汉斯·哈恩Hans Hahn

汉斯·哈恩的父母是Ludwig Benedikt 哈恩和Emma Blümel。哈恩曾在斯特拉斯堡、慕尼黑和哥廷根求学。他在维也纳大学师从Gustav Ritter von 古斯塔夫·冯·埃舍里希攻读博士学位。1902年，他凭借学位论文Zur Theorie der zweiten Variation einfacher Integrale Ⓣ（论简单积分的第二变分理论）获得博士学位。在维也纳期间，他与另外三位数学学生保罗·埃伦费斯特、Heinrich Tietze和古斯塔夫·赫格洛茨结下了亲密友谊。他们被称为“形影不离的四人的”。

哈恩于1905年提交教授资格论文（Habilitation）学位论文后，被任命为维也纳的privatdozent。在1905-06学年，哈恩在因斯布鲁克替代了奥托·施托尔茨。他从1909年到1914年在奥匈帝国的切尔诺维茨担任编外教授；切尔诺维茨在第一次世界大战后成为罗马尼亚的一部分时更名为切尔讷乌齐，然后在1940年成为苏联的一部分后更名为切尔诺夫策。哈恩在第一次世界大战中在奥匈军队服役并受了重伤。1916年，他搬到波恩，在那里被任命为编外教授，一直任教到1920年，并于1917年被任命为讲席。他于1921年回到维也纳担任讲席。他在维也纳最著名的三个学生是卡尔·门格尔，他于1924年获得博士学位，维托尔德·胡列维茨，他于1926年获得博士学位，以及库尔特·弗雷德里希·哥德尔，他于1929年获得博士学位。

卡尔·门格尔 论述了 哈恩 在第一次世界大战前的贡献：-

哈恩 的最初成果是对经典 变分法 的贡献。随后他转向实函数与集合函数的研究，特别是积分。他还发表了一篇关于非阿基米德系统的奠基性论文，并早早认识到 莫里斯·弗雷歇 抽象空间的意义。在一篇引入局部连通性的论文中，他刻画了一个点能在连续运动中遍历的集合；也就是说，一个时间区间或线段的连续像（现在常称为 朱塞佩·皮亚诺 连续统）。这篇论文是早期集合论几何的经典之作。

正如 卡尔·门格尔 所解释的，哈恩 是集合论和 泛函分析 的先驱。然而对许多数学家来说，他最令人铭记的是我们下面再次提到的 Hahn-Banach theorem。他还在变分法上做出了重要贡献，主要是在1903年至1913年间，发展了 卡尔·魏尔斯特拉斯 的思想。他对实分析感兴趣，在该领域撰写了各种不同主题的文章。他考虑了导数的性质、用定积分表示函数、半连续函数以及多变量的分别连续函数。1923年，他引入了今天被称为 哈恩 序列空间的概念。他还写了两本关于实函数的书，Theorie der reellen Funktionen Ⓣ（实函数论）（1921年）和 Reelle Funktionen Ⓣ（实函数）（1932年）。

哈恩写了四篇关于泛函分析的论文。这些包括他在1911年写的一篇关于积分方程的报告，他对恩斯特·黑林格二次型不变量理论的修改，其中他摒弃了恩斯特·黑林格积分的使用，以及他在Banach spaces中对偶性的工作，最终以他在1927年证明哈恩-斯特凡·巴拿赫定理而告终。他写了关于曲线理论的论文，其中一篇给出了简单闭合多边形的卡米耶·若尔当定理的严格证明，他将其建立在奥斯瓦尔德·维布伦的几何公理之上。该领域的其他论文刻画了作为线段连续像的拓扑空间，与此主题相关的是现在被称为哈恩-斯特凡·马祖尔凯维奇定理的东西。他还研究了有序阿贝尔群和有序域的理论，于1907年开创了该理论（例如见[6]）。

哈恩研究的另一个领域是测度论。在这个领域，他研究了将昂利·勒贝格积分构造为波恩哈德·黎曼和的极限，这是埃米尔·博雷尔在1910年左右提出的积分，并研究了抽象测度理论，特别是乘积测度。傅里叶分析也引起了哈恩的兴趣，他研究了奇异积分和正交展开，考察了马克-安托万·帕塞瓦尔关系在各种情况下的有效性。在一些论文中，他研究了广义调和分析（独立于诺伯特·维纳），他还写了一篇关于费耶尔可和性的短注。

