## Why do math?

- "I am interested in mathematics only as a creative art." - G. H. Hardy
- "If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy." - Alfréd Rényi
- "The only thing I am interested in using mathematics for is
to have a good time and to help others do the same." - Paul Lockhart, in
*A Mathematician's Lament* - "I am not really doing research, just trying to cultivate myself." - Alexander Grothendieck
- "To think deeply of simple things." - Arnold E. Ross
- "We must know, we will know!" - David Hilbert
- "The real satisfaction from mathematics is in learning from others and sharing with others." - Bill Thurston
- "The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present-day kaleidoscope of events." - Morris Kline
- "A typical mathematician does not actively try to be useful. Individual mathematicians are motivated primarily by a subtle mixture of ambition and intellectual curiosity, and not by a wish to benefit society, nevertheless, mathematics as a whole does benefit society." - Timothy Gowers
- "Mathematics is one of the few disciplines that teaches us about the power of thought as distinct from the power of authority." - Klaus Fischer
- "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." - Eric Temple Bell
- "Mathematics seems to endow one with something like a new sense." - Charles Darwin
- "We shall not cease from exploration

And the end of all our exploring

Will be to arrive where we started

And know the place for the first time."

- T. S. Eliot - "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…" - Andre Weil
- "The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." - Henri Poincaré
- "Life is good for only two things, discovering mathematics and teaching mathematics." - Siméon Poisson
- "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." - John von Neumann

## What is math?

- "A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." - G. H. Hardy
- "Doing mathematics should always mean finding patterns and crafting
beautiful and meaningful explanations." - Paul Lockhart, in
*A Mathematician's Lament* - "To do mathematics is to engage in an act of discovery and
conjecture, intuition and inspiration; to be in a state of confusion —
not because it makes no sense to you, but because you gave it sense and
you still don't understand what your creation is up to; to have a
breakthrough idea; to be frustrated as an artist; to be awed and
overwhelmed by an almost painful beauty; to be alive, damn it." -
Paul Lockhart, in
*A Mathematician's Lament* - "Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature." - Hermann Weyl
- "The product of mathematics is clarity and understanding. Not theorems, by themselves. … In short, mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new." - Bill Thurston
- "We often hear that mathematics consists mainly of proving theorems. Is a writer's job mainly that of writing sentences?" - Gian-Carlo Rota
- "Mathematics is no more computation than typing is literature." - John Allen Paulos
- "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." - Henri Poincaré
- "One of the biggest problems of mathematics is to explain to everyone else what it is all about. The technical trappings of the subject, its symbolism and formality, its baffling terminology, its apparent delight in lengthy calculations: these tend to obscure its real nature. A musician would be horrified if his art were to be summed up as 'a lot of tadpoles drawn on a row of lines'; but that's all that the untrained eye can see in a page of sheet music… In the same way, the symbolism of mathematics is merely its coded form, not its substance." - Ian Stewart
- "If one tries to describe it, one thinks of words like curiosity, urge for knowledge, love of games. So is mathematics a game, diffcult and elaborate? In a sense yes, but we know that because of its very serious consequences it is more than a game. In fact the motivation for mathematizing is very close to that of the artist. It is, like in the arts, not easy to decide and explain what results and which aspects are important, of great value, deep. Intensity of ideas, unity, beauty are some of the criteria, also the opening of new horizons. All this, however is accessible only to the circle of mathematicians with the adequate preparation. Thus the audience, the general public, so important for the artist, is missing in our case; our art is an invisible part of the general cultural tradition." - Beno Eckmann

