Yet I exist in the hope that these memoirs... may find their way to the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality.
Edwin A. Abbot, from Flatland.
1
 
The arithmetic of life does not always have a logical answer.
Inshirah Abdur-Rauf,
2
 
The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes.
Neils Henrik Abel, quoted in To Infinity and Beyond by Eli Maor.
651
 
Where is it proved that one obtains the derivative of an infinite series by taking the derivative of each term?
Abel, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1119
 
The following theorem can be found in the work of Mr. Cauchy: If the various terms of the series u_0 + u_1 + u_2 +... are continuous functions,… then the sum S of the series is also a continuous function of x. But it seems to me that this theorem admits exceptions. For example the series sin(x) - (1/2)sin(2x) + (1/3)sin(3x) - ... is discontinuous at each value (2m + 1)pi of x.
Abel, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1115
 
Cauchy is mad, and there is no way of being on good terms with him, although at present he is the only man who knows how mathematics should be treated. What he does is excellent, but very confused…
Abel, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1113
 
I shall devote all my efforts to bring light into the immense obscurity that today reigns in Analysis. It so lacks any plan or system, that one is really astonished that there are so many people who devote themselves to it -- and, still worse, it is absolutely devoid of any rigor.
N. H. Abel, quoted in Analysis by Its History by E. Hairer and G. Wanner
1163
 
There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general and it is extremely peculiar that such a procedure has led to so few of the so-called paradoxes.
Niels Henrik Abel, quoted in Understanding the Infinite by Shaughan Lavine.
691
 
If you disregard the simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.
Niels Henrik Abel, quoted in Calculus Gems, by G.F. Simmons.
3
 
It [analysis] lacks at this point such plan and unity that it is really amazing that it can be studied by so many people. The worst is that it has not at all been treated with rigor. There are only a few propositions in higher analysis that have been demonstrated with complete rigor. Everywhere one finds the unfortunate manner of reasoning from the particular to the general, and it is very unusual that with such a method one finds, in spite of everything, only a few of what many be called paradoxes. It is really very interesting to seek the reason. In my opinion that arises from the fact that the functions with which analysis has until now been occupied can, for the most part, be expressed by means of powers. As soon as others appear, something that, it is true, does not often happen, this no longer works and from false conclusions there flow a mass of incorrect propositions.
Niels Henrik Abel, quoted in Learn from the Masters, edited by F. Swetz
4
 
Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.
Neils Henrik Abel, quoted in Theory of Complex Functions, by Reinhold.
5
 
Five centuries ago the printing press sparked a radical reshaping of the nature of education. By bringing a master's words to those who could not hear a master's voice, the technology of printing dissolved the notion that education must be reserved for those with the means to hire personal tutors. Today we are approaching a new technological revolution, one whose impact on education may be as far-reaching as that of the printing press: the emergence of powerful computers that are sufficiently inexpensive to be used by students for learning, play and exploration. It is our hope that these powerful but simple tools for creating and exploring richly interactive environments will dissolve the barriers to the production of knowledge as the printing press dissolved the barriers to its transmission.
H. Abelson and A.A. diSessa, from Turtle Geometry: The Computer as a Medium for Exploring Mathematics.
6
 
Chaos was the law of nature; order was the dream of man.
Henry Adams, quoted in Uncertainty: Einstein, Heisenberg, Bohr and the Struggle for the Soul of Science by David Lindley.
1397
 
A teacher effects eternity; [s]he can never tell where [her] his influence stops.
Henry Adams,
597
 
If your actions inspire others to dream more, learn more, do more and become more, you are a leader.
John Quincy Adams,
842
 
I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture, in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain.
John Adams, quoted in Bartlett's Familiar Quotations.
7
 
The importance of group theory was emphasized very recently when some physicists using group theory predicted the existence of a particle that had never been observed before, and described the properties it should have. Later experiments proved that this particle really exists and has those properties.
Irving Adler, quoted in Out of the Mouths of Mathematicians, by R. Schmalz.
10
 
Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing.
Alfred Adler, from "Mathematics and Creativity," the New Yorker Magazine.
8
 
It is not only by the questions we have answered that progress may be measured, but also by those we are still asking. The passionate controversies of one era are viewed as sterile proccupations by another, for knowledge alters what we seek as well as what we find.
Freda Adler, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
9
 
If you play music kids like, they will dance.
Andrea Agostini,
679
 
Don't worry about people stealing your ideas. If your ideas are any good, you'll have to ram them down people's throats.
Howard Hathaway Aiken,
1063
 
The basic ideas and simplest facts of set-theoretic topology are needed in the most diverse areas of mathematics; the concepts of topological and metric spaces, of compactness, the properties of continuous functions and the like are often indispensable.
P. Alexandroff and H. Hopf, quoted in Theory of Complex Functions, by Reinhold Remmert.
11
 
Champions aren't made in gyms. Champions are made from something they have deep inside them--a desire, a dream, a vision. They have to have last-minute stamina, they have to be a little faster, they have to have the skill and the will. But the will must be stronger than the skill.
Mohammed Ali, from The Greatest
1153
 
I'm so fast that last night I turned off the light switch in my hotel room and was in bed before the room was dark.
Muhammad Ali,
1793
 
I don't have to be what you want me to be. I'm free to be whom I want.
Muhammad Ali,
869
 
He who is not courageous enough to take risks will accomplish nothing in life.
Muhammad Ali,
837
 
Do not call for black power or green power. Call for brain power.
Muhammad Ali,
836
 
I am America. I am the part you won't recognize. But get used to me. Black, confident, cocky; my name, not yours; my religion, not yours; my goals, my own; get used to me.
Muhammad Ali,
838
 
The inner circle of creative mathematicians have the well-kept trade secret that in a great many cases theorems come first and axioms second.
C.B. Allendorfer, quoted in the American Mathematical Monthly, December, 2008, p. 879.
1266
 
When students hear the story of Andrew J. Wiles' proof of Fermat's Last Theorem, it is not the result itself that stirs their emotions, but the revelation that a mathematician was driven by the same passion as any creative artist.
Jesefina Alvarez, from "Loving Math Infinitely," The Chronicle of Higher Education, 19 January, 2001.
1084
 
Mathematics appreciation is more than a course. It is an attitude that we should cultivate in every mathematics course.
Jesefina Alvarez, from "Loving Math Infinitely," The Chronicle of Higher Education, 19 January, 2001.
1083
 
If you've got one foot in the past and one foot in the future, you're pissing on the present.
Greg Ames, from Buffalo Lockjaw, p. 277.
1257
 
Life is short and we have not too much time for gladdening the hearts of those who are traveling the dark way with us. Oh, be swift to love! Make haste to be kind.
Henri Frederic Amiel,
1079
 
There is no smallest among the small and no largest among the large; but always something still smaller and something still larger.
Anaxagoras,
12
 
The best time to plant a tree is twenty years ago; the second best time is today.
Ancient Chinese Proverb,
1088
 
Science could not fill the hole in my soul.
Terry Anderson, at Westfield State College's 159th Commencement.
468
 
If you're going to spend a long time locked in somebody's basement, take a professor with you.
Terry Anderson, at Westfield State College's 159th Commencement.
467
 
Success is liking yourself, liking what you do, and liking how you do it.
Maya Angelou,
914
 
I've learned that people will forget what you said, people will forget what you did, but people will never forget how you made them feel.
Maya Angelou,
799
 
We really are 15 countries, and it's remarkable that each of us thinks we represent the real America. The Midwesterner in Kansas, the black American in Durham - both are certain they are the real American.
Maya Angelou,
1184
 
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
W. S. Anglin, quoted in The American Mathematical Monthly, vol. 102, no. 4, April 1995.
509
 
You can't plough a field by turning it over in your mind.
Anonymous,
1311
 
The saddest moment in a person's life comes but once.
Anonymous,
1435
 
A school is a building which has four walls and tomorrow inside.
Anonymous,
975
 
Mathematics is music for the mind; music is mathematics for the soul.
Anonymous, from A Dictionary of Quotations in Mathematics by Nowlan
1666
 
