**Leopold Löwenheim**'s father, Detmold Ludwig Löwenheim, was a mathematics teacher at the Krefeld Polytechnic at the time that his son Leopold was born. However, he was of independent means and three years later, in 1881, he gave up teaching so that he could concentrate on his research. Many years later Leopold revised and edited his father's unfinished work on Democritus. Leopold's mother was Elise Röhn who was a writer. Löwenheim attended the Königliche Luisen Gymnasium in Berlin, completing his secondary schooling there in 1896. At school he had a wide variety of interests, being particularly fascinated by mathematics and natural sciences. He also developed a liking for philosophy and, in the year he graduated from high school, he joined the Deutsche Gesellschaft für Ethische Kultur. His father had also been a member of this same Society showing that both had strong concerns for social thought and culture.

Having obtained qualifications to study at university, Löwenheim entered the Friedrich-Wilhelm University of Berlin in 1896 and studied mathematics and natural sciences there, and also at the Technische Hochschule in Charlottenburg, during the next four years. At the time when Löwenheim studied at these two universities they were in separate cities but now expansion has meant that Charlottenburg has become part of Berlin. Löwenheim was intending to qualify to teach in Gymnasiums and he wished to teach across a range of topics. Therefore, in 1901, he took the examinations to qualify to teach mathematics and physics at all high school grades, and the examinations to teach chemistry and mineralogy at middle school grades. He spent the next year doing postgraduate training, and the following year he taught as a probationary teacher, Then in 1904 he was appointed as a teacher to the Jahn-realgymnasium in Berlin. He joined the Berlin Mathematical Society in 1906, reading his first paper to the Society in the following year. He also became a reviewer for the *Jahrbuch über die Fortschitte der Mathematik*.

Despite war service in France, Hungary and Serbia between August 1915 and December 1916, he published a series of important papers on mathematical logic during the eleven years from 1908 to 1919, extending work by Charles Peirce, Schröder, and Whitehead. It was during this period, in fact in 1914, that he completed and published his father's work on Democritus. We will explain below some of the ideas he introduced into mathematical logic over this period, but at this point we will continue to sketch events in his life. He married Johanna Rassmussen in 1931 but his busy life as a teacher came to an abrupt end a few years later.

When Hitler became Chancellor of Germany in 1933, he immediately announced legal actions against Germany's Jews. On 7^{th} April 1933, Hitler introduced a law for the "Restoration of the civil service". This meant that all non-Aryans and Jewish civil servants were dismissed from their positions with the exception of those who either had fought in the Great War or had been in office since August 1914. The definition of non-Aryan included those with one grandparent of the Jewish religion and this was precisely the position that Löwenheim was in. However he should have escaped dismissal under the "Restoration of the civil service" law since he met both the other exemption clauses (either one of which should have allowed him to continue working) - he had fought in the Great War and he had been in office since August 1914. In line with what happened to others in a similar position, Löwenheim escaped immediate dismissal in 1933 but was forced to retire in 1934.

Löwenheim now somehow had to try to earn enough to support himself and his wife despite losing his job. He managed to get a post teaching eurythmy (harmony of bodily movement developed with the aid of music into an educational system) and geometry at the Anthroposophic School of Eurythmy in Berlin. The school had a philosophy based on the premise that the human intellect has the ability to contact spiritual worlds. Not only did Löwenheim have problems with Nazi discrimination, but he also almost lost his life in the British bombing raids on Berlin. On 23 August 1943 his home suffered a direct hit by a bomb. In fact bombing raids on Berlin had not been uncommon, with the first daylight raid in January 1943, but the hit on Löwenheim's home came before the major bombing assault on the capital began in November 1943. Löwenheim survived, but he lost his mathematical manuscripts, 1000 drawings, and his mathematical models. In fact he lost unpublished manuscripts on logic, geometry, music and the history of art.

Although Löwenheim survived, top mathematical logicians such as Bernays, Tarski and Heinrich Scholtz (all of whom had visited Löwenheim during the 1939s) had no contact with him after 1939 and presumed that he had died in the war. Quine translated *On making indirect proofs direct* from German into English and published it in *Scripta Mathematica* in 1946. Löwenheim, therefore, had a paper published that he was not aware of. Heyting reviewed the paper and wrote:-

Simple devices, some of which may be useful in investigating the intuitionistic validity of propositions. As the author remarks, a satisfactory definition of the concept of direct proof has not yet been given.

Let us now go back to look at his early work. The first significant publications by Löwenheim were the two papers, *Über das Auflösungsproblem im logischen Klassenkalkül* published in 1908 and *Über die Auflösung von Gleichungen im logischen Gebietekalkul* in 1910. In these [1]:-

Löwenheim analysed and improved upon the customary methods of solving equations in the calculus of classes or domains(that is, set theory in its Peirce-Schröder[Charles Peirce and Ernst Schröder]setting)and proved what is now known as Löwenheim's general development theorem for functions of functions.

In *Über Transformationen im Gebietekalkül* (1913) Löwenheim studied matrices of domains and, in a similar way to a modern linear algebra course, showed that transformations can be represented by matrices.

Let us now look at the result for which Löwenheim is most remembered, namely the Löwenheim-Skolem paradox (which Skolem pointed out is not a paradox!). This appears in the paper *Über Möglichkeiten im Relativkalkül* (On possibilities in the calculus of relatives) published in 1915. In this paper Löwenheim proved the remarkable result that for any set of sentences of standard predicate logic, if there is an interpretation in which they are true in some domain, there is also an interpretation that makes them true in a countable subset of the original domain. The result was called a paradox since it was believed that certain sets of axioms characterised the real numbers, and now Löwenheim's result showed that the same axioms must hold in a countable subset of the real numbers. It also seemed to make attempts to axiomatise set theory somewhat of a problem. Although it seems to contradict common sense (as do other results which depend on the Axiom of Choice), there is no paradox. The result implies that no uncountable mathematical system, such as those involved in analysis, geometry, and set theory, can be characterised up to isomorphism using only first-order sentences. If one examines the case of the real numbers more closely, then the axioms for an ordered field are all first-order sentences. Löwenheim's result then shows that the real numbers contain a countable ordered field which then cannot satisfy the least-upper-bound axiom which is a second-order sentence.

As we mentioned above, despite people believing that Löwenheim had died in the war, he had been able to begin teaching again in 1946. He taught for three years up to 1949 at the Pestalozzi-Schule and the Franz-Mehring-Schule both in the Lichtenberg district of Berlin. Johannes Teichert, the son of Löwenheim's wife by a previous marriage, inherited what remained of Löwenheim's unpublished manuscripts. Löwenheim was known to have submitted three papers to *Fundamenta Mathematicae* in 1939 and they were feared lost, but in 1978 Thiel [5] announced that one of the three papers had been found.

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