**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 ^{p}/_{q} -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 in1940still 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 the1890s.

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].

**Article by:** *J J O'Connor* and *E F Robertson*

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