Any examination of twentieth century mathematics shows the surprising depth, originality and quantity of Polish contributions to the discipline. Similarly, any list of important twentieth century mathematicians contains Polish names in a frequency out of proportion to the size of the country. How did such creativity and mathematical influence develop in a country that had little tradition in research, that was partitioned under foreign domination from 1795 until the end of World War I, and whose educational institutions were suppressed by foreign powers. Surprisingly, it was planned! What was to become known as the Polish School of Mathematics was established following a plan proposed by Zygmunt Janiszewski.

During the period 1795-1918 Poland was partitioned with parts under Russian, Prussian, and Austrian control. All three countries tried to suppress Polish nationalism with the Polish language and Polish educational institutions receiving special attention. The Polish people, however, had a strong belief in the eventual resurrection of Poland and supported Polish culture and Polish education in a necessarily informal and unobtrusive manner; consequently, Poland was called a `nation without a country'. The Mianowski Foundation, while not alone, did as much as possible to support Polish science and mathematics, even though the Russians restricted it to what was supposed to be a purely charitable role. With its considerable financial resources (its assets included an oilfield in the Caucauses) it supported doctoral students studying abroad, scholars engaged in significant research, student scholarships, and publications. For example, it supported a series of books called Poradnik dla Samoukow (Guidebooks for Self-Instruction). These were designed to get around the Russian and German educational restrictions and were written by prominent mathematicians including Janiszewski, Sierpinski, and Zaremba; they covered topics such as series, differential and integral equations, and topology. Another series it supported was Nauka Polska jej Potrzeby Organizacja I Rozwój (Sciences and Letters in Poland, Their Needs, Organization and Progress); the first issue (1917) contained two articles which were to be very important to the establishment of the Polish School of Mathematics.

The first article was by Stanislaw Zaremba, who noted that a number of secondary school teachers had the potential to be future scholars; he urged that a way be found to have them sent abroad for further study. The second article `On the Needs of Mathematics in Poland' by Zygmunt Janiszewski outlined the plan for what was to be the Polish School of Mathematics. He urged that a commission should be formed to undertake the organization of mathematics, that an effort should be made to identify and retain talented students as early as secondary school, and that a center for mathematical research should be created; he felt that it was important to create a stimulating mathematical atmosphere and to foster contacts between mathematical co-workers (One effect of this was that many Polish articles were multi-authored; today, Internet collaboration is having a similar effect). His two main recommendations, however, were especially innovative, and would eventually prove to be highly effective: first, that Poland should have the majority of its mathematicians concentrate their work in a single branch of mathematics; second, that there be created an international journal of mathematics (accepting papers in English, French, German, and Italian) concentrating in a narrow area of mathematics. This was almost revolutionary at the time, and even the sympathetic supporter Henri Lebesgue cautioned that it might not be possible to find sufficient papers in a single area to sustain the journal. The next year a third article appeared in Nauka Polska; it was written by Stefan Mazurkiewicz and was largely a supplement to Janiszewki's. In particular, Mazurkiewicz urged that there be two mathematical research centers instead of one, and that a monograph series should be published in addition to the journal. It is amazing how completely these plans were to be implemented.

Nationhood was restored to Poland at the end of World War I, and the University of Warsaw opened in 1918 with Janiszewski, Mazurkiewicz, and Sierpinski as professors of mathematics. The three initiated Janiszewki's proposals with Warsaw serving as the proposed mathematical research center, and with set theory, including related areas such as topology and parts of real analysis, being chosen as the area of concentration. The long unsatisfied demand for education in the new nation gave rise to a large enrollment of students, many of whom were mathematically talented; the combination of enthusiastic and talented faculty, mirrored by the same qualities in the students led to an active and lively mathematical atmosphere.

The second part of the plan came to fruition with the publication of the first volume of Fundamenta Mathematicae in 1920; unfortunately, Janiszewski died before the first issue appeared. While Fundamenta was conceived as an international journal, the first volume deliberately contained only papers by Polish authors. The original plan for Fundamenta as a specialized journal was that volumes would alternate concentration between set theory (and related fields), and logic and foundations (indeed, the choice of title was deliberate), but the number of papers in the set theoretic areas soon greatly exceeded those in logic and foundations and this part of the plan was abandoned. Consequently, many fundamental theorems in topology first saw print in Fundamenta. From our perspective, the choice of set theoretic fields as the area of concentration appears to have been an excellent, but natural, choice; however, it should be remembered that at the time these areas had not yet received full acceptance by the mathematical community. The choice reflected both insight and courage.

