The unique and peculiar character of mathematical reasoning is best exhibited in proofs of impossibility. When it is asserted that doubling the cube (i.e. in constructing the cube root of two) is impossible, the statement does not merely refer to a temporary limitation of human ability to perfom this feat. It goes far beyond this, for it proclaims that never , no matter what will anybody ever be able to construct the cube root of two...if the only instruments at his disposal are a straightedge and a compass. No other science, or for that matter no other discipline of human endeavor, can even contemplate anything of such finality.
Mark Kac and Stanislaw Ulam, from Mathematics and Logic.
1129
 
Science marches on blindly without regard to the real welfare of the human race or to any other standard, obedient only to the psychological needs of the scientists and of the government officials and corporate executives who provide the funds for research.
Theodore Kaczynski, quoted in "The Unabomber and the Bland Decade", in Scientific American, April, 1998.
614
 
You don't have to lay an egg to know if it tastes good.
Pauline Kael, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
191
 
A science of all these possible kinds of space [the higher dimensional ones] would undoubtedly be the highest enterprise which a finite understanding could undertake in the field of geometry... If it is possible that there could be regions with other dimensions, it is very likely that God has somewhere brought them into being.
Immanuel Kant, quoted in The Fourth Dimension: A Guided Tour of the Higher Universes by Rudy Rucker.
660
 
Science is organized knowledge. Wisdom is organized life.
Immanuel Kant, Critique of Pure Reason
1724
 
When a straight line, the mathematical evidence of a law, failed to appear on the paper, I suggested different kinds of logarithmic paper. If you cannot simplify the curve on one kind of paper, simplify the paper.
Theodore von Karman, quoted in Fractals: Form, Chance and Dimension by Benoit Mandelbrot, p. 273.
1755
 
Puzzles are made of the things that the mathematician, no less than the child, plays with, and dreams and wonders about, for they are made of things and circumstances of the world he [or she] live in.
Edward Kasner, quoted in Mathematics: A Human Endeavor, by Harold Jacobs.
192
 
We have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed.
Edward Kasner and James Newman,
193
 
Words of wisdom are spoken by children at least as often as scientists. The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it... At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex."
Edward Kasner and James Newman, from Mathematics and the Imagination.
688
 
The infinite in mathematics is alway unruly unless it is properly treated.
Edward Kasner and James Newman, from Mathematics and the Imagination.
670
 
There is a famous formula, perhaps the most compact and famous of all formulas -- developed by Euler from a discovery of de Moivre: e^(ipi) + 1 = 0... It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
Edward Kasner and James Newman, quoted in To Infinity and Beyond by Eli Maor.
652
 
Perhaps the greatest paradox of all is that there are paradoxes in mathematics.
Edward Kasner and James Newman, from Mathematics and the Imagination.
669
 
A person needs a little madness, or else they never dare cut the rope and be free.
Nikos Kazantzakis,
1019
 
Beauty is truth, truth beauty, - that is all
Ye know on earth, and all ye need to know.
John Keats, from Ode on A Grecian Urn
1506
 
Nothing you do for children is ever wasted.
Garrison Keillor,
1012
 
Tyrrany cannot defeat the power of ideas.
Helen Keller, quoted in the Holocaust Museu
971
 
The most pathetic person in the world is someone who has sight, but has no vision.
Helen Keller,
1457
 
I long to accomplish a great and noble task, but it is my chief duty to accomplish small tasks as if they were great and noble.
Helen Keller, quoted in Wisdom for the New Millennium edited by Helen Exley.
1076
 
I long to accomplish a great and noble tasks, but it is my chief duty to accomplish humble tasks as though they were great and noble. The world is moved along, not only by the mighty shoves of its heroes, but also by the aggregate of the tiny pushes of each honest worker.
Helen Keller,
1027
 
I can see, and that is why I can be happy, in what you call the dark, but which to me is golden. I can see a God-made world, not a manmade world.
Helen Keller,
1685
 
Security is mostly a superstition. It does not exist in nature.... Life is either a daring adventure or nothing.
Helen Keller,
1383
 
College is not the place to go for ideas.
Helen Keller,
1382
 
The highest result of education is tolerance.
Helen Keller, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
195
 
