Quotes from the Mathematical Quotations Server

Collected by Mark R. Woodard

Anotated by Jaume Gudayol

Adler, Alfred
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.
"Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.

• 'A mathematician is great or he is nothing' is a good point of view when one is trying to do mathematics. I mean that one has to be ambitious in order to get nice results. This said, the point of view of Adler is ludicrous. It is true that only a few of the mathematicians of any generation are going to produce any result or theory that will last into history. But where do these `happy few' come from? On this point, Adler seems to believe in 'spontaneous generation'. This position is difficult to sustain, whether in mathematics or in any other aspect of `creation'. Even Ramanujan needed two mathematics books to develop his abilities, and many people is still wondering what results would Ramanujan have obtained had he been provided with a proper education. On the other hand, Adler seems to have forgotten that, besides the `creative' aspect of mathematics, there is the `everyday' mathematics that is developed in order to provide scientists and engineers with useful tools, something which is not usually done by the more brilliant mathematicians. In other words, having a `mathematical community' and not only a few brilliant mathematicians, is useful both for the sake of mathematics and for the society that fosters it.

Adler, Alfred
The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then, little will ever be accomplished.
"Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.

• Facts are in fact harsher. The quality and, in particular, the originality of a mathematician's production tends to decay as age increases. This is due in part to the loss of mental faculties, but also to the excess of knowledge one accumulates. To much knowledge (that makes one an expert) lets less room for imagination, and thus for originality. A (partial) remedy for this problem is to keep changing the subjects of one's research periodically.

Adler, Alfred
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.
Reflections: mathematics and creativity, New Yorker, 47(1972), no. 53, 39 - 45.

• One should try, at least. I have tried to explain my (current) subjects of research to some of my non-mathematical friends, with some success.

Anglin, W.S.
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.
"Mathematics and History", Mathematical Intelligencer, v. 4, no. 4.

• This could be said of any science.

Ascham, Roger (1515-1568)
Mark all mathematical heads which be wholly and only bent on these sciences, how solitary they be themselves, how unfit to live with others, how unapt to serve the world.
In E G R Taylor, Mathematical Practitioners of Tudor and Stuart England, Cambridge: Cambridge University Press, 1954.

• This point has been widely exaggerated. A logical mind might not be the best tool to deal with the real world, but that's all there is to it.

Auden, W. H. (1907-1973)
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.
The Dyer's Hand, London: Faber & Faber, 1948.

• I wish this were true. In the way it is written, it applies mainly to the dead mathematicians. To judge the living, other standards are applied. (I cannot help believing that Wolff was always judged mainly by his looks.) Those which are in contact with our ereryday life in the world of science should be aware that we are not better (or worse) than the rest of human beings. Thus, quite often we judge others by their looks, their power, their ability to talk or to sell their results, or even by the power of their advisers, when they are young. And all of this applies only to a mathematician's reputation. If we begin to talk about permanent positions, grants, economical support or prices then things get worse, as usual.

St. Augustine (354-430)
If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?
De Magistro ch X, 23.

• This is nonsense. The formula `A=B' will tell us nothing if we knew neither `A' nor `B', of if we knew them both. But in general we know `A' but not `B', or viceversa, and we will now learn something about `B'.

Bagehot, Walter
Life is a school of probability.
Quoted in J. R. Newman (ed.) The World of Mathematics, Simon and Schuster, New York,1956, p. 1360.

• But most people fail to see it. Maybe our brain is blind to this kind of light, but most likely we are failing to teach them to see (that is, to compute) the probabilities as a light to things.

Bell, Eric Temple (1883-1960)
Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.

• They are also inspired by the ambition of outreaching their peers. On the motivations of scientists, few things can be added to what is said in `The double helix', by James Watson.

Besicovitch, A.S.
A mathematician's reputation rests on the number of bad proofs he has given.
In J. E. Littlewood A Mathematician's Miscellany, Methuen & Co. Ltd., 1953.

• If a mathematician gives a bad proof and someone else notices it, it means that they are already listening to him. On the other hand, I would rather have a nice (and correct!) theorem with a bad proof than a dim but correctly proved theorem.

Bolyai, János (1802 - 1860)
Out of nothing I have created a strange new universe.
[A reference to the creation of a non-Euclidean geometry.]

• This is the way non-Euclidean geometries were seen when created. Nowadays, however, non-Euclidean geometries are seen as perfectly normal mathematical objects. What has made the difference? Well, getting used to an idea is part of it, but the great change was produced when a way was found to visualize these geometries. In the same way, complex numbers were not fully accepted as mathematical entities until they were seen as points of the plane. We must be aware that the same could happen to other mathematical entities if we found some way to visualize them. If we succeeded in visualizing them, they would be more widely used and more popular.

Bourbaki
Structures are the weapons of the mathematician.

• The tools of the mathematician (I don't like these military terms).

Bridgman, P. W.
It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention.
The Logic of Modern Physics, New York, 1972.

• I agree insofar as it is said in the same way as `literature is a human invention'. I would rather say that it (as literature) is the sophistication of an ability (or rather of several abilities) nature has given us (because of its utility for natural selection). In this case the abilities would be that of counting (that comes with language), plus that of visualizing the three-dimensional metric space, plus some other ones.

Brown, George Spencer (1923 - )
To arrive at the simplest truth, as Newton knew and practised, requires years of contemplation. Not activity. Not reasoning. Not calculating. Not busy behaviour of any kind. Not reading. Not talking. Not making an effort. Not thinking. Simply bearing in mind what it is one needs to know. And yet those with the courage to tread this path to real discovery are not only offered practically no guidance on how to do so, they are actively discouraged and have to set about it in secret, pretending meanwhile to be diligently engaged in the frantic diversions and to conform with the deadening personal opinions which are continually being thrust upon them.
The Laws of Form. 1969.

• Yes but please let the inspiration catch me while I am working (Picasso said that, or something similar).

Butler, Bishop
To us probability is the very guide of life.
Preface to Analogy.

• But we always look forward to a miracle (thankfully).

Butler, Samuel (1612 - 1680)
... There can be no doubt about faith and not reason being the ultima ratio. Even Euclid, who has laid himself as little open to the charge of credulity as any writer who ever lived, cannot get beyond this. He has no demonstrable first premise. He requires postulates and axioms which transcend demonstration, and without which he can do nothing. His superstructure indeed is demonstration, but his ground his faith. Nor again can he get further than telling a man he is a fool if he persists in differing from him. He says "which is absurd," and declines to discuss the matter further. Faith and authority, therefore, prove to be as necessary for him as for anyone else.
The Way of All Flesh.

• Yes, but at least he can say exactly in what he believes. Moreover, he can claim that his axioms are `experimental facts'.

Carroll, Lewis
"When I use a word," Humpty Dumpty said, in a rather scornful tone, "it means just what I choose it to mean - neither more nor less."
"The question is," said Alice, "whether you can make words mean so many different things."
"The question is," said Humpty Dumpty, "which is to be master - that's all."
Through the Looking Glass.

• We the mathematicians are masters of our language. During a conference, we can use the symbol `f' (or the word function) for a smooth function, a continuous one, a measure or a distribution. This, which is not so obvious, means that inside our (human, not just a mathematicians' brain) there must be some way to dissociate the word from its meaning.

Carmichael, R. D.
A thing is obvious mathematically after you see it.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

• Yes, but then you have to prove it (or at least to see the proof).

