Old and new quotes

(Contributions may be sent to olsson@math.ku.dk )


I sometimes think the public is in awe of mathematics because it cannot comprehend how mathematicians can explore phenomena that are not detectable by our senses or by high-tech equipment we invent. And the public does not understand why it can take so long before mathematical developments result in practical applications.
What I am trying to say is that there is a unity in mathematics that we are connected to. We are entranced by its beauty and its power. It is frustrating not to be able to communicate that to the public. (Isadore Singer)

Aber der übliche Mathematikunterricht langweilt nicht nur, er unterfordert vor allem die Intelligenz der Schüler. Es scheint eine fixe Idee der Pädagogik zu sein, daß Kinder nicht in der Lage sind, abstrakt zu denken.(...) Es wäre allerdings unfair, wollte man die Mathematiklehrer allein für das Desaster verantwortlich machen. Diese bedauernswerten Menschen sind nicht nur mit den Vorgaben der Didaktiker und ihrer Moden geschlagen, sie müssen auch am Gängelband der Ministerialbürokratie operieren, die ihnen ganz brutale Lehrpläne und Lernziele vorschreibt.  (Hans Magnus Enzensberger)

What is mathematics and its purpose?

Mathematics is the study of analogies between analogies. All science is. Scientists want to show that things that don't look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding. (Gian-Carlo Rota, 'Indiscrete thoughts')

Mathematics in general is fundamentally the science of self-evident things. (Felix Klein, 'Anwendung der Differential- und Integralrechnung auf Geometrie', 1902)

The whole of mathematics consists in the organization of a series of aids to the imagination in the process of reasoning. (Alfred N. Whitehead, Preface to 'Universal algebra' , 1898)

We all believe that mathematics is an art. (Emil Artin, Bull. AMS, 1953, p. 479)

... mathematics is partly art and partly science (corresponding roughly to the traditional pure/applied distinction)  (Sir Michael Atiyah, Bull. AMS, 2006, p. 87)

Mathematics as an expression of the human mind reflects the active will, the comtemplative reason, and the desire for aestetic perfection. (Richard Courant, 'What is mathematics?')

The aim of mathematics is to explain as much as possible in simple terms. (Sir Michael Atiyah, Bull.LMS, vol 10, 1978)

The ultimate goal of mathematics is to eliminate any need for intelligent thought. (Alfred N. Whitehead)

The object of pure Mathematic (is) that of unfolding the laws of human intelligence. (JJ Sylvester, CW 3, 424)

Mathematics is the science of patterns, and nature exploits just about every pattern that there is. (Ian Stewart, 'Natures numbers')

Mathematics is often defined as the science of space and number . . . it was not until the recent resonance of computers and mathematics that a more apt definition became fully evident: mathematics is the science of patterns. (Lynn Arthur Steen )

Philosophically, mathematics is not a part of science. Mathematics studies patterns, science studies nature ( Lynn Arthur Steen )

The ubiquity of mathematics in science and human affairs derives from the fact that mathematics is the study of the collection of all possible patterns. (John Barrow)

It is true that Mr. Fourier had the opinion that the principal purpose of mathematics was the benefit of the society and the explanation of phenomena of nature; but a philosopher like he should know that the sole purpose of science is the honor of the human mind, and under this title, a question about numbers is as valuable as a question about the system of the world. (C. G. Jacobi, Letter to Legendre, 1830)

I have mentioned mathematics as a way to settle in the mind a habit of reasoning closely and in train; not that I think it necessary that all men should be deep mathematicians, but that having got the way of reasoning, which that study necessarily brings the mind to, they might be able to transfer it to other parts of knowledge as they shall have occasion. (John Locke: Of the Conduct of the Understanding)

The Mathematics are either pure or mixed. To the Pure Mathematics are those sciences belonging which handle quantity determinate, merely severed from any axioms of natural philosophy; and these are two, Geometry and Arithmetic; the one handling quantity continued, and the other dissevered. [--] Mixed hath for subject some axioms or parts of natural philosophy, and considereth quantity determined, as it is auxiliary and incident unto them. [--] For many parts of nature can neither be invented with sufficient subtilty, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity, without the aid and intervening of the mathematics; of which sort are perspective, music, astronomy, cosmography, architecture, enginery, and divers others. (Francis Bacon, The Advancement of Learning (1605). )

Mathematik ist die Wissenschaft von der Verknüpfung der Grössen.. (Hermann Grassmann, Lehrbuch der Arithmetik, 1861)

Die Mathematik ist die Lehre von möglichen Beziehungen zwischen möglichen Dingen. (Reinhold Baer in 'Hegel und die Mathematik' 1931)

Mathematik ist die Sprache der Götter (Novalis)

