Chinese Philosophy, Art, Life, and Mathematics
Preface
A mathematician might think I am writing math notes. A philosopher might
think I am advocating Confucianism or Taoism. A businessman might think I am
posting my ads. The title Chinese Philosophy, Art, Life, and Mathematics
sounds too general and too broad. This booklet mainly displays my attitude
towards mathematics. However, since the booklet can apply to all kinds of
fields, I hate to fill it with stiff rules, but prefer instead to provide simple
stories. You may like some of my thoughts, and you may disagree with others.
Perhaps you agree with part of my belief; perhaps you totally disagree. If you
are moved to say something about it, it would be my great honor to be your first
audience.
Li-Chung Wang
Chicago, Illinois
September 1987
1.
The best thing is to have no math and just keep our mind open. Confucius said
nature runs four seasons and makes creatures grow but nature never says
anything. The second best thing is to live with a mathematician instead of
learning math from him. It is more enjoyable to appreciate a piece of art than
to be a piece of art. If unfortunately we have to study math on our own, we like
to have a map at hand and know where we are. If we do not have any sense of
direction and just follow the logical steps of an argument, modern mathematics can be an enormous
mess.
2.
In 1984, I met Professor Lo-geng Hua. He told me he was surprised by my
interest in books about abstract math. I agree that it was difficult to study
math by this approach, but to memorize abstract definitions is a fun challenge
for me. If we could study in a more familiar field, we would have more concrete
resources to work with. Conversely, if we study in a more abstract field, the
direction becomes vague and the properties becomes limited. Some of the abstract
definitions have only one meaningful and concrete example. In these cases, we
should abandon the camouflage of the abstraction and study the concrete one
directly. I told professor Hua that all the modern math books are written as
abstractly as possible. Maybe it is a fad. Professor Hua sighed and made no
comment. Later on I became more selective when choosing which math books to
read.
3.
An introduction to a topic requires a direct contact with the main idea, even
if it is a very vague contact. We want to see the picture of the entire body,
not just a hand or a leg. If the idea is located on the peak of a mountain, and
it is hard for us to walk there, then let us take a helicopter in order to
explore.
If we cannot see the goal, then we do not know where we are going. After
covering a short distance, our guide will not automatically tell us what the
entire journey is all about. When one introduces a topic, one may place too much
emphasis on introductory basics instead of the big picture. These basics are the
ticket which enables us to study a subject. However, it is improper for an
introduction to a movie discussion to mention nothing but where the ticket
window is.
The most important aspect of an introduction is the goal. When we see our
goal, we see our direction: maybe there are many ways to approach a subject;
maybe we will never have the opportunities to go through the details, but at
least we have some idea of the subject if it is introduced with the goal in
mind.
A good introduction requires no experience or training. Ears are the only
qualification for music, and common sense for math
4.
I love birds.
When they come,
They always bring with them melodious songs.
When they stand beside me,
They are brave as soldiers,
Meek as lambs,
And lovely as flowers.
A photographer cannot capture their delicate souls.
A painter cannot express their vigor.
Only the eyes of our minds can appreciate their perpetual beauty.
5.
Very often when we see a statement that we cannot identify as true, we might
just assume it is true, and thereby suffer the consequences. We should try our
best to avoid quoting such axioms. The less often we quote something uncertain
in our theory, the higher its quality will be. One rat dropping can destroy an
entire pot of soup. For example, if we can use finite steps to reach a
conclusion, we should never prove its validity by the axiom of infinity.
6.
Sometimes we are not sure about the existence of a function, but we use an
axiom to assume its existence. Many people believe that in this way we plug the
hole of the theory as if the theory originally had a leak. Other people believe
that the theory is not completely perfect if we quote a dubious axiom, but in
terms of the difficult of understanding, we still remain at the first level of
hell,
provided we stick to quoting the same axioms even if many times. These are all
false concepts to evaluate the quality of a theory. My point is that every time
one quotes the same dubious axiom, one leads us to the next level of hell. For
example, when one quote such an axiom three times, one actually leads us to the
third level of hell, which is worse than the second. If one quotes a dubious
axiom indiscriminately in a theory, one leads us perpetually to further descend
in hell. One should not construct a function by an axiom if one could do it by
an exact formula. The best procedure is to keep the number of times we quote a
dubious axiom to a minimum. This is the reason why Weiner's original proof of
general Tauberian theorem is clear and valuable but others’ proofs are not: The
others unnecessarily quote the Hahn-Banach theorem (which is based on the axiom
of choice) too many times.
7.
It is said that in ancient times there was a great Chinese painter who
excelled at drawing dragons. He drew his dragons in a special order: the body
first, the head second, and the eyes the last. Since he drew dragons so
lifelike, once he finished drawing their eyes, the dragons on the paper would
fly into the sky. No wonder there is a saying: eyes are the window of the soul.
