恩斯特·施勒德Ernst Schröder

恩斯特·施勒德的父母是Heinrich Georg Friedrich Schröder和Karoline Walther。他们于1840年9月9日在巴伐利亚州丁根区的一个地区Haunsheim结婚。Heinrich Schröder于1810年9月28日出生在慕尼黑，在慕尼黑大学学习，并成为中学的物理学教授。他先在慕尼黑理工学院工作，然后在索洛图恩的中学，当施勒德出生时，他是曼海姆高等市民学校的物理学和化学教授。他在鼓励文理中学（Gymnasium）和该地区学院的科学教学方面非常有影响力。他的妻子Karoline是来自Haunsheim的一位牧师的女儿，她的父亲Johann Gottfried Walther（1785-1852）在施勒德年轻时辅导了他两年。事实上，施勒德在这两年里与他的祖父住在一起。Walther牧师给他的孙子提供了特别好的基础教育，特别强调拉丁语的学习。当然，施勒德的父亲对他的儿子也有很大的影响，鉴于我们所说的Heinrich Schröder在促进科学教育方面所做的工作，得知他对儿子从事科学事业有重大影响就不足为奇了。施勒德是他父母四个孩子中的长子：Clara生于1842年；Heinrich生于1845年，成为一家银行的董事；Walter生于1850年，成为一名商人。在祖父的辅导之后，施勒德在曼海姆的几所不同的学校学习，他在语言、化学和数学方面表现出非凡的能力。1856年，当他十五岁时，他进入曼海姆的中学，在那里学习了四年，于1860年毕业。

从曼海姆中学毕业后，施勒德进入海德堡大学。他在奥托·黑塞指导下学习数学，在古斯塔夫·基尔霍夫指导下学习物理，在Robert Bunsen指导下学习化学。此时的海德堡是一个令人兴奋的地方，古斯塔夫·基尔霍夫和Bunsen在分析元素光谱方面取得了根本性进展。古斯塔夫·基尔霍夫和Bunsen都曾是奥托·黑塞的学生。当施勒德在奥托·黑塞作为其导师的指导下进行研究时，其他几位很快将成名的学生也在海德堡跟随奥托·黑塞进行博士研究。例如，阿道夫·迈尔（1861年博士学位）和海因里希·马丁·韦伯（1863年博士学位）与施勒德同时是奥托·黑塞的学生，而奥劳斯·亨里齐于1862年抵达海德堡开始他的学习。鲁普雷希特-卡尔-海德堡大学于1862年授予施勒德博士学位，其论文为Über die Vielecke von gebrochener Seitenzahl oder die Bedeutung der Stern-Polygone in der Geometrie。Ⓣ（关于p/q边多边形或几何中星多边形的意义）在他的论文中，他写道：-

幂概念最初只与整数相关联，将其推广到有理分数在代数中成果丰硕；这提示我们，在几何中只要有机会，也应尝试做同样的事。

在分数幂的例子中，他接着定义了$\Large\frac{p}{q}\normalsize$边形。

奥托·黑塞 在被任命到海德堡之前曾是柯尼斯堡的教授，古斯塔夫·基尔霍夫 和本生都曾在柯尼斯堡做过他的学生。恩斯特·弗朗茨·诺伊曼 当时是柯尼斯堡的物理学教授，也教过 古斯塔夫·基尔霍夫 和本生。奥托·黑塞 教过 卡尔·诺伊曼，即 恩斯特·弗朗茨·诺伊曼 的儿子，因此有了 恩斯特·弗朗茨·诺伊曼 与海德堡教职员之间的这些紧密联系，施勒德 在获得博士学位后前往柯尼斯堡，跟随 恩斯特·弗朗茨·诺伊曼 学习数学物理、跟随 F J Richelot 学习数学分析两年，也就不足为奇了。

