The reason people find it so hard to be happy is that they always see the past better than it was, the present worse than it is, and the future less resolved than it will be.
Marcel Pagnol,
798
 
The world is my country, all mankind are my bretheren, and to do good is my religion.
Thomas Paine,
1258
 
The right to vote...is the primary right by which other rights are protected.
Thomas Paine,
1275
 
Such is the irresistible nature of truth: that all it asks, and all it wants, is the liberty of appearing.
Thomas Paine,
911
 
'Yes, they must understand, not merely know.' But this misses the capital point that being a mathematician, again like being a poet, or a composer or an engineer, means doing, rather than knowing or understanding.
Seymour Papert, from "Teaching Children to be Mathematicians vs. Teaching About Mathematics"
1199
 
The reason for the qualification is that the traditional branches of mathematics do not provide the most fertile ground for the easy, prolific growth of mathematical traits of mind.
Seymour Papert, from "Teaching Children to be Mathematicians vs. Teaching About Mathematics"
1201
 
For most children at school, the problem is not that they do not understand particular mathematical structures or concepts. Rather, they do not understand what kind of thing a mathematical structure is: they do not see the point of the whole enterprise. Asking them to learn it is like asking them to learn poetry in a completely unknown foreign language.
Seymour Papert, from "Teaching Children to be Mathematicians vs. Teaching About Mathematics"
1203
 
Before I was two years old I had developed an intense involvement with automobiles. The names of car parts made up a very substantial portion of my vocabulary... Years later... playing with gears became a favorite pastime... I became adept at turning wheels in my head and at making chains of cause and effect...Working with differentials did more for my mathematical development than anything I was taught in elementary school.
Seymour Papert, from "The Gears of My Childhood," preface to Mindstorms: Children, Computers and Powerful Ideas.
1243
 
Yes, one can use algebra as a vehicle for initiating students to the mathematical way of thinking. But, to do so effectively one should first identify as far as possible components of the general intellectual skills one is trying to teach; and when this is done it will appear that algebra (in any traditional sense) is not a particularly good vehicle.
Seymour Papert, from "Teaching Children to be Mathematicians vs. Teaching About Mathematics"
1202
 
For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.
Seymour Papert, from Mindstorms.
283
 
Here again, it is necessary, if we want any clarity, to ward off a too easy, superficial assent from math ed reformers who say, 'Yes, that's why we must use The Method of Discovery.' For, when 'Discovery' means discovery this is wonderful, but in reality 'Discovery' usually means something akin to the following fantasy about a poetry class: the discovery-method teacher has perfected a series of questions that lead the class to discover the line 'Mary had a little lamb.' My point is not that this would be good or bad, but that no one would confuse it with creative work in poetry.
Seymour Papert, from "Teaching children thinking," in The Computer in the School: Tutor, Tool, Tutee, edited by Robert Taylor.
1574
 
Children begin their lives as eager and competent learners. They have to learn to have trouble with learning in general and mathematics in particular.
Seymour Papert, from Mindstorms: Children, Computers, and Powerful Ideas, p. 40.
1337
 
Turtle geometry is a computational style of geometry... The goal of children's first experiences in the Turtle learning environment is not to learn formal rules but to develop insights into the way they move about in space... This method tries to establish a firm connection between personal activity and the creation of formal knowledge.
Seymour Papert, from Mindstorms: Children, Computers, and Powerful Ideas, pp. 55, 58-9.
1338
 
The generality of the idea of structured programming as a mathetic principle, that is to say an aid to learning, will become more apparent through the next example, which describes the process involved in learning another physical skill - juggling. We do not choose it at random. The Turtle circle was a good carrier for learning mathematics 'with one's body.' Juggling turns out to be an equally good carrier for learning a body skill 'with mathematics'.
Seymour Papert, from Mindstorms: Children, Computers, and Powerful Ideas, p. 105.
1339
 
It seems to be nobody's business to think in a fundamental way about science in relation to the way people think and learn it. Although lip service has been paid to the importance of science and society, the underlying methodology is like that of traditional education: one of delivering elements of ready-made science to a special audience. The concept of a serious enterprise of making science for the people is quite alien.
Seymour Papert, from Mindstorms: Children, Computers, and Powerful Ideas, p. 188.
1340
 
