21st Century COE Program : Mathematics of Nonlinear Structure via Singularity

Characteristics of Mathematics

Mathematics is not just a study of numbers, nor is it simply about calculations.

It is not about applying formulas, either. It can perhaps be better described as “a field of creation through accurate and logical thinking." Mathematics has a long, rich history and continues to grow rapidly. New findings are regularly presented at conferences in and outside of Japan and articles based on such findings are published in mathematics journals in countries around the world.

Mathematics is a very diverse filed. The appendix shows a list of subcategories classified under mathematics by the American Mathematical Society (the list is an excerpt from the AMS's Mathematical Reviews journal). Many of you must be surprised to see so many subfields of mathematics. "Algebra" and "Geometry", for instance, may be familiar subjects of mathematics in high school, but they are divided further into more specified categories in the list, which also includes combinations of subfields, such as "algebraic geometry."

Mathematics is an academic discipline of great depth, with a number of unsolved problems. It has progressed on cumulative contributions from countless mathematicians in the world who have tackled those problems while creating new areas of inquiry.

Some of the subfields in the appendix, such as fluid mechanics, quantum theory, information and telecommunication, and biology, may seem irrelevant to mathematics at first glance. These fields, however, take mathematical approaches to describing and analyzing phenomena under study.

They illustrate how a number of subfields of mathematics have benefited from and evolved through interactions with other disciplines. Mathematics, on the other hand, has made considerable contributions to the advancement of other academic fields. In fact, mathematics is often described as the foundation of scientific studies.

One of the major characteristics of mathematics is its general applicability.

One equation, for instance, can represent a particular phenomenon in physics as well as a certain logic in economics. This general nature of mathematical equations enables unified treatment of diverse phenomena in various academic fields. Furthermore, mathematical theorems are no respecter of age or seniority: any theorem has to be proven through appropriate mathematical procedures whether you are a novice student researcher or an eminent professor of mathematics. And once proven true, mathematical theorems will never be reversed. This "universality" of mathematics is another important feature that allows the discipline to transcend time and space.

21st Century COE Program : Mathematics of Nonlinear Structure via Singularity