Pure mathematics is, in its way, the poetry of logical ideas (Albert Einstein)

Die Mathematik muss man schon deswegen studieren, weil sie die Gedanken ordnet. (M. W. Lomonossow)

... the following conclusion is obtained, which is also known as the thesis of logicism concerning the nature of mathematics:
Mathematics is a branch of logic. It can be derived from logic in the following sense:
a. All the concepts ot mathematics, i.e., of arithmetic, algebra, and analysis, can be defined in terms of four concepts of pure logic.
b. All the theorems of mathematics can be deduced from those definitions by means of the principles of logic (including the axioms of infinity and choice).
(Carl G. Hempel, On the Nature of Mathematical Truth, Amercan Math. Monthly 1945)

Is mathematics useful?

Die Mathematik wäre tot, wenn sie auf Anwendungen warten wollte.  (Hubert Korzonek)

I have never done anything useful. No discovery of mine has made or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. (G H Hardy's 'Apology')

I have often reminded my students that the best mathematical achievements took place when the question, 'What is it for?' was not asked (Bhama Srinivasan)

...we've learned that it's hard to prophesy what piece of mathematics will have what particular applications. Mathematics has to advance as an organic whole, in ways that seem right to the people inside it. (Michael Freedman)

We don't make the products; we make them possible (W.Y. Velez)

Very little mathematics has direct applications - though fortunately most of it has plenty of indirect ones. (Gian-Carlo Rota, 'Indiscrete thoughts')

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? (Albert Einstein)

Mathematics is the language in which the laws of the universe are encoded, and without it, we would have no idea of where we are or how we got here. (Tim Radford)

In my opinion mathematicians must not succumb to this trend and say 'Yes, we will do useful things.' There are a few things which have direct  use. But one must present mathematics in its diversity, and be courageous to present oneself as well and sincerely as possible. (Matthias Kreck)

These days people are much more aware that mathematics is a communal endeavor: even the most brilliant idea gets meaning only from its relation to the whole.
(Dusa McDuff)

The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe. (Karl Pearson)

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. (Eugene Wigner:  The Unreasonable Effectiveness of Mathematics in the Natural Sciences)

It is natural to think that an abstract science cannot be of much importance in affairs of human life, because it has omitted from its consideration everything of real interest. (A.N. Whitehead)

In the history of human thought we can easily trace many examples of principal importance showing how mathematical thinking essentially contributed to the fundamental change of our basic concepts of the structure of natural phenomena. (A. Gieysztor, opening ceremony of ICM 1983)

In mathematical science, more than in all others, it happens that truths which are at one period the most abstract, and apparently the most remote from all useful application, become in the next age the bases of profound physical inquiries, and in the succeeding one, perhaps, by proper simplification and reduction to tables, furnish their ready and daily aid to the artist and the sailor. (Charles Babbage, ' Decline of Science in England ')

Mathematics contains much that will neither hurt one if one does not know it nor help one if one does know it. (J.B. Mencken)

Mathematics is purely hypothetical: it produces nothing but conditional propositions. (C.S. Pierce)

Mathematics, springing up from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearences and of concepts which relate totally disparate concrete ideas. (Saunders MacLane, American Math. Monthly, 1954)

The lack of real contact between mathematics and biology is either a tragedy, a scandal or a challenge, it is hard to decide which. (Gian-Carlo Rota, 'Discrete Thoughts')

Much of modern art - from cubism to the very newest - would be unthinkable without the original inspiration from non-Euclidian geometry... (Torkil Heiede)

Historically, architecture was part of mathematics, and in many periods of the past, the two disciplines were indistinguishable. (Nikos A. Salingaros)

Mathematics commands all my respect, but I have no use for engines. ( Joseph Conrad )

Defined broadly enough, mathematics encompasses everything. ( John Allen Paulos )

The professor has been a volunteer surf life-saver for more than 50 years. He said that mathematics had given him a decisive edge in many surf competitions. "I've been using various theorems - like Pythagoras - which I can use by standing on a beach before a surf race to decide which route I'll take. It's like an operations research problem." (From BBC)

Mathematics and sex are two words not usually seen in the same sentence. But mathematician, Dr Clio Cresswell, has written a whole book about these diverse topics.  Jenni speaks to her about the 12 relationship rule and how you can model the emotions of love using differential equations. (From BBC)

So I went to work: and here I must needs observe, that as reason is the substance and original of the mathematics, so by stating and squaring everything by reason, and by making the most rational judgment of things, every man may be in time master of every mechanic art. (Daniel Defoe, Robinson Crusoe )

All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA). (V. Arnold, in: Mathematics: Frontiers and Perspectives, AMS 2000)

Die Annahme, dass alle Naturwissenschaften eine mathematische Behandlungsweise zulassen müssen, ist als eine hypothetische anzusehen. (Issai Schur)

Ich behaupte aber, dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist. (Immanuel Kant)

Es ist nicht gelungen, die Psychologie auf exacter mathematischer Grundlage zu begründen. (E. Landau, CW I).

