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
 
Such is the irresistible nature of truth: that all it asks, and all it wants, is the liberty of appearing.
Thomas Paine,
911
 
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
 
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
 
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 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
 
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
 
’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
 
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
 
Bees…by virtue of a certain geometrical forethough…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.
789
 
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
 
Only wimps do the general case. True teachers tackle examples.
Parlett, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1120
 
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
 
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 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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
Computers are useless. They can only give you answers.
Pablo Picasso, quoted in A Primer of Mathematical Writing by Steven G. Krantz.
739
 
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
 
It takes a long time to become young.
Pablo Picasso, quoted in The Harper Book of Quotations, edited by Robert I. Fitzhenry.
625
 
Art is a lie that makes us realize the truth.
P. Picasso, quoted in Discovering Geometry, by M. Serra.
297
 
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
 
When you are insane, you are busy being insane - all the time... When I was crazy, that's all I was.
Sylvia Plath,
865
 
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
 
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
 
God ever geometrizes.
Plato,
304
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
[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
 
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
 
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
 
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
 
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
 
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
 
Beauty in mathematics is seeing the truth without effort.
George Polya, quoted in Exploring Elementary Mathematics, by Julian Weissglass.
315
 
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
 
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
 
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
 

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