The Scottish Book () was a thick notebook used by mathematicians of the

Lwów School of Mathematics The Lwów school of mathematics () was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland (since 1945 Lviv, Ukraine). The mathematicians often met at the famous Scottish Café to discuss mathematical problems, ...

Poland Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...

for jotting down problems meant to be solved. The notebook was named after the " Scottish Café" where it was kept. Originally, the mathematicians who gathered at the cafe would write down the problems and equations directly on the cafe's marble table tops, but these would be erased at the end of each day, and so the record of the preceding discussions would be lost. The idea for the book was most likely originally suggested by

Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...

's wife, Łucja Banach. Stefan or Łucja Banach purchased a large notebook and left it with the proprietor of the cafe.

History

The Scottish Café () was the café in

Lwów Lviv ( or ; ; ; see #Names and symbols, below for other names) is the largest city in western Ukraine, as well as the List of cities in Ukraine, fifth-largest city in Ukraine, with a population of It serves as the administrative centre of ...

(now ''Lviv'',

Ukraine Ukraine is a country in Eastern Europe. It is the List of European countries by area, second-largest country in Europe after Russia, which Russia–Ukraine border, borders it to the east and northeast. Ukraine also borders Belarus to the nor ...

) where, in the 1930s and 1940s, mathematicians from the Lwów School collaboratively discussed research problems, particularly in

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

and

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

Stanislaw Ulam Stanislav and variants may refer to: People *Stanislav (given name), a Slavic given name with many spelling variations (Stanislaus, Stanislas, Stanisław, etc.) Places * Stanislav, Kherson Oblast, a coastal village in Ukraine * Stanislaus County, ...

recounts that the tables of the café had

marble Marble is a metamorphic rock consisting of carbonate minerals (most commonly calcite (CaCO3) or Dolomite (mineral), dolomite (CaMg(CO3)2) that have recrystallized under the influence of heat and pressure. It has a crystalline texture, and is ty ...

tops, so they could write in pencil, directly on the table, during their discussions. To keep the results from being lost, and after becoming annoyed with their writing directly on the table tops,

's wife provided the mathematicians with a large notebook, which was used for writing the problems and answers and eventually became known as the ''Scottish Book''. The book—a collection of solved, unsolved, and even probably unsolvable problems—could be borrowed by any of the guests of the café. Solving any of the problems was rewarded with prizes, with the most difficult and challenging problems having expensive prizes (during the

Great Depression The Great Depression was a severe global economic downturn from 1929 to 1939. The period was characterized by high rates of unemployment and poverty, drastic reductions in industrial production and international trade, and widespread bank and ...

and on the eve of

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

), such as a bottle of fine brandy. For problem 153, which was later recognized as being closely related to Stefan Banach's " basis problem",

Stanisław Mazur Stanisław Mieczysław Mazur (; 1 January 1905 – 5 November 1981) was a Polish mathematician and a member of the Polish Academy of Sciences. Mazur made important contributions to geometrical methods in linear and nonlinear functional analysis ...

offered the prize of a live goose. This problem was solved only in 1972 by Per Enflo, who was presented with the live goose in a ceremony that was broadcast throughout Poland.Mauldin, ed.; Kaluza. The café building used to house the at the street address of 27

Taras Shevchenko Taras Hryhorovych Shevchenko (; ; 9 March 1814 – 10 March 1861) was a Ukrainian poet, writer, artist, public and political figure, folklorist, and ethnographer. He was a fellow of the Imperial Academy of Arts and a member of the Brotherhood o ...

Prospekt. The original cafe was renovated in May 2014 and contains a copy of the Scottish Book.

