D'Alembert, Jean Le Rond (1717-1783)

Just go on and faith will soon return.
[To a friend hesitant with respect to infinitesimals.]
In P. J. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.

Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this .... The imagination in a mathematician who creates makes no less difference than in a poet who invents.... Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer.
Discours Preliminaire de L'Encyclopedie, Tome 1, 1967. pp 47 - 48.

Dantzig

Neither in the subjective nor in the objective world can we find a criterion for the reality of the number concept, because the first contains no such concept, and the second contains nothing that is free from the concept. How then can we arrive at a criterion? Not by evidence, for the dice of evidence are loaded. Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiplied necessity that we call mathematics.
How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.

Darwin, Charles

Every new body of discovery is mathematical in form, because there is no other guidance we can have.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Mathematics seems to endow one with something like a new sense.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Davis, Philip J.

The numbers are a catalyst that can help turn raving madmen into polite humans.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.
Number, Scientific American, 211, (Sept. 1964), 51 - 59.

Davis, Philip J. and Hersh, Reuben

One began to hear it said that World War I was the chemists' war, World War II was the physicists' war, World War III (may it never come) will be the mathematicians' war.
The Mathematical Experience, Boston: Birkhäuser, 1981.

Dehn, Max

Mathematics is the only instructional material that can be presented in an entirely undogmatic way.
In The Mathematical Intelligencer, v. 5, no. 2, 1983.

De Morgan, Augustus (1806-1871)

[When asked about his age.] I was x years old in the year x^2.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

It is easier to square the circle than to get round a mathematician.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.
Transactions Cambridge Philosophical Society, vol. X, 1864, p. 184.

Descartes, René (1596-1650)

Of all things, good sense is the most fairly distributed: everyone thinks he is so well supplied with it that even those who are the hardest to satisfy in every other respect never desire more of it than they already have.
Discours de la Méthode. 1637.

Each problem that I solved became a rule which served afterwards to solve other problems.
Discours de la Méthode. 1637.

If I found any new truths in the sciences, I can say that they follow from, or depend on, five or six principal problems which I succeeded in solving and which I regard as so many battles where the fortunes of war were on my side.
Discours de la Méthode. 1637.

I concluded that I might take as a general rule the principle that all things which we very clearly and obviously conceive are true: only observing, however, that there is some difficulty in rightly determining the objects which we distinctly conceive.
Discours de la Méthode. 1637.

I thought the following four [rules] would be enough, provided that I made a firm and constant resolution not to fail even once in the observance of them. The first was never to accept anything as true if I had not evident knowledge of its being so; that is, carefully to avoid precipitancy and prejudice, and to embrace in my judgment only what presented itself to my mind so clearly and distinctly that I had no occasion to doubt it. The second, to divide each problem I examined into as many parts as was feasible, and as was requisite for its better solution. The third, to direct my thoughts in an orderly way; beginning with the simplest objects, those most apt to be known, and ascending little by little, in steps as it were, to the knowledge of the most complex; and establishing an order in thought even when the objects had no natural priority one to another. And the last, to make throughout such complete enumerations and such general surveys that I might be sure of leaving nothing out. These long chains of perfectly simple and easy reasonings by means of which geometers are accustomed to carry out their most difficult demonstrations had led me to fancy that everything that can fall under human knowledge forms a similar sequence; and that so long as we avoid accepting as true what is not so, and always preserve the right order of deduction of one thing from another, there can be nothing too remote to be reached in the end, or to well hidden to be discovered.
Discours de la Méthode. 1637.

When writing about transcendental issues, be transcendentally clear.
In G. Simmons Calculus Gems. New York: McGraw Hill Inc., 1992.

If we possessed a thorough knowledge of all the parts of the seed of any animal (e.g. man), we could from that alone, be reasons entirely mathematical and certain, deduce the whole conformation and figure of each of its members, and, conversely if we knew several peculiarities of this conformation, we would from those deduce the nature of its seed.

Cogito Ergo Sum. "I think, therefore I am."
Discours de la Méthode. 1637.

I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.
La Geometrie.

Perfect numbers like perfect men are very rare.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

omnia apud me mathematica fiunt.
With me everything turns into mathematics.

