**Adolf Lindenbaum**'s parents were Mowsza Henoch Lindenbaum and Emilja Krykus. Mowsza Lindenbaum owned several companies and made a movie in 1931 but no further details of the family seem to be known except that they were Jewish. There are no records of Adolf's early years although we can assume that he was brought up in Warsaw. Again it is not known whether he attended a primary school or whether he was educated at home for his first years. The first definite information that is known is September 1914 when, at the age of ten, he entered the Rocha Kowalskiego Gymnasium. After one year at this school, he transferred to the Jana Kreczmara Gymnasium (sometimes known as the Michała Kreczmara Gymnasium. The Jana Kreczmara Gymnasium was a private school founded in 1907 in Miodowa Street in Warsaw but moved to Śniadeckich Street. It is unclear exactly why he moved school after a year, but it is possible that he was living in Śniadeckich Street at this time. Since the Jana Kreczmara and Rocha Kowalskiego Gymnasiums are far apart, it is quite possible that his parents moved homes. Certainly, from 1915 to 1922 he attended the Jana Kreczmara Gymnasium and passed his matriculation examination on 27 May 1922 having been exempted from the oral part of the examination because he "showed good progress."

Of course, World War I began in 1914, in the year that Lindenbaum began his schooling. At the time that Lindenbaum was born, Poland did not exist as an independent country but had been partitioned between Austria-Hungary, Prussia and Russia. Warsaw was in the Russian sector and was, therefore, part of the Russian Empire. Attacked by German forces, the Russian sector of Poland was, by the end of 1915, captured by German troops so Warsaw came under German control. After the war ended in 1918, Poland was reestablished as an independent country and its borders set by the Treaty of Versailles in 1919. Conflict did not end there, however, for in 1919 the Polish-Russian war began. In the summer of 1920 Russian troops invaded Poland and advanced towards Warsaw. A huge war effort was required by the Poles and young men were enlisted to help. Lindenbaum, who was sixteen at the time, enlisted in an organisation set up by the Polish White Cross to help the soldiers. He ran errands for the organisation for four months until the peace treaty was signed in October 1920.

We should mention one of Lindenbaum's classmates at the Jana Kreczmara Gymnasium. This was Mojżesz David Kirszbraun (1903 or 1904-1942) who, like Lindenbaum, showed outstanding mathematical abilities at the school. He graduated at the same time as Lindenbaum and the two entered the University of Warsaw to study mathematics together. Today Kirszbraun is remembered for the Kirszbraun Theorem concerning extensions of Lipschitz maps between Hilbert spaces. Lindenbaum and Kirszbraun were close friends and undertook joint research projects while at university.

On 14 September 1922 Lindenbaum applied for admission to the Mathematics department in the Faculty of Philosophy of Warsaw University. He gave his address as Apartment 4 at 45 Złota Street. He wrote in his application (see, for example, [11]):-

Five days later his application was accepted and on 2 October 1922 he began his university studies. During the five years of his university course, Lindenbaum was taught by some outstanding academics. Among his mathematics lecturers were Kazimierz Kuratowski, Stefan Mazurkiewicz and Wacław Sierpiński while Stanisław Leśniewski and Jan Łukasiewicz lectured to him on logic. Alfred Tarski taught at the Polish Pedagogical Institute in Warsaw from 1922 to 1925 but began teaching at the University of Warsaw in session 1925-26 when he taught on cardinal numbers, a course which Lindenbaum attended. Since Lindenbaum also attended Tarski's elementary mathematics (plane geometry) course in 1926-27 it looks likely that he was attending courses by a friend to give him support rather than to learn new mathematics. Lindenbaum also attended a logic course by Tadeusz Kotarbiński (1886-1981). Kotarbiński had studied mathematics in Kraków before studying under Łukasiewicz at Lvov. He had been appointed as a lecturer in philosophy at the University of Warsaw in 1918. Lindenbaum also attended a course on the history of philosophy given by Władysław Tatarkiewicz (1886-1880) who had taught in Wilno and Poznań before taking up an appointment in Warsaw in 1923. Tatarkiewicz was also an expert on aesthetics and the history of art, and taught a course on French art which Lindenbaum attended. He also attended a course on psychology given by Władysław Witwicki (1878-1948), a course on linguistics by Karol Appel (1857-1930), and a course on the history and culture of Palestine given by Moses Schorr (1874-1941). Schorr was a rabbi who preached at the Great Synagogue of Warsaw from 1923 and lectured at the University from 1926.Always loving mathematics, I did not hesitate for a moment what to choose among the different advanced studies, having decided to apply for admission to that faculty where I will be able to study this science.

