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.
Wrestling with Maths Books
Most of times there is a feeling that we need to read many books of the same topics in mathematics to understand them fully along with solving problems of these books. Now, many of the concepts are defined in seemingly “different form” which illustrate the same concepts in different manner. (Add an example of closure of sets etc…necessity) Now, to reconcile these different forms for the same concepts, you need to understand the idea behind these concepts because the underlying notion of all these concepts are same and they define these notions in different manner. Just like programming problems, where you use different structures in programs for the same algorithm but in the end all correct programs lead to same result. So, understand the concepts and then also learn what notions they introduce to reach to such concepts and how they reach these concepts. This way you will learn how to formalize your concepts if you happen to make one in future.
4 important classes in maths
- Concepts and Idea for the topics or for its development
- Definitions and its examples (which is basically a formalization of concepts and ideas developed in 1)
- Theorems (lemma, corollary, conjectures) related those definitions and concepts
- Problems and exercises on these theorems.
Now you have to digest all these as well, you might stuff yourself with each of these component, but it won’t convert to full theory backed intuition unless you mix and digest them all.
When you define concepts, try to think of the properties that objects (following that definition) might follow and then go through mathematical rigors to check if your intuition about the objects were right or not?
For those theorems listed in the books, first see what they have to offer in plain English, convert all mathematical symbols in English and then analyze how do they look or what can you see through them.