A Prelude Over Mathematics

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:

\[\frac{1}{1^3} - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + ... = \frac{\pi^3}{32}\]

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 \(\frac{\pi}{32}\) This is accomplished through a clever use of Fourier series and the Euler-Maclaurin formula, which allow us to approximate the sum of the infinite series with remarkable accuracy. This clear formula is more than just a functional tool; it is a symbol of the deep interconnectedness of all things. It is a reminder that even the most seemingly disparate concepts can be brought together in a meaningful way, and that through the pursuit of knowledge and truth, we can experience the profound mysteries of existence. For the great minds of humanity, the pursuit of truth, knowledge, and beauty were all interconnected aspects of the human experience. It was through the exploration of these concepts that we were able to connect with the transcendent nature of reality, and to transcend the limitations of our own individual perspectives.

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:

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:

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)