But the primary learning experience is not one of memorizing facts or of practicing skills. Rather, it is getting to know the Turtle, exploring what a Turtle can and cannot do. It is similar to the child's everyday activities, such as making mudpies and testing the limits of parental authority - all of which is a component of 'getting to know'... Yet the Turtle is different - it allows children to be deliberate and conscious in bringing a kind of learning with which they are comfortable and familiar to bear on math and physics. And, as we have remarked, this is a kind of learning that brings the child closer to the mathetic practice of sophisticated adult learners.
Seymour Papert, from Mindstorms: Children, Computers, and Powerful Ideas, p. 136-7.
1341
 
The phrase 'technology and education' usually means inventing new gadgets to teach the same old stuff in a thinly disguised version of the same old way. Moreover, if the gadgets are computers, the same old teaching becomes incredibly more expensive and biased towards its dullest parts, namely the kind of rote learning in which measurable results can be obtained to treating children like pigeons in a Skinner box.
Seymour Papert, from "Teaching children thinking," in The Computer in the School: Tutor, Tool, Tutee, edited by Robert Taylor.
1573
 
The understanding of learning must be genetic. It must refer to the genesis of knowledge. What an individual can learn, and how he learns it, depends on what models he has available. This raises, recursively, the question of how he learned these models. Thus the "laws of learning" must be about how intellectual structures grow out of one another and about how, in the process, they acquire both logical and emotional form.
Seymour Papert, from Mindstorms: Children, Computers, and Powerful Ideas, p. vii.
1336
 
I like to see in Deborah's experience ["a sixth grader who had problems with school learning" and was introduced to LOGO] a small recapitulation of how the success of such thinkers as Copernicus and Galileo allowed people to break away from superstitious dependencies that had nothing in themselves to do with physics. In both cases -in Deborah's personal history and in the history of Western thought - the success of a mathematical theory served more than an instrumental role: It served as an affirmation of the power of ideas and the power of the mind.
Seymour Papert, from Mindstorms: Children, Computers and Powerful Ideas, pp. 118-119.
1619
 
If the styles of involvement with motorcycle maintenance [as described in Robert Pirsig's Zen and the Art of Motorcycle Maintenance] are so intricately interwoven with our psychological and social identities, one would scarcely expect this to be less true about the varieties of involvements of individuals with mathematics.
Seymour Papert, from "The Mathematical Unconscious," Epilogue to Mindstorms: Children, Computers and Powerful Ideas, pp. 205.
1621
 
Most people feel that they have no "personal" involvement with mathematics, yet as children they constructed it for themselves. Jean Piaget's work on genetic epistemology teaches us that from the first days of life a child is engaged in an enterprise of extracting mathematical knowledge from the intersection of body with environment. The point is that, whether we intend it or not, the teaching of mathematics, as it is traditionally done in our schools, is a process by which we ask the child to forget the natural experience of mathematics in order to learn a new set of rules.
Seymour Papert, from "The Mathematical Unconscious," Epilogue to Mindstorms: Children, Computers and Powerful Ideas, pp. 206-7.
1620
 
It is in fact easy for children to understand how the Turtle defines a self-contained world in which certain questions are relevant and others are not. The next chapter discusses how this idea can be developed by constructing many such "microworlds", each with its own set of assumptions and constraints. Children get to know what it is like to explore the properties of a chosen microworld undisturbed by extraneous questions. In doing so they learn to transfer habits of exploration from their personal lives to the formal domain of scientific theory construction.
Seymour Papert, from Mindstorms: Children, Computers and Powerful Ideas, p. 117.
1618
 
It is generally assumed in our society that every child should, and can, have experience of creative work in language and plastic arts. It is equally generally assumed that very few people can work creatively in mathematics. I believe that there has been an unwitting conspiracy of psychologists and mathematicians in maintaining this assumption. The psychologists contribute to it out of genuine ignorance of what creative mathematical work might be like. The mathematicians, very often, do so out of elitism, in the form of a deep conviction that mathematical creativity is the privilege of a tiny minority.
Seymour Papert, from "Teaching Children to be Mathematicians vs. Teaching About Mathematics"
1200
 
Bees...by virtue of a certain geometrical forethought...know that the hexagon is greater than the square and the triangle, and will hold more honey for the same expenditure of material.
Pappas, quoted in Agnesi to Zeno, by Sanderson Smith.
1194
 
What matters most is... one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.
Seymour Pappert, from Mindstorms: Children, Computers and Powerful Ideas.
737
 
Though God has given to men the best and most perfect understanding of wisdom and mathematics, He has allotted a partial share to some of the unreasoning creatures as well... This instinct is specially marked among bees. They prepare for the reception of the honey the vessels called honeycombs, with cells all equal, similar and adjacent, and hexagonal in form.
Pappus, quoted in Mathematics: A Human Endeavor, by Harold R. Jacobs.
284
 