1864年，在柯尼斯堡两年后，施勒德参加了资格考试，以获得在中学教授数学和自然科学的资格。他在巴登-巴登参加了这些国家考试，但随后前往苏黎世，于1865年向苏黎世联邦理工学院提交了他的教授资格论文（Habilitation）论文。Dipert [12]推测，他去苏黎世的原因可能不完全是学术性的，因为他是一位非常热情的登山者，在瑞士期间进行了多次无向导的艰难攀登。在获得讲师资格后，他在苏黎世联邦理工学院担任了一段时间的Privatdozent。回到德国后，他于1869年10月在巴登-巴登参加了进一步的国家教学考试，并在那里任教，直到1870年普法战争爆发。施勒德自愿参军，尽管视力不佳，他还是被录取了。然而，他的服役期相当短，因为在1870年底，巴登教育部要求他返回，担任巴登-巴登实科中学的数学和自然科学教授。黑森林与瑞士阿尔卑斯山相当不同，但施勒德充分利用了这个地区，在巴登-巴登的岁月里进行了许多长途徒步旅行。1874年，他被任命为达姆施塔特技术大学的正式教授。他在那里待了两年，于1876年转到卡尔斯鲁厄技术大学。几乎可以肯定，这一变动是因为雅各布·吕罗特。像施勒德一样，雅各布·吕罗特在曼海姆长大，两人在那里上学时成为朋友。雅各布·吕罗特于1869年被任命为卡尔斯鲁厄技术大学的数学教授，他的签名出现在施勒德的任命书上。施勒德在卡尔斯鲁厄度过了他余下的职业生涯，并于1890-91年被任命为技术大学的校长。

施勒德的重要工作是在代数、集合论和逻辑领域。他关于有序集和序数的工作是该学科的基础。然而，正如Peckhaus所指出的[22]，他从不认为自己是逻辑学家：-

他自己的研究对象是绝对代数，就其基本问题和基本假设而言。在施勒德的研究中，逻辑与代数之间有何联系？……人们可能会认为这些领域属于两个独立的研究领域，但事实并非如此。它们在他关于一般科学的启发式观念的框架中交织在一起。

事实上，施勒德起初对数学物理感兴趣，他转向逻辑只是试图深化其基础。在职业生涯早期，他写了一篇重要文章Über iterirte FunctionenⓉ（论迭代函数）（1871年），常被引用为现代动力系统理论的基础。现在可以看到施勒德随着Lehrbuch der Arithmetik und Algebra für Lehrer und StudierendeⓉ（供教师和学生使用的算术与代数教科书）于1873年由Teubner出版而转向逻辑。Ivor Grattan-Guinness[5]写道：-

在副标题中，他提到了“七种代数运算”：第一级的加法和减法，第二级的乘法和除法，第三级的幂、根和对数。……他提出数学是“数的学说”，而非量的学说；他通过寻求“绝对代数”——普通代数只是其一个例子——来强调代数倾向。

1874年，他出版了Normale Elemente der absoluten Algebra Ⓣ（绝对代数基础），这本书是为巴登-巴登的学校使用而写的（尽管很难相信他有能够欣赏这本小书中的思想的学生），在书中他继续发展先前出版物中的思想。1877年，他受乔治·布尔和赫尔曼·格拉斯曼的影响，写了他的第一部关于数学逻辑的著作Der Operationskreis des Logikkalkuls Ⓣ（逻辑演算的运算）。它首次包含了 duality principle 的表述，并强调了 conjunction（intersection）和 disjunction（union）的 duality，展示了如何找到 dual theorems。他是第一个使用术语“propositional calculus”的人，似乎也是第一个使用术语“mathematical logic”的人。事实上，他比较了代数和乔治·布尔的逻辑，说：-

这两种运算的对象肯定存在对比。它们完全不同。在算术中，字母是数字，但在这里，它们是任意概念。

在Vorlesungen über die Algebra der Logik Ⓣ（逻辑代数讲义）中，这是一部于1890年至1905年间出版的大型著作（在他去世后由Eugen Müller编辑并完成），施勒德详细阐述了代数逻辑，为阿尔弗雷德·塔斯基发展现代代数理论提供了来源，并给出了逻辑史的广泛参考书目。格理论也源于这项工作。Brady写道[3]：-

它首次阐述了抽象格理论，首次在理查德·戴德金之后阐述了理查德·戴德金的链理论，最全面地发展了关系演算，并在关系演算中处理了数学基础，利奥波德·勒文海姆在1940年仍然认为这像集合论一样合理。施勒德解决关系方程的概念是陶拉尔夫·斯科伦函数的前身，他启发了利奥波德·勒文海姆对著名定理的表述和证明，即每个具有无限模型的句子都有可数模型，这是现代逻辑的第一个真正定理。

