intelligence; else, failing to meet our basic needs, we would not survive. But we think only when some obstacle arises, preventing the satisfaction of our needs and throwing us into disequilibrium. In other words, intelligence organizes our experience, while thinking is our means for reflecting on that experience - in order to modify our intelligence through a chance of habits.
Donald L. Finkel, from Teaching with Your Mouth Shut, pp. 151.	1687

The journey for an education starts with a childhood question.
David L. Finn,	560

Unc, why do they call them stars when they are really round?
K. C. Fisher, at age 3	121

The effort of the economist is to 'see,' to picture the interplay of economic elements. The more clearly cut these elements appear in his vision, the better; the more elements he can grasp and hold in his mind at once, the better. The economic world is a misty region. The first explorers used unaided vision. Mathematics is the lantern by which what before was dimly visible now looms up in firm bold outlines. The old phantasmagoria disappear. We see better. We also see further.
Irving Fisher, quoted in The World of Mathematics, by J.R. Newman.	122

Read in order to live.
Gustave Flaubert,	997

'Believe' is what we think we need to do when our existential understanding [of anything] has to be supplemented to fill in the spaces of our ignorance. As long as we remember that, it’s not so dangerous. Those who think that 'believing' is knowing, or is a substitute for knowing (or learning), or takes priority over knowing are dangerous.
Lou Jean Fleron, personal communication, 3/8/2011.	1690

With one trillion dollars [one-fifth of the 1996 U.S. National debt], you could buy... a $100,000 house for every family in Kansas, Missouri, Nebraska, Oklahoma, and Iowa AND then you could put a $10,000 car in the garage of each one of those houses AND there would be enough left to build a $10 million library and a $10 million hospital for each of 250 cities in those states AND there would be enough left to build a $10 million dollar school in each of 500 communities in those states AND there would still be enough left to put in the bank, and from the interest alone, pay 10,000 nurses and teachers [indefinitely], plus give a $5,000 bonus to every family in those states.
Lou Jean Fleron,	120

Objectivity is the delusion that observations could be made without an observer.
Heinz von Foerster, Quoted in Radical Constructivism: A Way of Knowing and Learning by Ernst von Glasersfeld.	772

Mathematicians are like lovers... Grant a mathematician the least principle, and he will draw from it a consequence which you must grant him also, and from this consequence another.
Fontenelle,	123

A great obstacle to happiness is to expect too much happiness.
Bernard de Fontenelle,	1487

There are three things I have loved but never understood. Art, music and women.
Bernard de Fontenelle,	1488

Behold a Universe so immense that I am lost in it. I no longer know where I am. I am just nothing at all. Our world is terrifying in its insignificance.
Bernard de Fontenelle,	1489

I learned certain things at school that the teachers would say, 'now here's what we're going to learn. You'll learn it, and then I'll tell what you've learned.' Then I'll get out, and I'll be arguing about this point I learned in history, and I realize I know the facts, but I don't have the foggiest idea what I'm talking about. I can tell you all the dates, but I don't know why it's in my head. I don't know why I think it or how I ever reached those conclusions because they don't really go with my ideas at all.
Weinstein Fontini, from Toward Humanist Education.	396

Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.
Joseph Fourier, quoted in Single Variable Calculus, by James Stewart.	124

The profound study of nature is the most fertile source of mathematical discovery.
Joseph Fourier, quoted in The Magic of Mathematics, by Theoni Pappas.	491

The deep study of nature is the most fruitful source of mathematical discoveries.
Jean-Baptist-Joseph Fourier, quoted in Calculus and Analytic Geometry, by Philip Gillett.	551

The conquest of the actual infinite by means of set theory can be regarded as an extension to our scientific horizon whose importance is comparable to the importance of the Copernican system in astronomy and the theory of relativity and quantum mechanics in physics.
A. Fraenkel, quoted in In Search of Infinity by N. Ya Vilenkin	1491

How wonderful it is that nobody need wait a single moment before starting to improve the world.
Anne Frank,	1363

I don't think of all the misery but of the beauty that still remains.
Anne Frank, quoted in Wisdom for the New Millennium edited by Helen Exley.	1066