I didn't know what lay ahead of me, but I believed in myself. My deepest instincts told me I would not perish. Poverty and bigotry would still be around, but at last I could fight with them on even terms. The significant thing was a choice of weapons with which to fight them most effectively. That I would accept those of a mother who placed love, dignity and hard work over hatred was a fate that accompanied me from her womb.
Gordon Parks, Choice of Weapons, pp. 221-2.
1773
 
Only wimps do the general case. True teachers tackle examples.
Parlett, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1120
 
Justice and power must be brought together, so that whatever is just may be powerful, and whatever is powerful may be just.
Blaise Pascal,
1280
 
All human evil comes from a single cause, man's inability to sit still in a room.
Blaise Pascal,
1344
 
Faith is different from proof; the latter is human, the former is a Gift from God.
Blaise Pascal,
1345
 
If we examine our thoughts, we shall find them always occupied with the past and the future.
Blaise Pascal,
1346
 
Love has reasons which reason cannot understand.
Blaise Pascal,
1347
 
Men are so necessarily mad, that not to be mad would amount to another form of madness.
Blaise Pascal,
1348
 
Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.
Blaise Pascal,
1353
 
Nature is an infinite sphere, whose center is everywhere and whose circumference is nowhere.
Blaise Pascal, quoted in To Infinity and Beyond by Eli Maor.
657
 
When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there... now instead of then.
Blaise Pascal, quoted in Infinity and the Mind by Rudy Rucker.
517
 
When we cite authors we cite their demonstrations, not their names.
Blaise Pascal, quoted in Mathematical Thought from Ancient to Modern Times by Morris Kline.
1092
 
When I approach a child, he inspires in me two sentiments; tenderness for what he is, and respect for what he may become.
Louis Pasteur,
1016
 
Accept challenges, so that you may feel the exhilaration of victory.
George S. Patton,
1326
 
In order to leave a good idea, you must leave lots of ideas.
Linus Pauling,
1224
 
Mathematics is no more computation than typing is literature.
John Allen Paulos,
285
 
How many pizzas are consumed each year in the United States? How many words have you spoken in your life? How many different peoples names appear in the New York Times each year? How many watermelons would fit inside the U.S. Capital building? What is the volume of all the human blood in the world?
John A. Paulos, from Innumeracy.
286
 
There are too many people who get degrees and think that they're educated. In order to be a truly knowledgeable person one has got to be engaged in serious, systematic, lifelong learning.
Benjamin Payton, quoted in My Soul Looks Back, 'Less I Forget, by Dorothy Winbush Riley.
287
 
It is likely that biology will eventually be as full-panolplied with mathematically expressed theory as physics now is... There is no substitute for mathematics to state in rational shorthand the relations between natural phenomena or generalizations about them.
D. R. Pearl,
288
 
We have not the slightest idea of what this equation [i^i=1/sqrt(e^pi)] means, but we may be sure that it means something very important.
Benjamin Peirce, quoted in The Mathematical Universe, by William Dunham.
289
 
Dimension is not easy to understand. At the turn of the century it was one of the major problems in mathematics to determine what dimension means and which properties it has. And since then the situation has become somewhat worse because mathematicians have come up with some ten different notions of dimension: topological dimension, Hausdoff dimension, fractal dimension, self-similarity dimension, box-counting dimension, capacity dimension, information dimension, Euclidean dimension, and more. They are all related. Some of them, however, make sense in certain situations, but not at all in others, where alternative definitions are more helpful. Sometimes they all make sense and are the same. Sometimes several make sense but do not agreee. The detials can be confusing even for a research mathematician.
Heinz-Otto Peitgen, Hartmut Jurgens and Dietmar Sa, from Fractals in the Classroom.
751
 
It is a feeling not uncommon amongst artists, that in their greatest works they are revealing external truths which have some kind of prior ethereal existence.
Roger Penrose, from A Dictionary of Quotations in Mathematics by Nowlan
1665
 
What you leave behind is not what is engraved in stone monuments, but what is woven into the lives of others.
Pericles,
1022
 
The anceints devoted a lifetime to the study of arithmetic; it required days to extract a square root or to multiply two numbers together. Is there any harm in skipping all that, in letting the school boy learn multiplication sums, and in starting his more abstract reasoning at a more advanced point. Where would be the harm in letting the boy assume the truth of many propositions of the first four books of Euclid, letting him assume their truth partly by faith, partly by trial?
John Perry, quoted in Memorabilia Mathematica, by Robert E. Moritz.
457
 
