As 2024 arrived, Westlake welcomed its own new arrival—Prof. Huayi Chen, who joins the university as chair professor in mathematics.
Chen was previously dean of the Mathematics Institute of Jussieu–Paris Rive Gauche at Paris Cité University, and had also conducted research and taught at Westlake as a visiting scholar.
It might be hard to discern a human side to mathematicians, but for a group of scholars known for their rational thinking and devotion to abstract questions, math is not a discipline to be put on a pedestal. To Chen, math is no more than "a common understanding between you and me."
In this interview, Chen led us to the doorway to the world of math and encouraged us to take a peek.
Math is an abstract sign language that exists everywhere and is above everything. Chen's research interest is geometric algebra, which is closely related to number theory. Goldbach's conjecture is considered in both disciplines, but, although it sounds simple, it is still unsolved.
Is math simple or complex? How does it help us to expand the boundaries of our thinking?
Q: Most people only ever actually do elementary arithmetic in their daily lives after leaving university, so why should we still study math?
A: Mathematics isn't about teaching people to count. Take jump rope, for example. We maybe get slower as we grow older, but it is a great way to develop physical coordination when we are young. Mathematics training is the same; it trains our logical coordination. When someone tells you a story, do you blindly accept it at face value, or can you deduce a conclusion through logic? That's the difference.
In a way, mathematics is the most cost-efficient logic training obtainable. The basic principle of mathematics is to reach conclusions with as few assumptions as possible; the rest is logical deduction.
Q: Let's get abstract now. Let's say, for example, that dragonflies developed mathematics. Would it be different from our mathematics? After all, they have compound eyes, so they see the world in a completely different way from us.
A: That's actually a classic question that has attracted philosophers, mathematicians, and scientists throughout history: Did we invent mathematics, or did we simply discover it? On the philosophical level, this is a question about the nature of mathematics. Do they only exist subjectively in our minds, or are they part of the objective universe?
Personally, I agree with Grothendieck. He states that the structure of things cannot be invented. We can only enhance our understanding of it with patience and humility. In other words, we discover it. Such structures are never ours to empower or make them into how we think they should be. However, we do need to create a new language and vocabulary to accurately describe the hidden structures within mathematics as we explore them. Using this language, we create theories around what we understand and observe over time. There's a continuous back-and-forth between understanding a subject and describing what we understand, and the medium to record and communicate this is the ever-developing language that we need to perfect for our work.
Q: What do you think when people say mathematics is their nightmare?
A: Studying mathematics doesn't require a lot in the way of resources, but that means you are more likely to reach your limit and get frustrated. I believe that everyone has the potential to learn math well. It is a highly abstract subject, which presents a high bar for educators. We need to train math teachers well to make more people fall in love with mathematics and dig into their potential to learn it.
On the other hand, there is more than one way to learn mathematics, so I also believe that students should find their own preferred way to study math well.
Chen's Path
Mathematics has been Chen's life for nearly 30 years. As a teenager, he won the gold medal in the International Mathematical Olympiad (IMO) and went on to study and teach math in France. His research interests include the Adele curve, the Hodge index theorem, and the Harder-Narasimhan theory.
How did he manage to keep his passion for mathematics?
Q: Would you like to tell us about your high school years?
A: I am from Fujian, and I graduated from Fu'an No. 1 Middle School.
I'd always been drawn to abstract thinking. On one occasion in a chemistry class, I thought of removing impure ions in an unconventional way. It didn't work, but I was fascinated by how it failed. My chemistry teacher gave me an analogy: If a person wishes to clean himself, would he manage by taking a dip in the gutter and then rinsing off under a tap? That was when I realized the gap between theoretical models and reality.
After I had made it into senior high school, I was recruited to the IMO team, where my interest in mathematics flourished. Back then, I was also into music and literature, especially novels. However, that didn't last. After I began my life in research, I really became aware of my preference for abstract work, and the fun of appreciating concrete matters seemed to vaporize. It's a shame, really.
Q: In your work, you seem to go from one path to another. What makes that possible?
A: I guess I see doing math as going from one path to another. Since the origins of mathematics, breaking the boundaries of thinking has always been a priority, and a key part of breaking those boundaries is to find the common ground among seemingly different subjects. What that allows us to do in mathematics is come up with rules that often apply to many scenarios—rules that people can use to simplify complex situations.
Take counting, for example. The essence of counting is the one-to-one correspondence within sets, but, from there, we can apply counting to infinite sets so as to study their rich content.
Another example is the Cartesian coordinate system developed by René Descartes. Descartes created a way to associate objects in plane geometry with real numbers, which made plane geometry a calculable subject. Therefore, it's fair to say that the different disciplines in math constantly cross paths, and my work is no exception.
Q: Do you see mathematics as a game?
A: Mathematics certainly shares similarities with games—you need to be able to see the fun in order to be fully invested. Such fun sometimes comes as a surprise. You may receive a lot of rewards and positive feedback at the beginning. For example, one feels strong motivation to go further after successfully solving an equation.
However, as you proceed, such feedback becomes rarer. As your experience grows, you realize that there are so many more questions to be answered. In mathematics, a good question often isn't one with a clear solution.
