Professor

Faculty of Computer and Information Sciences

Hosei University, Tokyo, JAPAN

- Teaching Materials (in Japanese)
- Basic materials for our project
- Introduction to Category Theory - computing examples with Haskell, Nippon Hyoron sha (in Japanese)
- Data Structures in Category Theory (provisional, sequel to the book above)
- Transformation Groups for Beginners (in Japanese)
- Lessons in Algebra (Scientist Publishing, out of print)
- Lessons in Algebra - calculate first, Nippon Hyoron sha (in Japanese)

Born in January 1954 in Yachimata-cho, Inba-gun, Chiba Prefecture (currently Yachimata-shi, a famous watermelon production area, mother's family home). When I was in elementary school, I had difficulty in calculating the vertical division, kind of technique that most Japanese children (I am afraid that this is not always the case in the world.) must master. I was reluctant to go through calculation drills while being scolded by my mother. Now, I'm grown up and conviced that there are things that we have to master even if we don't understand the theory, such as multiplication tables and vertical division. One thing that I'm proud is that I, even a child, can understand that memorizing the upper or lower half triangle of the diagonal is enough. In the classroom I refused to recite the multiplication table completely, which led to my getting grade 2 out of full 5 in math, which disappointed my parents. Anyway, I didn't defy the teacher openly. I couldn't compete with serious children in the multiplication table recitation competition. But what happened? I'm unserious about that, and I've never had a problem with that. It means that the way people grow up is not a straight road. By the way, when my daughter was five years old, she used to recite the multiplication table by listening to her elder brother practice the reciting. I think there are quite a few such children. They imitate whatever their elder siblings do. I myself am the eldest son in my family. Therefore, I lost my chance of early learning. No problem!

When I was in elementary school, I was a big fan of Shigeo Nagashima (Yomiuri Giants), the greatest baseball star of Japan (before Shohei Ohtani, an MLB star). I was enthusiastic about his batting average floating game by game. At that time, Nagashima was fighting with Koba (another icon of Hiroshima Carp) for the leading hitter. Leading hitter, you know, has the highest batting average. I wanted to follow the batting average everyday which usually requires arithmetic operation of division. But I didn't like division, so I
made up the table of the reciprocal numbers from 1 to 500, which seemed sufficient to cope with the situation (I know this is not the case with MLB). With this table I was able to calculate averages without using division because
the number of hits divided by at-bats is equivalent to
the number of hits multiplied the reciprocal of at-bats.
What a beatiful world it was!

To get the table of reciprocals,
I took a greedy strategy: guess the reciprocal, multiply by the number in question, and compare the result with 1. Repeating guess, compare, and correct, the guess gradually approached the true square root. At some point, I stopped the procedure and put the recently obtained value in the table. I did't have to * work hard * because it's just a game and fun. Anyway, thanks to my lovely table, the batting average can be calculated solely by multiplication. I spared no effort for the sake of my beloved Nagashima. However, during the summer vacation of the 4th grade of elementary school, when I spent a week in the countryside of Chiba, where my uncle operated a farm, who is a Chunichi Dragons fan. He said to me, "It's easy to calculate the batting average." He taught me how to divide vertically and I absorbed his teaching immediately. What was it that differentiated between my study at school and my study at my uncle's. If you get involved in what you want to do, you will learn it in one shot. Even as a child, I realized that everything in the world was like that. It was an experience that will last a lifetime, both for my own metacognition and for my teaching career. After that, I threw away the beautiful table of reciprocals and my fear of
vertical division. I had been respecting my uncle as a master watermelon maker, but after this experience, I began to respect him even more realizing that he could utilize math in daily life.

When I was in the third year of junior high school, I learned about square roots at school. My father told me that he could show something more interesting, and taught me how to do square root calculations, which looks like a vertical division. As a junior high school student, I was surprised at the beautiful result; the given value was recoverd by squaring the calculated result. At that time I learned proofs in Euclidean geometry. It was fantastic, but I still didn't feel that all areas of math go like Euclid.

As soon as I entered high school, I was shocked by knowing that algebra is also built up with rigorous proofs. During the summer vacation of my first year, I attempted to prove that square root approximations can be calculated with arbitrary precision using the algorithm my father had taught me a year before. By trial and error, I found the reason and wrote several pages of paper (of course, unpublished. It was just an exercise). In retrospect, it was a discussion of coefficient comparison of infinite series expansion, but a high school student, who does not know such a technical framework yet, was able to write the details of the proof with his immature languages. At that time, I was convinced that I could prove the correctness of vertical division, which I struggled with when I was in elementary school, by using the same method in reasoning square root calculation. It was far easier. Naturally, square root calculation had been more difficult. As a result of this summer vacation enthusiasm, I decided to do more mathematics seriously. Dear high school students, why don't you give it a try?

After several years, I got married, and had kids. There was an accident. My computer was turned off (my son did it) while I was writing my thesis. It was a negligible unpappy event during my life with tremendous amount of happiness. Anyway, I submitted my doctoral dissertation. A long story is waiting to follow, but I decide to omit it and conclude my introduction. I just want to say thanks to Minako, my wife, and my whole family.