A Light that Illuminates the Sea of Mathematical Theory: Teiji Takagi and His Time

A Light that Illuminates the Sea of Mathematical Theory: Teiji Takagi and His Time

In 1875, Japan was in the throes of modernization. As the Edo shogunate collapsed and the new government actively introduced Western culture and learning, mathematics was also rapidly shifting from traditional Japanese mathematics to Western mathematics. Born in this era of change, Teiji Takagi was a mathematician who laid the foundation for pushing Japanese mathematics to a world-class level.

Born in Gifu Prefecture, Takagi excelled in mathematics from an early age, and after graduating from the Daisan High School (the predecessor of today's Kyoto University), he entered Tokyo Imperial University (now the University of Tokyo) in 1894. At that time, the Department of Mathematics at Tokyo Imperial University was introducing Western mathematics, with particular emphasis on French and German mathematics. After graduating from the university and becoming an assistant in 1898, he went to Germany in 1901 to further his studies. There, under the tutelage of the world-renowned mathematician Darfit Hilbert, he achieved important results in the field of algebraic number theory.

After returning to Japan, Takagi became a central figure in the full-fledged development of number theory research in Japan. His greatest achievement was the establishment of a system of number theory called the "theory of analogs. This theory classified the expansions of algebraic integers and analyzed their properties, and was a groundbreaking achievement in number theory. In particular, "Takagi's theory of analogues" was applied to number-theoretic problems such as Fermat's Last Theorem, and had a profound influence on later mathematical circles. His theory was also highly regarded worldwide, and in the 1920s, it was developed into the "Mutual Law" by German mathematician Emil Altin, and came to be established as the foundation of number theory.

However, the period in which he was active was overshadowed by war and turmoil: from the 1920s to the 1940s, Japan's international standing increased, but the country was also thrust into the Sino-Japanese War (1937) and World War II (1941). During this period, the Japanese mathematical community also could not avoid the effects of the war. Mathematical research stagnated, and priority was given to applications in military technology. Nevertheless, as president of the Mathematical Society of Japan, Takagi devoted himself to the development of mathematics and continued to protect the field of education and research.

After the war, Japan sought to recover from the devastation. In 1949, the Imperial University of Tokyo was renamed the University of Tokyo in accordance with the reform of the school system, and the mathematical research system was restructured. In the midst of the postwar turmoil, Takagi also devoted himself to the enlightenment of mathematics, writing "Introduction to Analysis" (1947). This book became widely popular as a textbook for students of mathematics and is still considered one of the most important books in mathematics education in Japan. He also promoted exchanges with the international mathematical community in the postwar period and devoted himself to the revival and development of mathematics in Japan.

In 1960, Teiji Takagi passed away after 85 years. However, the system of number theory he established still lives on as the foundation of modern mathematics. His research continues to illuminate the paths of many mathematicians like a light in the sea of number theory.