Indeed, only a few are mathematically gifted in the sense that they are endowed with the talent to discover new mathematical facts. But by the same token, only a few are musically gifted in that they are able to compose music. Nevertheless there are many who can understand and perhaps reproduce music, or who at least enjoy it. We believe that the number of people who can understand simple mathematical ideas is not relatively smaller than the number of those who are commonly called musical, and that their interest will be stimulated if only we can eliminate the aversion toward mathematics that so many have acquired from childhood experiences.
Hans Rademacher, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz
449
 
Courage doesn't always roar. Sometimes courage is the quiet voice at the end of the day saying, "I will try again tomorrow."
Mary Anne Radmacher,
801
 
Yes I know my enemies,
They're the teachers who taught me to fight me,
Compromise, conformity, assimilation, submission,
Ignorance, hypocrisy, brutality, the elite,
All of which are American dreams.
Rage Against the Machine,
989
 
An equation for me has no meaning unless it expresses a thought of God.
Ramanujan, quoted in The Man Who Knew Infinity by Rober Kanigel
1144
 
A computer lets you make more mistakes faster than any invention in human history -- with the possible exception of handguns and tequila.
Mitch Ratcliffe,
913
 
In an era when the world seems obsessed with speed, measured in nanoseconds, and computer power measured in gigabytes, very few of us take the time to think through what we are doing; to consider where we are going; to measure the effects of our actions on others.
Hunter R. Rawlings, III, from the 133rd Commencement Address at Cornell University, 5/27/01.
909
 
Which side are you on?
Florence Reese,
1801
 
The simplest schoolboy is now familiar with truths for which Archimedes would have sacrificed his life.
Ernest Renan, quoted in The Oxford Book of Quoations, 3d edition
443
 
In high-quality teaching, the process of inquiry, not merely "giving instruction," is the very heart of what teachers do.
Glenn Report, "Before It's Too Late: A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century," Sept. 2000.
1087
 
The basic teaching style in too many mathematics and science classes today [in the United States] remains essentially what it was two generations ago. By contrast, teaching innovation and higher student performance are well documented in other countries.
Glenn Report, "Before It's Too Late: A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century," Sept. 2000.
1086
 
...the deeper, intrinsic value of mathematical and scientific knowledge shapes and defines our common life, history, and culture. Mathematics and science are primary sources of lifelong learning and the progress of our civilization.
Glenn Report, "Before It's Too Late: A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century," Sept. 2000.
1085
 
Music is math without the formulas.
Tim Reynolds, from Highlights from Dave Matthews and Tim Reynolds Live at Luther College
1147
 
Teaching was the hardest work I had ever done, and it remains the hardest work I have done to date.
Ann Richards,
1024
 
Both class and race survived education, and neither should. What is education then? If it doesn't help a human being to recognize that humanity is humanity, what is it for? So you can make a bigger salary than other people?
Beah Richards, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
326
 
Crime is a social problem, and education is the only real deterrent. Look at all of us in prison; we were all truants and dropouts, a failure of the educational system. Look at your truancy problem, and you're looking at your future prisoners. Put the money there.
Wilbert Rideau, from The Angolite.
327
 
Look around me in this place. It's a graveyard, a human wasteland... America has embraced vengeance as its criminal-justice philosophy. People don't want solutions to crime, they only want to feel good. That's what the politicians are doing… offer[ing] them a plate of vindictiveness.
Wilbert Rideau, from The Angolite.
677
 
Crime is a social problem, and education is the only real deterrent. Look at all of us in prison: we were all truants and dropouts, a failure of the education system. Look at your truancy problem, and you"re looking at your future prisoners. Put the money there.
Wilbert Rideau, from Time Magazine, 23 August, 1993.
676
 
The recognition of the fact that infinite series fall into two classes [according to whether the limit is independent of the ordering of the terms or not] constitutes a turning-point in the conceptualization of the infinite in mathematics.
Bernhard Riemann, quoted in Theory of Complex Functions, by Reinhold Remmert.
328
 