尽管在广泛的数学主题上有如此丰富的深刻论文，许多人仍将哈恩视为一位数学哲学家。在1920年代，哈恩与菲利普·弗兰克、奥托·纽拉特、莫里茨·石里克一起创立了逻辑实证主义的维也纳学派，这是一个由才华横溢的科学家和哲学家组成的讨论小组，定期在维也纳聚会。门格尔在文章Hahn and the Vienna Circle中写到了哈恩在维也纳学派中的角色。

理查德·冯·米泽斯 也是该学派的成员：-

[哈恩] 坚持认为逻辑和数学本质上是重言式，对外部世界无所言说。然而他承认，选择公理 不是重言式；他说，我们接受还是否认它，取决于我们希望“集合”一词意味着什么。在他关于无穷存在的文章中，他说对于涉及无穷集合的公理系统，不可能有绝对的一致性证明。

哈恩 关于数学直觉的思想的一个例子，见于我们在 THIS LINK 条目文章中转载的摘录。

哈恩关于数学直觉的思想在曼德尔布罗于1982年写的一篇文章中受到强烈批评。他写道，哈恩讲座英译本的编辑（1956年出版）：-

……指出[哈恩]讨论了备受珍视的直觉能力、它在数学中的作用、它使我们陷入的悖论之巢，以及我们在爬出来方面有多成功。我的反应非常不同：分形几何表明哈恩大错特错。直觉并非一成不变，而是能够并且必须经过训练以执行新任务。

哈恩因其成就获得许多荣誉，包括1921年的Lieban奖。他当选为奥地利科学院（维也纳皇家科学院），并被授予加尔各答数学学会荣誉会员。

哈恩于1934年去世，他仍然设法成为Set Functions一书的作者，该书于1948年出版，即他去世十四年后。该书由Arthur Rosenthal合著，并继续发展最初出现在斯坦尼斯拉夫·萨克斯的Theory of the Integral（1937年）中的材料。保罗·哈尔莫斯写道：-

该书分为一个导论（包含集合论和点集拓扑学相关部分的阐述）和五章，标题为（I）加性与完全加性集函数，（II）测度，（III）可测函数，（IV）积分和（V）微分。作者的处理详尽、严谨且全面；除少数例外，它包括斯坦尼斯拉夫·萨克斯一书的前四章，与那些章节的主要区别在于细节的丰富。所研究的集函数要么定义在完全抽象的集合中，要么定义在度量空间（特别是欧几里得空间）中；局部紧空间、正规空间和拓扑群的中间情形没有讨论。

Hans Hahn's parents were Ludwig Benedikt Hahn and Emma Blümel. Hans was a student at Strasbourg, Munich and Göttingen. He undertook research for his doctorate at the University of Vienna under Gustav Ritter von Escherich. He was awarded a Ph.D. in 1902 for his thesis Zur Theorie der zweiten Variation einfacher Integrale Ⓣ . While in Vienna he formed a close friendship with three other students of mathematics, Paul Ehrenfest, Heinrich Tietze and Herglotz. They were known as the 'inseparable four'.

Hahn was appointed to the teaching staff in Vienna as a privatdozent in 1905 after submitting his habilitation thesis. In session 1905-06 Hahn substituted for Otto Stolz at Innsbruck. He taught at Czernowitz in Austria-Hungary from 1909 to 1914 as an extraordinary professor; Czernowitz was renamed Cernauti when it became part of Romania after World War I, and then Chernovtsy after it became part of the USSR in 1940. Hahn served in the Austro-Hungarian army in World War I and was severely wounded. In 1916 he moved to Bonn where he was appointed as an extraordinary professor teaching there until 1920 having been appointed to a chair in 1917. He returned to a chair in Vienna in 1921. His three most famous students at Vienna were Karl Menger who was awarded his doctorate in 1924, Witold Hurewicz who was awarded his doctorate in 1926, and Kurt Gödel who was awarded his doctorate in 1929.

Menger writes about Hahn's contributions before World War I:-

Hahn's first results were contributions to the classical calculus of variations. He then turned to the study of real functions and set functions, especially integrals. He further published a fundamental paper on non-Archimedean systems, and early recognised the significance of Fréchet's abstract spaces. In a paper introducing local connectedness he characterised the sets which a point can traverse in a continuous motion; this is, the continuous images of a time interval or a segment (now often called Peano continua). The paper is a classic of the early set-theoretical geometry.

