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[资料，科普] 量子力学的哥本哈根诠释（13）：希尔伯特 David Hilbert 与数学

已有 235 次阅读 2025-8-23 20:42 |个人分类:资料与科普|系统分类:科普集锦

[资料，科普] 量子力学的哥本哈根诠释（13）：希尔伯特 David Hilbert 与数学

图1 希尔伯特 David Hilbert, 1862-01-23 ~ 1943-02-14, 81

https://www.factsnippet.com/site/facts-about-david-hilbert.html

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、希尔伯特 David Hilbert

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

Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. His mathematical abilities were nicely summed up by Otto Blumenthal, his first student [30]:-

In the analysis of mathematical talent one has to differentiate between the ability to create new concepts that generate new types of thought structures and the gift for sensing deeper connections and underlying unity. In Hilbert's case, his greatness lies in an immensely powerful insight that penetrates into the depths of a question. All of his works contain examples from far-flung fields in which only he was able to discern an interrelatedness and connection with the problem at hand. From these, the synthesis, his work of art, was ultimately created. Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and of the classical great ones, Gauss, Galois, and Riemann. But when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert.

希尔伯特对数学的许多分支做出了贡献，包括不变量、代数数域、泛函分析、积分方程、数学物理和变分法。他的第一个学生Otto Blumenthal很好地总结了他的数学能力[30]：-

在分析数学天赋时，必须区分创造新概念的能力和感知更深层次联系和潜在统一的天赋。在希尔伯特的例子中，他的伟大之处在于他对问题的深刻洞察。他所有的作品都包含了来自遥远领域的例子，只有他能够辨别出与手头问题的相互关联和联系。从这些，合成，他的艺术作品，最终被创造出来。就新思想的创造而言，我会把闵可夫斯基放在更高的位置，在经典的伟大思想中，高斯、伽罗瓦和黎曼。但说到洞察力，只有少数最伟大的人能与希尔伯特相提并论。

二、Quotations, David Hilbert

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

Wir müssen wissen. Wir werden wissen.

[We must know. We will know.]

Speech in Königsberg in 1930, now on his tomb in Göttingen.

我们必须知道。我们会知道的。

[我们必须知晓。我们终将知道。]

I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations.

Report on Number Theory, 1897.

我试图避免冗长的数值计算，从而遵循黎曼的假设，即证明应该通过思想而不是大量的计算来给出。

Mathematics is a game played according to certain simple rules with meaningless marks on paper.

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

数学是一种根据某些简单规则在纸上做无意义标记的游戏。

How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.

The art of doing mathematics consists in finding that special case which contains all the germs of generality.

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

数学科学中根深蒂固的一点是，每一次真正的进步都伴随着更锋利的工具和更简单的方法的发明，这些工具和方法同时有助于理解早期的理论，并抛弃一些更复杂的发展。

数学的艺术在于找到包含所有普遍性萌芽的特例。

The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.

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

数学理论发展得越深入，其构建就越协调一致，迄今为止分离的科学分支之间的关系也就越明显。

One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.

Quoted in H Eves Mathematical Circles Revisited (Boston 1971).

一项科学工作的重要性可以通过它使早期出版物变得多余的数量来衡量。

Physics is becoming too difficult for the physicists.

Quoted in C Reid, Hilbert (London 1970)

物理学对物理学家来说变得太难了。

Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical.

Quoted in R Remmert, Theory of complex functions (New York, 1989).

每一门数学学科都经历了三个发展阶段：幼稚、形式和批判。

In mathematics ... we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials ... being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.

Geometry and the imagination (New York, 1952).

在数学方面。我们发现存在两种趋势。一方面，抽象化的趋势试图使材料迷宫中固有的逻辑关系具体化。..正在研究，并以系统有序的方式将材料关联起来。另一方面，直觉理解的趋势促进了对所研究对象的更直接掌握，可以说是与他们建立了一种实时的融洽关系，这强调了他们关系的具体意义。

A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.

一个数学理论只有在你把它弄清楚，可以向你在街上遇到的第一个人解释之前，才能被认为是完整的。

Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.

在我看来，数学科学是一个不可分割的整体，一个有机体，它的活力取决于其各部分的连接。

参考资料：

[1] David Hilbert, 2014-11, MacTutor History of Mathematics

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

[2] Quotations, David Hilbert, MacTutor History of Mathematics

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

[3] 2024-12-05，《古今数学思想》/Mathematical Thought from Ancient to Modern Times/王涛，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=576386&Type=bkzyb&SubID=63159

[4] 2023-04-28，希尔伯特，D. /David Hilbert/李文林，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=44260&Type=bkzyb&SubID=61734