They were brilliant mathematicians, but would they have been able to calculate the extent to which their creation, the Lwów School of Mathematics, would, for the next eight decades, inspire subsequent generations of artists, as well as scientists? Monuments are still being erected in honour of members of the School; plays about them are being produced, films made, and ever new poems, books and articles (such as this one) are being written.
At a marble café table, the Lwów mathematicians would endlessly multiple the alcohol strength percentage and divide cigarettes – until they soared, in clouds of smoke, into the world of mathematical theorems. In their hands, a simple notebook created so many problems that it became much more than that; a true book. The mathematicians themselves, though long dead, still zero death out, put time in brackets and condense space.
The Lwów School of Mathematics continues to be relevant, not just because the problems (or, as they preferred to call them, problemats) they posited remain salient, or because so many outstanding minds came together in one time and place, but primarily because the Lwów mathematicians elevated science to the status of play. The pleasure of mind wrestling (a pastime which requires brilliant companions) leads to a firm belief that one is part of something extraordinary and not universally accessible. The Lwów mathematicians were anything but self-effacing, romantic scientists – they were aware of their own genius to the point of self-regard.
Hugo Steinhaus, who inaugurated the Lwów School, stated outright in his Wspomnienia i zapiski [Recollections and Remarks] that: “All the chaos in the heads of half-educated people is due to the fact that really not everyone has been cut out to be a scientist. Scientific issues, and the scientific method, remain unavailable to the vast majority of people. It’s a similar story with poetry: I have known people who considered poetry a school exercise, meant to perfect young people’s style.” Steinhaus took the view that, for Stefan Banach, the play’s master of ceremonies, mathematics was a craft, and practising this craft held the same mystery as poetry.
By contrast, Stanisław Ulam, who made the Lwów School famous around the world, expressed admiration for the power of Banach’s mind. When disagreeing with his interlocutor, Banach would not counter them immediately (and sharply). Instead, he would calmly ask them a series of questions so brilliant the erring person would eventually discover the flaws in their reasoning themselves. Ulam recalled one could spend hours with Banach, whether at his university office or a café, discussing some mathematical problem or other. Throughout, Banach would smoke and drink alcohol or coffee, combining his cheerful disposition with a sceptical outlook on the world.
A monument with one unknown
The first bench dedicated to the Lwów School can be found in Kraków’s Planty Park. This (rather peculiar) monument brings to mind an equation with one unknown: Hugo Steinhaus himself. As far as the formation of the group was concerned, Steinhaus recalled the key moment in the summer of 1916 as follows: “Although Kraków was formally a fortress, people were free to roam the Planty in the evenings. During one such walk I overheard the words ‘Lebesgue measure’. I approached the park bench and introduced myself to the two young apprentices of mathematics. […] The youngsters were Stefan Banach and Otto Nikodym. From then on we would meet on a regular basis, and […] we decided to establish a mathematical society.”
The monument, unveiled a century after this encounter, shows Stefan Banach and Otto Nikodym sitting on a bench; it is anyone’s guess why Steinhaus is nowhere to be seen. His absence becomes even more surprising if we consider that, when asked to name his greatest mathematical discovery, Steinhaus would invariably reply: “Stefan Banach.”
Banach was left-handed and only partially sighted in his left eye; words burst out of his mouth at the speed of bullets. His colleagues recalled how he thought the world full of mathematics. Before he met Steinhaus, Banach felt there was no need to publish his mathematical illuminations or document his achievements – for instance, with an examination certificate. In secondary school, he was predicted to fail in eight subjects at the end of the school year. When he completed his secondary education in 1910, his name was not mentioned among those who graduated top of their class. There is no information about the several years following his graduation. In 1913, he emerged briefly at the Engineering Department at Lwów Technical University. Graduating in the following year with a so-called ‘half-diploma’, he found employment as a foreman during road construction works.
It was at that point that the three men met on a bench in Kraków, and the occasion marked a turning point in both Banach’s professional and personal life. In 1919, Steinhaus, Banach and Nikodym (along with 13 fellow mathematicians) founded the Polish Mathematical Society [Towarzystwo Matematyczne]. The Society was soon to expand, with branches established across the (newly independent) country.