It is not enough to have a good mind. The main thing is to use it well.
Discours de la Méthode. 1637.

If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things.
Discours de la Méthode. 1637.

De Sua, F. (1956)

Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. Quantum mechanics for example would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Diophantus

[His epitaph.]
This tomb hold Diophantus Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.
In Ivor Thomas Greek Mathematics, in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Dirac, Paul Adrien Maurice (1902- )

I think that there is a moral to this story, namely that it is more important to have beauty in one's equations that to have them fit experiment. If Schroedinger had been more confident of his work, he could have published it some months earlier, and he could have published a more accurate equation. It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory.
Scientific American, May 1963.

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.

In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

Disraeli, Benjamin

There are three kinds of lies: lies, damned lies, and statistics.
Mark Twain. Autobiography.

Donatus, Aelius (4th Century)

Pereant qui ante nos nostra dixerunt.
"To the devil with those who published before us."
[Quoted by St. Jerome, his pupil]

Doyle, Sir Arthur Conan (1859-1930)

Detection is, or ought to be, an exact sciences and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces much the same effect as if you worked a love story or an elopement into the fifth proposition of Euclid.
The Sign of Four.

When you have eliminated the impossible, what ever remains, however improbable must be the truth.
The Sign of Four.

From a drop of water a logician could predict an Atlantic or a Niagara.
A study in Scarlet 1929.

It is a capital mistake to theorize before one has data.
Scandal in Bohemia.

Dryden, John (1631-1700)

Mere poets are sottish as mere drunkards are, who live in a continual mist, without seeing or judging anything clearly. A man should be learned in several sciences, and should have a reasonable, philosophical and in some measure a mathematical head, to be a complete and excellent poet.
Notes and Observations on The Empress of Morocco. 1674.

Dubos, René J.

Gauss replied, when asked how soon he expected to reach certain mathematical conclusions, that he had them long ago, all he was worrying about was how to reach them!
In Mechanisms of Discovery in I. S. Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.

Dunsany, Lord

Logic, like whiskey, loses its beneficial effect when taken in too large quantities.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Dürer, Albrecht (1471-1528)

But when great and ingenious artists behold their so inept performances, not undeservedly do they ridicule the blindness of such men; since sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge, although with plenty of care and diligence. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who are themselves ignorant of this art.
The Art of Measurement. 1525.

Whoever ... proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured.
J Heidrich (ed.) Albrecht Dürer's schriftlicher Nachlass Berlin, 1920.

And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art...
Course in the Art of Measurement

Dyson, Freeman

I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.
Missed Opportunities, 1972. (Gibbs Lecture?)

For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created.
Mathematics in the Physical Sciences.

The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design.
"Freeman Dyson: Mathematician, Physicist, and Writer". Interview with Donald J. Albers, The College Mathematics Journal, vol 25, no. 1, January 1994.

Eddington, Sir Arthur (1882-1944)

Proof is the idol before whom the pure mathematician tortures himself.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about 'and'.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own.
Space, Time and Gravitation. 1920.

It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231, 425,076,185,631,031,296 protons in the universe and the same number of electrons.
The Philosophy of Physical Science. Cambridge, 1939.

To the pure geometer the radius of curvature is an incidental characteristic - like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the "cat without a grin" and the "grin without a cat" are equally set aside as purely mathematical phantasies.
The Expanding Universe..

Human life is proverbially uncertain; few things are more certain than the solvency of a life-insurance company.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

Edwards, Jonathon

When I am violently beset with temptations, or cannot rid myself of evil thoughts, [I resolve] to do some Arithmetic, or Geometry, or some other study, which necessarily engages all my thoughts, and unavoidably keeps them from wandering.
In T. Mallon A Book of One's Own. Ticknor & Fields, New York, 1984, p. 106-107.

Egrafov, M.

If you ask mathematicians what they do, yo always get the same answer. They think. They think about difficult and unusual problems. They do not think about ordinary problems: they just write down the answers.
Mathematics Magazine, v. 65 no. 5, December 1992.

Eigen, Manfred (1927 - )

A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant.
Jagdish Mehra (ed.) The Physicist's Conception of Nature, 1973.