Problems contributed by individual authors

A total of 193 problems were written down in the book.

contributed a total of 43 problems, 24 of them as a single author and 19 together with Stefan Banach. Banach himself wrote 14, plus another 11 with

Stanisław Ulam Stanisław Marcin Ulam ( ; 13 April 1909 – 13 May 1984) was a Polish and American mathematician, nuclear physicist and computer scientist. He participated in the Manhattan Project, originated the History of the Teller–Ulam design, Telle ...

and Mazur. Ulam wrote 40 problems and additional 15 ones with others. During the Soviet occupation of Lwów, several Russian mathematicians visited the city and also added problems to the book.

Hugo Steinhaus Hugo Dyonizy Steinhaus ( , ; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Univers ...

contributed the last problem on 31 May 1941, shortly before the German attack on the Soviet Union; this problem involved a question about the likely distribution of matches within a matchbox, a problem motivated by Banach's habit of

chain smoking Chain smoking is the practice of smoking several cigarettes in succession, sometimes using the ember of a finishing cigarette to light the next. The term chain smoker often also refers to a person who smokes relatively constantly, though not nec ...

cigarettes.

Continuity

After World War II, an English translation annotated by Ulam was published by

Los Alamos National Laboratory Los Alamos National Laboratory (often shortened as Los Alamos and LANL) is one of the sixteen research and development Laboratory, laboratories of the United States Department of Energy National Laboratories, United States Department of Energy ...

in 1957. After World War II, Steinhaus at the

University of Wrocław The University of Wrocław (, UWr; ) is a public research university in Wrocław, Poland. It is the largest institution of higher learning in the Lower Silesian Voivodeship, with over 100,000 graduates since 1945, including some 1,900 researcher ...

revived the tradition of the Scottish book by initiating ''The New Scottish Book'' in 1945-1958. The tradition of the Scottish Book continues to inspire not only mathematicians but also educators in other fields. Piotr Kowzan proposed a "goose method" as a pedagogical tool for marking open problems and encouraging future research. Inspired by the eccentric rewards in the Scottish Book, this approach aims to foster curiosity and knowledge-building across generations.

Associated people

The following mathematicians were associated with the

or contributed to ''The Scottish Book'': * P. Alexandroff (

Pavel Alexandrov Pavel Sergeyevich Alexandrov (), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote roughly three hundred papers, making important contributions to set theory and topology. In topol ...

?) *

Herman Auerbach Herman Auerbach (October 26, 1901, Tarnopol – August 17, 1942) was a Polish mathematician and member of the Lwów School of Mathematics. Auerbach was professor at Lwów University. During the Second World War because of his Jewish descent he ...

(murdered in

Bełżec extermination camp Belzec (English: or , Polish: , approximately ) was a Nazi German extermination camp in occupied Poland. It was built by the SS for the purpose of implementing the secretive Operation Reinhard, the plan to murder all Polish Jews, a major ...

) * A.F. Fermant ( :ru:Бермант, Анисим Фёдорович, i.e
Anisim Fedorovich Bermant
) * Bogolubov (

Nikolay Bogolyubov Nikolay Nikolayevich (Mykola Mykolayovych) Bogolyubov (; ; 21 August 1909 – 13 February 1992) was a Soviet, Ukrainian and Russian mathematician and theoretical physicist known for a significant contribution to quantum field theory, classi ...

?) *

(forced to live as a lice feeder, died of cancer in 1945) *

Karol Borsuk Karol Borsuk (8 May 1905 – 24 January 1982) was a Polish mathematician. His main area of interest was topology. He made significant contributions to Shape theory (mathematics), shape theory, a term which he coined. He also obtained important r ...

(imprisoned for working in the Underground Warsaw University) *

Meier Eidelheit Meier "Maks" Eidelheit (6 July 1910 – March 1943) was a Polish mathematician belonging to the Lwów School of Mathematics who worked in Lwów and was murdered in the Holocaust. Biography Meier Eidelheit left the Lwów Gymnasium in 1929 a ...

(murdered in the

Holocaust The Holocaust (), known in Hebrew language, Hebrew as the (), was the genocide of History of the Jews in Europe, European Jews during World War II. From 1941 to 1945, Nazi Germany and Collaboration with Nazi Germany and Fascist Italy ...

in 1943) *

Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...