The unintrusive Banach
Steinhaus was impressed with Banach, who, within several days, presented him with a solution to a rather complicated mathematical problem that Steinhaus himself had been grappling with for quite some time, even though he had plenty of expertise as a mathematician. Banach in turn was impressed with Łucja Braus, who worked as a stenotypist for Hugo’s cousin, Władysław Steinhaus – so much so that, in 1920, Banach and Braus married. They were very much in love – Łucja outlived her husband by 11 years, and she had the words ‘a mathematician’s wife’ engraved on her tombstone.
The fact that Steinhaus, by then an esteemed lecturer at the Jan Kazimierz University in Lwów, took Banach under his wing, enabled the latter to start publishing his papers. These earliest publications included “On the Independent Value of Orthogonal Functions” and “On the Functional Equation f(x+y) = f(x) + f(y)”. These papers gave rise to further ones, each more brilliant than the last. Given that all the works were published in French, news of Banach’s genius spread overseas.
At that moment in the early 20th century, independent Poland was being born, and lecturers were needed at Polish universities, including the one in Lwów. On the strength of a recommendation from Steinhaus, Antoni Łomnicki, a geometry and cartography specialist, became a professor at Lwów Technical University, offering Banach both the post of assistant and accommodation at his home. The latter was a temporary measure, meant as a helping hand until Banach settled into a place of his own. In return, Banach was to babysit for Łomnicki’s young daughter.
But there was a problem. An individual who produced (often at great speed) academic papers on real-valued functions, worthy of a professorship, went about in a darned suit and mended shoes. To make matters worse, he did not hold as much as a Master’s degree. Łomnicki arranged for special permission from the ministry – thus began Banach’s teaching career as an assistant lecturer. Because he skipped his Master’s, the Jan Kazimierz faculty employed a variety of methods to force Banach to complete his PhD. They were well aware that Banach already had all the relevant material, it was just that he was not writing his thesis.
After the war, Otto Nikodym, the bearded man from the Planty bench, recalled that while Banach enjoyed intellectual speculations, recording these speculations seemed tedious to him. In the end, Stanisław Ruziewicz, professor at Lwów Technical University, came up with an idea: he asked his assistant to accompany Banach to a café, ask him searching questions and write down his propositions and proofs. The notes were then handed to Banach for editing, meaning that in 1921, his doctoral thesis was completed. It was Banach’s seventh publication; it was also his first concerning functional analysis. It is titled “On Operations on Abstract Sets and their Application to Integral Equations”.
Banach’s PhD thesis granted him a place in the history of 20th-century mathematics. The dissertation is considered a breakthrough on a par with the Cartesian revolution; a foundation on which Banach and his students erected the magnificent edifice of functional analysis, a discipline that offered an entirely new outlook on mathematics. 12 years later, Banach brought out his (even more renowned) monograph Teoria operacji kluczowych [The Theory of Linear Operations], which featured the concept of ‘Banach space’, well-known to every aspiring scientist today.
After such a brilliant doctoral dissertation, reaching the next stage (that of an independent lecturer – a process known in Poland as habilitacja, or ‘habilitation’) proved a much smoother process. In 1922, aged just 30, Banach became a professor at Jan Kazimierz University in Lwów; he was awarded full professorship five years later. From bumping into Steinhaus at a bench in Kraków, six years was all it took for Banach’s genius to blossom and completely conquer the world of mathematics.
A period of intense activity began for Banach – indeed, it was his heyday. He received accolades from scientific organizations and invitations to conferences abroad; published new propositions, as well as school and academic textbooks. He gave lectures at the University and Technical University and was elected chairman of the Polish Society for Science. He was also awarded a science prize of 20,000 złotys, which was enough to buy four cars at the time. Yet Banach never received the prize: World War II broke out not long afterwards.
In 1929, despite resistance from a rival journal (Warsaw’s Fundamenta Mathematicae), Banach and Steinhaus founded Studia Mathematica, whose focus was functional analysis. The journal is still published today.
Chief, lead us!
