The fact that the editors of "Młody Technik" would ask someone to write an article on Polish mathematics was decided ... 60 years ago, when neither the journal nor the author of this text existed. It is then that something incredible occurred: in a few years Poland, a small and a rather poor country, became a world mathematical power and Polish mathematicians are among the most appreciated specialists to this day. That's exactly how it happens in science: outstanding minds inspire the minds of other people to great achievements, and the tide, once aroused, runs through the years. No one would take an interest in Polish mathematics, were it not for the activity of a dozen, and later of several dozens of greatly talented and vigorous people in the first years of Polish regained independence. The explosion of mathematics in Poland, a country with no significant tradition in this domain and in a particularly difficult situation after 123 years without sovereignty and four years of devastating war, was something marvellous. This is what the article is about.Till the end of the 19th century Polish contribution to world mathematics was practically unnoticeable, although Józef Maria Hoene-Wroński (1778-1853) made his way into the history of mathematics with a valuable application of some functional determinants to the theory of differential equations; today such determinants are known as wronskians. Poles took no active part in the all-round significant development of mathematics that marked the second half of the 19th century. How numerous were the chef d'oeuvres of Polish literature written in times when the language of Poles was subject to discrimination! By contrast, in the same period Polish mathematical production was indeed insignificant. Shortly before the outbreak of World War I the situation started to change. Scientists with broader knowledge made their appearance and serious editorial and organisational activities were developed, mostly inspired by Samuel Dickstein (1851-1939) and partly financed by him with his private money. Valuable papers were appearing in various domains of classical mathematics, as, in particular, those by Stanisław Zaremba (1863-1942) and Kazimierz Żorawski (1866-1953). In the Russian and Prussian sectors of partitioned Poland institutions of higher education being used for Russification and Germanisation of Polish students, they were boycotted by the patriotically-minded youth. Numerous gifted graduates of schools of secondary education emigrated from these two sectors of Poland - either to what was then Galicia (Cracow or Lvov) or westwards, to France, Belgium and Great Britain; students from the Russian sector also chose Germany. Studying abroad had significant impact on the interests, minds and maturity of young Polish scientists of that period. All the future founders of the Polish school of mathematics spent some time studying abroad: Mazurkiewicz, Steinhaus and Sierpiński in Göttingen, Janiszewski in Paris, Kuratowski in Glasgow. The abovementioned Polish mathematicians focused their scientific interests on a very recent branch of mathematics, emerged at the end of the 19th century, which was set theory and its applications, topology in the first place. Wacław Sierpiński (1882-1969), then a young assistant professor at the Lvov University, started his lectures in 1908. In 1909 he gave a one-year course on set theory, the first such course in the world, and few years later he published a text-book on the theory - again the first textbook ever. In 1912 Zygmunt Janiszewski (1888-1920) defended his PhD thesis in Paris, Stefan Mazurkiewicz (1888-1945) wrote his PhD thesis on a topological problem under Sierpiński's guidance in 1913. Some years later the three mathematicians played most prominent roles in the emergence of a strong mathematical school in the new revived Poland. Tsarist armies left Warsaw for good in August 1915 and as soon as November that year two Polish universities opened to students: the Warsaw University and the Warsaw Institute of Technology. Janiszewski, Mazurkiewicz and Sierpiński became lecturers in mathematics. Mazurkiewicz was both a great teacher and a very active researcher, Janiszewski was no second to him in knowledge and ingenuity, but prevailed in precision and orderly mind. Both worked mainly in topology. Sierpiński was already an acknowledged specialist in set theory and number theory. Dickstein, teaching algebra, successfully imparted enthusiasm and passion to young students of mathematics. Many young people had matured while studying abroad, and all those who created the atmosphere of those years, either students or teachers, realised that after a century long gap they were the first generation that was offered the privilege of learning and teaching in a Polish university. At the beginning of 1918 one could justifiably speak of a strong scientific centre in Warsaw, with set theory, topology and their applications in the focus of scientific activities. Young people, such as Bronisław Knaster, Stanisław Saks, Antoni Zygmund, Kazimierz Kuratowski, Alfred Tarski and Kazimierz Zarankiewicz, soon arrived at important results making a noticeable impact on European mathematics. Set theory is simply the theory of sets - but what is topology? Consider two similar figures, say, an arbitrary triangle and a smaller triangle of the same shape. Each of the two can be seen as a deformation or transformation of the other, the transformation consisting in either a uniform extension of the smaller triangle or in a contraction of the bigger one. If other transformations are allowed, say, projections, the triangle, even if remaining a triangle, may become quite different. But