About mathematicians:

I must admit that outside the university, it is difficult to be a pure mathematician. No one in my family understands what I do. My neighbors wonder why I spend all my time in my study scribbling with pencil on a yellow pad of paper instead of going outside to mow the lawn. (Isadore Singer)

It's true we pure mathematicians are connected to a different world. But it is a very real world nevertheless.  (Isadore Singer)

If the mathematicians were inclined toward parable and metaphor - and they most definitely are not - they might describe a vast wilderness, and in it a small society
of men and women whose business it is to lay railroad track. This has become an art, and they have become artists - artists of track, lovers of track, connoisseurs of track.  (James Gleick,  New York Times,  August 12 1986)

The scientific life of mathematicians can be pictured as an exploration of the geography of the “mathematical reality” which they unveil gradually in their own private mental frame. (Alain Connes)

Youngsters from seven countries, asked to come up with a portrait of the typical mathematician, showed a badly dressed, middle-aged nerd with no social life. (The Times, 3 January 2001)

...offenbar ist es allgemeine Ansicht, dass ein Mathematiker jemand ist, der 99 Formeln durcheinander mischt und daraus eine 100. Formel gebiert. (Martin Aigner, Mitt.DMV 3, 2003)

Ich kann Eurer Hoheit gar nicht sagen, wie schädlich es für einen Mann von Lebensart hierzulande ist, für einen Philosophen und Mathematiker gehalten zu werden. Die Damen meinen sofort, er müsse in den Mond oder die Venus oder etwas ähnlich Albernes verliebt sein. (Lorenzo Magalotti)

A lot of mathematicians are a little bit strange in one way or another. It goes with creativity. (Peter L. Duren, NYT,26.5.1996)

A mathematician is a blind man in a dark room looking for a black cat which isn't there. (Charles Darwin)

No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. (G.H. Hardy's Apology).

One of the many things that outrage me in G. H. Hardy `apology' is the sexist and agist statement: `Mathematics is a young man's game'. Speak for yourself, G.H.! Just because YOU were finished after forty (or whenever) does not mean that the rest of us are. (Doron Zeilberger)

...mathematicians are often thought of as working in isolation, and that is occasionally the case, as with Andrew Wiles and his solitary struggle to prove Fermat's Last Theorem. But usually mathematics is a highly social activity, with collaboration between two or more individuals the rule rather than the exception. (Robert Osserman)

An accomplished mathematician, i.e. a most wretched orator. (Isaac Barrow)

Geben wir zu, dass das Streben der Mathematiker ein göttlicher Wahnsinn des menschlichen Geistes ist, eine Zuflucht gegenüber der bedrängenden Enge der dem Zufall ausgelieferten Ereignisse.(Alfred North Whitehead)

Mathematicians -- for what they do -- are really poorly rewarded. And it's a very competitive field, almost as bad as being a concert pianist. (Gian-Carlo Rota)

Most mathematicians say they are in it for the math itself, for the delirious quest for patterns, the thrill of the detective chase and the lure of beautiful answers. (New York Times, 29.6.2003).

Mathematicians also make terrible salesmen. Physicists can discover the same thing as a mathematician and say 'We've discovered a great new law of nature. Give us a billion dollars.' And if it doesn't change the world, then they say, 'There's an even deeper thing. Give us another billion dollars.' (Gian-Carlo Rota)

Among mathematicians in general, three main categories may be distinguished;  and perhaps the names logicians, formalists, and intuitionists may serve to characterize them. (Felix Klein)

I have hardly ever known a mathematician who was able to reason. (Stephen Hawking)

Mathematicians [...] seem to have been working upward and outward among the branches instead of  digging down to the rootlets of the infinite tree of mathematical truth  (J.K. Ellwood, American Mathematical Monthly, 1894)

It is impossible to be a mathematician without being a poet in soul. (Sofia Kovalevskaya)

Even the best mathematicians make calculational errors! (Karen Hunger Parshall)

The mathematician has reached the highest rung on the ladder of human thought. (Havelock Ellis)

It may be true that people who are merely mathematicians have certain specific shortcomings; however, that is not the fault of mathematics but is true of every exclusive occupation. (Gauss)

Our faith in Mathematics is not likely to wane if we openly acknowledge that the personalities of even the greatest mathematicians may be as flawed as those of anyone else. (Gian-Carlo Rota, 'Indiscrete thoughts') 

Mathematicians are born, not made. (Henri Poincaré)

