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Alfred Tarski

Andrew McFarland, Joanna McFarland, and James T. Smith, editors
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
P. N. Ruane
, on

With its focal point being the life and work of Alfred Tarski, this big book provides an almost an encyclopaedic account of mathematical life in Poland in the period 1920 to 1945 (referred to here as the ‘golden age of Polish mathematics’).

Prior to reading it, I could only associate the name Tarski with the Banach-Tarski paradox — and I didn’t even know that both he and Stefan Banach were Polish. Moreover, I had no clearly defined ideas regarding the nature of Polish mathematics, nor (more excusably) the Polish education system. However, this gap has now been bridged by the pleasurable process of reviewing this book from the perspective of the general (mathematical) reader.

Within 500 large pages there is a wealth of biographical material and many newly translated versions of Tarski’s early mathematical works. Containing numerous photographs and other attractive illustrations, the book’s principal aims are:

  1. To provide translations of Tarski’s works previously only available in Polish.
  2. To describe the cultural and scientific background from which those works emerged.
  3. To update Givant’s 1986 bibliography of Tarski

Tarski, together with Aristotle, Gottlob Frege and Kurt Gödel, is rated as one of the four great logicians. Indeed, his ideas have underpinned much of the work on mathematical logic and set theory since 1950, but early in his career he published various works on the teaching of geometry. In addition to his duties as a schoolteacher, he researched the applications of measure theory to area, volume and the decomposition of point sets into congruent parts. Included also are outlines of his research on well-ordering and finiteness, cardinal arithmetic, the axiom of choice and further applications of set theory to geometry.

The authors’ first aim is achieved in the first half of the book, which contains the first English versions of Tarski’s early mathematical output. The high standard of translation must arise from the fact that Joanna McFarland’s native tongue is Polish and she and Andrew reside in Poland. Although James Smith is credited with translations from other languages, he must be credited with devising the book’s overall structure. Such qualities, together with the fact that Tarski wrote with great clarity, will enable readers with a basic knowledge of axiomatic set theory and measure theory to ascertain the basic nature of the mathematical content of the book.

The youthful Tarski emerged from a Polish ‘cultural and scientific background’, but what does that mean? Firstly, there was the complex geo-political situation arising from Poland’s domineering neighbours (USSR, Germany, Austria). There was also anti-semitism and the rise of Polish nationalism. Much insight into these factors can be gleaned from the central biographical narrative, but most colourfully from the myriad of biographical vignettes dispersed throughout the text. Nearly all of these relate to other mathematicians, for many of whom portraits are included. Stefan Banach, for instance, worked closely with Tarski in their formative years, but he was subsequently unable to escape Nazi persecution, and died a horrible death due to the necessity of partaking in medical experiments (he was a flea-tester). Tarski himself was lucky to be stranded in the USA when the Poland was invaded in 1939.

Because I haven’t looked at Givant’s 1986 bibliography of Alfred Tarski, and because I could find no summary of the authors’ process of updating it, I assume that their third aim is partly achieved in the second half of the book. Tarski’s career as a teacher (at many levels) is described in four chapters including one dealing with his employment and family life. Another chapter consists of many exercises he devised for use in schools and a translated version of his school geometry text written. Within all this, one gains insight into the changing nature of mathematical education in Poland in the 1920s and 1930s. But what also comes across are the very high (unrealistic) expectations that Tarski had of Polish high school pupils. In fact, the mathematical exercises that he devised were more like those used in Math Olympiads, and I give a few paraphrased examples:

  • What necessary and sufficient conditions must non-zero integers \(a\) and \(b\) satisfy so that \(a x^4 + b\) may be expressed as the product of four linear factors?
  • Show that \(\displaystyle\sum_{g=2}^n E(\log_g n) = \sum_{k=2}^{n} E(\sqrt[k]{n})\), where \(E(x)\) is the whole number \(p\) satisfying \(p \leq x \leq p+1\).

One of his easy exercises requires proving that \(\left| |a|-|b|\right| = |a+b| + |a-b| - |a| - |b|\). Well, his heart was in the right place!

In my view, this book is faultlessly constructed, and there is far more in it than can be described here. It has both a name index a subject index and a massive bibliography, and its clearly organized contents makes it easy to navigate. Despite being such a large book, one’s motivation to proceed through the extensive amount of material is enhanced by the liveliness of its presentation. In short, it should appeal to many readers and could form a reference work for those interested in the history of early 20th century mathematics.

Peter Ruane’s introduction to set theory came from Naïve Set Theory (Paul Halmos) and Patrick Suppes’ book on Zermelo-Fraenkel axiomatic set theory. But that was fifty years ago!

Foreword, by Ivor Grattan-Guinness
Maps of Central Europe

Part One: Debut
1 School, University, Strife
2 A Contribution to the Axiomatics of Well-Ordered Sets (1921)
3 Doctoral Research

Part Two: Geometry
4 Area, Volume, Measure
5 On the Equivalence of Polygons (1924)
6 On Decomposition of Point Sets into Respectively Congruent Parts (1924)
7 Degree of Equivalence of Polygons
8 Research Threads

Part Three: Teaching
9 Career and Family
10 Congress of Mathematicians of Slavic Countries (1929)
11 Circumference of a Circle (1932)
12 Exercises Posed by Tarski
13 Geometry for the Third Gimnazjum Class (1935)
14 Teaching and Logic — to the New World

Part Four: Supplement
15 Assorted Contributions
16 Posthumous Publications
17 Biographical Studies
18 Research Surveys

Index of Persons
Index of Subjects.