**Jerzy Łoś**'s parents were Zygmunt Łoś (5 March 1887 - 28 December 1936), son of August Łoś and Maria Pietruska, and Zofia Augusta Rostworowska (19 June 1900 - 4 March 1962), daughter of Władysław Rostworowski and Zofia Siemiginowska. Zygmunt, who was a 29 year old Doctor of Laws, and Zofia, who was 16 years old, were married in 1916. In 1919 Zofia, who had studied plant science, inherited a large agricultural estate in Torskie, a small town near Zalishchyky. Torskie was, at this time, in the Podolia region of Poland but it is now in the west of Ukraine. Zygmunt and Zofia Łoś had two children, Jerzy Maria Michał Łoś, the subject of this biography, and Władysław Maria Michał Łoś. The family moved to Torskie where the production of fruit on the estate was changed to make it a major wine producer in the 1930s. In addition to looking after the estates, Zofia Łoś was a local government councillor, president of the school council in Torskie, and held other roles in health insurance and agricultural organisations. In 1925, she was co-founder of a publishing company which published agricultural material.

Jerzy Łoś completed his primary and secondary education in 1937 but, one year before that, tragedy struck the family when his father was killed. On 28 December 1936 Zygmunt Łoś died in a plane crash near Susiec, a village in eastern Poland. The LOT Polish Airlines Lockheed 10A Electra had taken off from Lwów with ten passengers and two crew. In the crash, due to icing, Zygmunt Łoś, one other passenger Jósef Zimmermann and one of the two members of the crew Jósef Fronc, were killed.

In 1937 Łoś entered the Jan Kazimierz University of Lwów where he began studying medicine, then philosophy and chemistry. After two years of study at university, things changed very dramatically with the outbreak of World War II. 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. On 22 September, Russian troops entered Poland occupying Lwów. Łoś and his mother moved to Trzebień, a small village, where they lived with one of their relatives. Władysław Łoś, the brother of our mathematician, had fled on 17 September 1939, before the Russian invasion, and moved to France where he changed his name to Michel de Los.

In April 1940 Łoś and his mother moved from Trzebień to Tarnogóra, a village near Zamość, where they lived with another relative. After Poland surrendered in September 1939 it was partitioned between Germany and Russia. The Molotov-Ribbentrop non-aggression pact between Germany and the Soviet Union held until 22 June 1941 when Germany broke the non-aggression pact and invaded the Soviet Union. Łoś had been in a region controlled by the Soviets but, after the German invasion, it was German troops that controlled the region Łoś was living in. He had to earn his living, and he did this by taking a position as a clerk at the sugar plant in Lublin in 1942. He remained there until 1943 when he became a bookkeeper for an estate in Tarnogóra. By 1944 the Soviet armies had driven the Germans out and the Yalta Conference in February 1945 saw parts of Poland (such as Lwów) become part of the Ukraine while Lublin remained in Poland.

In 1945 Łoś married Maria Dux; they had two sons Zygmunt Łoś (born 10 June 1946) and Władysław Łoś (born 21 March 1955). With the end of World War II, he resumed his studies at the Maria Sklodowska-Curie University in Lublin in 1945 studying mathematics and philosophy, and in 1947 he was awarded a master's degree in philosophy after submitting his thesis *The foundations of a methodological analysis of Mill's method* (Polish). He had already published *An attempt to axiomatise traditional logic* (Polish) in 1946 in which, following the general ideas of Ivan Sleszyński in *The Theory of Proof* (1925), he analysed traditional logic, and especially laws of immediate inference, by means of relations between extensions of arguments.

