Rebecca Goldstein, translated by Tang Lu.

Starting from Incompleteness —— Godel's Proof and Paradox

Cognition of the Garden of Philosophy

Plato has always been very attractive to people who like mathematics, and Zarathustra himself likes mathematics better. Plato founded the Institute of Higher Education in Athens (probably the first university in Europe) and wrote on the door of the institute: "People who don't know geometry are not allowed to enter."

Plato was an opponent of relativity. This is his lifelong career. He not only thinks that moral truth is objective, but also bases his opinion on the objectivity of abstract reality, which can be understood not only through perception, but also through reason.

The most tenacious and dynamic field of Platonism is mathematics, or rather meta-mathematics. If a mathematician feels that he is discovering objective truth, not just building a theoretical system, he believes in Platonism. It is Platonism to think that objects such as image number and * * * are the prototypes of our system, and the system is real only because they describe the properties of objects such as image number and * * *.

For people interested in abstraction, meeting Plato for the first time is an extremely exciting experience. I remember my first time. ) it can be said that it is somewhat fascinating. Plato thinks that the beauty of the abstract kingdom goes far beyond any specific details, and it can and should ignite a * * that is far bigger than any beauty can ignite (those beauties are frivolous and imperfect creatures, and we can't expect them to return our love at all, and their beauty can't be compared with the transcendental, and will gradually disappear with the passage of time). Using "falling in love" to describe Godel's experience in college is imitating Plato himself. He once described the closeness and fascination of thinking with the beauty of abstract objectivity in extremely loving language.

My dear Socrates, this is the life that people should live. Study beauty itself. Once you see beauty itself, you will never be confused by gold, clothes, handsome boys and beautiful children. You will never pay attention to such beauty again. These beautiful women make you and many people like you miss each other so much. If possible, you will stay with your beloved all day, forget all about eating and sleeping, and never want to be apart for a moment, in pursuit of the greatest satisfaction. "She said," If one is lucky enough to see such natural and pure beauty, it is not the beauty of flesh and blood, but the beauty of sacred natural unity. If he can open his eyes and stare at this beauty and meditate on it until he is with it forever, do you still think his life is worthless? "

Drinking articles actually refers to the cocktail party in Plato's Athens, and Plato ironically used this name as the name of his dialogue collection. In the dialogue, he urged us to get rid of superficial intoxication, including our emotional love for young and beautiful things, and to indulge in the beauty of truth-the inevitable and eternal truth obtained through pure reason, of which mathematics is a model. One aspect of Plato's thought is to refuse to simply think that * * * is incompatible with reason. Plato urges us to approach and make people full of * * *, which is the reason for a higher intoxication. Of course, being able to understand the purely abstract beauty of what the mind loves means a higher sense of intoxication. "Gaze at this beauty and meditate on it until you are with it forever." Young Godel is unusually sentimental.

It seems that in the face of Plato's fascinating dream of truth, Godel's reaction is to decide to devote himself wholeheartedly to "really important" mathematics (reminiscent of Einstein's wording). That kind of mathematics will be meta-meaningful, philosophical and can clarify the objective source of all abstract truths.

Note: "Seminar" is the title of Plato's works quoted, which means "seminar" in modern English. In the Chinese translation of Plato's works, the ancient Greek semantics are generally taken directly and translated into "drinking articles".