Life is either daring adventure or nothing.
Helen Keller,
194
 
I have visited sweatshops, factories, and crowded slums. If I could not see it, I could smell it. The foundation of society is laid upon a basis of . . . individualism, conquest and exploitation . . . A social order such as this, built upon such wrong and basic principles, is bound to retard the development of all. The output of a cotton mill or a coal mine is considered of greater importance than the production of healthy, happy-hearted and free human beings.
We, the people, are not free.
Our democracy is but a name.
Helen Keller,
1030
 
Snowflakes are one of nature's most fragile things, but just look what they can do when they stick together.
Vesta M. Kelly, quoted in The Oxford Book of Quotations, 3rd edition.
470
 
Mathematics is the only good metaphysics.
Lord Kelvin,
196
 
...it is the greatest achievement of a teacher to enable his students to surpass him.
John Kemeny, from "Rigor Versus Intuition in Mathematics," in Mathematics: People, Problems and Results, by D. Campbell and J. Higgins.
197
 
A considreable portion of my high school trigonometry course was devoted to the solution of oblique triangles... I have still not had an excuse for using my talents for solving oblique triangles. If a professional mathematician never uses these dull techniques in a highly varied career, why must all high school students devote several weeks to the subject?
John Kemeny, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz
447
 
If you have a large number of unrelated ideas, you have to get quite a distance away from them to get a view of all of them, and this is the role of abstraction. If you look at each too closely you see too many details. If you get far away things may appear simpler because you can only see the large, broad outlines; you do not get lost in petty details.
John Kemeny, from "Rigor Versus Intuition in Mathematics," in Mathematics: People, Problems and Results, by D. Campbell and J. Higgins.
198
 
We must embrace pain and burn it as fuel for our journey.
Miyazawa Kenji,
891
 
As we express our gratitude, we must never forget that the highest appreciation is not to utter words but to live by them.
John Fitzgerald Kennedy,
834
 
Without belittling the courage with which men have died, we should not forget those acts of courage with which men...have lived. The courage of life is often a less dramatic spectacle than the courage of a final moment; but it is no less a magnificent mixture of triumph and tragedy.
John Fitzgerald Kennedy, Profiles in Courage
828
 
A man does what he must -- in spite of personal consequences, in spite of obstacles and dangers and pressures -- and that is the basis of all human morality.
John F. Kennedy, quoted in Wisdom for the New Millennium edited by Helen Exley.
1068
 
There can be no freedom without learning and learning without freedom is always in vain.
John F. Kennedy,
1239
 
When I was young, all of my radical friends were in reform school. Today, all of my radical friends are in school reform.
Dan Kennedy, from "A Tale of Two CD's," American Mathematical Monthly, Aug.- Sept., 1994.
199
 
Only those who dare to fail greatly can achieve greatly.
John F. Kennedy,
1263
 
The greater our knowledge increases, the greater our ignorance unfolds.
John F. Kennedy, quoted in A Teacher's Treasury of Quotations, by Bernard E. Farber.
200
 
We essentially spend twelve years getting our students ready for calculus, and when they get there, they discover that it is 300 years old, filled with the same calculations they hated in high school, and not exactly worth twelve years of anticipation. So they shinny down the mathematics tree and strike out into the forest, armed at least with those twelve rich years of valuable mathematics learning: trigonometric identities, the rational-root theorem, synthetic division, side-angle-side, FOIL, the commutative property of addition -- hey, you name it. Then, on the first day on the job out in the real world, someone notices on their transcript and says with relief, "At last, someone who knows some math! Come here and explain this spreadsheet to me." What will there twelve years do for them then?
Dan Kennedy, from "Climbing around on the tree of mathematics", The Mathematics Teacher, vol. 88, no. 6, Sept. 1995, pp. 460 - 465
1174
 
Every reader [of this article] can probably point with fondness to a mathematics teacher in the past who has made a difference in his or her life. However, I dare say that this fondness will arise because that teacher taught you about studying, or perseverance, or believing in yourself, or some such enduring lesson of human existence; it will probably not be because that teacher taught you how to rationalize the denominator or how to factor a trinomial -- even though that is what the two of you spent your time together doing.
Dan Kennedy, from "Climbing around on the tree of mathematics", The Mathematics Teacher, vol. 88, no. 6, Sept. 1995, pp. 460 - 465
1173
 
