Chinese Philosophy, Art, Life, and Mathematics


    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


    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.


    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.


    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


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.


    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.


    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.


    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.


    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?


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


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.


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."


    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.


    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.


    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.


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.


    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.


    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.


    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.


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.


    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.


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?