Question

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?

Standard community wiki rules apply: one quote per post.

Answer 1

"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it's the exact opposite!" -- Paul Dirac (some people attribute it to Franz Kafka!?)

Answer 2

Another quote from Dieudonné's "Foundations of Modern Analysis, Vol. 1":

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 prestiguous 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.

Answer 3

Dieudonné in "Foundations of Modern Analysis, Vol. 1":

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.

Answer 4

‘Life is complex: it has both real and imaginary components.” (I don't know who said this...)

Answer 5

Like many people, I am fascinated by the quote from Weyl (already listed here), that

In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.

But I can see why people are puzzled by the quote, so I'd like to add some more information (too much to put in a comment) as another answer.

First, what is the context? The quote occurs in Weyl's paper Invariants in Duke Math. J. 5 (1939), pp. 489--502, the first page of which can be seen here. This page includes most of what Weyl has to say on algebra v. geometry, though the quote itself does not occur until p.500. Then on p.501 Weyl explains his discomfort with algebra as follows

In my youth I was almost exclusively active in the field of analysis; the differential equations and expansions of mathematical physics were the mathematical things with which I was on the most intimate footing. I have never succeeded in completely assimilating the abstract algebraic way of reasoning, and constantly feel the necessity of translating each step into a more concrete analytic form.

Second, why the image of angel and devil? According to V.I Arnold, writing here, Weyl had a particular image in mind, namely, the Uccello painting "Miracle of the Profaned Host, Episode 6", which can be viewed here.

Arnold describes this painting as "representing an event that happened in Paris in 1290." "Legend" is probably a better word than "event," but in any case it is a very strange origin for a famous mathematical quote.

Answer 6

"Life is good for only two things, discovering mathematics and teaching mathematics." -- Simeon Poisson

Answer 7

Mathema est ars et scientia, discenda

Aquinas?

Answer 8

Jean Bourgain, in response to the question, "Have you ever proved a theorem that you did not know was true until you made a computation?" Answer: "No, but nevertheless it is important to do the computation because sometimes you find out that more is there than you realized."

Answer 9

Mathematicians are born, not made. -- Henri Poincare

Answer 10

At the risk of overloading an already bloated thread, I found a rather large collection here. Example:

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

Richard W. Hamming, in N. Rose's Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

Answer 11

"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." V.I.Arnold

http://pauli.uni-muenster.de/~munsteg/arnold.html

Answer 12

"The noblest ambition is that of leaving behind something of permanent value."

-G.H. Hardy, A Mathematicians Apology

Answer 13

Not famous yet, maybe from now on!

At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate.

Terence Tao

Answer 14

"Maybe at times I like to give the impression, to myself and hence to others, that I am the easy learner of things of life, wholly relaxed, "cool" and all that - just keen for learning, for eating the meal and welcome smilingly whatever comes with it's message, frustration and sorrow and destructiveness and the softer dishes alike. This of course is just humbug, an images d'Epinal which at whiles I'll kid myself into believing I am like. Truth is that I am a hard learner, maybe as hard and reluctant as anyone."

Grothendieck in Pursuing stacks (letters to Quillen).

Answer 15

"It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out."

-Emil Artin, Geometric Algebra

Answer 16

I once read, in an autobiographical piece, what the author said to his high-school teacher upon graduation; my recollection is:

"Poincaré has written that geometry is the art of making a correct argument from incorrectly drawn figures. For you, sir, it is the opposite."

I would love to know the correct quote, and an accurate source. I've seen a version attributed to Poincaré, but couldn't verify that.

Answer 17

Who can does; who cannot do, teaches; who cannot teach, teaches teachers.

Paul Erdos.

Answer 18

And each man hears as the twilight nears, to the beat of his dying heart, the devil tap on the darkening pane, "You did it, but is it art?"

Epigraph to Hille-Phillips, "Functional analysis and semigroups"

Answer 19

"In mathematics you don't understand things. You just get used to them."

John von Neumann

Answer 20

"There are, therefore, no longer some problems solved and others unsolved, there are only problems more or less solved, according as this is accomplished by a series of more or less rapid convergence or regulated by a more or less harmonious law. Nevertheless an imperfect solution may happen to lead us towards a better one."

Henri Poincare

Answer 21

“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 \Sha, which is not known to be finite.”

-John Tate on the Birch-Swinnerton-Dyer Conjecture

Answer 22

Do not ask whether a statement is true until you know what it means. -- Errett Bishop

Answer 23

"Mathematics consists of proving the most obvious thing in the least obvious way." - George Polya

Answer 24

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. —- Albert Einstein

Answer 25

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 a comment to a MO question

Answer 26

"Either mathematics is too big for the human mind or the human mind is more than a machine." - Kurt Gödel

A declaration of war by a Platonist.

Answer 27

Farkas Bolyai to his son Janos, speaking about attempts to study Euclid's Vth postulate on parallel lines:

"You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of the parallels alone... I thought I would sacrifice myself for the sake of truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction.... I turned back when I saw that no man can reach the bottom of the night. I turned back unconsoled, pitying myself and all mankind.

I admit that I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness - aut Caesar aut nihil."

Answer 28

Without Mathematics Earth is just a Big Zero - Chacko Mash

Answer 29

"We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things." - Anonymous quote from Bernt Øksendal's "Stochastic Differential Equations".

Answer 30

"Why is this a good idea?"

• Bill Ralph, on the most important question to ask yourself when doing (or studying) mathematics.