Home Page of Stefan Banach

"The New Scottish Book"

by Roman Duda

(translated by John J. Greczek)

[Published in: Emilia Jakimowicz and Adam Miranowicz (ed.),
"Stefan Banach. Niezwykłe życie i genialna matematyka"
(Stefan Banach - Remarkable life, Brilliant mathematics.),
2nd edition, Oficyna Wydawnicza "Impuls", Kraków 2009.]

(posted on this web-site with the written permission of the Author, Translator and Publishers)

The mathematics that originated in Wrocław after 1945 (since that year a part of Poland) had a strong connection to Lvov. Three of its four pioneers had spent a considerable amount of time in Lvov: Hugo Steinhaus received venia legendi there in 1917 and lived his most productive years there until 1941, Edward Marczewski spent the first two years of the war in Lvov, i.e., from 1939 to 1941, and Bronisław Knaster was there for the duration of the war from 1939 to 1945. Only the fourth, Władysław ¦lebodziński, was never in Lvov. He worked in Poznań before the war, completed his academic studies in Warsaw and spent most of the war as a prisoner in Auschwitz. Of these four the first to arrive in Wrocław was E. Marczewski [2]. Following the collapse of the Old Town enclave during the Warsaw uprising he came to Wrocław in October 1944 (as a political prisoner), survived the siege of the city, and in May 1945 joined Professor S. Kulczyński's Cultural-Scientific Group there. He stayed on in Wrocław and became the organiser and one of the leaders of a new centre of mathematics (German mathematicians left the city before the siege and the German Institute of Mathematics had been destroyed by fire during the siege [7]). He was a man of great energy who cared very much that the new centre would be a continuation of what he considered to be the best in Polish mathematics, namely, the Warsaw and Lvov traditions [5]. In pursuit of this goal he attracted pioneers and looked for potential successors who would be able to continue work.

As instances of Marczewski's efforts were his decisions to continue with the Lvov Scottish Book (which Łucja Banach, wife of Stefan Banach, brought to Wrocław as part of her luggage, after she and her son Stefan were deported from Lvov in 1946), and to establish a new mathematics periodical.

The first problems were entered into the New Scottish Book by H. Steinhaus at the beginning of July, 1946. These had a symbolic status in that he had entered the last problem in the Lvov Book (#193, 31 May, 1941) closing it, as it were, and starting the new Wrocław Book five years later. Several days after that Gustav Choquet, at that time an employee of the French Institute in Kraków [4], entered two new problems. Subsequent ones were entered by E. Marczewski, B. Knaster, and H. Steinhaus in 1946, and also in that year by an illustrious group of the best Polish mathematicians who had survived the war: A. Alexiewicz, S. Goł±b, A. Mostowski, W. Orlicz, W. Sierpiński, R. Sikorski, J. Szarski, T. Ważewski, Z. Zahorski and also others on the occasion of the Fourth Convention of Polish Mathematicians held in Wrocław from 12 to 14 December, 1946. Altogether 35 problems were entered that year and prizes were offered for solutions to some of them. For example, for solutions to two of his problems G. Choquet promised a bottle of champagne, to be consumed in Paris, if the answers were positive, and a bottle of Bordeaux if negative. V. Jarnik offered "Plze?ské pivo" (a Czech beer) in a quantity of 300 litres (!) for a negative answer to his problem number 33. Later H. Steinhaus promised a ration coupon for meat, at that time an attractive prize, for a solution to his problem number 175 (1952). (The prize was won by R. Sikorski who then insisted it had to be redeemed in Warsaw, which was no doubt more difficult than finding the solution in the first place.) For a solution to his so-called "Easter" problem number 269 (1955) H. Steinhaus offered an Easter egg (decorated by him personally) as the prize.