It was Banach (along with his best friend, the outstanding mathematician Stanisław Mazur) who was in charge of the scientific work going on in cafés as tonnes of cigarettes were smoked and cognac and coffee were consumed in excess. Best friends Banach and Mazur had a number of things in common: no degree, a predilection for extravagant behaviour, a strong aversion to the idea of having their discoveries published, and a tendency to debate things for hours. Both men became professors without holding a Master’s degree. After World War II, Mazur, by then a professor at the Institute for Mathematics at the Polish Academy of Arts and Sciences, would allegedly respond to published mathematical sensations with the comment: “There are still things they don’t know.” This was his way of implying Banach and himself had hit upon the idea in question much earlier, but chose not to publish their findings.
Stanisław Ulam, a student of Banach and Mazur, was the youngest participant of these feasts. It was also Ulam who later became the most famous member of the group – much of his fame was due to the fact that, in collaboration with Hungarian physicist Edward Teller, he constructed the American atomic bomb. Ulam’s memoirs, published in English and translated into many languages, paint so vivid a picture of the atmosphere of the Lwów gatherings that they inspired the feature film Geniuses, which opened in Polish cinemas in summer 2021.
The names of those who frequented the Lwów cafés comprise a long list: Herman Auerbach, Zygmunt Wilhelm Birnbaum, Leon Chwistek, Meier Eidelheit, Samuel Eilenberg, Władysław Hetper, Mark Kac, Stefan Kaczmarz, Bronisław Knaster, Joseph Kampé de Fériet, Kazimierz Kuratowski, Antoni Łomnicki, Józef Marcinkiewicz, John von Neumann, Władysław Nikliborc, Władysław Orlicz, Józef Pepis, Stanisław Ruziewicz, Stanisław Saks, Juliusz Paweł Schauder, Józef Schreier, Siergiej Sobolew, Włodzimierz Stożek, Edward Szpilrajn – and, after the war, Marczewski. There must have been others whom Steinhaus, Ulam and Mazur omitted to mention.
As Ulam tells us in his memoir, the merrymaking may have followed roughly this schedule: on 6th November 1936, a Friday, Włodzimierz Stożek, professor at Lwów Technical University, entered Banach’s office at Mikołaja Street in the city with his companions and said: “Chief, lead us!” Ulam recalled that Professor Stożek (whose name means ‘cone’ in Polish) looked more like a sphere than a cone. He was short, plump, completely bald and always cheerful. From time to time, he would be joined by his son-in-law, Leon Chwistek: a fellow mathematician, as well as an artist and philosopher, who, as a young man, was friends with the writer and artist Stanisław Ignacy Witkiewicz (Witkacy). More often than not, Stożek would play chess with Władysław Nikliborc; the pair were cheered on by other mathematicians, including the brilliant chess player Herman Auerbach. Shy and taciturn, Auerbach would occasionally show his dry wit.
The Scottish Café in Akademicki Square was about 200 metres away from Banach’s office, but it is possible that the group’s first stop was Pokój śniadań (‘The Breakfast Room’), a snack bar run by Zofia Teliczekowa and famous for its delicious sandwiches, a selection of herrings, tripe and chitterlings, as well as frankfurters served with horseradish. It was there that the mathematicians may have had a late breakfast (or what is known today as brunch). Initially, the group got together at the Roma café, but, in the end, Banach opted for the Scottish Café across the road.
This was how Józef Mayen, a Lwów-based journalist, described the clientele of that establishment in 1934: “University professors and lovers, old gossipy ladies and lone newspaper readers, bibliophiles and billiard players, Jewish intelligentsia and students from nearby accommodation – all stations and spheres, classes and races, religions and tastes lived there in harmony; not with each other, true, but alongside each other, filling, on average, half the room.”
The Scottish Café had marble tables, enabling the Lwów mathematicians to jot down their calculations in pencil, erase them quickly, and then continue jotting. According to Ulam, a bystander watching the Banach group from a distance would have been able to see them writing a few lines down on the table (an activity punctuated with laughter), suddenly engaging in short bursts of conversation, and then falling silent for longer periods, as they looked at each other wildly, in extreme concentration. One of these café sessions (featuring Ulam, Banach and Mazur) lasted for 17 hours, and was only broken up for meals.
