Recently I have the time and motivation to give math another try. Out of school, my day to day usage of math mostly involves arithmetics. The common use has been to estimate algorithm complexity, which is straightforward in most cases, (until it is not. e.g the coupon collector problem)

Math in school has been mostly learning by repetition, like how one trains ML models with supervised learning. Students are given bulk loads of problems and answers to solve until the problems themselves become familiar. That worked up to certain point. In fact, the math of my college freshmen year in Singapore was manageable since most questions were not so different from my high school math.

But can the subject be anything other than homework exercises just to get over with. Someone said that learning math is like learning a new language. To learn a language, one pickup the grammar and the vocabulary, word by word until the whole sentence starts to make sense. The vocabulary of math is the strange notations and symbols that can be daunting to look at. And mathematical materials are not always accessible to untrained eyes since math authors can have assumption of how much readers know. And it is not really their fault since after hundreds of years of development, mathematics has huge body of knowledge.

Also if like a new language, once understood, math can open up insights, meanings and other sort of pleasures. Something enjoyable. I remembered years ago when long English text started to become comprehensible. A new world of views, books, novels followed. So if that same realization can also be experienced with math, I want to give it another try. And who knows if I can become a cool dad once my kid reach school age. I can tell them one thing or two about mathematics that their teachers may not.

So far, I started with the book Programmer introduction to mathematics. The author covers a large breadth of topics with suggested keywords to look up even further. That was too many to take in. Since my purpose was to have an appreciation of math, not to know a lot of math, I needed to narrow it down to a smaller area. I ‘m trying another book Linear algebra done right (suggested by this quora post) and working through the first chapter on vector space.

The book does not start with the familiar cartesian planes and vectors as arrows in a 2D space, though it did mentioned that. Instead, vector spaces are developed by defining the notion of vector spaces as a set of objects (which can be lists as of Euclidean spaces, functions or objects like \(J(\infty)\)) and their addition and scalar multiplication that satsify certain properties like having additive identity, communtativity and associativity of addition…. With this approach, vector spaces are manipulatable. We can define a new vector space over an existing one by defining appropriate addition and scalar multiplication. And we can also have subspaces (a subset of a vector space which is also a vector space).

In a mathematician’s lament, the author argues that, math problems is hard creative work, which should be approached slowly and contemplatively. Thus, there need not be a goal to reach nor an immediate return on the effort. But given that we often push art aside for things more consequential in our day-to-day, that sounds challenging to keep up. Let see how long I can keep this personal project going.