After about ten years, with the growth in numbers, and with the goal of maintaining collegiality, it was decided to implement Mazurkiewicz's dditions to Janiszewski's plan. A second center concentrating on functional analysis was started in Lwow, where Banach and Steinhaus were professors, and the journal Studia Mathematica was founded in 1929, also devoted to functional analysis. The publication of Banach's thesis in Volume 3 of Fundamenta Mathematicae had marked the beginning of functional analysis as a discipline, and much of its development was recorded in Studia Mathematica. While both centers were very strong, cooperation between them was very good and their identities merged into the Polish School of Mathematics. For example, when the envisioned monograph series was initiated under the series title Monografie Matematyczne (Mathematical Monographs) in 1931, the editor was Waclaw Sierpinski in Warsaw, and the first volume was Théorie Opérations Linéares by Stefan Banach in Lwów. It, like many of the volumes to follow, became a widely recognized classic.

By 1936 the Polish Journals and Monographs had become widely read and highly respected, the achievements of Polish mathematicians had gained international recognition, and the country's mathematical community was spirited and active. While it was recognized that concentrating effort in the areas of specialization had been a major factor in the School's success, there was a growing feeling that the School was too narrow and it was time to broaden; areas such as abstract algebra and applied mathematics were severely underrepresented. In 1936 the Council of Exact and Applied Sciences met and a committee was appointed to study the state of mathematics in Poland; it consisted of Sierpinski (chair), Banach (deputy chair), Zaremba, Pogorzelski, Zygmund, and Banachiewicz (representing the Academy of Technical Sciences). World War II broke out before the committee's work was finished and before any of its partial recommendations could be implemented, so we can only guess how the School would have evolved without disruption. Poland was to suffer great losses to its mathematical community during the War.

For more information about the Polish School of Mathematics, see the interesting books by Kuratowski [Kur] and Kuzawa [Kuz].

We conclude with a discussion of mathematicians that Poland has honored on its postage stamps. As noted above, Janiszewski, along with Stefan Banach, Waclaw Sierpinski, and Stanislaw Zaremba were all instrumental in the development of the Polish School of Mathematics; Poland issued stamps in their honor on the occasion of the 1982 International Congress of Mathematicians, actually held in Warsaw in 1983. A recent Philamath article [T] gives an interesting discussion of the International Congresses of Mathematicians. The astronomer Thaddeus Banachiewicz was a member of the committee appointed in 1936 to evaluate the status of Polish mathematics after following Janiszewski's plan for nearly 20 years; a stamp was issued in his honor in 1983. While Poland has issued many stamps in honor of Copernicus, Jan Sniadecki seems to be the only other pre-twentieth century mathematician Poland has honored.

Banach was born in Cracow. He never knew his mother Katarzyna Banach, and his father Stefan Greczek, who worked for the railroad, never cared for him; his parents never married and he was reared by the Plowa family in Cracow. He had a difficult childhood and had to earn his living starting at the age of 14; colleagues describing him later referred to his `street urchin language'. At first he studied mathematics on his own, but he laterattended the Jagiellonian University for a short period of time (he may have attended lectures by Zaremba). He then entered Lwów Technical University, but his studies were interrupted by World War I. Apparently, no one recognized his extraordinary ability at this time. Hugo Steinhaus often bragged that Stefan Banach was his greatest mathematical discovery. One summer evening while walking in the Planty, a park that surrounds the central area of Cracow, Steinhaus overheard the words `Lesbesgue integral'. The words were so unexpected he went over to the nearby park bench to make the speaker's acquaintance; it was the self-taught Stefan Banach talking mathematics with Otto Nikodym. During this initial conversation Steinhaus told Banach about a problem on which he had spent some time; he was very surprised when Banach showed up a few days later with a solution (see [S]). Banach's career, however, didn't really start until four years later when he was offered an instructorship at the Lwów Institute of Technology. Shortly thereafter, the University of Lwów awarded him his Ph.D. without requiring him to fulfill all of the formal requirements; this was necessary, since being self-taught Banach lacked a college degree (in addition, he refused to take any exams, intensely disliking them). His thesis, largely recorded by friends during his coffee house conversations, Sur les Opérations dans des Ensembles Abstraits et leur Application aux Equations Intégrales, was to mark the start of functional analysis as a discipline. In 1922 he passed his qualifying exam as docent and in the same year was named a professor (at the University of Lwów). He quickly gained an international recognition and gave one of the plenary addresses at the 1936 International Congress of Mathematicians; Sierpinski called him the most distinguished mathematician produced by Poland.

Banach had a strong, but eccentric, personality; he hardly ever wrote letters and never addressed questions addressed to him by post, but in person he was a vibrant and energetic participant in mathematical discussions. He was a great facilitator of collaboration, and there were a number of co-authored papers written by mathematicians in Lwów. He had a great ability to pose interesting and illuminating questions; evidence of this is the famous Scottish Book (an annotated translation has appeared [M]). The Scottish Cafe was often frequented by mathematicians in Lwów; for example, after the Friday meetings of the mathematical society the group would adjourn to the cafe. It was decided that some record of the lively mathematical discussions at the cafe should be kept, and Banach bought a notebook,to be kept by the head waiter at the cafe, in which the group would record the interesting problems that they posed. That notebook, which survived the war, is now widely known as the Scottish Book.