The problem of Mazur’s goose
On 6th November 1936, thanks to Łucja Banach, the mathematicians could at last abandon writing on their table. The former stenotypist spent 2.5 złotys on an object deemed priceless by the world of mathematics: a thick, bound notebook, dubbed the ‘Scottish Book’. Hugo Steinhaus recalls the notebook was kept at the café, so that any member of the group could ask to see it and log a problem of their choice, complete with their signature and date. Steinhaus, a linguistic purist, insisted that the problem be referred to as problemat – the more precise term with regard to mathematics.
Problemat number one was logged in the notebook on 17th July 1935 (a Wednesday) by Stefan Banach. The issue at hand was metric space. On 6th November 1936, the day we are looking at here, Hugo Steinhaus, logged problemat no. 152: “A circle (shield) radius 1, covers at least two points on the integer grid (x,y), no more than 5. When the circle is moved by two nw vectors (n= 1,2,3,…..), where w has two irrational components in irrational proportion, numbers 2,3,4 recur an infinite number of times. What is the frequency of these occurrences for n→∞? Is there such a frequency?” Steinhaus offered an award for calculating the frequency: 100 grams of red caviar. For proof that the frequency exists: a small beer. For a counterexample: a small black coffee.
On the same day, the notebook went to Stanisław Mazur, who logged problemat no. 153, pertaining to the issue of a positive (or negative) solution to the question of basis in Banach spaces. As an award, Mazur offered a live goose.
It was not until 36 years later that Per Enflo, a Swedish mathematician specializing in functional analysis, came up with a solution. In 1972, the Swede arrived in Warsaw, and Mazur handed him the live goose in a basket, with multiple TV cameras filming the occasion. Other prizes on offer at the Scottish Café included: bottles of whisky, wine and cognac, lunch at Cambridge, fondue in Geneva, and a kilo of pork fat. This last item was an award for solving problemat no. 184 (logged by Stanisław Saks); at the time, it may have been worth more than a lunch at Cambridge. Some of the problems remain unsolved till this day.
Manuscripts buried in a football pitch
The Scottish Book concludes with an entry by Hugo Steinhaus, dated 31st May 1941: a combinatorics task involving two boxes of matches. By then, Soviet troops were patrolling the streets of Lwów. Following the Soviet invasion of Poland on 17th September 1939, the patron of the University in Lwów was changed: Jan Kazimierz, a former king of Poland, was replaced with the Ukrainian poet Ivan Franko. The Polish mathematicians (Banach, Mazur and Stożek included) collaborated with their Soviet colleagues enthusiastically: they travelled to Moscow regularly and got to work on Russian-language textbooks. Their aim was to survive. This is why, from 1940, the Scottish Book features problemats logged by Soviet mathematicians including Sergei Sobolev and Pavel Alexandrov.
And yet, it became more and more difficult to survive – mathematics offered no protection from the war. Lwów was about to face another occupation, this time by Nazi Germany. For Polish mathematicians of Jewish heritage, this occupation would equal death. Ulam and his brother made it to the US; Steinhaus fled Lwów and went into hiding in the countryside, where he used the name Grzegorz Krochmalny. Their colleagues Jan Paweł Schauder and Meir Eidelheit were shot in Lwów by a German patrol. Herman Auerbach, the differential geometry specialist, took his own life in the Lwów Ghetto, while Ulam’s closest friend, Józef Schreier, killed himself after the Nazis discovered his hiding place: a bunker in the town of Drohobycz. Władysław Hepter starved to death in a Soviet labour camp. Professors Antoni Łomnicki, Włodzimierz Stożek and Stanisław Ruziewicz were executed by the Nazis in the 1941 massacre of Lwów professors at Wzgórza Wuleckie. Stefan Kaczmarz, who specialized in linear equation systems, was killed during the defence of Poland in September 1939. Stefan Banach only survived the war because he had been employed as a louse feeder at the Lwów Institute for Typhus and Virus Research (known as the Weigl Institute). But even Banach did not live very long after the war had finished: he died of cancer in August 1945. Władysław Nikliborc, who looked after him until the very end, was persecuted by communist Poland’s secret police and committed suicide in 1948. Stanisław Mazur, who had always had communist leanings, survived the war in the Soviet Union, and, after it ended, made his contribution to the new Polish People’s Republic.
