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数学有数学的实在,数学的实在不同于物理的实在。两个实在的“莫名其妙的”契合,是一个奇迹——就是爱老师说的,世界上最难理解的事情就是世界竟然是可以理解的。
那天引用了爱老师关于数学与实在的关系(借辩证法的说法,可以说“辩证关系”),似乎有的同学感兴趣,于是多说几句。
我说过,那段话有点儿“暧昧”,以前我甚至分不清谁是“确定的”,谁是“不确定的”——现在大概能分清了。分清了才敢多说。
先溯源。爱老师的那句话,是他1921年1月27日在普鲁士科学院的一个演讲(Geometry and Experience)里说的,是为了回答他的一个设问:仅靠思维能认识实在吗?
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?
接着他就说了那句名言:In my opinion the answer to this question is, briefly, this:as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
既然问题是形而上的,那句话所说的当然就不是技术上的数学的适用性——那似乎就没有答案了,怎么想都行,而且肯定是越想越糊涂。最好是不想——那或许是另一种(更现实的?)态度。
12年后,爱老师在牛津Rhodes House做Herbert Spencer演讲,就是有名的《论理论物理的方法》(on the method of theoretical physics),从另一个角度讲了理论我联系中的想象的发现:
To the discoverer in this field the products of his imagination appear so necessary and natural that he regards them, and would have them regarded by others, not as creations of thought but as given realities.【后面这句话,有的译本作…natural that he is apt to treat them not as …没管别人怎么看了】
想象的东西是那么自然而必然地出现了,它们似乎根本不是思想的创造,而就是老天的实在(既存事实)。前者呼应着数学,而后者还是实在——所以,这句话说的还是前面那个老问题。
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