A Prelude Over Mathematics
This section was labeled under, or is related to mathematics.
The following is an inverted part from a long note I’ve been working on recently named “Notes on Mathematics”, however not ready yet to be published (can’t be sure if it will ever be), but I felt the need to publish this part however.
It is not so common to have feelings towards one of the most, or actually the most, abstractive branch of science that ever existed, yet I’m always experiencing that collection of hunting and curiousness feelings whenever I contemplate a beautiful notation of mathematics, this has been occurring with me more frequently since the last month when I’ve seen the following equation which was posted originally by the 級数bot:

Every time I look at it I imagine the elegance of equations, the harmony of geometric shapes, for those with the eyes to see, mathematics reveals a glimpse into the very fabric of the universe itself. It is the very essence of numbers themselves that hold a certain allure. So let’s consider that formula:
I’d like first to talk about how does this work, the subtle interplay of numbers that give rise to its elegance and precision. At its core, the formula is a representation of the famous Basel problem, which seeks to find the sum of the reciprocals of the squares of all positive integers. While this may seem like a daunting task, the formula above provides a simple yet stunningly effective solution.
By manipulating the alternating series of terms, we are able to transform it into a more
manageable form, ultimately arriving at the value of
I don’t experience these feelings only in my personal reading and studies, but I also notice this in some writings, especially from Bertrand Russell writings, actually he says in his autobiography:
There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics. Russell, Bertrand. The Autobiography of Bertrand Russell. London: George Allen and Unwin, 1971. p. 66.
While people can imagine it’s killing to spend your time doing proofs for things that were given to you before, they just don’t understand the joyfulness. And notice that I didn’t say “they can not feel it”, because I don’t really believe it’s possible, and I clearly do not believe that there are people who are smarter than others so they don’t experience these feelings, or as Descartes puts it:
Those who have this knowledge and awareness of themselves convince themselves readily that all other human beings can have the same knowledge and awareness of themselves: because there is nothing in all this that depends on anybody other than oneself. Descartes René. The Passions of the Soul. Charlottesville, VA: InteLex, n.d. arc. 45.
I’m yet not able to find a more precise description better than what Descartes uses, which is knowledge. Though I can’t abstract my feelings towards it at all, since it’s very mixed with nostalgia and appreciations. It’s like a blocked channel in a water flow, or even a minor who can’t understand an adult description of the joyfulness of sex.
La Géométrie is one of the most famous examples for that feelings (In which Descartes is having a similar experience in doing some proofs that he delivered to his students and us as incomplete.), if you have ever heard of this book, it’s known for being ’unclear’, but actually this was very intended by him when he said “I did not undertake to say everything, in order to give others the pleasure of discovering [it] for themselves” Boyer, Carl B. (2004) [1956], History of Analytic Geometry, Dover. pp. 103-104. . I’ve actually seen this as recklessness, but I couldn’t help imagining myself doing the same thing if I’ve been through very pleasing journey. I really anticipate to be that the reason why many mathematicians use the infamous “the proof is left to the reader as an exercise”, it’s an invite to share the joyfulness.
It was not merely a tool for understanding the physical world, but rather a means of exploring the deepest aspects of human consciousness and existence.
(Added 13:18, 07 Apr 2023)
