奥托·赫尔德Otto Hölder

奥托·赫尔德研究Fourier series的收敛性，并于1884年发现现在以他命名的不等式。他通过利奥波德·克罗内克和菲利克斯·克莱因对group theory产生兴趣，并证明了合成列中factor groups的唯一性。他的父亲是赫尔德（1811-1890），斯图加特理工学院的法国文学教授，是Christian Gottlieb 赫尔德（1776-1847）的儿子，后者是斯图加特文理中学的教授，以及Friederike Karoline Gaupp（1784-1813）。本传记的主人公赫尔德的母亲是Pauline Christiane Ströbel（1821-1894），August Friedrich Ströbel和Friederika Christiane Moser的女儿。赫尔德和Pauline于1845年5月17日在斯图加特结婚。他们有三个儿子：Alfred，生于1846年1月2日，Eduard 赫尔德，生于1847年11月27日，以及Ludwig 赫尔德，本传记的主人公，生于1859年12月22日，并于1860年2月2日受洗。1864年，全家住在斯图加特Hermannstrasse 18号。三个儿子都成为了教授。Eduard 赫尔德在图宾根学习法律，并在几所不同的大学成为法学教授。

赫尔德在斯图加特的一所文理中学学习，实际上是在最早专攻科学的文理中学之一，1876年毕业。随后他进入斯图加特的理工学院学习工程学，但发现自己对数学的兴趣超过了对工程学的兴趣。他父亲的一位同事提出，斯图加特理工学院并非学习数学的最佳场所，赫尔德最好去柏林大学，那里拥有世界上最顶尖的数学学派之一。1877年，他进入柏林大学，开始在那里学习数学。他与卡尔·龙格是同学，并听了卡尔·魏尔斯特拉斯、利奥波德·克罗内克和恩斯特·爱德华·库默尔的讲座。在入学的第一年，他听了卡尔·魏尔斯特拉斯关于函数论的讲座，内容涵盖了分析的基础。卡尔·魏尔斯特拉斯给年轻的赫尔德留下了深刻的印象，他的影响在赫尔德的整个学术生涯中都有所体现。卡尔·龙格[9]：——

……很久以后回忆说，卡尔·魏尔斯特拉斯的讲座[给赫尔德]留下了深刻而持久的印象，尽管这些讲座并不精炼，结构也不够完善。卡尔·魏尔斯特拉斯有时会在即兴给出证明时陷入混乱，但下一次又能镇定自若地把它纠正过来。而卡尔·魏尔斯特拉斯则是一位富有同情心的导师，他认真倾听学生的意见，并真正对问题作出回应……

赫尔德对代数的兴趣部分源于此时利奥波德·克罗内克的影响，而利奥波德·克罗内克对严谨性的喜爱几乎肯定对赫尔德后来在代数方面的工作产生了深远影响。当被问及一个问题时，利奥波德·克罗内克 [9]：-

……无法让他倾听，而总是立即改变话题去谈论自己的工作。另一方面，利奥波德·克罗内克是一个平易近人得多的人，许多年轻人会被邀请到他好客的家中。

在柏林学习后，赫尔德去了埃伯哈德-卡尔斯蒂宾根大学，在那里他由Paul du Bois-Reymond指导。他于1882年向蒂宾根大学提交了他的学位论文，Beiträge zur Potentialtheorie Ⓣ（对势论的贡献），该论文通过算术平均研究解析函数和求和程序。这些求和程序现在被称为“赫尔德求和法”。他的学位论文还包含体积密度的连续性条件，现在被称为函数的“赫尔德条件”。获得博士学位后，赫尔德去了莱比锡。菲利克斯·克莱因当时在那里，但在他待在那里的两年里，两人之间似乎很少有互动。此时赫尔德仍然对函数论感兴趣，尽管菲利克斯·克莱因对赫尔德后来的职业生涯有强烈影响。他被拒绝在莱比锡habilitate的机会，因此他搬到了哥廷根。