What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of mathematics?
Benjamin Franklin,	125

One today is worth two tomorrows.
Benjamin Franklin,	934

Games lubricate the body and the mind.
Benjamin Franklin,	1252

People who are willing to give up freedom for the sake of short term security deserve neither freedom nor security.
Benjamin Franklin,	926

We have already pointed out and will recognize throughout this book the importance of compact sets. All those concerned with general analysis have seen that it is impossible to do without them.
Frechet, quoted in Analysis by Its History by E. Hairer and G. Wanner.	1124

I have here a geometer who is a big Cyclops…who has only one eye left, and a new curve, which he is presently computing, could render him totally blind.
Frederick II, quoted in Analysis by Its History by E. Hairer and G. Wanner.	1110

A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. In this position I was put by a letter from Mr. Bertrand Russell as the work was nearly through the press.
Gottlob Frege,	534

In the small matters trust the mind, in the large ones the heart…
Sigmund Freud, from Words I Wish I Wrote by Robert Fulgham	953

Perhaps much of the magic - the mystery - lies in this: a good photograph convinces us that it does indeed render everything; we would not know how to add to its completeness. And yet half a dozen photographs made in a given room at a given hour will each describe a different reality. None renders everything, and the best are those that describe without evasion or vagueness a single, coherent perception: not everything, but the precise intuition of a truth. Photography has enabled us to explore the richness of that swarming, shifting, four-dimensional continuum that we choose to call reality. It has shown us that what is 'out there' is not a catalogue of discrete facts and Ptolemaic measurements but a sea of changing relationships.
Lee Friedlander, from E.J. Bellocq: Storyville Portraits, p. 12, Museum of Modern Art (New York), 1970	1804

Man's main task is to give birth to himself.
Eric Fromm,	918

The best way out is through.
Robert Frost, quoted in Wisdom for the New Millennium edited by Helen Exley.	1073

I am not a teacher, I am an awakener.
Robert Frost,	126

Education is the ability to listen to almost anything without losing your temper or your self confidence.
Robert Frost,	922

Using mathematics to tell stories and using stories to explain mathematics are two sides of the same coin. They join what should never have separated: the scientist's and the artist's ways of uncovering truths about the world.
William Frucht, from the Introduction to Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems, and Musings	1045

I'm not often aware that I am happy. But I often remember that I have been happy. Especially when I sit in my kitchen wrapped in an invisible patchwork quilt made of the best moments of yesterdays. These precious things -- these leftovers from living on -- remain to serve as survival rations for the heart and soul. You can't entirely live off them. But life is not worth living without them.
Robert Fulgham, from Uh-Oh.	920

If you don't play the game, you can't know enough to make the rules. If you are not engaged in the sweaty work of the world, you should not be in charge of the deodorant concession.
Robert Fulgham, from Words I Wish I Wrote	943

It has been said that the hottest places in Hell are reserved for those who, in times of moral crisis, maintain their neutrality. And I say the moral crisis of the times is continuous. Knowing and understanding and being are not enough. One must do. To gain the world and give nothing to it is to lose you soul.
Robert Fulgham, from Words I Wish I Wrote	944

But nothing's better for you [then old, leftover meatloaf]. It's a matter of mental health. I've never heard anybody say he was depressed by eating a cold meatloaf sandwich.
Robert Fulgham, from Uh-Oh	921

I have great hope for tomorrow. My hope lies in three things: truth, youth, and love.
Buckminister Fuller,	832

Wherefore I realized that All the words in all dictionaries Are the consequent tools Of all men"s conscious And conscientious attempts To communicate All their experiences-- Which is of course To communicate Universe.
Buckminster Fuller, from "How little I know", excerpted in The Art of Science Writing by D. Worsley and B. Mayer.	742

There is nothing in a caterpillar that tells you it's going to be a butterfly.
R. Buckminster Fuller,	1769

When I am working on a problem, I never think about beauty. I only think of how to solve the problem. But when I am finished, if the solution is not beautiful, I know it is wrong.
Buckminster Fuller,	127

60 quotes found and displayed.