intuition

A wonderful post about formalizing the intuition. Multiple quotes:

“Often you write something down to clarify your thoughts. In particular when you’ve reached a real impasse, when there’s a real problem that you want to overcome, then the routine kind of (mathematical) thinking is of no use to you. Leading up to that kind of new idea there has to be a long period of tremendous focus on the problem without any distraction. You have to think about nothing but that problem – just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over, and it’s during that time that some new insight comes.” -- Andrew Wiles

 “One thing is to prove it by equations; the other is to check it by calculations. I have mathematically proven to myself so many things that aren’t true. I’m lousy at proving things — I always make a mistake. … So I always have to check with calculations; and I’m very poor at calculations — I always get the wrong answer. So it’s a lot of work in these things. … If [we] can do a real physical problem with real physical things in them, then I’m sure we have the right method.” - Feynman

“Our brains are complicated devices, with many specialized modules working behind the scenes to give us an integrated understanding of the world. Mathematical concepts are abstract, so it ends up that there are many different ways that they can sit in our brains. ”
“A given mathematical concept might be primarily a symbolic equation, a picture, a rhythmic pattern, a short movie—or best of all, an integrated combination of several different representations. These non-symbolic mental models for mathematical concepts are extremely important, but unfortunately, many of them are hard to share.”
“Mathematics sings when we feel it in our whole brain. People are generally inhibited about even trying to share their personal mental models. People like music, but they are afraid to sing. You only learn to sing by singing.”  -- Bill Thurston
 
I particular love the last one - perhaps because it relates to singing :)
 
 Finally, “Intuition is natural ideas, concretely instantiated, that sing in our minds.” -- John Sidles

A nice lecture here.