Many valuable mathematical problems and solutions originated in a smoke filled Scottish Café in Lviv. Unfortunately, many of them were lost for this reason for ever. Written with charcoal pencil on a table, they didn’t survive the confrontation of a janitor’s cleaning cloth.

Accidental guests of this café, which was situated on Akademicki square close to the Univeristy of Jan Kazimierz, had to be quite surprised by the sight of a group of men sitting around a table, drawing something on its marble top – it was extremely suitable for such writing – or just keeping deep silence, drinking coffee and looking at each other with a vacant stare. This is how it remembers the youngest of those men, Stanislaw Ulam, one of the only two students who had the priviledge of being a member of this select group. The body consisted of members of Lviv’s mathematical school, with the cleverest representatives with the names of Stefan Banach, Hugo Steinhaus, Stanisław Mazur, Władyslaw Orlicz, Juliusz Paweł Schauder, Antoni Łomnicki, Marek Kac, Stanislaw Ruziewicz, Wlodzimierz Stożek, Stefan Kaczmarz, Stanislaw Saks and Ulam.

Belonging today to Ukraine, Lviv was at that time, in thirties last century, the third largest city of Poland (the bigger were only Warsaw and Lodz) and it was the Poland’s big centre of cultural and scientific life, with its four renowned academies, including the University of Jan Kazimierz (today University of Ivan Franko), existing since 1661. The university was very close to the Scottish Café.

Math of café

It was just here, in the café, not behind the walls of respectable academies, where the basis of modern maths had originated. Why in a café? Mathematicians used to work during official meetings of Mathematical Society, of course, but the most interesting discussions took places during the consumption of coffee and alcohol. Initially they attended the café Roma, but as the owner refused them credit, they moved to the Scottish Café, which was friendlier and with more understanding to an intellectual meaning of their sessions. Such café activity was specific for Lviv after I world war. Lviv was then a city several hundred years old, full of magnificent relics and its inhabitants liked to have fun and enjoy life. It was a tradition to go to an exclusive restaurant “George” after faculty meetings. “Scottish Café” was a place of meetings not only for mathematicians, but also philosophers of a world renown, such as Tadeusz Kotarbinski, Wladyslaw Tatarkiewicz and Roman Ingarden.

The heart of the mathematical table and “primus inter pares” was Professor Stefan Banach, the creator of quite new then and very modern functional analysis, which was one of the most important subjects of those discussions. Banach was not only a mathematical genius (closer look to his scientific achievements is a theme for separate article), but also a very interesting character. An illegitimate son of a highlander woman (he never met his mother) and a clerk, he had not completed any academic courses (he was a self-taught) yet became a full professor at the age of 35!

He could not work successfully in a quiet spot – he needed noise. It often happened that after meetings in “Scottish Café” he went to a bar at the railway station to continue his meditations! He hated making scrupulous notes so many of his writings had been lost. One of his scientific works came from the notes made by his assistant – Banach only approved the final. His mind used to work especially quickly, like a computer.  It’s not surprising that Americans wanted to recruit him, but he was very attached to Lviv and he refused even the most lucrative proposals. His response to an offer by Norbert Wiener, the father of cybernetics, given to him by John von Neumann, is legendary. Neumann gave him a cheque with a number one asking for adding as much zeros as he wanted, but Banach had said it was too little to leave out.

The offer of going to USA had been accepted by Ulam (on the photo), who was then one of the members of the Los Alamos team working on the nuclear bomb. The genesis of his mathematical successes stems from his works in the Scottish Café. A very interesting person was also professor Hugo Steinhaus, who loved not only maths but was also an admirer of the Polish language and creator of brilliant aphorisms. Those men liked to discuss not only maths but also physics, astronomy and politics, play chess and listen to the music. Meetings washed down by alcohol, called jokingly by mathematicians from Warsaw “banachalia” (it refers to the antic bacchanalia), lasted a few or even several hours. – It was very difficult to make Banach feel tired and drink more than he could – remembered Ulam. He said also that he had only experienced such intense brainstorming atLos Alamos. No one of those people were jealous of the successes of his colleagues, they wanted only to have a great mathematical adventure.

Some mathematical problems originated during those meetings, but very often Banach, Mazur or Ulam brought them to the café. Many of them were connected with functional analysis. As the time was passing by, the marble table top was awash with more and more small mathematical signs and symbols and complicated thesis and proofs, understandable only to those chosen ones. But written on a table or paper serviettes, they have been lost forever. The proof of very valuable thesis connected with a space of Banach, the result of a meeting lasting seventeen hours, was lost among them!

The birth of the “Scottish Book”

Even mathematical genius can’t be compared with a woman’s wisdom and cleverness. It was Lucja Banach, a wife of Stefan, who bought an exercise book for recording some of the questions and solutions. This was the genesis of the legendary Scottish Book, not only a collection of interesting ideas and problems – some of them found it’s solutions many years later – but also a testimonial of time, interest and way those mathematicians from Lviv in 30. XX century used to work. The book was under the care of the headwaiter who kept it in a secret place and was given to mathematicians on their request.