(fled Europe for

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

in 1939) *

Maurice René Fréchet Maurice may refer to: *Maurice (name), a given name and surname, including a list of people with the name Places * or Mauritius, an island country in the Indian Ocean *Maurice, Iowa, a city * Maurice, Louisiana, a village * Maurice River, a tr ...

Leopold Infeld Leopold Infeld (20 August 1898 – 15 January 1968) was a Polish physicist who worked mainly in Second Polish Republic, Poland and Canada (1938–1950). He was a Rockefeller family, Rockefeller fellow at University of Cambridge, Cambridge Univers ...

(returned from

Cambridge Cambridge ( ) is a List of cities in the United Kingdom, city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 Unit ...

to Lwów in 1935 but left again for

in 1936 under the Nazi threat) * Joseph Kampé de Fériet * Marek Kac (went to study in the US in 1938; his family who stayed were murdered in

Krzemieniec Kremenets (, ; ; ) is a city in Ternopil Oblast, western Ukraine. It is the administrative center of the Kremenets Raion, and lies north-east of the Pochaiv Lavra. The city is situated in the historic region of Volhynia and features the 12th-ce ...

in 1942) * Stefan Kaczmarz (died in unclear circumstances in 1939 after being called up for military service) *

Bronisław Knaster Bronisław Knaster (22 May 1893 – 3 November 1980) was a Polish mathematician; from 1939 a university professor in Lwów and from 1945 in Wrocław. In 1945, he completed a project in collaboration with Karol Borsuk and Kazimierz Kuratowski con ...

(lived as a lice feeder during the occupation of Lwów) *

Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. He worked as a professor at the University of Warsaw and at the Ma ...

(worked in the Underground Warsaw University) * Antoni Łomnicki (murdered in the Massacre of Lviv professors) *

Lazar Lyusternik Lazar Aronovich Lyusternik (also Lusternik, Lusternick, Ljusternik; ; 31 December 1899 – 22 July 1981) was a Soviet mathematician. He is famous for his work in topology and differential geometry, to which he applied the variational principle. U ...

(participated in the persecution of his teacher in 1936) *

Józef Marcinkiewicz Józef Marcinkiewicz (; 30 March 1910 in Cimoszka, near Białystok, Poland – 1940 in Kharkiv, USSR) was a Polish mathematician. Life and career He was a student of Antoni Zygmund; and later worked with Juliusz Schauder, Stefan Kaczmarz ...

(believed killed in the

Katyn massacre The Katyn massacre was a series of mass killings under Communist regimes, mass executions of nearly 22,000 Polish people, Polish military officer, military and police officers, border guards, and intelligentsia prisoners of war carried out by t ...

; his manuscripts, entrusted to his parents, were lost) *

John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...

(moved to

in 1930) * Władysław Nikliborc ( pl, died 1948) * Cyril Offord * Władysław Orlicz * Stanisław Ruziewicz (murdered in the Massacre of Lviv professors) *

Stanisław Saks Stanisław Saks (30 December 1897 – 23 November 1942) was a Polish mathematician and university tutor, a member of the Lwów School of Mathematics, known primarily for his membership in the Scottish Café circle, an extensive monograph on the t ...

(joined the

Polish underground The Polish Underground State (, also known as the Polish Secret State) was a single political and military entity formed by the union of resistance organizations in occupied Poland that were loyal to the Government of the Republic of Poland ...

, executed in 1942 by

Gestapo The (, ), Syllabic abbreviation, abbreviated Gestapo (), was the official secret police of Nazi Germany and in German-occupied Europe. The force was created by Hermann Göring in 1933 by combining the various political police agencies of F ...

) *

Juliusz Schauder Juliusz Paweł Schauder (; 21 September 1899 – September 1943) was a Polish mathematician known for his work in functional analysis, partial differential equations and mathematical physics. Life and career Born on 21 September 1899 in Lwów ...

