The book offers an excellent answer to the questions of what mathematical structures are, how they were discovered and how they were adopted in mathematical research. The book is divided into two parts. The first part contains five chapters. Chapter 1 contains information on algebraic research in the late nineteenth century. From Chapter 2 to Chapter 5, the beginning of structural approach and first concepts of modern algebra are discussed. The main aim of these chapters is to show the development of the ideal theory in the period between the work by R. Dedekind and E. Noether. The author describes the most important discoveries of Dedekind, Hilbert, Hensel, Steinitz, Loewy, Fraenkal and Noether. The reader can see the development of algebraic number theory (ideal prime numbers, algebraic invariants, theory of p-adic numbers) and the beginning of axiomatic approach to algebra (the Hilbert axiomatic approach, structural image of algebra, axioms for p-adic systems, theory of rings and ideals).

The second part deals with three different mathematical theories and their historical roots. The first one is the so-called Oystein-Ore lattice developed between 1935 and 1945. The second attempt to create modern structural algebra is associated with the Bourbaki works. Details of Bourbaki's ideas in set theory, algebra, general topology and commutative algebra are discussed in next chapters. The author describes an interpretation of relations between a general non-formalized idea of a mathematical structure and its axiomatic formalization. The third theme described in this part is category theory. The author analyses roots of the theory crystallizing in the USA between 1943 and 1960. Its development in the next decade is described at the end of the book. An extensive bibliography and subject index are included. The book offers excellent information on the treated topics and can be recommended to historians of mathematics and mathematicians who are interested in the development of mathematics in the twentieth century. It could also be of interest to philosophers and historians of sciences studying the development of scientific ideas.

Reviewer:

mnem