The Pursuit of Meaning: A Humanities Student's Journey Through Mathematics

The Pursuit of Meaning: A Humanities Student's Journey Through Mathematics (Translated from Chinese by ChatGPT)

As a child, I cherished literature and history, happily spending afternoons meandering through the bookshelves, familiarizing myself with every major bookstore and library in Taiyuan. The sublime aesthetic experience brought by exquisite prose and the allure of stories from different times and places always captivated me. Thus, during my middle school years, I was decidedly a student of the humanities. I understood the value of studying these subjects: learning was an exploration of life and the world, a beautifully enlightening pleasure.

However, compared to literature and history, I struggled to find the meaning in studying mathematics. I couldn't grasp the significance of the numbers, equations, polynomials, and trigonometric functions presented in class. Nor could I be convinced by the meager and uninspiring real-world applications outlined in the math textbooks that studying mathematics deserved an effort comparable to that of the humanities. Adding to my confusion, my friends, who seemed unenthusiastic and expressed little in their studies, effortlessly scored much higher than I did.

Consequently, mathematics became the subject I performed worst in during my middle school years. Without seeing its meaning, I found it hard to invest genuine effort.

Nonetheless, I did not want to limit myself to one area. After spending extensive time with humanities books, I suddenly felt that I had entered a "comfort zone," where there was no significant progress for a while. Many things seemed to be caught in a loop. I speculated that to surpass my former self and uncover deeper mysteries, I needed to address my weaknesses and ensure balanced development across my abilities. Moreover, I believed that every discipline inherently holds value; the less I understood the meaning of mathematics, the more my curiosity drove me to delve deeper.

Driven by the pursuit of meaning, time flew by, and one day, I found myself sitting in a classroom of the Mathematics Department at Beijing Normal University.

Looking back now, I realize that the meaning of mathematics is perhaps the hardest to seek, especially given the current textbooks and curriculum system. My first college course was Mathematical Analysis, a crucial foundational class. Professor Huang started off by saying that this course was like climbing a cliff: challenging to learn, but if you made it to the top, the rest would be easier. However, I encountered more and more concepts and theorems, more and more things that should have meaning, but I couldn’t grasp it; I could only learn and memorize almost mechanically. After three semesters, when Mathematical Analysis concluded, I was still confused; in the fourth semester, due to various reasons, my GPA collapsed, and I cried in front of my parents.

Yet, these setbacks were meaningful. The first epiphany is always the hardest, and my previous inefficient learning was like the early accumulation phase of a learning curve. It was during the summer after the semester when my GPA plummeted, after long contemplation, that I had a breakthrough and understood things I should have realized earlier, and everything suddenly became clear. It was an exciting yet calming feeling: excited because I finally strongly felt the beauty of mathematics, finally found the meaning of studying it, and began to establish connections between mathematics and other familiar things. Calm because the things that had troubled me—GPA, peer pressure, directions for future development—began to dissipate. I felt pleasure and a sense of accomplishment, yet there was almost no one to share it with. (Of course, this realization wasn’t instantaneous, but a gradual process.)

After reading Fung Yu-lan's "A Short History of Chinese Philosophy" and its discussion on the concept of "unity of heaven and man," I found many resonances, particularly when he referred to a passage from Plato's "Republic." In it, Plato states that philosophers must ascend from the cave of the sensory world to the realm of reason. Once philosophers reach the realm of reason, they become one with the universe and, in this unity, transcend reason itself.

"Earlier chapters have already informed us that Chinese philosophy tends to emphasize that to become a sage, one does not need to perform out of the ordinary. A sage cannot and does not need to perform miracles. He does what ordinary people do, but due to a high level of awareness, his actions carry a different meaning. In other words, he acts in a state of enlightenment, while others act in ignorance. In Zen Buddhism, it is said, "Awareness is the source of all wonders." The meaning derived from awareness constitutes the highest realm of human life."

The world studied by mathematics is not our living reality but a world of ideas, an infinitely vast existence. The real world seems like a projection of this ideal world. In it, the infinite connections hidden among all things are revealed; seemingly distant and unrelated things are tightly connected at certain levels (as the saying goes, "Seeking the elephant, distances shrink from thousands of miles to mere inches"). Moreover, this world is entirely open to exploration. Merely recognizing the existence of this world can reshape one's worldview. Each exploration can transform our understanding of the world, truly showing that truth is infinite. Exploring this world of ideas allows a person to connect themselves with the entire world, the entire universe, potentially reaching what Fung Yu-lan describes as the "realm of heaven and earth."

In this diagram, each point represents a scientific paper, and the connections between them signify the relationships or links between these papers. 

Before the truth, everyone is equal; in the world of mathematics, all are equal, and the exploration of ideas requires only a clear mind, where the various constraints of the material world vanish, leaving only infinite space. It is said that Einstein's research required no special equipment; his method of work was simply to think and calculate. "Even confined in a nutshell, I consider myself the king of an infinite universe."

In such a world, people return to their true selves—sincere, peaceful, and serene. In this world, we begin to truly appreciate the meaning of all our actions. The benevolent are not troubled, the wise are not confused, and the brave do not fear. "As Heaven's movement is ever vigorous, so must a gentleman ceaselessly strive along; As Earth's condition is receptive devotion, so must a gentleman carry the outer world with broad virtue."

I am fond of the anthem of Nanjing University, which begins with the lyrics: "Great indeed is sincerity which moves heaven; with the tripod's three legs that stand for wisdom, benevolence, and courage." My high school motto was "Sincerity, Integrity, Enlightenment, Resoluteness." I used to wonder, while honesty is indeed a fine quality, could it really move "all under heaven"? Thus, the "sincerity" mentioned here far exceeds its usual connotation. The so-called "sincerity" is the genuine frankness exhibited by a person who is aware of the true meaning of their existence. He is "awakened," hence fearless, nothing can shake him, he is the embodiment of freedom and philosophy, he is "the king of an infinite universe."

