The merit of painting lies in the exactness of reproduction. Painting is a science and all sciences are based on mathematics. No human inquiry can be a science unless it pursues its path through mathematical exposition and demonstration.
Leonardo da Vinci,
59
 
He who does not understand the supreme certainty of mathematics is wallowing in confusion.
Leonardo da Vinci,
1744
 
No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof.
Leonardo da Vinci, quoted in Concepts of Mathematical Modeling by Walter J. Meyer.
793
 
You should look at certain walls stained with damp, or at stones of uneven color. If you have to invent some backgrounds you will be able to see in these the likeness of divine landscapes, adorned with mountains, ruins, rocks, woods, great plains, hills and valleys in great variety; and expressions of faces and clothes and an infinity of things which you will be able to reduce to their complete and proper forms. In such walls the same thing happens as in the sound of bells, in whose stroke you may find every named word which you can imagine.
Leonardo da Vinci,
1803
 
Mechanics is the paradise of mathematical science because here we come to the fruits of mathematics.
Leonardo da Vinci, quoted in The Magic of Mathematics, by Theoni Pappas.
497
 
Algebra is generous; she often gives more than is asked of her.
D'Alembert, quoted in A History of Mathematics, by Carl Boyer.
58
 
The hottest places in hell are reserved for those who, in a time of great moral crisis, maintain their neutrality.
Dante, quoted in Harper's Quotations.
60
 
This easy-to-state example [how to best assign 70 men to 70 different jobs] illustrates why up to 1947, and for the most part even to this day, a great gulf exists between man's aspirations and his actions. Man may wish to state his wants in complex situations in terms of a general objective to be optimized but there are so many different ways to go about it, each with its advantages and disadvantages, that it would be impossible to compare all the cases and choose which among them would be the best. Invariably, man in the past has turned to a leader whose 'experiences' and 'mature judgment' would guide the way.
George Bernard Dantzig, from "Linear Programming" in History of Mathematical Programming: A Collection of Personal Reminiscences, edited by J.K. Lenstra, A.H.G. Rinnooy Kan and A. Schrijver, Elsevier, 1991.
1247
 
Linear programming can be viewed as part of a great revolutionary development which has given mankind the ability to state general goals and to lay out a path of detailed decisions to take in order to "best" achieve its goals when faced with practical situations of great complexity.
George Bernard Dantzig, from "Linear Programming" in History of Mathematical Programming: A Collection of Personal Reminiscences, edited by J.K. Lenstra, A.H.G. Rinnooy Kan and A. Schrijver, Elsevier, 1991.
1246
 
They [the mathematicians of the Enlightenment] defined their terms vaguely and used their methods loosely, and the logic of their arguments was made to fit the dictates of their intuition. In short, they broke all the laws of rigor and of mathematical decorum.
The veritable orgy which followed the introduction of the infinitesimals... was but a natural reaction. Intuition had too long been held imprisoned by the severe rigor of the Greeks. Now it broke loose, and there were no Euclids to keep its romantic flight in check.
Tobias Dantzig, quoted in Calculus: A Liberal Art by W.M. Priestley.
724
 
By one of those insights of which only the greatest minds are capable, the famous geometer [Riemann] generalizes the concept of the definite integral…
Darboux, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1116
 
Every new body of discovery is mathematical in form, because there is no other guidance we can have.
Charles Darwin,
62
 
In October 1838, I happened to read for amusement 'Malthus on Population,' and being very well prepared to appreciate the struggle for existence which every where goes on, from long continued observations of the habits of animals and plants, it at once struck me that under these circumstances favorable variations would tend to be preserved, and unfavorable ones to be destroyed. The result would be the formulation of a new species. Here then I had a theory by which to work.
Charles Darwin, from The Autobiography of Charles Darwin, Harcourt Brace, 1959.
61
 
A mathematician is a blind man in a dark room looking for a black cat which isn't there.
Charles Darwin, quoted in "A Quote a Day Educates" by Monte Zerger, Mathematical Intelligencer, vol. 20, no. 2, Spring 1998.
682
 
Mathematical modeling of natural phenomena is hardly new. Nevertheless, advances in numerical analysis and the development of the computer have made it possible in ways that are much more complex and more realistic than ever before. Mathematical modeling in partnership with the computer is rapidly becoming a third element of the scientific method, coequal with more traditional elements of theory and experiment.
E.E. David, quoted in "Mathematics: an integral part of our culture" by Harald M. Ness, Jr., from Essays in Humanistic Mathematics.
504
 
