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[资料，科普] 量子力学的哥本哈根诠释（12）：庞加莱 Jules Henri Poincaré 与数学

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[资料，科普] 量子力学的哥本哈根诠释（12）：庞加莱 Jules Henri Poincaré 与数学

图1 庞加莱 Jules Henri Poincaré, 1854-04-29 ~ 1912-07-17, 58

https://ids.si.edu/ids/deliveryService?id=SIL-SIL14-p005-03&iframe=true&container.fullpage

https://library.si.edu/image-gallery/74033

以下对应英文的汉语，来自机器翻译。

机器翻译的某些术语，有待您进一步自行核对。

一、庞加莱 Jules Henri Poincaré

https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/

Poincaré's popular works include Science and Hypothesis (1901), The Value of Science (1905), and Science and Method (1908). A quote from these writings is particularly relevant to this archive on the history of mathematics. In 1908 he wrote:-

The true method of foreseeing the future of mathematics is to study its history and its actual state.

Finally we look at Poincaré's contributions to the philosophy of mathematics and science. The first point to make is the way that Poincaré saw logic and intuition as playing a part in mathematical discovery. He wrote in Mathematical definitions in education (1904):-

It is by logic we prove, it is by intuition that we invent.

In a later article Poincaré emphasised the point again in the following way:-

Logic, therefore, remains barren unless fertilised by intuition.

McLarty [119] gives examples to show that Poincaré did not take the trouble to be rigorous. The success of his approach to mathematics lay in his passionate intuition. However intuition for Poincaré was not something he used when he could not find a logical proof. Rather he believed that formal arguments may reveal the mistakes of intuition and logical argument is the only means to confirm insights. Poincaré believed that formal proof alone cannot lead to knowledge. This will only follow from mathematical reasoning containing content and not just formal argument.

庞加莱的通俗著作包括《科学与假设》（1901年）、《科学的价值》（1905年）和《科学与方法》（1908年）。这些著作中的一句话与这个数学史档案特别相关。1908年，他写道：-

预测数学未来的真正方法是研究它的历史和现状。

最后，我们来看看庞加莱对数学和科学哲学的贡献。首先，庞加莱认为逻辑和直觉在数学发现中起着重要作用。他在《教育中的数学定义》（1904）中写道：-

我们用逻辑证明，用直觉发明。

在后来的一篇文章中，庞加莱再次强调了这一点：-

因此，除非直觉滋养，否则逻辑仍然是贫瘠的。

McLarty[119]举了一些例子来证明庞加莱并没有费心去严谨。他数学方法的成功在于他充满激情的直觉。然而，当庞加莱无法找到逻辑证明时，他并没有使用直觉。相反，他认为形式论证可能会揭示直觉的错误，逻辑论证是确认见解的唯一手段。庞加莱认为，仅凭形式证明无法带来知识。这只能从包含内容的数学推理中得出，而不仅仅是形式论证。

二、Quotations, Henri Poincaré

https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/quotations/

Mathematicians are born, not made.

In the old days when people invented a new function they had something useful in mind. Now, they invent them deliberately just to invalidate our ancestors' reasoning, and that is all they are ever going to get out of them.

How is an error possible in mathematics? A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations. Need we add that mathematicians themselves are not infallible?

Quoted in D MacHale, Comic Sections (Dublin 1993)

数学家是天生的，不是后天培养的。

在过去，当人们发明一种新功能时，他们心里有一些有用的东西。现在，他们故意发明它们，只是为了推翻我们祖先的推理，这就是他们能从中得到的。

数学中怎么可能出现错误？一个理智的人不应该犯逻辑谬误，但也有一些非常优秀的人无法理解数学论证。我们还需要补充一点吗，数学家本身并不是绝对可靠的？

The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another.

Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988).

值得研究的数学事实是那些通过与其他事实的类比，能够引导我们了解物理定律的事实。它们揭示了其他事实之间的亲缘关系，这些事实早已为人所知，但被错误地认为彼此陌生。

Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.

Thought is only a flash between two long nights, but this flash is everything.

Quoted in J R Newman, The World of Mathematics (New York 1956).

数学家研究的不是物体，而是物体之间的关系。因此，只要关系保持不变，他们就可以自由地用其他对象替换某些对象。内容对他们来说无关紧要：他们只对形式感兴趣。

思想只是两个长夜之间的一道闪光，但这道闪光就是一切。

The mind uses its faculty for creativity only when experience forces it to do so.

Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.

Les faits ne parlent pas

Facts do not speak.

American Journal of Physics 12 (1890) 211.

只有当经验迫使大脑发挥创造力时，它才会发挥创造力。

数学发现，无论大小，都不是自发产生的。它们总是以一片播种了初步知识并由有意识和潜意识的劳动精心准备的土壤为前提。

事情已经过去了

事实不会说话。

参考资料：

[1] Jules Henri Poincaré, 2003-10, MacTutor History of Mathematics

https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/

[2] Quotations, Henri Poincaré, MacTutor History of Mathematics

https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/quotations/