Cauchy, Augustin-Louis (1789 - 1857)
Men pass away, but their deeds abide.
[His last words (?)]
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt, 1971.

• Really? Did he say that? How witty, there lying in bed, having difficulties for controlling his arsehole but still saying great things. I don't buy it. For me all these `Famous last words' are setups. Old men die and then someone comes and puts some words in their mouth. (Sorry, Cauchy, it's not your fault.)

Cayley, Arthur
As for everything else, so for a mathematical theory: beauty can be perceived but not explained.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• It can be shown (at least to an educated mind).

Cayley, Arthur
Projective geometry is all geometry.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• At least is the only geometry we perceive. Then why are we used to think in terms of Euclidean geometry? Because, for obvious practical reasons, a metric is useful in terms of survival.

Chekov, Anton (1860 - 1904)
There is no national science just as there is no national multiplication table; what is national is no longer science.
In V. P. Ponomarev Mysli o nauke Kishinev, 1973.

• But some countries have invested some of their national pride in science. And it seems that it produces returns, to the countries (and to science). I wish I had lived in one of such countries.

Chesterton, G. K. (1874 - 1936)
Poets do not go mad; but chess-players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination.
Orthodoxy ch. 2.

• Mathematicians are creative artists (and they go mad as often as poets do).

Chesterton, G. K. (1874 - 1936)
You can only find truth with logic if you have already found truth without it.
The Man who was Orthodox. 1963.

• Or, in other words, you can prove something only if you already know it to be true.

Churchill, Sir Winston Spencer (1874-1965)
I had a feeling once about Mathematics - that I saw it all. Depth beyond depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

• That must have been the Whisky. I recall one seeing the fourth dimension clearly, visualizing objects there by moving them in time and seeing their trace. But this was after my second Irish Coffee and I also let it go. (I have tried the same without alcohol, but it lacks the feeling of complete truth.)

Coolidge, Julian Lowell (1873 - 1954)
[Upon proving that the best betting strategy for "Gambler's Ruin" was to bet all on the first trial.]
It is true that a man who does this is a fool. I have only proved that a man who does anything else is an even bigger fool.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

• Maybe, but people do not play for money, they play for the thrill.

Crick, Francis Harry Compton (1916 - )
In my experience most mathematicians are intellectually lazy and especially dislike reading experimental papers. He (René Thom) seemed to have very strong biological intuitions but unfortunately of negative sign.
What Mad Pursuit. London: Weidenfeld and Nicolson, 1988.

• The problem is that, for a mathematician, experimental papers are mostly just special cases of more or less known facts (sounds pretentious, but it is not so, or at least I do not mean it to be). A description of a typhoon is just another dynamical system, disregarding the details. (A mathematician always oversimplifies.)

Crowe, Michael
Revolutions never occur in mathematics.
Historia Mathematica. 1975.

• Here is a short list of some revolutions that have occurred in mathematics. And observe that none of the following events had any logical connection with the previous setting, and that any of them could perfectly never have happened. The list is: the introduction of coordinates, the introduction of the imaginary unit, the complex plane, Fourier series, Cantor's set theory, and Godel's incompleteness theorem. We usually take too much for granted, once we know it. But another intelligent species, with a brain built in a different way, could perfectly never have mixed geometry with numbers, thus distorting mathematics (as we know it) into two completely different subjects.

Dantzig
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight.

• Gammon. No good designer of garments works without having in mind to whom his garments may fit, and no good mathematician works without having in mind (somehow) to what purpose his theorems might serve. There is always, deep in the mind, some model, related to the reality, which helps to fill of content the collection of symbols he has on the paper. Besides, all of our thinking is made with a human mind, shaped by the evolution of our species. Thus our mathematical objects are shaped by a human mind, and thus they are human objects. There are some objects we will never consider, because they are out of connection with our reality.

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.

• The concept of numbers is subjective. If it was not in our minds, we would not be writing about it.

Darwin, Charles
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.

• Mathematics is a (sometimes clever) cocktail of senses.

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.

• But then why we don't do it that way? Most of the teaching under graduate studies consists in collections of well established truths, leaving no room to discovery.

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.

• Cute. x=43, x^2=1849. (In case you were in a lazy mood.)

Descartes, René (1596-1650)
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.

• There is no way to deduce cubics from quadrics, though. Not at least a geometric way.

Descartes, René (1596-1650)
When writing about transcendental issues, be transcendentally clear.
In G. Simmons Calculus Gems. New York: McGraw Hill Inc., 1992.

• Or maybe be always clear, just in case any of the issues turned out to be transcendental.

Descartes, René (1596-1650)
Cogito Ergo Sum. "I think, therefore I am."
Discours de la Méthode. 1637.

• Well, (ah, ah) I have always thought that this was a redundancy. Inside the `I' there is already the idea of existence. Thus "I, therefore, am" would be more proper. Moreover, since there is an idea of existence embedded in our brains, and that idea includes us in the existents, we exist in that sense. (Uuuugh! Confronting Descartes! Extremely pretentious!)

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.

• Oh, Godel, you will be remembered ever after Euclides is forgotten.

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.

• One should publish the (hopefully) beautiful theory contrasted to the facts, and let the science community check it out.

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.

• There are some limits, because the human mind has them (we are not gods).

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]

• This is a good suggestion to do to any research student when he begins. Or, in other words, "do not read more bibliography that the strictly needed (if you want some originality). Most people wouldn't agree with me, would they?

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.

• There are skilled provers and skilled guessers.

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?)

• They are having some fun right now. By the way, that article is a piece of gold.

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.

• Proof is the first check of a conjecture. A physical theory is void until it has been seen that it can predict the results of some experiments. In the same way, a mathematical conjecture is void until it has been proved. It does not mean that a conjecture is useless. The strive made to prove conjectures is what pushes mathematics forward. But again in the same way as a physical theory, albeit true, can be superseded by another one which is much more fit, or elegant, or whatever, a theorem (that is, a proved conjecture) can, with time, become irrelevant, a special case, or unfit. So proof is a means, not the ultimate goal.

Eddington, Sir Arthur (1882-1944)
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.

• We cannot but find ourselves in everything we see. Despite our partially successful attempts at objectivity, we see every aspect of reality with human eyes.

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.

• A theory is always somehow a model of some reality. And a model (thus also a theory) might become irrelevant, even if previously it wasn't.

Einstein, Albert (1879-1955)
Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.

• Simple things have usually a peculiar way of being simple.

Einstein, Albert (1879-1955)
I don't believe in mathematics.
Quoted by Carl Seelig. Albert Einstein.

• But he surely trusted them.

Einstein, Albert (1879-1955)
Imagination is more important than knowledge.
On Science.

• Moreover, Knowledge tampers with imagination.

Einstein, Albert (1879-1955)
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?

• Maybe it is that reality is also a product of the human thought?

Einstein, Albert (1879-1955)
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.

• Right. But as far as they can predict the results of an experiment with a given accuracy, I don't care.

Einstein, Albert (1879-1955)
The human mind has first to construct forms, independently, before we can find them in things.

• But then we have to have in mind that the human mind has been shaped in this way by evolution.

Einstein, Albert (1879-1955)
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.

• We no longer look for certainty in concepts. We look for usefulness.

Einstein, Albert (1879-1955)
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.

• But there is always more truth in the formulas than can be translated into words.

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.

