What is Mathematical Culture?

Does mathematics in general, and particularly modern mathematics, give people a cold clinical image? And do people avoid it precisely because it is so difficult? To a certain extent, the answer to both these question is yes, with many discoveries today the result of highly complex and abstract research, albeit challenging and exciting at the same time. However, the depth and breadth of such excitement and thrills are not impartial; rather, they are deeply related to the Mathematical Culture that exists in the background.

If a person is enchanted by the appeal of mathematics as a young child and decides to pursue a career in mathematics, he/she may be lucky enough to produce successful results at the end of an arduous period of training. In other words, they may be lucky enough to discover and prove a theorem, or even to establish a new theory. However, compared to rushing towards the cutting-edge (or, as some would say disparagingly, the minor details) of each segment of the field, a great difference arises in mathematical understanding when focusing on taking a broader and more fundamental view of the same topics. This is greatly influenced by the background of Mathematical Culture, an area which I would like to share with you in this article. Additionally, I would like to introduce the activity ENCOUNTERwithMATHEMATICS, whose task it is to foster and disseminate the Mathematical Culture.

Is there only one answer to mathematical questions?

High school students, who aspire to enter university mathematics departments, often say that mathematics is enjoyable since there is only one correct answer and no confusion. Actually, there is also a belief in general that math problems have only one answer. This may be true in the sense that answers are very clear, and certainly it is for simple computational problems. In order to facilitate grading of entrance examinations, there is a trend to create problems that have only one answer. However, generally speaking there are many cases in which problems with more than one answer make better questions. For example, consider a question that asks what kind of triangle is an equilateral triangle. There are at least two correct answers to this question - a triangle with three sides of equal length or a triangle with three angles of equal degree. This is true because a theorem states that equal lengths and equal angles are equivalent conditions. Now, consider a question that asks for a function that is drawn as a straight line. Would it be best to answer with a function that becomes zero after it has been differentiated twice? Or would it be best to answer with a function of degree 1?

Furthermore, a polynomial function of degree 3, which is zero up to the second degree, is the same as a function satisfying f(ax)=a³f(x), where "x" is a variable and "a" is any constant (polynomials that provide such a function are called homogeneous of degree three). The best expression depends on the particular case.

So, is there one answer or two answers to the next question? Or perhaps I should suggest that no answer exists at all! This problem deals with the size of infinite sets. Both ℝ, the set composed of all real numbers, and ℚ, the set composed of all rational numbers, are infinite sets. However, the size of "infinity" in each set is completely different. When taking ℚ away from ℝ and then placing the remainder in a row on the real line, it is true that irrational numbers are missing. However, rational numbers can be distributed in any given place. This means that ℚ exists deep within ℝ. Conversely, when aligning ℕ, the set of all integers, along the real line, the integers exist discretely or extremely infrequently when compared to the rational numbers. Still, both ℕ and ℚ have the same size as infinite sets. For example, assume that positive rational numbers are presented as irreducible fractions. When numbers are assigned in order, beginning from the smallest sum of numerators and denominators, a response of 1-to-1 occurs between the positive rational numbers and the natural numbers (assigned numbers). However, the same method cannot be used to assign numbers to real numbers. This is because there is a much greater amount of real numbers. Even the closed interval of [0,1] is the same size as ℝ. The symbol # is often used to express the size of sets. In the case of a finite set A, #A represents the number of elements. Similarly, ♯φ=0 is used for the empty set φ. The process explained above is there for expressed as ♯ℕ=♯ℚ＜♯ℝ.

So, are ♯ℚ and ♯ℝ infinities that exist adjacent to each other? Let's look at this question in more detail. If they are not adjacent, it must mean that an infinite set X exists for which ♯ℚ＜♯X＜♯ℝ. If the problem is formulated in this way, many mathematicians will respond that there is no answer (actually, it depends on their position). What in the world does this mean?

In terms of ZF or ZFC, which are axiomatic systems (or logical frameworks for discussion) that are currently used as standards in modern mathematics, this problem is by no means unanswerable. More specifically, it has been proven that no contradiction occurs in axiomatic systems regardless of which answer, namely Yes or No, is given (Kurt Gödel 1940, Paul Cohen 1963). This discovery caused quite a shock to mathematicians, as well as to many parts of the rest of the world, in the middle of the 20th century.

My argument has gone off on a tangent, however. In the case of proof problems, it is inherently clear that more than one method often exists for proving a certain fact. However, depending on the type of problem the question of whether multiple answers exist for a mathematical problem may be meaningless. Normally, it is best to assume that many answers exist. Indeed, the ability to state a mathematical phenomenon in many different ways increases the richness of that phenomenon. The mathematics that provides an understanding of this phenomen is also enriched (as are the humans who have understood it). This fosters richness and an organic nature in mathematics.