My thesis advisor suggested that the entire body of Mathematical Knowledge was very much like a tree. The main body was this big trunk of general knowledge, from which emerged smaller branches of specialization, from which finally sprouted twigs of truly arcane trivia... Doctoral dissertations, in other words, were not about branches; they were about twigs.
Dan Kennedy, from "Climbing around on the tree of mathematics", The Mathematics Teacher, vol. 88, no. 6, Sept. 1995, pp. 460 - 465
1172
 
But why, some say, the moon? Why choose this as our goal? And they may well ask why climb the highest mountain? Why, 35 years ago, fly the Atlantic? Why does Rice play Texas? We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not only because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win . . .
John F. Kennedy, Jr., Rice University, 12 September 1962
1142
 
The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.
Johannes Kepler,
1477
 
There must be some definite cause why, whenever snow begins to fall, its initial formations invariably display the shape of a six-cornered starlet. For if it happens by chance, why do they not fall just as well with five corners or with seven? Why always six?
Johannes Kepler, from On the Six-Cornered Snowflake
901
 
Where there is matter, there is geometry.
Johannes Kepler,
201
 
Mother Astronomy would certainly starve if daughter Astrology did not earn the bread of both.
Johannes Kepler, quoted in Calculus Gems, by George F. Simmons.
202
 
I [was] embarrassed by my discourtesy in having appeared before you without a New Year's present... Just then by a happy chance water-vapour was condensed by the cold into snow, and specks of down fell here and there on my coat, all with six corners and feathered radii. 'Pon my word, here was something smaller than any drop, yet with a patterns; here was the ideal New Year's gift for the devotee of Nothing, the very thing for a mathematician to give, who has Nothing and receives Nothing, since it comes down from heaven and looks like a star.
Johannes Kepler, from On the Six-Cornered Snowflake
900
 
The only people for me are the mad ones, the ones who are mad to live, mad to talk, mad to be saved... the ones who never yawn and say a commonplace things, but burn, burn, burn, like the fabulous yellow roman candles exploding like spiders across the stars.
Jack Kerouac, quoted in The Harper Book of Quotations, 3rd edition.
924
 
You can count how many seed are in the apple, but not how many apples are in the seed.
Ken Kersey, from Words I Wish I Wrote by Robert Fulgham
942
 
Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks or hewing wood, no more than painting is the art of mixing colors on a pallette, no more than the science of geology is the art of breaking rocks, or the science of anatomy the art of butchering.
C.J. Keyser, quoted in Memorobilia Mathematica, by Robert Moritz
452
 
The most difficult thing that teaching has to do is to give worthy sense of the meaning and scope of a great idea.
C. J. Keyser,
203
 
Instruction begins when you, the teacher, learn from the learner; put yourself in his place so that you may understand... what he learns and the way he understands it.
Kierkegaard,
1028
 
It is possible to provide some heuristic "arguments" for this phenomenon [negative times negative equals positive] but they are mostly unconvincing and mainly serve only for the amusement of the mathematician who invents them... The law of signs is actually easy to comprehend provided you were taught mathematics by... someone who believed in teaching, and not patronizing, the students.
Jerry P. King, from The Art of Mathematics.
206
 
And there are three dimensions of any complete life to which we can fitly give the words of this text: length, breadth, and height. Now the length of life as we shall use it here is the inward concern for one’s own welfare. In other words, it is that inward concern that causes one to push forward, to achieve his own goals and ambitions. The breadth of life as we shall use it here is the outward concern for the welfare of others. And the height of life is the upward reach for God. Now you got to have all three of these to have a complete life.
Martin Luther King Jr, from "The Three Dimensions of a Complete Life"
1725
 
The means by which we live have outdistanced the ends for which we live. Our scientific power has outrun our spiritual power. We have guided missiles and misguided men.
Martin Luther King, jr., quoted in The Oxford Book of Quotations
442
 
Change does not roll in on the wheels of inevitability, but comes through continuous struggle.
Martin Luther King, Jr.,
1698
 
Outside of the closed circle of professional mathematicians, almost nothing is known of the true nature of mathematics or mathematics research.
Jerry P. King, from The Art of Mathematics.
205
 
I have a dream... I have a dream today... And if America is to be a great nation this must become true.
Martin Luther King, Jr.,
210
 