## Math is more than rigor

- "The wonderful thing about formal mathematical proof is that it eliminates the use of intuition in an argument. And the trouble with formal mathematical proof is that it eliminates the use of intuition in an argument." - Eugenia Cheng
- "In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life." - Michael Atiyah
- "Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition." - Morris Kline
- "In mathematics we don't deduce things from axioms. Rather we try
to capture a certain idea by introducing axioms, check which theorems follow
from the axioms and compare these results against the idea we are trying to
capture. If the results agree we are happy. If the results disagree, we
change the
*axioms*. The ideas we try to capture transcend the deductive system. The deductive system is there to help us find consequences from the axioms, but it does not tell us how to gauge the validity of results against the idea we try to capture, nor how to adjust the axioms." - Ittay Weiss - "Everything useful in mathematics has been devised for a purpose. Even if you don't know it, the guy who did it first, he knew what he was doing. Banach didn't just develop Banach spaces for the sake of it. He wanted to put many spaces under one heading. Without knowing the examples, the whole thing is pointless." - Sir Michael Atiyah
- "People think of axioms as laws you have to follow, or true things you have to assume, and I think neither of these perspectives is correct. It's more accurate to think of axioms as a way to agree that we're talking about the same thing." - Qiaochu Yuan
- "The world of ideas is not revealed to us in one stroke; we must both permanently and unceasingly recreate it in our consciousness." - René Thom
- "The moving power of mathematical invention is not reasoning but imagination." - Augustus de Morgan
- "It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better." - Henri Poincaré
- "… if one were to refuse to have direct, geometric, intuitive insights, if one were reduced to pure logic, which does not permit a choice among every thing that is exact, one would hardly think of many questions, and certain notions … would escape us completely." - Henri Lebesgue
- "Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs." - Felix Klein
- "No, you're not thinking, you're just being logical." - Niels Bohr
- "Mathematics is not a deductive science - that's a cliché. 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." - Paul Halmos
- "If one must choose between rigour and meaning, I shall unhesitatingly choose the latter." - René Thom
- "Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere." - W.S. Anglin
- "Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It's just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas." - Bill Thurston
- "How often people speak of art and science as though they were two entirely different things, with no interconnection. An artist is emotional, they think, and uses only his intuition; he sees all at once and has no need of reason. A scientist is cold, they think, and uses only his reason; he argues carefully step by step, and needs no imagination. That is all wrong. The true artist is quite rational as well as imaginative and knows what he is doing; if he does not, his art suffers. The true scientist is quite imaginative as well as rational, and sometimes leaps to solutions where reason can follow only slowly; if he does not, his science suffers." - Isaac Asimov
- "Logic moves in one direction, the direction of clarity, coherence and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles. […] It is the interaction between these different aspects that gives mathematics its power." - William Byers

## Learning and teaching

- "Each day learn something new, and just as important, relearn something old." - Robert Brault
- "Keep in mind that there are millions of theorems but only thousands of proofs, hundreds of proof blocks, and dozens of ideas. Unfortunately, no one has figured out how to transfer the ideas directly yet, so you have to extract them from complicated arguments by yourself." - Fedja Nazarov
- "What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises." - G. C. Lichtenberg
- "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?" - Paul Halmos
- "The only way to learn mathematics is to do mathematics." - Paul Halmos
- "A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos
- "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
- "Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. … A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution." - George Polya
- "If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it." - George Polya
- "One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That's so unlike the true nature of mathematics." - Leon Henkin
- "We encourage children to read for enjoyment, yet we never encourage them to 'math' for enjoyment. We teach kids that math is done fast, done only one way and if you don't get the answer right, there's something wrong with you. You would never teach reading this way." - Rachel McAnallen
- "Mathematics is about problems, and problems must be made
the focus of a student's mathematical life. Painful and creatively
frustrating as it may be, students and their teachers should at all
times be engaged in the process - having ideas, not having ideas,
