Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience.
As a primary school student, I was drawn to mathematics by the abstract beauty of formal manipulation, and the remarkable ability to repeatedly use simple rules to achieve non-trivial answers.
As a high school student, competing in mathematics competitions, I enjoyed mathematics as a sport, taking cleverly designed mathematical puzzle problems and searching for the right “trick” that would unlock each one.
As an undergraduate, I was awed by my first glimpses of the rich, deep, and fascinating theories and structures which lie at the core of modern mathematics today.
As a graduate student, I learnt the pride of having one’s own research project, and the unique satisfaction that comes from creating an original argument that resolved previously open question.
Upon starting my career as a professional research mathematician, I began to see the intuition and motivation that lay behind the theories and problems of modern mathematics, and was delighted when realizing how even very complex and deep results are often at heart be guided by very simple, even common-sensical, principles. The “Aha!” experience of grasping one of these principles, and suddenly seeing how it illuminates and informs a large body of mathematics, is a truly remarkable one.
And there are yet more aspects of mathematics to discover; it is only recently for me that I have grasped enough fields of mathematics to begin to get a sense of the endeavour of modern mathematics as a unified subject, and how it connects to the sciences and other disciplines.
I don’t have any magical ability. I look at a problem, and if it looks something like one I’ve done before; I think maybe the idea that worked before will work here. If nothing’s working out; then you think of a small trick that makes it a little better but still is not quite right.
I play with the problem, and after a while, I figure out what’s going on. Most people, faced with a math problem, will try to solve the problem directly. Even if they get it, they might not understand exactly what they did.
Before I work out any details, I work on the strategy. Once you have a strategy, a very complicated problem can split up into a lot of mini-problems. I’ve never really been satisfied with just solving the problem. I want to see what happens if I make some changes; will it still work?
If you experiment enough, you get a deeper understanding. After a while, when something similar comes along, you get an idea of what works and what doesn’t work. It’s not about being smart or even fast.
It’s like climbing a cliff: If you’re very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there.
Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools. You still need a plan — that’s the hard part — and you have to see the bigger picture.