Unarmed truth is the most powerful thing in the universe.
Martin Luther King, Jr.,
209
 
It's far easier to go from the real numbers to the complex numbers than... from the rationals to the reals.
Jerry P. King, from The Art of Mathematics.
207
 
It is a tortuous logic that views the tragic results of segregation and discrimination as an argument for the continuation of it.
Martin Luther King, Jr., From the speech "The American Dream" 6 June, 1961 at Lincoln University
817
 
One's intellectual and aesthetic life cannot be complete unless it includes an appreciation of the power and the beauty of mathematics. Simply put, aesthetic and intellectual fulfillment requires that you know about mathematics.
Jerry P. King, from The Art of Mathematics.
204
 
The beautiful thing about learning is nobody can take it away from you.
B.B. King,
1055
 
The function of education is to teach one to think intensively and to think critically. Intelligence plus character - that is the goal of true education.
Martin Luther King, Jr.,
208
 
Darkness cannot drive out darkness; only light can do that. Hate cannot drive out hate; only love can do that. Hate multiplies hate, violence multiplies violence, and toughness multiplies toughness in a descending spiral of destruction... The chain reaction of evil -- hate begetting hate, wars producing more wars -- must be broken, or we shall be plunged into the dark abyss of annihilation.
Martin Luther King, Jr.,
973
 
Non-violence is a powerful and just weapon. It is a weapon unique in history, which cuts without wounding and enobles the man who wields it. It is a sword that heals.
Martin Luther King, jr., quoted in the The Harper Book of Quotations, 3rd ed.
757
 
I keep six honest serving-men
They taught me all I know;
Their names are What and Why and When
And How and Where and Who.
Rudyard Kipling, quoted in The Harper Book of Quotations.
972
 
The world is so full of a number of things, that I'm sure we should all be happy as kings.
Kipling,
1208
 
Words, of course, are the most powerful drug known to [hu]mankind.
R. Kipling,
957
 
Numbers are friends, for me, more or less. It doesn't mean the same for you, does it - 3,844? For you it's just a three and an eight and a four and a four. But I say, "Hi! 62 squared."
Wim Klein, quoted in "The Twins" by Oliver Sacks.
218
 
Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.
Felix Klein,
213
 
The influence of physics in stimulating the creation of such mathematical entities as quaternions, Grassmann's hypernumbers, and vectors should be noted. These creations became part of mathematics.
M. Klein, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1123
 
I should like to bring up again for emphasis... points, in which my exposition differs especially from the customary presentation in the textbooks:
1. Illustration of abstract considerations by means of figures.
2. Emphasis upon its relation to neighboring fields, such as calculus or differences and interpolation...
3. Emphasis on historical growth. It seems to me extremely important that precisely the prospective teacher should take account of all of these.
Felix Klein, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1095
 
It is well known that the central problem of the whole of modern mathematics is the study of transcendental functions defined by differential equations.
Felix Klein, quoted in Memorabilia Mathematica, by R.E. Moritz.
212
 
The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.
Felix Klein, quoted in Single Variable Calculus, by James Stewart.
211
 
Can we not at least have a better appreciation of students' difficulties... having witnessed mathematicians of the first rank make mistakes, "prove" erroneous theorems, and often come to the right conclusions for insufficient or invalid reasons?
Israel Kleiner, from "Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)," Mathematics Teacher, Oct. 1988.
219
 
A elegantly executed proof is a poem in all but the form in which it is written.
Morris Kline, quoted in Exploring Elementary Mathematics by Julian Weissglass.
214
 
God exists since mathematics is consistent and the devil exists since we cannot prove the consistency.
Morris Kline, from Mathematical Thought from Ancient to Modern Times.
215
 
The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present- day kaleidoscope of events.
Morris Kline,
216
 
The shortest path between two truths in the real domain passes through the complex domain.
Morris Kline, quoted in "Thinking the Unthinkable: The Story of Complex
217
 
Indeed, one of the interesting questions that the history answers is what survives in mathematics. History makes its own and sounder evaluations.
Morris Kline, from Mathematical Thought from Ancient to Modern Times.
1089
 
