About Mathematics

More than 20 years ago, in a calculus class, the lecturer told us during the last lesson of the semester: "Now we've learned a year of calculus. But in the real world, you will probably not do it the way we do our homework anymore, as there are lots of software products such as MATLAB to perform these tasks. But I think it is still necessary to understand the theories and concepts in order to use those software tools properly."

Then he paused for a moment and said, "It is also important to note that thesesoftware tools do not work the same way you do for homework or examinations. They don't use formulas for differentiation or integration. Instead, the programs simulate the fundamental core of calculus by approximating the area under an irregular curve with a large number of rectangular bars. The result will be more accurate if you cut it into more pieces, with the trade-off of a higher computational cost."

I am grateful to have had such a young and excellent mathematics teacher during my first year of university. I still consider him to be one of the best teachers I've ever had. He focused on explaining the concepts and theirmotivation in a very intuitive way.

Whenever I contacted him to ask questions, he always tried his best to explain everything patiently. He reassured me that no matter how poor my foundation was at the beginning, given that I had passed the entrance exams to enroll in the university, my classmates who seemed to achieve much better marks did not actually have a distinctive advantage. "Four years is not a short time. By continuously studying hard, you can reach a level equal to anyone in your class."

He highlighted the generality of an approach rather than focusing on methods that only applied to very specific situations, which are unlikely to occur in the real world.

I didn't fully understand his words back then. Several years later, when I read the mathematical analysis textbook authored by Professor Terence Tao, I was drawn to the in-depth discussion of how definitions are constructed and how to derive theories from them, as well as how the system would change if we defined a concept differently. After reading his textbook and many posts on his blog, I think I've gained a basic idea of how to construct mathematical models by abstracting physical reality.

Then I wondered why one plus one equaled two. It doesn't have to be that way! In a binary system, one plus one equals "10"!

Then I kept thinking: why do we need a binary system**?** It is because an electrical switch is a robust way to represent data in a digital system. Can we find a way nowadays to represent data directly in a more complex manner? Would it be beneficial because we could skip many intermediate steps in computations? Yes, that is the motivation behind quantum computers!