After the award of his Master's Degree, Łoś moved to Wrocław, where a very active group of mathematicians, including Hugo Steinhaus, Bronisław Knaster and Edward Marczewski, started to build a new department. Steinhaus, in particular, greatly influenced the direction that Łoś's research would take. Łoś was appointed first as an assistant in the Department of Physics at Wrocław and then as an assistant in the Department of Logic there. He undertook research for his Ph.D. with Jerzy Słupecki (1904-1987) as his advisor. Słupecki had been a student of Jan Łukasiewicz and had completed his Ph.D. just before the outbreak of World War II. Łoś was awarded his Ph.D. in 1949 for his thesis *On logical matrices* (Polish). This was published as a 42-page monograph which was reviewed by Jan Kalicki [3]:-

In the same year, 1949, Łoś joined the Real Functions group in the Mathematical Institute of the Polish Academy of Sciences and continued to hold various positions in the Institute, in parallel to his university posts, until he retired in 1991. He remained at Wrocław until 1952 when he was sent to the Nicolaus Copernicus University in Toruń to create an algebra group there. In the same year he was appointed as Head of the Algebra group in the Mathematical Institute of the Polish Academy of Sciences. His main contributions to algebra are in the area of Abelian groups. While working at Toruń, he was awarded a doctorate (equivalent to a D.Sc. or habilitation in standard) in 1955 from the Mathematical Institute of the Polish Academy of Sciences. For this degree he submitted two papers,The monograph is divided into four chapters: I. Introductory definitions and theorems. Theory of matrices of Lindenbaum. II. Some consequences of the theory of Lindenbaum's matrices. III. On definability of logical functors in terms of each other. Theory of functional matrices. IV. Algebraic matrices. .... It is not easy to give an adequate account of the results contained in Łoś's monograph in a short review. It seems impossible to give any account of the author's interesting methods. Even in the space of forty-two pages the author was compelled to give only a sketchy account of many of his remarkable results. This makes the reading of his work not easy. ...[It]impressed the reviewer as substantial progress in the study of logical and algebraic matrices. In the reviewer's opinion Łoś's work is of very real value.

*The algebraic treatment methodology of elementary deductive systems*(1955) and

*On the extending of models*(1955). One of the mathematical concepts which has proved extremely fruitful is that of an ultraproduct. This was introduced by Łoś in the first of these two papers in which [1]:-

Łoś married his second wife, Maria Wycech (1929-2011), the daughter of Czesław Wycech (1899-1977) and Anna Kasprzyk (1899-1989). Czesław Wycech was a Member of the Polish Parliament from 1947 to 1971. Maria, who used the name Wycech-Łoś after her marriage, was a writer and mathematician who wrote a number of joint papers with her husband:... the general reduced product construction is introduced and the fundamental theorem relating the theory of ultraproduct to the theories of factors is(implicitly)stated.

*Remarks on the efficiency frontiers in von Neumann models*(1976),

*Extremal properties of equilibria in von Neumann models*(1977), and

*Many agents in a von Neumann model*(1979). They also co-edited conference proceedings:

*Mathematical Methods in Economics*(1974),

*Computing Equilibria: How and Why?*, (1976), and

*Warsaw Fall Seminars in Mathematical Economics*(1976).

In 1961 Łoś moved from Toruń to Warsaw where he lectured at the University and worked at various Institutes of the Polish Academy of Sciences: the Institute of Philosophy; the Institute of Mathematics; and, towards the end of his career, the Institute of Computer Science. His move to Warsaw also saw a major change in his research interests. Up to that time he had worked on the foundations of mathematics and on algebra. However, the move to Warsaw saw him become interested in the applications of mathematics, particularly to economics. Let us look briefly at his contributions to these areas.

His first work was on the foundations of mathematics and he published papers such as: *Many-valued logics and the formalization of intensional functions* (Polish) (1948) which gives an example of a consistent intensional system of logic; (with C Ryll-Nardzewski) *On the application of Tychonoff's theorem in mathematical proofs* (1951) which uses the Tychonoff product theorem (the product of compact spaces is compact) as a substitute for the axiom of choice; and *An algebraic proof of completeness for two-valued propositional calculus* (1951) which shows the completeness of the two-valued propositional calculus with an algebraic proof.

