Many a times, you will hear that a titan in an area is the one who has got extraordinary intuition in that area. Name it maths, physics, psychology, philosophy etc. Until you have the intuition, it’s very hard to learn the subject at higher level. There is an article by Terence Tao, on “there is more to maths than rigours and proof “. While learning maths as a child, we have a very fuzzy notion of everything that we do and do not delve into the correctness of the notion. The emphasis is more on computation than theory.
But as soon as we enter into the world of higher mathematics, learning why a particular method that we followed in early undergraduate years was right, we move into the realm of proofs and leave all intuition behind us. This stage is important as it gives us the way to move forward logically from one step to another step. But some guys tend to remain at this stage and get lost in the mathematical rigours. It is equally important to get back your intuition since the real aim behind walking into the realm of mathematical rigour was to remove false intuition. After this stage, you will a “big picture” of what is theory you are working on, why it works in some cases and why it fails in some.
So, the question comes to mind that how do we develop this big word intuition in our learning. First of all remember that maths is a language, though a difficult language, but still a language. Some learn it fast some learn it slow but key is perseverance. So, when you start to learn any mathematical subfield. Start slow and work on each aspect quite meticulously, but one key point is to be in “Learning subject space”. What this means, if you are learning vector spaces, you must let your mind wander in vector spaces. Try to get a feel that you are in vector space and every object around you is properties, operators in vectors space. Then ask yourself those dumb question that terry tao mentioned in his article.
What I feel, is important in developing intuition for the subject to let your mind float in that subject realm and then ask yourself those dumb questions and Yes! Practice a lot. The great mathematician S. Ramanujan is one of the undisputed genius of all time and it’s all because he practised so much that everything he did became his nature.