##
**Ladislav Svante Rieger (1916–1963).**
*(Czech)*
Zbl 1349.01003

Dějiny Matematiky / History of Mathematics 36. Prague: Matfyzpress (ISBN 978-80-7378-047-0). 333 p., open access (2008).

The monograph is devoted to the eminent mathematician Ladislav Svante Rieger (1916–1963), who is appreciated as the founder of mathematical logic and axiomatic set theory in Czechoslovakia and who has also gained international credits. The first chapter of the book introduces the most important members of the famous Rieger family, including e.g. L. S. Rieger’s great-grand father František Ladislav Rieger (1818–1903), one of the leading characters of the Czech national revival. Then, a detailed biography of L. S. Rieger based on numerous archival materials is presented. The attention is paid to his personal life as well as to his pedagogical and scientific carreer path that led from the position of an assistant (1945) and associated professor (1951) at the Technical University in Prague to the position in the Mathematical Institute of the Czechoslovak Academy of Sciences (1958), where he spent the rest of his life. Since 1951, Rieger also held lectures and seminars at Charles University in Prague; they were focused above all on mathematical logic and axiomatic set theory and some of their participants became later Rieger’s followers in the mentioned branches (e.g., Jiří Bečvář, Petr Vopěnka or Petr Hájek).

The following four chapters are devoted to particular domains of Rieger’s scientific interest. The first of them discusses Rieger’s contribution to group theory, which represented the main domain of his interest in the period 1941–1952. For example, Rieger was the first to study cyclically ordered groups and he is considered the founder of their theory. The next chapter analyzes Rieger’s research in the field of lattice theory and mathematical logic, performed in the years 1949–1957. Rieger’s publications from this area belong to his most cited works. They bring a lattice-theoretical interpretation of the Heyting calculus, a new proof of Gödel’s completeness theorem of first-order predicate calculus and other results. Since 1954, Rieger concentrated his attention above all on axiomatic set theory. His achievements in this domain are presented in the next chapter. For example, he gave a new proof of the relative consistency of the axiom of choice and the generalized continuum hypothesis. The following chapter discusses other scientific treatises of L. S. Rieger that appeared after the year 1958 and concern areas on the margin of mathematical logic. The most valuable of them is the posthumously published monograph [L. Rieger, Algebraic methods of mathematical logic. Translated from Czech by Michal Basch. Prague: Academia-Publishing House of the Czechoslovak Academy of Sciences (1967; Zbl 0218.02001)]. Other publications concern e.g. machine learning and algorithmic and numerical methods associated with the development of first computers.

The book ends by a conclusion, a detailed English summary, a list of Rieger’s publications supplemented with references to the respective journals, a list of Rieger’s contributions in the Mathematical Reviews, a survey of his pedagogical activities at Czech Technical University and Charles University in Prague, a survey of his lectures in the Union of Czech Mathematicians and Physicists and in the Prague Mathematical Community, a list of references, a collection of photographs and photocopies of interesting documents, a list of abbreviations and a name index.

The following four chapters are devoted to particular domains of Rieger’s scientific interest. The first of them discusses Rieger’s contribution to group theory, which represented the main domain of his interest in the period 1941–1952. For example, Rieger was the first to study cyclically ordered groups and he is considered the founder of their theory. The next chapter analyzes Rieger’s research in the field of lattice theory and mathematical logic, performed in the years 1949–1957. Rieger’s publications from this area belong to his most cited works. They bring a lattice-theoretical interpretation of the Heyting calculus, a new proof of Gödel’s completeness theorem of first-order predicate calculus and other results. Since 1954, Rieger concentrated his attention above all on axiomatic set theory. His achievements in this domain are presented in the next chapter. For example, he gave a new proof of the relative consistency of the axiom of choice and the generalized continuum hypothesis. The following chapter discusses other scientific treatises of L. S. Rieger that appeared after the year 1958 and concern areas on the margin of mathematical logic. The most valuable of them is the posthumously published monograph [L. Rieger, Algebraic methods of mathematical logic. Translated from Czech by Michal Basch. Prague: Academia-Publishing House of the Czechoslovak Academy of Sciences (1967; Zbl 0218.02001)]. Other publications concern e.g. machine learning and algorithmic and numerical methods associated with the development of first computers.

The book ends by a conclusion, a detailed English summary, a list of Rieger’s publications supplemented with references to the respective journals, a list of Rieger’s contributions in the Mathematical Reviews, a survey of his pedagogical activities at Czech Technical University and Charles University in Prague, a survey of his lectures in the Union of Czech Mathematicians and Physicists and in the Prague Mathematical Community, a list of references, a collection of photographs and photocopies of interesting documents, a list of abbreviations and a name index.

Reviewer: Magdalena Hykšová (Praha)

### MSC:

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

01A70 | Biographies, obituaries, personalia, bibliographies |

01A60 | History of mathematics in the 20th century |

03-03 | History of mathematical logic and foundations |

20-03 | History of group theory |