Modern man is a prisoner who thinks he is free because he refrains from touching the walls of his dungeon.
Nicolas Gomez Davila,
1720
 
One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.
P.J. Davis, quoted in "The Role of Paradoxes in the Evolution of Mathematics", by I. Kleiner and N. Movshovitz-Hadar, American Mathematical Monthly, vol. 101, no. 10, December 1994.
553
 
The ideal mathematician's work is intelligible only to a small group of specialists, numbering a few dozen or at most a few hundred. This group has existed only for a few decades and there is every possibility that it may become extinct in another few decades. However, the mathematician regards his work as part of the very structure of the world, containing truths that are valid forever, from the beginning of time, even in the most remote corner of the universe.
Philip Davis and Reuben Hersh, from The Mathematical Experience.
64
 
Mathematics, in one view, is the science of infinity.
Philip Davis and Reuben Hersh, from The Mathematical Experience.
63
 
My first reaction was, "Wonderful! How did they do it [prove the four-color theorem]?" I expected some brilliant new insight, a proof which had in its kernel an idea whose beauty would transform my day. But when I received the answer, "They did it by breaking it down into thousands of cases, and then running them all on the computer, one after another," I felt disheartened. My reaction was, "So it just goes to show, it wasn't a good problem after all."
P.J. Davis and R. Hersh, quoted in From Here to Infinity, by Ian Stewart.
673
 
Besides, I acknowledge that I owe very much to the bright minds of the Bernoulli brothers, especially to the young one presently Professor in Groningen. I have made free use of their discoveries...
Marquis de L'Hospital, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1118
 
The extent of this calculus is immense: it applies to curves both mechanical and geometrical; radical signs cause it no difficulty, and even are often convenient; it extends to as many variables as one wishes; the comparison of infinitely small quantities of all sorts is easy. And it gives rise to an infinity of surprising discoveries concerning curved or straight tangents, questions De maximis & minimis, inflexion points and cusps of curves, envelopes, caustics from reflexion or refraction, &c. as we shall see in this work.
Marquis de L'Hospital, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1107
 
My trade and my art is living.
Michel Eyquem de Montaigne, quoted in Uh-Oh by Robert Fulgham
919
 
My opinion of mankind is founded upon the mournful fact that, so far as I can see, they find within themselves the means of believing in a thousand times as much as there is to believe in.
Augustus De Morgan, A Budget of Paradoxes, vol. 1, p. 115, quoted in The Trisectors by Underwood Dudley, p. vii.
1710
 
Common integration is only the memory of differentiation... The different artifices by which integration is effected, are changes, not from the known to the unknown, but from forms in which memory does not serve us to those in which it does.
A. De Morgan, quoted in Memorabilia Mathematica, by R.E. Moritz.
68
 
How glorious it is, and also how painful, to be an exception.
Alfred de Musset,
923
 
Music is the arithmetic of sounds as optics is the geometry of light.
Claude Debussy, from A Dictionary of Quotations in Mathematics by Nowlan
1667
 
Numbers are the free creation of the human mind.
Richard Dedekind, quoted in Sweet Reason by J. Henle and T. Tymoczko.
65
 
The splendid creations of this theory [complex function theory] have excited the admiration of mathematicians mainly because they have enriched our science in an almost unparalleled way with an abundance of new ideas and opened up heretofore wholly unknown fields to research. The Cauchy integral formula, the Riemann mapping theorem and the Weierstrass power series calculus not only laid the groundwork for a new branch of mathematics but at the same time they furnished the first and until now the most fruitful example of the intimate connections between analysis and algebra. But it isn't just the wealth of novel ideas and discoveries which the new theory furnishes; of equal importance on the other hand are the boldness and profundity of the methods by which the greatest difficulties are overcome and the most recondite of truths, the mysteria functiorum, are exposed to the brightest light.
Richard Dedekind, quoted in Theory of Complex Functions, by Reinhold Remmert.
66
 
I see it, but I don't believe it.
[On Cantor's proof that the points in the unit interval were in one-to-one correspondence with points in the unit square.]
Richard Dedekind,
564
 
At its heart, music is all higher mathematics.
Mos Def, from Rolling Stone, 25 June, 2009.
1314
 