施勒德说他的目标是（例如见[22]）：-

……将逻辑设计为一门计算学科，特别是提供对相对概念的精确处理，并且从那时起，通过摆脱自然语言的常规主张，也从哲学领域的“陈词滥调”中撤走任何肥沃土壤。这应该为一种科学通用语言奠定基础，这种语言与像Volapük[一种像世界语一样的通用语言，当时在德国非常流行]这样的语言努力大相径庭，看起来更像是一种符号语言而不是声音语言。

施勒德 对 查尔斯·桑德斯·皮尔士 评价很高。两人有通信往来，但 查尔斯·桑德斯·皮尔士 对 施勒德 的态度则较为复杂，有时称赞他，而在其他场合又对他提出严厉批评。Brady 写道 [3]：-

施勒德 将 查尔斯·桑德斯·皮尔士 的关系演算发展得远比 查尔斯·桑德斯·皮尔士 更为深入、更为系统。施勒德 在一阶和高阶逻辑中考察了量词（或者至少是对于固定域而言等价于量词的和与积）。他明白，有些概念如可数性超出了关系演算的范围（也超出了一阶谓词逻辑的范围）。

Dipert [12] 对 施勒德 的性格作了有趣的描述，并将其与 Peirce 的性格进行了比较：-

就 施勒德 的个性而言，施勒德 显然是一个极为平和温厚的人。他所有的传记作者都证实了这一点，这些品质在他与 查尔斯·桑德斯·皮尔士 的通信中以及他对 克莉丝汀·拉德-富兰克林 及其幼女 Margaret 的慷慨中表现得尤为明显。《Vorlesungen》一书在当时颇为罕见地注意标注他人的工作，从不含糊地将实际上属于他人的成果据为己有。尽管 Peirce 总体上称赞 施勒德，但他有时仍会在出版物和私人通信中猛烈攻击他。然而 施勒德 崇敬 Peirce，并且充分具备 查尔斯·桑德斯·皮尔士 承认自己所缺乏的品质：自制力。

Putman 展示了 施勒德 在完成 [25] 这项工作一百年后所受到的尊重：-

当我开始追溯逻辑学的后期发展时，我做的第一件事就是查看 施勒德 的《Vorlesungen über die Algebra der Logik》，……[其]第三卷讨论关系逻辑（Algebra und Logik der Relative，1895）。这三卷立即成为最著名的高级逻辑教材，体现了 1890 年代任何对逻辑研究感兴趣的数学家所应当知道、或至少应当有所了解的内容。

然而，正如Wussing在[1]中所写，这种对施勒德的尊重在他自己的时代远没有那么明显：-

施勒德参与了19世纪下半叶数学逻辑作为一门独立学科的发展。这是他的真正成就，尽管他的贡献直到20世纪初才被认可。有三个因素导致了这种延迟：他生前该领域的不成熟状态；他风格上的某种冗长；以及最重要的是，他在技术学院的教学所造成的孤立。结果，他成了一个局外人，在选择术语、概述论证以及判断数学逻辑能取得什么成就方面处于不利地位。

施勒德有许多体育爱好：骑自行车、徒步旅行、游泳、滑冰、骑马和园艺。因为他总是被人看到在卡尔斯鲁厄周围骑自行车，所以当地人称他为“自行车教授”。他甚至60岁时开始滑雪。他从未结婚，但似乎发现他在卡尔斯鲁厄技术高等学校的职责极其繁重，也许是因为他非常认真地承担这些职责。当然，他发现很难找到必要的时间来完成他的三卷本主要著作Vorlesungen über die Algebra der Logik Ⓣ（逻辑代数讲义）。施勒德的父亲海因里希于1873年退休，一年后他的妻子卡罗琳于1875年去世，他搬到卡尔斯鲁厄靠近他的儿子施勒德和海因里希；Heinrich Schröder于1885年去世。鉴于施勒德是一个多么健康的人，他在60岁时去世令人惊讶。他在去世前几天还在滑雪和骑自行车，但感冒了。这在几天内似乎恶化了，根据死亡证明，他死于“脑热”。他的许多朋友认为，他60岁时剧烈的体育活动导致了他的过早死亡。他被埋葬在卡尔斯鲁厄的主要墓地，离他住过的公寓很近。由于没有近亲继续照料他的坟墓，按照通常的习俗，30年后它被重新使用，所以今天墓地里没有留下任何记录。

