ken atchity door to door: YOSHI MATHEMATICUS

YOSHI MATHEMATICUS

Mathematical Culture and ENCOUNTER with MATHEMATICS

Yoshihiko Mitsumatsu
Professor of Geometry (Specializing in Differential Topology), Dynamic Systems and Mathematical Physics,
Faculty of Science and Engineering, Chuo University

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.

Yoshihiko Mitsumatsu Professor of Geometry (Specializing in Differential Topology), Dynamic Systems and Mathematical Physics, Faculty of Science and Engineering, Chuo University: Born in Tokyo in 1957. Graduated from the Department of Mathematics, University of Tokyo, in 1980.
In 1985, completed the Doctoral Program of the Graduate School of Science, University of Tokyo. Holds a PhD in science (University of Tokyo, 1985).
After serving as a Full-Time Instructor and Assistant Professor at the Faculty of Science and Engineering, Chuo University, assumed his current position in 1999. Has served as the Director of ENCOUNTERwithMATHEMATICS since 1996.
Current research focuses mainly on dynamic differential topology, such as foliations and contact structures. Also conducts research on theoretical mathematical physics.
His written work, Topology of 3D Contact Structures (The Mathematical Society of Japan; Mathematical Memoirs Vol. 1, 2001), contains many illustrations and is accessible to the non-specialist, even though it deals with extremely abstract and difficult material.
Among the different fields of geometry, contact structure theory in particular lends itself to geometric intuition. Conversely, it is a complex field in which even experts often make mistakes.

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Yoga Gives Back

Written by Kayoko Mitsumatsu

From Posture To Power: Microcredit Changes Lives On Both Sides of the Ocean

We can talk about making change. But affecting real change in the world comes from impacting the lives of individuals. It comes from empowering people to be independent and able to make a living on their own while at the same time being fully integrated in our interdependent global community. Sometimes it takes something that seems to be radical to make this change, but these shifts can be deceptively small. They can be even as small (or as large) as the cost of a Los Angeles Yoga class.

How could this be? In the practice of microcredit, lending seemingly small amounts of money to small-scale entrepreneurs allows them to succeed and change their circumstances. This is the fundamental basis of microcredit’s transformative power; it’s a revolutionary concept in finance. This practice shifts the emphasis from financial transactions being focused only about large loans and large scale businesses, which are out of reach of many one-person operations, institutions and organizations – to lending practices that actually make a positive difference in a person’s life. Microcredit practices based on social justice and empowerment operate through a combination of small-scale loans based on trust and community support, with structured repayment plans and programs that encourage savings. In these situations, people actually have the opportunity to solidify long-term change in their lives.

Dr. Muhammad Yunus and the Yoga Gives Back Team: Joel Bender, Leslie Hendry, Dr. Muhammad Yunus and Kayoko Mitsumatsu (left-right)

In 2006, economist Dr. Muhammad Yunus and the institution he founded, the Grameen Bank, were awarded the Nobel Peace Prize. This award was in recognition of their revolutionary efforts to create this micro-credit model, support sustainable economic practices, and provide an alternative to what were exploitive norms. When a person is caught in a cycle of poverty, they may become trapped in the system, particularly since large banking institutions have historically ignored people without assets or credit histories, who are just starting out and who may not have already proven themselves. Most banks prefer to lend large sums of money that represent a greater profit margin and are set up only for large loans in order to make a profit themselves. And even making it more challenging, people at the bottom of the economic scale worldwide have had to rely on moneylenders charging usurious terms and interest rates to obtain funds to plant the next crop, buy inventory of new equipment, or upgrade a business in any way. Institutions such as the Grameen Bank and their affiliates are offering another approach. It’s one that has significantly impacted millions of people.

As Dr. Yunus described in an exclusive interview with Yoga Gives Back in May, 2009, “We started in 1976, and today within Grameen Bank in Bangladesh, eight million people borrow micro loans, ninety-seven percent of them women.” This represents an enormous potential for change and empowerment that just did not exist on a large scale in this form prior to 1976.

There is a social cycle in economics, and Dr. Yunus explained this: “Poverty is not inside the person. Poverty is forced from outside. Poor people are as capable [a] human being as anybody else. They have the same power and energy, same potential. But they are denied such facilities. They cannot express themselves. Money is one of the most important items missing in their lives. You need a dollar to catch a dollar. You cannot catch a dollar with an empty hand. So, that first dollar, nobody has given to her. So she remains dependent on someone else. She has to be hired, to slave; she has to work to make food available on the table. What we have done, we have put that first dollar into her hands as a banking proposition. Conventional banks never come to the poor people. They go to rich people to make more money. The person who does not have a dollar – she cannot get it. So we did it. We did it in a way that is affordable to her. Because we do not insist on collateral. There is no collateral.”

