叶航：带给上帝的问题——答“物含妙理总堪寻”在我的上一篇博文后，“物含妙理总堪寻”留下两段评论——
“我愿意引述一段我所喜爱的爱因斯坦的话，出自于他在普朗克六十岁生日庆祝会上的讲话。科学的庙堂里有各式各样的人。根据动机的不同，爱因斯坦把这些人分成三类。一类人是为了追求一种知性的愉悦，科学对于他们其实是一种游戏和娱乐；一类人则完全是出于功利目的。这两类人中有许多杰出的科学家。如果上帝派天使把这两类人逐出科学的庙堂，那么聚集在那里的人将大为减少，但不会空空荡荡，仍有一部分人留在那里，这就是那第三类人，普朗克就是其中之一。爱因斯坦说：‘有一点我可以肯定：如果庙堂里只有天使所驱逐的那两类人，那么这座庙堂决不会存在，正如只有蔓草就不成其为森林一样’。”
“我的一位同事说：‘我死时要带上两道难题去问上帝。’————有一个版本说这句话是海森堡说的，他早年曾研究过湍流。海森堡说：“我要带两个问题去问上帝，一个是量子力学，一个是湍流。我估计第一个问题是有答案的。”也就是说，海森堡觉得“湍流”问题连上帝都不一定知道答案。做学问的人大抵都有一个‘终身问题’，我想问叶航老师，你带给上帝的问题是什么呢？利他？很愿意听你说说。”
答：对我而言，利他问题已经解决了。如果我能够进天堂，我想问上帝的问题是“你究竟掷不掷骰子？”你呢？想问上帝什么问题？请诸位留下你们的问题
：
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下面是我的回答：
我想问上帝的问题是“你究竟掷不掷骰子？”
————J Ford 有个回答：“上帝掷骰子，只不过掷的是一只灌铅的骰子。”当然严格说这不是一个回答，只是对科学及其问题的一个解释，但我还是很喜欢这个美妙的解释。
看不到答案的问题才是最有嚼头的，我的问题小一点，是：
数学家何以能够“闭门造车，出门合辙”？
这个现象被叫做“Wigner 原理”，Wigner 在一篇长文中说数学在物理上有一种“不可思议的有效性（Unreasonable Effectiveness）”，比如黎曼几何之与爱因斯坦的广义相对论，陈省身的纤维丛之与杨振宁的规范场论，等等。
这也就是所谓的“预定和谐”之谜。
这个问题上承接“毕达哥拉斯--柏拉图”传统，下与开普勒、爱因斯坦的研究风格一致，我很早就被这个问题深深迷住了。
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下面是 Wigner 那篇精彩的文章，可惜没有翻译：
数学在自然科学中不可思议的有效性
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
by Eugene Wigner
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
New York: John Wiley & Sons, Inc. Copyright © 1960 by John Wiley & Sons,Inc.
Mathematics, rightly viewed, possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.
--BERTRAND RUSSELL, Study of Mathematics
THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with
the circumference of the circle."
Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense. I was even more confused when, not many days later, someone came to me and expressed his bewilderment [1 The remark to be quoted was made by F. Werner when he was a student in Princeton.] with the fact that we make a rather narrow selection when choosing the data on which we test our theories. "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?" It has to be admitted that we have no definite evidence that there is no such theory.
The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.
Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, "What is mathematics?", then, "What is physics?", then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work
WHAT IS MATHEMATICS?
Let us consider a few examples of "false" theories which give, in view of their falseness, alarmingly accurate descriptions of groups of phenomena. With some goodwill, one can dismiss some of the evidence which these examples provide. The success of Bohr's early and pioneering ideas on the atom was always a rather narrowone and the same applies to Ptolemy's epicycles. Our present vantage point gives an accurate description of all phenomena which these more primitive theories can describe. The same is not true any longer of the so-called free-electron theory, which gives a marvelously accurate picture of many, if not most, properties of metals, semiconductors, and insulators. In particular, it explains the fact, never properly understood on the basis of the "real theory," that insulators show a specific resistance to electricity which may be 1026 times greater than that of metals. In fact, there is no experimental evidence to show that the resistance is not infinite under the conditions under which the free-
If viewed from our real vantage point, the situation presented by the free-electron theory is irritating but is not likely to forebode any inconsistencies which are unsurmountable for us. The free-electron theory raises doubts as to how much we should trust numerical agreement between theory and experiment as evidence for the correctness of the theory. We are used to such doubts.
A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world. Mendel's laws of inheritance and the subsequent work on genes may well form the beginning of such a theory as far as biology is concerned. Furthermore,, it is quite possible that an abstract argument can be found which shows that there is a conflict between such a theory and the accepted principles of physics. The argument could be of such abstract nature that it might not be possible to resolve the conflict, in favor of one or of the other theory, by an experiment. Such a situation would put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called "the ultimate truth." The reason that such a situation is conceivable is that, fundamentall
y, we do not know why our theories work so well. Hence, their accuracy may not
prove their truth and consistency. Indeed, it is this writer's belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted.
Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of earning.
Merci W. Cooper
Histoire et philosophie des mathmatiques Le quasi-empirisme en philosophie des mathmatiques. Une presentation Liens mathmatiques en relation indirecte avec le quasi-empirisme Retour la page d'accueil
魏老师是我大学最爱的老师，没有之一。
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