Learning Mathematics From the Air

学集合论的时候读到了Ulam深刻优美的theorem，结果学统计的时候他的名字又在Monte-Carlo simulation的时候出现了。“这是一个人吗？”我以前以为Cramer-von Mises criterion是那个经济学家Ludwig von Mises业余搞统计的时候顺手搞的，结果发现统计学家von Mises是Ludwig的弟弟。在搜索之后，我惊讶地发现the set theorist Ulam就是the statistician Ulam,甚至还是the physicist Ulam, the inventor of the H-bomb Ulam. “又是像冯 诺伊曼那样的精力十足的天才吧。”我一直非常羡慕历史上这些想干什么都行的全才。他们不光自己精力充沛，也生活在了自由自在，做什么都能获得支持的国度和时代。“能够发展成自己最棒的样子：”这也一直是我自己的目标。

我一直觉得Ulam的名字像一种贝类，似乎更适合当海洋生物学家。在一次和教授的见面中谈到了Ulam的传奇经历，没想到教授居然对他非常熟悉，在长吁短叹后，他对Ulam表现出了深深的遗憾。“这是一个没能实现上帝给他的神召的天才，”对人对己都很严厉的教授少见地动情地说。

“他还要怎么干才能算是实现上帝给他的神召？”我简直不敢相信自己的耳朵。见面结束后我就拐去了图书馆，去搜这个贝类的名字，找到了他的自传。

一口气读完后，看着Ulam那用最狡捷的目光看着观者的照片，的确感觉内心空落落的，遗憾得想快跑出图书馆，逃脱命运之手的摆弄。的确是太遗憾了，太遗憾了，上帝的宠儿被流放到了新大陆，头脑的艺术家当了工程师，亲友几乎全部在x浩x劫x中xsix去，他咬牙切齿地算着帐：

Always in good humour and joking incessantly, he loved to consume frankfurters liberally smeaered with horseadish, a dish which he maintained cured melancholy. Stozek was one of the professors murdered by the Germans in 1941.

Antoni Lomnicki, a mathematician of aristocratic features who specialised in probability theory and its applications to cartography, had office hours in these rooms. He too was murdered by the Germans in Lwów in 1941.

Kaczmarz, tall and thin, who later was killed in military service in 1940...

Although Lwów was a remarkable center for mathematics, the number of professors both at the Institute and at the University was extremely limited and their salaries were very small. People like Schauder had to teach in high school in order to supplement a meager income as lecturer or assistant. Schauder was murdered by the Germans in 1943.

Lwów had frequent and lively interaction with other mathematical centers, especially Warsaw. From Warsaw Sierpinski would come occasionally, so would Mazurkiewicz, Knaster, and Tarski. In Lwów they would give short talks at the meetings of the mathematical society on Saturday evenings. Sierpinski especially liked the informal Lwów atmosphere, the excursions to inns and taverns, and the gay drinking with Banach, Ruziewicz, and others. (Ruziewicz was murdered by the Germans on July 4, 1941.)

Schreier was murdered by the Germans in Drohobycz in April, 1943.

有血有肉的、完整的Ulam永远地留在了欧洲。在欧洲他是Scottish Café里的神童，在白色的大理石桌面上洋洋得意地高谈阔论“整个宇宙”。

In mathematical discussions, or in short remarks he made on general subjects, one could feel almost at once the great power of his mind. He worked in periods of great intensity separated by stretches of apparent inactivity. During the latter his mind kept working on selecting the statements, the sort of alchemist's probe stones that would best serve as focal theorems in the next field of study.

He enjoyed long mathematical discussions with friends and students. I recall a session with Mazur and Banach at the Scottish Café which lasted seventeen hours without interruption except for meals. What impressed me most was the way he could discuss mathematics, reason about mathematics, and find proofs in these conversations.

Since many of these discussions took place in neighbourhood coffee houses or little inns, some mathematicians also dined there frequently. It seems to me now the food must have been mediocre, but the drinks were plentiful. The tables had white marble tops on which one could write with a pencil, and, more important, from which notes could be easily erased.

There would be brief spurts of conversation, a few lines would be written on the table, occasional laughter would come from some of the participants, followed by long periods of silence during which we just drank coffee and stared vacantly at each other. The café clients at neighboring tables must have been puzzled by these strange doings. It is such persistence and habit of concentration which somehow becomes the most important prerequisite for doing genuinely creative mathematical work.