It is just as dubious whether 1.111111111... where the 1's are repeated indefinitely, has any finite meaning. It is a curious thing that people on the whole do not boggle over an infinite decimal of this kind, but when they look at an infinite addition like this: 1 + 1/10 + 1/100 + 1/1000 + 1/10000 + … ad infinitum, although this is just another way of writing the other. But I am not surprised at their looking aghast at the latter, though rather surprised that they accept the former.
Rosza Peter, from Playing with Infinity: Mathematical Explorations and Excursions, p. 104.
1752
 
I am inclined to believe that one of the origins of mathematics is man's playful nature, and for this reason mathematics is not only a Science, but to at least the same extent also an Art.
Rosza Peter, from Playing with Infinity: Mathematical Explorations and Excursions, p. 1.
1753
 
I have received a great deal from the arts and I would now like in my turn to present mathematics and let everyone see that mathematics and the arts are not so different from each other. I love mathematics not only for its technical applications, but principally because it is beautiful; because man has breathed his spirit of play into it, and because it has given him his greatest game - the encompassing of the infinite. Mathematics can give to the world such worthwhile things - about ideas, about infinity; and yet how essentially human it is - unlike the dull multiplication table, it bears on it for ever the stamp of man's handiwork.
Rosza Peter, from Playing with Infinity: Mathematical Explorations and Excursions, p. v.
1754
 
I love mathematics...principally because it is beautiful; because man has breathed his spirit of play into it, and because it has given him his greatest game - the encompassing of the infinite.
Rozso Peter, quoted in Out of the Mouths of Mathematicians, by R. Schmalz.
290
 
To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas.
Ivars Peterson, from The Mathematical Tourist.
291
 
Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
Ivars Peterson, from A Mathematical Mystery Cruise.
292
 
A child[s]... first geometrical discoveries are topological... If you ask him to copy a square or a triangle, he draws a closed circle.
Jean Philips, from "How Children Form Mathematical Concepts."
296
 
When school children study analytic geometry, they should be made aware that his seemingly trivial and esoteric subject exists to us only because of the heroic efforts of a succession of brilliant minds, culminating in the work of Descartes. Its depth, originality, and profundity are lost on students. It has been carefully polished and refined so exquisitely, presented so elegantly and simply, that students myopically receive it as a trifle.
J. D. Philips, from "Mathematics as an Aesthetic Discipline," in The Humanistic Mathematics Network Journal, No. 12, Oct. 1995
295
 
The notion that anyone other than a scientist will ever use even the most elementary trigonometry or algebra is laughable. Imagine the absurdity of being in a car or on a plane when suddenly the need arises to solve a quadratic equation or to graph a trigonometric function. But this is precisely the scenario that the traditional defense has coerced us into accepting as realistic. Clearly this is absurd. And so is our complicity.
J. D. Philips, from "Mathematics as an Aesthetic Discipline," in The Humanistic Mathematics Network Journal, No. 12, Oct., 1995.
293
 
Students must learn that mathematics is the most human of endeavors. Flesh and blood representatives of their own species engaged in a centuries long creative struggle to uncover and to erect this magnificent edifice. And the struggle goes on today. On the very campuses where mathematics is presented and received as an inhuman discipline, cold and dead, new mathematics is created. As sure as the tides.
J. D. Philips, from "Mathematics as an Aesthetic Discipline," in Humanistic Mathematics Network Journal, No. 12, Oct. 1995.
294
 
A child['s] ... first geometrical discoveries are topological … If you ask him to copy a square or a triangle, he draws a closed circle.
Jean Piaget,
1467
 
If there were only one truth, you couldn't paint a hundred canvases on the same theme.
Pablo Picasso,
1421
 
Everything you can imagine is real.
Pablo Picasso,
1605
 
Computers are useless. They can only give you answers.
Pablo Picasso, quoted in A Primer of Mathematical Writing by Steven G. Krantz.
739
 
Inspiration exists, but it has to find you working.
Pablo Picasso,
1422
 
If only we could pull out our brain and use only our eyes.
Pablo Picasso,
1420
 
I paint objects as I think them, not as I see them.
Pablo Picasso,
1419
 
I am always doing that which I cannot do, in order that I may learn how to do it.
Pablo Picasso,
1418
 
When I was a child I could draw like Rafeal. Yet it took me a lifetime to learn to draw like a child.
Picasso,
1205
 
Everything you can imagine is real.
Pablo Picasso,
1633
 
Art is a lie that makes us realize the truth.
P. Picasso, quoted in Discovering Geometry, by M. Serra.
297
 
