Problems worthy of attack prove their worth by fighting back.Piet Hein

A good stack 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 always the case, with mathematics, that a little direct experience of thinking over things on your own can provide a much deeper understanding than merely reading about them.Roger Penrose

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it's all illuminated and you can see exactly where you were. Then you enter the next dark room…Andrew Wiles

I prided myself in reading quickly. I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what I'd just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments.William Thurston

We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.George Pólya

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

The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases.Paul Halmos

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. … It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind — we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.G. H. Hardy

Mathematicians need proofs to keep them honest. All technical areas of human activity need reality checks. It is not enough to believe that something works, that it is a good way to proceed, or even that it is true. We need to knowwhyit's true. Otherwise, we won't know anything at all.Ian Stewart

It is an error to believe that rigor in proof is the enemy of simplicity. On the contrary, we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to discover simpler methods of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigor.David Hilbert

If my false figures came near to the facts, this happened merely by chance … These comments are not worth printing. Yet it gives me pleasure to remember how many detours I had to make, along how many walls I had to grope in the darkness of my ignorance until I found the door which lets in the light of the truth … In such manner did I dream of the truth.Johannes Kepler

The best way to learn is to do; the worst way to teach is to talk.Paul Halmos

If you really wish to learn then you must mount the machine and become acquainted with its tricks by actual trial.Wilbur Wright

It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.Henri Poincaré

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where where ½ proof = 0, and it is demanded for proof that every doubt becomes impossible.Carl Friedrich Gauss

The best of ideas is hurt by uncritical acceptance and thrives on critical examination.George Pólya