• Nowadays we can teach infinitesimal calculus to any sixteen years old person, without any problem. What changes have made it possible? First, a good formalization of the concepts involved. And a good visual model of the derivative.

Euler, Leonhard (1707-1783)
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.

• Why should there be any order in the sequence of prime numbers? I have never expected to find one, nor seen the need of it. Number theory abounds in that kind of nonsenses.

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.

• See above.

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.

• A way to visualize the (boundary of) the Mandelbrot set, or of any other closed set with non-integer dimension, is to show a collection of full sets that are at a given distance from the set. (There are many other ways, though.)

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.

• There should be a way of changing this habit.

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.

• Problem solving, however, is one step below research. True research, and its teaching, begins when one is faced with a problem without even knowing whether a reasonable solution for it can be found.

Fisher, Irving
The effort of the economist is to "see," to picture the interplay of economic elements. The more clearly cut these elements appear in his vision, the better; the more elements he can grasp and hold in his mind at once, the better. The economic world is a misty region. The first explorers used unaided vision. Mathematics is the lantern by which what before was dimly visible now looms up in firm, bold outlines. The old phantasmagoria disappear. We see better. We also see further.
Transactions of Conn. Academy, 1892.

• Mathematization has brought economy great successes, and it seems wise to follow this path. This said, one should be aware of the limitations mathematization has for a science that deals with human beings. My first objection to it is of purely scientifical concern. I simply do not believe that a mathematical model (or a set thereof) could ever fit the reality of everyday economics.

  Let us assume we have a model that makes useful predictions with respect to the evolution of the economy. Then it is unavoidable that someone uses it with a twist to maximize his/her own results. This feedback is bound to destroy the validity of the model in the long run. Thus some kind of theorem on the incompleteness of any mathematical model for economy could be proven. Also one should be able to derive some kind of undecidibility of some economical propositions.

  But there is not only the mathematical point of view to economics. We must have always in mind that economics is also related to politics, that is, to the will of the people. Those who sustain that there is only one way to do things, as many economists nowadays do, are simply liers. To begin with, they claim that we should always maximize the GNP. This is already a political choice, and a reactionary one. The greatest economist of the twentieth century (of history, maybe) built its `The General Theory …' with the idea of maximizing employment. And it made much more sense, at least for any society that calls itself democratic.

Fisher, Ronald Aylmer (1890 - 1962)
Natural selection is a mechanism for generating an exceedingly high degree of improbability.

• It must needs be. The `survival of the fittest' is a mechanism to fight against probability, and no living being in an hostile environment would survive without it.

Flaubert, Gustave (1821-1880)
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?

• Old enough.

Frayn, Michael
For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.
Constructions. 1974.

• A false sense.

Frege, Gottlob (1848 - 1925)
A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
In Scientific American, May 1984, p 77.

• The work of Frege is, in any case, still valuable. And his sincerity is still more so.

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.

• c.f. above. In any case, this sentence leaves the impression that Galbraith, however brilliant, is unable to fully comprehend mathematical results.

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.

• Those were the days. By that time, and up to Einstein's General Relativity, it was expected to find, sooner rather that later, a General Theory Of Everything. The first disappointment was to discover that any useful description of reality at an atomic level would have to be probabilistic in nature. This fact added up to the knowledge that any experiment would modify the reality that was being analyzed. Now, though, when we study the evolution of Physics in the XXth century, the idea that remain is that we are building an endless sequence of theories, coming closer and closer to a precise (in numerical terms) description of the results of experiments. Nowadays, only fools (and grant-appliers) expect to find a theory that can describe any physical reality. Thus, either there is no language fit to describe the universe, or we are not using the right one. What is true is that the (our) mathematical language is the best one (as far as we know now) to help us understand the universe. So that the key word in Galileo's sentence is humanly.

Galilei, Galileo (1564 - 1642)
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.

• Nowadays it is widely believed that mathematics and physics are sciences so cut out from reality that the proper ambiance of freedom that science has always required is no longer necessary for them. Galileo's life proves otherwise. Moreover, the more abstract these sciences become, the more dependent on the opinions of the academic authorities. Few people realize the dangers of this fact, specially having in mind that authorities are by definition conservative.

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.

• Two hundred years later, the situation in this aspect has rather worsened than improved. Few authors dare to recognize in their papers their failures.

Galton, Sir Francis (1822-1911)
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.

• How beautiful it is, and how often it is misunderstood.

Gardner, Martin
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.

• This sounds a bit overreasuring. Y would say that the only things we can see in the universe are those that can be expressed in mathematical terms. I am not talking about ESP (I do not believe in that kind of things) but about the fact that both our vision of the universe and our mathematics are limited by the way we have evolved.

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.

• I think in the same way. The fact that a proposition has remained unproved through many years and attempts is not what makes trying to prove it appealing to me.

Gauss, Karl Friedrich (1777-1855)
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.

• Well, no, at least not me.

Gauss, Karl Friedrich (1777-1855)
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.

• Especially since he always said as much as possible.

Gauss, Karl Friedrich (1777-1855)
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.

• As I see it, number is the product of language, while space is the product of our vision and orientation system. Thus, it is difficult to translate space to numbers (hey, but we did it!).

Gauss, Karl Friedrich (1777-1855)
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.

• He was the only one ho could say such a thing unboastingly.

Gauss, Karl Friedrich (1777-1855)
[attributed to him by H.B Lübsen]
Theory attracts practice as the magnet attracts iron.
Foreword of H.B Lübsen's geometry textbook.

• And vice versa (should be).

Gauss, Karl Friedrich (1777-1855)
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.

• He was talking about the impossibility of proving the parallels axiom, I suppose.

Gibbs, Josiah Willard (1839-1903)
Mathematics is a language.

• Indeed.

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.

• I am afraid many mathematicians spend their days proving theorem after theorem, being aware that none of these theorems is going to last. I would be happy to make just one theorem (or a formula, or else) if I knew this one was going to stay for long (nothing lasts for ever, though).

Goedel, Kurt
I don't believe in natural science.
[Said to physicist John Bahcall.]
Ed Regis, Who Got Einstein's Office? Addison Wesley, 1987.

• Sorry? I need someone to explain me what he meant by this.

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.

• This common belief that identities say nothing new denotes a complete misunderstanding of knowledge and of our relation to it. Is not what is true, but what we know to be true what matters.

Graham, Ronald
It would 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.

• This could happen. But then the next step would be to build a better computer (one that explains itself in a more useful way to us).

Haldane, John Burdon Sanderson (1892-1964)
In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind. The catch in this criterion lies in the world "simplest." It is really an aesthetic canon such as we find implicit in our criticisms of poetry or painting. The layman finds such a law as dx/dt = K(d^2x/dy^2) much less simple than "it oozes," of which it is the mathematical statement. The physicist reverses this judgement, and his statement is certainly the more fruitful of the two, so far as prediction is concerned. It is, however, a statement about something very unfamiliar to the plain man, namely, the rate of change of a rate of change.
Possible Worlds, 1927.

• I had always wondered what these J.B.S. in J.B.S. Haldane stood for.

Halmos, Paul R.
Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

• I recall that during my mathematics studies I once, in an examination, lied down the hypothesis and began to chain statements one after the other without thinking, and I arrived to the conclusion after two pages of reasoning. That's my limit of non-thinking. And, by the way, there was a shorter (3/4 of a page) way of proving the same thing.