Thus, we should clarify our theme as the painter animated his dragon by drawing
the eyes.
This story teaches us that when we study a subject, we should grasp the key
point. Otherwise, we might miss its real value, hence the entirety.
Our brains are quite different from computer files because we can learn,
classify, judge, deduct, discover, and invent. However, we are also have some
weakness: If a book does not express ideas in a manner conducive to learning, we
may spend a time studying, but gain very little. Therefore, a book- especially
one on mathematics- should not be written as a pile of files.
Some author love to put ten to twenty equivalent statements
together to show rich relations. From statement seven to statement three, there
might be a much simpler and more direct proof. However, if we follow the
author's logical
reasoning, we may have to climb giant mountain (lengthy theorems) only to
accomplish a clumsy proof and thereby blur the key point.
Consequently, when we write about a topic, we must determine which position
we should take, which lenses we should use and how we should weave the lights
and shadows in order to accentuate the main theme.
8.
Mathematics cannot separate from philosophy. If one tries to separate them, one may
have to waste numerous pages in expressing a simple and direct intuition. The
concepts of pasting, category theory, Riemann surfaces, etc. can be easily
understood philosophically, but they become awkward when one tries to describe
them exclusively in mathematical symbols. If one has wings to fly anywhere one
desires, why should one struggle like an earthworm digging a tunnel through the
center of earth?
9.
Studying classical math is like fighting in an ancient war. There are too
many computations, and we, like ancient soldiers, have swords as our only
weapons, the power of which is very limited. The battlefield is vast. Every
incident involves multiple actions and contributes to great epics. In contrast,
modern math is like modern war, focuses on the method. These methods are similar
to a nuclear bomb that only targets the enemy's military bases. The power and
precision of the method resolve the war quickly.
10.
A good theorem is always simple and natural, not tricky or artificial. After
we fell a large tree, we value its trunk instead of its twigs. If we have to
move sideways or to creep along a tunnel to trace the elusive footprints of some
game, we should think about whether it is worthwhile to continue this search. It
might be a great technical achievement to spend tremendous energy, money and
equipment capturing a one-thousandth of a second's existence of some material.
However, an important principle is seldom derived or developed that way. All
human civilization has been a struggle to facilitate our study rather than
complicate it.
11.
Sometimes we criticize a person prematurely and judge him only after a few
superficial encounters. Later, we discover that a person can neither be perfect
nor be devoid of value. Confucius said, "If three of us are walking together, at
least one of the other two is good enough to be my teacher." We may idolize the
person we learn from, but we should not raise him to the level of a god.
12.
There are millions of books in the libraries, but I cannot find a single
statement in any of them that is precisely what I want to say. It is I who can
decide how I shall convey my thoughts into words. Different roads lead in
different directions. It is I who should decide how to clear a path through the
woods.
13.
I had a hard time concentrating on my reading because it was too cold. Our
heat is not on yet. One might think we would save a lot of money this way. In
order to resist the cold, we must eat more. I have also found my roommates take
baths more often than usual, not because they like to be clean, but because they
will feel much warmer if they take a hot shower. We spend the money we save on
heat for food, gas, and water.
14.
When I read a novel, I have a difficult time reading it all in one sitting. I
do not know why. When the story describes some character, an image pops into my
mind and stays with me for a while. Then it disappears and this is a signal to
continue my reading. After a while the plot invites another seemingly unrelated
image to imprint itself in my mind and tell a story of its own. Most of the
story reminds me of my past. The dreamlike vision resembles a bubble: when it
bursts, I am brought to reality again. A book seems to guide me on a imaginary
journey. It allows me to see the wonderland for a while, then I have to prepare
for the next stop.
15.
Oftentimes, when we meet a theorem whose proof is at hand, we just take the
construction for granted, and do not think of it as the most difficult part
contributed by the inventor. We are used to deducting and modifying, but not
used to discovering or inventing. This quick but superficial study of a great
art often neglects the philosophy which supports it from behind. Unfortunately,
we pay little attention to the historical background, the venture, the design,
the spirit and the inspiration of these great thinkers. We just save the
skeletons of their achievements; we do not let our eyes look deeply into the
hidden treasure, and thus might never recognize the real value of what we have
received.
16.
A tea pot and a few cups are easy to make, but how you put them together and
make it look nice is what makes it an art. A few flowers are easy to pick, but
it is how you arrange them in a vase for enjoyment that makes it an art. An
untrained hand might just know the structure, the quantity, the speed, or the
efficiency, but an experienced eye could see the nature as well as the beauty,
and resonate with the mind of the creator.
17.
