一位数学家的辩白 —— 第二十一章

It will probably be plain by now to what conclusions I am coming; so I will state them at once dogmatically and then elaborate them a little. It is undeniable that a good deal of elementary mathematics—and I use the word ‘elementary’ in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus—has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are just the parts which have the least aesthetic value. The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’ (and this is as true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.
But here I must deal with a misconception. It is sometimes suggested that pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications. The imputation is usually based on an incautious saying attributed to Gauss, to the effect that, if mathematics is the queen of the sciences, then the theory of numbers is, because of its supreme uselessness, the queen of mathematics—I have never been able to find an exact quotation. I am sure that Gauss’s saying (if indeed it be his) has been rather crudely misinterpreted. If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.
我将要做出的结论现在似乎已经显而易见了，所以我先武断地将它一次性陈述出来，然后再慢慢详述。不可否认的是初等数学的很大一部分——我用的“初等”一词是职业数学家用的那个意思，比如微积分的应用知识——被认为具有一定的实用价值。这部分数学总体上来说是相当枯燥的，它们正是最没有美学价值的那部分。“真正”数学家们的“真正”数学，比如费马、欧拉、高斯、阿贝尔和黎曼的数学，几乎完全是无用的（这对于“应用”数学和“纯”数学同样适用）。从工作的“实用性”上来评判任何天才职业数学家的一生都是不可能的。
但这里我要纠正一种误解，有人认为纯数学家们以他们工作的无用性为荣，并自夸其没有实际的用途。这种责难常常基于高斯的一句无心之词，大意是如果数学是科学中的皇后，那么数论因为其极端的无用性是数学中的皇后。我从来没有能够找到这句话的确切出处。我相信高斯的这句话（如果的确是他说的）不过是被粗鲁地曲解了。如果数论可以被用于任何实用且高尚的用途，如果它能够被直接用于增加人类的幸福或减轻人类的痛苦，如同生理学或甚至化学那样，高斯和其他数学家都不会愚蠢到对这样的用途感到伤心或遗憾的。但是科学既可以为善也可以为恶（尤其在战争时期），那么高斯和其他一些数学家就应该为不管如何有一门科学，就是他们的科学，由于其远离人类的日常活动而保持了其优雅性和纯洁性。