From 1954 his work on the foundations of mathematics took him towards algebra with papers such as *Sur le théorème de Gödel pour les théories indénombrables* Ⓣ (1954) which proves, without using the axiom of choice, the equivalence between the Gödel-Malcev completeness theorem and the existence of prime ideals in Boolean algebra; and (with C Ryll-Nardzewski) *Effectiveness of the representation theory for Boolean algebras* (1954) which examines when the existence of prime ideals in Boolean algebras is equivalent to the axiom of choice.

By 1955 Łoś was publishing work entirely within the area of algebra such as *On the complete direct sum of countable abelian groups* (1955) which proves:-

In the same year of 1955 Łoś publishedA complete direct sum of countable abelian torsion-free groups is decomposable into a discrete direct sum of countable groups if and only if every summand, with at most finitely many exceptions, is a divisible group.

*On the axiomatic treatment of probability*which looks at the mathematical foundations of probability. However, he continued to work on the foundations of mathematics publishing in the same year the major 62-page paper

*The algebraic treatment of the methodology of elementary deductive systems*which contains a careful look at Gödel's completeness theorem as well as some related new results and new proofs of known results.

Although Łoś worked in different areas, there were common threads to his approach [1]:-

One of the areas of mathematical economics that Łoś studied was associated with von Neumann models consisting of pairs of matrices containing production process data. For example he publishedA characteristic feature of the entire Łoś's research activity is his ability to find very natural links between problems and methods of various domains of mathematics. This tendency can be seen clearly also in his later works devoted to mathematical methods of economy.

*A simple proof of the existence of equilibrium in a von Neumann model and some of its consequences*(1971),

*Extended von Neumann models and game theory*(1976), and

*Mathematical theory of von Neumann economic models. Report on recent results*(1978).

He made long visits to several universities abroad including: the University of California at Berkeley (1959-60) when, in collaboration with Alfred Tarski, he ran a model theory seminar; again the University of California at Berkeley (1962-3) when, in collaboration with Bjarni Jónsson, he ran a universal algebra seminar; Aarhus University in Denmark (1967) when he lectured on mathematical methods in economics; the Poincaré Institute in Paris (1969) when he lectured on the theory of economic models; Yale University (1973) when he lectured on applications of mathematics to economics; and the University of Wisconsin (1978-79) when again he lectured on applications of mathematics to economics.

Leszek Pacholski tells us something of Łoś's character in [4]:-

Similar sentiments are recorded by the authors of [1]:-Łoś was a gifted mathematician with many original ideas, and a talented organizer. He was also a man of strong convictions, convictions that he was not willing to compromise for the sake of political expedience. He spoke out against decisions he felt were wrong, even when many others chose to keep quiet. When "Solidarity" started its fight for freedom and independence in Poland, he often provided workers with advice and ideas. With his passing Poland has lost a focal point of its scientific and intellectual life, and a forceful advocate for human rights.

Łoś received many honours during his career including the Cross of Merit, Polonia Restituta, the Award of the Secretary of the Polish Academy of Sciences for achievements in mathematical economics, Sierpinski Medal, and the Jurzykowski Award for activity in the Solidarity movement. In 1995 he received an honorary doctorate from the University of Hagen in Germany. In 1964 he was elected a corresponding member of the Polish Academy of Sciences and in 1983 he became a full member.Professor Łoś profoundly influenced his students, colleagues and all those who had the chance to meet him. His enthusiasm for scientific work turned out to be contagious, as can be seen with his numerous followers, many of them professors in mathematics or economics themselves by now. A broadminded man, he was known by his independent and courageous standpoints, which brought him to actively support the "Solidarity" movement in the early1980's; his opinions and views were highly appreciated by many of those who met him, even if they differed from their own.

In the spring of 1996 Łoś suffered a stroke which he survived but lived with a serious condition for the final two years of his life.

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

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