As Menger explains, Hahn was a pioneer in set theory and functional analysis. However to many mathematicians he is best remembered for the Hahn-Banach theorem which we mention again below. He also made important contributions to the calculus of variations, mostly between 1903 and 1913, developing ideas of Weierstrass. He was interested in real analysis, writing on a variety of different topics in that area. He considered properties of the derivative, the representation of functions by definite integrals, semicontinuous functions, and separately continuous functions of several variables. In 1923 he introduced what today is known as the Hahn sequence space. He also wrote two books on real functions, Theorie der reellen Funktionen Ⓣ (1921), and Reelle Funktionen Ⓣ (1932).

Hahn wrote four papers on functional analysis. These include a report on integral equationS he wrote in 1911, his modification of Hellinger's theory of invariants of quadratic forms, in which he dispensed with the use of the Hellinger integral, and his work on duality in Banach spaces, culminating with his proof of the Hahn-Banach theorem in 1927. He wrote papers on the theory of curves including one which gave a rigorous proof of the Jordan's theorem for simple closed polygons which he based on Veblen's geometrical axioms. Other papers in this area characterise topological spaces that are continuous images of a line segment and related to this topic is what is now known as the Hahn-Mazurkiewicz theorem. He also studied the theory of ordered abelian groups and ordered fields, initiating the theory in 1907 (see for example [6]).

Another area on which Hahn did research was measure theory. In this area he studied a construction of the Lebesgue integral as a limit of Riemann sums, an integral proposed by Borel around 1910, and worked on the theory of abstract measures, in particular product measures. Fourier analysis also interested Hahn, and he looked at singular integrals and orthogonal expansions investigating the validity of the Parseval relation in various circumstances. In some papers he looked at generalised harmonic analysis (independently of Norbert Wiener), and he also wrote a short note on Fejér summability.

Despite this wealth of deep papers on a wide range of mathematical topics, many people think of Hahn as a mathematical philosopher. During the 1920s Hahn, together with Philipp Frank, Otto Neurath, Moritz Schlick, founded the Vienna Circle of Logical Positivists, a discussion group of gifted scientists and philosophers who met regularly in Vienna. Menger writes on Hahn's role in the Vienna Circle in the article Hahn and the Vienna Circle.

Von Mises was also a member of the Circle:-

[Hahn] maintains that logic and mathematics are essentially tautological and say nothing about the external world. He admits, however, that the axiom of choice is not a tautology; whether we accept it or deny it, he says, depends on what we want the word "set" to mean. In his essay on the existence of infinity he says that there can be no absolute proof of consistency for an axiomatic system concerned with infinite sets.

An example of Hahn's ideas on mathematical intuition are given in extracts we reproduce in the article at THIS LINK.

However, Hahn's ideas on mathematical intuition are strongly criticised by Benoit Mandelbrot in an article he wrote in 1982. He writes that the editor of the English translation of Hahn's lectures (published in 1956):-

... states that [Hahn] discusses the cherished faculty of intuition, its role in mathematics, the nest of paradoxes it got us into, and how successful we have been in crawling out. My reaction is very different: Fractal geometry demonstrates that Hahn was dead wrong. Intuition is not invariable but can and must be trained to perform new tasks.

Hahn received many honours for his achievements including the Lieban Prize in 1921. He was elected to the Austrian Academy of Sciences (Kaiserliche Akademie der Wissenschaften in Wien), and was made an honorary member of the Calcutta Mathematical Society.

Although Hahn died in 1934, he still managed to be an author of the book Set Functions which was published in 1948, fourteen years after his death. The book was co-authored by Arthur Rosenthal and continued to develop material which first appeared in Saks's Theory of the Integral (1937). Halmos writes:-

The book is divided into an introduction (containing an exposition of the relevant parts of set theory and point set topology) and five chapters, entitled (I) Additive and totally additive set functions, (II) Measure, (III) Measurable functions, (IV) Integration and (V) Differentiation. The authors' treatment is thorough, rigorous and exhaustive; it includes, with minor exceptions, the first four chapters of Saks's book and differs from those chapters mainly in the wealth of detail. The set functions studied are defined either in a perfectly abstract set, or else in metric spaces (and, in particular, Euclidean spaces); the intermediate cases of locally compact spaces, normal spaces and topological groups are not discussed.