赫尔曼·闵可夫斯基Hermann Minkowski

赫尔曼·闵可夫斯基的父母是商人Lewin 闵可夫斯基和Rachel Taubmann。雅各布·赫尔曼是他们父母的第三个儿子。赫尔曼的大哥Max（1844-1930）接管了家族生意，但他也是一位艺术收藏家和柯尼斯堡的法国领事。二哥Oskar（1858-1931）是一名医生，以糖尿病研究最为著名，也是天体物理学家Rudolph Minkowski（1895-1976）的父亲。除了Max和Oskar，闵可夫斯基还有一个姐姐Fanny（1863-1954）和一个弟弟Toby（1873-1906）。Lewin和Rachel 闵可夫斯基是德国人，尽管他们的儿子赫尔曼出生时他们住在俄罗斯。当赫尔曼八岁时，全家回到德国并定居在柯尼斯堡，Lewin 闵可夫斯基在那里经营他的生意。

闵可夫斯基在柯尼斯堡的文理中学学习时首次展现出数学天赋。在这个教育阶段，他已经在阅读理查德·戴德金、约翰·彼得·古斯塔夫·勒热纳·狄利克雷和卡尔·弗里德里希·高斯的著作。他当时表现出的杰出能力在海因里希·马丁·韦伯（当时在柯尼斯堡大学）于1881年写给理查德·戴德金的一封信中被提及（见[14]）。他在柯尼斯堡大学学习，于1880年4月入学。他在柏林大学度过了三个学期，例如在1882-83学年的冬季学期在那里度过。他在柯尼斯堡期间与大卫·希尔伯特成为亲密朋友，因为大卫·希尔伯特与闵可夫斯基同时是本科生。1884年，当他还是一名柯尼斯堡的学生时，阿道夫·赫维兹被任命为教职员。学生闵可夫斯基很快与新任命的学者阿道夫·赫维兹成为亲密朋友。他于1885年从柯尼斯堡获得博士学位，学位论文题为Untersuchungen über quadratische Formen, Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enthält Ⓣ（关于平方型的研究，确定包含给定亏格的不同形式的数目）闵可夫斯基在大学学习早期就对二次型产生了兴趣。1881年，Academy of Sciences（巴黎）宣布，将于1883年颁发的数学科学大奖将授予解决将一个整数表示为五个平方数之和的表示数问题。费迪南·艾森斯坦在1847年给出了这种表示数的公式，但他没有给出该结果的证明。事实上，Academy为大奖设置了一个已经被解决的问题，因为亨利·约翰·斯蒂芬·史密斯在1867年发表了一个证明概要。然而，当奖项主题被设定时，Academy并不知道亨利·约翰·斯蒂芬·史密斯的贡献。

费迪南·艾森斯坦在1847年发表他未经证明的公式时，一直在研究具有整数系数的$n$元二次型，但由于他当时已经生病，细节从未发表。闵可夫斯基虽然当时只有十八岁，但重建了费迪南·艾森斯坦的二次型理论，并为大奖问题提供了一个漂亮的解答。亨利·约翰·斯蒂芬·史密斯重新整理了他早先的证明，增加了细节，并将其提交给科学院。决定是奖项由闵可夫斯基和亨利·约翰·斯蒂芬·史密斯共享，但这对闵可夫斯基的数学职业生涯来说是一个惊人的开端。1883年4月2日，Academy将数学大奖联合授予了职业生涯初期的年轻闵可夫斯基和职业生涯末期的年长亨利·约翰·斯蒂芬·史密斯。闵可夫斯基于1885年提交的博士论文是这项获奖工作的延续，涉及他对形式亏格的自然定义。获得博士学位后，他继续在柯尼斯堡进行研究。

1887年，波恩大学的一个教授职位空缺，闵可夫斯基申请了该职位；根据德国大学的规定，他必须向院系口头提交一篇原创论文，作为Habilitationsschrift。闵可夫斯基提交了Räumliche Anschauung und Minima positiv definiter quadratischer FormenⓉ（空间直观与正定二次型的最小值），该论文当时未发表，但1991年这篇演讲发表在[12]中。让·迪厄多内写道：-

这篇演讲特别有趣，因为它包含了闵可夫斯基几年后在其著名的“数的几何”中发展的方法的第一个例子。

闵可夫斯基从1887年起在波恩任教，1892年晋升为助理教授。两年后，他回到柯尼斯堡，在那里任教两年，然后被任命到苏黎世联邦理工学院。在那里，他成为朋友阿道夫·赫维兹的同事，后者在费迪南德·格奥尔格·弗罗贝尼乌斯于1892年离开苏黎世前往柏林后，被任命填补其讲席。阿尔伯特·爱因斯坦是他所授几门课程的学生，两人后来对相对论中的类似问题产生了兴趣。闵可夫斯基于1897年在斯特拉斯堡与奥古斯特·阿德勒结婚；他们有两个女儿，莉莉生于1898年，露丝生于1902年。