(had no paper to record his results after 1941, killed by

in 1943) *

Józef Schreier Józef Schreier (; 18 February 1909, Drohobycz, Austria-Hungary – April 1943, Drohobycz, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, group theory and combinatorics. He was a member ...

(took his life in the Drohobycz Ghetto in 1943 to avoid capture) *

Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions ...

(house burned by Nazis in 1944, worked in the Underground Warsaw University) *

Sergei Sobolev Prof Sergei Lvovich Sobolev, FRSE (; 6 October 1908 – 3 January 1989) was a Soviet Union, Soviet mathematician working in mathematical analysis and partial differential equations. Sobolev introduced notions that are now fundamental for severa ...

(spent World War II in hiding, teaching in illegal underground education) * Ludwig Sternbach (taken to

) *

Simion Stoilow Simion Stoilow or Stoilov ( – 4 April 1961) was a Romanian mathematician, creator of the Romanian school of complex analysis, and author of over 100 publications. Biography He was born in Bucharest, and grew up in Craiova. His father, Colonel S ...

* Edward Szpilrajn (later changed his name to

Edward Marczewski Edward Marczewski (15 November 1907 – 17 October 1976) was a Polish mathematician. He was born Szpilrajn but changed his name while hiding from Nazi persecution. Marczewski was a member of the Warsaw School of Mathematics. His life and work aft ...

to escape Nazi persecution) *

(left Poland for the US in 1939, worked on the

Manhattan Project The Manhattan Project was a research and development program undertaken during World War II to produce the first nuclear weapons. It was led by the United States in collaboration with the United Kingdom and Canada. From 1942 to 1946, the ...

) *

Rolin Wavre Rolin-Louis Wavre (25 March 1896 in Neuchâtel – 9 December 1949 in Geneva) was a Swiss mathematician. Wavre studied at the Sorbonne and received his Ph.D. in 1921 from the University of Geneva, where he became a professor extraordinarius in 19 ...

Antoni Zygmund Antoni Zygmund (December 26, 1900 – May 30, 1992) was a Polish-American mathematician. He worked mostly in the area of mathematical analysis, including harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...

(emigrated to the US in 1940)

References

External links

Scottish book preface
* At the

Adam Mickiewicz University in Poznań The Adam Mickiewicz University (; Latin: ''Universitas Studiorum Mickiewicziana Posnaniensis'') is a research university in Poznań, Poland. Due to its history, the university is traditionally considered among Poland's most reputable institution ...

website (archived by the

Wayback Machine The Wayback Machine is a digital archive of the World Wide Web founded by Internet Archive, an American nonprofit organization based in San Francisco, California. Launched for public access in 2001, the service allows users to go "back in ...

): ** (typescript of the English version) ** at ** ** , an article by Roman Duda
Photograph and description of the Scottish café (Kawiarnia Szkocka)
at the MacTutor archive
Review of Roman Kaluza’s 1996 book ''The Life of Stefan Banach''
from ''

American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...

'' 104 (1997), 577-579. * (page in

Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Polish people, people from Poland or of Polish descent * Polish chicken * Polish brothers (Mark Polish and Michael Polish, born 1970), American twin ...

, with unarchived PDF links)
Lviv Scottish Book
- A new Scottish Book at the original café following the tradition of the original Scottish Book * Books: ** ** (Includes selected papers presented at the ''Scottish Book'' Conference held at

North Texas State University The University of North Texas (UNT) is a public university, public research university located in the Dallas–Fort Worth metroplex. Its main campus is in Denton, Texas, Denton, with a satellite campus in Frisco, Texas, Frisco. It serves as the ...

, Denton, Tex., May 1979) {{Coord, 49, 50, 09, N, 24, 1, 57, E, type:landmark_region:UA, display=title Birkhäuser books Education in the Second Polish Republic History of Lviv History of mathematics Mathematics manuscripts Education in Poland during World War II Lwów in World War II