It takes a long time to become young.
Pablo Picasso, quoted in The Harper Book of Quotations, edited by Robert I. Fitzhenry.
625
 
Every child is an artist. The problem is how to remain an artist after he(she) grows up.
P. Picasso, quoted in Discovering Geometry, by M. Serra.
298
 
True mathematical reasoning is so much more evident than it is possible to render any doctrine of logic proper - without just such reasoning - that an appeal in mathematics to logic could only embroil a situation.
C.S. Pierce, quoted in What is Mathematics, Really? by Reuben Hersch, p. 199.
1255
 
Mathematics is purely hypothetical: it produces nothing but conditional propositions.
C. S. Pierce, quoted in A History of Mathematics, by Carl Boyer.
299
 
The notion of infinity is our greatest friend; it is also the greatest enemy of our piece of mind.
James Pierpont,
300
 
Mathematicians study structure independent of context, and their science is a voyage of exploration through all the kinds of structure and order which the human mind is capable of discerning.
Charles Pinter, from A Book of Abstract Algebra.
301
 
The nerds are running the world now.
Joe Piscapo,
302
 
All matter originates and exists only by virtue of a force… We must assume behind this force the existence of a conscious and intelligent Mind. This Mind is the matrix of all matter.
Max Planck, Nobel Prize-winning Father of Quantum Theory
1767
 
When you are insane, you are busy being insane - all the time... When I was crazy, that's all I was.
Sylvia Plath,
865
 
Be kind, for everyone you meet is fighting a hard battle.
Plato,
1431
 
The study of mathematics develops and sets into operation a mental organism more valuable than a thousand eyes, because through it alone can truth be apprehended.
Plato, quoted in Euclidean and Non-Euclidean Geometries by Greenberg
877
 
Geometry existed before the creation.
Plato,
873
 
He is unworthy of the name of man who is ignorant that the diagonal of a square is incommensurate with its side.
Plato, quoted in Memorabilia Mathematica, by R. E. Moritz.
306
 
Let no one ignorant of geometry enter here.
Plato, Inscription above Plato's academy.
305
 
God ever geometrizes.
Plato,
304
 
There should be no element of slavery in learning. Enforced exercise does no harm to the body, but enforced learning will not stay in the mind. So avoid compulsion, and let your children's lessons take the form of play.
Plato, from "The Republic."
303
 
The wildest colts make the best horses.
Plutarch,
1405
 
I don't need a friend who changes when I change and who nods when I nod; my shadow does that much better.
Plutarch,
988
 
The mind is not a vessel to be filled, it is a fire to be kindled.
Plutarch,
307
 
What we achieve inwardly will change outer reality.
Plutarch,
1406
 
Few persons can be made to believe that it is not quite an easy thing to invent a method of secret writing that shall baffle investigation. Yet it may be roundly asserted that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve.
Edgar Allen Poe, quoted in "Cryptology: From Caesar Ciphers to Public-key Cryptosystems," by D. Luciano and G. Prichett, in The College Mathematics Journal, Jan. 1987.
308
 
Men have called me mad. But the question is not yet settled, whether madness is or is not the loftiest intelligence--whether much that is glorious--whether all that is profound--does not spring from disease of thought--from moods of mind exalted at the expense of the general intellect.
Edgar Allan Poe,
864
 
Not in knowledge is happiness, but in the acquisition of knowledge.
Edgar Allan Poe, through the character Agathos from 'The Power of Words'
1577
 
It is here whispered that, of this infinity of matter, the sole purpose is to afford infinite springs, at which the soul may allay the thirst to know, which is for ever unquenchable within it - since to quench it, would be to extinguish the soul's self.
Edgar Allan Poe, through the character Agathos from 'The Power of Words'
1578
 
No thought can perish, so no act is without infinite result. We moved our hands, for example when we were dwellers on the earth, and, in so doing, gave vibration to the atmosphere which engirdled it. This vibration was indefinitely extended, till it gave impulse to every particle of the earth's air, which thenceforward, and for ever, was actuated by the one movement of the hand.
Edgar Allan Poe, through the character Agathos from 'The Power of Words'
1579
 
This wild star - it is now three centuries since, with clasped hands, and with streaming eyes, at the fee of my beloved - I spoke it - with a few passionate sentences - into birth.
Edgar Allan Poe, through the character Agathos from 'The Power of Words'
1580
 
The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.
Henri Poincare, quoted in A Radical Approach to Real Analysis, by Bressoud.
310
 
