一位数学家的辩白 —— 第十八章

There is still one point remaining over from §11, where I started the comparison between ‘real mathematics’ and chess. We may take it for granted now that in substance, seriousness, significance, the advantage of the real mathematical theorem is overwhelming. It is almost equally obvious, to a trained intelligence, that it has a great advantage in beauty also; but this advantage is much harder to define or locate, since the main defect of the chess problem is plainly its ‘triviality’, and the contrast in this respect mingles with and disturbs any more purely aesthetic judgement. What ‘purely aesthetic’ qualities can we distinguish in such theorems as Euclid’s or Pythagoras’s? I will not risk more than a few disjointed remarks.
In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.
A chess problem also has unexpectedness, and a certain economy; it is essential that the moves should be surprising, and that
every piece of the board should play its part. But the aesthetic effect is cumulative. It is essential also (unless the problem is too
simple to be really amusing) that the key-move should be followed by a good many variations, each requiring its own individual answer. ‘If P-B5 then Kt-R6; if .... then ....; if .... then ....’—the effect would be spoilt if there were not a good many different replies. All this is quite genuine mathematics, and has its merits; but it is just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly) which a real mathematician tends to despise.
I am inclined to think that I could reinforce my argument by appealing to the feelings of chess-players themselves. Surely a chess master, a player of great games and great matches, at bottom scorns a problemist’s purely mathematical art. He has much of it in reserve himself, and can produce it in an emergency: ‘if he had made such and such a move, then I had such and such a winning combination in mind.’ But the ‘great game’ of chess is primarily psychological, a conflict between one trained intelligence and another, and not a mere collection of small mathematical theorems.
当我在第十一章中进行“真正的数学”和象棋之间的比较时，遗留下来了一个问题。我们现在会想当然地认为真正的数学定理在实质上，在严谨性上，在重要性上的优势都是压倒性的。对于一个受过训练的智者来说，数学定理在美上的优势也同样明显。但这个优势是很难定义和定位的，因为棋题的主要缺陷明显在于它的“琐碎”，这方面的对比于更加纯粹的美学评论参杂在一起。我们能在欧几里德或毕得哥拉斯的定理中分辨出什么样的“纯粹美学的”特质来呢？我在这里只敢谈几个零碎的观点。
在两个定理的证明中，都有很高程度的不可预见性，并辅以必然性和简约性。论据的形式如此奇怪和令人意外，和遥不可及的结论相比运用的武器确实如此的幼稚和简单，却必然到达最后的结果。细节上没有什么复杂之处，每个例子里只用了一次攻击；对于很多更为复杂的定理也同样如此，但是完全地欣赏这些需要很熟练的技术水平。在数学定理的证明中我们并不需要太多的“变数”，“各种情况的列举”实际上是一种更无趣的数学论证方式。数学证明应该如同简单而清晰的星座，而不是银河中散乱的星簇。
一个棋题也有其不可预测性和简约性；象棋的走法必须是令人惊讶的，棋盘中的每个棋子也必须有其扮演的角色。但其美学效果却是累计性的，关键一招必定伴随着众多变数（除非这个棋题太过简单而没有什么乐趣），每一可能性都有不同的答案。“如果P-B5然后Kt-R6，如果怎样就怎样，如果怎样就怎样”——如果不是有这么多不同的应对其效果就会大打折扣。这一切都是异常天才的数学，也有其有益之处；但这只是“通过列举情况的证明”,这是真正的数学家所努力回避的。
我想我可以通过求证于棋手们自己来加强我的论证。一个象棋大师，一个伟大棋局中的棋手，当然会从根本上鄙视一个棋题设计家的纯数学艺术。他储备了很多这样的想法，而且紧急时候也可以造出来：“如果他走了这样然后这样一步，我脑子里就有这样加这样的组合来制胜。”象棋的“伟大棋局”重要是心理上的，是一位受过训练的智者与另一位的抗争，而不只是众多小小数学定理的一个组合。