Mathematicians are a species of Frenchmen: if you say something to them they translate it into their own language and presto! It is something entirely different. (J.W. Goethe)

If they would, for example, praise the beauty of a woman, or any other animal, they describe it by rhombs, circles, parallelograms, ellipses, and other geometrical terms... (Jonathan Swift, ' Gullivers travels ', Part III, ch. 2 )

We mathematicians are used to the fact that our subject is widely misunderstood, perhaps more than any other subject (except perhaps linguistics).  (Keith Devlin)

You don't tell a football player who just lost the world cup final, "It's only a game" and you don't tell a mathematician hot on the trail of a new result, "It's only mathematics". (Brian Hayes, The MSRI Emissary   Fall 1999)

That the principles laid down by mathematicians are true, and their way of deduction from those principles clear and incontestible, we do not deny; but, we hold there may be certain erroneous maxims of greater extent than the object of Mathematics, and for that reason not expressly mentioned, though tacitly supposed throughout the whole progress of that science; and that the ill effects of those secret unexamined errors are diffused through all the branches thereof. To be plain, we suspect the mathematicians are as well as other men concerned in the errors arising from the doctrine of abstract general ideas, and the existence of objects without the mind. (George Berkeley, ' A Treatise Concerning the Principles of Human Knowledge , 1710)

If you ask somebody who is not a mathematician to give you a geometric fact it is very probable that you will get the answer that the sum of the angles in a triangle is 180 degrees. But if you put the same question to a mathematician it is possible that you will get another question in return: What do you mean by a geometric fact? ... The non-mathematician is ready to give you a mathematical fact, while the mathematician puts the whole issue of mathematical facts in question. (Torkil Heiede)

Many people still tend to fall to their knees and decry their own poor mathematics ability when making the acquaintance of a mathematics teacher, mathematician, engineer, programmer, or dollar-bill exchange machine. ( Stan Hartzler )

Ja, sehen Sie, wir sind nämlich Mathematiker und gelten sogar in unseren Kreisen als verrückt. (Hubert Cremer, quoted by Werner Fenchel)

Ein Onkel von mir behauptet, dass man nur irgend jemand aus dem Tollhause oder Irrenhause zu nehmen brauchte, das würde gewiss der beste Mathematiker sein. (Eisenstein)

Now, many of the famous mathematicians were indeed mentally ill, but there is no reason to necessarily believe that mental illness runs high among mathematicians in general. (Janelle L. Wilson and Carmen M. Latterell )

Alle göttlichen Gesandten müssen Mathematiker sein. (Novalis)

Die Mathematiker sind die einzig Glücklichen. (Novalis)

Nein! Wir Sind Dichter! (Kronecker, quoted by JJ Sylvester, CW IV, 625)

Mathematicians are rarely main characters in literature. But in stories where they do feature as protagonists, they are troubled individuals. (Janelle L. Wilson and Carmen M. Latterell )

Er ist ein Mathematiker und also hartnäckig. (Goethe)

Our dried voices, when
We whisper together
Speak utter nonsense
To sensible people
But to us they betray
An unknown beauty

(From a poem in the Rota Festschrift 1998)

I always like to make explicit the fact that before I went off not too long ago to fight in the trenches, I was a mathematician by profession. I don't like people to get the idea that I have to do this for a living. I mean, it isn't as though I had to do this, you know, I could be making, oh, 3000 dollars a year just teaching.  ( Tom Lehrer during a performance of his songs)

Based on the kinds of images of mathematicians promulgated in our popular culture, we have to wonder how encouraging this is for drawing young people into the field of mathematics (Janelle L. Wilson and Carmen M. Latterell )

The teaching of mathematics:

Some — perhaps many — people are not smart enough, or interested enough, to solve hard problems that demand the use of algebra and geometry. No amount of "educational reform" can change this. (Amy L. Wax, New York Times,  July 8, 2003)

I am opposed to textbooks... I find it hateful to give a course where I have to plough my way through chapter after chapter of a given book. The liveliness of the lecture, which is meant to give an impetus to the sudents, would suffer tremendously. (Emil Artin , Tata institute, 1960)

One of the first things that would have struck you if you had walked into one of Moore 's graduate classes was that there were no textbooks. On each student's desk you would see the student's own notebook and nothing else! ( Keith Devlin )

The most useful pice of advise I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is both simple and non-trivial. (Sir Michael Atiyah, Bull.LMS, vol 10, 1978)

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts. (Paul Halmos)

The completion of a rigorous course in mathematics -- it is not even necessary that the student does well in such a course -- appears to be an excellent means of sharpening the mind and developing mental skills that are of general benefit. (Keith Devlin )

The best way to learn is to do; the worst way to teach is to talk. (P.R. Halmos)

