Early on I noticed that mathematicians live in a world inaccessible to common mortals ... They are a special breed possessed by an intense cerebral life; simultaneously living on two distinct levels of consciousness, they are at once present and able to carry on normally and yet are immersed in the abstractions that form the core of their lives.
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.
Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously, and in which the unconscious (however one interprets that word) seems to play a role.
It's not only the question, but the way you try to solve it.
I don’t have any particular recipe [for developing new proofs]... It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.
What is research? What does research mean? The pursuit of truth? A quest into the unknown? Well, to start, itmeans finding a problem and solving it. It could be a tiny step of a small puzzle, (by standing at the shoulder of a giant!). Or, it could also be a glimpse at a bigger picture (eventually). Since any worthwhile problem won't be easy to solve, the road can be long and windy with twists and turns.
I often heard, "Math is too big. Wait until I have learned enough." The fact is that nobody knows everything. Don't wait. Just get started.
For nine years I studied mathematics in school, sometimes by myself and sometimes following a mentor's suggestions. After school I was supposed and encouraged to do mathematics independently. I set myself the challenge of finding my theorem completely on my own. Finding a new theorem is like finding a gem deep in earth. No one knows where the gem is, and even if one knows where it is, one does not know how to get at it. It is either an almost impossible task or a very exciting and rewarding one, depending on one's ability and attitude toward the gem, mathematics.