1. Mathematics: A field receiving little recognition as culture

Mathematics is a subject studied from junior high school. Even so, many people dislike mathematics and it is an academic field that receives little understanding from the general public. “I had no problem with arithmetic in elementary school, but I couldn’t deal with factorization from junior high school.”—Such experience has led many people to be wary of mathematics. I also sometimes hear comments which indicate a complete lack of interest in mathematics. For example, some people don’t see any meaning in being able to solve quadratic equations. After graduating high school, many people rarely have a chance to use the algebra, trigonometric functions, exponential functions, vectors, matrices, and infinitesimal calculus which they learned in high school. Furthermore, I am sometimes asked how mathematics benefits society. Compared to music, art and literature, it is difficult to see how mathematics contributes to society. In that respect, mathematics receives little recognition as culture and tends to be avoided. In this article, I will discuss the significance of mathematics while considering why we must study the subject. I will also discuss research methods for mathematics, particularly research on the partial differential equations which is my specialty.

2. Mathematics is culture

Imagine a world without music, art or literature. How would mankind evolve (or devolve) in such a world? I can envision people being driven by desire; a world of endless conflicts without the concept of respecting human life. The idea of valuing other people is the most fundamental and essential aspect of mankind. Art and literature are the intellectual creative activities of mankind, producing countless works which should be preserved for future generations. Art can be described as the human activity which best embodies the trends and values of the public at a certain point in time. In actuality, mathematical researchers are constantly pursuing mathematical beauty. Mathematical research is a creative human activity. From this perspective, mathematical research is fundamentally the same as the activities of artists.

In other words, mathematics itself is a form of culture. Its role goes beyond simply providing knowledge for scientific technology; indeed, it is an intellectual creative activity and an essential art for mankind. Although it is extremely difficult to realize (mathematical) beauty, this is the unwavering goal of researchers. We constantly seek to supplement incomplete knowledge, sometimes taking it upon ourselves to create the required tools and incorporate them into proof in the form of a lemma. When we reach an impasse, we maintain our concentration until we are struck with an idea, sometimes even engaging in frenetic behavior. As a result, mathematics researchers often seem eccentric to the general public. Let’s take a moment to consider the theorem that there are infinitely many prime numbers. The term prime number refers to a natural number greater than 1 that has no positive divisors other than 1 and itself (2, 3, 5, 7, 11, 13, …). This theorem has been well known since the era of ancient Greece and was already proven in Euclid’s famous book Elements which was written around 300 BC. At first glance, prime numbers appear to be in random order. However, the distribution of these numbers was finally clarified in the 19th century. “OK, so there are infinitely many prime numbers and you understand their distribution. So what?”—I have no answer to such a question! This understanding possesses its own beauty and was not conceived to be of benefit to society.

Such mathematical research does not pursue theory that is immediately useful. Instead, in this kind of research, mathematical researchers themselves select a subject having a sense of inherent beauty and express that beauty as theorem. Such research requires that researchers have demonstration ability and computing power for logically considering matters. In some cases, proof by contradiction is used. In other words, when it is difficult to directly prove a proposition to be substantiated, an existing theory can be contradicted by deducing while denying the conclusion. This means that the conclusion of the original proposition is correct. In this demonstration of logic, one must become acclimated to denial of the proposition. For example, let’s attempt to deny the proposition that “all people are inherently good.” I will give the answer to this proposition later. In many cases, the researcher considers denial of the proposition as part of logical demonstration for proving difficult problems. Proof is obtained through a thought process which is difficult yet enjoyable. In my opinion, the term “theorem” is not appropriate when referring to a mathematical consequence obtained through calculation only. I believe that “theorem” should refer to proposition which has been proven through a combination of logic and calculation. The answer to the aforementioned proposition is that some evil people exist in the world, an answer which can be expressed in a variety of ways.

3. The Kirchhoff equations—equations filled with riddles

I have conducted research specializing in the mathematical field of mathematical analysis. In particular, I have focused on asymptotic behavior and the existence of solutions for wave equations. Recently, I have been working with the Kirchoff equations. I would now like to discuss my research as an introduction to one element of mathematical research. These equations were announced in 1883 in the famous work Mathematical Physics, written in German by Kirchhoff. These equations are known as the second-order hyperbolic partial differential equations, proposing an elastic string vibration for which the transversal motion is greater than the longitudinal motion. As such, the Kirchhoff equations resemble the wave equations, yet they are still filled with riddles.

In 1940, the Russian mathematician S. Bernstein finally proved the existence of a time global solution for real analytic solutions to the Kirchhoff equations. Since then, despite opportunities for extending the real analytic class, a satisfactory solution has not yet been found. On a related note, researchers have yet to find the existence of a time global solutions for the Navier-Stokes equations, which are fundamental equations in fluid dynamics. This unsolved problem has gained fame as a Millennium prize problem for which the Clay Mathematics Institute offered a cash prize. Some mathematicians say that the Kirchhoff equations should also be designated as a Millennium prize problem.

However, unlike the Navier-Stokes equations, some scholars doubt whether the Kirchhoff equations can be classified as a practical problem. The Kirchhoff equations possess a conserved quantity known as the law of energy conservation. The greater the conserved quantity in an equation gets, the easier it is to analyze that equation. Unfortunately, just like the wave equations, the Kirchhoff equations have only a single law of energy conservation. I am currently working on this kind of unsolved problem and have come up with various ideas. The proof relies on proof by contradiction. I will announce my solution once the proof has been completed.

4. Mathematics as language

As discussed previously, there are many people who dislike mathematics. On the other hand, there are numerous people who love mathematics while recognizing one field forming the foundation of natural sciences. At the Department of Mathematics, there are freshmen who want to be a teacher or get the job in IT industry. Most of them have the desire to get the job related to Mathematics. At the Department of Mathematics in the Faculty of Science and Engineering, we have composed a curriculum based on the broad subjects of algebra, geometry, and mathematical analysis. We also offer courses in mathematical statistics and computer mathematics. Instruction focuses on instilling students with logical thinking ability and high-level mathematical knowledge.

There is more to mathematics than simply performing correct calculations. Indeed, a highly beneficial way to understand mathematics is to learn mathematical theorems based on a view of the subject as a language for describing natural phenomena and mathematical economic theory. As I stated earlier, the most important thing for studying mathematics is the desire to acquire logical thinking skills and demonstration ability. We professors at the Department of Mathematics hope that students will recognize the culture of mathematics. In closing, it is my heartfelt wish that many young people with the desire to actively obtain high-level mathematical knowledge will enroll in the Department of Mathematics at Chuo University.