Almost everyone knows that mathematics serves the very practical purpose of dictating engineering design. Fewer people seem to be aware that mathematics carries the main burden of scientific reasoning and is the core of the major theories of physical science. It is even less widely known that mathematics has determined the direction and content of philosophical thought, has destroyed and rebuilt religious doctrine, has supplied substance to economics and political theories, has fashioned major painting, musical, architectural, and literary styles, has fathered our logic, and has furnished the best answers we have to fundamental questions about the nature of man and his universe. As the embodiment and most powerful advocate of the rational spirit, mathematics has invaded domains ruled by authority, custom, and habit, and supplanted them as the arbiter of thought and action. Finally, as an incomparably fine human achievement mathematics offers satisfactions and aesthetic values at least equal to those offered by any other branch of our culture. Despite these by no means modest contributions to our life and thought, educated people almost universally reject mathematics as an intellectual interest. This attitude toward the subject is, in a sense, justified. School courses and books have presented 'mathematics' as a series of apparently meaningless technical procedures. Such material is as representative of the subject as an account of the name, position, and function of every bone in the human skeleton is representative of the living, thinking, and emotional being called man. Just as a phrase either loses meaning or acquires an unintended meaning when removed from its context, so mathematics detached from its rich intellectual setting in the culture of our civilization and reduced to a series of techniques has been grossly distorted… Consequently, a subject that is basic, vital, and elevating is neglected and even scorned by otherwise highly educated people. Indeed, ignorance of mathematics has attained the status of a social grace.
Morris Kline, from the Introduction to Mathematics in Western Culture.
1414
 
Much research for new proofs of theorems already correctly established is undertaken simply because the existing proofs have no aesthetic appeal. There are mathematical demonstrations that are merely convincing; to use a phrase of the famous mathematical physicist, Lord Rayleigh, they "command assent." There are other proofs "Which woo and charm the intellect. They evoke delight and an overpowering desire to say, Amen, Amen." An elegantly executed proof is a poem in all but the form in which it is written.
Morris Kline, from Proofs Without Words II
1211
 
The usual courses in mathematics are... deceptive in a basic respect. They give an organized logical presentation which leaves the impression that mathematicians go from theorem to theorem almost naturally, that mathematicians can master any difficulty, and that the subjects are completely thrashed out and settled. The succession of theorems overwhelms the student, especially if he is just learning the subject... The polished presentations in the courses fail to show the struggles of the creative process, the frustrations, and the long arduous road mathematicians must travel to attain a sizable structure. Once aware of this, the student will not only gain insight but derive courage to pursue tenaciously his own problems and not be dismayed by the incompleteness or deficiencies in his own work.
Morris Kline, from Mathematical Thought from Ancient to Modern Times.
1094
 
The back-to-basics movement, in part a reaction to the new math, is not the solution to decent mathematics education. It means to me the meaningless drill in techniques that was common twenty and more years ago. That type of education failed, as is evidence by the attitude of most intelligent, educated people towards mathematics. It will almost surely fail again. It may not be worse that the new math but it is surely not better.
Morris Kline, Quoted in "Interview of Morris Kline" by G.L. Alexanderson, from Mathematical People edited by D.J. Albers and G.L. Alexanderson.
776
 
This work [Mathematical Thought from Ancient to Modern Times] is part of my long-time efforts to humanize the subject of mathematics... [Depart] from the traditional dry-as-dust mathematics textbook.
Morris Kline, from Mathematical Thought from Ancient to Modern Times.
1090
 
If I could come back after five hundred years and find that the Riemann hypothesis or Fermat's last "theorem" was proved, I would be disappointed, because I would be pretty sure, in view of the history of attempts to prove these conjectures, that an enormous amount of time had been spent on proving theorems that are unimportant to the life of man... When medicine has discovered how to cure or prevent cancer, heart troubles, birth defects, mental disorders, and other diseases, I would be so overjoyed that I would be more tolerant of even that large part of mathematical research that is useless.
Morris Kline, Quoted in "Interview of Morris Kline" by G.L. Alexanderson, from Mathematical People edited by D.J. Albers and G.L. Alexanderson.
777
 
Instead of presenting mathematics as rigorously as possible, present it as intuitively as possible. Accept and use without mention any facts that are so obvious that students do not recognize that they are using them. Students will not lose sleep worrying about whether a line divides the plane into two parts. Prove only what the students think requires proof. The ability to appreciate rigor is a function of the age of the student and not the age of mathematics.
Morris Kline, Quoted in "Interview of Morris Kline" by G.L. Alexanderson, from Mathematical People edited by D.J. Albers and G.L. Alexanderson.
774
 