奇怪的是，哥廷根不承认赫尔德的图宾根博士学位，因此，在1884年，他提交了一篇论文以在哥廷根获得第二个博士学位，并在同一年在哥廷根大学完成了特许任教资格。他的特许任教资格论文考察了一个既不被假定为连续也不被假定为有界的函数的约瑟夫·傅里叶级数的收敛性。起初在哥廷根，他继续研究约瑟夫·傅里叶级数的收敛性。在哥廷根开始工作后不久，他发现了现在以他命名的不等式，该不等式出现在他的论文Über einen MittelwerthsatzⓉ（《关于一个中值定理》）（1889年）中。看来赫尔德在哥廷根期间对群论产生了兴趣，部分是通过与瓦尔特·冯·戴克的讨论，部分是通过菲利克斯·克莱因，后者正在讲授伽罗瓦理论。哥廷根大学的教职员工想给赫尔德提供一个助理讲师职位，但这样的任命只能由普鲁士文化部做出，尽管一再请求，他们觉得他没有足够的授课经验来担任这样的职位。赫尔德于1889年5月被提供了图宾根的一个职位，但不幸的是，他遭受了精神崩溃。在埃尔兰根的一家诊所接受治疗时，他无法回复图宾根的提议，因此他的兄弟爱德华，当时是图宾根的法律教授，代表他的兄弟接受了这个提议。当图宾根的教职员工得知赫尔德生病并在诊所时，他们不确定该如何进行，但经过多次讨论，他们保持了对赫尔德的信心。他稳步康复，于1890年6月发表了他的就职演讲。

菲利克斯·克莱因在哥廷根关于埃瓦里斯特·伽罗瓦理论的讲座引起了赫尔德的兴趣，他开始研究方程的埃瓦里斯特·伽罗瓦理论，并从那里被引导去研究群的合成列。赫尔德证明了合成列中因子群的唯一性，这个定理现在被称为卡米耶·若尔当-赫尔德定理，并于1889年在Mathematische Annalen中发表了结果，论文为Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen Ⓣ（将任何代数方程循环到方程链中）。尽管赫尔德不认为他发明了因子群的概念，但这一概念在赫尔德的这篇论文中首次清晰地出现。他澄清了这个概念，他声称这既不新颖也不困难，但没有得到足够的重视 [13]：-

让·迪厄多内评论卡米耶·若尔当-赫尔德定理时说，只有到赫尔德时它才会呈现出确定的形式。这同样适用于商群的概念：在这两种情况下，卡米耶·若尔当的思想都是早期的表述，后来让位于公认的“标准”形式。

在群论和埃瓦里斯特·伽罗瓦理论方法的帮助下，赫尔德于1891年回到对吉罗拉莫·卡尔达诺-尼科洛·塔尔塔利亚公式中cubic不可约情况的研究。我们对以下引自[9]的引文做了小的更正：-

赫尔德是最早严格处理著名经典情形的人之一，即分裂域不是根式扩张的情形：有理数域上具有三个实根的三次不可约方程，尽管如此仍必须添加单位复根。赫尔德对这一结果的证明长期以来受到怀疑，与此同时还有另外三人的论述几乎同时出现，并在欧根·尼托的书Ⓣ《代数讲义》（Vorlesungen über Algebra）(1900)第二卷中作了概述。

赫尔德对群论还作出了许多其他贡献。他寻找有限单群，并在1892年的论文Die einfachen Gruppen im ersten und zweiten Hundert der OrdnungszahlenⓉ（序数前一百和第二百年中的单群）中，在Mathematische Annalen里证明了直到200阶的所有单群都已被知道。他的方法以类似于今天解决该问题的方式使用彼得·卢德维格·梅德尔·西罗定理。赫尔德还研究了阶为$p^{3}, pq^{2}, pqr$和$p^{4}$的群，其中$p, q, r$为素数，并于1893年发表了他的结果。他的证明同样严重依赖于彼得·卢德维格·梅德尔·西罗定理的使用。

由赫尔德引入的概念包括内自同构和外自同构。1895年，他写了一篇关于群扩张的长论文。人们常说Otto Schreier是开创群扩张研究的人，但茹利亚 Nicholson写道[2]：-