For most problems found in mathematics textbooks, mathematical reasoning is quite useful. But how often do people find textbook problems in real life? At work or in daily life, factors other than strict reasoning are often more important. Sometimes intuition and instinct provide better guides; sometimes computer simulations are more convenient or more reliable; sometimes rules of thumb or back-of-the-envelope estimates are all that is needed. ( Lynn Arthur Steen )

Fortunately we still have our common sense to guide mathematics education. Unfortunately (...) present trends towards discovery-based learning and constructivist pedagogy seem as little rooted in mathematicians' common sense as they are rooted in education research.  (Bastiaan Braams (NYU))

The endless cycle of fashion in mathematics education-first new math, then back to basics, then problem solving, now constructivism - leads many observers to question whether the subject  harbors any enduring principles or convictions ( Lynn Arthur Steen ).

To a mathematician, mathematics is singular-a Platonic paradigm in which there are simple, unquestionable criteria for distinguishing right from wrong and true from false. But to mathematics educators, mathematics is plural. Mathematics, among other things, offers a lens through which one can look at the world. In mathematics education the direction is reversed-one looks at mathematics through the lens of learners (and teachers). ( Lynn Arthur Steen ).

Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice. (George Polya, 'Mathematical discovery')

Let me put this in bad English: This series doesn't NOT converge... it's like saying, `Yes, we have no bananas.' (Michael J. Best in an undergraduate lecture)

Given the brief -- and generally misleading -- exposure most people have to mathematics at school, raising the public awareness of mathematics will always be an uphill battle. ( Keith Devlin )

It is a mistake to suppose that requiring the nonmathematical to take more advanced math courses will enhance their understanding and not merely exacerbate their sense of inadequacy. (William Raspberry)

The student of arithmetic who has mastered the first four rules of his art, and successfully striven with money sums and fractions, finds himself confronted by an unbroken expanse of questions known as problems. (Steven Leacock)

Golden rule for math teachers: You must tell the truth, and nothing but the truth, but not the whole truth. (Anonymous)

But in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer. (Tom Lehrer )

Bitte vergiss alles, was Du auf der Schule gelernt hast, denn Du hast es nicht gelernt. (Emil Landau)

Liegt in dem System, dass die Mathematik der Hochschule und die der Schule keine rechte Beziehung haben, nicht etwas Widersinniges? (Felix Klein, Vorträge über den Mathematischen Unterricht ,  1907)

"Wir wollen und brauchen keine Schule", weil in unseren absolut klaren Wissenschaft jede neue Entdeckung die bisherige Schulweisheit wertlos machen kann.  (L. Kronecker, Jahresbericht der DMV, 1891)

Dass auch schon für den ersten wissenschaftlichen Unterricht in der Mathematik die mögligst strengste Methode vor jeder andern den Vorzug verdiene, werden wohl wenige bestreiten. (Hermann Grassmann, Lehrbuch der Arithmetik, 1861)

Es ist unglaublich, wie unwissend die studirende Jugend auf Universitäten kommt, wenn ich nur 10 Minuten rechne oder geometrisire, so schläft 1/4 derselben sanfft ein. (G. C. Lichtenberg)

In the mathematical discipline of set theory, the letter Ø represents an empty set. Some have suggested that this constitutes Denmark's sole contribution to the science of mathematics! (BBC: Peculiarities of the Danish Alphabet)

I'm not the sort of person who does my mathematics writing on paper. I do that at the last stage of the game. I do my mathematics in my head. I sit down for a hard day's work and I write nothing all day. I just think. And I walk up and down because that helps keep me awake, it keeps the blood circulating, and I think and think. (Sir Michael Atiyah)

Any good theorem should have several proofs, the more the better (Sir Michael Atiyah)

I glory in the diversity of mathematics and the lack of a uniform straightjacket. (Sir Michael Atiyah, Bull. AMS, 2006, p. 87)

Mere proof won't convince me. (Caption to a cartoon by James Thurber)

If the people do not believe that mathematics is simple, it is only because they do not realize how complicated life is (J. von Neumann)

Do mathematics only if you are passionate about it, only if you would do it even if you had to find the time for it after a full day’s work in another job.
Like poetry and music, mathematics is not an occupation but a vocation. (Béla Bollobás)

Young man, in mathematics you don't understand things. You just get used to them (J. van Neumann)

It is all too familiar. You go to a colloquium talk, supposedly meant for a "general mathematical audience" and "graduate students", and you don't understand a word  (...) I recently came back from ICM 2002, in Beijing, and once again was struck by the fact that the Tower of Babel hit us mathematicians particularly hard. In addition to the intrinsic compartimization, over-specialization and splintering of math, most mathematicians (even, or perhaps especially, the most prominent) have no idea how to give a general talk. (Doron Zeilberger)

