BOOK REVIEW
in
Science & Technology
March 25, 2025
Foundations of Geometry: A Rigorous Exploration of Euclidean Principles and Their Philosophical Underpinnings
by Franz Rothe
Reviewed by Jhona Kahuila
Book Browse Reflection
A Course in Old and New Geometry II: Basic Euclidean Geometry meticulously outlines mathematical experts' theories for Euclidean geometry, including axioms of incidence, order, congruence, parallelism, and continuity. Each is presented with clarity, and remarks are included to provide additional context or highlight logical separations. This systematic presentation reflects the rigor of the author’s approach and its influence on modern mathematics.
Moreover, it situates known works of math experts within a broader historical and philosophical framework. It references Euclid's Elements as one, then, the foundation of axiomatic geometry and contrasts it with the formalist program, which emphasizes the independence of axioms from intuitive truths. The inclusion of correspondence with other math experts further enriches the discussion, shedding light on the philosophical debates surrounding the nature of axioms and definitions.
Then, it covers a wide range of topics, including affine planes, ordered fields, algebraic expression, Pythagorean fields, Euclidean fields, and real closed fields. It also delves into mathematical axioms, congruence, rigid motions, and the fundamental theorems of geometry. This breadth ensures that the reader gains a deep understanding of the subject.
The book is rigorous, providing precise definitions, detailed proofs, and clear logical reasoning. The proofs of the context of Pythagorean planes are thorough and leave little room for ambiguity.
Overall, it provided a highly technical and mathematical exposition of the properties of Pythagorean planes, their congruence relations, and their connection to mathematical axioms and Euclidean geometry. It focuses on its structure, content, and clarity.