Series 2: Mathematics, King of the World: Casual Remarks on Foundational Issues in Modern Mathematics!

2006/3/4 23:04:34

Mathematics cannot exist without imagination, yet an idiot cannot construct any mathematical system. An idiot may have imagination beyond that of ordinary people, but his chances of writing even a five-character quatrain are slim. A five-character quatrain has 20 characters, and with tens of thousands of Chinese characters—let us take the most common 10,000—the number of random combinations of 20 characters is 10 to the 80th power, a task that no computer in this world can currently complete, let alone an idiot. Where human reason surpasses computers is precisely in transcending this terrifying randomness. A computer may beat a human at chess, but it cannot write Shakespeare's tragedies, compose Beethoven's symphonies, or construct a mathematical system—this, at least, holds true in our present world.

Of course, an idiot has a chance of writing down a mathematical formula, but without a system of contemplation, the idiot cannot understand what he has written. To him, the most exquisite mathematical formula is no different from a doodle. The meaning of a formula exists only within a system; a formula in isolation is merely a doodle. This is true not only for idiots but for anyone, including those who do mathematics.

The axiomatic method is fundamentally a way of playing the game so that it seems to have meaning. But what is even more interesting is that a formula written by an idiot is not necessarily meaningful to any constructed mathematical system—and this holds for non-idiots as well. For example, for over a thousand years, Euclid's "the sum of the three interior angles of a triangle equals 180 degrees" was taken as truth. Such a thing, if it has any meaning in a non-Euclidean system, can only mean nonsense. Conversely, a formula having meaning seems to a priori belong to some particular system or framework.

Anyone with even a basic knowledge of relativity knows that "the sum of the three interior angles of a triangle equals 180 degrees" corresponds to a flat world. If an idiot in a non-flat world happened to write down "the sum of the three interior angles of a triangle equals 180 degrees" and then proclaimed it a truth of the world, he would merely be creating a new pretext for his idiocy. But if he also happened to sign his name, and then this thing traveled through some special wormhole or other bizarre means and suddenly landed before Euclid, Euclid would probably exclaim "a divine oracle from God!" He would read the idiot's name, and it would become a holy name to be invoked in future prayers. It seems that the God of one world may well be the idiot of another.

But the proposition "a formula having meaning seems to a priori belong to some particular system or framework" itself also belongs to something that could be either idiotic or divine. Any a priori belonging is actually the result of some preset game. Of course, one can continue playing this game: treating all mathematical systems containing a certain proposition as a single object of study, or using it as a basis for classification—dividing all possible worlds into categories. For example, "the sum of the three interior angles of a triangle equals 180 degrees," "the sum is greater than 180 degrees," and "the sum is less than 180 degrees" can divide possible worlds into three categories. One could even examine the relationships among all mathematical systems containing a certain proposition, to see whether there exists a smallest or largest one, and so forth. All these perspectives are games commonly played in mathematics, but games are just games. If you try to find something necessarily eternal within them, you have simply been played by playing too long. (Note: this series is the vernacular version of a chapter from this woman's book Chán Zhōng Shuō Chán, which is written entirely in classical Chinese. The vernacular version of one chapter is excerpted here first.)