Artists who are interested in four dimensional space are not motivated by a desire to illustrate new physical theories, nor by a desire to solve mathematical problems. We are motivated by a desire to complete our subjective experience by inventing new aesthetic and conceptual capabilities. We are not in the least surprised, however, to find physicists and mathematicians working simultaneously on a metaphor for space in which paradoxical three dimensional experience are resolved only by a four dimensional space. Our reading of the history of culture has shown us that in the development of new metaphors for space artists, physicists, and mathematicians are usually in step.
Tony Robbin, quoted in The Fourth Dimension and non-Euclidean Geometry in Modern Art by Linda Dalrymple Henderson.
668
 
So you think that you're a failure, do you? Well, you probably are. What's wrong with that? In the first place, if you've any sense at all you must have learned by now that we pay just as dearly for our triumphs as we do for our defeats. Go ahead and fail. But fail with wit, fail with grace, fail with style. A mediocre failure is as insufferable as a mediocre success.
Tom Robbins,
1425
 
We waste time looking for the perfect lover, instead of creating the perfect love.
Tom Robbins,
1333
 
Those who expect to reap the blessings of freedom must undergo the fatigue of supporting it.
Tom Robbins, from Words I Wish I Wrote by Robert Fulgham
948
 
All this attention has been gratifying but also embarassing. What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered as a mathematician should, simply for the theorems I have proved and the problems I have solved.
Julia Robinson, quoted in Modern Mathematicians, by Harry Henderson.
486
 
I just don't want to die without knowing the answer [to Hilbert's 10th problem].
Julia Robinson,
1407
 
Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc... But the next quite logical step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus.
Abraham Robinson,
439
 
The infinitely small and infinitely large numbers of a non-standard model of Analysis are neither more nor less real than, for example, the standard irrational numbers.
Abraham Robinson, quoted in Mathematical Expeditions by R. Laubenbacher and D. Pengelley.
961
 
I think that I have always had a basic liking for the natural numbers. To me they are the one real thing. We can conceive of a chemistry that is different from ours, or a biology, but we cannot conceive of a different mathematics of numbers. What is proved about numbers will be a fact in any universe.
Julia Robinson,
1206
 
The collection of all number systems is not a finished totality whose discovery was complete around 1600, or 1700, or 1800, but that it has been and still is a growing and changing area, sometimes absorbing new systems and sometimes discarding old ones, or relegating them to the attic.
Abraham Robinson, quoted in Mathematical Expeditions by R. Laubenbacher and D. Pengelley.
963
 
[In justifying his ideas on infinitesimals, Leibniz relied] on the fact that "tout se gouverne par raison" [everything is controlled by ratio]-- hardly a consistency proof in our sense. However, even in the twentieth century no mathematician is known to have changed his profession because Godel showed that no conclusive proof for the consistency of arithmetic is possible. Nor can it be said that a theory of infinitesimals is less intuitive than the delta, epsilon procedure and its ramifications and developments.
Abraham Robinson, quoted in Mathematical Expeditions by R. Laubenbacher and D. Pengelley.
962
 
All finite things reveal infinitude.
Theodore Roethke, quoted in To Infinity and Beyond: A Cultural History of the Infinite by Eli Maor.
1497
 
In any moment of decision, the best thing you can do is the right thing, the next best thing is the wrong thing, and the worst thing you can do is nothing.
Theodore Roosevelt,
829
 
The future belongs to those who believe in the beauty of their dreams.
Eleanor Roosevelt,
878
 
Far and away the best prize that life offers is the chance to work hard at work worth doing.
Theodore Roosevelt,
830
 
It is not the critic who counts;
not the man who points out how the strong man stumbles,
or where the doer of deeds could have done them better.
The credit belongs to the man who is actually in the arena,
whose face is marred by dust and sweat and blood,
who strives valiantly;
who errs and comes short again and again;
because there is not effort without error and shortcomings;
but who does actually strive to do the deed;
who knows the great enthusiasm,
the great devotion,
who spends himself in a worthy cause,
who at the best knows in the end the triumph of high achievement and who at the worst,
if he fails,
at least he fails while daring greatly.
So that his place shall never be with those cold and timid souls who know neither victory nor defeat.
Theodore Roosevelt,
1778
 
The reason that fiction is more interesting than any other form of literature, to those who really like to study people, is that in fiction the author can really tell the truth without humiliating himself.
Eleanor Roosevelt,
1002
 
The school is the last expenditure upon which America should be willing to economize.
Franklin D. Roosevelt,
596
 
When you come to the end of your rope, tie a knot and hang on.
Franklin D. Roosevelt,
1443
 
