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Backgrounds of arithmetic and geometry. An introduction. (English) Zbl 0913.51001

Series in Pure Mathematics. 23. Singapore: World Scientific. xii, 286 p. (1995).
The book of R. Miron and D. Brânzei enables an excellent introduction into the backgrounds of mathematics. The internationally recognized reputation of the graduates from the mathematical faculties in Romania finds in this book its confirmation and motivation. We believe that any departments in this field, all over the world, would benefit from trying to correlate their syllables with this book. The backgrounds of arithmetics are completely presented, but maybe too concisely. The constructive method allows the introduction and structuring the sets of cardinal and ordinal, natural, integer, rational, real and complex numbers, but it is also correlated with the presentation of Peano’s axiomatics. The great qualities of the book refer to the backgrounds of geometry, a domain in which both of the authors are specialists. The algebraic backgrounds of geometry are based on a Weyl-like axiomatic system, for the first time minimalized by Miron. An interesting structure of almost affine space, generalizing the notion of affine space in a sense seemingly very convenient for theoretical physics is reached here. By successive addition of independent axioms the affine and then the Euclidean spaces are reached. The axiomatic backgrounds of geometry clearly, precisely and minutely present Hilbert’s and Birkhoff’s axiomatics together with the main aspects of their metatheory.
A chapter about geometrical transformations goes along with a modern, elegant and clever presentation of projective, hyperbolic and absolute geometries from the viewpoint of the Erlangen Programme; various connections and remodelling of these geometries are pointed out here. A chapter written by Francisc Rado evokes some other links between algebra and geometry by developing Bachmann’s axiomatic system.
Eight of the nine chapters are followed by problems whose indications are to be found at the end of the book; many of these problems may constitute a promoting base for certain personal investigations. An ample bibliography, 579 titles (out of which 564 are books), allows the reader to go further into some other aspects only tangentially referred to in the book: logic, axiomatic theory of sets, philosophical hints.
We also appreciate the authors’ original but efficient idea of inserting after the index a summary of the figures in the book that enables a quick finding of all geometrical considerations.
The book is useful not only for students and teachers in mathematics and researchers in this field, but also for those who have as their hobby some beautiful arithmetical and geometrical problems.

MSC:

51-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
03B30 Foundations of classical theories (including reverse mathematics)
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