"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, 1929 - 2019

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.

~The Simpleton Symposium Blog

I love to do research, I want to do research, I have to do research, and I hate to sit down and begin to do research—I always try to put it off just as long as I can. ... Isn’t there something I can (must?) do first? Shouldn’t I sharpen my pencils, perhaps? In fact I never use pencils, but pencil sharpening has / become the code phrase for anything that helps to postpone the pain of concentrated creative attention. It stands for reference searching in the library, systematizing old notes, or even preparing tomorrow’s class lecture, with the excuse that once those things are out of the way I’ll really be able to concentrate without interruption.

~ Paul Halmos

"The Axiom of Choice is obviously true, the Well-ordering theorem obviously false, and who can tell about Zorn’s lemma?"

~ Jerry Bona

"Young man, in mathematics you don't understand things, you just get used to them." 

~John Von Neumann

"God exists since mathematics is consistent, and the Devil exists since we cannot prove it." 

~ André Weil

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

~ John von Neumann

“While math has many applications, it also has many problems you might not care about. Life also has problems you don’t care about that you’ll have to solve anyways. Math can prepare you for that.”

~ ???

“Let no one ignorant of geometry enter here.”

~ entrance to Plato’s Academy

"If I have seen further it is by standing on the shoulders of Giants."

~ Newton

“The introduction of the digit 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...”

~Grothendieck

"To take an example in the other direction, passionate love is, also, driven by the quest for discovery. It provides us with a certain kind of understanding known as 'carnal' which also restores itself, blossoms forth and grows in depth. These two impulses -that which animates the mathematician at his desk ( let's say), and that which impels the lover towards the loved one - are much more closely linked than is commonly believed, or, let us say, people are inclined to want to believe"

~ Grothendieck

"In those critical years I learned how to be alone.  [But even] this formulation doesn’t really capture my meaning.  I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation [1945-1948], when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law..By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member. or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like” volume” which was “obviously self-evident”, “generally known,” ”in problematic” etc…it is  in this gesture of ”going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one-it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone."

~ Grothendieck

"It might be possible to masturbate over the incredible mathematical regularity of physics in the universe, but beware of the fact that it wasn’t particularly mathematical or regular until we picked out those theories that fit the universe’s behavior very closely. Those theories have predictive power because that is the nature of the selection criteria we used to find them; if they lacked that power, they would be discarded and replaced until a theory emerged meeting the selection criteria. To be clear, mathematical models can be written to describe anything you want, including the color of your bong haze, but they only have power because of their self consistency. If the universe does something to deviate from what the math says it should, the math is simply wrong, not the universe.  Every time you find neutrino mass, God help your massless neutrino Standard Model!  Wonderful how the math works… until it doesn’t."

~ Poetry in Physics blog

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

~ G.H. Hardy

The primes are the raw material out of which we have to build arithmetic, and Euclid’s theorem assures us that we have plenty of material for the task.

~ G. H. Hardy

A traveller who refuses to pass over a bridge until he personally tests the soundness of every part of it is not likely to go far; something must be risked, even in mathematics. 

~ Horace Lamb

Writing papers was the punishment we had to endure for the thrill of discovering new mathematics.

~Anon

Though he was by no means a sloppy type, he was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this, and tried in vain to imitate him, but found it quite difficult to make good mistakes.

~ Shimura (on Taniyama)

Good mathematicians see analogies. Great mathematicians see analogies between analogies.

~ Stefan Banach

Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.

~ Sir Michael Atiyah

I don’t know. A proof is a proof. What kind of a proof?  It’s a proof. A proof is a proof, and when you have a good proof, it’s because it’s proven.

~ Jean Chrétien, former Prime Minster of Canada

Let's assume we know nothing, which is a reasonable approximation.

~ David Kazhdan

I can assure you, at any rate, that my intentions are honourable and my results invariant, probably canonical, perhaps even functorial.

~ Andre Weil, purporting to be Lipschitz writing from Hades

"If they're hiring number theorists, I'm a number theorist, if they're hiring representation theorists, I'm a representations theorist, if they're hiring geometers, I'm a geometer"
~ Misthy Ray

"That without a recognition of logical fictions, without a comparison of reality with the purely IMAGINED world of the absolute and immutable, without a constant counterfeiting of the world by means of numbers, man could not live"
~  Beyond good and evil

 
"And down under all those piles of stuff, the secret was written: We build our computers the way we build our cities -- over time, without a plan, on top of ruins."
~ Ellen Ullman

“It is easy to speak with precision upon a general theme. Only, one must commonly surrender all ambition to be certain. It is equally easy to be certain. One has only to be sufficiently vague.”
~  C.S. Peirce 

"The map is not the territory"
~ Alfred Korzybski