It is said of Jordan's writings that if he had 4 things on the same footing (as a,b,c,d ) they would appear as a, M3', ε2, ∏''1,2.
---Littlewood's Miscellany, p60.
It is well known that to describe a result as well-known without giving either the proof or a reference is neither pleasing nor helpful to the reader.
---J.W.P. Hirschfeld, Bull. Amer. Math. Soc. 27 (1992), p331.
Two women were in a SoHo boutique when they were approached by an eager salesman.
"Ladies," he said. "If there's anything you need, I'm Nick."
"And if we don't need anything, who are you then?" one of the "ladies" asked.
---New York Times (Metropolitan Diary), July 19, 1998.
"It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out."
E. Artin (Geometric Algebra, p. 14)
"Z is complicated."
Eric Babson (MSRI, 12/18/08)
"To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples..."
John B. Conway (Subnormal Operators, Pitman Advanced Publishing Program, 1981)
"It's bad to have too good a memory if you want to be a mathematician." - Andrew Wiles
"Groups, as men, will be known by their actions".
-Guillermo Moreno.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." --- John von Neumann. (From a 1947 ACM keynote, recalled by Alt in this 1972 CACM article.)
"Mathematics is the art of giving the same name to different things." Henri Poincaré.
"If others had reflected upon mathematical truths as deeply and continuously as I have, they would have made my discoveries." -Gauss
"Young man, in mathematics you don't understand things. You just get used to them." --John von Neumann [source]
"I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives." - Charles Hermite
“The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial.” Lucien Szpiro during Algebra 1 lecture.
Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he's scribbled on, scrunches them all up, and throws them in the trash can. --J. von Neumann's housekeeper, describing her employer.
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona
"The art of doing mathematics is finding that special case that contains all the germs of generality." -- David Hilbert
Algebra is the offer made by the devil to the mathematician...All you need to do, is give me your soul: give up geometry --Michael Atiyah
"I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance."
There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.
---Grotehndieck
"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."
---Pierre de Fermat
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions", F. Klein
The introduction of numbers as coordinates is an act of violence.
---Hermann Weyl, Philosophy of Mathematics and Natural Science.
The purpose of computing is insight, not numbers.
— Richard Hamming (1962)
"Manifolds are a bit like pornography: hard to define, but you know one when you see one." -S. Weinberger
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.
Only Dirichlet, Not I, not Cauchy, not Gauss, knows what a perfectly rigourous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain; I prefer not to go into these delicate matters.
C. G. J. Jacobi, writing to von Humboldt, in 1846.
“Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions.”
– Th. Bröcker & K. Jänich, Introduction to differential topology (p.25)
"You don't have to believe in God, but you should believe in The Book." --- Paul Erdős. describing the Book held by the God that contains the most beautiful proofs to all the theorems
"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
"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
"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
"Each day learn something new, and just as important, relearn something old." - Robert Brault
"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
"Each problem that I solved became a rule which served afterwards to solve other problems." - Rene Descartes
"The student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight." - Gila Hanna
"Geometry is the art of correct reasoning from incorrectly drawn figures." - Henri Poincaré
"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
"Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial." - Peter Freyd
"All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove." - G.H. Hardy
In describing the natural understanding we have of factoring, the famous mathematician Paul Erdos would have said, "Every baby knows that any integer greater than one can be factored into a product of primes." While Erdos often exaggerated what babies know, it is certainly true that most grade school children know it.
"Q: What's yellow, linear, normed, and complete? A: A Bananach space."