The concept for this work was inspired by The Tao of Programming, by Geoffrey James. It can be found at http://www.canonical.org/~kragen/tao-of-programming.html. Inspiration was also taken from the Tao de Ching, by Lao Tzu. A translation of that work can be found at http://www.sacred-texts.com/tao/taote.htm.

Some of the koans in this work were inspired by koans from the Tao of Programming, as well as the M.I.T. AI koans, some of which can be found at http://www.catb.org/~esr/jargon/html/koans.html.

I wish to thank the authors of all of these works, without whom this would have never come to be.

This work is also not complete; it may never be, if I continue to think of additions or improvements, or if I learn something or something happens that should be addressed.

-aristotle2600

The Tao of Mathematics

The Tao of Mathematics

The master mathematician speaks: “Seek the Tao in all mathematics, for all mathematics contain the Tao.”

The Tao gave birth to Set Theory.

Set Theory gave birth to Number Theory.

Number Theory gave birth to Algebra.

Now there are hundreds of fields, thousands of sub-fields, and millions of problems.

Each has its place in the Tao, one singular purpose or many unconnected.

The Master was teaching, and one of his students asked what the Tao was like. “It is like a great marsh just before dawn,” the Master responded, “filled with many secrets, much life, and great beauty and complexity. When caught in it, there is much despair. A great fog clouds your vision. Once you have found your way out, you are relieved…much becomes clear, as you look back within the fog and see a glimpse of the beauty of the marsh, and marvel. Then, if you wait a little while, the Sun rises, burning away the fog, and laying the marsh bare for you to see, splendid in its majesty.”

The Emperor came to the Master, seeking his great technique, or the formula which gave him mastery over Mathematics. On hearing this, the Master showed him the technique of counting. The Emperor was displeased, and insisted the Master not mock him. The Master gave the formula 1=1. The Emperor, enraged, demanded his secret. “What would you have me say?” asked the Master. “Every technique has its place in Mathematics, but the Tao cannot be defined by technique. Every formula serves a purpose, but no formula, however profound, can contain the Tao. I have no mastery because there is no mastery, only Tao.” Ashamed, the Emperor returned to his palace.

The fool hears about Tao and laughs.

The novice hears about Tao and listens.

The initiate hears about Tao and calculates.

The wise one hears about Tao and seeks.

The enlightened one hears about Tao and understands.

The Master hears about Tao and smiles.

A great proof is like an epic poem, or a masterwork painting. It speaks to the heart, and is an expression of the beauty of the world. But while a painting speaks emotionally, a proof speaks logically. Unlike a poem, shorter is always better.

A student asked about the complexity of the Tao. “Consider the sky,” the Master said in reply. “Would you say it is simple?” “Of course it is simple,” came the answer. “But surely, you see the clouds? Note their diverse shapes, how they move together in an intricate dance.” The student conceded that there was much complexity in the clouds, but that that was irrelevant. “Perhaps the clouds are irrelevant,” the Master agreed. “But what of the birds? See that flock, and this flock? There are also many insects which we cannot see from here because we are but human. Surely these things make the sky less simple?” “Yes,” the student said. “Life in all its forms is a marvel of complexity, and when life interacts, it creates the beauty of the flock. But this is not the sky.” The Master tried once again, saying “And yet, invisible to us, the air above us is filled with currents, and systems, and fronts, and air masses. This must make the sky complicated indeed!” The student objected, saying “But it is just the sky! All these things are within the sky, not truly part of it!” “Ah,” said the Master, smiling. “You begin to understand.”

Mathematical Intuition

The master mathematician speaks: “Follow the voice of the Tao, for it is the voice of Truth.”

A young mathematician went to a conference. The keynote speaker was widely praised and renowned. His presentations, though, had many flaws. He skipped proofs, did not define his symbols, nor did he explain the assumptions he made. He also spoke with excessive haste. Nonetheless, all dutifully listened. When the mathematician returned to his Master, he asked about how these could be so. The Master replied, “All you have told me has the same explanation. The keynote is fully enveloped in the Tao. He has transcended symbol, for no symbol can contain the Tao. He has no need for proof, since the Tao is the source of all proof. His only assumption was that the Tao is good. He probably wanted the conference over with so he could go back to his work.”