Einstein, Albert (1879-1955)

[During a lecture:] This has been done elegantly by Minkowski; but chalk is cheaper than grey matter, and we will do it as it comes.
[Attributed by Pólya.]
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co. Ltd., 1953.

Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.

I don't believe in mathematics.
Quoted by Carl Seelig. Albert Einstein.

Imagination is more important than knowledge.
On Science.

The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.
What I Believe.

The bitter and the sweet come from the outside, the hard from within, from one's own efforts.
Out of My Later Years.

Gott würfelt nicht.

Common sense is the collection of prejudices acquired by age eighteen.
In E. T. Bell Mathematics, Queen and Servant of the Sciences. 1952.

God does not care about our mathematical difficulties. He integrates empirically.
L. Infeld Quest, 1942.

How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?

[About Newton]
Nature to him was an open book, whose letters he could read without effort.
In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

What is this frog and mouse battle among the mathematicians?
[i.e. Brouwer vs. Hilbert]
In H. Eves Mathematical Circles Squared Boston: Prindle, Weber and Schmidt, 1972.

Raffiniert ist der Herr Gott, aber boshaft ist er nicht. God is subtle, but he is not malicious.
Inscribed in Fine Hall, Princeton University.

Nature hides her secrets because of her essential loftiness, but not by means of ruse.

The human mind has first to construct forms, independently, before we can find them in things.

Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
In A. Sommerfelt "To Albert Einstein's Seventieth Birthday" in Paul A. Schilpp (ed.) Albert Einstein, Philosopher-Scientist, Evanston, 1949.

Do not worry about your difficulties in mathematics, I assure you that mine are greater.

The truth of a theory is in your mind, not in your eyes.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

These thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words afterward.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

A human being is a part of the whole, called by us "Universe," a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the resta kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

The world needs heroes and it's better they be harmless men like me than villains like Hitler.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiousity of inquiry.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

Everything that is really great and inspiring is created by the individual who can labor in freedom.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

The search for truth is more precious than its possession.
The American Mathematical Monthly v. 100 no. 3.

If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.
Address at the Sorbonne, Paris.

We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think nothing. Already the distance-concept is logically arbitrary; there need be no things that correspond to it, even approximately.
"Space-Time." Encyclopaedia Britannica, 14th ed.

Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.
The Evolution of Physics.

Science without religion is lame; religion without science is blind.
Reader's Digest, Nov. 1973.

Ellis, Havelock

The mathematician has reached the highest rung on the ladder of human thought.
The Dance of Life.

It is here [in mathematics] that the artist has the fullest scope of his imagination.
The Dance of Life.

Erath, V.

God is a child; and when he began to play, he cultivated mathematics. It is the most godly of man's games.
Das blinde Spiel. 1954.

Euler, Leonhard (1707 - 1783)

If a nonnegative quantity was so small that it is smaller than any given one, then it certainly could not be anything but zero. To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be. These supposed mysteries have rendered the calculus of the infinitely small quite suspect to many people. Those doubts that remain we shall thoroughly remove in the following pages, where we shall explain this calculus.

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

[upon losing the use of his right eye]
Now I will have less distraction.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Everett, Edward (1794-1865

In the pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven.
Quoted by E.T. Bell in The Queen of the Sciences, Baltimore, 1931.

Eves, Howard W.

A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well placed internal charges.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

One is hard pressed to think of universal customs that man has successfully established on earth. There is one, however, of which he can boast the universal adoption of the Hindu-Arabic numerals to record numbers. In this we perhaps have man's unique worldwide victory of an idea.
Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

Ewing, John

If the entire Mandelbrot set were placed on an ordinary sheet of paper, the tiny sections of boundary we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only mathematicians have microscopes fine enough to actually observe them.
"Can We See the Mandelbrot Set?", The College Mathematics Journal, v. 26, no. 2, March 1995.

de Fermat, Pierre (1601?-1665)

[In the margin of his copy of Diophantus' Arithmetica, Fermat wrote]
To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.

And perhaps, posterity will thank me for having shown it that the ancients did not know everything.
In D. M. Burton, Elementary Number Theory, Boston: Allyn and Bacon, Inc., 1976.

Feynman, Richard Philips (1918 - 1988)

We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work.
Nobel Lecture, 1966.