Schreier对扩张理论的探讨和发展似乎直接承继自赫尔德：赫尔德的许多方法被Schreier借用。他们工作之间的区别在于，Schreier把每个想法都推到了逻辑结论，而赫尔德最初的动机是希望对特定种类的群进行分类，因此是带着这一固定目标来发展该理论的。

在[22]中，巴特尔·伦德特·范德瓦尔登写道：-

……一遍又一遍地阅读赫尔德的论文是一种深刻的思想享受。

1889年，赫尔德被任命为图宾根大学数学特聘教授。1896年，他离开图宾根，被任命为柯尼斯堡大学正教授。1899年，赫尔德与海伦（1871-1927）结婚，她是律师、银行董事和政治家卡尔·恩斯特·劳滕施拉格（1828-1895）及其妻子索菲·威廉明娜·法贝尔（1831-1902）的女儿。海伦和赫尔德有四个孩子：恩斯特赫尔德（生于1901年）、夏洛特·索菲（生于1902年）、伊尔姆加德·路易丝（生于1904年）和沃尔夫冈·卡尔（生于1906年）。Ernst Hölder成为一名主要研究数学物理领域的数学家。他于1926年获得莱比锡大学博士学位，学位论文为Gleichgewichtsfiguren rotierender Flüssigkeiten mit Oberflächenspannung Ⓣ（具有表面张力的旋转流体的平衡图形）。他的学位论文导师是利昂·利希滕斯坦。1899年，也就是他结婚的同一年，赫尔德被任命为莱比锡大学数学正教授，接替索菲斯·李曾占据的讲席。他于1912-13年担任莱比锡大学文学院院长，并于1918年担任莱比锡大学校长。尽管他留在莱比锡，但他确实于1902年申请了柏林的讲席。大卫·希尔伯特排名第一，弗里德里希·赫曼·肖特基排名第二，赫尔德排名第三。弗里德里希·赫曼·肖特基被任命为该讲席。

从1900年起，赫尔德开始对射影直线的几何学和哲学问题产生兴趣，这些兴趣贯穿他的整个职业生涯，并开始发挥重要作用。现在让我们看看其中一些作品。也许我们应该从赫尔德于1892年发表的论文开始，他在文中对Robert Grassmann的Die Zahlenlehre oder Arithmetik - streng wissenschaftlich in strenger Formelentwicklung Ⓣ（数论或算术——在严格公式发展中严格科学）（1891）作出了反应。Mircea Radu写道[16]：-：-

赫尔德的论文至少因三个原因而重要：首先，它代表了可称为赫尔德关于数学基础的研究宣言，包含大量思想，赫尔德在其一生中通过各种出版物逐渐发展这些思想。其次，赫尔德对Robert Grassmann基础思想的分析提供了对雅各布·赫尔曼和Robert Grassmann对算术公理化作出的贡献的重要评估，这一贡献虽然经常被提及，但本身仍未得到广泛承认，也未得到充分理解。第三，揭露Robert Grassmann思想中弱点的努力促使赫尔德提出形式公理学面临的主要问题：公理的独立性、一致性、完备性，以及纯数学与其应用之间关系的问题。

赫尔德于1899年被任命为莱比锡的讲席，他发表了就职演讲《几何学中的直观与思维》，该演讲于1900年出版。这部作品给出了一个证明，表明阿基米德公理可以从理查德·戴德金的连续性概念推导出来。评论[14]指出：-

对数学、力学和精确自然科学中所采用的推理方法进行的彻底研究，使赫尔德教授确信，那里所用的演绎方法是由一系列具有相当独特形式的连锁结论组成的，因此这些科学有一种独特的自己的方法和逻辑。并不是说这些科学中的推理与其他思想和生活领域中的推理绝对不同；所讨论的独特性完全在于主题的性质以及相关智力活动组合的风格。在霍尔德教授看来，只有通过数学家和哲学家的合作，才能达到关于数学程序的正确哲学，而不是像迄今为止那样，在这一领域进行孤立和片面的劳动。他自己对所涉及思维模式所作的评论非常好，结合他在讨论后附上的详尽注释，将会对学生有所帮助。

