Archive for October, 2008

A review of Through a reporter’s eyes: The life of Stefan Banach by Roman Kaluza. Translated and edited by Ann Kostant and Wojbor Woyczynski. Boston: Birkhaüser, 1996.

In the preface to this book, Roman Kaluza states that “this book contains little of mathematical character,” and hence this is not a work on the important mathematical developments of Stefan Banach during his productive life. Nevertheless, it offers a detailed and comprehensive look inside the Polish mathematical and educational communities in the time between the wars. Although Banach left no correspondence or memoirs from which to examine his life, Kaluza does an admirable job reconstructing his career through a comprehensive study of university archives, newsletters, interviews with students and acquaintances, and the recollections of Banach’s contemporaries.

Most mathematicians initially become familiar with the name of Stefan Banach through their studies of functional analysis, topology (the Banach space), or through the paradoxical Banach-Tarski theorem, yet they learn nothing about the man himself. This role is well served by Kaluza’s book, which may be roughly divided into two parts. The first half of the book addresses the academic development and mathematical work of Banach. The second half concerns the personal and social aspects of Banach’s life. Joining these two parts is a fascinating chapter on the Scottish Café, which has attained legendary status in Polish mathematical history, where Banach and other mathematicians struggled with a wide range of important mathematical topics over coffee and cognac.

The first four chapters of the book concentrate on the education and subsequent mathematical contributions of Banach during his career. As a child, he attended the Henryk Sienkiewwicz Gymnasium in Cracow, a school concentrating in the humanities. An excellent mathematics student, he tutored from the age of 15 and was largely self-taught in the more advanced aspects of his subject. Upon graduation, we learn that he decided to enter into engineering, believing that “mathematics was so highly developed, it would be impossible to do anything new.” Enrolling in Lvov Polytechnic, Banach soon realised that a future did exist in mathematics. His meeting with Huglo Steinhaus in Lvov represented a turning point in Banach’s life. This meeting is colourfully described by Kaluza using excerpts from the memoirs of Steinhaus. We learn that not only did Banach become acquainted and begin to collaborate with one of Poland’s leading mathematicians, it was through Steinhaus that he meets his wife.

The years from 1919 to 1929 represented Banach’s most prolific period of research. After the publication of his first paper (in collaboration with Steinhaus) in 1918, Banach produced a remarkable string of results. One of his earliest papers remains his most influential. In “Sur l’équation functionelle f(x + y) = f(x) + f(y),” he demonstrates that measureable solutions to this equation are necessarily continuous, and thus linear. Banach followed this paper with four more in a similar vein, providing the necessary background for his developments in the theory of functional analysis. These early papers, published while Banach was still a student, lead to his Ph.D. thesis on complete normed metric spaces, known now as Banach spaces. Despite his emphasis on the non-mathematical nature of Banach’s life, Kaluza does an admirable job in conveying to the reader a sense of the content of Banach’s work. He considers the mathematical content of these early papers individually, albeit briefly. At this point the inclusion of a few simple equations instead of unnecessarily long descriptions would be beneficial.

Banach went on to receive his Ph.D. in 1921, and by 1924 was promoted to the “highest possible rank in the [Polish] scientific community” – Professor Ordinarus of mathematics.

Kaluza moves from this most productive period to consider the rest of Banach’s mathematical work, from his founding (with Hugo Steinhaus) of the Studia Mathematica in 1929 to his death in 1945. The Studia Mathematica was an important journal that conveyed the recent Polish successes in functional analysis and set theory. Kaluza shows that the Studia was the first mathematical journal to specialise in these areas, thus attracting not only Polish but internationally renowned scholars. In 1932 Banach published his influential monograph The Theory of Linear Operations. This treatise consolidated his published results on linear functional analysis along with a substantial amount of new theorems and applications. During the late 1930s, Banach’s output declined sharply, primarily due to the geopolitical situation in Europe. His last (non-posthumous) papers, on non-divergent series of orthogonal functions, appeared in 1940.

To connect the largely intellectual history of the first part of the book to Banach’s personal and social life described in the second half, Kaluza provides a fascinating chapter on the legendary Scottish Café. This café, situated near the Lvov institute, became the centre of Polish mathematics during the 1930s. Frequented by Banach, Steinhaus, Ernst Zermelo, and other prominent mathematicians on a regular basis, Kaluza describes the heated discussions on mathematics, politics, and just about anything else occurred between patrons of the café. A major product of these discussions was what became known as The Scottish Book, a large notebook kept at the café where mathematicians could enter problems and offer solutions. This famous notebook, contained entries by Banach and the Polish community, as well as luminaries such as Henri Lebesgue, John von Neumannm, and René Fréchet, was eventually published by Birkhäuser in 1981.

The final half of Through a reporter’s eyes concerns the personal and social life of Banach. Kaluza draws heavily upon interviews with Banach’s former students and the recollections of his contemporaries. We learn that he was a rather unconventional academic, prone to cancelling large blocks of classes at a time, and spending up to twelve hours a day in the cafés of Lvov. However, he was greatly admired by the student body and regarded as highly talented teacher who could convey the most difficult of concepts with great clarity. The ending of the book describes Banach’s life in Lvov under the Nazi and Soviet occupations of World War II. By that time, the thriving Polish academic community of the 1920s and 1930s had disbanded; many sought refuge in England or the United States, and many others were jailed at home. In a chilling description of life under the Nazi occupation, Kaluza details how Banach spent a few months in prison and the remainder of the time working as a “feeder of lice in the Rudolf Weigl Bacteriological Institute,”, the standard work-duty for many former academics. Banach suffered greatly under these conditions, and was described by Jadwiga Hallaunbrenner, a Lvov mathematician, as being “exhausted, starved, and wasted away.” With the end of the war, Banach was offered the position of Minister of Education in the new Soviet-controlled Polish government, but died soon thereafter of cancer on the 31st of August 1945.

Three appendices included in this work warrant mention. The first, entitled “Mathematics in Stefan Banach’s Time,” provides a short description of the emergent and increasingly abstract mathematics of the early twentieth century. It provides some background for the non-mathematician in the areas of mathematical logic and set theory to accompany the exposition of Banach’s results in the first half of the book. The second appendix is a list of selected publications of Banach. This list includes all his major mathematical papers and textbooks, and thus represents an invaluable source for the historian of mathematics. Finally, a selected bibliography provides a list of articles and reminisces about Banach. Unfortunately, for the English reader, almost all of these are in Polish.

In conclusion, Through a reporter’s eyes provides the reader with a short and engaging version of Banach’s life. Although this book is described by the author as having a journalistic rather than scientific character, an honest attempt is made to convey the import of Banach’s most significant results. The method of summarizing the results of Banach’s papers individually is especially useful as anyone with further interest is provided with an exact reference. There are a few minor errors in the first half (some of which may be the result of translation) that mostly concern background material. For example, at one point Kaluza seems to suggest that Cauchy was the creator of the δ-ε definition of a limit (it was Weierstraß), and at another point refers to French as being the lingua franca of early twentieth century mathematics, when it was in fact German. Nevertheless, especially considering the fact that Banach left no correspondence, Kaluza does an excellent job in digging through the archives to reconstruct Banach’s life. I would recommend this book to anyone interested in early twentieth century mathematics and the European scientific education system. It will also prove to be an important companion to any future historical study of Polish mathematics.

Stefan Banach on a Polish postage stamp

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