discovering patterns, making conjectures, constructing examples and
counterexamples, devising arguments, and critiquing each other's work."
- Paul Lockhart, in
*A Mathematician's Lament* - "What I cannot create, I do not understand." - Richard Feynman
- "He who does not know the lemma does not know the theorem." - Old Romanian saying
- "An expert is a man who has made all the mistakes that can be made, in a very narrow field." - Niels Bohr
- "If you want to build a ship, don't drum up people together to collect wood and don't assign them tasks and work, but rather teach them to long for the endless immensity of the sea." - Antoine de Saint-Exupery
- "The mind is not a vessel to be filled but a fire to be kindled." - Plutarch
- "The years of searching in the dark for a truth that one feels but cannot express, the intense desire and the alternations of confidence and misgiving until one breaks through to clarity and understanding, are known only to him who has experienced them himself." - Albert Einstein
- "An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." - Howard Gardner
- "While one person hesitates because he feels inferior, another is busy making mistakes and becoming superior." - Henry C. Link
- "No one who cannot rejoice in the discovery of his own mistakes deserves to be called a scholar." - Donald Foster
- "It is impossible for a man to learn what he thinks he already knows." - Epictetus
- "If you want to put the holes in your knowledge on display, try teaching someone." - Alan Cox
- "Spoon feeding, in the long run teaches us nothing but the shape of the spoon." - E. M. Forster
- "It isn't enough just to learn - one must learn how to learn, how to learn without classrooms, without teachers, without textbooks. Learn, in short, how to think and analyze and decide and discover and create." - Michael Bassis
- "Dealing with failure is easy: Work hard to improve. Success is also easy to handle: You've solved the wrong problem. Work hard to improve." - Alan Perlis
- "The student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight." - Gila Hanna
- "My advisor in college was interviewed for a scholarship when he was a student. The mathematicians asked, 'How quickly do you read?' 'Sometimes it takes me weeks to read one line.' He got the scholarship." - Douglas Zare
- "Do not ask whether a statement is true until you know what it means." - Errett Bishop
- "Each problem that I solved became a rule which served afterwards to solve other problems." - Rene Descartes
- "The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver." - I. N. Herstein
- "Mental acuity of any kind comes from solving problems
yourself, not from being told how to solve them." -
Paul Lockhart, in
*A Mathematician's Lament* - "As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of the structure, not just familiarity with a particular tour through the structure." - Bill Thurston
- "… it can often be profitable to try a technique on a problem even if you know in advance that it cannot possibly solve the problem completely." - Terence Tao
- "If you can't solve a problem, then there is an easier problem you can solve: find it." - George Polya
- "[Mathematical maturity is] fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse - but crisp and exact - language that mathematicians use to communicate ideas." - Larry Denenberg
- "To state a theorem and then to show examples of it is literally to teach backwards." - E. Kim Nebeuts
- "Suppose that you want to teach the 'cat' concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I'll bet not. You probably show the kid a lot of different cats, saying 'kitty' each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience." - R. P. Boas
- "It is true that you can differentiate and integrate many functions without knowing what a limit is. It is not true that differentiating and integrating many functions is the same as calculus." - Qiaochu Yuan
- "The value of an education… is not the learning of many facts but the training of the mind to think something that cannot be learned from textbooks." - Albert Einstein
- "Any fool can know. The point is to understand." - Albert Einstein
- "Just as any sensitive human being can be brought to appreciate beauty in art, music or literature, so that person can be educated to recognize the beauty in a piece of mathematics. The rarity of that recognition is not due to the 'fact' that most people are not mathematically gifted but to the crassly utilitarian manner of teaching mathematics and of deciding syllabi and curricula, in which tedious, routine calculations, learned as a skill, are emphasized at the expense of genuinely mathematical ideas, and in which students spend almost all their time answering someone else's questions rather than asking their own." - Peter Hilton
- "The title which I most covet is that of teacher. The writing of a research paper and the teaching of freshman calculus, and everything in between, falls under this rubric. Happy is the person who comes to understand something and then gets to explain it." - Marshall Cohen
- "In a completely rational society, the best of us would be teachers and the rest of us would have to settle for something less, because passing civilization along from one generation to the next ought to be the highest honor and the highest responsibility anyone could have." - Lee Iacocca