Do what you can, with what you have, where you are.
Theodore Roosevelt, quoted in Wisdom for the New Millennium edited by Helen Exley.
1067
 
To announce that there must be no criticism of the President, or that we are to stand by the President, right or wrong, is not only unpatriotic and servile, but is morally treasonable to the American public.
Theodore Roosevelt,
849
 
It is not the critic who counts, not the man who points out how the strong man stumbled, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena; whose face is marred by dust and sweat and blood.
Theodore Roosevelt,
330
 
Any question becomes unanswerable if we do not permit ourselves a universe large enough to deal with the question. Ax = B is generally unsolvable in a universe of positive integers. Likewise, generic angles become untrisectable, cubes unduplicatable, and so on, in a universe limited by rulers and compasses.
I claim that Godelian noncomputability results are a symptom, arising within mathematics itself, indicating that we are trying to solve problems in too limited a universe of discourse. The limits in question are imposed in mathematics by an excess of "rigor," and in science by cognate limitations of "objectivity" and "context independence." In both cases, our universes are limited, not by the demands of problems that need to be solved but by extraneous standards of rigor. The result, in both cases, is a mind-set of reductionism, of looking only downward toward subsystems, and never upward and outward.
Robert Rosen, from Essays on Life Itself.
1282
 
It is extremely hard for mathematicians to do expository writing. It is not in our nature. In fact, the very nature of mathematical meaning and grammar militates against it. However, this puts us at a distinct disadvantage relative to other sciences... Good exposition should be valued, not only for the success in communication but also as evidence of real mathematical insight. It is no accident that among our greatest mathematicians are our greatest teachers and expositors.
Hugo Rossi, from "From the Editor", Notices of the American Mathematical Society, vol. 42, no. 1, January 1995.
720
 
The goal of teaching is learning, not teaching.
Hugo Rossi,
331
 
God created infinity, and man, unable to understand infinity, had to invent finite sets.
Gian Carlo Rota, quoted in "A Quote a Day Educates" by Monte Zerger, Mathematical Intelligencer, vol. 20, no. 2, Spring 1998.
681
 
A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.
Gian-Carlo Rota, quoted in Excursions in Calculus, by Robert M. Young.
332
 
The facts of matheamtics are verified and presented by the axiomatic method. One must guard, however, against confusing the presentation of mathematics with the content of mathematics. As axiomatic presentation of a mathematical fact differs from the fact that is being presented as medicine differs from food. It is true that this popular medicine is necessary to keep the mathematician at a safe distance from the self-delusions of the mind. Nonetheless, understanding mathematics means being able to forget the medicine and enjoy the food.
Gian-Carlo Rota, from Indiscrete Thoughts.
780
 
Those few colleagues who are successful at teaching undergraduate [mathematics] courses should earn our thanks as well as our respect.
Gian-Carlo Rota, from Indiscreet Thoughts
1543
 
One can spend a lifetime working on mathematics without ever having any idea whether mathematical items exist, nor does one have to care about such a question. The existence of mathematical items is a chapter in the philosophy of mathematics that is devoid of consequence. We do not object to anyone who chooses to worry about existence… Discussions of 'existence' are motivated by deep-seated emotional cravings for permanence which are of psychiatric rather than philosophical interest. Verifying the existence of anything is like asking for an authorization to proceed while failing to realize that no such authorization is required.
Gian-Carlo Rota, from Indiscreet Thoughts
1542
 
Mathematics can save the world from the flakes by unmasking them and by contributing some hard thinking… This is the biggest chance we have had in a long while to make a lasting contribution to the well-being of Science. Let us not botch it as we did with the few other chances we have had in the past.
Gian-Carlo Rota, from Indiscreet Thoughts
1541
 
Grade school teachers, high school teachers, administrators and lobbyists are as much mathematicians as you or Hilbert… They contribute to the well-being of mathematics as much as or more than you or other mathematicians. They are right in feeling left out by snobbish research mathematicians who do not know on which side their bread is buttered. It is our best interest, as well as the interest of justice, to treat all who deal with mathematics in whatever way as equals. By being united we will increase the probability of our survival.
Gian-Carlo Rota, from Indiscrete Thoughts.
760
 
