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The Scottish book (2nd ed.)
Mauldin R., Birkhäuser Basel, New York, NY, 2015. 322 pp. Type: Book (978-3-319228-96-9)
Date Reviewed: Oct 28 2016

The Scottish book is a legend among mathematicians and computer scientists. It has nothing to do with Scots or Scotland, and could rather be called the Polish book, as it took its name from the Scottish Café located in the town of Lwów, in Poland (now Lviv, in Ukraine), frequented in the late 1930s by a group of mathematicians from Jan Casimir University. The content of the original Scottish book, from the 1930s, is a collection of mathematical problems then unsolved, handwritten primarily in Polish, posed by the participants of meetings, which in today’s terms would be called seminars, except that they were held at the table in a café rather than in an academic environment. Its contributors are also the legends of mathematics and include, among others, Stefan Banach, who started the book with Problem #1, Stanislaw Ulam, who formulated jointly with Banach problems #2 and #3 and wrote extensively about the book after WW II, Hugo Steinhaus, who restarted the book in a different setting after the war, and many others, with a cameo by John von Neumann, who contributed Problem #163.

The book’s original manuscript survived the war, first in hiding and then in the hands of the Banach family. It was Stan Ulam who translated the book into English and privately published his manuscript in a limited edition in 1957 [1], and later as an official report of the Los Alamos Lab [2]. The current edition is an extended version of the first widely distributed edition published by Birkhäuser in 1981. The new edition has, in fact, three parts. In addition to the original Scottish book (Part 2), containing problems from the 1930s and amended with commentaries and solutions that appeared over the years, the book is enhanced by the Scottish Book Conference Lectures (Part 1), from the 1979 conference at North Texas State University (which was also included in the 1981 edition), but also with the New Scottish Book (Part 3), originated by Hugo Steinhaus in the University of Wroclaw, where many of the Lwów faculty transferred after the war.

The problems seem arcane to non-mathematicians and can be hardly described in a short review, so let me only mention just one, Problem #2 (Banach and Ulam), which reads as follows: “Can one define, in every compact metric space E, a measure (finitely additive) so that Borel sets which are congruent should have equal measure?” As you read this, it is clear that terms such as “metric space,” “compact space,” “finitely additive,” “Borel sets,” “congruent sets,” and “measure” all require further elaboration, for which there is no room in this review. Thus, I will only add that almost every single problem, and there are 193 total, has a commentary, which significantly facilitates the reading. There are more that 80 authors of various commentaries, so it would make sense to add another index, like there are indexes for problem authors and problems themselves.

Not all problems use sophisticated mathematical language. The last problem in the original book, posed by Steinhaus, can be briefly described as a question: What is the expected number of matches a smoking mathematician has in any of his two matchboxes he carries (implicit reference to Banach as a chain smoker)? Continuing on a lighter note, one should mention that some problems were promising an associated prize to those who solved them. Most were bottles of beer, wine, brandy, or champagne, and in one case a live goose, while John von Neumann advertised “a bottle o whiskey of measure greater than zero.”

The book is extremely well thought out by its editor Dan Mauldin, and congratulations to him for keeping it alive. The prefaces to both editions and Ulam’s and Kac’s articles on the history of the Scottish book give such a good background, filled with anecdotes, that I, as a reviewer, cannot add much besides pointing the reader to an interview with Kac and Ulam in Los Alamos Science [3]. Overall, it is a fascinating read that is very inspiring--sort of a treasure on a bookshelf.

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Reviewer:  Janusz Zalewski Review #: CR144886 (1702-0112)
1) Ulam, S. The Scottish book (original English translation by Stanislaw Ulam). Los Alamos Scientific Laboratory, Los Alamos, NM, 1957, http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf.
2) Ulam, S. M. The Scottish book: a LASL monograph. Los Alamos Scientific Laboratory, Los Alamos, NM, 1977.
3) Feigenbaum, M. Reflections of the Polish masters: an interview with Stan Ulam and Mark Kac. Los Alamos Science Fall (1982), 54–65.
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