While that may seem like a radical concept, the Grameen Bank has proven that their success is much more than ideological. Dr Yunus cited statistics for us, “The repayment rate is almost 100%, without collateral or anything. This is the strength of microcredit which is now spread all over the world. There are by now at least 150 million borrowers in the world today. Bu it is still very small comparing to the need. We need more microcredit organizations in the world so that more people can be reached.”

Yoga Gives Back is a 501(c)(3) nonprofit organization begun in LA, with the goal to provide a way for Western Yoga practitioners to give back to India, the birthplace of Yoga, which has given us so much. We’re heeding Dr. Yunus’ call to reach more people through expanded opportunities for and awareness of microcredit and have targeted India and Indian families, women, and would-be entrepreneurs, who would be otherwise stuck in the cycle of poverty that Dr. Yunus described. Yoga Gives Back supports microfinancing operations in India, through the Grameen Foundation, headquartered in Washington, DC. Donations go directly to Grameen Koota, Bangalore, South India and our YGB awareness campaign focuses on the real stories of how poor women are changing their lives with small loans from Grameen Koota.

Ms. Vinatha M Reddy

Grameen Koota was started by visionary teacher and philanthropist, Ms. Vinatha M Reddy (pictured, above) in 1999, who learned about Nobel Peace Recipient Dr Muhammad Yunus’ microfinancing and immediately began exploring ways of replicating his work in her hometown. In 2000, she started providing small microcredit loans, averaging $25.00 a month, to fifty local women. Today, Grameen Koota’s work is reaching out to over 280,000 women in Bangalore, Mysore and many areas of Karnataka state, in South India, making real impact on these women’s lives as well as on the local economy.

Yoga Gives Back

The Los Angeles-based Yoga Gives Back has a worldwide reach and began out of the inspiration and dedication of my Yoga mentor Joel Bender and me. Our group has suddenly started growing almost daily and our mission is shared by a quickly expanding and loving Yoga community.

When I was growing up in Japan, which is a very middle-class society, I had never experienced or seen any real poverty, especially growing up in Tokyo in 1960s and 1970s, when Japan's post-war economic miracle was taking place. Then, I had a chance to live in Brazil in the late 1970s for two years, where as a teenager, I saw the divide between poverty and wealth, the reality of the real slums, kids on the streets begging every time you stopped the car at the traffic light, all very much like India. Social injustice hit me hard.

Once I started Yoga practice and feeling my life being so enriched by it , I felt a need to use my resources to help others. I was producing a documentary on Social Entrepreneurship and learned about microfinancing. I shared this information with Joel [featured in the teacher profile in this month’s issue of LA YOGA] who had been looking for the way he could give back to India since his first visit there – so we teamed up.

When I think about the growth of Yoga Gives Back thus far, I give a lot of credit to the entrepreneurship and volunteer spirit of Americans, willing to embrace and support a new grass roots campaign like ours.

Kayoko Mitsumatsu, co-founder of Yoga Gives Back, is a documentary filmmaker whose films deal with social justice issues, including First They Killed My Father a memoir of a Cambodia’s Killing Fields’ survivor.

A Life Changing Story of Microcredit: Jayashree

Jayashree

Jayashree lives in the suburbs of Bangalore and is married and the mother of two children. We visited her at her humble dirt floor one room home for the first time in 2007. We were touched by her warm smile and most generous hospitality, serving us hot tea and cookies.

Jayashree was born into a very poor family in Bangalore. When she was a child, her family’s financial situation became so severe that she had to drop out of school and work at a garment factory. She toiled throughout her childhood and after she got married at fifteen, her meager earnings had to support her ailing in-laws. She had two children. Her husband’s rickshaw business was not enough to pay her sons’ education or secure food for the family.

In 2007, Jayashree learned from her neighbors about Grameen Koota’s microcredit program and applied for her first loan. She was granted a first loan of 7,000 rupees ($175), and immediately bought her husband’s auto rickshaw. Owning their own rickshaw business increased their income significantly.

Jayashree was able to pay back her entire loan in one year and became eligible for her second loan in 2008. Having proven her diligence, her second loan amount doubled to 15,000 rupees ($375). This time, she bought a sewing machine and started stitching bags. The family’s earnings have now tripled and she is earning 150 rupees ($3.75) a day. In addition, she opened a small shop next to her house to sell her products which is earning her an additional 100 rupees ($2.50) a day.

She is sending her elder son to a medical school. Her son dreams of becoming a doctor and thanks his mother’s business for providing his education, their own home and daily food.

To participate in a Yoga practice where the proceeds support Yoga Gives Back and directly impact the transformation of not only your own life, but that of someone in India, start your Thanksgiving week at Yogaglo, Sunday, November 22, 10:00 A.M. - 12 noon. Some of the teachers include John Sahakian, Felicia Tomasko, Alexandra Aiken and Ally Hamilton; more to be announced. Yogaglo.com; yogagivesback.com.

For more information on Yoga Gives Back or to host a Yoga Gives Back class at your own studio or location, visit: yogagivesback.org or contact yogagivesback@earthlink.net.