A first fact should surprise us, or rather would surprise us if we were not used to it. How does it happen there are people who do not understand mathematics? If mathematics invokes only the rules of logic, such as are accepted by all normal minds...how does it come about that so many persons are here refractory?
Henri Poincare, quoted in The World of Mathematics, by J.R. Newman.
312
 
Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long the relations don't change. Matter is not important, only form interests them.
Henri Poincare, quoted in Contemporary Abstract Algebra, by J. Gallian.
309
 
There are no solved problems; there are only problems that are more or less solved.
Henri Poincare, quoted in Excursions in Calculus, by Robert M. Young.
313
 
The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.
Henri Poincare, quoted in More Joy of Mathematics, by Theoni Pappas.
499
 
But for harmony beautiful to contemplate, science would not be worth following.
Henri Poincare, quoted in Exploring Elementary Mathematics: a Small Group Approach for Teaching by Julian Weisglass.
311
 
A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws no longer had any secret for us, we could still only know the initial state approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But this is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.
Poincare, from "How Chaotic Things Work" by William Mercier, College Mathematics Journal, vol. 28, no. 2, March 1997
1227
 
What we call objective reality is, in the last analysis, what is common to many thinking beings, and could be common to us all; this common part, we shall see, can only be the harmony expressed by mathematical laws. It is this harmony then which is the sole objective reality, the only truth we can obtain.
Henri Poincare,
1763
 
A hundred years ago such a function [an integrable function due to Riemann which is discontinuous on an everywhere dense set of points] would have been considered an outrage on common sense.
Poincare, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1122
 
If we wish to foresee the future of mathematics our proper course is to study the history and present condition of the science.
Henry Poincare, quoted in Mathematical Thought from Ancient to Modern Times by Morris Kline.
1091
 
If we wish to foresee the future of mathematics our proper course is to study the history and present condition of the science.
Henri Poincare, quoted in Mathematical Thought from Ancient to Modern Times by Morris Kline.
903
 
Mathematical discoveries, small or great, are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labor, both conscious and subconscious.
Henry Poincare,
812
 
Point set topology is a disease from which the human race will soon recover.
Jules H. Poincare,
1466
 
It is only the affirmation of the power of the mind which knows it can conceive of the indefinite repetition of the same act, when the act is once possible.
Henri Poincare, quoted in To Infinity and Beyond by Eli Maor.
649
 
What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates are false. One geometry cannot be more true than another; it can only be more convenient.
Henri Poincare, from Sceince and Hypothesis
1483
 
Life is good for only two things: discovering mathematics and teaching mathematics.
Simeon Poisson,
314
 
Mathematics is the abstract key which turns the lock of the physical universe.
John Polkinghorne, quoted in Mathematics: The Science of Patterns by Keith Devlin.
539
 
We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning.
George Polya, from Induction and Analogy in Mathematics, Volume 1 of Mathematics and Plausible Reasoning
1214
 
Certainly, let us learn proving, but also let us learn guessing.
George Polya, from Induction and Analogy in Mathematics, Volume 1 of Mathematics and Plausible Reasoning
1215
 
If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.
George Polya, from Induction and Analogy in Mathematics, Volume 1 of Mathematics and Plausible Reasoning
1216
 
Mathematics presented with rigor is a systematic deductive science but mathematics in the making is an experimental inductive science.
George Polya, How to Solve It, p. 117.
1424
 
Beauty in mathematics is seeing the truth without effort.
George Polya,
1417
 
Induction is the process of discovering general laws by the observation and combination of particular instances. It is used in all sciences, even in mathematics. Mathematical induction is used in mathematics alone to prove theorems of a certain kind. It is rather unfortunate that the names are connected because there is very little logical connection between the two processes. There is, however, some practical connection; we often use both methods together.
George Polya, from How to Solve It, p. 114.
1423
 
Strictly speaking, all our knowledge outside mathematics and demonstrative logic (which is, in fact, a branch of mathematics) consists of conjectures. There are, of course, conjectures and conjectures. There are highly respectable and reliable conjectures as those expressed in certain general laws of physical science. There are other conjectures, neither reliable nor respectable, some of which may make you angry when you read them in a newspaper. And in between there are all sorts of conjectures, hunches, and guesses.
George Polya, from Induction and Analogy in Mathematics, Volume 1 of Mathematics and Plausible Reasoning
1213
 
Teaching the mechanical performance of routine mathematical operations and nothing else is well under the level of the cookbook because kitchen recipes do leave something to the imagination and judgment of the cook but mathematical recipes do not.
George Polya, How to Solve It, p. 172.
1748
 