在他们第二个女儿出生的那一年，全家离开了苏黎世，因为闵可夫斯基于1902年接受了哥廷根大学的讲席。正是大卫·希尔伯特安排专门为闵可夫斯基设立了这一讲席，他终身担任此职。在哥廷根，他对数学物理产生了兴趣，从大卫·希尔伯特及其合作者那里获得了热情。他参加了1905年关于电子理论的讨论班，学习了电动力学的最新结果和理论。

闵可夫斯基发展了新的空间和时间观，并为相对论奠定了数学基础。到1907年，闵可夫斯基意识到亨德里克·洛伦兹和阿尔伯特·爱因斯坦的工作最好在非欧几里得空间中理解。他认为以前被认为独立的空间和时间，在四维“时空连续统”中耦合在一起。闵可夫斯基提出了电动力学的四维处理。他在这一领域的主要著作是Raum und ZeitⓉ（空间与时间）（1907年）和Zwei Abhand lungen über die Grundgleichungen der ElektrodynamikⓉ（关于电动力学基本方程的两篇贡献）（1909年）。莫里斯·克莱因在评论[10]时写道：-

这篇论文的一个关键点是数学物理学家与理论物理学家在处理物理问题上的方法差异。在1908年发表的一篇论文中，闵可夫斯基通过引入四维（时空）非欧几何重新表述了阿尔伯特·爱因斯坦1905年的论文，当时阿尔伯特·爱因斯坦对此举并不以为然。但更重要的是闵可夫斯基、大卫·希尔伯特——闵可夫斯基曾与其共事数年——菲利克斯·克莱因和赫尔曼·外尔所秉持的态度或哲学，即：在接纳新的物理事实时，应让纯粹的数学考量，包括思想的和谐与优雅，占据主导地位。可以说，数学是主人，物理理论则可以被驯服以服从主人。换言之，理论物理学是数学物理学的一个子领域，而数学物理学又是纯数学的一个子学科。在这种观点下，闵可夫斯基追随了儒勒·昂利·庞加莱，后者的哲学是：与理论物理学相对，数学物理学能够提供新的物理原理。这种哲学似乎是对十八世纪观点的一种延续（当然经过了修改），即世界是按数学设计的，因此世界必须服从数学家所揭示的原理和定律，如皮埃尔·莫佩尔蒂、约瑟夫·拉格朗日和威廉·哈密顿的最小作用量原理。阿尔伯特·爱因斯坦是一位理论物理学家，对他而言，数学必须适应物理学。

这个时空连续统为后来所有相对论中的数学工作提供了框架。这些思想被阿尔伯特·爱因斯坦用于发展广义相对论。事实上，正如Corry在[7]中指出的，闵可夫斯基对阿尔伯特·爱因斯坦产生了重大影响：-

在其科学事业的早期，阿尔伯特·爱因斯坦认为数学仅仅是服务于物理直觉的工具。到了晚年，他开始将数学视为科学创造力的根本源泉。这一转变的一个主要动因是两位杰出的德国数学家的影响：大卫·希尔伯特和闵可夫斯基。

我们在这部传记中已多次提到，闵可夫斯基和大卫·希尔伯特是亲密的朋友。不太为人所知的是，闵可夫斯基实际上曾向大卫·希尔伯特建议，他应该把什么作为其1900年巴黎著名演讲的主题。闵可夫斯基在1900年1月5日写给大卫·希尔伯特的一封信中写道：-

最具影响力的做法是尝试对未来作一预览，即勾勒出未来数学家应当致力于解决的问题。这样，你或许可以确保人们在未来几十年里都会谈论你的演讲。

时间无疑证明了闵可夫斯基是正确的！

第一届国际数学家大会于1897年在苏黎世举行。[8] 闵可夫斯基于1896年12月加入组织委员会——他可能尚未在苏黎世参加7月的预备会议。他加入了娱乐委员会，并被任命为负责选择演讲者的分委员会成员。他建议邀请大卫·希尔伯特作报告，以防菲利克斯·克莱因无法出席，而事实上，菲利克斯·克莱因确实出席了大会，但大卫·希尔伯特没有。闵可夫斯基也提出自己在一个分组会议上作报告，但由于会议记录中未解释的原因，他最终没有作。在大会上，他主持了第一组：算术与代数。

闵可夫斯基在1900年巴黎国际数学家大会上担任秘书之一，并在1904年海德堡国际数学家大会的第一组作了报告，题为Zur Geometrie der Zahlen（论数的几何）。此时他代表哥廷根大学，同样在1908年罗马国际数学家大会上也是如此。