这次就职演讲是赫尔德的量公理化理论的起点，他于1901年将其出版为Die Axiome der Quantität und die Lehre vom Mass Ⓣ（量的公理和质量学说）。Joel Michell写道[12]：-

他的论文是测量理论的分水岭，划分了古典时代（从欧几里得延伸）和现代时代（延伸至Luce等人，1990年）。他的关注点部分属于古典时代。他将古典的量概念公理化（使用理查德·戴德金的连续性概念），使得量值之比（如欧几里得的《几何原本》第五卷中所理解）可以表示为正实数（如艾萨克·牛顿所暗示）。重要的是，他实现这一结果的清晰度是其他在古典框架内工作的人所未能达到的（例如，赫尔曼·冯·亥姆霍兹，1878年；戈特洛布·弗雷格，1903年；以及阿尔弗雷德·诺思·怀特海和伯特兰·罗素，1913年）。然而，他的其他关注点更为现代。在论文的第二部分，他表明一个表面上非加性结构——直线段的公理，蕴含线性距离满足第一部分给出的量公理。他将非加性结构与定量结构联系起来的关注预示了一种明显的现代兴趣……

他哲学著作的亮点是Die Mathematische Methode Ⓣ（数学方法）（1924年）。H R Smart写道[21]：-

撰写这样一本书的首要前提是大量的勇气、决心和耐心。作者承担了一项任务，为了哪怕勉强成功地完成它，还需要在他的指尖上拥有同样大量的学识。最后，最重要的是，需要一种非凡的能力来实现赫尔德博士为自己设定的目标，即自觉地表达数学推理的逻辑。因为对于科学家来说，可能正如对于诗人一样，他们通常完全无法连贯地描述他们‘疯狂中的方法’。正如Holder教授本人所指出的，伟大的数学家很少能够说出他们最初是如何真正取得划时代成果的。……赫尔德博士（他是莱比锡的数学教授）希望他的工作被视为主要对数学逻辑的贡献，并谦虚地将把他的发现纳入更全面论著的问题留给其他逻辑学家。

H Wieleitner [23]写道，这本书：-

……是一部广泛、组织良好、写作清晰的著作，对于所有数学和哲学问题，特别是在应用方面，是一本非常有价值的参考书。

让我们引用巴特尔·伦德特·范德瓦尔登为赫尔德所写的讣告[22]来结束这篇传记：-

一位真正伟大的科学家离开了我们，他是世纪之交为现代数学指明方向的人物之一：从形式到批判，从计算到概念。他的努力始终不渝地指向思维与表达的逻辑精确性。

Otto Hölder worked on the convergence of Fourier series and in 1884 he discovered the inequality now named after him. He became interested in group theory through Kronecker and Klein and proved the uniqueness of the factor groups in a composition series. His father was Otto Hölder (1811-1890), professor of French at the Polytechnikum in Stuttgart, the son of Christian Gottlieb Hölder (1776-1847), who was a professor at the Gymnasium in Stuttgart, and Friederike Karoline Gaupp (1784-1813). The mother of Otto, the subject of this biography, was Pauline Christiane Ströbel (1821-1894), the daughter of August Friedrich Ströbel and Friederika Christiane Moser. Otto and Pauline were married on 17 May 1845 in Stuttgart. They had three sons: Alfred, born 2 January 1846, Eduard Otto, born 27 November 1847 and Ludwig Otto, the subject of this biography, who was born on 22 December 1859 and baptised on 2 February 1860. In 1864 the family were living at 18 Hermannstrasse, Stuttgart. All three sons became professors. Eduard Otto Hölder studied law in Tübingen and became a professor of law at several different universities.