Halmos, Paul R.
...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

• We should remember this when teaching. Teachers use to present general theories to their students because they seem nicer and more complete. And students like that out of laziness. But is the example, the particular case that we can visionate and not the general theory what makes us believe something to be true.

Halmos, Paul R.
Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

• I have precisely the opposite recommendation. Be lazy. Read it, and try to remember more or less what is it about, but do forget the details. Then when (if) you ever need it, try to deduce it by yourself. If you are lucky enough, you might to it better that the author. (And what's the point in spending to much time doing things you can not publish?)

Halmos, Paul R.
To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

• Born and bred. A family without economical problems in a first world country also helps.

Hamming, Richard W.
Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

• Neither should I. But obviously this is not the point. The point is that Lebesgue integral is more useful as a tool than Riemmann's, and thus it is more likely to provide the necessary tools that allow us to check and test the plane's security. Engineers use the corona `theorem' to check the security of planes, so a counterexample to it built with the help of Lebesgue's integral could question the security of planes. And we should always remember the Tacoma bridge.

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.

• I do not like it much, though. I use it whenever necessary, but it gives me a feeling of being cheating. It is a question of different tastes. However, I still have the (completely wrong) impression that a direct proof would remain useful should the axioms prove contradictory.

Hardy, Godfrey H. (1877 - 1947)
Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

• This sentence is true if we substitute `on the whole' (which is meaningless) by `on the long term'. Then again, as Keynes said, on the long term we will all be dead.

Hardy, Godfrey H. (1877 - 1947)
There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
A Mathematician's Apology, London, Cambridge University Press, 1941.

• I deeply disagree with that. The fact that men who explain are usually second rate minds does not mean that a first rate mind should not try to explain his/her own works. Whenever we find it done, we are filled with a sensation of illumination.

Hardy, Godfrey H. (1877 - 1947)
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations.
A Mathematician's Apology, London, Cambridge University Press, 1941.

• Mathematical reality lies deep inside our heads. We have evolved in a way that has made our mathematical reality useful to us. If we were small enough to be influenced by the randomness of quantum mechanics our mathematical reality would be much closer to probabilities. On the other hand, if we were the size of a solar system our conception of a `straight line' would be influenced by general relativity. And there might be concepts we will never be able to grasp.

Hardy, Godfrey H. (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.

• So what? the most ancient person we can remember lived no more than 5000 years ago.

Harris, Sydney J.
The real danger is not that computers will begin to think like men, but that men will begin to think like computers.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

• Economists are already thinking like computers. Moreover, they are thinking people behave as computers (which they do not, in case you doubted).

Heath, Sir Thomas
[The works of Archimedes] are without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
A History of Greek Mathematics. 1921.

• If this is true (which I assume it is) then why they are never seen by the present days students?

Heaviside, Oliver (1850-1925)
[Criticized for using formal mathematical manipulations, without understanding how they worked:]
Should I refuse a good dinner simply because I do not understand the process of digestion?

• I completely agree with this. Physicists should use symbols regardless of their mathematical precision. If their formal manipulations lead to useful predictions, it is then the job of the (wise) mathematicians to give them a formal meaning.

Heisenberg, Werner (1901-1976)
An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them.
Physics and Beyond. 1971.

• To me, an expert is someone who knows so much on his subject that the is no longer able to have original ideas about it.

Hempel, Carl G.
The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as 'All bachelors are unmarried', but they also share the complete lack of empirical content which is associated with that certainty: The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter.
"On the Nature of Mathematical Truth" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• This is true. The mathematical propositions are not propositions about facts. They are propositions about ideas. Thus they are full of ideal content. (But our ideas are related to the world, somehow.)

Hermite, Charles (1822 - 1901)
There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation.
In The Mathematical Intelligencer, v. 5, no. 4.

• No, this world does not exist. On the other hand if it existed, it would be, unlike the physical world, infinite. In fact, it would be full of contradictions, and thus useless.

Hilbert, David (1862-1943)
Galileo was no idiot. Only an idiot could believe that science requires martyrdom - that may be necessary in religion, but in time a scientific result will establish itself.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1971.

• Nor am I.

Hilbert, David (1862-1943)
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

• This is not so. Our mathematics is a game played by human beings according to certain simple (to humans) rules with marks on paper and a human brain, shaped by evolution.

Hilbert, David (1862-1943)
Physics is much too hard for physicists.
C. Reid Hilbert, London: Allen and Unwin, 1970.

• Physics is much too hard for anyone.

Hilbert, David (1862-1943)
How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.

• How optimistic and ingenuous this sounds.

Hilbert, David (1862-1943)
One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and Schmidt,1971.

• And one can measure the unimportance of a scientific work by the number of earlier publications that appear cited in it.

Hilbert, David (1862-1943)
Mathematics knows no races or geographic boundaries; for mathematics,the cultural world is one country.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

• This is not true. Everybody knows that your nationality will affect the way your work is evaluated.

Hobbes, Thomas
To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad.
["This" is that the volume generated by revolving the region under 1/x from 1 to infinity has finite volume.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

• The only think required for this is to have received a course on integral calculus before one turns thirty years old. Happily `intuition' changes very fast.

Holmes, Oliver Wendell
Certitude is not the test of certainty. We have been cocksure of many things that are not so.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

• We are cocksure of many things that are not so.

Hofstadter, Douglas R. (1945 - )
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law.
Gödel, Escher, Bach 1979.

• And even some more time for snacks.

Hughes, Richard
Science, being human enquiry, can hear no answer except an answer couched somehow in human tones. Primitive man stood in the mountains and shouted against a cliff; the echo brought back his own voice, and he believed in a disembodied spirit. The scientist of today stands counting o ut loud in the face of the unknown. Numbers come back to him - and he believes in the Great Mathematician.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• It is often forgotten that the human mind is built up to see what it expects to see. Anything else is treated by the brain as a SEP, and can only be catched by surprise out of the corner of one's eye. (Not original, but nevertheless true.)

Huxley, Aldous
If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood, and many other desirable things. As a man he was a failure; as a monster he was superb.
Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934.

• Well I am glad that every now and then some of these monsters exist, even though I wouldn't like to be (or to be seen as) one of them.

Ibn Khaldun (1332-1406)
Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence.
The Muqaddimah. An Introduction to History.

• Nonsense. The brain has no perfect model of the world inside it. The geometrical vision inside the brain is of a practical kind, and needs not be orderly. It is orderly only insofar as being so has some evolutive advantage.

Jacobi, Carl
It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honour of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• The human mind is more honoured by love or friendship than by science. I do science because I believe it will be ultimately useful.

Jacobi, Carl
One should always generalize.
(Man muss immer generalisieren)
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.

• Yes, but there are always many possible generalizations. It is not so simple to find the right (or useful) generalization.

Jacobi, Carl
It is often more convenient to possess the ashes of great men than to possess the men themselves during their lifetime.
[Commenting on the return of Descartes' remains to France]
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

• Convenient is not the word. The right word in this sentence is `comfortable'.

Jevons, William Stanley
It is clear that Economics, if it is to be a science at all, must be a mathematical science.
Theory of Political Economy.

• I cannot but disagree with this. I am not against mathematical economics, indeed I like the subject. But economy can never be just a mathematical science. This is so for two reasons. the first one is purely humanitarian: we can not make a science that, when applied, influences the way people feels about life without having these feelings into account. The other reason is more scientific. The mathematization of economics should begin by the choice of a set of axioms, and this choice cannot be but a political one. Moreover, as any finite set of axioms would be affected by Godel's incompleteness theorems, every now and then we would be forced to make another (political) choice.