A student was working on a problem and the Master looked at his paper. Equations and definitions and limits were scattered about on the tablet. The Master asked what was the meaning of this. “I am growing closer to the Tao, not letting my mind be clouded by artificial organization.” The Master replied “You cannot find the Tao by imitating it, only by understanding it. Organization and structure is a means to aid that understanding. By eschewing it, you hinder your own enlightenment.” Humbled, the student began to rewrite his problem.

A student of the house of Qing came to the master, saying “I have heard that there is nothing to your so-called ‘Mathematics,’ that it is simple intuition. I wish to learn.” “Certainly,” said the Master. He assigned the student to the advanced group, at the student’s insistence. The students there were unlike any the student from Qing had ever seen. They finished each other’s sentences, only wrote half-equations on the whiteboard before starting over, and seemed to skip from one line to another, with no apparent relation between the two. The student asked what was the meaning of all this, and the students responded: “We are enveloped in the Tao, and our intuition guides us. Come, we have heard that you are a great one for intuition.” The student from Qing, his pride injured, returned to his house, never to be heard from again.

Symbols and Meaning

The master mathematician speaks: “Symbols make things easier, not harder.”

A student asked the Master, “The Tao is not constrained by symbols, but we must know symbols nonetheless. Why is this?” The Master responded, asking “Must one be a great poet to be a great lover?”

A great magician was showing his magic to a crowd of around 20-30 people. The Master observed, and pointed to the ∇ symbol, asking what it meant. “Gradient” came the reply. “But what does it mean?” persisted the Master. Angrily, the magician said “Grad!” The Master then pointed to the ÷ and asked the meaning. “Surely even the simplest man knows that means to divide!” the magician laughed. “And what does it mean?” the Master asked patiently. “I just told you, it means to divide!” snapped the magician. Finally, the Master pointed to a 7 near the end of the magician’s board. “Pray tell me, great magician, for I am but a simple man. What is the meaning of this symbol?” “Why, that is a seven; surely, even such a simple man as you knows this?” At that, the Master turned to the crowd. “Those who wish to be magicians should remain here. Those that wish enlightenment should follow me.” He gained 2 new followers that day.

The Master and a novice were walking in the forest. The novice asked, “Master, why must we be precise in our symbols? Surely the Tao transcends symbols.” The Master made to reply, but then exclaimed “A wild gourd! Run!!” and ran with much haste the way they had come. Curious, the novice stared at the retreating Master. He then turned, and cursed the Master’s imprecision with language, and ran after the Master, with much haste.

Those who hate mathematics see its many diverse symbols and see chaos. But the Tao is not chaos, it is the Tao. Symbols allow the Tao to be expressed.

Problem Solving

The master mathematician speaks: “A great problem is a great gem. Solve one, and hear the Tao sing to you.”

The Master said: Breathing, eating, walking, sensing, loving, talking. These are all techniques of life, but they are not life.

A prince from the House of Chang observed the Master working on a great problem. The Master’s pen flew across the page, barely stopping to re-ink, like a swift river reworking the countryside. As the prince watched, the Master was able to write a theorem he had proven, and decided to take a break from the great problem. The prince clapped his hands, exclaiming “Excellent! Your technique is flawless, master mathematician.” But the Master waved him off, saying “I go merely where the Tao takes me. Every definition, every rule or formula, follows from those before. Free to go where it will, my spirit progresses on the problem without effort; in short, the problem solves itself. It is true that at times, there are obstacles; contradictions even. I see these problems, I understand them, I watch them. Then I write a single identity, and harmony is restored. Finishing my theorem, I can then continue on my way.” After hearing this, the prince said “Would that all the mathematicians in the kingdom were as wise.”

