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.
Stunning: Droplets of water bead on the head of this blue dragonfly as it slumbers on a leaf
Dragonflies, flies and beetles also take on an unearthly quality as the water droplets form on them.
These remarkable photographs were taken by physiotherapist Miroslaw Swietek at around 3am in the forest next to his home.
Using a torch, the 37-year-old amateur photographer hunts out the motionless bugs in the darkness before setting up his camera and flash just millimetres from them.
Close up: Amateur photographer Miroslaw Swietek captured this common fly slumbering on top of a plant as the water condenses on its body
Bejewelled: This month looks like it has been encrusted in diamonds as it rests on a twig
Mr Swietek said: 'I took up photography as a relaxing hobby two and a half years ago and I particularly like taking pictures of insects and lizards.
'I photograph them in their natural environment in the forest next to my village.
'They all are covered in dew because I go to the forest in the morning at around 3am.
'At 3am to 4am insects are sleepy and taking photos of them is easy, but it is very difficult to find them.
Close up: The insects appear to be completely drenched in water as they rest while the sun is down
Shower time: Mr Swietek gets up at 3am to capture the insects while they are less active in a forest near his home
'You must be very fast taking the photos because the dew quickly disappears.
'It is very satisfying getting a good shot of an insect which I have had to hunt out.
'I have books which help my identify insects but because they are all covered in dew I find it almost impossible to know which types they are.'
Although insects do not 'sleep' in the same sense as humans, they enter a state of torpor where they are virtually immobile and much less sensitive to external stimuli.
Mr Swietek lives with his wife and teenage son in Jaroszow, a village in Poland around 30 miles from the city of Wroclaw.
Clinging on: The amateur photographer searches for the insects using a torch and then sets up his camera and flash right next to them
Hobby: Another dragonfly enjoys an early morning wash. Mr Swietek only took up photography two and a half years ago
Read more: http://www.dailymail.co.uk/sciencetech/article-1260946/The-stunning-pictures-sleeping-insects-covered-early-morning-dew.html#ixzz0jt1PcUAP
Yoshihiko Mitsumatsu