Doing research in mathematics is frustrating and if being frustrated is something you cannot get used to, then mathematics may not be an ideal occupation for you. (Peter Sarnak)

Pure mathematics offers no mercenary inducements to its followers, who is attracted to it by the importance and beauty of the truths in contains; and the complete absence of any material advantage to be gained by means of it, adds perhaps another charm to its study. (J.W.L. Glaisher)

(It is true that) mathematics "makes sense" in a very profound manner. But we might also note that mathematics is as distantly removed from the senses as a field of scientific knowledge can be. As a physicist, I find that while mathematics is intellectually fascinating, it is not sensuously so. (Caroline Herzenberg, New York Times, July 2, 2003)

Pure mathematics enjoys the luxury of studying its constructions, whether finite or infinite, in complete independence of all questions of efficiency. (J. Schwartz)

Bei manchen Definitionen ..., die man in mathematischen Schriften findet, möchte man an den Rand schreiben: 
"Was man nicht recht beweisen kann,
Das sieht man als Erklärung an."
(G. Frege, Jahresbericht DMV, 1903)

There is no clear-cut distinction between example and theory (Sir Michael Atiyah)

Lest men suspect your tale untrue, Keep probability in view (John Gay)

A witty statesman said, you might prove anything by figures. (Thomas Carlyle, Chartism)

I firmly believe that mathematics does not exist outside of humans. It is something we, as a species, invent. (I don't see what else it could be.) (K. Devlin )

In mathematics you don't understand things. You just get used to them. (Johann von Neumann)

For in all sorts of reasoning every single argument should be managed as a mathematical demonstration (John Locke: Of the Conduct of the Understanding)

When faced with a problem that seems intractable, the best strategy is sometimes to formulate what appears to be an even harder problem. By expanding one's horizons, one may find an unanticipated route that leads to the goal. (Robert Osserman)

...for mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning. (John Locke, Second Reply to the Bishop of Worcester )

Unter allen Fachsprachen ist diejenige der Mathematik am höchsten entwickelt, was nicht bedeutet, dass sie auch die verständlichste ist. (D. Blessenohl )

Outside of the closed circle of professional mathematicians, almost nothing is known of the true nature of mathematics or mathematics research. (Jerry P. King)

Die Mathematik allein kann nicht glücklich machen, und sie ist so schwierig, dass man nur selten die Genugthuung hat, einen Schritt vorwärts zu kommen. (Eisenstein)

One thing that really fascinates me about mathematics is its very real permanence. It is essentially immortal...Once a theorem always a theorem would
summarize this viewpoint adequately. ( A.B. Mingarelli )

The whole history of  the development of mathematics has been a history of the destruction of old definitions. old hobbies, old idols. (David Eugene Smith, American Mathematical Monthly, 1894)

...wer Probalititäten als Wahrheit gibt, der wird, mit oder ohne Willen, ein Irreführer.(Jac. Berzelius)

A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.  (J.E: Littlewood)

Eine falsche Hypothese ist besser als gar keine (Goethe, Analyse und Synthese)

Some people have been found to regard all mathematics, after the 47'th proposition of Euclid, as a sort of morbid secretion, to be compared only with the pearl said to be generated in the deceased oyster (JJ Sylvester, CW II, 658)

Diese Ueberzeugung von der Lösbarkeit eines jeden mathematischen Problems ist uns ein kräftiger Ansporn während der Arbeit; wir hören in uns den steten Zuruf: Da ist das Problem, suche die Lösung. Du kannst sie durch reines Denken finden; denn in der Mathematik giebt es, kein Ignorabimus!  (D. Hilbert: Mathematische Probleme 1900)  (Translation: ... We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.) 

Mathematik ist die perfekte Methode, sich selbst an der Nase herum zu führen. (A. Einstein)

Because of mathematics' precise, formal character, mathematical arguments remain sound even when they are long and complex. In contast, common sense arguments can generally be trusted only if they remain short; even moderately long nonmathematical arguments rapidly becomes farfetched an dubious. (Jacob T. Schwartz, 'Discrete thoughts')

The problem with simple arguments is that they may be difficult to explain. (Karin Erdmann)

In recent years we have become much more preoccupied with streamlining and organizing our subject than with maintaining its overall vitality. If we are not careful, a great adventure of the mind will become yet another profession. (Mark Kac, 'Discrete thoughts')

The pendulum of mathematics swings back and forth towards abstraction and away from it with a timing that remains to be estimated. (Gian-Carlo Rota, 'Indiscrete thoughts')

The collected literature of mathematics is treated as a sacred text to be guarded against corruption and dilution. (Brian Hayes, The MSRI Emissary , Fall 1999)