This brings to mind the looks in various films and literature: D'Artagnan's gaze at the Cardinal in "The Three Musketeers," the defender of Jerusalem's look in "Kingdom of Heaven," Robert Frobisher's gaze as he looks out at the sunset from the Scott Monument in "Cloud Atlas," and the Mushroom sitting under the Bodhi tree narrating ancient Indian myths in "Billy Lynn's Long Halftime Walk." "When you see it, you will be able to tell others."

Of course, studying mathematics itself may not necessarily make one aware of a greater "meaning" (which is actually the pursuit of philosophy), but the study of mathematics can spark such a process of pursuit. This is because the abstract symbols and language are so far removed from the real world that any learner will instinctively try to establish connections. This process requires emotive (or perhaps trans-rational) thinking.

I personally enjoy opening a mathematics book after listening to Beethoven or reading some humanities classics. At such times, I feel that behind the abstract symbols lies an incredibly romantic and beautiful world. It's a world inhabited by mathematicians who, like poets and philosophers of yore, in another corner of the Earth, filled with warmth and immense effort, conceived a structure, wrote down an equation. Today, as you ponder their profound insights, you are in dialogue with them. In these moments, one feels exhilarated and free, transcending the limitations of one's own life. With moist eyes, I cannot help but quote two passages:

"Sixsmith， I climb the steps of the Scott Monument every morning and all becomes clear. Wish I could make you see this brightness. Don't worry, all is well. All is so perfectly, damnably well. I understand now that boundaries between noise and sound are conventions. All boundaries are conventions, waiting to be transcended. One may transcend any convention if only one can first conceive of doing so. Moments like this, I can feel your heart beating as clearly as I feel my own, and I know that separation is an illusion. My life extends far beyond the limitations of me."

— Movie Cloud Atlas

Alas! I find myself at odds with the times, yet my aspirations extend the path. I set my heart above the ages, sending my thoughts to a thousand generations below!

— From "Wenxin Diaolong"

嗟乎！身与时舛，志共道申！标心于万古之上，而送怀于千载之下！

— 《文心雕龙》

Mathematics is not so "magical." It's just one method of exploring the world, a perspective of abstraction and unification. As my classmate studying theoretical physics puts it, theoretical physics provides him with the nutrients to understand the world and "grasp the way." To counter nihilism: compared to the vast universe, our efforts might seem insignificant, but the reflection and cognition that follow these efforts can lead us to a much grander realm. The meaning lies in the thinking itself.

The ways of heaven are geometric, all forms naturally maintain their order;

Variational principles are infinite, a solitary heart in measurement finds homotopy.

天道几何，万品流形先自守；

变分无限，孤心测度有同伦。

Today's words are reflections I've gathered over a period, serving as a preface for the content organized for my discussion course "The Spirit of Science: A Panorama of Mathematics and Natural Sciences" in Kaili. Details from each session will gradually be updated to this public account. In August 2020, I participated in the "Gē Zhì Project," where I conducted a ten-day discussion course for high school students in Kaili, Guizhou, discussing the integration of science and humanities. Those ten days incorporated my long-term contemplations and served as a summary of my past experiences. Every ending always contains a new beginning. Now, as I begin my graduate studies, I will integrate my new insights and thoughts as I review the discussion content. I feel like Bilbo Baggins in "The Lord of the Rings," who returns from an adventure back to the Shire, where everything is so beautiful. However, a new adventure is about to begin.

Returning to my journey in mathematics, I've previously mentioned the difficulties and experiences I've encountered. From my own experience, the discussion courses I designed aim to think about mathematics and natural sciences from a different perspective than traditional school lessons, hoping my students can avoid the struggles and pain I once endured. From start to finish, never giving up the pursuit of meaning.

My own understanding of the significance of elementary mathematics came only after I had learned much about higher mathematics. It appears that mathematics is a rigorously established system of symbolic axioms; without a solid foundation, it is impossible to understand profound conclusions. However, I believe that many truly beautiful ideas should be something anyone can (at least partially) appreciate their brilliance, even without much preparatory knowledge. Only by feeling the beauty of these ideas will there be a real motivation to study mathematics, to enhance one's thinking ability, and thereby to reap the greatest benefits from mathematics. Indeed, in Kaili, my students were also able to appreciate many ideas from higher mathematics (including functional analysis, algebraic topology, etc.).

This appreciation also comes with a foundation. Starting from some real-world applications such as artificial intelligence, quantum computing, and other cutting-edge technological fields that are of interest to the humanities, the application of mathematics in these areas is most thrilling and can attract students' enthusiasm and their innate nature to explore the essence, thereby transcending the complex system of mathematical symbols to glimpse the spirit of modern mathematics.

If we can transcend the limitations of the foundational system, first gaining some understanding of the discipline's core before starting the rigorous learning process, we are sure to engage the power of active thinking and achieve excellent results. Research in neuroscience suggests that the human ability to understand knowledge is innate, and the learning process may be the brain actively adjusting its own structure based on external information. This idea was profoundly considered by Plato in "Phaedrus," and I won't expand further here.

Several of my friends are about to start studying mathematics, natural sciences, or engineering. It is my greatest honor and pleasure if my words can bring you help and inspiration. I couldn't have written today's text without the enlightenment from some teachers and friends. Though far apart, I am deeply grateful and extend my sincerest thanks.

Princeton ist ein Ort nicht. Wo bin ich, wo das ist.  Das ist in mein Herz.