## Math itself

- "The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas." - David Hilbert
- "As long as algebra and geometry were separated, their progress was slow and their use limited; but once these sciences were united, they lent each other mutual support and advanced rapidly together towards perfection." - Joseph-Louis Lagrange
- "If geometry lets us see what we are thinking about, algebra enables us to talk precisely about what we see, and above all to calculate. Moreover it tends to organize our calculations and to conceptualize them; this in turn can lead to further geometrical construction and algebraic calculation." - M.W. Hirsch
- "Before functoriality, people lived in caves." - Brian Conrad
- "Structures are the weapons of the mathematician." - Bourbaki
- "… commutative algebra is a lot like topology, only backwards." - John Baez
- "It's better to work with a nice category containing some nasty objects, than a nasty category containing only nice objects." - John Baez
- "Geometry is the art of correct reasoning from incorrectly drawn figures." - Henri Poincaré
- "The full beauty of the subject of generating functions emerges only from tuning in on both channels: the discrete and the continuous." - Herbert Wilf
- "It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out." - Emil Artin
- "The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps…" - Alexander Grothendieck
- "Mathematical objects are determined by - and understood by - the network of relationships they enjoy with all the other objects of their species." - Barry Mazur
- "Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial." - Peter Freyd
- "So a bundle of sets over $I$ is "essentially just" a function with codomain $I$. The two are not of course identical conceptually. To construe a function as a bundle is to offer a new, and provocative, perspective." - Robert Goldblatt
- "The isolation and explication of the notion of
*adjointness*is perhaps the most profound contribution that category theory has made to the history of general mathematical ideas." - Robert Goldblatt - "Dimension 4 is the most difficult dimension. It is too old to spank, the way we might deal with the little dimensions 1, 2, and 3; but it is also too young to reason with, the way we deal with the grown-up dimensions 5 and higher." - R H Bing
- "… I see the reification of the function as one of the most important intellectual advances of the past half-millenium, with important implications not only in mathematics but also in computer science, philosophy, and to a lesser extent the humanities in general." - Alexander Woo
- "I just love sheaves. They have algebra this way (and he sliced his hand up
and down) and topology this way (and he sliced his hand left to right)."
- D. C. Spencer, as related in Krantz's
*Mathematical Apocrypha* - "This remarkable conjecture relates the behaviour of a function $L$, at a point where it is not at present known to be defined, to the order of a group Ш, which is not known to be finite." - John Tate, on the Birch-Swinnerton-Dyer Conjecture
- "Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate." - David Mumford
- "All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove." - G.H. Hardy
- "Manifolds are a bit like pornography: hard to define, but you know one when you see one." - Shmuel Weinberger
- "The real irony is that the view of infinity as some forbidden zone or road to insanity - which view was very old and powerful and haunted math for 2000+ years - is precisely what Cantor's own work overturned. Saying that infinity drove Cantor mad is sort of like mourning St. George's loss to the dragon: it's not only wrong but insulting." - David Foster Wallace
- "The shortest path between two truths in the real domain passes through the complex domain." - Jacques Hadamard
- "Topology is precisely the mathematical discipline that allows the passage from local to global…" - René Thom
- "… when we are set to work and we take $\mathsf{ZF}$ to be our foundational theory, then we fix one universe of set theory that we work in. … When you are done working with this universe you throw it in the bin, and get another when you need to. Or you can save that universe in a scrapbook if you like." - Asaf Karagila
- "Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven't. You get the feeling that the result you have discovered is forever, because it's concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don't get a feeling of having done an honest day's work. Don't get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques." - Gian-Carlo Rota
- "The reader will probably observe the conspicuous absence of a time-honored topic in calculus
courses, the 'Riemann integral'. It may well be suspected that, had it not been for its prestigious
name, this would have been dropped long ago, for (with due respect to Riemann's genius) it is
certainly quite clear for any working mathematician that nowadays such a 'theory' has at best the
importance of a mildly interesting exercise […]. Only the stubborn conservatism of academic
tradition could freeze it into a regular part of the curriculum, long after it had outlived its
historical importance." Dieudonné's
*Foundations of Modern Analysis, Vol. 1* - "For general continuous curves, it's not that a simple proof [of the Jordan curve theorem] is not possible, it's that it's not desirable. The true content of the result is homology theory, which proves the separation result in n dimensions. There are special proofs in 2D that are simpler, but every such proof that I have seen feels like a one-night stand." - Greg Kuperberg
- "In the study of commutative Noetherian rings, localization at a prime followed by completion at the resulting maximal ideal is a way of life." - Mel Hochster
- "… the notions
*category*and*functor*were not formulated or put in print until the idea of a*natural transformation*was also at hand." - Saunders Mac Lane - "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." - E. J. McShane