The important thing in Science is not so much to obtain new facts as to discover new ways of thinking about them. (Sir William Lawrence Bragg, 'Beyond

An abstract definition needs at least two interesting examples to justify its existence. (James A. Green)

I know numbers are beautiful. If they aren't beautiful, nothing is. (Paul Erdös)

Mathematicians often speak of the unity of their subject, whether to praise it or lament its passing. (Jeremy Gray)

The unity of mathematics is often invoked in a rather imprecise way. (Jeremy Gray)

Manche Menschen haben einen Gesichtskreis vom Radius Null und nennen ihn ihren Standpunkt. (David Hilbert)

The method of solving problems by honest confession of one's ignorance is called Algebra ( Mary Everest Boole , Philosophy and fun of algebra, p. 14 )

The Study of Algebra may be pursued in three very different schools, the Practical, the Philological, or the Theoretical, according as Algebra itself is accounted an Instrument, or a Language, or a Contemplation. (W.R.Hamilton, In 'Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time ', 1837).

Truth comes out of error more easily than out of confusion. (Francis Bacon)

How should I answer the question whether Euclidian Geometry is true? It has no sense!.... Euclidian Geometry is, and will remain, the most convenient. (Poincare, according to Henry Adams )

The purpose of computation is insight, not numbers. (Richard Hamming )

You feed yourself. Make sure you have all the information, whether it's aesthetic, scientific, mathematical, I don't care what it is. Then you walk away from it and let it ferment. You ignore it and pretend you don't care. Next thing you know, the answer comes. (Ray Bradbury)

Beware of the man who won't be bothered with details. (William Feather)

I think that mathematics can benefit by acknowledging that the creation of good models is just as important as proving deep theorems. (David Mumford, The Berlin Intelligencer,  ICM 1998)

My own view is that no major advance was ever found as a result of a committee's recommendation... We should be honest in telling these agencies we often don't know where some ideas are going to lead, but we hope they are going to clarify a problem. (David Mumford, The Berlin Intelligencer,  ICM 1998)

Historically speaking, it is of course quite untrue that mathematics is free from contradiction; non-contradiction appears to be a goal to be achieved, not a God-given quality that  has been granted to us all. (Nicholas Bourbaki)

The human brain finds it extremely hard to cope with a new level of abstraction. This is why it was well into the eighteenth century before mathematicians felt comfortable dealing with zero and with negative numbers, and why even today many people cannot accept the square root of minus-one as a genuine number. (Keith Devlin )

It is the man not the method that solves the problem. (H. Maschke)

One of the guiding principles of modern mathematics is that the main difficulty in any subject should be to find the right way to look at it. Once this proper viewpoint is found the theorems become obvious and their proofs routine if not trivial. Make the right definitions and the rest will take care of itself!  (Everett C. Dade)

The art of doing mathematics consists in finding that special case which contains all the germs of generality. (David Hilbert)

Mathematics 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. (Norbert Wiener, Ex-Prodigy: My childhood and youth)

Belief is no substitute for arithmetic. (Henry Spencer)

Logic is a system whereby one may go wrong with confidence. (Charles Kettering)

Die Logik ist die Hygiene, deren sich der Mathematiker bedient, um seine Gedanken gesund und krftig zu erhalten. (H. Weyl)

Wer ein mathematisches Buch nicht mit Andacht ergreift und es wie Gottes Wort liest, der versteht es nicht. (Novalis)

Be wise! Generalize! (Piccayne Sentinel)

To generalize is to be an idiot. (William Blake)

There is no permanent place in the world for ugly mathematics. (G.H. Hardy's Apology)

The axiomatic method has many advantages over honest work. (Bertrand Russell)

I don't have any solution but I certainly admire the problem. (Ashleigh Brilliant)

Logic is like the sword - those who appeal to it shall perish by it. (Samuel Butler)

There are three kinds of lies: Lies, Damn Lies, and Statistics. (Benjamin Disraeli)

The best that any mathematician can really hope for is to prove some first class theorems and have them understood by a few good Mathematicians. Everything else is politics. ( M.P.Schutzenberger, according to G.E.Andrews )

Do not be troubled by your difficulties with Mathematics, I can assure you mine are much greater. (Albert Einstein)

Nature imitates mathematics. (Gian-Carlo Rota, 'Indiscrete thoughts')

Krumme Linie - Sieg der freien Natur über die Regel. (Novalis)

Chaos is lawless behavior governed entirely by law. (Ian Stewart, 'Does God Play Dice?').

