2025-05-16 , 12933 , 896 , 227
美国数学家信息论创始人克劳德香农在贝尔实验室演讲: 天才的特质与思维方式-4
4/6. 对问题进行统筹归纳
Another mental gimmick for aid in research work, I think, is the idea of generalization.
This is very powerful in mathematical research.
The typical mathematical theory developed in the following way to prove a very isolated, special result, particular theorem – someone always will come along and start generalization it. He will leave it where it was in two dimensions before he will do it in N dimensions; or if it was in some kind of algebra, he will work in a general algebraic field; if it was in the field of real numbers, he will change it to a general algebraic field or something of that sort. This is actually quite easy to do if you only remember to do it. If the minute you've found an answer to something, the next thing to do is to ask yourself if you can generalize this anymore – can I make the same, make a broader statement which includes more – there, I think, in terms of engineering, the same thing should be kept in mind.
As you see, if somebody comes along with a clever way of doing something, one should ask oneself, “Can I apply the same principle in more general ways? Can I use this same clever idea represented here to solve a larger class of problems? Is there any place else that I can use this particular thing?”
我认为，帮助研究工作的另一个智力技巧是归纳。这在数学研究中非常有用。
典型的数学理论是这样发展的：若要来证明一个非常孤立的、特殊结果的、特殊的定理的话——那么总会有人出现（尝试上手），并开始（一般化地）推广它。
（比如）他会先把它放在二维的地方进行研究，然后再做N维化的推广；或者，如果它是出现在某种代数分支的话，他会让其出现在一般的代数数域进行研究；如果它是出现在实数域的话，那么他会把它泛化到一般的代数领域或类似的地方中去。
这其实很容易做到，只要你有意识记下你要这么去做。如果你找到了某个问题的答案，接下来要做的就是问问自己——你是否还能把它一般化？我能不能做同样的、更广泛的陈述？包括更多，我认为，在工程方面，同样的事情也应该记住（得去这样做）。
正如你所看到的，如果有人提出一种聪明的方法来做某事，那么一个人则更应该问自己：
“我可以以更普遍的方式应用这样的原则吗？我可以基于这里表述的巧妙想法来解决更广的一类问题吗？”
5/6. 结构化的分析证明法
Next one I might mention is the idea of structural analysis of a problem.
Suppose you have your problem here and a solution here.
You may have two big a jump to take. What you can try to do is to break down that jump into a large number of small jumps. If this were a set of mathematical axioms and this were a theorem or conclusion that you were trying to prove, it might be too much for me try to prove this thing in one fell swoop. But perhaps I can visualize a number of subsidiary theorems or propositions such that if I could prove those, in turn I would eventually arrive at this solution. In other words, I set up some path through this domain with a set of subsidiary solutions, 1, 2, 3, 4, and so on, and attempt to prove this on the basis of that and then this one the basis of these which I have proved until eventually I arrive at the path S.
Many proofs in mathematics have been actually found by extremely roundabout processes. A man starts to prove this theorem and he finds that he wanders all over the map. He starts off and prove a good many results which don't seem to be leading anywhere and then eventually ends up by the back door on the solution of the given problem; and very often when that's done, when you've found your solution, it may be very easy to simplify; that is, to see at one stage that you may have short-cutted across here and you could see that you might have short-cutted across there. The same thing is true in design work. If you can design a way of doing something which is obviously clumsy and cumbersome, uses too much equipment; but after you've really got something you can get a grip on, something you can hang on to, you can start cutting out components and seeing some parts were really superfluous. You really didn't need them in the first place.
下一个我可能会提到的想法是对问题进行结构化分析。
假设这是你的问题和解决方案，（为了解决这个问题）你可能要跳两大步。
你可以试着把跳跃分解成许多小的跳跃。
假如这是一组数学公理，而这是你要证明的定理或结论，我可能很难一下子证明它；但或许，我可以设想这样一组子定理或子命题——如果我可以证明它们，那么我最终会相应地得到原问题。解换句话说，我用一组子解，比如， {1,2,3,4,...} ，等等——在这个定义域上建立一条路径、尝试在这个路径的基础上证明这个子命题、在这些路径的基础上证明这些子命题之子命题，直到我最终，才可以得到证明命题的路径 S。
许多数学证明实际上是通过极其迂回的过程找到的。
一个人开始证明这个定理，他发现他在地图上到处游荡。他开始并证明了许多似乎在任何地方都没有领先的结果，但最终这些成果，成为了解决某个给定问题的后门；很多时候，当你找到你的解决方案时，它可能很容易简化；也就是说，在某个阶段看到可以在这里有捷径，或者在那个有捷径。在设计工作中也是如此。
如果你设计了一种方法来完成了一件工作，但这个方法明显笨拙和繁琐，使用了过多的设备——但是当你真正掌握了一些你可以抓住的东西之后，你就可以开始切割组件，看到一些零件确实是多余的，以及一开始，你也是真的不需要它们。这时候，你就可以对实现方法进行简化。
6/6. 如何看待问题有反转？
Now one other thing I would like to bring out which I run across quite frequently in mathematical work is the idea of inversion of the problem.