Kant, Emmanual (1724 - 1804)
The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.
The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.

• Here Kant seems to misuse the word `experience'. Mathematicians may not resort to physical experience, even if they sometimes do (or should do). But they manipulate mentally objects and formulas until they get used to them and can somehow predict their behaviour. Is this experience? I think so.

Kaplan, Abraham
Mathematics is not yet capable of coping with the naivete of the mathematician himself.
Sociology Learns the Language of Mathematics.

• Nor it will ever be.

Kaplansky, Irving
We [he and Halmos] share a philosophy about linear algebra: we think basis-free, we write basis-free , but when the chips are down we close the office door and compute with matrices like fury.
Paul Halmos: Celebrating 50 Years of Mathematics.

• One thing modern education is forgetting is to teach the ability to compute. I am not thinking of being able to compute the square root of 78 with 8 correct digits, in which a computer will always be better. But I find the ability of manipulating formulas, regardless of their meaning, useful to a mathematician in many occasions, and it is being forgotten by modern mathematics. Moreover, this ability can better (or even only) learned when one is quite young.

Kasner, E. and Newman, J.
Mathematics is man's own handiwork, subject only to the limitations imposed by the laws of thought.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

• Which are many more than we use to think of.

Kasner, E. and Newman, J.
Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

• One often forgets that common sense is a social construction, and changes from generation to generation.

Kasner, E. and Newman, J.
When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.

• I would never consider logical paradoxes as setbacks, but as mathematics' greatest success. Godel's results will in the future be considered as the main production of 20th century mathematics.

Kasner, E. and Newman, J. R.
The testament of science is so continually in a flux that the heresy of yesterday is the gospel of today and the fundamentalism of tomorrow.
E. Kasner and J. R. Newman, Mathematics and the Imagination, Simon and Schuster, 1940.

• But we keep on burning heresiarchs every now and then.

Keller, Helen (1880 - 1968)
Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important... The knowledge doesn't make life any sweeter or happier, does it?
The Story of My Life. 1903.

• Yes, indeed it does.

Kepler, Johannes (1571-1630)
Nature uses as little as possible of anything.

• And this includes brains.

Keynes, John Maynard
It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability.
The Application of Probability to Conduct.

• As far as the odds are 10^12 to one, I don't care (I was going to write 10^6 to 1 but this is less than the number of seconds in a fortnight).

Koestler, Arthur (1905- )
In the index to the six hundred odd pages of Arnold Toynbee's A Study of History, abridged version, the names of Copernicus, Galileo, Descartes and Newton do not occur yet their cosmic quest destroyed the medieval vision of an immutable social order in a walled-in universe and transformed the European landscape, society, culture, habits and general outlook, as thoroughly as if a new species had arisen on this planet.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

• This means that Toynbee completely failed to understand history.

Koestler, Arthur (1905- )
Nobody before the Pythagoreans had thought that mathematical relations held the secret of the universe. Twenty-five centuries later, Europe is still blessed and cursed with their heritage. To non-European civilisations, the idea that numbers are the key to both wisdom and power, seems never to have occurred.
The Sleepwalkers. 1959.

• This fact shows than the evolution of knowledge that we take for granted is often much more random than we think, and could in any moment have taken a completely different turn.

Kovalevsky, Sonja
Say what you know, do what you must, come what may.
[Motto on her paper "On the Problem of the Rotation of a Solid Body about a Fixed Point."]

• And run when you must.

Kronecker, Leopold (1823 - 1891)
God made the integers, all else is the work of man.
Jahresberichte der Deutschen Mathematiker Vereinigung.

• Language made the integers, but our eyes made the straight lines.

LaGrange, Joseph-Louis
The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure.
Preface to Mécanique Analytique.

• His methods might not require figures, but maybe his readers would have been grateful to him if he had used them for illustration.

Landau, E.
[Asked for a testimony to the effect that Emmy Noether was a great woman mathematician, he said:]
I can testify that she is a great mathematician, but that she is a woman, I cannot swear.
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co ltd., 1953.

• He also said she was very plain.

Lehrer, Thomas Andrew (1928- )
In one word he told me the secret of success in mathematics: plagiarize only be sure always to call it please research.
Lobachevski (A musical recording.)

• The standard way of plagiarizing in mathematics consists in taking one (good) article and making small improvements, usually by taking away some hypothesis, thus producing a new article. This practice is very usual, and more than half the articles in mathematics are written in this way (in Spanish it is called `fusilar', to execute by shooting). Even worse, often the plagiarizer is the same author of the plagiarized first article. This has nothing to do with research, and in fact falls short of being plain fraud. It is often believed, however, that when you do the same beginning with two articles instead of one, then this is research. False. Research is related to the number of original ideas introduced, and the ratio of original ideas with respect to the number of published articles is decreasing.

Leibniz, Gottfried Whilhem (1646-1716)
Nothing is more important than to see the sources of invention which are, in my opinion more interesting than the inventions themselves.
J. Koenderink, Solid Shape, Cambridge Mass.: MIT Press, 1990.

• Our way of writing articles as perfect constructions rather than as works in progress makes us hide the sources of invention. Something should be done about this.

Leibniz, Gottfried Whilhem (1646-1716)
The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.

• This (something nonexistent but useful) was the way complex numbers were viewed before we had some way (the complex plane) of relating them to our mental construction of reality.

Leibniz, Gottfried Whilhem (1646-1716)
In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labour of thought is wonderfully diminished.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

• Picturing it is often the whole point.

Leibniz, Gottfried Whilhem (1646-1716)
Although the whole of this life were said to be nothing but a dream and the physical world nothing but a phantasm, I should call this dream or phantasm real enough, if, using reason well, we were never deceived by it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• That's real enough for me, too.

da Vinci, Leonardo (1452-1519)
No human investigation can be called real science if it cannot be demonstrated mathematically.

• >No human investigation can be called real science if it cannot produce useful predictions. There is no mathematical proof of the fact that the sun will raise tomorrow (we need to believe in Newton's Laws, at least). A lot of mathematics into some science makes the science more believable, though, at least for me.

da Vinci, Leonardo (1452-1519)
Inequality is the cause of all local movements.

• That's a law.

Lichtenberg, Georg Christoph (1742 - 1799)
All mathematical laws which we find in Nature are always suspect to me, in spite of their beauty. They give me no pleasure. They are merely auxiliaries. At close range it is all not true.
In J P Stern Lichtenberg, 1959.

• Any time I hear someone saying that he fails to see the beauty or to get pleasure from mathematics I cannot help but thinking that that person is misformed, as if anyone fails to get pleasure from poetry or music. (that's harsh on my side, I know.)

Lichtenberg, Georg Christoph (1742 - 1799)
I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, `Oh, I see. That's how it must be.'; This is a sign that they explain it to themselves from within their own system.

• Precisely. Inner knowledge is when one gets some new (to him) idea and gets it to fit into their previously built system. It is a dangerous thing, because all checks on the truth of the idea are then omitted.But when someone interiorizes an idea in this way, he is then able to use it in a more efficient way. (It is like when some response becomes reflex.)

Lippman, Gabriel (1845-1921)
[On the Gaussian curve, remarked to Poincaré:]
Experimentalists think that it is a mathematical theorem while the mathematicians believe it to be an experimental fact.
In D'Arcy Thompson On Growth and Form, 1917.