After an auspicious beginning the New Scottish Book experienced a few very lively decades, as witnessed by the large numbers of problems, comments and solutions that were entered in it. In the absence of a repository for it like the Scottish Café in Lvov, the Wrocław Book resided at the Mathematics Seminary of the University and Polytechnic, which was for many years the centre of Wrocław's mathematical life. Later it was kept in the Library of Wrocław University's Mathematics Institute. The Book's pioneers and caretakers made sure it was readily available for use and were always especially ready to put it in front of any mathematician visiting Wrocław.

From 1946 to 1955 some 286 problems were entered into the Book, but its best decade was from 1956 to 1965 when 464 were entered. Then in the decade from 1966 to 1975 157 problems were entered. Starting in 1976 (notably the year of E. Marczewski's death) interest in the Book and in its significance began to decline, as manifested by the rapidly decreasing number of problems and comments that were being entered in it. Only one problem was entered in 1982, and none during the next four years, and finally in 1987 K. Głazek added the last two problems that he had presented to no avail during some conferences; he received no response to them in the Book either. Thus the New Scottish Book lasted for over forty years and it had a dynamic and expansive quality in contrast to the Scottish Book from Lvov whose existence was shorter, limited to the years from 1935 to 1941, and whose dynamism declined relatively quickly [3]. During the time of its existence altogether 968 problems were entered into the New Book, an average rate of 24 annually (its Lvov predecessor had 193, an average of 27 annually). In reality, however, it was somewhat more than that because some problems consisted of several questions, and there were also ones that were not numbered. Unlike its predecessor [6], the New Book has not so far been more extensively analyzed and studied. Nevertheless, it may be confidently asserted that for four decades it was a very integral part of mathematical life in Wrocław, a support and inspiration for many and a contemporary chronicle for all.

Another of E. Marczewski's initiatives was the founding of a new periodical. This was a bigger challenge than purchasing a thick notebook for the New Book, but already in 1948 the first volume of Colloquium Mathematicum made its appearance with four pioneers as its editors. In the same year Studia Mathematica was reborn in Wrocław (the editors of the first Wrocław volume, which was also the tenth sequential volume, were H. Steinhaus, E. Marczewski and B. Knaster). Studia confined itself to "the theory of operators and its applications", that is to functional analysis, whereas Marczewski wanted his Colloquium to serve all of the branches of mathematics that might appear in Wrocław. The new periodical styled itself after the tradition of the Warsaw Fundamenta Mathematicae and the Lvov Studia Mathematica but also possessed its own individual characteristics. To the extent the others presumed to be specialized [1], Colloquium took on a more general character. Moreover, Colloquium contained a chronicle section Chronique, which today is a veritable storehouse of information about Polish mathematical life during those times. It also had a section devoted solely to problems Probl?mes which survived for many years and was used to revisit problems previously formulated in articles published in Colloquium, and included some of the more interesting ones from the New Scottish Book. It also contained letters to the editor, various comments and solutions to problems, partial or complete, and if complete a notation that a particular problem had been solved.

A more detailed review and discussion of these problems, their destiny and influence on mathematics, is beyond the scope of this article. Let us only note, however, that the Probl?mes section survived until 1990 and during that time published 1384 problems, of which about a fourth came from the New Scottish Book. The most prolific, in that respect, was the 1948-1972 period during which a total of 312 problems came from the Book. Subsequently, for a time, only single problems were abstracted from it until 1982 when the last two were published in Volume 46. Altogether 335 problems from the New Scottish Book appeared in Colloquium Mathematicum.

Anecdotally let us note problem P 1217 (Q) in Colloquium Mathematicum 44 (1981) which went as follows:

S. Manhart (Sany)

P 1217 (Q). Consider a random walk of extreme element Hint = H(t) of the solid category S. The process develops within a rectilinear 3-cell N whose boundary ∂N is connected and closed. Estimate the expectation of τ_ϵ = inf {t > 0:H(t) ∉ N}.

Letter of January 4, 1982

P 1217 (Q), R1. In the Manhart case, τ_ϵ turned to be 2⁵+1 (letter of February 6, 1982). In other cases the problem is still open.