In fact Lindenbaum's student career ended in a slightly strange way. He was undertaking research on point set topology for his doctorate advised by Wacław Sierpiński. In October 1926 he applied for exemption from the course requirement of session 1926-27 to give him time to complete his dissertation. He had already published two papers, namely *Contributions à l'étude de l'espace métrique* Ⓣ (1926) and (with Alfred Tarski) *Communication sur les recherches de la théorie des ensembles* Ⓣ (1926). His request was refused and he continued to attend courses in 1926-27.

Let us say a little about these papers. The first examines isometric mappings of point sets between metric spaces. These results would eventually become part of his doctoral thesis. The second, written jointly with Tarski, is a 2-page paper which lists a large number of theorems which they had proved, the majority being on cardinal numbers, but the paper gives no proofs.

He submitted his thesis *On metric properties of point sets* (Polish) in the autumn of 1926 but in order to remain registered as a student and therefore avoiding military service, he applied in September 1927 to become a student of psychology and pedagogy. This was accepted so in 1927-28 he was again a first year student but he only attended two courses, one being on the Polish poet Adama Mickiewicz given by Józef Ujejski (1883-1937). The oral examination on his thesis was on 22 June 1928 and, being successful, he became a Doctor of Philosophy. He remained enrolled as a student until June 1929.

From 1929 to 1934 Lindenbaum worked on his habilitation thesis. During these years he had no employment so one can only assume that his parents continued to provide him with financial support. In fact the habilitation thesis was a collection of papers that he had published over these years. After the two papers listed above, Lindenbaum published the following papers between 1928 and 1934: *Sur quelques propriétés des fonctions de variable réelle* Ⓣ (1928); *Remarques sur une question de la méthode axiomatique* Ⓣ (1930); (with A Kozniewski) *Sur les opérations d'addition et de multiplication dans les classes d'ensemble* Ⓣ (1930); *Sur les ensembles ordonnés* Ⓣ (1931); *Bemerkung zu den vorhergehenden "Bemerkungen ..." des Herrn J. v. Neumann* Ⓣ (1931); *Sur les constructions non-effectives dans l'arithmétique élémentaire* Ⓣ (1932); *Sur les figures convexes* Ⓣ (1932); *La projection comme transformation continue la plus générale* Ⓣ (1932); *Sur un ensemble linéaire extrêmement non homogène par rapport aux transformations continues et sur le nombre des invariants de ces transformations* Ⓣ (1932); *Sur les ensembles localement dénombrables dans l'espace métrique* Ⓣ (1933); *Sur les superpositions des fonctions représentables analytiquement* Ⓣ (1933); *Sur les ensembles dans lesquels toutes les équations d'une famille donnée ont un nombre de solutions fixé d'avance* Ⓣ (1933); *Sur la théorie de l'ordre multiple* Ⓣ (1934); *Sur les superpositions des fonctions représentables analytiquement* Ⓣ (1934).

After presenting his habilitation thesis, Lindenbaum became a docent in the Faculty of Mathematics and Natural Sciences. He taught in the faculty for two semesters before being appointed as an assistant professor in the Philosophical Seminar run by Łukasiewicz. This Seminar was a unit in the Faculty of Mathematics and Natural Sciences. Courses taught by Lindenbaum were on set theory, measure theory, algebra, actuarial mathematics and the foundations of mathematics. In addition to the research that he was publishing, Lindenbaum and Tarski collaborated on writing the book *Theorie der eineinendeutigen Abbildungen* Ⓣ. They must have essentially completed this work since a publication date in 1938 was announced. This, however, never happened since events leading up to the outbreak of war in 1939 disrupted life in Poland.

As one might imagine, being Jewish, Lindenbaum had problems. In 1935 Polish universities adopted an anti-Semitic policy which meant that Lindenbaum had little chance of being promoted beyond docent. However, he believed in left-wing policies and he was certainly a member of the Polish Communist Party by 1935, and he may have joined earlier than that. This party came under suspicion and was accused of having agents of the Polish Regime among its members. Many of its leaders were taken to Moscow, tried and murdered during the period of political repression in the Soviet Union in 1936-39. The Polish Communist Party was dissolved by Stalin in 1938.