Solving problems is the specific achievement of intelligence, and intelligence is the specific gift of mankind: solving problems can be regarded as the most characteristically human activity.
George Polya, from Mathematical Discovery, vol. 1, p. v.
1297
 
Solving problems is a practical art, like swimming, or skiing, or playing the piano... if you wish to become a problem solver you have to solve problems.
George Polya, from Mathematical Discovery, vol. 1, p. v.
1299
 
It so happens that one of the greatest mathematical discoveries of all times was guided by physical intuition. I mean Archimedes' discovery of that branch of science that we call today the integral calculus. Archimedes found the areas of the parabolic segment, the volume of the sphere and about a dozen similar results by a uniform method in which the idea of equilibrium plays an important role. As he says himself, he 'investigated some problems in mathematics by means of mechanics.'
George Polya, from Induction and Analogy in Mathematics, Volume 1 of Mathematics and Plausible Reasoning
1219
 
You must know that Hardy had a running feud with God. In Hardy's view God had nothing more important to do than frustrate Hardy. This led to a sort of insurance policy for Hardy one time when he was trying to get back to Cambridge after a visit to [Herald] Bohr in Denmark. The weather was bad and there was only a small boat available. Hardy thought there was a real possibility the boat would sink. So he sent a postcard to Bohr saying, "I proved the Riemann Hypothesis. G.H. Hardy." That way if the boat sank, everyone would think that Hardy had proved the Riemann Hypothesis. God could not allow so much glory for Hardy so he could not allow the boat to sink.
George Polya, quoted in Out of the Mouths of Mathematicians, by R. Schmalz.
317
 
If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.
George Polya, quoted in Calculus: A Liberal Art by W.M. Priestley.
721
 
I am intentionally avoiding the standard term which, by the way, did not exist in Euler's time. One of the ugliest outgrowths of the "new math" was the premature introduction of technical terms.
George Polya, quoted in "The Euler Characteristic and Polya's Dream," by P Hilton and J. Pederson, American Mathematical Monthly, Feb. 1996.
320
 
To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.
George Polya, quoted in "Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)," by Israel Kleiner, Mathematics Teacher, Oct. 1988.
319
 
In the "commentatio" (note presented to the Russian Academy) in which his theorem on polyhedra (on the number of faces, edges and vertices) was first published Euler gives no proof. In place of a proof, he offers an inductive argument: he verifies the relation in a variety of special cases. There is little doubt that he also discovered the theorem, as many of his other results, inductively.
George Polya, quoted in "The Euler Characteristic and Polya's Dream," by P. Hilton and J. Pederson, American Mathematical Monthly, Feb. 1996.
318
 
[Hilbert] once had a student in mathematics who stopped coming to his lectures, and [he] was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: "I never thought he had enough imagination to be a mathematician."
George Polya, quoted in Mathematics Magazine, vol. 60, no. 5.
316
 
Mathematics has two faces. Presented in finished form, mathematics appears as a purely demonstrative science, but mathematics in the making is a sort of experimental science. A correctly written mathematical paper is supposed to contain strict demonstrations only, but the creative work of the mathematician resembles the creative work of the naturalist: observation, analogy, and conjectural generations, or mere guesses, if you prefer to say so, play an essential role in both. A mathematical theorem must be guessed before it is proved. The idea of a demonstration must be guessed before the details are carried out.
George Polya,
1150
 
Beauty in mathematics is seeing the truth without effort.
George Polya, quoted in Exploring Elementary Mathematics, by Julian Weissglass.
315
 
As the small pebble stirs the peaceful lake;
The centre mov'd, a circle straight succeeds,
Another still, and still another spreads.
Alexander Pope, quoted in Mathematically Speaking: A Dictionary of Quotations edited by C.C. Gaither and A.E. Cavazos-Gaither.
1244
 
If your goals are to make people more alike, to prepare them to be docile functionaries in some bureaucracy, and to prevent them from being vigorous, self-directed learners, then the standards of most schools are neither high nor low. They are simply apt.
Postman and Weingartner, From Teaching as a Subversive Activity
1615
 
[Paraphrasing Heisenberg] We have to remember that what we observe children doing in schools is not what they are, but children exposed to our methods of teaching. We have to remember that what we observe children doing in schools is not what they are, but children exposed to our methods of teaching.
Postman and Weingartner, from Teaching as a Subversive Activity
1617
 
The only places one finds such standards is in a school syllabus. They do not exist in natural, human learning situations, since they have nothing to do with the conditions of learning - with what the learner needs to be and to do in order to learn about learning, or indeed about anything.
Postman and Weingartner, From Teaching as a Subversive Activity
1616
 