This is mathematical gibberish, not easily identified as such by a non-mathematician, but it has the following hidden message:

The alleged S. Manhart (Sany) is S. Hartman (Nysa) whose supposed letter of 4 January reminds the reader that since that day he is on "a random walk (...) inside a rectilinear 3-dimensional cell N, whose boundary ?N is connected and closed" in the internment camp in Nysa . The time of his internment was to be deduced from "τ_ϵ = inf {t > 0: H(t) ∉ N}."

In an update it could be noted that in his case the time was 2⁵+1 (=33 days) but "in other cases the problem is still open".

The letter made it past the censors and French friends in Paris understood the problem....

Nothing lasts for ever, but it is also possible to point to some more direct and proximate causes that led to the demise of the New Book and in its footsteps likewise the disappearance of the section Problemes in Colloquium Mathematicum:

1) The passing in the seventies of the generation of pioneers.

2) The move by the Mathematics Institute of Wrocław University away from its shared quarters with the Mathematics department of the Wrocław Polytechnic, resulting in a splintering of a hitherto common life of these two bodies.

3) The evolution of the Mathematics Faculty of the Polytechnic into the Mathematical Institute of Wrocław Polytechnic, a progression from which was the establishment by the Polytechnic of its own mathematics courses as part of the Faculty of Basic Technology.

4) A diminution of the importance of the Polish Mathematical Society and specifically the reduction of the number and frequency of its meetings and conferences.

5) The tendency to a more internal focus by both Institutes leading to a lessening of common interests in mathematics.

Today the New Scottish Book, in the form of three thick notebooks containing a multitude of entries attesting to its onetime frequent use, is a historical relic carefully preserved in the Library of the Mathematical Institute of Wrocław University. It is an important document and record of the work of the Wrocław mathematicians in the years from 1945 to 1987 and of the colleagues who visited them during that time. It further merits to be well remembered and studied because Wrocław mathematics was then widely known and influential in some world centers of mathematics.

Bibliography

[1] R. Duda, Fundamenta Mathematicae, Studia Mathematica, Acta Arithmetica - pierwsze trzy specjalistyczne czasopisma matematyczne, Zeszyty Naukowe Politechniki ¦l±skiej, Matematyka-Fizyka 76, 1995, s. 47-80; R. Duda, Fundamenta Mathematicae and the Warsaw school of mathematics, w: C. Goldstein, J. Gray, J. Ritter (eds.), L'Europe mathématique - Mythes, histoires, identitités, Paris 1996, s. 479-498.

[2] R. Duda, ¦l±zacy z wyboru - pionierzy matematyki w powojennym Wrocławiu, w: M. Hałub, A. Mańko-Matysiak (red.), ¦l±ska Republika Uczonych, Wrocław: Oficyna Wydawnicza Atut, 2006, s. 450-471.

[3] R. Duda, Lwowska szkoła matematyczna, Wrocław: Wydawnictwo Uniwersytetu Wrocławskiego, 2007.

[4] A. Gulisashwili, Gustave Choquet rozmawia o swoim pobycie w Polsce po II wojnie ¶wiatowej, Wiadom. Mat. 39 (2003), s. 157-164.

[5] E. Marczewski, Pocz±tki matematyki wrocławskiej, Wiadom. Mat. 21.1 (1969), s. 63-76.

[6] R.D. Mauldin (ed.), The Scottish Book. Mathematics from the Scottish Café, Boston: Birkhäuser, 1981.

[7] W. Narkiewicz, Matematycy Wrocławscy 1900-1945, Wiadom. Mat. 39 (2003), s. 107-115.

Acknowledgements

We deeply thank Prof. dr. hab. Roman Duda for his permission to post all the works of Stefan Banach on this website. We also thank John J. Greczek for this English translation.

Emilia Jakimowicz and Adam Miranowicz

Questions or comments about this page can be sent to Emilia Jakimowicz or Adam Miranowicz. We would also appreciate every link from your pages to our Home Page of Stefan Banach.

File translated from T_EX by T_THgold, version 4.00.
On 04 Jan 2012, 18:50.