Logic is the art of going wrong with confidence.
Morris Kline, quoted in The Magic of Mathematics, by Theoni Pappas.
494
 
Basically, art and mathematics speak the same language.
Konrad Knopp, from A Dictionary of Quotations in Mathematics by Nowlan
1663
 
I wanted [the novelette Surreal Numbers] to provide some material that would help to overcome one of the most serious shortcomings in our present educational system, the lack of training for research work; there is comparatively little opportunity for students to experience how new mathematics is invented, until they reach graduate school.
I decided that creativity can't be taught using a textbook, but that an 'anti-text' such as this novelette might be useful… My aim was to show how mathematics can be 'taken out of the classroom and into life,' and to urge readers to try their own hands at exploring abstract mathematical ideas.
Donald Knuth, from the postscript of Surreal Numbers: How Two Ex-students Turned on to Pure Mathematics and Found Total Happiness.
1787
 
In my opinion the two greatest weaknesses in our present mathematics education are the lack of training in creative thinking and the lack of practice in technical writing.
Donald Knuth, from the postscript of Surreal Numbers: How Two Ex-students Turned on to Pure Mathematics and Found Total Happiness.
1751
 
Science is what we understand well enough to explain to a computer. Art is everything else we do.
Donald Knuth, from the Introduction to A=B by M. Petkovsek, H. Wilf, and D. Zeilberger.
703
 
[The Euclidean algorithm is] the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day.
Donald Knuth, quoted in "Media Highlights," The College Mathematics Journal, March 1996.
220
 
There are young people out there cutting raw cocaine with chemicals from the local hardware store. They are manufacturing new highs and new products buy soaking marijuana in ever changing agents, and each of these new drugs is more addictive, more deadly and less costly than the last. How is it that we have failed to tap that ingenuity, that sense of experimentation? How is it that these kids who can measure grams and kilos and can figure out complex monetary transactions cannot pass a simple math or chemistry test?
Senator Kohl, from the U.S. Senate Hearing: "Crisis in Math and Science Education."
221
 
Data worship results in a myopic view of what the world could and should be. Children, we might remind corporate America, are more than math and science scores. While math and science play important roles in our lives, there are other scores we might help children increase: their creativity score, their empathy score, their resiliency score, their curiosity score, their integrity score, their thoughtfulness score, their take-initiative score, their innovation score, their critical thinking score, their passion score, their problem-solving score, their refusal to follow leaders who lie to them score, their democratic engagement score...and so forth.
Philip Kovacs, from "Gates, Buffett and the Corporatization of Children" available at http://www.commondreams.org/views06/0628-30.htm .
1245
 
It is impossible to be a mathematician without being a poet in soul.
Sofia Kovalevskaia, quoted in From Agnessi to Zeno, by Sanderson Smith.
222
 
Say what you know, do what you must, come what may.
Sofia Kovalevskaia,
223
 
Many who have never had the occasion to discover more about mathematics consider it a dry and arid science. In reality, however, it is a science which demands the greatest imagination.
Sofia Kovalevskaia, quoted in The Mathematical Universe, by W. Dunham.
224
 
The poor learn to live with patience, and the rich with guilt. Few of us ever learn to live at peace with our own conscience.
Jonathan Kozol, from The Night is Dark and I am Far From Home
1133
 
In New York, the difference is twice that high. The kids up in the Bronx that I write about get a little over $11,000 per pupil, per year. But lift up one of them in your grown-up arms and plunk her down 10 miles away in the Westchester suburb of Bronxville, and she'd be getting $19,000 every single year.
Jonathan Kozol,
1460
 
The percentage of black children who now go to integrated public schools is at its lowest level since 1968. If you took a photograph of the classes I visit in New York, Chicago or St. Louis, it would look exactly like a class from Alabama in the 1940s.
Jonathan Kozol,
1459
 
If you could lead through testing, the U.S. would lead the world in all education categories. When are people going to understand you don't fatten your lambs by weighing them?
Jonathan Kozol,
225
 
To live guilt-free and conscience-clean within a world of pain that is, in large part at the very least, of our own making, is not freedom worth respect.
Jonathan Kozol, from The Night is Dark and I am Far From Home
1134
 