The poor, no less than the rich, stay tuned in to the Dream Machine in bad times as well as good....By 1995, millions of the poor were left without housing, medical care; jobs, or educational opportunity; six million children--one of every four kids under 6 years of age in America--were officially poor. Mired in Third-World conditions of poverty while video-bombarded with First-World dreams, rarely has a population suffered a greater gap between socially cultivated appetites and socially available opportunities.
Charles Derber, from The Wilding of America.
769
 
In order to reach the Truth, it is necessary, once in one's life, to put everything in doubt -- so far as possible.
Rene Descartes, quoted in Mathematics and the Imagination by Edward Kasner and James Newman.
690
 
I believe that I can now give an account of [atmospheric phenomena], and I have decided to write a small treatise that will include an explanation of the cause of the rainbow, the matter that has given me the greatest difficulty.
Rene Descartes, quoted in The Rainbow Bridge, p. 182.
1674
 
The rainbow is such a remarkable phenomenon of nature, and its cause has been so meticulously sought after by inquiring minds throughout the ages, that I could not choose a more appropriate subject for demonstrating how, with the method I am using, we can arrive at knowledge not possessed at all by those whose writings are available to us.
Rene Descartes, attributed in The Rainbow Bridge (p. 182) to De l'arc-en-ciel from Discours de la methode.
1673
 
When I consider this carefully, I find not a single property which with certainty separates the waking state from the dream. How can you be certain that your whole life is not a dream?
Rene Descartes,
1672
 
Human wisdom remains always one and the same although applied to the most diverse objects and it is no more changed by their diversity than the sunshine is changed by the variety of objects which it illuminates.
Rene Descartes, attributed to Rule I Oeuvres, vol. X, p. 360 in Polya's Mathematical Discovery, vol. 1, p. 115.
1300
 
We call infinite that thing whose limits we have not perceived, and so by that word we do not signify what we understand about a thing, but rather what we do not understand.
Rene Descartes, quoted in "Torricelli's Infinitely Long Solid and Its Philosophical Reception in the Seventeenth Century," by P. Mancosu and E. Vailati, Isis, vol. 82, 1991.
675
 
Neither the true nor the false roots are always real; sometimes they are imaginary; that is, while we can always imagine as many roots for each equation as I have assigned, yet there is not always a definite quantity corresponding to each root we have imagined.
Rene Descartes, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1104
 
The reading of all good books is like a conversation with the finest people of past centuries.
Rene Descartes,
825
 
Here I beg you to observe in passing that the scruples that prevented ancient writers from using arithmetical terms in geometry, and which can only be a consequence of their inability to perceive clearly the relation between these two subjects, introduced much obscurity and confusion into their explanations.
Descartes, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1100
 
Cogito, ergo sum. [I think, therefore I am.]
Rene Descartes,
69
 
Although my knowledge grows more and more, nevertheless I do not for that reason believe that it can ever be actually infinite, since it can never reach a point so high that it will be unable to attain any greater increase.
Rene Descartes,
70
 
Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.
Rene Descartes,
824
 
And I dare say that this [the determination of the tangent line] is not only the most useful and most general problems in geometry that I know, but even that I ever desired to know.
Descartes, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1108
 
It can be argued that the mathematics behind these images [of the orbit diagram for quadratic functions and the Mandelbrot set] is even prettier than the pictures themselves.
Robert L. Devaney, from "The Orbit Diagram and the Mandelbrot Set," College Mathematics Journal, Vol. 22, no. 1, Jan. 1991.
559
 
It has been said that the three great develpments in twentieth century science are relativity, quantum mechanics, and chaos. That strikes me the same as saying that the three great developments in twentith century engineering are the airplane, the computer, and the pop-top aluminum can. Chaos and fractals are not even twentieth century ideas: chaos was first observed by Poincare and fractals were familiar to Cantor a century ago, although neither man had the computer at his disposal to show the rest of the world the beauty he was seeing.
Robert L. Devaney, from "Introduction: Special Issue on Dynamical Systems," College Mathematics Journal, Vol. 22, no. 1, Jan. 1991.
558
 
Modern dynamical systems theory has a relatively short history. It begins with Poincare (of course)... [to whom] a global understanding of the gross behavior of all solutions of the system was more important than the local behavior of particular, analytically-precise solutions.
R.L. Devaney, from An Introduction to Chaotic Dynamical Systems.
463
 