A good book will not tell us how to think, instead it helps us learn to think
independently. The purpose of true art is not to collect a bunch of precious
flowers to decorate ourselves, but rather to enhance our sensitivity to
appreciate the spiritual side of routine things.
18.
She is so dependable and trustworthy that whenever I have questions, I always
come to her. She never complains and never avoids the trouble. Her words
suddenly disperse all the mysterious fog out of my mind like music from heaven
¾ clear pictures take its place one after another.
She is like a magnet, and I am like a nail: When I get lost, I know where I
should go.
19.
Reading literature changes a person's temperament. According to my
experience, the more a person studies literature, the more proper is his
attitude toward other people. Perhaps you find someone who reads a lot, but
somehow still has defective manners toward others. I would say he might not have
digested what he has learned very well. Perhaps he has other reasons for his
improper manners; perhaps they can be corrected by training. However, it is
certain that the more you learn and digest, the more you understand your fellow
human beings and, consequently, the better you treat them.
20.
Good-heartedness has no price tag. We cannot explain why she gives favor to
others. Her kindness is nourishment for our courage, or a match to ignite our
hopes. She may think she is doing what she should, and never ask if we can
reward her somehow. When this brave angel touches our heart with her magic wand,
there are really not enough words to express our gratitude.
21.
When our territory does not satisfy us, we may explore or even create a new
one. We also like to project our own experiences and consequent viewpoints onto
this new world. As expected, there remain some loose links between the old and
new worlds. The new world might give us a clearer and broader point of view
which might help to solve our problems in the old world, but in most cases, the
answers to the old problems still remain unknown. Repeating a problem a hundred
times, in new or old settings, does not automatically produce a solution.
22.
Xian-zhong Zhang was a great rebel commander at the end of China's Ming
dynasty. It is said that when he was a child, he could direct flies in the air
to land on his classroom desk and order them to fall into line. Then he made the
flies drill just like a sergeant does with his recruits. When we write a
mathematics book, we should follow the example of this military genius and order
all the ideas flying around in our mind to come to attention. Then we could
order the most important ones to step forward, reserve the second rank for later
consideration, and dismiss the least important ones. Perhaps there is abundance
of source materials in our mind, but in order to build them into a directed
powerful force, we should make a wise selection among our ideas, and order them
appropriately.
23.
Which world does a person live in, a picture world or a language world? I
would say a picture world. When a beautiful girl tells me something, it is
easier to look at her face than it is to pay attention to what she says and to
remember it. Her pretty face will be the main picture in my memory, her voice
and expression are just dead decorations. If I hear her voice in my mind, the
words seldom come from my memory of her speech, but emerge irrepressibly from my
complicated feelings.
24.
What is new? During a tornado a house survives. It has lost its roof, doors
and windows, and it will never endure another slight onslaught, but it still
looks like a house, though of a new sort. When we change the wrapping paper of a
gift, it becomes superficially new. When one adopts a new name which has a music
ring, one may seem to be a new person. If a carton of orange juice has been
diluted and tastes like water, then it is new. When we select our viewpoints,
change our settings, and add our special style, then we make the appearance of
an object seem new, but its content really isn't. These changes only make it
difficult for us to recognize an object's origin and essence. Facing such
difficulties, we may wonder whether we feel more comfortable living in this new
kind of world. Does it confuse us more? Or does it make us understand things
better? If not, then there is nothing new at all except for the time. We waste a
lot of time.
25.
When we discuss a subject, we might study it from different perspectives, or
focus on a small part of it and study its details. We should always remember
their connection to the core of the subject when we restrict ourselves to
details. Although a hand or foot has its important function for the whole body,
its meaning and value shrink to nothing when it has been cut or separated from
the body.
26.
A broad view of countryside stimulates my mind; a piece of chalk and a
blackboard make my thoughts flow. When I write on the blackboard, I feel like a
farmer cultivating fields. It seems there are tremendous treasures buried in the
blackboard before the chalk in my hand touches it.
27.
Writing outdoors is more comfortable than writing indoors, even though it is
somewhat cool for me now. I sit at the landing of a fire escape, facing the
backyard, and can see the ground, the trees, the lawn, the parking lot, and the
orange sunset light reflected on the large wall of a school building. Fresh air
and a broad view inspire my imagination much more than being shut in a room and
facing a wall or the ceiling. There are some noises in the deep shadows of the
trees; perhaps they come from crickets, locusts, or birds. Once in a while a car
starts, the engine roars and emits a jet of exhaust fumes; the car flees because
it does not belong to this quiet green picture; it actually ruins it. Everyone
is getting tired and ready to rest; the beacon on the spire of the Sears Tower
seems like an early star hanging in the sky as the sunlight abandons it.
28.