• It is both, at least nowadays. No mathematical theorem without relation with some experimental facts would ever have gained that recognition the Gaussian curve has.

Littlewood, J. E. (1885 -1977)
A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.

• Why? Because, by showing us things from a different angle, it teaches us a lot more.

Littlewood, J. E. (1885 -1977)
I recall once saying that when I had given the same lecture several times I couldn't help feeling that they really ought to know it by now.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.

• I had the same feeling, specially when I was giving, each week, the same lecture four times in a row.

Littlewood, J. E. (1885 -1977)
It is possible for a mathematician to be "too strong" for a given occasion. He forces through, where another might be driven to a different, and possible more fruitful, approach. (So a rock climber might force a dreadful crack, instead of finding a subtle and delicate route.)
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.

• This is another instance of the fact that to much expertise does not necessarily lead to a better research.

Littlewood, J. E. (1885 -1977)
The infinitely competent can be uncreative.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

• The infitely competent will be uncreative, because with all the knowledge will not leave room to imagination (I am thinking of Funes `the memorious').

Littlewood, J. E. (1885 -1977)
We come finally, however, to the relation of the ideal theory to real world, or `real' probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: `If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician'. In practice he is apt to say: `try this; if it works that will justify it'. But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.
A Mathematician's Miscellany, Methuen Co. Ltd, 1953.

• But since there cannot be mathematical evidence on anything, inductive experience is enough for me (that's all you can ever get, boy.)

Lobatchevsky, Nikolai
There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• There might be. But `natural selection' is also applicable to ideas.

Mach, Ernst (1838-1916)
The mathematician who pursues his studies without clear views of this matter, must often have the uncomfortable feeling that his paper and pencil surpass him in intelligence.
"The Economy of Science" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• But what is really happening is that he is using the intelligence of his ancestors (the ones who created that mathematical language).

Maistre Joseph Marie de (1753 - 1821)
The concept of number is the obvious distinction between the beast and man. Thanks to number, the cry becomes a song, noise acquires rhythm, the spring is transformed into a dance, force becomes dynamic, and outlines figures.

• Poetic, but not necessarily true. There might be some animal that, because of the conditions of its evolution, has the concept of number inside its brain.

Mann, Thomas (1875-1955)
Some of the men stood talking in this room, and at the right of the door a little knot had formed round a small table, the centre of which was the mathematics student, who was eagerly talking. He had made the assertion that one could draw through a given point more than one parallel to a straight line; Frau Hagenström had cried out that this was impossible, and he had gone on to prove it so conclusively that his hearers were constrained to behave as though they understood.
Little Herr Friedemann.

• But Frau Hagenström was right, or at least more right than the student. We may not live on an Euclidean universe, but this is what we see.

Mathesis, Adrian
The greatest unsolved theorem in mathematics is why some people are better at it than others.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

• Simply because all persons are born different. Is the fact that each person is his/her own world what makes any of us invaluable, and thus makes us all equal.

Matthias, Bernd T
If you see a formula in the Physical Review that extends over a quarter of a page, forget it. It's wrong. Nature isn't that complicated.

• Or otherwise the formula is just a particular case of some other formula.

Maxwell, James Clerk (1813-1879)
... that, in a few years, all great physical constants will have been approximately estimated, and that the only occupation which will be left to men of science will be to carry these measurements to another place of decimals.
Scientific Papers 2, 244, October 1871.

• It scares me to see that even the greatest of men end up, with age, with the wrong belief that nothing is left to be discovered.

Mayer, Maria Goeppert (1906 -1972)
Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man.
J. Dash, Maria Goeppert-Mayer, A Life of One's Own.

• The mind of man is closer to us than nature and, in any case, nature (or rather or view of it) might be just a construction of our mind.

McShane, E. J.
There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.
Bulletin of the American Mathematical Society, v. 69, p. 611, 1963.

• What is really annoying is the fact that mathematicians tend to believe that the constrains someone has put in some definition in order to make things simpler are real constrains and not simplifications.

Mermin, N. David (1935 -)
Bridges would not be safer if only people who knew the proper definition of a real number were allowed to design them.
"Topological Theory of Defects" in Review of Modern Physics, v. 51 no. 3, July 1979.

• I doubt it.

Mordell, L.J.
Neither you nor I nor anybody else knows what makes a mathematician tick. It is not a question of cleverness. I know many mathematicians who are far abler than I am, but they have not been so lucky. An illustration may be given by considering two miners. One may be an expert geologist, but he does not find the golden nuggets that the ignorant miner does.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

• Yet again another proof of the fact that an excessive amount of knowledge or expertize can put limits to the insights.

Napoleon (1769-1821)
A mathematician of the first rank, Laplace quickly revealed himself as only a mediocre administrator; from his first work we saw that we had been deceived. Laplace saw no question from its true point of view; he sought subtleties everywhere; had only doubtful ideas, and finally carried the spirit of the infinitely small into administration.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc.,1988.

• A good mathematician will hardly ever be a good administrator. The skills required are not only different, but often opposite to one another. Moreover, mathematicians are not very able at dealing with people, in particular with subordinates.

Nebeuts, E. Kim
To state a theorem and then to show examples of it is literally to teach backwards.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

• I agree.

Neumann, Franz Ernst (1798 - 1895)
The greatest reward lies in making the discovery; recognition can add little or nothing to that.

• Recognition adds little, but not nothing. We all need some recognition to convince ourselves that our work is interesting.

von Neumann, Johann (1903 - 1957)
In mathematics you don't understand things. You just get used to them.
In G. Zukav The Dancing Wu Li Masters.

• That's true for all the sciences. But see also the comment on Lichtenberg's on the ways of getting used to some idea.

Newman, James, R.
The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• I think I remember there was some luck involved, also. Certainly there was luck in the finding of Pluto, at least.

Newman, James R.
To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• In mathematical economics it is also important to be absolutely clear on what you are assuming. As an example, it is always assumed that an increase in the GNP or on the per capita income are good things, regardless of how they are distributed. Ultimately, the problem with economy is that it claims to be related to welfare, when it is only partly so.

Newman, James R.
Games are among the most interesting creations of the human mind, and the analysis of their structure is full of adventure and surprises. Unfortunately there is never a lack of mathematicians for the job of transforming delectable ingredients into a dish that tastes like a damp blanket.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• Well, for us mathematicians, playing and doing mathematics are not necessarily different things.

Newton, Isaac (1642-1727)
The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.
Principia Mathematica.

• Nowadays we take a set of axioms and then call anything satisfying them a line.

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, p19.

• I am not so sure of this whole business to be that rational. There is more than reason to be a good mathematician.

Pascal, Blaise (1623-1662)
Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.
Pensees. 1670.

• Thankfully.

Pascal, Blaise (1623-1662)
Our notion of symmetry is derived form the human face. Hence, we demand symmetry horizontally and in breadth only, not vertically nor in depth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.

• This is a good point. It shows how much the way we have evolved affects the way we think, even mathematically. Moreover, we always look for symmetries, even when there are (or need not be) any around. To a given problem, we always find first the solutions that have some symmetries, and often ignore the other possible solutions.

Pascal, Blaise (1623-1662)
Everything that is written merely to please the author is worthless.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1966.

• Everything that is written merely to please someone is worthless (except in love letters).

Pascal, Blaise (1623-1662)
The sole cause of man's unhappiness is that he does not know how to stay quietly in his room.
Pensees. 1670.