## Observations about math

- "I believe that in mathematics nothing is a trick if seen from a sufficiently high level." - Qiaochu Yuan
- "A great deal more is known than has been proved." - Richard Feynman
- "The art of doing mathematics is finding that special case that contains all the germs of generality." - David Hilbert
- "The purpose of computation is insight, not numbers." - Richard Hamming
- "Groups, as men, will be known by their actions." - Guillermo Moreno
- "The introduction of numbers as coordinates is an act of violence." - Hermann Weyl
- "When asked what it was like to set about proving something, the mathematician likened proving a theorem to seeing the peak of a mountain and trying to climb to the top. One establishes a base camp and begins scaling the mountain's sheer face, encountering obstacles at every turn, often retracing one's steps and struggling every foot of the journey. Finally when the top is reached, one stands examining the peak, taking in the view of the surrounding countryside and then noting the automobile road up the other side!" - Robert J. Kleinhenz
- "Sometimes it's necessary to go a long distance out of the way in order to come back a short distance correctly." - Edward Albee
- "… most problems are not solved by having mastery of a big machine that is then applied to the problem at hand. Rather, they typically reduce to concrete questions in linear algebra, calculus, or combinatorics. One part of the difficulty in solving a problem is finding this kind of reduction (this is where machines can sometimes be useful) …" - Matt Emerton
- "The higher arithmetic presents us with an inexhaustible store of interesting truths - of truths, too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties." - Carl Friedrich Gauss
- "In mathematics you've always got the Scylla of showing that the objects of interest exist at all on the one hand, and the Charybdis of showing that said objects also aren't trivial or exhaustive." – Steve Huntsman
- "We humans have a wide range of abilities that help us perceive and analyze mathematical content. We perceive abstract notions not just through seeing but also by hearing, by feeling, by our sense of body motion and position. Our geometric and spatial skills are highly trainable, just as in other high-performance activities. In mathematics we can use the modules of our minds in flexible ways - even metaphorically. A whole-mind approach to mathematical thinking is vastly more effective than the common approach that manipulates only symbols." - Bill Thurston
- "All problems in mathematics are psychological." - Pierre Deligne
- "Old theorems never die; they turn into definitions." - Edwin Hewitt
- "One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories." - P.J. Davis
- "The price of metaphor is eternal vigilance." - Norbert Wiener
- "When things get too complicated, it sometimes makes sense to stop and wonder: 'Have I asked the right question?'" - Enrico Bombieri
- "To ask the right question is harder than to answer it." - Georg Cantor
- "It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial." - Henry Whitehead
- "Fundamental progress has to do with the reinterpretation of basic ideas." - Alfred North Whitehead
- "If things are nice there is probably a good reason why they are nice; and if you do not know at least one reason for this good fortune, then you still have work to do." - Richard Askey
- "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.)" - J. E. Littlewood
- "When you prove something by contradiction, all you learn is that the statement you wanted to prove is true. When you prove something directly, you learn every intermediate implication you had to prove along the way." - Qiaochu Yuan
- "A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise." - George Polya
- "A poet might compare the evening to an etherized patient and leave it there, and in poetry this is fine, for in poetry we revel in mystery, allusion, in half-knowledge. But in math, we can't stand these things, and so we must grab our things and run to the nearest hospital, examining all the gurneys we can in the hope of better understanding the twilight." - Paul VanKoughnett
- "There are many questions which fools can ask that wise men cannot answer." - George Polya
- "If you cannot - in the long run - tell everyone what you have been doing, your doing has been worthless." - Erwin Schrödinger
- "The question you raise, 'How can such a formulation lead to computations?' doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered." - Alexander Grothendieck
- "As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure… The day dawns when the illusion vanishes; intuition turns to certitude; the twin theories reveal their common source before disappearing; as the Gita teaches us, knowledge and indifference are attained at the same moment. Metaphysics has become mathematics, ready to form the material for a treatise whose icy beauty no longer has the power to move us." - Andre Weil
- "Nothing is more important than to see the sources of invention which, in my opinion, are more interesting than the inventions themselves." - Gottfried von Leibniz
- "In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure." - Herman Henkel
- "The outcome of any serious research can only be to make two questions grow where only one grew before." - Thorstein Veblen
- "There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else - but persistent." - Raoul Bott
- "Computers are incredibly fast, accurate, and stupid. Human beings are incredibly slow, inaccurate, and brilliant. Together they are powerful beyond imagination." - Leo Cherne
- "Anyone, anywhere along the line, can fill in the details and check them. The fact that a computer can run through more details in a few hours than a human could ever hope to do in a lifetime does not change the basic concept of mathematical proof. What has changed is not the theory but the practice of mathematics." - Wofgang Haken
- "I've tried to learn the hidden beauty in various things, but still for many areas the source of interest is for me a complete mystery. My theory is that too often people project their human weaknesses/properties onto their mathematical activity. There are obvious examples on the surface: for example, the idea of a classification of some objects is an incarnation of collector instincts, the search for maximal values is another form of greed, computability/decidability comes from the desire of a total control. Fascination with iterations is similar to the hypnotism of rhythmic music." - Maxim Kontsevich
- "It is hard to communicate understanding because that is something you get by living with a problem for a long time. You study it, perhaps for years, you get the feel of it and it is in your bones. You can't convey that to anyone else. Having studied the problem for five years you may be able to present it in such a way that it would take somebody else less time to get to that point than it took you. But if they haven't struggled with the problem and seen all the pitfalls, then they haven't really understood it." - Michael Atiyah
- "This common and unfortunate fact of the lack of adequate presentation of basic ideas and motivations of almost any mathematical theory is probably due to the binary nature of mathematical perception. Either you have no inkling of an idea, or, once you have understood it, the very idea appears so embarrassingly obvious that you feel reluctant to say it aloud…" - Mikhail Gromov
- "Now what makes a mathematical statement a true Lemma? First, it should be
applicable to a wide variety of instances, even seemingly unrelated problems.
Secondly, the statement should, once you have seen it, be completely obvious.
The reaction of the reader might well be one of faint envy: Why haven't I noticed
this before? And thirdly, on an esthetic level, the Lemma - including its proof -
should be beautiful!" - Martin Aigner and Günter Ziegler, in
*Proofs from THE BOOK* - "Very few people realize the enormous bulk of contemporary mathematics. Probably it would be easier to learn all the languanges of the world than to master all mathematics at present known." - W. W. Sawyer

## Math and reality

- "'Imaginary' universes are so much more beautiful than this stupidly constructed 'real' one." - G. H. Hardy
- "Numbers are the free creation of the human mind." - Richard Dedekind
- "The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter." - Carl G. Hempel
- "The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." - John William Navin Sullivan
- "[Math is] not at all like science. There's no experiment I can do with test tubes and
equipment and whatnot that will tell me the truth about a figment of my imagination. The
only way to get at the truth about our imaginations is to use our imaginations…"
- Paul Lockhart, in
*A Mathematician's Lament* - "What
*is*the fundamental hypothesis of science, the fundamental philosophy? We stated it in the first chapter:*the sole test of the validity of any idea is experiment*. … If we are told that the same experiment will always produce the same result, that is all very well, but if when we try it, it does*not*, then it does*not*. We just have to take what we see, and then formulate all the rest of our ideas in terms of our actual experience." - Richard Feynman - "Numbers exist only in our minds. There is no physical entity that is
the number 1. If there were, 1 would be in a place of honor in some great
museum of science, and past it would file a steady stream of mathematicians
gazing at 1 in wonder and awe." - from
*Linear Algebra*by Fraleigh and Beauregard - "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality." - Albert Einstein
- "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." - Georg Christoph Lichtenberg
- "[Mathematics] is an independent world

Created out of pure intelligence."