闵可夫斯基最初的数学兴趣在于纯数学，他花费大量时间研究二次型和连分数。然而，他最原创的成就是他在1890年开创的“数的几何”。Geometrie der Zahlen Ⓣ（数的几何）于1910年首次出版，但前240页（共256页）作为第一部分于1896年问世。Geometrie der Zahlen于1953年由纽约切尔西出版社重印，并于1968年再次重印。闵可夫斯基于1907年出版了Diophantische Approximationen: Eine Einführung in die Zahlentheorie Ⓣ（丢番图逼近：数论导论）。该书对他的数的几何工作及其在丢番图逼近和代数数理论中的应用作了初等阐述。数的几何的工作进而引出了对凸体的研究以及关于堆积问题的问题，即给定形状的图形如何放置在另一个给定图形内。

在年仅44岁时，闵可夫斯基因阑尾破裂突然去世。

Hermann Minkowski's parents were Lewin Minkowski, a businessman, and Rachel Taubmann. Hermann was his parents' third son. Hermann's oldest brother Max (1844-1930) took over the family business, but he was also an art collector and the French consul in Königsberg. The second brother Oskar (1858-1931) was a physician, best known for his work on diabetes, and father of astrophysicist Rudolph Minkowski (1895-1976). Apart from Max and Oskar, Minkowski also had an older sister, Fanny (1863-1954) and a younger brother, Toby (1873-1906). Lewin and Rachel Minkowski were Germans although their son Hermann was born while they were living in Russia. When Hermann was eight years old the family returned to Germany and settled in Königsberg where Lewin Minkowski conducted his business.

Minkowski first showed his talent for mathematics while studying at the Gymnasium in Königsberg. Already at this stage in his education he was reading the work of Dedekind, Dirichlet and Gauss. The outstanding abilities he showed at this time were noted in a letter that Heinrich Weber, then at Königsberg University, wrote to Dedekind in 1881 (see [14]). He studied at the University of Königsberg, entering the university in April 1880. He spent three semesters at the University of Berlin, for example spending the winter semester of the academic year 1882-83 there. His became close friends with Hilbert while at Königsberg, for Hilbert was an undergraduate at the same time as Minkowski. In 1884, while he was a student at Königsberg, Hurwitz was appointed to the staff. The student Minkowski soon became close friends with the newly appointed academic Hurwitz. He received his doctorate in 1885 from Königsberg for a thesis entitled Untersuchungen über quadratische Formen, Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enthält Ⓣ Minkowski became interested in quadratic forms early in his university studies. In 1881 the Academy of Sciences (Paris) announced that the Grand Prix for mathematical science to be awarded in 1883 would be for a solution to the problem of the number of representations of an integer as the sum of five squares. Eisenstein had given a formula for the number of such representations in 1847, but he had not given a proof of the result. In fact the Academy had set a problem for the Grand Prix which had already been solved, for Henry Smith had published an outline of a proof in 1867. However the Academy were unaware of Smith's contributions when the prize topic was set.

Eisenstein had been studying quadratic forms in $n$ variables with integer coefficients at the time he published his unproved formula in 1847 but as he was already ill by this time details were never published. Minkowski, although only eighteen years old at the time, reconstructed Eisenstein's theory of quadratic forms and produced a beautiful solution to the Grand Prix problem. Smith reworked his earlier proof, adding detail and submitted that to the Academy. The decision was that the prize be shared between Minkowski and Smith but this was a stunning beginning to Minkowski's mathematical career. On 2 April 1883 the Academy granted the Grand Prize in Mathematics jointly to the young Minkowski at the start of his career and the elderly Smith at the end of his. Minkowski's doctoral thesis, submitted in 1885, was a continuation of this prize winning work involving his natural definition of the genus of a form. After the award of his doctorate, he continued undertaking research at Königsberg.

In 1887, a professorship became vacant at the University of Bonn, and Minkowski applied for that position; according to the regulations of German universities, he had to submit orally to the faculty an original paper, as an Habilitationsschrift. Minkowski presented Räumliche Anschauung und Minima positiv definiter quadratischer Formen Ⓣ which was not published at the time but in 1991 the lecture was published in [12]. Dieudonné writes:-

This lecture is particularly interesting, for it contains the first example of the method which Minkowski would develop some years later in his famous "geometry of numbers".

Minkowski taught at Bonn from 1887, being promoted to assistant professor in 1892. Two years later he moved back to Königsberg where he taught for two years before being appointed to the Eidgenössische Polytechnikum Zürich. There he became a colleague of his friend Hurwitz who had been appointed to fill Frobenius's chair after he left Zürich for Berlin in 1892. Einstein was a student in several of the courses he gave and the two would later become interested in similar problems in relativity theory. Minkowski married Auguste Adler in Strasbourg in 1897; they had two daughters, Lily born in 1898 and Ruth born in 1902.