Otto Hölder studied at a Gymnasium in Stuttgart, in fact at one of the earliest Gymnasiums specialising in science, graduating in 1876. He then entered the polytechnic in Stuttgart to study engineering but found himself more attracted to mathematics than to engineering. One of his father's colleagues suggested that Stuttgart Polytechnikum was not the best place to study mathematics and that Hölder would be better to go to the University of Berlin which had one of the top mathematics schools in the world. In 1877 he entered the University of Berlin and began to study mathematics there. He was a fellow student of Carl Runge and he attended lectures by Karl Weierstrass, Leopold Kronecker and Eduard Kummer. In his first year of study he attended Weierstrass's lectures on the theory of functions which covered the fundamentals of analysis. Weierstrass made a marked impression of the young Hölder and his influence showed on Hölder throughout his career. Runge [9]:-

... recalled much later that Weierstrass's lectures left a deep and lasting impression [on Hölder], even though they were not polished and well constructed. Weierstrass would sometimes get in a muddle improvising a proof, only to put it right imperturbably next time. But Weierstrass was a sympathetic tutor, who listened attentively to his students and really responded to the question ...

Hölder's interest in algebra came partly through the influence of Kronecker at this time and Kronecker's liking for rigour almost certainly was to have a profound influence on Hölder's later work in algebra. When asked a question, Kronecker [9]:-

... could not be made to listen but always changed the subject straight away to talk about his own work. On the other hand, Kronecker was a far more approachable person, and many young people would be invited to his hospitable home.

After studying in Berlin, Hölder went to the Eberhard-Karls University of Tübingen where he was advised by Paul du Bois-Reymond. He presented his dissertation, Beiträge zur Potentialtheorie Ⓣ, which investigates analytic functions and summation procedures by arithmetic means, to the University of Tübingen in 1882. These summation procedures are now the known as the "Hölder summation method". His dissertation also contains the continuity condition for volume density which now is known as the "Hölder condition" on a function. After the award of his doctorate, Hölder went to Leipzig. Felix Klein was there at the time but there seems to have been little interaction between the two during the two years that he was there. Hölder at this time was still interested in function theory, although Klein had a strong influence on Hölder later in his career. He was denied the opportunity to habilitate at Leipzig so he moved to Göttingen.

Strangely, Göttingen did not recognise Hölder's Tübingen doctorate so, in 1884, he submitted a thesis for a second doctorate at Göttingen and, in the same year, habilitated at the University of Göttingen. His habilitation thesis examined the convergence of the Fourier series of a function that was not assumed to be either continuous or bounded. At first at Göttingen he continued to work on the convergence of Fourier series. Shortly after be started working at Göttingen he discovered the inequality now named after him which appeared in his paper Über einen Mittelwerthsatz Ⓣ (1889). It appears that Hölder became interested in group theory while at Göttingen, partly through discussions with Walther von Dyck and partly through Felix Klein who was lecturing on Galois theory. The university faculty at Göttingen wanted to offer Hölder an assistant lectureship but such appointments could only be made by the Prussian Ministry of Culture and, despite repeated requests, they felt that he did not have sufficient lecturing experience for such a post. Hölder was offered a post in Tübingen in May 1889 but unfortunately he suffered a mental collapse. Receiving treatment in a clinic in Erlangen, he was in no position to reply to Tübingen's offer so his brother Eduard, who was by this time a professor of law at Tübingen, accepted the offer on behalf of his brother. The faculty at Tübingen were unsure how to proceed when they learnt that Hölder was ill and in a clinic but, after much discussion, they kept their confidence in Hölder. He made a steady recovery, giving his inaugural lecture in June 1890.

Klein's lectures on Galois theory at Göttingen had interested Hölder who began to study the Galois theory of equations and from there he was led to study composition series of groups. Hölder proved the uniqueness of the factor groups in a composition series, the theorem now called the Jordan-Hölder theorem, and published the result in Mathematische Annalen in 1889 in the paper Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen Ⓣ. Although Hölder did not consider that he invented the notion of a factor group, the concept appears clearly for the first time this paper of Hölder's. He clarified the concept which he claimed was neither new nor difficult but was not sufficiently appreciated [13]:-

Dieudonné commented on the Jordan-Hölder Theorem that it would assume its definitive form only with Hölder. This could also be said of the concept of quotient group: in both cases Jordan's ideas were early formulations which later gave way to the accepted 'standard' forms.

