Mathematics is not a spectator sport!
There exist a lot of questions that the fools can ask, and the intelligent cannot answer.
You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it...
John von Neumann was the only student I was ever afraid of.
Euclid ’s manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.
Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.
In order to solve this differential equation you look at it until a solution occurs to you.
What is the difference between method and device? A method is a device which you use twice.
Geometry is the science of correct reasoning on incorrect figures.
Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself.
When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
Quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous. The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better.
The result of the mathematician’s creative work is demonstrative reasoning, a proof, but the proof is discovered by plausible reasoning, by GUESSING.
The best of ideas is hurt by uncritical acceptance and thrives on critical examination.
Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side.
I am too good for philosophy and not good enough for physics. Mathematics is in between.
Mathematics consists in proving the most obvious thing in the least obvious way.
The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.
The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.
If there is a problem you can’t solve, then there is an easier problem you can’t solve: find it.
If you wish to learn swimming you have to go into the water and if you wish to become a problem solver you have to solve problems.