有趣的是，1913年6月诺伯特·维纳向哈佛大学提交了他的学位论文；当时诺伯特·维纳18岁。该学位论文致力于比较施勒德和伯特兰·罗素的逻辑系统，特别关注他们对关系的不同处理。在[15]中对这一点及后续发展有有趣的讨论。

Ernst Schröder's parents were Heinrich Georg Friedrich Schröder and Karoline Walther. They were married on 9 September 1840 in Haunsheim, a district of Dillingen in Bavaria. Heinrich Schröder was born in Munich on 28 September 1810, studied at the University of Munich and became a professor of physics in high schools. He worked first at the Polytechnic School in Munich, then the Lyceum in Solothurn and, at the time Ernst was born, he was professor of physics and chemistry at the Higher Bürgerschule in Mannheim. He was very influential in encouraging the teaching of science in the gymnasiums and colleges in the district. His wife Karoline was the daughter of a pastor from Haunsheim and her father, Johann Gottfried Walther (1785-1852), tutored Ernst for two years when he was young. In fact Ernst lived with his grandfather during these two years. The Rev Walther gave his grandson an exceptionally good basic education with particular emphasis on the study of Latin. Of course Ernst's father was also a big influence on his son and, given what we have said about the work Heinrich Schröder did in fostering science education, it will come as no surprise to learn that he was a major influence on his son taking up a career in science. Ernst was the eldest of his parents four children: Clara was born in 1842; Heinrich was born in 1845 and became the director of a bank; and Walter was born in 1850 and became a businessman. After tutoring by his grandfather, Ernst studied at several different schools in Mannheim where he showed exceptional abilities in languages, chemistry, and mathematics. In 1856, when he was fifteen years old, he entered the Lyceum in Mannheim and he studied there for four years, graduating in 1860.

After graduating from the Mannheim Lyceum, Schröder entered the University of Heidelberg. He studied mathematics under Otto Hesse, physics under Gustav Kirchhoff and chemistry under Robert Bunsen. Heidelberg was an exciting place at this time with Kirchhoff and Bunsen making fundamental advances analysing the spectrum of elements. Both Kirchhoff and Bunsen had been students of Hesse. While Schröder was undertaking research with Hesse as his advisor, several other students, who would soon become famous, were also undertaking doctoral studies with Hesse at Heidelberg. For example, Adolph Mayer (doctorate in 1861) and Heinrich Weber (doctorate in 1863) were students of Hesse at the same time as Schröder, while Olaus Henrici arrived in Heidelberg to begin his studies in 1862. The Ruprecht-Karls-Universität Heidelberg awarded Schröder a doctorate in 1862 for his thesis Über die Vielecke von gebrochener Seitenzahl oder die Bedeutung der Stern-Polygone in der Geometrie. Ⓣ In his thesis he writes:-

The extension of the power concept, originally only associated with integers, to rational fractions has been very fruitful in algebra; this suggests that we should try to do the same thing in geometry whenever the opportunity presents itself.

Among examples of fractional powers, he goes on to define $\Large\frac{p}{q}\normalsize$ -sided polygons.

Hesse had been a professor at Königsberg before being appointed to Heidelberg and Kirchhoff and Bunsen had both been his students at Königsberg. Franz Neumann had been the professor of physics at Königsberg at that time and had also taught Kirchhoff and Bunsen. Hesse had taught Carl Neumann, Franz Neumann's son, so with these strong links between Franz Neumann and the staff at Heidelberg, it is little surprise that after the award of his doctorate Schröder went to Königsberg to spend two years studying mathematical physics with Franz Neumann and mathematical analysis with F J Richelot.