then even more significant transformations may be conceived. For example, squeezing a circle may transform it into an ellipse or some other complicated figure, while extending a sphere one can end up with an ovoid surface. In fact, there are cases when the difference between a circle and an ellipse may be neglected and a sphere can be replaced by a surface that resembles an egg. In the 1850's a German mathematician, Karl Riemann, worked on problems in the theory of complex functions. To represent such functions he introduced a class of surfaces that are today called Riemann surfaces. He proved that properties of these functions are strictly related to geometric properties of some corresponding surfaces, the only important feature being the "general shape" of the surface. For example, a circle, an ellipse and any other closed curve that does not intersect itself are all equivalent from his point of view, just like a sphere and an egg. On the other hand, a circle and a curve in the form of the numeral eight are not interchangeable, and neither are a sphere, a round cracknel and a pretzel. Therefore Riemann concentrated his attention on transformations that allow to extend, bend, squeeze or twist a figure. Tearing it apart, for example, was however forbidden. That is how topology was born - an area of mathematics that investigates figure properties that remain unchanged under such admissible transformations. The emergence of a group of mathematicians active in a single area, so recent and important, was precisely one of the factors that made possible the upgrowth of the Polish school of mathematics. That this "school" could be brought into being was due not only to the fortunate meeting of so many talented mathematicians, but also to their will and ability to cooperate and to put forward sound organisational ideas, and to the fact that a clear program of action had been designed and that research was focused on a then very prospective and promising branch of mathematics. The man with most merits in organisational matters was Janiszewski. In 1917 the Mianowski Fund, an institution that greatly supported Polish mathematicians, particularly those coming from the Russian sector of partitioned Poland, sent out an inquiry asking about the needs of science in Poland. Moreover, in 1918, at the dawn of newly recovered independence, the Fund published a volume under the title "Polish science, its needs, its organisation and its development" containing answers to the inquiry. In particular, there was a 7-page article by Janiszewski about the needs of Polish mathematics. It proved to be very true and prophetic. Janiszewski clearly defined the main goal which Polish mathematicians should strive to achieve in the independent country: a centre of creative mathematical work with international renown. One of the basic means suggested by Janiszewski to reach this goal was precisely a concentration of efforts on a relatively small part of mathematics, where results of Polish mathematicians had already been winning international recognition. The most novel idea of Janiszewski was that of publishing a journal covering only those branches of mathematics that were to determine the main research stream in Poland, and where papers would be presented in international languages, mostly German, French, English and Italian. "To gain significant position in the world of science we must come up with some initiative of our own", he wrote. Janiszewski's ideas could be qualified as revolutionary due to two main factors. First, up to date there were no journals whatsoever that would present selected branches of mathematics only. Most scientists in Poland and abroad considered that such a journal had no chance to survive as a scientific communication medium because of the limited number of good quality publishable papers. For instance, such doubts were expressed by one of the best known French mathematicians Lebesgue in his letter to Sierpiński. It later turned out that the development of mathematics in general and of topology in particular brought a rapid improvement of the quality of papers submitted for publication, caused by the growing rivalry between authors and the fame that the Polish journal, given the name of Fundamenta Mathematicae, quickly acquired. To this day it still plays the role intended for it and a publication in the Fundamenta is considered a great distinction by every mathematician. Publication of papers in international languages rather than in Polish, even by Polish authors, was the second important novel element in Janiszewski's ideas. It was rather hard work to convince everyone that Poles could and should publish their results in other languages - the more so that was a period when the Polish nation could finally use its own language without discrimination! Nevertheless, the mathematical community came to understand that this was an absolutely necessary measure. To support the idea there was the example of Zaremba and Żorawski, whose many valuable papers could not duly influence world science just because they were published in Polish. Quite frequently theorems first proved by Poles came to be rediscovered in other countries and the new discoverers acquired fame of pioneers. Janiszewski fell victim of a particularly bad influenza that ravaged Europe in 1920. He did not live to see the first volume of Fundamenta Mathematicae come out of print. Nevertheless, his ideas, supported by Sierpiński, Mazurkiewicz and many others, found immediate response all over the country. In 1919 The Mathematical Society was founded in Cracow and its range soon covered all of Poland. Research centres appeared in Vilnius and Poznań, the mathematical