- William Wordsworth - "There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain." - Alfred North Whitehead
- "Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." - Nikola Tesla
- "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." - Shlomo Sternberg
- "The electron is a theory we use; it is so useful in understanding the way nature works that we can almost call it real." - Richard Feynman
- "Mathematics is a game played according to certain simple rules with meaningless marks on paper." - David Hilbert
- "The map is not the territory." - Alfred Korzybski
- "All models are wrong, but some are useful." - George Box
- "There is something in statistics that makes it very similar to astrology." - Gian-Carlo Rota
- "The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with reasonable success." - Paul Dirac
- "Although this may seem a paradox, all exact science is based on the idea of approximation. If a man tells you he knows a thing exactly, then you can be safe in inferring that you are speaking to an inexact man." - Bertrand Russell
- "It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts." - Sir Arthur Conan Doyle

## Math humor

- "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" - Jerry Bona
- "God exists since mathematics is consistent, and the Devil exists since we cannot prove it." - Andre Weil
- "Free jazz: the jazz from which all other jazz can be derived by adding relations." - Watson Ladd
- "Let my name be Ishmael, let the captain's name be Ahab, let the boat's name be Pequod, and let the whale's name be as in the title." - Barry Cipra
- "A projective module is the splittin' image of a free module." - Allen Hatcher
- "You can't add apples and oranges."

"False. You can in the free abelian group generated by an apple and an orange." - Tony Varilly - "If the value $f(a)$ agrees with our expectations, then we would like to say that $f$ is continuous at $a$. But perhaps someone from the planet Zog would always guess that the graph of $f$ had a jump at $a$." - Martin Crossley
- "The expression "$+ \cdots$" is not uncommonly used in mathematical writings to mean something which the writer proposes to ignore, in the hope that it does not really matter very much." - E.C. Titchmarsh
- "MacPherson told me that my theorem can be viewed as blah blah blah Grothendieck blah blah blah, which makes it much more respectable." - Jim Propp
- "You either believe in the law of the excluded middle, or you don't." - Lew Lefton
- "A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that '$E$ is dense in $E$' does not mean the same thing as '$E$ is dense in itself'." - J. E. Littlewood
- "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." - J. E. Littlewood