It's easier to get forgiveness than permission.
Anonymous,
906
 
The new painters do not propose, any more than did their predecessors, to be geometers. But it may be said that geometry is to the plastic arts what grammar is to the art of the writer. Today, scholars no longer limit themselves to the three dimensions of Euclid. The painters have been lead quite naturally, one might say by intuition, to preoccupy themselves with the new possibilities of spatial measurement which, in the language of the modern studios, are designated by the term fourth dimension.
Guillaume Apollinaire, quoted in The Fourth Dimension and non-Euclidean Geometry in Modern Art by Linda Dalrymple Henderson.
665
 
Now and then it's good to pause in our pursuit of happiness and just be happy.
Guillaume Apollinaire,
930
 
At this point the program began to surprise us... It would work out compound stategies based on all the tricks it had been "taught", and often these approaches were far more clever than those we would have tried. Thus it began to teach us things about how to proceed that we never expected. In a sense it had surpassed its creators in some aspects of the "intellectual" as well as the mechanical parts of the task.
Kenneth Appel and Wofgang Haken , quoted in From Here to Infinity, by Ian Stewart.
672
 
Mathematical knowledge adds vigour to the mind, frees it from prejudice, credulity, and superstition.
John Arbuthnot, quoted in "Is Mathematics Necessary?", by Underwood Dudley, Mathematics Education Dialogues, March 1998.
707
 
The mathematics are the friends of religion, inasmuch as they charm the passions, restrain the impetuosity of the imagination, and purge the mind of error and prejudice.
John Arbuthnot, quoted in "Is Mathematics Necessary?", by Underwood Dudley, Mathematics Education Dialogues, March 1998.
706
 
I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
Archimedes, quoted in Calculus Gems, by George F. Simmons.
13
 
The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all.
Aristotle, quoted in Geometry of Four Dimensions, by Henry Parker Manning.
571
 
Our account does not rob the mathematicians of their science... In point of fact they do not need the infinite and do not use it.
Aristotle, quoted in Using History to Teach Mathematics: An International Perspective, edited by Victor Katz
1136
 
We are what we repeatedly do. Excellence, then, is not an act, but a habit.
Aristotle,
1586
 
It is the mark of an educated mind to be able to entertain a thought without accepting it.
Aristotle,
826
 
There is no transfer into another kind, like the transfer from length to area and from area to a solid.
Aristotle, quoted in Geometry of Four Dimensions, by Henry Parker Manning.
572
 
Happiness is the exercise of vital powers along lines of excellence in a life affording them scope.
Aristotle,
1717
 
The mathematical sciences particularly exhibit order, symmetry, and limitations; and these are the greatest forms of the beautiful.
Aristotle,
787
 
ALL men by nature desire to know.
Aristotle, opening sentence of Metaphysics.
1240
 
I didn't fully see, until the cancer, how we fight every day against the creeping negatives of the world, how we struggle daily against the slow lapping of cynicism. Dispiritedness and disappointment, these were the real perils of life, not some sudden illness or cataclysmic millennium doomsday. I knew now why people fear cancer: because it is a slow and inevitable death, it is the very definition of cynicism and loss of spirit. So, I believed.
Lance Armstrong, from It's Not About the Bike: My Journey Back to Life
1169
 
The truth is, if you asked me to choose between winning the Tour de France and cancer, I would choose cancer. Odd as it sounds, I would rather have the title of cancer survivor than winner of the Tour, because of what it has done for me as a human being, a man, a husband, a son, and a father.
Lance Armstrong, from It's Not About the Bike: My Journey Back to Life
1170
 
To believe in the face of utter hopelessness, every article of evidence to the contrary, to ignore apparent catastrophe -- what other choice was there? We do it every day, I realized... To believe, when all along we humans know that nothing can cure the briefness of this life, that there is no remedy for our basic mortality, that is a form of bravery.
Lance Armstrong, from It's Not About the Bike: My Journey Back to Life
1168
 
