Home Page of Stefan Banach
# 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 Ślebodziński, 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. Gołąb,
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 2^{5}+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
2^{5}+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 Śląskiej,
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, Ślązacy z wyboru - pionierzy matematyki w
powojennym Wrocławiu, w: M. Hałub, A. Mańko-Matysiak (red.),
Śląska 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 światowej, Wiadom. Mat. 39 (2003), s. 157-164.
[5] E. Marczewski, Początki 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.

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On 04 Jan 2012, 18:50.