Are mathematical ideas invented or discovered? This question has been repeatedly posed by philosophers through the ages and will probably be with us forever. (Gian-Carlo Rota, 'Indiscrete thoughts')

It seems unfair to crow about the successes of a theory and sweep all its failures under a rug. (Richard Brauer)

Science is what we understand well enough to explain to a computer. Art is everything else we do. (Donald E. Knuth, foreword to the book ``A=B'' )

Computers are finite objects (Dave Benson)

Ich gestehe, dass der Fermat'sche Theorem als isolirter Satz für mich wenig Interesse hat, denn es lassen sich eine Menge solche Sätze leicht aufstellen, die man weder beweisen, noch widerlegen kann. (Gauss, 1816)

L'algèbre n'est qu'une géométrie écrite, la géométrie n'est qu'une algèbre figurée. (Sophie Germain)

I am never forget the day I first meet the great Lobachevsky. In one word he told me secret of success in mathematics: Plagiarize! ( Tom Lehrer )

If you know the solution to a problem, then you don't have to solve it. (Heard in a lecture)

Thus the problem is not really clearly formulated. And this is what makes it difficult. (Heard in a lecture)

In a colloquium lecture 1/3 should be understood by everyone, 1/3 by the experts, 1/3 by the lecturer and 1/3 by none. (Heard in a lecture)

No brain cells have been killed in the making of these proofs (T.W. Körner in a lecture)

Maths is the purest science in that you don't need any test tubes or animal testing to do it. All the other sciences eventually boil down to maths, apart from biology, which boils down to soup.  (Guy Browning)

In the real world, when letters meet numbers, you get a bus route. In maths, you get an equation. With equations, mathematicians can work out the distance to the planets, the direction of evolution and the mind of God. They miss their bus, but that's the price you pay for hot maths action. (Guy Browning)

This book fills a well needed gap in the literature (Heard in a lecture)

Watch this proof carefully - it looks like a confidence trick (Heard in a lecture)

If you start off with 72 elements and take away half, you're left with the other half (Heard in a lecture)

Tough guys don't do math. Tough guys fry chicken for a living! (From the movie "Stand and deliver" )

The only thing that makes me truly happy is mathematics. Snow, ice, and numbers. (From the movie "Smilla's Sense of Snow" )

This one's tricky. You have to use imaginary numbers, like eleventeen ...  (Calvin)

Mathematical graffiti (anonymous):

We leave the obvious generalizations to the reader (Israel Herstein)

There was a young fellow named Guesser
Whose knowledge grew lesser and lesser
 He finally one day
 When it vanished away
Became a mathematics professor.

A mathematician I know,
says: It cannot be so,
My wife, he begins,
is about to have twins,
but two into one doesn't go.

There once was a maiden named Emma
Who had a peculiar dilemma.
She had so many beaus
That to choose, heaven knows,
She had to appeal to Zorn's Lemma.

A conjecture both deep and profound
Is whether a circle is round.
 In a paper of Erdös
 Written in Kurdish
A counterexample is found.

A mathematician named Ray
Says extraction of roots is child's play.
 You don't need equations
 Or long calculations;
Just hot water to run on the tray.

A mathematician confided
That the Möbius band is one-sided
 And you'll get quite a laugh
 If you cut one in half
'Cause it stays in one piece when divided.

A Dozen, a Gross and a Score,
plus three times the square root of four,
 divided by seven,
 plus five times eleven,
equals nine squared and not a bit more.

(PROOF: ((12 + 144 + 20 + (3 * 4^(1/2))) / 7) + (5 * 11) = 9^2 + 0

A graduate student at Trinity
Computed the square of infinity.
 But it gave him the fidgets
 To put down the digits,
So he dropped math and took up divinity.

A fair maid from North Minneapolis
was an expert at drawing parabolas.
She extended their arms
and showed off their charms
and they looked positively fabulous.

There was a Young Lady from Bath,
In love she was -and deeply- with Math;
 She married a Fraction,
 But died of Subtraction,
That algebraic Young Lady from Bath.

'Tis a favorite project of mine
A new value of pi to assign.
 I would fix it at 3
 For it's simpler, you see,
Than 3 point 1 4 1 5 9.

A challenge for many long ages
Had baffled the savants and sages.
 Yet at last came the light:
 Seems old Fermat was right--
To the margin add 200 pages.

           Problems  worthy
             of attack
             prove their worth
             by hitting back.

                        Piet Hein

There's a delta for every epsilon,
It's a fact that you can always count upon.
There's a delta for every epsilon
        And now and again,
        There's also an N.

                                  Tom Lehrer (American Mathematical Monthly, 81 (1974) 612)

Take all that you've got
Multiply by zero
Divide by the time you've got left
That's x
Add it up write it down tell me what you've found


Calvin: As far as I'm concerned, if something is so complicated that you can't explain it in 10 seconds, then it's probably not worth knowing anyway.

            I hate quotations. Tell me what you know. --Ralph Waldo Emerson