The family left Zürich in the year that their second daughter was born for Minkowski accepted a chair at the University of Göttingen in 1902. It was Hilbert who arranged for the chair to be created specially for Minkowski and he held it for the rest of his life. At Göttingen he became interested in mathematical physics gaining enthusiasm from Hilbert and his associates. He participated in a seminar on electron theory in 1905 and he learnt the latest results and theories in electrodynamics.

Minkowski developed a new view of space and time and laid the mathematical foundation of the theory of relativity. By 1907 Minkowski realised that the work of Lorentz and Einstein could be best understood in a non-euclidean space. He considered space and time, which were formerly thought to be independent, to be coupled together in a four-dimensional 'space-time continuum'. Minkowski worked out a four-dimensional treatment of electrodynamics. His major works in this area are Raum und Zeit Ⓣ (1907) and Zwei Abhand lungen über die Grundgleichungen der Elektrodynamik Ⓣ (1909). Kline, reviewing [10] writes:-

A key point of the paper is the difference in approach to physical problems taken by mathematical physicists as opposed to theoretical physicists. In a paper published in 1908 Minkowski reformulated Einstein's 1905 paper by introducing the four-dimensional (space-time) non-Euclidean geometry, a step which Einstein did not think much of at the time. But more important is the attitude or philosophy that Minkowski, Hilbert - with whom Minkowski worked for a few years - Felix Klein and Hermann Weyl pursued, namely, that purely mathematical considerations, including harmony and elegance of ideas, should dominate in embracing new physical facts. Mathematics so to speak was to be master and physical theory could be made to bow to the master. Put otherwise, theoretical physics was a subdomain of mathematical physics, which in turn was a subdiscipline of pure mathematics. In this view Minkowski followed Poincaré whose philosophy was that mathematical physics, as opposed to theoretical physics, can furnish new physical principles. This philosophy would seem to be a carry-over (modified of course) from the Eighteenth Century view that the world is designed mathematically and hence that the world must obey principles and laws which mathematicians uncover, such as the principle of least action of Maupertuis, Lagrange and Hamilton. Einstein was a theoretical physicist and for him mathematics must be suited to the physics.

This space-time continuum provided a framework for all later mathematical work in relativity. These ideas were used by Einstein in developing the general theory of relativity. In fact Minkowski had a major influence on Einstein as Corry points out in [7]:-

In the early years of his scientific career, Albert Einstein considered mathematics to be a mere tool in the service of physical intuition. In later years, he came to consider mathematics as the very source of scientific creativity. A main motive behind this change was the influence of two prominent German mathematicians: David Hilbert and Hermann Minkowski.

We have mentioned several times in this biography that Minkowski and Hilbert were close friends. Less well known is the fact that Minkowski actually suggested to Hilbert what he should take as the theme for his famous 1900 lecture in Paris. Minkowski, in a letter to Hilbert written on 5 January 1900, writes:-

What would have the greatest impact would be an attempt to give a preview of the future, i.e. a sketch of the problems with which future mathematicians should occupy themselves. In this way you could perhaps make sure that people would talk about your lecture for decades in the future.

Time has certainly proved Minkowski correct!

The first International Congress of Mathematicians was held in Zürich in 1897. [8] Minkowski joined the organising committee in December 1896 -- he might not yet have been in Zürich for the preliminary meeting in July. He joined the amusement committee and was appointed to the sub-committee that was responsible for choosing the speakers. He suggested inviting Hilbert to give a talk in case Klein could not attend, as it was, Klein did attend the congress but Hilbert did not. Minkowski also offered to give a talk himself in one of the section meetings, but for reasons that are not explained in the minutes he did not after all. At the congress, he chaired section I: Arithmetic and Algebra.

Minkowski acted as one of the secretaries at the 1900 ICM in Paris, and gave a talk in section I at the 1904 ICM in Heidelberg, entitled Zur Geometrie der Zahlen (On the Geometry of Numbers). At this point he represented the University of Göttingen, likewise at the 1908 ICM in Rome.

Minkowski's original mathematical interests were in pure mathematics and he spent much of his time investigating quadratic forms and continued fractions. His most original achievement, however, was his 'geometry of numbers' which he initiated in 1890. Geometrie der Zahlen Ⓣ was first published in 1910 but the first 240 pages (of the 256) appeared as the first section in 1896. Geometrie der Zahlen was reprinted in 1953 by Chelsea, New York, and reprinted again in 1968. Minkowski published Diophantische Approximationen: Eine Einführung in die Zahlentheorie Ⓣ in 1907. It gave an elementary account of his work on the geometry of numbers and of its applications to the theories of Diophantine approximation and of algebraic numbers. Work on the geometry of numbers led on to work on convex bodies and to questions about packing problems, the ways in which figures of a given shape can be placed within another given figure.

At the young age of 44, Minkowski died suddenly from a ruptured appendix.