Teachers are my heroes; they are the most courageous people in this country.
Jonathan Kozol, at Westfield State College's 157th Commencement.
519
 
Getting a Rhodes scholarship was easy compared to being a first year teacher in the Boston schools... Teaching little kids is hard work.
Jonathan Kozol, at Westfield State College's 157th Commencement.
511
 
We are victims of a desperate fraud if we believe that we can turn the tables in this land without the price of powerful confrontation and upheaval.
Jonathan Kozol, from The Night is Dark and I am Far From Home
1135
 
I'm not Catholic. I'm not Protestant. I happen to be Jewish. So I can't take communion. But I feel I take communion every day and every night when I'm with little children in the South Bronx because there is something sacred in their presence. These children are my religion.
Jonathan Kozol, at Westfield State College's 157th Commencement.
513
 
There is not even a universally accepted definition of the term "fractal". It seems that if one does not prove theorems (as, evidently, fractal geometers do not), then one does not need definitions. One notable difference between fractal geometry and calculus is that fractal geometry has not solved any problems. It is not even clear that it has created any new ones.
Steven Krantz, from "Fractal Geometry," in The Mathematical Intelligencer, vol. 11, no. 4, 1989.
474
 
Mathematicians today have paid too much attention to their beloved specialties and not enough to good writing, good exposition, incisive judgement, and the overall health of the discipline.
Steven Krantz, from "See no evil, hear no evil, speak no evil" in Notices of the American Mathematical Society, vol. 45, no. 9, October 1998.
743
 
God made the integers, all else is the work of man.
Leopold Kronecker,
226
 
What good your beautiful proof on [the transcendence of] pi: Why investigate such problems, given that irrational numbers do not even exist?
Leopold Kronecker, quoted in Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, by Manfred Schroder.
569
 
Analysis does not owe its really significant successes of the last century to any mysterious use of sqrt(-1), but to the quite natural circumstances that one has infinitely more freedom of mathematical movement if he lets quantities vary in a plane instead of only on a line.
Leopold Kronecker, quoted in Theory of Complex Functions, by Reinhold Remmert.
227
 
All results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of the integers.
Leopold Kronecker,
228
 
Our concern must be to live while we are alive… to release our inner selves from the spiritual death that comes with living behind a facade designed to conform to external definitions of who and what we are.
Elisabeth Kubler-Ross, quoted in The Scandalous Gospel of Jesus, by Peter Gomes, p. 45
1737
 
The success of a paradigm… is at the start largely a promise of success discoverable in selected and still incomplete examples. Normal science consists in the actualization of that promise, an actualization achieved by extending the knowledge of those facts that the paradigm displays as particularly revealing, by increasing the extent of the match between those facts and the paradigm's predictions, and by further articulation of the paradigm itself.
Thomas Kuhn, The Structure of Scientific Revolutions, p. 23-4.
1799
 
The study of paradigms… is what mainly prepares the student for membership in the particular scientific community with which he will later practice. Because he there joins men who learned the bases of their field from the same concrete models, his subsequent practice will seldom evoke overt disagreement over fundamentals. Men whose research is based on shared paradigms are committed to the same rules and standards of scientific practice.
Thomas Kuhn, The Structure of Scientific Revolutions, p. 11.
1765
 
Paradigms gain their status because they are more successful than their competitors in solving a few problems that the group of practitioners has come to recognize as acute.
Thomas Kuhn, The Structure of Scientific Revolutions, p. 23.
1798
 
Few people who are not actually practitioners of a mature science realize how much mop-up work of this sort a paradigm leaves to be done or quite how fascinating such work can prove in the execution. And these points need to be understood. Mopping-up operations are what engage most scientists throughout their careers. They constitute what I am calling normal science… That enterprise seems an attempt to force nature into the preformed and relatively inflexible box that the paradigm supplies. No part of the aim of normal science is to call for new sorts of phenomena; indeed those that will not fit the box are often not seen at all. Nor do scientists normally aim to invent new theories, and they are often intolerant of those invented by others.
Thomas Kuhn, The Structure of Scientific Revolutions, p. 24.
1711
 
Indeed, the only true serious questions are the ones that even a child can formulate.
Milan Kundera,
1335
 

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