The path of 20th century mathematics

Starting in the second half of the 20th century, the appreciation of the beauty and richness of 19-th century mathematics started to develop. Thanks to the discovery of calculus in the second half of the 17th century, the stagnation of mathematics experienced a sudden spurt in growth and has made great strides in the last 350 years. From today's perspective, the stringency of logic was quite dubious until the 19th century. However, at the very least, a handful of great geniuses served as leaders who advanced mathematics in the right direction, constructing theory closely related to mathematical objects and avoiding rampant fragmentation of mathematics into separate fields.

Then, upon entering the 20th century axiomatism was introduced. Although, logical stringency began to thoroughly permeate all areas of mathematics, the effect of growth was to split the field into different directions. As a result, internal advancements within each field became conspicuous. Ultimately, it seems that mathematics fragmented into many fields and grew at the same time, while losing its organic nature. In particular, a mathematics group called Bourbaki appeared in France. The group worked to safely preserve the important advances of modern mathematics of axiomatization and abstraction (mathematical principles) for future generations to the greatest extent possible. Moreover, a pronounced trend among western universities after WWII was that mathematics and physics existed within a framework of separate undergraduate schools and departments. Before this, mathematics and physics had been indivisible. On the other hand, Russia did not divide mathematics and physics to the same degree as western countries. At that time, Japan was working feverishly to catch up with (and even surpass) western culture, so the system of axiomatization and segmentation was readily accepted. In terms of general mathematics in Japan, it was natural to attempt to reach the latest advances in each field as quickly as possible.

As an adherent of western culture, we first became aware of the gap with Russia in the last quarter of the 20th century. The mathematics education that I received at universities (graduate schools) from the 1970s to 1980s was segmented into different fields and was admittedly composed of a curriculum full of abstract theory. The same situation continues to stubbornly exist at almost all universities today. Of course, there are an infinite number of different study methods and teaching methods depending on the individual mathematician. One example is Mr. Tatsuru Takakura (a colleague of mine in the Department of Mathematics). Takakura is 7 years younger than me, but studied almost the same curriculum at the same university. Even so, Takakura has already recovered from his segmented and abstract mathematics education as a student and has gone on to acquire an organic and beautiful mathematical perspective.

Trends in France

In France there is the Bourbaki Seminar, which was started long ago by the aforementioned mathematician group Bourbaki. Recent outstanding successes from the world of mathematics are introduced by mathematicians who had nothing to do with the mathematician who developed the theory itself. In addition, the Ecole Normale Supérieure de Lyon (which I was affiliated with during the mid 1990s) began a series of meetings at the end of the 1980s known as Les Rencontres Mathématiques (Mathematical Get Togethers). At these meetings, experts and non-expert mathematicians and also young researchers spent a day-and-a-half discussing designated themes. These meetings were unlike anything in Japan and were extremely well received by audiences. It is astounding that the French put so much effort into cultivating the organic nature of mathematics!

All of the students that I saw at the Ecole Normale in Lyon had a solid understanding of classical analytical mechanics. I realized that they naturally absorbed and understood the origin of important problems in modern mathematics. I reached the conclusion that French mathematicians had developed their bold approach to preserving mathematics through abstraction because in the past they had already been familiar with and had substantially contributed to the other side, namely that of the organic and rich Mathematical Culture.

ENCOUNTERwithMATHEMATICS

When I returned to Japan in the autumn of 1995, I wanted to create get togethers in Japan similar to the Bourbaki Seminar and Les Rencontres Mathématiques. Upon consulting with many individuals, the general opinion seemed to be that, unfortunately, get togethers like the Bourbaki Seminar were still not possible in Japan. If such meetings were to take place, it would have been the responsibility of The Mathematical Society of Japan. Conversely, people seemed to feel that Rencontres Mathématiques could be held, although it would be difficult. Therefore, beginning from November 1996, I started a get together named ENCOUNTERwithMATHEMATICS in the Department of Mathematics of Chuo University. I try to hold meetings four times a year. For meetings, we decide a theme and recruit a lecturer for an audience of non-expert professional mathematicians. A large number of expert mathematicians also join the audience, but we have our lecturers proceed without too much worry. These meetings discuss major academic trends and explicit topics that cannot be heard at normal lectures or research meetings. This makes the meetings very meaningful for young graduate students, who are close to becoming experts. Each time we have a large audience from all over Japan (please visit the webpage http://www.math.chuo-u.ac.jp/ENCwMATH/).

Back in Lyon, Mr. Etienee Ghys (the founder of Les Rencontres Mathématiques) has expanded his activities even further. Unfortunately, the get togethers were dissolved in order to form a better organization. Whenever I meet Etienne somewhere in the world, I ask him how to have these mathematical activities carried forward by others. His answer is always the same - "If your activities are progressing well, then keep at it, even if it is tough." Following his advice, we are continuing our activities through cooperation with graduate students and parties both inside and outside the Department of Mathematics. At the end of January 2012, I will visit Lyon again after many years to pursue my research. Although no ENCOUNTERwithMATHEMATICS will be held until autumn, I will start them once again in September.