The linear-programming was -- and is -- perhaps the single most important real-life problem.
Keith Devlin, from Mathematics: The New Golden Age.
605
 
Indeed, nowadays no electrical engineer could get along without complex numbers, and neither could anyone working in aerodynamics or fluid dynamics.
Keith Devlin, from Mathematics: The New Golden Age.
586
 
Given the brief -- and generally misleading -- exposure most people have to mathematics at school, raising the public awareness of mathematics will always be an uphill battle. But if you believe, as I do, that one of the main reasons why our country's schoolchildren consistently perform poorly in international comparisons of mathematical ability is the attitude toward mathematics they pick up from society, then it"s a battle we should engage in.
Keith Devlin, from "What's Going On During Mathematics Awareness Month", April 1999 column on MAA Online.
788
 
Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent to a pair of ears. Mathematics can only be "seen" with the "eyes of the mind". It is as if we had no sense of hearing, so that only someone able to sight-read music would be able to appreciate its patterns and harmonies.
Keith Devlin, from Mathematics: the Science of Patterns.
538
 
The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers.
Keith Devlin, from Mathematics: The New Golden Age
73
 
There can be very little of present-day science and technology that is not dependent on complex numbers in one way or another.
Keith Devlin, from Mathematics: The New Golden Age.
72
 
Sure, some [teachers] could give the standard limit definitions, but they [the students] clearly did not understand the definitions - and it would be a remarkable student who did, since it took mathematicians a couple of thousand years to sort out the notion of a limit, and I think most of us who call ourselves professional mathematicians really only understand it when we start to teach the stuff, either in graduate school or beyond.
Keith Devlin, from "The Calculus Ultrafilter," Focus, Dec. 1994.
71
 
The increased abstraction in mathematics that took place during the early part of this century was paralleled by a similar trend in the arts. In both cases, the increased level of abstraction demands greater effort on the part of anyone who wants to understand the work.
Keith Devlin, from Mathematics: the Science of Patterns.
540
 
Calculus works by making visible the infinitesimally small.
Keith Devlin, from The Math Gene: How Mathematical Thinking Evolved and Why Numbers are Like Gossip.
898
 
What is mathematics? Ask this question of person chosen at random, and you are likely to receive the answer "Mathematics is the study of number." With a bit of prodding as to what kind of study they mean, you may be able to induce them to come up with the description "the science of numbers." But that is about as far as you will get. And with that you will have obtained a description of mathematics that ceased to be accurate some two and a half thousand years ago!
Keith Devlin, from Mathematics: The Science of Patterns
1177
 
In fact, the answer to the question "What is mathematics?" has changed several times during the course of history... It was only in the last twenty years or so that a definition of mathematics emerged on which most mathematicians agree: mathematics is the science of patterns.
Keith Devlin, from Mathematics: The Science of Patterns
1178
 
As the science of abstract patterns, there is scarcely any aspect of our lives that is not affected, to a greater or lesser extent, by mathematics; for abstract patterns are the very essence of thought, of communication, of computation, of society, and of life itself.
Keith Devlin, from Mathematics: The Science of Patterns
1179
 
[Mathematics is] intellectual practice, a way of thinking - a 'game,' if you like - that our species (Homo sapiens) has developed over many centuries. A fun game, to be sure. A difficult game that sometimes yields an unexpected outcome. Moreover, a game that, more often than I can fully explain to myself, turns out to have significant practical applications. But a game for all that, a product of the collective human mind, which reflects more about the human mind than the world around us. Insofar as our mathematics tells us 'about the physical world,' it describes what the human mind encounters the world to be.
Keith Devlin, from the Foreword to A Certain Ambiguity.
1749
 
Education is not a preparation for life, education is life itself.
John Dewey,
792
 
That education is not an affair of "telling" and being told, but an active and constructive process, is a principle almost as generally violated in practice as conceded in theory. Is not this deplorable situation due to the fact that the doctrine is itself merely told? It is preached; it is lectured; it is written about.
John Dewey, quoted in Teaching With Your Mouth Shut by Donald L. Finkel
1623
 
We cannot think of ourselves save as to some extent social being. Hence, we cannot separate the idea of ourselves and our own good from our idea of others and their good.
John Dewey,
75
 
Failure is instructive. The person who really thinks learns quite as much from his failures as from his successes.
John Dewey, quoted in Contemporary Abstract Algebra, by J. Gallian.
76
 
