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
 
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
1198
 
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
 
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
 
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
 
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
 
Every new body of discovery is mathematical in form, because there is no other guidance we can have.
Charles Darwin,
62
 
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 numberical 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
 
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
 
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
 
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
 
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
 
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
 
Numbers are the free creation of the human mind.
Richard Dedekind, quoted in Sweet Reason by J. Henle and T. Tymoczko.
65
 
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
 
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
 
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
 
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
 
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
 
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
 
The reading of all good books is like a conversation with the finest people of past centuries.
Rene Descartes,
825
 
Cogito, ergo sum. [I think, therefore I am.]
Rene Descartes,
69
 
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
 
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
 
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
 
Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.
Rene Descartes,
824
 
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
 
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 centry 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
 
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
 
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
 
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
 
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
 
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
 
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.
1193
 
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
 
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
 
Education is not a preparation for life, education is life itself.
John Dewey,
1197
 
Every great advance in science has issued from a new audacity of imagination.
John Dewey,
593
 
Education is not a preparation for life, education is life itself.
John Dewey,
792
 
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... 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
 
We only think when confronted with a problem.
John Dewey, quoted in Harper's Quotes.
78
 
[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, quoted in Journey Through Genius, by W. Dunham.
322
 
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
 
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
 
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
 
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
 
If there's no struggle, there's no progress.
Frederick Douglass,
86
 
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
 
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
 
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
 
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
 
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.
93
 

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