Lindenbaum had married his fellow logician Janina Hosiasson (1899-1942) who was also a Jew. The marriage took place either late in 1935 or early in 1936. The couple had known each other from the time they were both studying at Warsaw University. Janina had received a doctorate from Warsaw in 1926 advised by Tadeusz Kotarbiński who had also taught Lindenbaum. Both she and Lindenbaum had delivered papers at the First Congress of Mathematicians from Slavic Countries held in Warsaw in September 1929. She was an active researcher who published around 20 papers. After their marriage they lived at apartment 34 in 16 Krasińskiego Street. Lindenbaum did other things which would later put his life in danger. In 1936 he had signed a petition demanding that the journalist and pacifist Carl von Ossietzky, who had been imprisoned by the Nazis, be allowed to travel to Oslo to received the Nobel Peace Prize which had been awarded to him. He also signed a letter protesting at the massacre in April 1936 in Lwów when nineteen people were killed by the Polish police during the funeral of Władysław Kozak who had been killed by the police on 14 April during a protest by the unemployed.

On 1 September 1939 German troops entered Poland and the German Luftwaffe began bombing all strategically important sites. By 4 September they were within 60 km of Warsaw which they encircled with two pincer movements coming from the north and south, one to the east of Warsaw the other to the west. Lindenbaum realised the danger he was in both as a Jew and as a known Communist. He knew that the Germans had lists of names of Polish Communists and other critics of Hitler's regime which, particularly because of the petitions and letters he had signed, terrified Lindenbaum. On 6 September, he and his wife went to Vilna (now known as Vilnius) where she took up residence but Lindenbaum continued on to Białystok. He could have escaped to the West or, perhaps more likely because of his left-wing views, gone to Moscow. He chose to do neither since he hoped to help build a Socialist Poland after the war. On 22 September, Russian troops entered Poland occupying Białystok and five days later Warsaw fell to the Germans and, following the Molotov-Ribbentrop non-aggression pact between Germany and the Soviet Union, Poland was partitioned between these two powers. Białystok, however, now became part of Byelorussian Soviet Socialist Republic. The Soviets set up the Białystok Pedagogical Institute and Lindenbaum was appointed as a docent at the Institute. He was offered a position in Moscow but turned it down.

Things changed dramatically on 22 June 1941 when Germany broke the non-aggression pact and invaded the Soviet Union. German armies advanced on Białystok and the battle for the city and surrounding area lasted until 3 July when the Germans took control. Lindenbaum does not appear to have made any efforts to escape which is slightly strange given that he was fully aware of the danger he was in from the Germans. He may have felt that all options were equally dangerous. In September 1941 he was arrested by the Gestapo and taken to Vilnius. Lindenbaum's wife Janina, who was in Vilnius, was also arrested by the Gestapo in September 1941. It appears that by this time the two were separated, for earlier that year Janina had written to Tarski saying that they had separated at the beginning of World War II. After seven months imprisonment, Janina was taken to Ponary (now known as Paneriai) just outside Vilnius, and shot near the railway station. The site was chosen by the Germans to carry out executions since the Soviets had begun to construct oil storage facilities which the Germans used to dispose of the bodies. It is not known exactly when Lindenbaum was executed but it is believed that he too was murdered at Ponary soon after his arrest.

Jan Woleński writes in [8]:-

After his early work on topology, he moved towards logic and the foundations of mathematics. He did important work on set theory, studied axioms equivalent to the axiom of choice and proved that the axiom of choice was independent of the other axioms of Zermelo-Fraenkel set theory. Fraenkel had claimed to prove this but Lindenbaum and Mostowski in a 1938 paper, claim that there are errors and obscurities in Fraenkel's proof which they correct. Lindenbaum also worked on the propositional calculus and its connection to logical matrices. Two other significant achievements are what is today called the Lindenbaum algebra and the maximalisation theorem sometimes today called Lindenbaum's Lemma.Lindenbaum's brilliant personality, charming style of life and powerful mathematical skills fascinated the Warsaw scientific community. Not surprisingly, he was commonly considered one of the most gifted Polish mathematicians of his generation. Tarski ... described Lindenbaum as "a man of unusual intelligence". Mostowski once called Lindenbaum the most lucid mind in the foundations of mathematics. Legendary stories told by Lindenbaum's friends and colleagues document many cases of theorems discovered by him but proved by someone else, as he had no time to complete his ideas. Yet the list of his scientific contributions is quite long; it comprises more than40papers, abstracts and reviews ..., mostly published in German and French.

Lindenbaum was a member of the Polish Mathematical Society from 1926, becoming treasurer of the Warsaw branch of that Society in 1938. He was elected as secretary at the foundational meeting of the Polish Logical Society in 1936. At the same meeting Jan Łukasiewicz was elected chairman and Alfred Tarski as vice-chairman.

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

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