"Tell me one last thing," said Harry. "Is this real? Or has this been happening inside my head?"
"Of course it is happening inside your head, Harry, but why on earth should that mean that it is not real?"
J.K. Rowling, Harry Potter and the Deathly Hollows, Part II.
1726
 
Unlike any other creature on this planet, humans can learn and understand, without having experienced. They can think themselves into other people's minds, imagine themselves into other people's places… We do not need magic to change the world, we carry all the power we need inside ourselves already: we have the power to imagine better.
J.K. Rowling, from her 2008 Harvard Commencement Address entitled "The Fringe Benefits of Failure, and the Importance of Imagination"
1401
 
Not only is another world possible, she is on her way. On a quiet day, I can hear her breathing.
Arundhati Roy,
1361
 
The Cube is an imitation of life itself - or even an improvement on life.
Erno Rubik,
1372
 
People always ask me if I will surpass the Cube. What can I answer? I did not plan to make the Cube. I did not plan the success. I wanted nothing else than to make the object as perfect as possible. Now, after the Cube, I still don't have any plans to make anything like it. I'm still the same person, thinking the same way, so it's possible I will invent something. But to want to repeat the Cube--that is not the way to live.
Erno Rubik,
813
 
We turn the Cube and it twists us.
Erno Rubik,
1373
 
The problems of puzzles are very near the problems of life... Our whole life is solving puzzles. If you are hungry, you have to find something to eat. But everyday problems are very mixed--they're not clear. A good puzzle, it's a fair thing. Nobody is lying. It's very clear, and the problem depends just on you. You can solve it independently. But to find happiness in life, you're not independent. That's the big difference.
Erno Rubik, Discover, March 1986, "The Perplexing Life of Erno Rubik" by John Tierney
814
 
Intellectually, perspective [drawing] is a breakthrough, because here, for the first time, the physical space we live in is being depicted as if it were an abstract, mathematical space. A less obvious innovation due to perspective is that here, for the first time, people are actually drawing pictures of infinities.
Rudy Rucker, from Mind Tools: The Five Levels of Mathematical Reality.
530
 
Instead of worrying if we really can squeeze a number like 1/omega in above zero but below 1/2, 1/3, 1/4, 1/5, etc., Conway suggests that we relax and just define 1/omega in terms of the gap between zero and all the 1/n.
Rudy Rucker, from Mind Tools: The Five Levels of Mathematical Reality.
529
 
But so great is the average person's fear of the infinite that to this day calculus all over the world is being taught as a study of limit processes instead of what it really is: infinitesimal analysis.
Rudy Rucker, from Infinity and the Mind.
526
 
The infinite normally inspires such feelings of helplessnes, fatility, and despair that the natural human impulse is to reject it out of hand.
Rudy Rucker, from Infinity and the Mind.
525
 
Logic and set theory are the tools for an exact metaphysics.
Rudy Rucker, from Infinity and the Mind.
516
 
The study of infinity is much more than a dry academic game. The intellectual pursuit of the absolute infinity is, as Georg Cantor realized, a form of the souls quest for God. Whether or not the goal is ever reached, an awareness of the process brings enlightenment.
Rudy Rucker, from Infinity and the Mind.
333
 
Mathematics is obviously something that women should be able to do very well. It's very intuitive. You don't need a lot of machinery, and you don't need a lot of physical strength. You just need stamina, and women often have a great deal of stamina. So why do not more women become mathematicians?... I think that for some reason, probably sociological, girls are refusing to look -- they simply won't try something that they view as a hard problem in mathematics. But boys for some reason are willing and eager to look at the hard problems.
Mary Ellen Rudin, quoted in More Mathematical People, by D.J. Albers, G.L. Alexanderson, and C. Reid.
616
 
We [as children] had a lot of time to develop games. We had few toys. There was no movie house in town. We listened to things on the radio. That was our only contact with the outside world. But our games were very elaborate and purely in the imagination. I think actually that that is something that contributes to making a mathematician - having some time to think and being in the habit of imagining all sorts of complicated things.
Mary Ellen Rudin, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz.
458
 
Every human is an artist. The dream of your life is to make beautiful art
dom Miguel Ruiz,
917
 
We cannot hope that many children will learn mathematics unless we find a way to share our enjoyment and show them its beauty as well as its utility.
Mary Beth Ruskai, from "From the Editor", Notices of the American Mathematical Society, vol. 42, no. 7, July 1995.
704
 
