2

2

I propose to put forward an apology for mathematics; and I may be told that it needs none, since there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy. This may be true; indeed it is probable, since the sensational triumphs of Einstein, that stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation. A mathematician need not now consider himself on the defensive. He does not have to meet the sort of opposition described by Bradley in the admirable defence of metaphysics which forms the introduction to Appearance and Reality.

A metaphysician, says Bradley, will be told that ‘metaphysical knowledge is wholly impossible’, or that ‘even if possible to a certain degree, it is practically no knowledge worth the name’. ‘The same problems,’ he will hear,‘the same disputes, the same sheer failure. Why not abandon it and come out? Is there nothing else more worth your labour?’ There is no one so stupid as to use this sort of language about mathematics. The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steam-engines and dynamos, obtrude themselves on the dullest imagination. The public does not need to be convinced that there is something in mathematics.

All this is in its way very comforting to mathematicians, but it is hardly possible for a genuine mathematician to be content with it. Any genuine mathematician must feel that it is not on these crude achievements that the real case for mathematics rests, that the popular reputation of mathematics is based largely on ignorance and confusion, and that there is room for a more rational defence. At any rate, I am disposed to try to make one. It should be a simpler task than Bradley's difficult apology.

I shall ask, then, why is it really worth while to make a serious study of mathematics? What is the proper justification of a mathematician's life? And my answers will be, for the most part, such as are to be expected from a mathematician: I think that it is worth while, that there is ample justification. But I should say at once that my defence of mathematics will be a defence of myself, and that my apology is bound to be to some extent egotistical. I should not think it worth while to apologize for my subject if I regarded myself as one of its failures.

Some egotism of this sort is inevitable, and I do not feel that it really needs justification. Good work is not done by ‘humble’ men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking ‘Is what I do worth while?’ and ‘Am I the right person to do it?’ will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.

2

我打算替数学做一次辩白。也许有人会和我说,数学根本不需要这些,因为当下很少有研究工作能像数学一样,无论出于什么原因,都能被公认为是有益的,并且也值得称道。这也许是真的。事实上,由于爱因斯坦 1 激动人心的成果,在大众眼里,可能只有恒星天文学和原子物理学的地位会比数学高。数学家不必认为自己正处于守势,也不需要面对像布拉德利 2 在维护形而上学时所做的辩白里描述的那种敌意,那份令人钦佩的辩白就是《现象与实在》的引言。

1Albert Einstein(1879.03.14—1955.04.18),德国 / 瑞士 / 美国科学家、物理学家。

2Francis Herbert Bradley(1846.01.30—1924.09.18),英国哲学家。

据布拉德利说,人们会对形而上学家说,“形而上学的知识是根本不存在的”,或是“即便在某种情况下它们是存在的,但实际上它们还是没有什么名副其实的内容”。还有人会说:“同样的问题,同样的争论,同样的彻底溃败。为什么不另起炉灶呢?难道没有别的事情值得去做了吗?”没有人会愚蠢到对数学说这种话。大量数学真理的权威性是明摆着的。它的实际应用随处可见,桥梁、蒸汽机和发电机都是例子。不用唠叨,人们就知道数学很有用。

在某种程度上,所有这些都能让数学家感到欣慰,但真正的数学家几乎不可能会对此感到满意。任何一位真正的数学家一定会认为,数学的口碑所仰仗的并不是这些朴素的实际应用成果,它在很大程度上是出于人们的无知和不解,所以还有更合理的辩词。无论如何,我打算试一试。相较于布拉德利艰难地为形而上学辩白,这应该会简单些。

那么我得问,为什么认真研究数学的确是值得的呢?数学家存在的意义又是什么呢?在很大程度上,我的答案就是数学家的答案:我认为数学研究是值得的,数学家的存在也是有充分理由的。但同时我还要说明,我为数学的辩白也是在为自己说话,这份辩白在某种程度上必然会很本位。如果我认为自己在数学上很失败,那就不会认为有必要为它辩白。

这种本位主义是不可避免的,我不认为真的需要为此辩解。优秀的成果不是由那些“谦虚”的人做出来的。无论什么学科,教授的首要职责之一,便是把他教的课程以及自己在其中的重要性稍作夸大。一个人若总是问自己“我做的事值得吗?”“我是研究这个的合适人选吗?”,那就永远做不好自己,也会让别人情绪低落。他必须不要太在意,稍微拔高一下学科和自身。这点并不难做到,不盲目把它们吹嘘得荒唐可笑才是更难的。

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