No thought, no idea, can possibly be conveyed as an idea from one person to another. When it is told it is to the one to whom it is told another fact, not an idea. The communication may stimulate the other person to realize the question for himself and to think out a like idea, or it may smother his intellectual interest and suppress his dawning effort at thought. But what he directly gets cannot be an idea. Only by wrestling with the conditions of the problem at first hand, seeking and finding his own way out, does he think.
John Dewey,
1688
 
Every great advance in science has issued from a new audacity of imagination.
John Dewey,
593
 
No thought, no idea, can possibly be conveyed as an idea from one person to another. When it is told it is to the one to whom it is told another fact, not an idea... Only by wrestling with the conditions of the problem at first hand, seeking and finding his own way out, does he think.
John Dewey,
77
 
The important thing is that thinking is the method of an educative experience. The essentials of method are therefore identical with the essentials of reflection. They are first that the pupils have a genuine situation of experience - that there be a continuous activity in which he is interested for his own sake; secondly, that a genuine problem develops within this situation as a stimulus to thought; third, that he possess the information and makes the observations needed to deal with it; fourth, that suggested solutions occur to him which he shall be responsible for developing in an orderly way; fifth, that he have the opportunity and occasion to test his ideas by application, to make their meaning clear and to develop for himself their validity.
John Dewey,
1779
 
We only think when confronted with a problem.
John Dewey, quoted in Harper's Quotes.
78
 
This therefore is Mathematics, she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth.
Proculus Diadochus,
321
 
[On Archimedes mathematical results:] It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanation... No investigation of yours would succeed in attaining the proof, and yet, once seen you immediately believe you would have discovered it.
Proculus Diadochus, quoted in Journey Through Genius, by W. Dunham.
322
 
One of the most important concepts in all of mathematics is that of function.
T.P. Dick and C.M. Patton, from Calculus of a Single Variable.
79
 
I don't like that sort of school... where the bright childish imagination is utterly discouraged... where I have never seen among the pupils, whether boys or girls, anything but little parrots and small calculating machines.
Charles Dickens, 11/5/1857 speech.
80
 
The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying.
Jean Dieudonne, from Mathematics - The Music of Reason.
515
 
You got to look outside- your eyes- you got to think outside- your brain- you got to walk outside- your life- to where the neighborhoods change.
Ani DiFranco, from "Willing to Fight," on the album Puddle Dive.
82
 
But I suppose like anybody
I had to teach myself to see…
Teach myself to smile and stop and talk
To a whole other color kid
Teach myself to be new in an instant
Like the truth is accessible at any time
Teach myself it's never really one or the other
There's a paradox in every paradigm.
Ani DiFranco, from Paradigm
1708
 
When I was four years old they tried to test my IQ. They showed me a picture of three oranges and a pear. They asked me, "which one is different and does not belong?" They taught me different was wrong.
Ani DiFranco, from "My IQ," on the album Puddle Dive.
81
 
A book on the new physics, if not purely descriptive of experimental work, must essentially be mathematical.
P.A.M. Dirac,
83
 
It was not until some weeks later that I realized there is no need to restrict oneself to 2 by 2 matrices. One could go on to 4 by 4 matrices, and the problem is then easily soluable. In retrospect, it seems strange that one can be so much held up over such an elementary point. The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.
P.A.M. Dirac, "Pretty Mathematics," International Journal of Theoretical Physics, Vol. 21, Ns. 8/9, 1982, pp. 603-5.
1411
 
A good deal of my research work in physics has consisted in not setting out to solve some particular problems, but simply examining mathematical quantities of a kind that physicists use and trying to get them together in an interesting way regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later that the work does have an application. Then one has had good luck.
P.A.M. Dirac, "Pretty Mathematics," International Journal of Theoretical Physics, Vol. 21, Ns. 8/9, 1982, pp. 603-5.
1410
 
We know that the evaluation or even only the reduction of multiple integrals generally presents very considerable difficulties…
Dirichlet, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1128
 
Our greatest natural resource is the minds of our children.
Walt Disney,
84
 
Success isn't permanent, and failure isn't fatal.
Mike Ditka, quoted in The Fourth, and By Far the Most Recent, 637 Best Things Anybody Ever Said, by Robert Byrne.
85
 