A mathematics book should not be written like a dictionary. Its style should
be like a piece of literature. It should not say too much or too little. It
should say just the exact amount a topic or a subject needs to be properly
covered. It may be easy to become a scholar, but it is difficult to become a
Shakespeare.
29.
Go forward and take her veil,
I want to see her pretty face.
Clouds! Make way!
Let sunshine color the world and light our minds.
Perhaps there is violent clashing under the ground;
Maybe there is tremendous heat to release.
No matter how catastrophic the change will be,
Everything will return to equilibrium.
30.
How wonderful it is that they are twins!
I rub my eyes to ensure that I am awake-
The same persons exist at the same time!
It saves time foe a painter: When one model is busy,
There is another one available.
It is also convenient for a writer:
One description can apply to two people.
But no one is happier than their mother,
Since she hit's the jackpot:
"Buy one, get one free."
31.
When I get excited, I feel like there are a thousand babies screaming in my
mind. I cannot quiet them. If I sit and try to read, they bellow like thunder. I
cannot concentrate on my work. Soon or later they will make me shoot out of my
chair. I have to walk around and around trying to soothe them until they fall
asleep at last.
32.
When I am depressed, my thoughts are like a snail squeezing its tight body
and withdrawing to its shell when sprinkled with salt: they cannot stretch out
and relax for a long time after. My thoughts are also like a mimosa, which
droops like a dead weed when touched. If we try to straighten its leaves, it is
as difficult as opening an unopenable broken umbrella.
33.
When I feel miserable, all the things that happen to me seem to have
unfavorable meanings. Any event can be interpreted in a good or bad way. When I
am too close to an event, I am unable to interpret it without bias. I have to
wait until my nerves relax and my mind resumes its flexibility. Then the truth
will unfold itself.
34.
Every grain, every tear;
Every brick, every drop of blood;
When a storm suddenly engulf all I have,
I do not know what to say;
I do not know what to do.
I wish I was like a street drunkard
Falling into his dreams.
35.
When I was a student, I used to focus on studying texts. I seldom did any of
the practice exercises. Now that I have graduated from school and have become a
tutor, I help others do exercises all the time. It is good for me to have
another view of these materials. I need not memorize any formulas or reread all
the text’s details. Whenever I meet a problem, all I have to do is think about
how to grasp its essence, how to design my strategy, and how to find the
necessary information in the text. Somehow this approach is more fun than
preparing an exam.
36.
I was asked to fill out a questionnaire about my experiences with writer's
block.
It was funny as if there had been a writer's block hiding somewhere and we
were going to bag it.
You, the writer, may be the source of the problem. Perhaps you create too
much tension for yourself. For example, you want to compose golden lines. Not
having enough material to write about is not a real problem. You can always
write as long as you think. A tiny grain of sand could be a door to an enormous
world. The same dull thing always has thousands of feelers to touch you. If you
write on a topic 365 times, unless you intentionally copy yourself, not two of
these 365 articles will be exactly the same.
Perhaps I encountered Writer's block before, but I do not have much
impression of it. It must be a king of noble blood. The more serious you treat
it, the more it will crush you. The best policy is to keep your door open: it
may be one of thousands of your guests, but it is not your favorite one.
37.
Sometimes it is quiet and I have plenty of time, but I just do not have much
to write about. However, sometimes it is noisy and I am under heavy pressure,
but suddenly an inspiration comes into my mind and carries with it a rich story.
When we have money in the bank, we can withdraw it whenever we desire. When
we read books, it seems that we store a lot of writing materials in our minds.
However, writing is somewhat different from withdrawing money from a bank. Even
if we have wisdom, abilities, and experiences, if our minds are not working,
then we will feel as though a spell were keeping it locked and have a hard time
breaking into our writing safe.
38.
The footsteps of the jinn that shuttles between the dreamlands,
And the jinn that explores the gold mines,
Can hardly be heard during the day.
But at night,
When the ground is tinted with moonshine
And all the lights have been extinguished,
There is no noise flicking
Except for your sword which shines like a laser knife.
The jinn all awaken and come out.
They are as busy as midwives
Cerebrating the birth of a crystal child.
39.
Like a caterpillar that spins silk and weaves its cocoon in order to become a
butterfly, by studying related material and training ourselves to check
carefully all the steps in our reasoning in order to neatly present our ideas to
others, we can reap equal rewards. Thus, our growth, our freedom, and our
achievements are born from hard work.
40.
There are thousands of eyes in a crystal mind;
Not even the slightest change could escape from its delicate screen.
There are tons of dynamite wrapped in a sensitive heart;
A tiny spark could ignite its emotional burst.
A crystal mind is like ice.
A sensitive heart is hot like fire.
Unless we endure the extreme cold
And suffer the intolerable heat,
How can we believe
They are different faces of the same type of art?