The Lvov-Warsaw School. Past and Present

The Lvov-Warsaw School. Past and Present is a collection of research papers on logic and its history. These papers are written by specialists from around the globe. These papers range from historical considerations to new philosophical and logical developments — these topics range from Computer Science, Mathematics, Metalogic, Scientific and Analytic Philosophy, and the Theory of Models and Linguistics. While this school is not well known outside of Poland, it is one of the most important schools in the history of logic. (I am a Polish American and never heard of it until picking up the book for review.) In the preface, the editor-in-chief, Jean-Yves Beziau, hopes the book will shed new light on the importance of the school for logicians and philosophers and help promote new ideas in logic.

The book is divided into three major parts for the papers, Part I: Twardowski’s School: The Period of Crystallization of Lvov-Warsaw School (LWS), Part II: Warsaw School of Logic, Its Main Figures and Ideas: The Period of Prosperity, and Part III: The War and Post-War Period. In total, there are 55 papers.

The first paper, “Introduction. The School: Its Genesis, Development and Significance” by Urszula Wybraniec-Skardowska, begins in the fall of 1895 with founder Kazimierz Twardowski (1886–1938), a student of Franz Brentano. Twardowski wanted Lvov to be a center of a scientific circle. As pointed out in the abstract: “The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name — Warsaw School of Logic (WSL).”

During the time period of 1918–1939, WSL grew out of Twardowski’s influence, with the two leaders being Stanisław Leśniewski and Jan Łukasiewicz, later joined by their student Alfred Tarski. As pointed out on page 8, unprecedented results were achieved. Heinrich Scholz from Münster said, “Warsaw became the main centre of logical studies.” In the lifetime of one generation, Polish logic grew from the ground up to the international stage. A. Fraenkel, Y. Bar-Hillel, and A. Levy pointed out, “Probably no other country, taking into account the size of its population, has contributed so greatly to the development of mathematical logic and foundations of mathematics as Poland… this curious fact should be explained sociologically.”

The last year of the WSL is regarded to be 1939. The invasion of Poland by Germany on September 1, 1939 and by the Soviet Union on September 17, 1939 began World War II. Poland was divided again under the German-Soviet pact of August 23, 1939 and the city of Warsaw was completely destroyed. This drove many members of the LWS School to leave Warsaw and many emigrated from Poland. In Poland, an underground resistance movement was taking shape and teaching was conducted in secret. After WWII, however, Polish logic never regained the strength it once had.

In reading the papers, one can’t but think how much dedication and love of mathematics, logic, and philosophy these individuals had. In reading the paper: “Alfred Tarski: Auxiliary Notes on His Legacy” by Jan Zygmunt, we learn about the massive amount of monographs (18), papers (59), abstracts (6), and publications as an editor (3) he produced. In one of his papers, number 30 on page 451, we learn that Tarski gave an effective proof — without using the Axiom of Choice — that for any infinite set \(E\) the following two statements were equivalent:

1. There exists a finitely additive two-valued measure m on the family of all subsets of \(E\) such that \(m(E)=1\) and for every finite subset \(F\subset E\) we have \(m(F)=0\).
2. There exists a maximal ideal in the field of all subsets of \(E\).

Tarski’s remarkable result was: In order for an \([E, I, G]\)-measure to exist, it is necessary and sufficient that there be no paradoxical decompositions of \(I\) relative to \(G\).

These particular achievements in the foundations of geometry were to make the readers understand the idea of a system of geometry based on the first-order logic with identity. He called this system elementary geometry.

The works presented in these papers truly show the advanced logic developed by the school. Personally, it is overwhelming and heartwarming to learn that such a school existed with all the events that took place in Warsaw. While school may not exist today, the spirt of the Polish people lives on and their work is honored in this book. If you are a mathematical historian, please read this book. It will make you proud of the bravery of these individuals and how dedicated they were to learning.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He can be reached at pto2@psu.edu. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.