• Hence the sentence `happy as a clam'.

Pascal, Blaise (1623-1662)
There are two types of mind ... the mathematical, and what might be called the intuitive. The former arrives at its views slowly, but they are firm and rigid; the latter is endowed with greater flexibility and applies itself simultaneously to the diverse lovable parts of that which it loves.
Discours sur les passions de l'amour. 1653.

• Are they really opposite?

Passano, L.M.
This trend [emphasizing applied mathematics over pure mathematics] will make the queen of the sciences into the queen of the sciences.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

• Nonsense. It is important to realize that the whole thing of pure versus applied is cyclical, with a period of some 50 years (two generations). Now we are into a period of applied, due partly to the influence of computers. If we really want to profit from this, though, we should put more discrete mathematics into our studies.

Peirce, Charles Sanders (1839-1914)
The one [the logician] studies the science of drawing conclusions, the other [the mathematician] the science which draws necessary conclusions.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• Aren't they the same thing? All the conclusions are necessary, given a set of axioms.

Peirce, Charles Sanders (1839-1914)
...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• Mathematics may need no ethics, but mathematicians do.

Pedersen, Jean
Geometry is a skill of the eyes and the hands as well as of the mind.

• The eyes and the hand have their models in the mind. Maybe Pedersen has a restricted view of the mind, seeing only its `linguistic' side.

Plato (ca 429-347 BC)
Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.

• This was long ago. On the other hand, any science properly done increases one's ability to question things, and is thus dangerous to the state.

Plato (ca 429-347 BC)
I have hardly ever known a mathematician who was capable of reasoning.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• I have never known a philosopher as overpraised as Plato. He is a fascist, and his reasonings are almost always misleading and tricky. (Aristotle is a better philosopher that Plato.)

Poe, Edgar Allen
To speak algebraically, Mr. M. is execrable, but Mr. G. is (x + 1)- ecrable.
[Discussing fellow writers Cornelius Mathews and William Ellery Channing.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

• Brilliant.

Poincaré, Jules Henri (1854-1912)
Mathematics is the art of giving the same name to different things.
[As opposed to the quotation: Poetry is the art of giving different names to the same thing].

• In other words, mathematics is the art of extracting from each set of objects the properties that are relevant in a given problem, and then deducing from these properties the behaviour of these objects regardless of their other properties.

Poincaré, Jules Henri (1854-1912)
Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• I would like to know when did he say so. Sounds much more like Hilbert than like himself, so he must have been influenced by Hilbert at that time. In any case, this `automatic proving' would have the same interest as `automatic writing', that is, a rather tangential interest.

Poincaré, Jules Henri (1854-1912)
A scientist worthy of his name, about 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 NC:Rome Press Inc., 1988.

• That is the feeling of having created something beautiful. This feeling must have been originally a byproduct of reproduction, but it turned out to be useful for the evolution not of the man himself but of the community in which he lives.

Poincaré, Jules Henri (1854-1912)
The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• Physical, in this case, means related to reality, not just related to the physical science.

Poincaré, Jules Henri (1854-1912)
Mathematicians do not study objects, but 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.

• This also sounds like Hilbert's (but is truer than the previous one).

Poincaré, Jules Henri (1854-1912)
The mind uses its faculty for creativity only when experience forces it to do so.

• This point of view is rather deterministic, and was in the spirit of his age. Nowadays we would rather say that creativity is something that happens at random, and then its products get used for a purpose when there is a need for them.

Poincaré, Jules Henri (1854-1912)
...by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.
Science and Method.

• True. Observe that we have adopted a Cartesian coordinates as a standard. But in Cartesian coordinates there is a privileged point, the origin of coordinates, that does not exist in space, but in our minds.

Polyá, George (1887, 1985)
Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
D. J. Albers and G. L. Alexanderson, Mathematical People, Boston: Birkhäuser, 1985.

• I would add a library.

Polyá, George (1887, 1985)
When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
The American Mathematical Monthly, v. 100, no. 3.

• That happens because learning something is not the same as understanding it. The definition of a continuous function may be useful to the students, but is it much more important for them to understand (or to assimilate) the notion of continuity.

Quine, Willard Van Orman
Just as the introduction of the irrational numbers ... is a convenient myth [which] simplifies the laws of arithmetic ... so physical objects are postulated entities which round out and simplify our account of the flux of existence... The conceptional scheme of physical objects is [likewise] a convenient myth, simpler than the literal truth and yet containing that literal truth as a scattered part.
In J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990.

• This point of view (with respect to physical objects, not w.r.t. irrational numbers) is not completely incorrect, but it has in any case no practical effects, and thus it is useless.

Renan, Ernest
The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.
Souvenirs d'enfance et de jeunesse.

• Moreover, these facts are today's common sense.

Richardson, Lewis Fry (1881 - 1953)
Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematicians' stock of formulae. Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless.
Mathematics of War and Foreign Politics.

• A mathematical statement may be wrong or not. Most nonscientists would finish their analysis an that point. But once some statement has been proved right, the most difficult part is to decide which statements are relevant.

Rosenlicht, Max (1949)
You know we all became mathematicians for the same reason: we were lazy.

• Thankfully. A mathematician which were not lazy would produce long proofs and lots of irrelevant articles.

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.

• Mathematics consists of distilling concepts from reality and then using the rules of logic to produce conclusions that can be useful.

Russell, Bertrand (1872-1970)
Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• Given any possible world, there will be there a set of axioms to whom this world must conform. But our mathematics are related to our world in that our set of axioms has been chosen by us.

Russell, Bertrand (1872-1970)
Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.
The Scientific Outlook, 1931.

• That is why I refuse most of the physics when it is translated into words as a mystification.

Russell, Bertrand (1872-1970)
At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• He is obviously talking about the Axiom of Choice. First of all, I have to say that I like this axiom. I like the idea of being able to make a possibly uncountable number of choices at once, and I apply it whenever I consider it necessary, or even convenient. This said, I am not sure the Axiom of Choice is something obvious. I think I accept it simply because I have been educated in a culture that has embedded the idea of `free will'. I do not think that a culture whose main idea were that of `predestination' would ever have accepted such an axiom. Thus in a sense the Axiom of Choice is accepted on a cultural basis. Moreover, in a life we can only make a finite number of choices, and even if we lived forever, we could only make a denumerable number of choices, provided that we needed some finite time to make each choice. Thus no result related to the Axiom of Choice will ever have any physical significance.

Russell, Bertrand (1872-1970)
Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• It turned out to be just `anything as small as you can take'.

Russell, Bertrand (1872-1970)
The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
Principles of Mathematics. 1903.

• This is only part of the truth. Mathematics is symbolic logic, but each time we define a new concept we are making a choice that restricts the parts of the symbolic logic we are studying.

Russell, Bertrand (1872-1970)
Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives' mouths.
The Impact of Science on Society, 1952.

• The important question is always: how many of the things we are taking for granted now are scientific truths and how many are just prejudices?

Russell, Bertrand (1872-1970)
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 I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. 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 no 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.
Portraits from Memory.

• He did quite a lot, though.

Sanford, T. H.
The modern, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers not because it is abstract, but because it is a representation of actual things.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• We should teach mathematics with this idea in mind, but it seems to me that the contact with actual things was lost somewhere on the way.