The first note comes from 17^th July 1935 and was made by Banach. The question is:

a)      When can a metric space (possibly of type (B) ) be so metrised that it will become complete and compact, and so that all the sequences converging originally should also converge in the metric?

b)      Can, for example, the space [Co] be so metrised?

Mathematicians tried to select problems avoiding trivial ones but they were not able to protect the book from some of them being entered. So fundamental questions are mixed with few problems on a lighter note. Some of the problems were solved immediately, some had to wait many years for a result. In over 20 cases happy men who managed to solve some problems were granted awards by authors of those problems. The first awards were quite modest: few bottles of beer or a bottle of wine. As time was passing by they became more original: fondue in Geneva, 100 grams of caviar, a dinner at “George’s” or at “Dorothy’s” in Cambridge (the founder was an Englishman A.J. Ward) or a bottle of whiskey of measure >0 (the founder was van Neumann). The most original came from Stanislaw Mazur – a live goose! The goose went finally 36 years later to a Swedish mathematician Per Enfel for solving a problem number 153 and it was Mazur who gave him this award.

“The Scottish Book” consists of 193 problems. Some of them come from foreigners, like the above mentioned Ward, van Neumann and Maurice Fréchet, Kampé de Fériet, Offord, Russians Bogolubow, P. Alexandrow, Siergiej Sobolew and Lazar Lusternik.

The war

The outbreak of the second world war surprised scientists in Lviv, although surely they could have guessed something would happen. In the summer of 1939 Mazur and Ulam, who used to visit Lviv every year, were discussing different ways of protecting the “Scottish Book”. They even considered burying it in a soccer pitch. According to a Ribbentrop-Molotow pact Lviv became a part of the Soviet Unionon the 23^rd September 1939 and it was at the time of making notes in the “Scottish Book” by some Russians.  On the 29^th June 1941 the city was occupied by the Germans who decided to exterminate Polish intelligence. In the first days of July more then 40 professors lost their lives, shot by the nazis. Many members of the Lviv mathematical school didn’t survive the war.

Stefan Kaczmarz was the first victim – being a Polish lieutenant, was shot in 1940 in Katyn by Russians along with thousands of other Polish officers. Prof. Antoni Łomnicki, Prof. Stanislaw Ruziewicz and Wladyslaw Stozek were shot in Lviv on the 4^th July 1941. Herman Auerbach who was forced to live in a ghetto committed suicide in 1942 avoiding a cruel ending in a “death car” inside of which Jews had been poisoned by fumes. Stanislaw Saks was murdured by the gestapo in 1942 in Warsaw, and in 1943 Juliusz Schauder was shot by Germans. Banach, who decided to cooperate with Ukrainian authorities and during German’s occupation was a lice-feeder, managed to survive but only to die of lunge cancer (he smoked too much) on the 31^st August 1945.

The men who survived and managed to continue their scientific work in Polish academies were Steinhaus, Mazur and Orlicz. Ulam and Kac saved their lives thanks to immigrating to USA before the war.

After war

The “Scottish Book” has managed to survive thanks to Lucja Banach, who brought it to Wroclaw, where Steinhaus also lived. Here the typewritten copy was made and a copy was sent to Ulam in 1956, who – when the existence of this book became known to many mathematicians in the world – had been inundated by requests to publish it. After consulting with Steinhaus he decided to translate it into English to make people aware of its existence. As many of its authors were dead he decided to keep those notes without any improvements. – At least the collection gives some picture of the interests of a compact mathematical group, an illustration of the mode of their work and thought; and reflects informal features of life in a very vital mathematical center– Ulam explained in the introduction. In 1977 the typewritten copy had been reprinted and in 1981 Birkhaus edited “The Scottish Book: Mathematics from The Scottish Café” by R. Daniel Mauldin – a version with commentaries and lectures from a special conference. The “Scottish Book” is no longer in use – it is kept in safe by Stefan Banach’s family – but the idea has not been lost. Although Steinhaus was not successful in creating “New Scottish Book”, which existed in Wroclaw only few years previous, many academies in the world have snatched the concept and now have their own “Scottish Books”.

What’s now? The building has survived well and looks outside the same, but the café  became a bar, then bank, and now restaurant and hotel is there. Luckily the new owners keep the memory of the famous past and the guests of the hotel can see the copy of the “Scottish Book”. 

source: Wikimapia.org 

This article does not intend to analyse the work of the Lviv mathematical school and the content of “Scottish Book”. The intention is to remind us of unusual times, unusual people and an unusual way of working with mathematics. The best description we find in the memories of Stanislaw Ulam who wrote in 1969 in his recollection: – Sessions with Banach, and more often with Banach and Mazur, created an atmosphere that only existed in Lviv. Such intimate cooperation was probably something new in mathematical life, or at least in such scale and intensity.

 Renata Głuszek

Read also: Problem 59 by a Dutchman

The article was previously published in Nieuwe Wiskrant (June 2010) and on the website Kennislink

Foto: Wikipedia, Wikimapa.org