Today I am thinking about problem-solving and how to generally look at a problem.
All mathematical fomulas were written down at some point in history by some people who wanted to solve a problem but lacks a way of thinking about it. Formulas gives out relations among expressions for us to think about problems within their contexts, so that we can reduce the problem and eventually generate an answer to it if lucky enough. Problems can be solved with certainty or probability. There are many standard methods to solve it, exhaustion for example, is one. Looking at all the possibilities and find the right answer. Or one can employ their human intuition to try out the ones that look most likely to be true, so as to reduce the time invested in the process. If for a certain kind of problem there is a codified flow of trying out possible answers, then it can be programmed into computer to try them out for us as algorithms. It can reduce our mannual work, but without that intuition, some flows can be extremely time and energy consuming.
Mathematical problems are extremely elegant, in that it lacks any complexity we have to deal with in daily life about an ordinary issue. We deal with problems all day long, and our pattern-finding can be absurdly wrong, because there are so many factors involved in any issue that we cannot know for sure whether the factors we observe are contributers. Mathematics is not something like that. It is a self-sufficient world where all the factors are in it. The initial factors are very simple, like throwing in a couple of basic lego blocks in different shapes, which can be replicated infinitely. With different amount of those blocks people can build complex relations that solve their own problem and facilitate others’ discovery. Coming up with problems and solutions are both creative processes, and both require an insight into the nature of the problems, tools at hand and other resources that can be manipulated into a more handy and specialized tool. Sometimes, however, we have to take a step back and look at our tools with its compartments, and think whether we need to wield such a contraption for this kind of problem. Sometimes the instrument is too much for our problem, and we need to reduce our tool first.
In any case, I think one should stay pleasant during any problem-solving. Intuition is only at work when one’s brain is active and full of cuiriosity and whimsy. Generally, I come up with more solutions after a good laugh. And laughing also gives me a good appetite.