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The Scottish Book: Mathematics from The Scottish Café, with Selected Problems from The New Scottish Book

R. Daniel Mauldin
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Michael Berg
, on

Joint review of

Hugo Steinhaus, the subject of one of the books under review, was a major figure in twentieth century mathematics and one of the mathematicians responsible for the development of the Polish school of mathematics that flourished between the World Wars and was largely decimated under the Nazi occupation of Poland. One of the premier gathering places for a number of these Polish scholars was the Scottish Café in Lwów, where in short order a marked tradition evolved, as well as a wonderful style of doing mathematics: the café was equipped with marble table tops which were well-suited to pencil doodling and were easily cleaned after the mathematicians in question were done with them, and the scholarly customers’ work (or play) was guided by problems posed and collected in a journal, the “Scottish Book,” which figures in the title of the other of the two books dealt with in this review.

The Scottish Book, then, is an annotated presentation of the problems, discussions, and solutions comprising the notebook in the title, which, as per the Preface to the 1957 edition of the present book, was instituted at the behest of Stefan Banach, arguably the father of modern functional analysis. This recollection, modulo the phrase, “[i]f I remember correctly,” came from none other than Stanislav Ulam, not only a well-known set-theorist but also one of the main players (with Edward Teller) in the building of the hydrogen bomb. This Preface, in fact, belongs to “the Limited Los Alamos Edition of 1957,” which is of course telling in its own right: it was penned there by Ulam in 1977. His remarks are revealing and poignant:

The mathematical life was very intense in Lwów. Some of us met practically every day, informally in small groups, at all times of the day to discuss problems of common interest … A large notebook was purchased by Banach and deposited with the headwaiter of the Scottish Coffee House, who, upon demand, would bring it out of some secure hiding place, leave it at the table, and after the guests departed, return it to its secret location.

Ulam goes on:

After the start of war between Germany and Russia, the city was occupied by German troops and the inscriptions ceased. The fate of the Scottish book during the remaining years of war is not known to me. According to Steinhaus, this document was brought back to … Wroclaw by Banach’s son …

And then Ulam recounts a conversation with Mazur:

… ‘A world war may break out. What shall we do with the Scottish Book and our joint unpublished papers? You [Ulam] are leaving for the United States shortly, and presumably will be safe. In case of a bombardment of the city, I shall put all the manuscripts and the Scottish Book into a case which I shall bury in the ground.’ We even decided upon a location … near the goal post of a football field outside the city. It is not known to me whether anything of the sort really happened. Apparently the Scottish Book survived in good enough shape to have a typewritten copy made, which Professor Steinhaus sent to me last year (1956).

Wow. Worthy of an Alfred Hitchcock movie.

Ulam then served as the translator and promulgator of the book: “I consulted my teacher and friend (and senior member of the group of authors of the problems), Professor Steinhaus, about the propriety of circulating this collection. With his agreement I have translated the original text …” The book under review is centered on and built around this original work by Ulam:

In the 35 years since the first edition of this book, many problems have been solved or partially solved. But even today, quite a few remain unsolved. In view of this, I [R. Daniel Mauldin] decided to gather new commentaries and update some of the old ones. The appendices … include a list of the unsolved and partially solved problems together with those than have no commentary, a list of unsolved prize problems [adult beverages being favored, it seems], a list of problems posed by each author, and a list of problems by subject area.

So there it is; a compendium, and a lot more, of problems, solutions, open questions, as well as a record of mathematical battles fought about three quarters of a century ago by some of the major players in the game in Poland. By the way, a quibble: on p. 123 we find a photograph of John Oxtoby (author of Measure and Category), Stanislav Ulam, and Donald A. (or Tony) Martin, to which the transposition (12) has to be applied. One other (color) photograph in the book occurs on p. 49, evidently taken at the 1979 Scottish Book Conference in Denton, Texas; the photo features, among others, Martin, Ulam, Kac, Erdős, and Zygmund, and indicates the sweep and influence of the material in the Scottish Book, from set theory through measure theory and analysis.

The other book under review is properly bracketed with The Scottish Book, given that, as Ulam stated, Hugo Steinhaus was one of the founding members of the Scottish Café enterprise and of course a major Polish mathematician. Mathematician for All Seasons is, as its subtitle conveys, a set of recollections and notes by Steinhaus covering the interval from 1887 to 1945, thus including the heyday of the Scottish Café. Not only Ulam, but Marc Kac and Stefan Banach, already mentioned above, counted Steinhaus as their teacher, the latter two actually being his PhD students. Of course Banach was a law unto himself — his biography, The Life of Stefan Banach — Through a Reporter’s Eyes, by Roman Kaluza, is an excellent account of his unique life and very unusual nature, even among mathematicians — and Steinhaus was known to refer to him as his “greatest discovery.” Steinhaus evidently overheard a conversation on the at the time avant garde subject of Lebesgue measure Banach was engaged in on a park bench with a friend and took it upon himself to streamline the iconoclastic young scholar into Polish mathematical life.

In any event, Steinhaus was a major scholar, responsible for a great deal of serious mathematics, and a figure of international significance: we read, in the words of Marc Kac, that “it was [Steinhaus] who, perhaps more than any other individual, helped to raise Polish mathematics from the ashes to which it had been reduced by the Second World War to the position of new strength and respect which it now occupies. He was a man of great culture and in the best sense of the word a product of Western Civilization.”

So it is that Mathematician for All Seasons is split up into two parts, covering, respectively, Steinhaus’s youth and formation (he earned his PhD in 1911 at Göttingen from none other than David Hilbert) and his life as a very major player in Polish mathematics. The second part is particularly poignant in that it includes chapters titled, “The First Occupation” and “The Second Occupation.” Part II also contains three sections titled “Flashes of Memory,” providing brief accounts of episodes connected to the occupations of Poland, first by the Germans, then by the Soviets. These offset the text proper, this being Steinhaus’s very well-written and fascinating account of his life in the mathematical circles he was part of, including, of course, Göttingen and Lvóv, the events in Poland in one of the most volatile and turbulent eras in modern history, and any number of mathematicians in his wide orbit.

Steinhaus expansively covers various aspects and facets of life in Europe as the twentieth century took shape and proceeded to unfold, marked and marred by wars on a scale never imagined and revolutions that affected Poland, in particular, in dramatic and often tragic ways. Accordingly these memoirs by Steinhaus possess an important historical dimension provided by someone who was, so to speak, on the front lines. It is an important book on many counts, including modern mathematical as well as cultural history. It is indeed a wonderful complement to The Scottish Book in that it puts everything, specifically the mathematics presented in its pages by flesh and blood scholars soon to be under attack, into a broad and human context. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

See the table of contents in the publisher's webpage.