The Scottish Book was preserved. Ulam recalled Mazur was far more certain than his colleagues that war was imminent. This was how Ulam described the conversation with his colleague in his memoir: “Mazur said to me: ‘A world war may break out. What shall we do with the Scottish Book and our joint unpublished papers? You are leaving for the United States and presumably will be safe. In case of a bombardment of the city, I shall put the manuscripts and the book in a case, which I shall bury in the ground.’”
In 1946, after her husband died, Łucja Banach brought the Scottish Book from Lwów [which by then was part of the Soviet Union and renamed Lviv, the name it still holds as part of today’s independent Ukraine – trans. note] to Wrocław [the former German city of Breslau, which became part of Poland in the post-war, and where many of Lwów’s former Polish inhabitants were resettled – trans. note]. Today, the original is held by Banach’s descendants. In 1955, Steinhaus sent a copy to Ulam in the US. Ulam translated the Scottish Book into English and sent 300 copies all around the world. The Scottish Book caused a sensation, and continues to do so till this day. A second edition was published in English in 2015. So far, the Scottish Book has not had a single edition in Polish – surely this is an unknown even more enigmatic than Hugo Steinhaus’s absence from the monument-bench at Planty Park in Kraków.
They were no idealists
The Scottish Book continues to fire people’s enthusiasm. My son, Mikołaj, a mathematics student at University College London, recently brought it home to me just how strong that enthusiasm is. Professor Sergei Tabachnikov of Pennsylvania State University, head of the Institute for Computational and Experimental Research in Mathematics, wrote a paper where the Lwów School of Mathematics is portrayed as a contemporary (and truly remarkable) phenomenon. Exclusively for this article, my son and I asked Professor Tabachnikov how someone who has dedicated their entire life to mathematics views the Lwów School of Mathematics (and the Scottish Book) today. Tabachnikov replied:
“This was a really spectacular school – it made a major contribution to functional analysis and convex geometry, and its culture of problem posing and problem solving is admirable. I wouldn’t call it idealistic, but [its creators] clearly enjoyed what they were doing. I think that the Scottish Book is something to be proud of; it is a memorial to the School, sadly now extinct. I believe that most of the problems in the Scottish Book are not solved. Also, not every problem is formulated in such a way that one can give a ‘yes’ or ‘no’ answer. Many are open-ended, such as Stanisław Ulam’s problem 19 – whether a solid of uniform density floating in water in any position is a sphere. It can be considered in various dimensions, but even in two-dimensional space the problem is not fully resolved, let alone other dimensions. Thus, some of the problems have contributed and are still contributing to the emergence and development of new mathematical theories.”
The Lwów School may also be an inspiration to those who do not practise mathematics as a profession. For American poet Susana H. Case, a 9/11 survivor who published her collection The Scottish Café / Kawiarnia Szkocka a year after the atrocity, the story from Lwów brought the end of a world to mind – a situation not unlike the New York attack. In 2016, a production of The Mathematical Opera, or the Paradoxical Distribution of the Sphere opened in Wrocław, directed by Roman Kołakowski.
In the old building at Akademicki Square (today: Shevchenko Prospect) in contemporary Lviv, the Scottish Café opened once again in 2015. Patrons can not only see a copy of the Scottish Book, but also treat themselves to some eel salad and cognac. Banach would have loved it.
The article was written in collaboration with Mikołaj Górlikowski. In addition, while working on the piece, I referenced the following sources: R. Daniel Mauldin (Ed.), The Scottish Book: Mathematics from The Scottish Café, with Selected Problems from The New Scottish Book, ed. II; Hugo Steinhaus, Wspomnienia i zapiski [Memories and Recollections]; Roman Kałuża, Stefan Banach; Stanisław M. Ulam, Przygody matematyka: autobiografia [The Adventures of a Mathematicians: an Autobiography]; Mariusz Urbanek, Genialni. Lwowska szkoła matematyczna [The Brilliant Ones: the Lwów School of Mathematics].
Translated from the Polish by Joanna Błachnio
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