Finkel, Benjamin Franklin

The solution of problems is one of the lowest forms of mathematical research, ... yet its educational value cannot be overestimated. It is the ladder by which the mind ascends into higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem.
The American Mathematical Monthly, no. 1.

Flaubert, Gustave (1821-1880)

Poetry is as exact a science as geometry.

Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?

Frankland, W.B.

Whereas at the outset geometry is reported to have concerned herself with the measurement of muddy land, she now handles celestial as well as terrestrial problems: she has extended her domain to the furthest bounds of space.
Hodder and Stoughton, The Story of Euclid. 1901.

Galbraith, John Kenneth

There can be no question, however, that prolonged commitment to mathematical exercises in economics can be damaging. It leads to the atrophy of judgement and intuition...
Economics, Peace, and Laughter.

Galilei, Galileo (1564 - 1642)

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
Opere Il Saggiatore p. 171.

Measure what is measurable, and make measurable what is not so.
Quoted in H. Weyl "Mathematics and the Laws of Nature" in I Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.

And who can doubt that it will lead to the worst disorders when minds created free by God are compelled to submit slavishly to an outside will? When we are told to deny our senses and subject them to the whim of others? When people devoid of whatsoever competence are made judges over experts and are granted authority to treat them as they please? These are the novelties which are apt to bring about the ruin of commonwealths and the subversion of the state.
[On the margin of his own copy of Dialogue on the Great World Systems].
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 733.

Galois, Evariste

Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Galton, [Sir] Francis (1822-1911)

Whenever you can, count.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

[Statistics are] the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man.
Pearson, The Life and Labours of Francis Galton, 1914.

I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error." The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 1482.

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.

Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure.
Gardner on Gardner: JPBM Communications Award Presentation. Focus-The Newsletter of the Mathematical Association of America v. 14, no. 6, December 1994.

Gauss, Karl Friedrich (1777-1855)

I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.
[A reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem.] In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 312.

If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 326.

There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 314.

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.

God does arithmetic.

We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.
Letter to Bessel, 1830.

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.
In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.

I have had my results for a long time: but I do not yet know how I am to arrive at them.
In A. Arber The Mind and the Eye 1954.

[His motto:]
Few, but ripe.

[His second motto:]
Thou, nature, art my goddess; to thy laws my services are bound...
W. Shakespeare King Lear.

[attributed to him by H.B Lübsen]
Theory attracts practice as the magnet attracts iron.
Foreword of H.B Lübsen's geometry textbook.

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.
Letter to Bolyai, 1808.

Finally, two days ago, I succeeded - not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect...geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.
Quoted in J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990.

Gay, John

Lest men suspect your tale untrue,
Keep probability in view.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. p. 1334.

Gibbs, Josiah Willard (1839 - 1903)

One of the principal objects of theoretical research in my department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.

Mathematics is a language.

Gilbert, W. S. (1836 - 1911)

I'm very good at integral and differential calculus, I know the scientific names of beings animalculous; In short, in matters vegetable, animal, and mineral, I am the very model of a modern Major-General.
The Pirates of Penzance. Act 1.

Glaisher, J.W.

The mathematician requires tact and good taste at every step of his work, and he has to learn to trust to his own instinct to distinguish between what is really worthy of his efforts and what is not.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

Goethe

It has been said that figures rule the world. Maybe. But I am sure that figures show us whether it is being ruled well or badly.
In J. P. Eckermann, Conversations with Goethe.

Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956, p. 1754.

Goodman, Nicholas P.

There are no deep theorems -- only theorems that we have not understood very well.
The Mathematical Intelligencer, vol. 5, no. 3, 1983.

Gordon, P

This is not mathematics, it is theology.
[On being exposed to Hilbert's work in invariant theory.]
Quoted in P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.

Graham, Ronald

It wouild be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, 'Yes, it is true, but you won't be able to understand the proof.'
John Horgan. Scientific American 269:4 (October 1993) 92-103.

Grünbaum, Branko (1926 - ), and Shephard, G. C. (?)

Mathematicians have long since regarded it as demeaning to work on problems related to elementary geometry in two or three dimensions, in spite of the fact that it it precisely this sort of mathematics which is of practical value.
Handbook of Applicable Mathematics.

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