Sarton, G.
The main duty of the historian of mathematics, as well as his fondest privilege, is to explain the humanity of mathematics, to illustrate its greatness, beauty and dignity, and to describe how the incessant efforts and accumulated genius of many generations have built up that magnificent monument, the object of our most legitimate pride as men, and of our wonder, humility and thankfulness, as individuals. The study of the history of mathematics will not make better mathematicians but gentler ones, it will enrich their minds, mellow their hearts, and bring out their finer qualities.

• I do believe that some knowledge of the history of mathematics makes better researchers and better teachers. It makes better researchers because it helps them to relativize things. And it sure makes better teachers because by understanding how some concepts evolved one can better introduce these concepts in the minds of his/her students.

Smith, Henry John Stephen (1826-1883)
It is the peculiar beauty of this method, gentlemen, and one which endears it to the really scientific mind, that under no circumstance can it be of the smallest possible utility.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

• Utility ends up finding its way, however unexpectedly. Few people would ever expect to see how number theory would be found useful.

Spengler, Oswald (1880 -1936)
The mathematic, then, is an art. As such it has its styles and style periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes form epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) side-glance at contemporary mathematics.
The Decline of the West.

• In mathematics there are styles and fashions. For the last 50 years mathematics has been done in the Bourbaky style, but this is slowly changing now. Among the subjects that have been in fashion lately, I could mention nonstandard analysis, fractals, and catastrophe theory. Styles last some 50 years, whereas the subjects in fashion last some 5 years.

Steinmetz, Charles P.
Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional.
In E. T. Bell Men of Mathematics, New York: Simona and Schuster, 1937.

• They are true with respect to an Axioms System. How a given particular Axioms System relates to the reality is another question.

Sternberg, S.
Kepler's principal goal was to explain the relationship between the existence of five planets (and their motions) and the five regular solids. It is customary to sneer at Kepler for this. It is instructive to compare this with the current attempts to "explain" the zoology of elementary particles in terms of irreducible representations of Lie groups.

• I hope that the willingness that theoretical physics has shown lately to apply whatever mathematical idea that becomes at hand does not denote a lack of original ideas.

Sullivan, John William Navin (1886-1937)
Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.
Aspects of Science, 1925.

• By informing us of the nature of our own minds mathematics informs us of the nature of the world that has shaped them.

Tietze
The story was told that the young Dirichlet had as a constant companion all his travels, like a devout man with his prayer book, an old, worn copy of the Disquisitiones Arithmeticae of Gauss.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

• It is always worth to read the masters.

Titchmarsh, E. C.
It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.

• There are many things for which it is useful to know that pi is irrational. The main utility is, however, that by knowing that it is not rational we do not spend our time in searching for its rational expression.

Todhunter, Isaac (1820 - 1910)
[Asked whether he would like to see an experimental demonstration of conical refraction]
No. I have been teaching it all my life, and I do not want to have my ideas upset.

• This sounds as if he rejected the experimental method. No good scientist should allow himself to be heard saying this.

Veblen, Thorstein (1857-1929)
The outcome of any serious research can only be to make two questions grow where only one grew before.
The Place of Science in Modern Civilization and Other Essays.

• By increasing the area of knowledge we also enlarge its boundaries.

Voltaire (1694-1778)
Vous avez trouve par de long ennuis
Ce que Newton trouva sans sortir de chez lui.
[Written to La Condamine after his measurement of the equator.]
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• Yes, but we should always check experimentally our theoretical results. Otherwise they are useless.

Voltaire (1694-1778)
He who has heard the same thing told by 12,000 eye-witnesses has only 12,000 probabilities, which are equal to one strong probability, which is far from certain.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

• It is not certain, but it is not that far from certain.

Voltaire (1694-1778)
There are no sects in geometry.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking Press, 1962.

• There have been many sects, among them the sect of the projectivists.

Weil, Andre (1906 -1998)
Every mathematician worthy of the name has experienced ... the state of lucid exaltation in which one thought succeeds another as if miraculously... this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work...
The Apprenticeship of a Mathematician.

• Being in this state of mind is a very nice experience, but then one has to check carefully all the details.

Weyl, Hermann (1885 - 1955)
Our federal income tax law defines the tax y to be paid in terms of the income x; it does so in a clumsy enough way by pasting several linear functions together, each valid in another interval or bracket of income. An archeologist who, five thousand years from now, shall unearth some of our income tax returns together with relics of engineering works and mathematical books, will probably date them a couple of centuries earlier, certainly before Galileo and Vieta.
The Mathematical Way of Thinking, an address given at the Bicentennial Conference at the University of Pennsylvania, 1940.

• I have always believed that an income tax as 0.4*(income)/(income +2*minimum_wage) would be more elegant, but then again the public, and in particular the politicians, use to like linear expressions.

Weyl, Hermann (1885 - 1955)
Without the concepts, methods and results found and developed by previous generations right down to Greek antiquity one cannot understand either the aims or achievements of mathematics in the last 50 years.
[Said in 1950]
The American Mathematical Monthly, v. 100. p. 93.

• That is to say that mathematical concepts have evolved in the same way as all other scientific concepts.

Whitehead, Alfred North (1861 - 1947)
Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about "any" things or about "some" things, without specifications of definite particular things.

• They proved propositions about straight lines and circles, or about areas of land. Thus their propositions had a connexion to reality.

Whitehead, Alfred North (1861 - 1947)
No Roman ever died in contemplation over a geometrical diagram.
[A reference to the death of Archimedes.]
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.

• I do not believe that Archimedes had decided to die in that way. He was unaware of the danger he was in.

Whitehead, Alfred North (1861 - 1947)
Life is an offensive, directed against the repetitious mechanism of the Universe.
Adventures of Ideas, 1933.

• Life increases entropy.

Whitehead, Alfred North (1861 - 1947)
It is more important that a proposition be interesting than that it be true. This statement is almost a tautology. For the energy of operation of a proposition in an occasion of experience is its interest and is its importance. But of course a true proposition is more apt to be interesting than a false one.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.

• In other words, it is more amusing to take an interesting proposition and to check out when it is true that to take a true proposition and see whether in any case this proposition is interesting.

Wiener, Norbert (1894-1964)
The modern physicist is a quantum theorist on Monday, Wednesday, and Friday and a student of gravitational relativity theory on Tuesday, Thursday, and Saturday. On Sunday he is neither, but is praying to his God that someone, preferably himself, will find the reconciliation between the two views.

• But what if there is no such reconciliation?

Wiener, Norbert (1894-1964)
Progress imposes not only new possibilities for the future but new restrictions.
The Human Use of Human Beings.

• Any evolution of things that does not bring more freedom to humanity should not be called progress.

Wiener, Norbert (1894-1964)
The Advantage is that mathematics is a field in which one's blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one's best moments that count and not one's worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician's reputation.
Ex-Prodigy: My Childhood and Youth.

• This used to be true, and it still is in the long term. For one's immediate career, though, it is the number of papers what counts.

Wilder, R. L.
There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned.
Introduction to the Foundations of Mathematics.

• The whole point of it is that we abstract from something.

Wittgenstein, Ludwig (1889-1951)
Mathematics is a logical method ... Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics.
Tractatus Logico Philosophicus, New York, 1922, p. 169.

• We do think in terms of lines, circles, and numbers. Thus mathematical propositions express thoughts.

Zeeman, E Christopher (1925 - )
Technical skill is mastery of complexity while creativity is mastery of simplicity.
Catastrophe Theory, 1977.

• Technical skill is mastery of details.