## From my professors

- "I know some of you thought the last problem set was a little difficult. To those students, all I can say is… wait until you see this week's problem set!" - Professor Benoist, Fall 2008
- "I don't know if the police of naming statements would agree with this." - Professor Abramovich, Fall 2008
- "I'll just say this - you won't know what it means, but I'll say it." - Professor Lichtenbaum, Spring 2009
- "The polynomial $x^n-1$ is a force of nature." - Professor Abramovich, Fall 2009
- "If the radius is less than 1, the manifold gets off on a technicality." - Professor Goodwillie, Spring 2010
- "You could think of it as like… someone gave you a vector bundle for your birthday, and it was all in these little pieces and you wanted to put it back together again." - Professor Goodwillie, Spring 2010
- "Either $J$ is maximal, or it isn't. You see, we're using logic." - Professor Rosen, Spring 2010
- "How are the germs made into a ring? By adding and multiplying." - Professor Abramovich, Fall 2010
- "We'll only use as much category theory as is necessary. Famous last words…" - Professor Abramovich, Fall 2010
- "Not everything you can write down is true." - Professor Lichtenbaum, Spring 2011
- "Am I confusing people or boring people at this point?" - Professor Kapouleas, Spring 2011
- "Have I made any mistakes yet? No? That's surprising…" - Professor Kapouleas, Spring 2011
- "I have so many folders on my computer named 'polynomials mod $p$'!" - Professor Rosen, Fall 2011
- "Are you friends with this definition?" - Professor Kahn, Fall 2011
- "You're in Massachusetts and I'm in Rhode Island, and I want to send you a curvature tensor." - Professor Kahn, Fall 2011
- "The summation sign is left out. I'm one of the cool kids." - Professor Kahn, Fall 2011
- "Suppose you have something like a knot. Something so much like a knot, it is a knot." - Professor Brock, Fall 2011
- "I have an assignment for all of you: figure out if one of these windows opens." - Professor Brock, Fall 2011
- "There's a question as to whether these spaces appear in real life, where 'real life' is what you do your Ph.D. thesis on." - Professor Brock, Fall 2011
- "We'll do the finite-dimensional case, then mumble about the infinite-dimensional case at the end." - Professor Brock, Fall 2011
- "This is exact on the left, and also exact on… the other left." - Professor Kottke, Spring 2012
- "You don't have to use this sign convention, but if you don't you'll be unconventional and no one will want to talk to you." - Professor Kottke, Spring 2012
- "An abelian group is a fancy-pants way of saying a number." - Professor Farb, Fall 2012
- "I always ask myself, with every theorem I prove, would Poincaré like this?" - Professor Farb, Fall 2012
- "Lefschetz proved his theorems with no hands." - Professor Farb, Fall 2012
- "We denote this homomorphism by… well, we don't denote it at all actually." - Professor Ginzburg, Fall 2012
- "If I could be good at basketball just by watching TV, I wouldn't be here, I would be in the NBA. I would be Michael Jordan." - Professor Souganidis, Winter 2013
- "The only difference is that it's not the same." - Professor Souganidis, Winter 2013
- "Given the choice between me going in circles and you going in circles, I'll let you guys do it." - Professor Nori, Winter 2013
- "It's only when a manifold is compact that life is beautiful." - Professor Nori, Winter 2013
- "'Dedekind domain' is a good notion; a PID is just a simple-minded Dedekind domain. Alternatively, you may think 'PID' is a good notion, and that a Dedekind domain is a sick PID." - Professor Kato, Winter 2013
- "Quadratic reciprocity is the song of love in the land of prime numbers." - Professor Kato, Winter 2013
- "The prime ideal is a princess of the world of ideals. Her father is the prince 'Point' in the world of geometry. Her mother is the princess 'Prime Numbers' in the world of numbers. She inherits the purity from her parents." - Professor Kato, Winter 2013
- "I'm a young guy called 'commutative ring', but I was originally 'the ring of continuous functions on a compact Hausdorff space'. Now I am an algebraic object, so I must say goodbye to my home village, the space, but I will always keep it in my heart as a set of maximal ideals." - Professor Kato, Winter 2013
- [after proving the equivalence of different definitions of Noetherian ring] "This proof is a little dangerous. It may take 30 years. We may have to tell our children that, when we were young, we were in a course called 'Algebra 2' where we tried to find a maximal ideal in $\Phi$, but that we still are not done. This is not such a good thing for the family. We need to use some sort of axiom of choice." - Professor Kato, Winter 2013
- "It never happens that, when we go home and open the refrigerator, we see all infinitely many prime numbers there." - Professor Kato, Winter 2013
- "If we drop money, we are usually very sad if the money is big. But for example, if we drop $3^{10}$ dollars, we can relax, because this is very small in the 3-adics." - Professor Kato, Winter 2013
- "$\mathbb{R}$ is like the sun, and the $p$-adics are like the stars. The sun blocks out the stars during the day, and humans are asleep at night and don't see the stars, even though they are just as important." - Professor Kato, Winter 2013
- "In mathematics, a cigar is never just a cigar." - Professor Farb, Spring 2013
- "When I write a constant $C$, I really mean the supremum over all the constants I've used so far in the proof." - Professor Silvestre, Winter 2014

## Non-math-related quotations

- "Do not seek to follow in the footsteps of the wise. Seek what they sought." - Matsuo Basho
- "Cultivate solitude and quiet and a few sincere friends, rather than mob merriment, noise and thousands of nodding acquaintances." - William Powell
- "Sometimes I think and other times I am." - Paul Valéry
- "Man is most nearly himself when he achieves the seriousness of a child at play." - Heraclitus
- "Did you ever stop to think, and forget to start again?" - Winnie the Pooh
- "You have brains in your head. You have feet in your shoes. You can steer yourself in any direction you choose. You're on your own. And you know what you know. You are the guy who'll decide where to go." - Dr. Seuss
- "We are social creatures to the inmost centre of our being. The notion that one can begin anything at all from scratch, free from the past, or unindebted to others, could not conceivably be more wrong." - Karl Popper
- Alice: "I don't believe there's an atom of meaning in it."