When we read novels, we hope for an ending, perhaps a happy one. But what if there isn't one? We spend a lot of effort trying to find out why things happen in a certain way. Our lives move on linearly, so it's hard to tell when an ending happens. When I was young, I thought it was such a shame that Xueqin Cao didn't get to finish "Dream of the Red Chamber." Now if you were to ask me if it needs to have an ending, I actually wouldn't be sure.
Looked at on this level, perhaps it's easier to let go of our obsession with finding an answer.
The Lean
Prof. Huayi Chen officially started at Westlake University on the first working day of 2024, and he delved straight into preparing a short-term course on the introduction to Lean. The course will be open to all faculty, all Ph.D. and undergraduate students at Westlake, and researchers from other institutes. Lean is a proof assistant and programming language that is used to test the accountability of mathematical proofs.
As the name suggests, Lean has been developed for people to "lean"on for testing and verification. After translating mathematical propositions and proofs into code, researchers feed them into Lean to test whether they stand up. With the development of AI, computers are playing an ever-bigger role in math studies.
Q: These days, we see more mathematicians teaming up on projects. In the past, they seemed to normally work alone.
A: That is a common misunderstanding. Mathematics is a systematic subject in which everything is interrelated. Most mathematical work, even the newest, is based on the findings of those who came before us. There are many published papers with one author, but that doesn't mean they are one-man research projects.
There are many more papers that are co-authored. Collaboration allows mathematicians from different specialties to work together and combine their strengths. The merging and collision of different perspectives and methods gives us new inspiration and creativity.
After all, mathematics is a common understanding between you and me.
Q: If we think of mathematics as a common understanding, are there mathematical theories that are only recognized by some?
A: All human knowledge can be seen as part of our common understanding, and mathematical theories are no exception. However, I'm not sure that there's a definitive answer to what you asked.
For example, intuitive mathematicians think of math as a creative activity, where the existence of the subject relies on the heart and mind of the mathematician and is not an external object. This school of math believes that just because we can't prove a proposition to be false doesn't make it true, for such proof isn't constructive. Intuitive mathematics is still a minority discipline—most mathematicians still believe in the classic logical principles—but it has its supporters.
Q: Are there times when math stops being reliable? Historically, there have been three crises in mathematics, though they were ultimately resolved.
A: I would be careful with the word "crises." Though math is known for being rigorous, its development hasn't been entirely lineal or logical—it has had more of a bouncy, back-and-forth development pattern. These so-called crises prove that the progress of mathematical thinking is out in front of "mechanical" mathematics.
Such "crises" challenge mathematics as a construct and force us to review it. Thanks to that, new methodologies and perspectives are born, which greatly advance math as a whole.
To give a few examples: The concept of infinitesimal numbers led to the birth of limit theory, which laid the foundations for modern calculus; set-theoretic paradoxes forced mathematicians to come up with the axiomatic set theory; and Gödel's incompleteness theorems pushed mathematicians to reconsider the entire nature and goals of mathematic theorems, driving the development of mathematical logic.
Q: Do you think computers can conduct mathematical research on their own in the future?
A: David Hilbert once tried to establish a mathematical system with formal mathematical methods where the existence of an algorithm could determine the correctness of mathematical propositions. However, Gödel's incompleteness theorems prove that if such a mathematical system contains basic arithmetical functions, then there are always some propositions that they cannot judge to be authentic. However, that is not to say that mathematics stops working. It only means that purely mechanical judgement isn't always going to work.
AI represents another possibility. It is bionic, and in the future, the automation of mathematics may involve bionic brains.
To make mathematics artificially intelligent, I believe we need to resolve two challenges: The first is to automatically translate mathematics from natural language to form language; the second is to automatically generate human language proofs using AI. Mathematics would look very different if we were to resolve these two issues.
Humans may also need an element of intuition as the foundation for the study of math. Computers, on the other hand, can freely test across different axioms.
Q: If that's the future of mathematics, where will mathematicians be?
A: In the future, mathematicians will use AI tools to study math more efficiently, and they will invest their energy into new hypotheses and new systems. Computers can be used to undertake some of the proving and testing.
AI is a human creation and should be a natural extension of human civilization rather than a "terminator." There are many sci-fi movies in which, once AI has developed to a certain level, it starts to have its own will and sets out to destroy mankind. I think that's fantasy, and I am not at all sure it will come true.
From the broader perspective of evolution, we are already a world apart from the earliest forms of life. In the future, humans may also evolve into very different life forms, perhaps even silicon-based life forms that live within computers. I think our future form doesn't have to exclude our current one. Even though they may overlap in time, they are likely to occupy different spaces in the ecosystem and might need very different resources.
Q: One last question: What made you decide to join Westlake?
A: I was invited by Prof. Gang Tian of the Institute for Theoretical Sciences and President Yigong Shi to be part of the mathematics department. Our fundamental goal is to cultivate talent in math research and design curricula for undergraduate and graduate student training. This is different from any other institution that I've worked for.
I previously spent nearly a year at the Institute for Theoretical Sciences as a visiting scholar, and I was really taken by the vigorous, free, and meticulous research atmosphere. Because of that, I am happy to take on the challenge and be part of the journey to construct a brand-new mathematics department.