With the help of group theory and Galois theory methods Hölder returned to a study of the irreducible case of the cubic in the Cardan-Tartaglia formula in 1891. We have made minor corrections to the following quote from [9]:-

Hölder was one of the first to give a rigorous account of the famous classical case where a splitting field is not a radical extension: the irreducible cubic equation over the rationals with three real roots, where it is nonetheless necessary to adjoin complex roots of unity. Hölder's proof of this result, which had long been suspect, was accompanied by three accounts by other people which appeared around the same time and were summarised in the second volume of Netto's book 'Vorlesungen über Algebra' Ⓣ (1900).

Hölder made many other contributions to group theory. He searched for finite simple groups and in the 1892 paper Die einfachen Gruppen im ersten und zweiten Hundert der Ordnungszahlen Ⓣ in Mathematische Annalen he showed that all simple groups up to order 200 are already known. His methods use the Sylow theorems in a similar way to how the problem would be solved today. Hölder also studied groups of orders $p^{3}, pq^{2}, pqr$ and $p^{4}$ for $p, q, r$ primes, publishing his results in 1893. His proofs again heavily rely on the use of Sylow theorems.

Concepts which were introduced by Hölder include inner and outer automorphisms. In 1895 he wrote a long paper on extensions of groups. Often Otto Schreier is said to be the one to have initiated the study of extensions of groups but Julia Nicholson writes [2]:-

Schreier's approach to and development of the theory of extensions seems to follow on directly from that of Otto Hölder: much of Hölder's methodology was borrowed by Schreier. The difference between their work is that Schreier drew out each idea to its logical conclusion, whereas Hölder had been motivated initially by the wish to classify particular sorts of groups and therefore developed the theory with this fixed aim in mind.

In [22] van der Waerden writes:-

... reading Hölder's papers again and again is a profound intellectual treat.

In 1889 Hölder was appointed as an Extraordinary Professor of Mathematics at the University of Tübingen. He left Tübingen in 1896 when he was appointed as a full professorship at the University of Königsberg. In 1899 Hölder married Helene (1871-1927), the daughter of the attorney, bank director, and politician Karl Ernst Lautenschlager (1828-1895) and his wife Sophie Wilhelmina Faber (1831-1902). Helene and Otto Hölder had four children: Ernst Otto (born 1901), Charlotte Sophie (born 1902), Irmgard Luise (born 1904), and Wolfgang Carl (born 1906). Ernst Hölder became a mathematician working mainly in the field of mathematical physics. He was awarded a doctorate by the University of Leipzig in 1926 for his thesis Gleichgewichtsfiguren rotierender Flüssigkeiten mit Oberflächenspannung Ⓣ. His thesis advisor was Leon Lichtenstein. In 1899, the same year that he married, Hölder was appointed as an Ordinary Professor of Mathematics at the University of Leipzig, succeeding to the chair that had been occupied by Sophus Lie. He was Dean of the Faculty of Arts of the University of Leipzig in 1912-13 and Rector of the University of Leipzig in 1918. Although he remained at Leipzig, he did apply for the chair in Berlin in 1902. David Hilbert was ranked first, Friedrich Schottky was ranked second and Hölder was ranked third. Schottky was appointed to the chair.

From 1900 Hölder became interested in the geometry of the projective line and philosophical questions, which had interested him throughout his career, began to play a prominent role. Let us now look at some of these works. Perhaps we should begin by looking at the paper by Hölder, published in 1892, in which he gives his reaction to Robert Grassmann's Die Zahlenlehre oder Arithmetik - streng wissenschaftlich in strenger Formelentwicklung Ⓣ (1891). Mircea Radu writes [16]:-:-

Hölder's paper is important for at least three reasons: First, it represents what might be called Hölder's research manifesto on the foundations of mathematics, containing a wealth of ideas which Hölder gradually developed in a variety of publications until the end of his life. Second, Hölder's analysis of Robert Grassmann's foundational ideas provides an important assessment of the contribution of Hermann and Robert Grassmann to the axiomatisation of arithmetic, a contribution which, though often mentioned, is itself still not widely acknowledged and not fully understood. Third, the effort of exposing the weak spots in Robert Grassmann's ideas led Hölder to formulate the main problems confronting formal axiomatics: independence of the axioms, consistency, completeness, and the issue of the relationship between pure mathematics and its applications.