Banach's physical condition greatly deteriorated during World War II; this partly was due to the general famine, but also due his job as a lice feeder at he Rudolf Weigl Bacteriological Institute. He died of cancer on August 31, 1945.

There is an excellent new biography of Banach by Roman Kaluza [K].

Banachiewicz was born in Warsaw and received his bachelor's degree in Astronomy from Warsaw University in 1904. He did his graduate work in Göttingen and Moscow and became an assistant at the Engelhardt Observatory near Kazan. Upon the reunification of Poland in 1918 he became Professor of Astronomy at the Unversity of Cracow where he remained for the rest of his life. He was a natural choice to be the representative of the Academy of Technical Sciences on the 1936 committee since much of his work in theoretical astronomy was very mathematical; his work on the three body problem was presented to the Paris Academy by Poincaré.

While Janiszewski was born in Warsaw, he received his secondary education in Lwów; he then went abroad to study in Zurich, Munich, and Göttingen before receiving his degree in Paris in 1911 with a thesis written under Lebesgue. He became a docent at Lwów in 1913, but left to join the Polish Legion (a Polish unit attached to the Austrian Army) believing that service in the Legion could hasten Polish independence. He left in 1916, when he refused to take a required oath to the Austrian government; he then financed, organized, and ran an orphanage in the city of Radom. Near the end of the war Janiszewski authored the aforementioned paper that was to serve as the blueprint for the building of the Polish School. In 1919 he joined the faculty of the University of Warsaw and started work as one of the editors of Fundamenta Mathematica; he died from influenza in 1920 before the first issue appeared and before most of his plans could be implemented. From all accounts he had a very strong and willful personality, was very creative, and enthusiastically expressed his nationalism in terms of service. He asked that after his death his belongings be given to charity and that his body be given to science. It is reported that his brain was preserved like Broca's [Ma].

In contrast to the all too short life of Janiszewski, Sierpinski had a long and extremely productive life, publishing over 700 papers, many of them seminal in nature. He was born in Warsaw and received his undergraduate education from the Russian Warsaw University; after teaching secondary school for a period of time, he moved to Cracow receiving his Ph.D. at the Jagiellonian University in 1906. He then moved back to Warsaw and was active in mathematics education and research. During World War I he was interned in Moscow, where he was well received by the mathematical community and started joint research with Egorov and Lusin. After the war he became one of the original Professors of Mathematics at the newly opened University of Warsaw and co-editor of Fundamenta Mathematica. While Sierpinski held many official positions in the Polish mathematical community throughout his career, including service as the President of the Polish Mathematical Society, his immense influence on Polish mathematics was due more to his internationally recognized scholarship and even more to the many students he taught and mentored. Sierpinski's papers at both the beginning and end of his career were devoted to number theory, but he had already turned to set theory by 1909. It is likely that set theory and topology was chosen as an area of concentration for the Polish School because of the successes of Sierpinski, together with the presence of Janiszewski and Mazurkiewicz working in the area.

While Poland did not have a strong tradition of mathematical research, there had been important Polish mathematicians in the past. Copernicus is, of course, of great importance and there have been many stamps issued in his honor. Adam Kochanski and Józef Maria Hoene-Wronski were well-known throughout Europe during their lives and Wronski is today remembered for the Wronskian determinant that carries his name. Jan Sniadecki was educated in Poznan and Cracow where he received his Ph.D. He was given support to study abroad and his resulting studies took him to Göttingen, Leyden and Paris; upon his return to Poland he held the chair of mathematics and astronomy at Cracow. His contributions were largely educational.

Zaremba was born in the Ukraine and was first educated as an engineer. He went to Paris for graduate school and received his degree from the Sorbonne in 1889. He stayed in France until 1900, when he joined the faculty at the Jagiellonian University in Cracow. His years in France enabled him to establish a strong bridge between Polish mathematicians and those in France. His research in classical analysis, particulary on harmonic analysis, was widely recognized. He was an older, established, mathematician who contributed to the success of the Polish School through his teaching and organizational skills as well as through his research. Zaremba wrote a number of textbooks and served as President of the Polish Mathematical Society.

References 

[K] R. Kaluza, Through a Reporter's Eyes: The Life of Stefan Banach, Birkhäuser, Boston, 1996.

[Kur] K. Kuratowski, A Half Century of Polish Mathematics, Pergamon, New York, 1980.

[Kuz] M. G. Kuzawa, Modern Mathematics: The Genesis of a School in Poland, College and University Press, New Haven, 1968.

[M] R. D. Mauldin (editor), The Scottish Book: Mathematical Problems from the Scottish Cafe, Birkhäuser, Boston, 1981.

[Ma] J. Mazur, Zygmunt Janiszewski: A Polish Mathematician, Polish Science and Learning II (1943), 38-39.

[S] H. Steinhaus, Stefan Banach, Studia Math. (Seria Specjalna, Z. I., 1963), 7-15.

[T] M. Tanoff, International Congresses of Mathematicians, Philamath XVII (July, 1995), 4-10.