People learn by playing, thinking, and amazing themselves. They learn while they're laughing and they learn while they are wondering "What the heck is this?"
Sandra Dodd,
1463
 
The formula "two and two make five" is not without its attractions.
Fyodor Dostoevsky, Notes from the Underground, 1864
1628
 
It's easier to build strong children than repair broken men.
Frederick Douglas,
1260
 
Educate your sons and daughters, send them to school and show them that beside the cartridge box, the ballot box, and the jury box, you have also the knowledge box.
Frederick Douglass, quoted in My Soul Looks Back, 'Less I Forget, by Dorothy Winbush Riley.
87
 
If there's no struggle, there's no progress.
Frederick Douglass,
86
 
How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?
Sir Arthur Conan Doyle,
88
 
The test of literature is, I suppose, whether we ourselves live more intensely for the reading of it.
Elizabeth Drew,
992
 
You will find as you look back upon your life that the moments when you have really lived are the moments when you have done things in the spirit of love.
Henry Drummond,
841
 
A man should be learned in several sciences, and should have a reasonable, philosophical and in some measure a mathematical head, to be a complete and excellent poet.
John Dryden,
1381
 
Whether you like it or not the millions are here, and here they will remain. If you do not lift them up, they will pull you down... Education must not simply teach work - it must teach life.
W.E.B. DuBois, quoted in My Soul Looks Back, 'Less I Forget, by Dorothy Winbush Riley.
89
 
Since I found that one could make a case shadow from a three-dimensional thing, any object whatsoever -- just as the projecting of the sun on the earth makes two dimensions -- I thought that by simple intellectual analogy, the fourth dimension could project an object of three dimensions, or, to put it another way, any three-dimensional object, which we see dispassionately, is a projection of something four-dimensional, something we are not familiar with.
Marcel Duchamp, quoted in The Fourth Dimension and non-Euclidean Geometry in Modern Art by Linda Dalrymple Henderson.
666
 
I have forced myself to contradict myself in order to avoid conforming to my own taste.
Marcel Duchamp,
1682
 
I am still a victim of chess. It has all the beauty of art -- and much more. It cannot be commercialized. Chess is much purer than art in its social position.
Marcel Duchamp,
1680
 
The chess pieces are the block alphabet which shapes thoughts; and these thoughts, although making a visual design on the chess-board, express their beauty abstractly, like a poem... I have come to the personal conclusion that while all artists are not chess players, all chess players are artists.
Marcel Duchamp,
1679
 
I am interested in ideas, not merely in visual products.
Marcel Duchamp,
1678
 
It is time to stop claiming that mathematics is necessary for jobs. It is time to stop asserting that students must master algebra to be able to solve problems that arise every day, at home or at work. It is time to stop telling students that the main reason they should learn mathematics is applications. We should not tell our students lies. They will find us out, sooner or later.
Underwood Dudley, from "Is Mathematics Necessary?", Mathematics Education Dialogues, March 1998.
708
 
It demeans mathematics to justify it to appeals to work, to getting and spending... Can you recall why you fell in love with mathematics? It was not, I think, because of its usefulness in controlling inventories.
Underwood Dudley, from "Is Mathematics Necessary?", Mathematics Education Dialogues, March 1998.
709
 
How is it that little children are so intelligent and men so stupid? It must be education that does it.
Alexandre Dumas, quoted in My Soul Looks Back, 'Less I Forget, by Dorothy Winbush Riley.
91
 
One's work may be finished some day, but one's education never.
Alexandre Dumas, quoted in My Soul Looks Back, 'Less I Forget, by Dorothy Winbush Riley.
90
 
Sixty years ago, I knew everything; now I know nothing; education is a progressive discovery of our own ignorance.
Will Durant,
889
 
It takes immense genius to represent, simply and sincerely, what we see right in front of us.
Edmond Duranty,
92
 
Geometry is the foundation of all painting.
Albrecht Durer, quoted in The Magic of Mathematics, by Theoni Pappas.
489
 
The geometry of innocent flesh on the bone
Causes Galileo's math book to get thrown
At Delilah who sits worthlessly alone
But the tears on her cheeks are from laughter.
Bob Dylan, from "The Tombstone Blues".
1313
 
The Bourbaki program was the extreme expression of the Cartesian style. It narrowed the scope of mathematics by excluding the beautiful flowers that Baconian travelers might collect by the wayside… For me, the main thing missing in the Bourbaki program is the element of surprise.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009, p. 213.
1302
 