Thinking very hard about the same problem for several hours can produce a severe fatigue, close to a breakdown. I never really experienced a breakdown, but have felt "strange inside" two or three times during my life. Once I was thinking hard about some mathematical constructions, one after the other, and at the same time trying to keep them all simultaneously in my mind in a very conscious effort. The concentration and mental effort put an added strain on my nerves. Suddenly things started going round and round, and I had to stop.

It was difficult to outlast or outdrink Banach during these sessions. We discussed problems proposed right there, often with no solution evident even after several hours of thinking. The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed in the meantime. If they were not polished or even not quite correct, Mazur would frequently put them in a more satisfactory form.

Needless to say such mathematical discussions were interspersed with a great deal of talk about science in general (especially physics and astronomy), university gossip, politics, the state of affairs in Poland; or, to use one of John von Neumann's favorite expressions, the "rest of the universe." The shadow of coming events, of Hitler's rise in Germany, and the premonition of a world war loomed ominously.

在美国，人们不再向彼此敞开心扉：

I had my meals at Adams House, and the lunches there were particularly agreeable. We sat at a long table—young men and sometimes great professors; the conversations were very pleasant. But often, towards the end of a meal, one after the other would gulp his coffee and suddenly announce: "Excuse me, I've got to go to work!" Young as I was I could not understand why people wanted to show themselves to be such hard workers. I was surprised at this lack of self assurance, even on the part of some famous scholars. Later I learned about the Puritan belief in hard work—or at least in appearing to be doing hard work. Students had to show that they were conscientious; the older professors did the same. This lack of self confidence was strange to me, although it was less objectionable than the European arrogance. In Poland, people would also pretend and fabricate stories, but in the opposite sense. They might have been working frantically all night, but they pretended they never worked at all. This respect for work appeared to me as part of the Puritan emphasis on action versus thought, so different from the aristocratic traditions of Cambridge, England, for example.

心中的Arcadia：

The Cambridge architecture, the medieval buildings, the beautiful courtyards, the walks I took through the town, some with L. C. Young (now a professor at the University of Wisconsin), are still among the strongest visual impressions of my life. Like my walks through the Paris of the French Revolution, these have somehow influenced my tastes, associations, readings, and studies to this day.

Early in 1935, I returned from Cambridge to Poland. It was now time to think seriously about a university career, although those were, difficult times in which to find even a modest "docent" position. A series of accidental letters was to change this; in one of them, luckily for me, I received an invitation to visit the United States.

Ulam的生活被一个潮湿、闷热的夜晚一分为二。

Adam and I were staying in a hotel on Columbus Circle. It was a very hot, humid, New York night. I could not sleep very well. It must have been around one or two in the morning when the telephone rang. Dazed and perspiring, very uncomfortable, I picked up the receiver and the somber, throaty voice of my friend the topologist Witold Hurewicz began to recite the horrible tale of the start of war: "Warsaw has been bombed, the war has begun," he said. This is how I learned about the beginning of World War II. He kept describing what he had heard on the radio. I turned on my own. Adam was asleep; I did not wake him. There would be time to tell him the news in the morning. Our father and sister were in Poland, so were many other relatives. At that moment, I suddenly felt as if a curtain had fallen on my past life, cutting it off from my future. There has been a different color and meaning to everything ever since.

从古希腊时期开始，谈话一直就是教育的重要方式。希尔伯特的学习主要靠谈话，Ulam也不例外。（这就是为什么我有时觉得《娱乐至死》的作者有些太悲观了。数学算是最需要fixed to the paper的科目了吧，但是依然，最热情的数学，就像音乐一样，来自空气）：

As for myself, ever since I started learning mathematics I would say that I have spent -- regardless of any other activity -- on the average two to three hours a day thinking and two to three hours reading or conversing about mathematics. Sometimes when I was twenty-three I would think about the same problem with incredible intensity for several hours without using paper or pencil. (By the way, this is infinitely more strenuous than making calculations with symbols to look at and manipulate.)

On the whole, I still find conversation with or listening to other people an easier and pleasanter way of learning than reading. To this day I cannot read "how to" instructions in printed form. Psychologically, these are indigestible for me.

Some people prefer to learn languages by the rules of grammar rather than by ear. This can be said to be true of mathematics—some learn it by "grammar" and others "from the air." I learned my mathematics from the air.