The King: "If there's no meaning in it, that saves a world of trouble, you know, as we needn't try to find any." - "Why do I act as I do? To tell you the truth, I have absolutely no idea why. It is simply my nature to act as I act, and that's all I can say." - Raymond Smullyan
- "People want economy and they will pay any price to get it." - Lee Iacocca
- "It is the certainty that they possess the truth that makes men cruel." - Anatole France
- "Of all manifestations of power, restraint impresses men most." - Thucydides
- "Sticks and stones may break my bones, but words can make me think I deserved it." - Randall Munroe
- "It is not only the prisoners who grow coarse and hardened from corporal punishment, but those as well who perpetrate the act or are present to witness it." - Anton Chekhov
- "A person usually has two reasons for doing something: a good reason and the real reason." - Thomas Carlyle
- "Several excuses are always less convincing than one." - Aldous Huxley
- "Of those who say nothing, few are silent." - Thomas Neill
- "Never confuse motion with action." - Benjamin Franklin
- "What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet." - Woody Allen
- "Write a wise saying and your name will live forever." - Unknown
- "A pedestal is as much a prison as any small space." - Gloria Steinem
- "Perfection is attained, not when there is nothing left to add, but when there is nothing left to take away." - Antoine de Saint-Exupery
- "A little nonsense now and then, is relished by the wisest of men." - Roald Dahl
- "The real danger is not that computers will begin to think like men, but that men will begin to think like computers." - Sydney J. Harris
- "Drawing on my fine command of language, I said nothing." - Robert Benchley
- "A harmless hilarity and a buoyant cheerfulness are not infrequent concomitants of genius; and we are never more deceived than when we mistake gravity for greatness, solemnity for science, and pomposity for erudition." - Charles Caleb Colton
- "People… being persuaded to spend money we don't have, on things we don't need, to create impressions that won't last, on people we don't care about." - Tim Jackson
- "You probably wouldn't worry about what people think of you if you could know how seldom they do." - Olin Miller
- "A hungry man is not a free man." - Adlai Stevenson
- "In its majestic equality, the law forbids rich and poor alike to sleep under bridges, beg in the streets and steal loaves of bread." - Anatole France
- "Every gun that is made, every warship launched, every rocket fired signifies, in the final sense, a theft from those who hunger and are not fed, those who are cold and are not clothed." - Dwight D. Eisenhower
- "The less justified a man is in claiming excellence for his own self, the more ready he is to claim all excellence for his nation, his religion, his race, or his holy cause. A man is likely to mind his own business when it is worth minding. When it is not, he takes his mind off his own meaningless affairs by minding other people's business." - Eric Hoffer
- "The skylines lit up at dead of night, the air-conditioning systems cooling empty hotels in the desert and artificial light in the middle of the day all have something both demented and admirable about them. The mindless luxury of a rich civilization, and yet of a civilization perhaps as scared to see the lights go out as was the hunter in his primitive night." - Jean Baudrillard
- "The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk: the notion that the future is more than a whim of the gods and that men and women are not passive before nature." - Peter L. Bernstein
- "There is no absurdity so palpable but that it may be firmly planted in the human head if you only begin to inculcate it before the age of five, by constantly repeating it with an air of great solemnity." - Arthur Schopenhauer
- "In certain kinds of writing, particularly in art criticism and literary criticism, it is normal to come across long passages which are almost completely lacking in meaning." - George Orwell
- "Philosophy: A route of many roads leading from nowhere to nothing." - Ambrose Bierce
- "Absurdity, n. A statement or belief manifestly inconsistent with one's own opinion." - Ambrose Bierce
- "The riddle does not exist. If a question can be put at all, then it can also be answered." - Ludwig Wittgenstein
- "Upon the whole, I am inclined to think that the far greater part, if not all, of those difficulties which have hitherto amused philosophers, and blocked up the way to knowledge, are entirely owing to our selves. That we have first raised a dust, and then complain, we cannot see." - George Berkeley
- "We cannot define anything precisely. If we attempt to, we get into that paralysis of thought that comes to philosophers, who sit opposite each other, one saying to the other, 'You don't know what you are talking about!'. The second one says, 'What do you mean by know? What do you mean by talking? What do you mean by you?'" - Richard Feynman
- "Consider the concepts referred to in the words 'where', 'when', 'why', 'being', to the elucidation of which innumerable volumes of philosophy have been devoted. We fare no better in our speculations than a fish which should strive to become clear as to what is water." - Albert Einstein
- "Our greatest pretenses are built up not to hide the evil and the ugly in us, but our emptiness. The hardest thing to hide is something that is not there." - Eric Hoffer
- "All human situations have their inconveniences. We feel those of the present but neither see nor feel those of the future; and hence we often make troublesome changes without amendment, and frequently for the worse." - Benjamin Franklin
- "The difference between something that can go wrong and something that can't possibly go wrong is that when something that can't possibly go wrong goes wrong it usually turns out to be impossible to get at or repair." - Douglas Adams
- "You can know the name of that bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bird. You'll only know about humans in different places, and what they call the bird. … I learned very early the difference between knowing the name of something and knowing something." - Richard Feynman
- "I was not; I have been; I am not; I do not mind." - Epicurus