The novice mathematician, when given a problem, sees the multitude of numbers, words, and symbols, and is filled with despair.

He cannot see the forest for the trees.

The average mathematician, when given a problem, sees the many constituents and problem types, and applies the techniques of solving, but is dismayed.

He thinks the names of the fish make the ocean.

The wise mathematician, when given a problem, sees how all the parts form patterns and make the whole, and is filled with awe.

He thinks the shape of the flock, the course of migration, make the sky.

The Master, when given a problem, sees only the problem, for he is filled with the Tao.

He attaches no folly to what he sees, only seeing what is there.

The Tao is an embodiment of freedom, like a bird of the sky. Because the sparrow has the Tao, it knows techniques, but has no intrinsic need to use them. Instead, it always simply does what is needed. It is not bound to any thing.

A student went to the Master, a mighty problem statement in his hand. “Master,” he said, dejected, “I cannot see the way to the solution.” The Master smiled kindly, asking “The way to all solutions is Tao. Surely you can see that?” “Yes, that is well and good, Master, but I cannot apply it here!” The Master asked him “So you know nothing about this problem then?” “Of course I know about the problem; I have read it many times!” answered the distressed student. “Good!” exclaimed the Master. “That is a beginning! Now, go forth, and discover other things about the problem. There are limitless gems and secrets concealed within that page in your hand.” “But how will I know that I am on the right path?” asked the student. “There is only one path, and that is Tao,” said the Master. “All solutions lie on that path.”

Word came to the monastery of a wondrous man, in the north, who was said to be a magician, and who could solve any math problem. On hearing this, the Master sent some of his students to observe the man and his technique. When they returned, they were distressed, saying the man had no technique, but merely flitted from one discipline to another, like a squirrel looking for nuts. The Master sent them out again, admonishing them to look harder, and again they returned, more distressed than before. Thrice more this happened, with no progress on the technique. Finally, the Master sent them with a letter, asking the magician to show them his techniques. But the magician did not understand the meaning of the letter, and looked kindly on the students, thinking them simple, and showed them long division, partial fraction expansion, and many other techniques, but solved not a single problem. Incensed, the students returned for the 6th time, and refused to go again. Upon hearing them refuse, the Master asked why. They replied that not only did they know all the magician had to teach them, but that he had been unkind to them. The Master replied thus: “You asked for technique, he gave you technique, how is that unkind? You wish for a technique of problem solving? Merely this: do what needs to be done. The magician was doing what needed to be done to solve his problems. How can you fault him for that?”

Learning and Teaching Math

The master mathematician speaks: “One must always get the answer a right way.”

If Great Master Euclid stands before you, giving you ill advice, whether for learning, teaching, or understanding, pay him no heed. If he interferes, strike him dead.

A student may discover an answer by chance.

A student may discover an answer by base intuition.

A student may discover an answer when errors offset errors.

This is not the Tao.

A student may miss a question because he applies his knowledge incompletely.

A student may miss a question because his mathematical intuition is not developed enough.

A student may miss a question because he lacks the background.

Truly, this is the Tao.

The teacher that paradoxically cannot teach is common

The teacher that can teach but does not see the Tao is still a great find.

The teacher that has found the Tao but cannot convey it is nonetheless a rare gem.

The teacher that understands the Tao and can teach it well is a priceless treasure you may meet once in a lifetime.

Learning math is an experience, like climbing Kilimanjaro. A mountain guide does not climb the mountain for his charges, nor can he.

If you treat your students like automatons, expect them to act like automatons.

If you treat your students like reasoned beings, expect them to act like reasoned beings.

The Master left the dojo one day, and went into one of the schools in a nearby village. There he found a teacher of math who was renowned for his expertise. It was said that there was no subject he did not have mastery of. He entered the classroom, and quietly observed the lesson. He spoke at great length, demonstrating theorems, proving postulates, and working examples. His students dutifully took note of his lesson with reserve and respect. When the lesson was over, he and the Master spoke. When the teacher asked the Master for his evaluation of his lesson, the Master considered, and said “You talk too much.”