When Hölder was appointed to the chair in Leipzig in 1899, he delivered the inaugural lecture 'Anschauung und Denken in der Geometrie' which was published in 1900. This work presents a proof that the Archimedean axiom can be derived from Dedekind's notion of continuity. The review [14] states:-

A thorough study of the methods of ratiocination employed in mathematics, mechanics, and the exact natural sciences has led Professor Hölder to the conviction that the deductive method there employed is made up of series of concatenated conclusions of quite characteristic form, and that consequently these sciences have a peculiar method and logic of their own. Not that the reasoning in these sciences is absolutely different from the reasoning in other departments of thought and life ; the peculiarity in question resides entirely in the nature of the subject-matter and in the style of the combinations of the intellectual acts concerned. A correct philosophy of mathematical procedure is to be reached, in Professor Holder's opinion, only by the cooperation of mathematicians and philosophers, and not, as has been heretofore the case, by isolated and one-sided labours in this do main. The review which he himself gives of the modes of thought concerned is a very good one, and taken together with the exhaustive notes which he has suffixed to his discussion, will be found to be of assistance to students.

This inaugural lecture was the starting point for Hölder's axiomatic theory of quantity which he published as Die Axiome der Quantität und die Lehre vom Mass Ⓣ in 1901. Joel Michell writes [12]:-

His paper is a watershed in measurement theory, dividing the classical (stretching from Euclid) and the modern (stretching to Luce et al ., 1990) eras. His concerns belonged, in part, to the classical era. He axiomatised the classical concept of quantity (using Dedekind's concept of continuity) in such a way that ratios of magnitudes (as understood in Book V of Euclid's 'Elements') could be expressed as positive real numbers (as intimated by Newton). Importantly, he achieved this result with a clarity that was not attained by others also working within the classical framework (e.g., Helmholtz, 1878; Frege, 1903; and Whitehead and Russell, 1913). However, other of his concerns were more modern. In Part II of his paper, he showed that axioms for an apparently nonadditive structure, stretches of a straight line, entail that linear distances satisfy the axioms for quantity given in Part I. His concern to relate nonadditive structures to quantitative ones anticipates a distinctly modern interest ...

The highlight of his philosophical writings is the book Die Mathematische Methode Ⓣ (1924). H R Smart writes [21]:-

The first prerequisites to the writing of such a book as this are a vast amount of courage, resolution and patience. The author has undertaken a task which requires, furthermore, for its even tolerably successful accomplishment, the possession, at his very finger-tips, of an equally vast store of learning. Finally, and above all, there is required an unusual ability in order to realize the goal which Dr Hölder has set before himself, namely to bring to self-conscious expression the logic of mathematical inference. For it is probably just as true of scientists as of poets that they are usually quite unable to give a coherent account of the 'method in their madness.' As Professor Holder himself points out, the great mathematicians have seldom been able to tell how, in the first instance, they really attained their epoch-making results. ... Dr Hölder (who is professor of mathematics at Leipzig) desires his work to be regarded as primarily a contribution to the logic of mathematics, and modestly leaves to other logicians the problem of incorporating his findings in more comprehensive treatises.

H Wieleitner [23] writes that the book:-

... is an extensive, well organized, clearly written work, and is a very valuable reference book for all mathematical and philosophical questions, particularly in regard to the applications..

Let us end this biography by quoting from van der Waerden's obituary of Hölder [22]:-

A truly great scientist has left us, one of those men who, at the turn of the century, pointed the way for modern mathematics: from the formal to the critical, from computation to concept. His efforts were directed incessantly toward logical accuracy in thinking and expression.