In 1864, after his two years in Königsberg, Schröder took the examinations to qualify him to teach mathematics and natural sciences in gymnasiums. He took these state examinations in Baden-Baden, but then went to Zürich where he submitted his habilitation thesis to the Eidgenössische Technische Hochschule in 1865. Dipert [12] speculates that his reasons for going to Zürich may not have been entirely academic ones since he was a very enthusiastic mountain climber and made a number of difficult ascents without a guide during his time in Switzerland. Having qualified as a lecturer, he taught for a while as a Privatdozent at the Eidgenössische Technische Hochschule. Returning to Germany, he took further state teaching examinations in Baden-Baden in October 1869 and was teaching there when the Franco-Prussian War broke out in 1870. Schröder volunteered for the army and, despite his poor eyesight, he was accepted. His period of active service was, however, quite short, for towards the end of 1870 the Ministry of Education in Baden requested he return to take up an appointment as professor of mathematics and the natural sciences at the Realgymnasium in Baden-Baden. The Black Forest was rather different from the Swiss Alps, but Schröder took full advantage of the area making many long hikes during his years in Baden-Baden. In 1874 he was named a full professor at the Technische Hochschule in Darmstadt. He remained there for two years, moving to the Technische Hochschule in Karlsruhe in 1876. It is almost certain that this move came about because of Jacob Lüroth. Like Schröder, Lüroth grew up in Mannheim and the two became friends while at school there. Lüroth had been appointed professor of mathematics at the Technische Hochschule in Karlsruhe in 1869 and his signature is on Schröder's letter of appointment. Schröder remained at Karlsruhe for the rest of his career, being made Director of the Technische Hochschule for the year 1890-91.

Ernst Schröder's important work is in the area of algebra, set theory and logic. His work on ordered sets and ordinal numbers is fundamental to the subject. However, he never considered himself to be a logician, as Peckhaus points out [22]:-

His very own object of research was absolute algebra in respect to its basic problems and fundamental assumptions. What was the connection between logic and algebra in Schröder's research? ... one could assume that these fields belong to two separate fields of research, but this is not the case. They were intertwined in the framework of his heuristic idea of a general science.

In fact Schröder started out being interested in mathematical physics, and his move towards logic was simply an attempt to deepen its foundations. Early in his career he wrote an important article Über iterirte Functionen Ⓣ (1871) often cited as a basis of modern dynamical systems theory. Now one sees Schröder moving towards logic with Lehrbuch der Arithmetik und Algebra für Lehrer und Studierende Ⓣ published by Teubner in 1873. Ivor Grattan-Guinness [5] writes:-

In the subtitle he mentioned 'the seven algebraic operations': addition and subtraction at the 'first level', multiplication and division at the second, and exponentiation, roots and logarithms at the third. ... he put forward mathematics as 'the doctrine of numbers', rather than of magnitudes; and he stressed the algebraic bent by seeking 'absolute algebra' of which common algebra was an example.

In 1874 he published Normale Elemente der absoluten Algebra Ⓣ which was written for use in the school in Baden-Baden (although it is difficult to believe that he had students capable of appreciating the ideas in this small book) and in it he continued to develop the ideas from the previous publication. He wrote his first work on mathematical logic Der Operationskreis des Logikkalkuls Ⓣ, influenced by George Boole and Hermann Grassmann, in 1877. It contained, for the first time, the formulation of the duality principle, and emphasised the duality of conjunction (intersection) and disjunction (union) showing how dual theorems could be found. He was the first to use the term 'propositional calculus' and seems to be the first to use the term 'mathematical logic'. In fact he compares algebra and Boole's logic saying:-

There is certainly a contrast of the objects of the two operations. They are totally different. In arithmetic, letters are numbers, but here, they are arbitrary concepts.

In Vorlesungen über die Algebra der Logik Ⓣ, a large work published between 1890 and 1905 (it was edited and completed by Eugen Müller after his death), Schröder gave a detailed account of algebraic logic, provided a source for Alfred Tarski to develop the modern algebraic theory and gave an extensive bibliography of the history of logic. Lattice theory also grew out of this work. Brady writes [3]:-

It offers the first exposition of abstract lattice theory, the first exposition of Dedekind's theory of chains after Dedekind, the most comprehensive development of the cakculau of relations, and a treatment of the foundations of mathematics in relation calculus that Löwenheim in 1940 still thought was as reasonable as set theory. Schröder's concept of solving a relational equation was a precursor of Skolem functions, and he inspired Löwenheim's formulation and proof of the famous theorem that every sentence with an infinite model has a countable model, the first real theorem of modern logic.

