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Student Voices

# Understanding the mind by understanding mathematics

KEN Eitetsu

2nd-year doctoral student, Graduate School of Mathematical Sciences

TO

Charles University

Prague, Czech Republic

## A very special math teacher

In my second year of junior high school, our math teacher gave us a series of lectures on Euclidean geometry. He proved every statement very rigorously using Euclid’s axioms. For example, he proved the statement that given two distinct circles on a plane, they share at most two points in common. Previously I had thought that it was the most natural thing, but he gave rigorous proof of it. It was my first chance meeting with mathematics.

Back then, I liked asking why questions since elementary school. I just kept asking “Why?” again and again. (I must have been a very annoying kid!) So, the encounter above was a bolt from the blue. The structure of mathematics seemed very graceful to me. It admits some fundamental statements as axioms. We can and must ask why over and over again in mathematics, but at some point, we must admit something as valid. My math teacher was outstanding. He successfully implemented his ideas and his philosophy of education in our class.

## Pursuing mathematics professionally

Since my first contact with mathematics, many questions have arisen in my mind. For example, in my third year in junior high school, I was unable to execute substitution; yes, if x=y, then f(x)=f(y). We accept this as true but don't give proof of it. Later, I learned that this was one of the axioms of modern mathematics. Or another, and maybe more serious, example: If we know that statement A or B holds and if we know that B does not hold, then it implies that A holds. Why is this logic valid?

The latter question, the validity of logic, I knew it was not a mathematical question. Rather it was a philosophical one. So, when I was an undergraduate student, in addition to mathematics, I also tried to broaden my scope of education, taking courses in philosophy, linguistics, and neuroscience. At some point when I was a junior in college, I came to swallow the bitter pill that we are just such creatures, accepting our logic as valid and taking it for granted. But then the following questions arose: What are the limits of our mathematical knowledge? What kind of mathematical statements are not provable? And what is the efficiency of our logic and proof systems in comprehending mathematics?

These questions allow us to take multiple approaches in philosophy, linguistics, and other disciplines without being restricted to mathematics. But I thought mathematics was the most suitable for me. When I imagined myself presenting my results, I felt the most confident when I thought about presenting mathematics. That is why I got into mathematical logic, and my area of expertise became proof complexity.

## Proof complexity: mathematics from a bird’s-eye view

Mathematical logic, or maybe metamathematics is the right word this time, is a field of mathematics in which we study mathematics itself. Proof complexity is a subfield of mathematical logic in which we study the lengths of mathematical proofs. We establish “mathematical models” of our “natural language” (restricted so that its expressive power is necessary and sufficient to carry out mathematical arguments), “theory,” and “proof.” Note that there are various types of “mathematical models” of proofs, each of which is called a “proof system.” Every time we fix a proof system, we can analyze the lengths of the proofs based on it. That is a brief description of proof complexity. It might sound very abstract, and it is, but it does have applications such as evaluating the capabilities of certain types of rule-based AIs.

## Alone or together: mathematical research has many forms

When we tackle creative work, we have to work a lot alone. But, at the same time, I believe we have to communicate with our colleagues, asking them good questions and being asked good questions in return. This way, communication facilitates natural progress.

The balance is up to the individual. In my case, I love communicating with my colleagues, so I enjoyed my visit to Charles University, meeting many new colleagues. I worked alone during the pandemic and learned that working exclusively alone is not for me. I realized that I had learned a lot from my surroundings when I was an undergraduate student, surrounded physically by my classmates and my colleagues. Thus, I tried to participate in classes at Charles University and stayed at the office as much as possible. It worked great for me.

## Supported by FoPM

I am a FoPM student, which creates the perfect environment for me. FoPM is short for “Forefront Physics and Mathematics Program to Drive Transformation.” It is a program that gathers students from many scientific disciplines. The program provides economic support for its students; actually, it supported the latter part of my visit to Charles University. Moreover, it has a monthly academic meeting where students from various fields can gather and discuss research, which is a great opportunity for students to meet new colleagues. In my case, I got to have great conversations with students intensively working in neuroscience, one of the fields I drifted away from. They told me about the structure of fly brains, as the field is working hard to create a complete map of their circuit structure. A map like this would contribute to understanding computational processes in the brain in general. It was fascinating to hear about the kind of computational models the scientists were considering.

FoPM also organizes career seminars, including international ones for students. The theme often varies, but it includes tips for studying abroad, job hunting, and international communication. Thanks to these seminars, I learned that there was a JSPS program for studying abroad, which I relied on during the first part of my visit.

Not officially, but students are also creating their communities within FoPM where they can exchange information, for example, tips for bureaucracy in academia.

Eitetsu in Vienna standing next to the bust of physicist Erwin Schrodinger © Eitetsu Ken

## Visiting research in the mecca of proof complexity

I spent about one year visiting Charles University in Prague, the center of proof complexity. There are so many influential researchers there. One of them wrote three monographs out of five existing in the field. He was my host, and I learned much from him and his PhD students. I could also meet many renowned, active researchers at the numerous colloquiums and weekly seminars. I encountered many of the latest questions of various subfields of proof complexity. So, I learned what kinds of problems were attracting interest and was able to collaborate with others.

Surprisingly, for me, the culture in the Czech Republic somehow felt similar to Japan. The lack of a culture shock was shocking in some sense. But one unexpected thing was that in Czech, “hello” and “bye” are the same words “ahoj.”

Eitetsu in Prague in the Czech Republic during his year visiting Charles University