It turns out that the Schrodinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009, p. 213.
1303
 
Mind and intelligence are woven into the fabric of our universe in a way that altogether surpasses our understanding.
Freeman Dyson,
1733
 
If we take a Baconian point of view, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009, p. 215.
1304
 
The Besicovitch style is architectural. He builds out of simply elements a delicate and complicated architectural structure, usually with a hierarchical plan, and then, when the building is finished, the completed structure leads by simple arguments to an unexpected conclusion. Every Besicovitch proof is a work of art, as carefully constructed as a Bach fugue.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009, p. 216.
1305
 
It may come to a surprise to Western readers that he [Manin; in Mathematics as Metaphor] writes with equal eloquence about other subjects… To his compatriots in Russia, such many-sided interests and expertise would come as no surprise. Russian intellectuals maintain the proud tradition of the old Russian intelligentsia, with scientists and poets and artists and musicians belonging to a single community. They are still today, as we see them in the plays of Chekhov, a group of idealists bound together by their alienation from a superstitious society and a capricious government. In Russia, mathematicians and composers and film-producers talk to one another, walk together in the snow on winter nights, sit together over a bottle of wine, and share each others' thoughts.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009, p. 222.
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At the beginning of the seventeenth century, two great philosophers, Francis Bacon in England and Rene Descartes in France, proclaimed the birth of modern science. Descartes was a bird, and Bacon was a frog. Each of them described his vision of the future. Their visions were very different. Bacon said, "All depends on keeping the eye steadily fixed on the facts of nature." Descartes said, "I think, therefore I am." According to Bacon scientists should travel over the earth collecting facts, until the accumulated facts reveal how Nature works. The scientists will then induce from the facts the laws that Nature obeys. According to Descartes, scientists should stay at home and deduce the laws of Nature by pure thought. In order to deduce the laws correctly, the scientists will need only the rules of logic and knowledge of the existence of God. For four hundred years since Bacon and Descartes led the way, science has raced ahead by following both paths simultaneously. Neither Baconian empiricism nor Cartesian dogmatism has the power to elucidate Nature's secrets by itself, but both together have been amazingly successful.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009, p. 212.
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One factor that has remained constant through all the twists and turns of the history of physical science is the decisive importance of the mathematical imagination.
Freeman Dyson, quoted in "Reading the Master: Newton and the Birth of Celestial Mechanics" by Bruce Pourciau, American Mathematical Monthly, Vol. 88, 1981.
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[On Richard P. Feynman's live demonstration of the rigidity of the O-rings when cold that doomed the space shuttle Challenger, killing seven astronauts:] The public saw with their own eyes how science is done, how a great scientist thinks with his hands, how nature gives a clear answer when a scientist asks her a clear question.
Freeman Dyson, from From Eros to Gaia, p. 312.
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Edward Lorenz discovered in 1963 that the solution of the equations of meteorology are often chaotic… Lorenz was a meteorologist and generally regarded as the discoverer of chaos. He discovered the phenomena of chaos in the meteorological context and gave them their modern names. But in fact I had heard the mathematician Mary Cartwright, who died in 1998 at the age of 97, describe the same phenomena in a lecture in Cambridge in 1943, twenty years before Lorenz discovered them. She called the phenomena by different names, but they were the same phenomena… I heard all about chaos from Mary Cartwright seven years before I heard von Neumann talk about weather control, but I was not far-sighted enough to make the connection.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009.
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[Von Neumann] said, "The computer will enable us to divide the atmosphere at any moment into stable and unstable regions. Stable regions we can predict. Unstable regions we can control"… Von Neumann, speaking in 1950, said it would take only ten years to build computers powerful enough to diagnose accurately the stable and unstable regions of the atmosphere. Then, once we had accurate diagnosis, it would take only a short time for us to have control. He expected that practical control of the weather would be a routine operation within the decade of the 1960's. Von Neumann, of course, was wrong.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009.
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Some mathematicians are birds, and others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time… Mathematics needs both birds and frogs. Mathematics is rich and beautiful because birds give it broad visions and frogs give it intricate details. Mathematics is both great art and important science, because it combines generality of concepts with depth of structures.
Freeman Dyson, from "Birds and Frogs," Notices of the AMS, Vol. 56, No. 2, February, 2009, p. 212.
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