A student embarked on a quest to learn a new math topic, and so enrolled in a class. He asked the Master how he should proceed. The Master asked him why he was taking the class. “To learn, of course,” came the student’s reply. The Master asked if he also wished to understand the subject, and the student replied, “Are they not the same thing?” The Master sent him on his way with his approval. Later that year, the student came to the Master once again, distressed “I am falling behind! I cannot grasp the understanding of the material! What shall I do?” The Master asked if he was learning well, and being taught well. “Very well!” the distressed student said. “If that were true,” the Master countered, “you would have understanding.”

The wise student will not be seduced by being right, but only seeks to better himself. Those who wish to be right do not see the Tao, for they are blinded by what is in front of them.

A true athlete does not take herbs or hidden ways.

A good businessman does not use cleverness to deceive his customers.

A righteous magistrate applies the law as it is meant.

A loving parent gives his children more than they need.

A mighty samurai keeps his blade sharp and does not rely on fear.

A wise student gives his answers honestly, and does not read his teacher.

Correctness, Sanity, and Truth

The master mathematician speaks: “The essence of mathematics is truth.”

Two men came to the master, an insurmountable dispute between them. After imploring the Master to use his wisdom and resolve the dispute, they gained his assent. They then presented their arguments. After he had listened, the Master held a coin by its edges, so that one side faced left, and the other to the right. “Is this coin heads or tails?” he asked the men. “It is obviously heads, as that is what faces to the right, the primary direction!” “No,” said the other. “See how the light catches the tails side, illuminating it. Clearly it is tails!” The men set to bickering over the question. The Master interrupted, saying “When you can agree on this, I will settle your dispute.” The men were not heard from again.

A student asked the Master as he was on his afternoon walk one day, “Which is better, to approach mathematics with a clear, intuitive mind, or to be sure in one’s knowledge?” The Master responded, “Go to the forest. Retrieve the 2 leaves that represent the forest.” Baffled, the student went away, and did as he was asked. Upon presenting and explaining his choices, the Master asked him “Which is the forest?” The student did not understand. “If I had never seen a forest, which leaf would you use to explain one to me?” Upon hearing this, the student was enlightened.

The Master said to his students: “Do not be seduced by this technique, or that method. Do not be distracted by some analogy, or some other shortcut. If you do not have the Tao, you have nothing.”

Mathematical Correctness and Mathematical Sanity met one another on a road one day. “Come,” said Correctness, “Individually we are but simple, and cannot turn Men’s hearts, but together we may conquer the world!” Enthusiastically, Sanity agreed, and they went on their way. After a little way, they met a disheveled beggar, Mathematical Truth, who called out to them. “You would be nothing without me! The Tao speaks to me, does it speak to you? You cannot hear it, because you cannot see how your selves have blinded you to it. Abandon this quest, for it is folly!” But the pair heeded his admonishment not, and continued on their way, their plans to conquer the world together forming as they went.

The Master asked his students, “Who knows the language with only nouns?” His students replied “There can be no such language! The world is more than simply this thing, and that thing. The many objects must interact and coexist in harmony.” The Master thought a moment, then wondered “But surely, there is a language with only verbs?” “Surely not!” replied the students. “Such a language would describe a nonsense world, of action and chaos but no matter, no objects with which to anchor oneself.” The Master smiled, pleased. “So it is with the Tao.”

A student was walking with the Master, and asked which was more important, to be right or be rational. The Master stopped, ignoring him, and picked a flower from a nearby bush. “I asked the royal experts about this flower. One was neurotic, almost raving, about xylem and roots, petals and stamens. Many facts came of his mouth. He is renowned for his expertise. But he did not seem a man, rather a book of knowledge. Another spoke poetically, alluding to its beauty, the shape of the stem and how pleasing it was. He is renowned for his poetry. But he did not seem to be talking about the flower I held in my hand.” He then looked at the student. “Which expert should I listen to?”