community in Lvov, already known before World War I, was now developing at a new pace. Polish mathematicians received important mathematical journals from all over the world in exchange for Fundamenta Mathematicae, which greatly fostered an intense exchange of ideas with mathematicians abroad. In the rather poor country which Poland was in the early 1920's the acquisition of all the significant journals was hardly affordable. In 1922 the annals of the Polish Mathematical Society were first published in French and soon other publications were edited in French, too. The position of Poles in the mathematical world was becoming strong. However, the golden age of Polish mathematics started at the very moment when a new "eruption" of creative mathematical thinking occurred in Lvov in the middle of the 1920's. Mathematicians in Lvov, just like those in Warsaw, reduced their activity to one domain of mathematics and now this was functional analysis - not very close to the Warsaw choice, though not unrelated. The work of mathematicians such as Hugo Steinhaus (1887-1972), Stefan Banach (1892-1945), Stanisław Mazur (1905-1981), Władysław Orlicz (1903-1990), Juliusz Schauder (1896-1943) and their disciples resulted in an admirable development of this branch of mathematics. Even if the basic notions were known at the beginning of the 20th century, and often earlier, it is due to Banach's splendid work that functional analysis became one of the central domains of modern mathematics. Nowadays not only is it useful to mathematicians: it is given particular importance in physics, and in quantum mechanics in particular, as well as in applications of physics. In 1929 Banach wrote a monograph on "linear operators". This first textbook ever on functional analysis established the renown of Banach and the scientific circle he represented for the years to come. The leading idea of functional analysis consisted in a geometrisation of mathematical analysis. We learn in school that pairs or triples of numbers can be viewed as points of the plane or space; in a similar way some more complicated constructions, like infinite sequences, functions etc. can be viewed as certain points. Clearly, not as points of the usual three-dimensional space; they belong to some special infinitely-dimensional spaces. Banach gave a description of a certain class of such spaces which are now known under the name of Banach spaces to every mathematician in the world. This is one of the most frequently used mathematical notions and someone who knows not what a Banach space is cannot claim to be a mathematician. Due to the aforementioned geometrisation various purely analytical theorems, e.g. about differential equations, can be rigorously proved using purely geometrical methods. The fact that at least one point on a stretched rubber band dropped loose does not change its position can be translated into a proof of a theorem that claims the existence of solutions for a wide class of differential or integral equations. It should be added that Banach spaces also allow an application of algebraic methods to analytical problems. This is due to the fact that laws of the usual vectorial calculus can be applied to "points" in such spaces. This clever combination of algebraic, topological and geometric methods is, in fact, the characteristic feature of Banach's method. Banach's was not the first attempt to build the theory that he eventually erected. Many great and small mathematicians intended to do it before. The future father of cybernetics, Norbert Wiener, seems to be the one who came closest to success. Independently of Banach he achieved some promising results in creating the foundations for functional analysis. So notable were his results that for some time the pertinent spaces were given the name of Banach-Wiener spaces. However, some time later Wiener concluded that the theory is just pure formalism that cannot play any meaningful role in mathematics. Following that conclusion, he shifted his scientific interest elsewhere. In his autobiography published in 1956 he admitted how wrong he was. "Banach's theory is right now beginning to show its full efficiency as a scientific method", he wrote. The way Stefan Banach had been discovered for mathematics seems to fit into either a 19th century novel or a modern science-fiction story. This is how Hugo Steinhaus described it: ,,... One summer evening in 1916, as I was walking along the Planty, I heard a conversation, or rather only a few words. I was so struck by the words "the Lebesgue integral" that I approached the bench on which the speakers were sitting and, then and there, I made their acquaintance. The speakers, Stefan Banach and Otto Nikodym, were discussing mathematics. They told me they had another friend - Wilkosz. (...) Insecurity of the future, the difficulty of earning one's living, the lack of contacts not only with foreign scientists, but even with the Polish ones - such was the pervasive atmosphere of Cracow in 1916. But all that did not prevent the three young men from spending a lot of time in cafes discussing problems in mathematics amidst a noisy crowd. Banach did not mind the noise; for some reasons, known only to himself, he liked to sit quite near the orchestra". Steinhaus invited the two young men to his flat and in a long conversation explained all the problems he had been unsuccessfully working on for a long time. A few days later Banach returned with a complete solution. Even before completing his studies Banach became assistant at the Lvov University in 1920. In the same year he made his PhD in mathematics, still without having completed his studies, of course. In 1922 he made his habilitation (a thesis for the position of assistant professor) and was immediately granted professorship, again with a special permission granted by state authorities and against academic tradition. Two years later he was already an associate member of the Polish Academy of Sciences. By 1939 he had published over 50 papers belonging to various domains of mathematics. Banach died on August 31, 1945, exhausted by his painful war experience. One of the peculiarities of the Lvov school of mathematics was social life concentrated in cafés, which Banach was fond of in earlier years, too. More generally, café life played a definitely positive and inspiring role in what used to be Galicia. For example, Boy-Żeleński initiated his writer's career in the "Green Goose" literary cabaret at Michalik's café in Cracow. In fact, the term "café intelligentsia" had nothing pejorative about it. Lvov mathematicians assembled in cafés close to the University, first at "Roma", then at "The Scottish Café". Among the numerous attractions of the place two were particularly worth mentioning: delicious pastries, which, as the owner used to claim, were sent to Warsaw by plane every day, and ... marble table-tops, which it was easy to write upon and, even more important, to quickly erase what had been previously written. Discussions went on for hours and hours in an atmosphere of perseverance, excitement, concentration and communion of thought. One of the most eminent representatives of the Lvov school of mathematics, Stanisław Ulam, well known in the United States for his participation in the atomic bomb construction project after World War II (and later in the construction of computers) wrote in 1963: "The only time when I encountered a communion of interests, frequency of discussions and intensity of intellectual community life similar to what I had seen in Lvov was the time of my research on nuclear energy during the war." The table, where Stanisław Ulam, Stanisław Mazur and Stefan Banach used to sit at, was one of the "strongest" at "The Scottish Café". The results of a discussion used to be written on the table-top in pencil and then, on the following day, the disputants, now sober both in the literal and metaphoric sense of the word, returned with a sheet of paper, intending to decipher yesterday's scrawls and to order them into a logical whole. Sadly, because of Banach's and his disciples' carelessness many of their valuable results were lost for Polish science. In fact, were it not for Banach's assistants, much of his work would never reach the printers, so untidy his manuscripts were. One autumn the mathematical session at the "Scottish Café" lasted till the early morning of the next day (today they would have certainly been kindly but firmly asked to leave the place). The session ended up with a proof of an important theorem in the theory of Banach spaces. However, when the happy participants of the session had written their ideas on the table-top and tottering took their way home, the charwoman, totally unaware, scrupulously cleaned the table. The argument was lost, never to be recovered. It was Banach's wife's great merit to buy a thick notebook with hard covers and leave it with the café head waiter. He was told to hand it to every mathematician upon request. In a few years the notebook became the well known "Scottish Book", containing a collection of problems posed by Lvov mathematicians to themselves and to the entire world; solutions were written into the book, too. Each author of a problem was expected to contribute with a prize for the mathematician who would solve it. The variety of prizes included a cup of coffee, a bottle of wine or a live goose. "The Scottish Book" fortunately survived the war and is now kept at the Mathematical Institute of the Polish Academy of Sciences in Warsaw. The idea of keeping a notebook to write mathematical problems and their solutions into has since been accepted in many academic centres all around the world. Such a notebook is usually called "The Scottish Book" (which makes Scots particularly proud). The achievements of the Warsaw school of mathematics were no lesser. Wacław Sierpiński made his name on works in number theory and set theory, without neglecting other branches of mathematics. In his lifetime he published 724 papers and communications, 50 books and an important number of popular, historical or survey articles, together with seven secondary school textbooks. Kazimierz Kuratowski is widely known as the architect of the foundations of modern topology, whereas Karol Borsuk gained renown with his papers on the theory of retracts and fixpoints. Today Karol Borsuk is considered the originator of the topological theory of shape, one of the greatest recent achievements in topology. Andrzej Mostowski, recently deceased, made his way into the history of mathematics with his research on foundations of mathematics. Polish mathematics, just as our entire science and culture, suffered huge losses in the last war. The Nazi occupants had murdered many excellent Polish scientists. Banach and Mazurkiewicz died exhausted by the war, many professors settled abroad. An accidental bomb initiated a fire that completely devoured the library of the Mathematical Seminar in Warsaw, Warsaw mathematicians' private collections were destroyed during the uprising in 1944. Very few of those who succeeded to survive the war had a book or reprint left. Nevertheless, Polish mathematics revived and returned to the highest world level. In the first post-war years someone said that Poland exported coal and mathematical theorems. This remains true today, too.

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On 04 Jan 2012, 18:51.