Teaching is painful, continual and difficult work to be done by kindness, by watching and by praise, but above all by example.
John Ruskin,
334
 
Pure mathematics is one of the highest forms of art; it has a sublimity quite special to itself, and an immense dignity derived from the fact that its world is exempt from change and time.`
Bertrand Russell, quoted in American Mathematical Monthly, May 2009, p. 422.
1264
 
It has been customary when Euclid, considered as a textbook, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test… The value of his work as a masterpiece of logic has been very grossly exaggerated.
Bertrand Russell, quoted in Mathematical Thoughts from Ancient to Modern Times by Morris Kline
1039
 
But as the work proceeded I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was not more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
Bertrand Russell, quoted in Excursions in Calculus, by Robert M. Young.
338
 
The others, however, found me suitable for teasing... I remained, however, profoundly unhappy. There was a footpath leading across fields to New Southgage, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.
Bertrand Russell, from pp. 31-2 of his autobiography.
1558
 
The examination system, and the fact that instruction is treated mainly as training for a livelihood, leads the young to regard knowledge from a purely utilitarian point of view, as the road to money, not as the gateway to wisdom.
Bertrand Russell, quoted in A Teacher's Treasury of Quotations, by Bernard E. Farber.
335
 
Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of paintings or music, yet sublimely pure and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learned as a task but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the real and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from with all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the natural world.
Bertrand Russell, from The Study of Mathematics: Philosophical Essays.
337
 
In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philophers pronounced him to be a mere ingenious juggler, and his arguments one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance, by a German professor who probably never dreamed of any connexion between himself and Zeno. Weierstrass, by strictly banishing all infinitesimals, has at last shown that we live in an unchanging world, and that the arrow, at every moment of its flight, is truly at rest. The only point where Zeno probably erred was in inferring (if he did infer) that, because there is no change, the world must be in the same state at one time as at another. This consequence by no means follows, and in this point the German professor is more constructuve than the ingenious Greek. Weierstrass, being able to embody his opinions in mathematics, where familiarity with truth eliminates the vulgar prejudices of common sense, has been able to give to his propositions the respectable air of platitudes; and if the result is less delightful to the lover of reason than Zeno"s bold defiance, it is at any rate more calculated to appease the mass of academic mankind.
Bertrand Russell, from The Principles of Mathematics.
610
 
Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and the unbearable pity for the suffering of mankind.
Bertrand Russell, from The Study of Mathematics.
339
 
Mathematics takes us...into the region of absolute necessity, to which not only the actual world, but every possible world must conform.
Bertrand Russell, from The Study of Mathematics
340
 
Education, which was at first made universal in order that all might be able to read and write, has been found capable of serving quite other purposes. By instilling nonsense it unifies populations and generates collective enthusiasm.
Bertrand Russell, quoted in A Teacher's Treasury of Quotations, by Bernard E Farber.
341
 
Zeno was concerned with three problems...These are the problem of the infinitesimal, the infinite, and continuity...From his to our own day, the finest intellects of each generation in turn attacked these problems, but achieved broadly speaking, nothing...Weierstrass, Dedekind, and Cantor,...have completely solved them. Their solutions...are so clear as to leave no longer the slightest doubt or difficulty. This achievement is probably the greatest of which the age can boast...The problem of the infinitesimal was solved by Weierstrass, the solution of the other two was begun by Dedekind and definitely accomplished by Cantor.
Bertrand Russell, in International Monthly, Vol. 4(1901).
342
 
I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere... But after some twenty years of arduous toil, I came to the conclustion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
Bertrand Russell, quoted in Bridges to Infinity by Michael Guillen.
536
 
This time was one of intellectual intoxication. My sensations resembled those one has after climbing a mountain in a mist, when, on reaching the summit, the mist suddenly clears, and the country becomes visible for forty miles in every directions. For years I had been endeavouring to analyse the fundamental notions of mathematics, such as ordinal and cardinal numbers. Suddenly, in the space of a few weeks, I discovered what appeared to be definitive answers to the problems which had baffled me for years... Intellectually, the month of September 1900 was the highest point of my life.
Bertrand Russell, quoted in Understanding the Infinite by Shaughan Lavine.
544
 
The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast.
Bertrand Russell,
336
 

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