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Modern algebra and the rise of mathematical structures. 2nd revised ed. (English) Zbl 1044.01008

Basel: Birkhäuser (ISBN 3-7643-7002-5/pbk). 451 p. (2004).
The book under review is a very thorough research monograph on the history and philosophy of that undefined branch of modern mathematics which is usually associated with the general expression “mathematical structures”.
This concept is frequently used in various contexts,and it has nearly become a commonplace to characterize the mathematical developments of the twentieth century as the rise and triumphant progress of mathematical structures. Certainly, this comprehension bears quite a grain of truth. However, as the author detects, the idea of a mathematical structure, in spite of its ubiquity and predominance in twentieth-century mathematics, is a phenomenon whose actual nature, meaning and impact are not clearly understood, and have scarcely been thoroughly analyzed or systematically discussed. In fact, the term “mathematical structure” has been used, in the entire vast literature, in different contexts by different authors, with different meanings and interpretations, or with no cognizable meaning at all. With this perception in mind, the author has conducted ample research, preponderantly on the history and philosophy of algebra, in order to try to answer some of the principal epistemological questions, among those the more general ones such as: How can the structural approach to mathematics be characterized? What really is a mathematical structure? When, and as an outcome of what foregoing developments, did structures enter the scene? When did they begin to be accepted and adopted in mathematical research and applications? When and how did mathematicians realize the crucial role of structures,and how did they respond to this new fashion?
The present book reflects the author’s discoveries, in this regard, and offers profound answers to some of these questions, at least partially.
The first edition of this highly enlightening, historically utmost significant treatise appeared in 1996 (Zbl 0858.01022) and has been exhaustively reviewed back then. The book under review is the second edition of the original text, without considerable changes in the conception or in greater details. This means that the extremely thorough and detailed review of the first edition does not need any up-dating, and that the interested reader should refer to this earlier review as for both the conception and the precise contents of the book. However, the new edition has been revised to the effect that recent developments in this field of research have been pointed out, be it in the footnotes or in the bibliography. Also, the author has improved the prose of some passages in order to attain more clarity, and he has corrected many typos and other minor errors in the first edition.
After these improvements and up-datings, this important contemporary analysis of the history and philosophy of the structural thinking in mathematics will become even more useful and attractive to its prospective customers. It should be added that not only mathematical historians or philosophers will find the author’s research highly enlightening and valuable, but also active mathematicians can learn a lot about their field of research from the author’s spreading reflections.

MSC:

01A60 History of mathematics in the 20th century
01-02 Research exposition (monographs, survey articles) pertaining to history and biography
00A30 Philosophy of mathematics
03B30 Foundations of classical theories (including reverse mathematics)
06-03 History of ordered structures
08-03 History of general algebraic systems

Citations:

Zbl 0858.01022
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