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.
Many a times, we are faced with the situation of finding the necessary condition and sufficient condition for some hypothesis to be true. Let’s consider a case (though it’s quite a specific case but you can generalize it) of scale space theory. We have to construct a scale-space representation of signal keeping in mind that fine scale information should be suppressed when the scale parameter increases. We also want that inherent property of the signal is retained at coarser scale. Along with it, we want no introduction of spurious structure in the process of smoothing. So, we have to formalize this concept. While formalizing this concept, we must know what conditions are absolutely necessary for construction of such space-scale representation of the signal. While deriving other conditions, we may find out that list goes on and on. So, the next question comes is where should it end? How should we describe sufficiency. We must know what is the condition for necessity and sufficiency. For this, we have to take the route for logic.
Let’s consider the statement, If “A number is divisible by ” then “it is divisible by 2”. Let’s factorize this statement. S1: A number is divisible by ; S2: it is divisible by 2. The clause S1 is called Hypothesis. It is what we are given, or what we may assume. The clause S2 is called Conclusion. Given the hypothesis, it is the statement that we must prove. When the If-then sentence is true, we say that the hypothesis is a sufficient condition for the conclusion. Thus it is sufficient to know that a number is divisible by in order to conclude that it is divisible by 2. The conclusion is then called a necessary condition of that hypothesis; or necessary for that hypothesis to happen or be true. Because if a number is divisible by , it necessarily follows that it will be divisible by 2. (Equivalently, it is necessary for a number to be divisible by 2, if the number is to be divisible by ).
Many a times, while solving mathematical problems (specially proof ), you come across a situation where existence of some quantity pops up. Now, in such situation we try to prove the existence and stops there whereas we should proceed further to find out what that quantity or entity is ? We should ask ourselves can I find any method to figure out that quantity, because it may happen that method will turn into some algorithm. Other times we should look other properties of that entity i.e. does it satisfy all criteria of being a normal entity in the system or does it violate certain properties which gives you some new elements to define in that system.
"A good stock 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
"Don't just read it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? Where does the proof use the hypothesis?" - Paul Halmos
China may be seen as a superpower in world economy but as far as world politics and domination is concerned it has yet to a start some course of action let alone the covering of the grounds. Here seems to be an article presenting the case in the wake of recent killing of Chinese citizen by IS.