You are trying to obtain the solution S on the basis of the premises P and then you can’t do it. Well, turn the problem over supposing that S were the given proposition, the given axioms, or the given numbers in the problem and what you are trying to obtain is P. Just imagine that that were the case. Then you will find that it is relatively easy to solve the problem in that direction. You find a fairly direct route. If so, it’s often possible to invent it in small batches. In other words, you’ve got a path marked out here – there you got relays you sent this way. You can see how to invert these things in small stages and perhaps three or four only difficult steps in the proof.
Now I think the same thing can happen in design work. Sometimes I have had the experience of designing computing machines of various sorts in which I wanted to compute certain numbers out of certain given quantities. This happened to be a machine that played the game of nim and it turned out that it seemed to be quite difficult. If took quite a number of relays to do this particular calculation although it could be done.
But then I got the idea that if I inverted the problem, it would have been very easy to do – if the given and required results had been interchanged; and that idea led to a way of doing it which was far simpler than the first design.
The way of doing it was doing it by feedback; that is, you start with the required result and run it back until – run it through its value until it matches the given input. So the machine itself was worked backward putting range S over the numbers until it had the number that you actually had and, at that point, until it reached the number such that P shows you the correct way. Well, now the solution for this philosophy which is probably very boring to most of you. I’d like now to show you this machine which I brought along and go into one or two of the problems which were connected with the design of that because I think they illustrate some of these things I’ve been talking about. In order to see this, you’ll have to come up around it; so, I wonder whether you will all come up around the table now.
UfqiLong
最后，我想提出的另一件事，是我在数学工作中经常遇到的问题，那就是问题出现的反转。
你试图在前提 P 的基础上得到解 S ，然而你做不到。把问题翻过来假设 S 是给定的命题，给定的公理，或者问题中给定的数字你想要得到的是 P ，想象一下，当时情况就是那样的。
然后你会发现，在那个方向上解决问题，是相对容易的，你会找到一条相当直接的路线。如果是这样，你甚至通常可以小批量地复制它。换句话说，你在这里标出了一条路径，然后你把中继的思路发送到了这个方向。你可以在小的阶段看到如何反转这些思路，而且也许在证明中只会有三四个困难的步骤。
现在，我认为在设计工作中，也会发生同样的事情。我有设计各种计算设备的经验，我想基于某些给定数量计算出某些数字。比如这是一台玩nim游戏[4]的机器，结果的计算似乎相当困难。尽管它可以完成，但它需要相当多的继电器来执行此计算。
但后来我想到，如果我把问题倒过来，那会很容易做到——如果给定的条件和要求的结果互换了；这个想法会引导出一种比第一个设计简单得多的方法。这样做的方式是通过反馈来实现。也就是说，你从所需的结果开始并运行它，直到通过它的值来运行，直到它与给定的输入匹配。
因此，机器本身向后工作，将范围 S 放在数字上，直到它到达你一开始的数字，也就是 P ，此时展现的就是正确的方法了。
这个解决方案对你们大多数人来说可能很无聊。因此，我现在想向你展示我今天带来的这台机器，并讨论与它的设计有关的一两个问题。我认为，它们说明了我一直在谈论的这些事情。
为了看到这一点，你必须围绕着它。
所以，我想知道你们现在，是否会围绕我坐在我的桌子旁。
最后，香帅就在实验室开始当场霸气展示他那天所带去的智能小发明了
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参考
^贝尔实验室（Bell Labs），全称贝尔电话实验室（Bell Telephone Laboratories, Inc.），是美国历史上最著名的研究与开发机构之一。
它成立于1925年，最初是作为美国电话电报公司（AT&T）的研究和发展部门，旨在推动电话和电信技术的发展。贝尔实验室同样也为二战正义方作出了巨大的贡献。
^艾伦·麦席森·图灵（Alan Mathison Turing，1912年6月23日~1954年6月7日），英国数学家、逻辑学家，计算机科学之父、人工智能之父。
1931年图灵进入剑桥大学国王学院，毕业后到美国普林斯顿大学攻读博士学位，第二次世界大战爆发后回到剑桥，后曾协助军方破解德国的著名密码系统Enigma，帮助盟军取得了二战的胜利。图灵是香农的好友。二人一起为计算机科学奠基。
^费兹·华勒（Fats Waller），美国爵士乐钢琴师、作曲家。
30年代初期，他与Fletcher Henderson以及Jack Teagarden等爵士乐团录制唱片。1934年首度灌录“Fats Waller And His Rhythm”的系列唱片，从此建立了爵士乐的传奇，他的音乐是通向现代爵士的重要桥梁。
^Nim游戏是博弈论中最经典的模型之一，它有着十分简单的规则和无比优美的结论。
通常的Nim游戏的定义是这样的：有若干堆石子，每堆石子的数量都是有限的，合法的移动是“选择一堆石子并拿走若干颗（不能不拿）”，如果轮到某个人时所有的石子堆都已经被拿空了，则判负（因为他此刻没有任何合法的移动）。
^闪电战又名闪击战，是第二次世界大战纳粹德国使用的一种战术，创建者是德国名将海因茨·威廉·古德里安。
它充分利用飞机、坦克和机械化部队的快捷优势，以突然袭击的方式制敌取胜，用机械化部队来快速切割敌军主力来达到预期效果。笔者注。
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