It's a fact that children with cancer have higher cure rates than adults with cancer, and I wonder if the reason is their natural, unthinking bravery... Adults know too much about failure; they're more cynical and resigned and fearful.
Lance Armstrong, from It's Not About the Bike: My Journey Back to Life
1171
 
These facts [from a Goursat version of classical calculus] capture the imagination so much that (even given without any proofs) they give a better and more correct idea of modern mathematics than whole volumes of the Bourbaki treatise.
Vladamir Arnold, from "On Teaching Mathematics".
1354
 
Attempts to create ‘pure’ deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation – model – investigation of the model – conclusions – testing by observations) and its substitution by the scheme: definition – theorem – proof. It is impossible to understand an unmotivated definition but this does not stop the criminal algebraists-axiomatisators… It is obvious that such definitions and such proofs can only harm the teaching and practical work.
Vladamir Arnold, from "On Teaching Mathematics".
1355
 
It is only possible to understand the commutativity of multiplication by counting and re-counting soldiers by ranks and files or by calculating the area of a rectangle in the two ways. Any attempt to do without this interference by physics and reality into mathematics is sectarianism and isolationism which destroy the image of mathematics as a useful human activity in the eyes of all sensible people.
Vladamir Arnold, from "On Teaching Mathematics".
1356
 
When I teach people, I marry them.
Sylvia Ashton-Warner, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
14
 
Self-education is, I firmly believe, the only kind of education there is.
Isaac Asimov,
1395
 
The most exciting phrase to hear in science, the one that heralds new discoveries, is not "Eureka!" (I found it!) but rather, "hmm…. That's funny…."
Isaac Asimov,
1396
 
If theory is the role of the architect, then such beautiful proofs are the role of the craftsman. Of course, as with the great renaissance artists, such roles are not mutually exclusive. A great cathedral has both structural impressiveness and delicate detail. A great mathematical theory should similarly be beautiful on both large and small scales.
Michael Atiyah, from "Mathematics: Art and Science", Bulletin of the American Mathematical Society, vol. 43, no. 1, pp. 87-8.
1294
 
I dislike frontiers, political or intellectual, and I find that ignoring them is an essential catalyst for creative thought. Ideas should flow without hindrance in their natural course.
Sir Micheal Francis Atiyah,
1518
 
Mathematicians depend ultimately on society for their livelihood and the privilege of working on something that passionately interests them. In return we must, in various ways, repay this debt and encourage our fellow citizens to take a friendly and tolerant view of our strange profession.
Sir Micheal Francis Atiyah,
1519
 
Everyone else my age is an adult, whereas I am merely in disguise.
Margaret Atwood,
1279
 
You can't. Do you hear me? You think you've figured something out? You run over here so pleased with yourself because you changed your mind. Now you're certain. You're so… sloppy. You don't know anything. The book, the math, the dates, the writing, all that stuff you decided with your buddies, it's just evidence. It doesn't finish the job. It doesn't prove anything.
David Auburn, Catherine, main character from Proof, a play by David Auburn, p. 81
1625
 
How happy the lot of the mathematician. He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve.
W.H. Auden, from The Dyer's Hand
1226
 
Computers are composed of nothing more than logic gates stretched out to the horizon in a vast numerical irrigation system.
Stan Augarten,
15
 
Infinity is a fathomless gulf, into which all things vanish.
Marcus Aurelius, quoted in To Infinity and Beyond: A Cultural History of the Infinite by Eli Maor.
1496
 
And thou wilt give theyself relief, if thou doest every act of thy life as if it were the last.
Marcus Aurelius, quoted in The Teacher's Quotation Book, edited by Wanda Lincoln and Murray Suid.
631
 
Every infinity… is made finite to God.
St. Autustine, quoted in Godel Meets Einstein: Time Travel in the Godel Universe by Palle Yourgrau.
1057
 

80 quotes found and displayed.