My Math

I consider the fact of mathematics is to understand it in my own mind language. The textbook and teachers will tell me how they see the concepts and theorems, but I use and must use my own language to interpret them. Mathematics is an absolutely correct thing: no matter how and where people see it, it’s correct itself. However, people have their own “glasses”, which are their mind languages, to see math, to acknowledge math and to teach others math. For example, I learn calculus from the language in textbook, and I’ll teach others by the language in textbook. But in my subconscious, I’m highly possible using another personal language: to record “what is an integral”, to use it for learning advanced concepts. This is more obvious when I see a difficult and complex theorem: I’d better leave it alone for several days when I couldn’t understand at the first glance. Studying in other people’s languages - looking at the correct proof - is often useless. During this stalemate, I kept trying to use my mathematics intuition and language to break down this theorem. When I grasp it eventually and look back at the recondite proof, I feel so natural and intuitive it is. I won’t ask questions like “how did the textbook notice this statement”, which usually happens when learning by rote. I consider that mathematics is very independent. My answers seldom matched the standard solutions to the problems I encountered from my primary school to high school. I believe in the importance of originality, so I would ask my teachers to think in my intuition rather than using the standard methods to solve and explain the problem. When my method fails, I need to understand the weakness, the gap in my mind language. When my method justifies, even win over, my teacher, I would feel proud, just like a real mathematician making a huge step. What is this tenacity? I found the similarity in student testimonials of HCSSIM: the core of “using a unique language” is the ability of building a valid, self-justified system. For instance, the textbook says, “the definition of inner product is...”, but why devise a brand new concept in that way? I asked myself, if I want to prove Cauchy-Schwarz Inequality, could I solve it in my own way instead of using “inner product”? More generally, I’m enthusiastic to discover better alternatives which could replace current math tools and structures, even if those alternatives make tiny or inconspicuous changes. In HCSSIM, rather than pure teaching and guidance, it bestows on us the ability to build math, suggesting that people with similar ideas may construct their own “mathematics space” here. In my opinion, “mathematics space” is a type of imagination. It symbolizes an independent structure, a world that contains only mathematical intuition. Most importantly, it’s abstract, requiring only imagination and deduction in “mathematics” intuition, with no need to combine theorems with any “real world” intuition. An abstract, imagined world that is independent of the real world, contains no imperfections. I love this imagination’s certainty, its plasticity; and most importantly, it’s built and modified by myself. This conclusion came from my interest in philosophy, in music, even in imagination itself. For many years, I usually doubted myself learning mathematics because of lack of talent. I want to confirm the essence of mathematics that is attracting me, and answer: why my passion continually appears and what impels me to learn up to now? Philosophy gave me the answer: I had never read it before, but its “abstraction” and philosophers’ self-justified knowledge structures, like independent imagined worlds, strongly appealed to me. Those are the most important properties of a “mathematical space”.