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Culture and Education

What Do Mathematicians Do?

Yasunari Nagai
Professor, Faculty of Science and Engineering, Waseda University

In a world where matters related to mathematics rarely make it into the news, the recent talk surrounding the abc conjecture appearing in newspapers is still fresh in our memory. It is certainly out of the ordinary even within the community of mathematicians that the peer review of Mochizuki's papers took five years since its preprint (an unfinalized draft) was made available in 2012. Judging from what I am hearing, it is very likely that his work is extraordinary in many ways including the volume of the papers and the grandness of the new theory. However, since I am not an expert in this subject, I unfortunately cannot add more to the conversation than to offer this simple impression.

Yet, how newspapers and other media outlets have covered the subject does bother me . I find it somewhat disappointing that the focus has entirely been on how difficult it is to solve the abc conjecture, on how complicated Mochizuki's papers are, and on how brilliant Mochizuki is with supporting facts about his personal career and history that paint him as a typical mathematical genius. This is what I came away with from the coverage: First and foremost, a mathematical problem exists, and mathematicians compete to see who can solve it. Mathematicians are a strange lot—they alienate themselves from society and devote their entire lives to solving mathematical problems, and the value of their work is measured by the difficulty of the problems they solve. It seems to me that such banal perspective is too often behind such stories.

In my opinion, this understanding is only half correct. Perhaps for many people, mathematics is indeed all about solving mathematical problems. Since we start learning basic arithmetic in elementary school, all the way to when we study for our university entrance exams, or even when studying college-level mathematics, the focus is always on the mathematical manipulations for solving problems. If this is how we experience mathematics for a stretch of over 10 years, it is only natural that we view mathematics as merely a problem-solving contest. However, as is the case with the abc conjecture or any other mathematical conjectures, such conjectures and unresolved problems do not just fall from the sky without anyone making an effort. All major conjectures or unresolved problems have been discovered by somebody. How did they discover such problems? Were they just suddenly struck with these ideas without any logical train of thought?

Actually it is extremely difficult for a mathematical conjecture to remain a problem to be solved. This is because the conjecture has to be something that cannot be solved right away, nor can be immediately disproven, yet presented in a way that will convince many people of its validity. It cannot be a mathematical “conjecture” if the proposer cannot give examples that make it seem like it holds true or provide proof for special cases under specific conditions, though it cannot be completely proven. Mathematicians call this type of mathematical discussion that gives validity to conjectures as "evidence."

The fact that evidence is required in order to raise a mathematical conjecture or a problem means discovery of a conjecture or a problem depends on discovering the evidence. And this is also not something that can be done by just anyone. A certain kind of mathematical ability or effort—such as an extraordinary ability to perform calculations, exceptional intuition, or many years of diligent mathematical inquiry—is necessary in order to discover mathematical problems. It is well-known that mathematicians like Euler and Gauss possessed an almost superhuman ability to carry out calculations, and discovered new mathematical theorems based on evidence derived from colossal calculations. Modern mathematicians often aim to make new mathematical discoveries with the help of computers. To discover a mathematical truth that no one has ever noticed before is something that brings altogether different kind of happiness to a mathematician than solving a problem that already exists.

On second thought, solving mathematical problems and discovering mathematical problems are two sides of the same coin. When we solve a mathematical problem that looks complex at first glance, we most often repeat the process of replacing the problem with other problems and tying them together, and boil it down to a problem that can be easily solved. This is the same approach one takes when solving problems that appear on university entrance exams. In other words, even when we are trying to solve mathematical problems, we are constantly creating new mathematical problems in order to find a solution. In that sense, it is not an exaggeration to say that mathematicians are spending most of their research time creating mathematical problems. For those who make a living studying mathematics, the success of their research solely depends on how good of a problem they can find. Once found, a good problem will be solved more or less “naturally.” The creativity of a mathematician is fully exhibited when a good problem is found, or when a new problem is structured in an attempt to solve a problem that could not be solved before.

This may sound like a trite statement, but I believe it's fitting to describe the creative efforts of mathematics as an artistic pursuit. For example, if we compare it to music, the task of discovering mathematical problems can be likened to the job of a composer. When describing some of the greatest masterpieces in history, composers are often said to have received divine inspiration. The discovery of mathematical problems can be described in a similar way. Since mathematics is a sort of logical game played amid a defined set of rules, it may seem that all meaningful questions and their conclusions are fixed from the very beginning. I view mathematical discoveries as an effort to dig up theorems that have always existed, but have never been noticed by anybody because they have been buried under land and sea.

When you hear stories about mathematics, I hope you also give thought to the people who found the problems and in what ways they have made such mathematical discoveries. That is where the magnificent romance of mathematics lies, different from acrobatically solving questions with tools that are already known.

Yasunari Nagai
Professor, Faculty of Science and Engineering, Waseda University

In 2005, Yasunari Nagai graduated from the Graduate School of Mathematical Sciences at the University of Tokyo where he obtained a Ph.D. in mathematical sciences.
Before taking his post as a full-time lecturer in the Faculty of Science and Engineering at Waseda University in 2011, he worked as a research fellow at the Japan Society for the Promotion of Science, a researcher at the Korea Institute for Advanced Study, a researcher at the Institute of Mathematics at the Johannes Gutenberg University Mainz, and a temporary assistant professor at the University of Tokyo, Graduate School of Mathematical Sciences.
He became an assistant professor in the same faculty at Waseda University, and later a professor in 2017.