Mathematical Quotations

Some of these quotations are funny or quirky, some are profound, some are informative, some are about the beauty in mathematics, and many are combinations of two or more of these. This confounds my desire about what to list first; many of the quotations tug at me to be listed before others. So they are in no particular order. You will have to read them all to assure yourself of getting the ones you like, and they are all worth reading. For less serious thoughts, try my math jokes page.

My favorite quotations from Mark Woodard's collection.

Darwin, Charles.
Mathematics seems to endow one with something like a new sense.
(In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh, North Carolina: Rome Press Inc., 1988.)
Dirac, Paul Adrien Maurice (1902-).
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
(In P. J. Davis and R. Hersh, The Mathematical Experience, Boston: Birkhäuser, 1981.)
Ellis, Havelock.
The mathematician has reached the highest rung on the ladder of human thought.
(The Dance of Life.)
Gardner, Martin.
Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals—the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.
(In G. Simmons, Calculus Gems, New York: McGraw Hill, 1992.)
Halmos, Paul R.
The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me—both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension.
(I Want to be a Mathematician, Washington: MAA Spectrum, 1985.)
Hardy, Godfrey H. (1877-1947).
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
(A Mathematician's Apology, London, Cambridge University Press, 1941.)
Hertz, Heinrich.
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser that we are, wiser even than their discoverers, that we get more out of them than was originally put into them.
(Quoted by Eric T. Bell in Men of Mathematics, New York, 1937.)
Holmes, Oliver Wendell.
Descartes commanded the future from his study more than Napoleon from the throne.
(In G. Simmons, Calculus Gems, New York: McGraw Hill Inc., 1992.)
Kepler, Johannes (1571-1630).
A mind accustomed to mathematical deduction, when confronted with the faulty foundations of astrology, resists a long, long time, like an obstinate mule, until compelled by beating and curses to put its foot into that dirty puddle.
(In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.)
Mackay, Alan Lindsay.
"Like the ski resort full of girls hunting for husbands and husbands hunting for girls, the situation is not as symmetrical as might seem."
Oakley, C.O.
The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality.
(The American Mathematical Monthly, 56, 1949, p. 19.)
Poincaré, Jules Henri (1854-1912).
Mathematics is the art of giving the same name to different things.
Poincaré, Jules Henri (1854-1912).
A scientist worthy of his name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
(In N. Rose Mathematical Maxims and Minims, Raleigh, North Carolina: Rome Press Inc., 1988.)
Rossi, Hugo.
In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.
("Mathematics Is an Edifice, Not a Toolbox," Notices of the AMS 43, no. 10, October 1996.)
Rota, Gian-carlo.
We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?"
(In preface to P. Davis and R. Hersh, The Mathematical Experience, Boston: Birkhäuser, 1981.)
Shaw, J. B.
The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth.
(In N. Rose Mathematical Maxims and Minims, Raleigh, North Carolina: Rome Press Inc., 1988.)
Wordsworth, William (1770 - 1850).
[Geometry is] an independent world
Created out of pure intelligence.
Young, J. W. A.
Mathematics has beauties of its own—a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found only in the works of the greatest beauty. When this subject is properly … presented, the mental emotion should be that of enjoyment of beauty, not that of repulsion from the ugly and the unpleasant.
(In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.)

My favorite quotations from Mathematically Speaking by C. C. Gaither, A. E. CAvazos-Gaither, and Andrew Slocombe.

Eric T. Bell.
A circle no doubt has a certain appealing simplicity at first glance, but one look at a healthy ellipse should have convinced even the most mystical of astronomers that the perfect simplicity of the circle is akin to the vacant smile of complete idiocy. Compared to what an ellipse can tell us, a circle has nothing to say.
Jerry P. King.
A mathematician, however, almost always works alone … When a mathematician works at mathematics he sits alone in his study staring at equations scribbled on his blackboard or at a dog-eared reprint of the research paper whose results he is trying to extend. It is quiet work, like writing poetry, and includes lots of "dead time" when the mathematician, like the poet, does nothing but sit and stare at the blank page. When you walk in on a research mathematician and find him reclining with his feet up, gazing wistfully out the window, what you say is: "Sorry, I didn't know you were working." Because he probably is.
Jerry P. King.
A mathematician, like everyone else, lives in the real world. But the objects with which he works do not. They live in that other place—the mathematical world. Something else lives here also. It is called truth.
Godfrey Harold Hardy (1877-1947).
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
(A Mathematician's Apology, London, Cambridge University Press, 1941.)
H. J. S. Smith.
[Arithmetic] is one of the oldest branches, perhaps the very oldest branch, of human knowledge; and yet some of its most abstruse secrets lie close to its tritest truths.
Blaise Pascal.
But dull minds are never either intuitive or mathematical.
David Cort.
But suspicion is a thing very few people can entertain without letting the hypothesis turn, in their minds, into fact . . .
Albert Einstein.
Don't worry about your difficulties in mathematics; I can assure you that mine are still greater.
George Eliot.
Every man who is not a monster, a mathematician, or a mad philosopher, is the slave of some woman or other.
Niels Bohr.
How wonderful that we have met with a paradox. Now we have some hope of making progress.
Serge Lang.
I must also add that I do mathematics also because it is difficult, and it is a very beautiful challenge for the mind. I do mathematics to prove to myself that I am capable of meeting this challenge, and win it.
Alfred Adler, "Reflections: mathematics and creativity," New Yorker 47, no. 53 (1972): 39-45.
In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is.
Unknown author, quoted by Eli Maor.
Infinity is a floorless room without walls or ceiling.
Pierre Boutroux.
Logic is invincible because in order to combat logic it is necessary to use logic.
Henri Poincaré.
Mathematicians do not deal in objects, but in relations between objects, thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.
Richard J. Trudeau.
Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere—Archimedes did it in a bathtub. It is dramatic, challenging, endless, and full of surprises.
Henri Poincaré.
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.
Harry Emerson Fosdick.
… when life ceases to be a fraction and becomes an integer.

From other sources.

Old Romanian saying.
He who does not know the lemma does not know the theorem.
(Quoted by Abiteboul, Hull, and Vianu in Foundations of Databases, Reading, Massachusetts: Addison-Wesley, 1995.)

Compilation and portions © copyright 1998 by Eric Postpischil.