Schröder said his aim was (see for example [22]):-

... to design logic as a calculating discipline, especially to give access to the exact handling of relative concepts, and, from then on, by emancipation from the routine claims of natural language, to withdraw any fertile soil from "cliché" in the field of philosophy as well. This should prepare the ground for a scientific universal language that, widely differing from linguistic efforts like Volapük [a universal language like Esperanto, very popular in Germany at the time], looks more like a sign language than like a sound language.

Schröder had a very high opinion of Charles Sanders Peirce. The two corresponded but Peirce showed a more mixed attitude towards Schröder, sometimes praising him while on other occasions he was highly critical. Brady writes [3]:-

Schröder developed Peirce's relative calculus much further and much more systematically than did Peirce. Schröder considered quantifiers (or, at least, sums and products equivalent to quantifiers for a fixed domain) in first- and higher-order logic. He understood that there are notions such as countability that are beyond relative calculus (and also beyond first-order predicate logic).

Dipert [12] gives an interesting account of Schröder's character which he compares with that of Peirce:-

As concerns Schröder's personality, Schröder was apparently an extremely even-tempered and gentle man. All his biographers attest to this fact, and these qualities are shown conspicuously in his correspondence with Peirce, and his generosity toward Christine Ladd-Franklin and her young daughter, Margaret. The Vorlesungen is, rather unusual for the times, careful to note the work of others and never takes vague credit for what was in fact others' work. While Peirce overall praised Schröder, he nevertheless sometimes ferociously attacked him, in print and in private correspondence. Schröder venerated Peirce, however, and had in abundance what Peirce acknowledged he lacked: self-control.

Putman shows the respect which Schröder was held in a hundred years after he completed the work [25]:-

When I started to trace the later development of logic, the first thing I did was to look at Schröder's 'Vorlesungen über die Algebra der Logik', ... [whose] third volume is on the logic of relations (Algebra und Logik der Relative, 1895). The three volumes immediately became the best-known advanced logic text, and embody what any mathematician interested in the study of logic should have known, or at least have been acquainted with, in the 1890s.

However, as Wussing writes in [1], this respect for Schröder was not nearly so evident in his own day:-

Schröder participated in the development of mathematical logic as an independent discipline in the second half of the nineteenth century. This is his real achievement, although his contribution was not recognised until the beginning of the twentieth century. Three factors account for the delay: the imature state of the field during his lifetime; a certain prolixity in his style; and, above all, the isolation imposed by his teaching in technical colleges. As a result he was an outsider, at a disadvantage in chosing terminology, in outlining his argumentation, and in judging what mathematical logic could accomplish.

Schröder had many sporting hobbies: cycling, hiking, swimming, ice-skating, horseback riding, and gardening. Because he was always seen riding his bicycle around Karlsruhe he was known locally as the 'Bicycle-professor'. He even took up skiing when he was sixty years old. He never married but seemed to find that his duties in the Technische Hochschule Karlsruhe extremely demanding, perhaps because he undertook them very conscientiously. Certainly he found it very difficult to find the necessary time to complete his major three volume work Vorlesungen über die Algebra der Logik Ⓣ. Schröder's father Heinrich retired in 1873 and, a year after his wife Karoline died in 1875, he moved to Karlsruhe to be close to his sons Ernst and Heinrich; Heinrich Schröder died in 1885. Given what a fit man Ernst Schröder was, it is surprising that he died at the age of 60. He was skiing and cycling days before his death, but caught a cold. This seemed to worsen over the course of a few days and he died of "brain fever" according to the death certificate. Many of his friends felt that his strenuous sporting life at the age of sixty had led to his premature death. He was buried in the main cemetery of Karlsruhe quite close to the apartment in which he had lived. Having no close relatives to continue tending his grave, following the usual custom it was reused after a period of 30 years so today there is no record remaining in the cemetery.

It is interesting to note that in June 1913 Norbert Wiener presented his doctoral thesis to Harvard University; Wiener was then 18 years old. The thesis was devoted to a comparison of the logical systems of Ernst Schröder and Bertrand Russell, with special attention to their different treatments of relations. There is an interesting discussion of this and subsequent developments in [15].