1. Declare a five-year moratorium on the use of all textbooks
2. Have "English" teachers "teach" Math, Math teachers English, Social Studies teachers science, Science teachers Art, and so on.
3. Transfer all elementary teachers to high school and vice versa.
4. Require every teacher who thinks he knows his "subject" well to write a book on it.
5. Dissolve all "subjects", "courses", and "course requirements'.
6. Limit each teacher to three declarative sentences per class, and 15 interrogatives.
7. Prohibit teachers from asking any questions they already know the answers to.
8. Declare a moratorium on all tests and grades.
9. Require all teachers to undergo some form of psychotherapy as part of their inservice training
10. Classify teachers according to their ability and make the lists public.
11. Require all teachers to take a test prepared by students on what the students know.
12. Make every class an elective and withhold a teacher's monthly check if his students do not show any interest in going to next month's classes.
13. Require every teacher to take a one-year leave of absence every fourth year to work in some other "field" other than education.
14. Require each teacher to provide some sort of evidence that he or she has had a loving relationship with at least one other human being.
15. Require that all the graffiti accumulated in the school toilets be reproduced on large paper and be hung in the school halls.
16. There should be a general prohibition against the use of the following words and phrases:
Teach, syllabus, covering ground, I.Q., makeup, test, disadvantaged, gifted, accelerated, enhancement, course, grade, score, human nature, dumb, college material, and administrative necessity.
Postman and Weingartner, from Teaching as a Subversive Activity
1034
 
Science is not about control. It is about cultivating a perpetual condition of wonder in the face of something that forever grows one step richer and subtler than our latest theory about it. It is about reverence, not mastery.
Richard Powers, quoted in Keys to Infinity, by Clifford Pickover.
483
 
But dreaming just comes natural
Like the first breath of a baby
Like sunshine feeding daisies
Like the love hidden deep in your heart.
John Prine, from Donald and Lydia
1774
 
I emulate the Pythagoreans who even had a conventional phrase to express what I mean 'a figure and a platform, not a figure and a sixpence,' by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.
Proclus,
1225
 
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 to light out intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth…
Proclus,
1415
 
The real voyage of discovery consists not in seeking new landscapes but in having new eyes.
Marcel Proust,
323
 
All the flowers of tomorrow are in the seeds of yesterday.
Proverb,
1010
 
The Mean Value Theorem is the midwife of calculus - not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance.
E. Purcell and D. Varberg, from Calculus with Analytic Geometry.
619
 
Happiness, which is not the usual lot of a college president these days, may well be a worthy goal. You may think me frivolous in wishing your President happiness. But I do. I have seen so many unhappy college presidents. An unhappy president means an unhappy college. An unhappy college means a miserable experience for us all. You should wish him happiness. He can make your life joyful... [Being president] is a spiritual job.
Nathan M. Pusey, quoted by Peter Gomes, at the Investiture of Evan S. Dobelle as President of Westfield State College on 27 September, 2008.
1810
 
I'm sick to death of leadership. I'm not interested in leadership. I would like a great investment in a keener, sharper, more critical and more discerning followership. I would like to raise up a generation of decent followers who will not be seduced by the appearances of leadership. Who will have a higher command of the greatest of expectations. If we are producing any more Presidents of this and Chairman of that and innovators here and there, I wonder for the safety and well being of the world. Since 1642 Harvard has been producing leaders, and look at the state of the world. Have we not enough? Should we not have something else? Maybe your great vocation in this College is to produce a generation of critical followers who say 'I'll go this far and not further. I do not trust you. You are a fraud and a phony. I like you. I am willing to invest in you. We'll see.' That is what is really called for today. And I am hoping that the kind of liberal arts... will be the thing we need to put ourselves forward as a country.
Institutions are not built on the back of their Presidents... They are built on the backs of everybody here.
What makes a college good, in short, is its magic power, perennially renewed, to widen experience, and in so doing to work those transformations, even exaltations, in young minds and hearts - indeed in all of us - drawing us into fuller and deeper life and engendering in us processes of learning which, when sustained, enable us later, in less favorable circumstances, still to care, when the bloom of fresh excitement shall have passed.
Nathan M. Pusey, quoted by Peter Gomes, at the Investiture of Evan S. Dobelle as President of Westfield State College on 27 September, 2008.
1693
 
There is geometry in the humming of the strings.
There is music in the spacing of the spheres.
Pythagoras, from A Dictionary of Quotations in Mathematics by Nowlan
1669
 

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