Delving Deep into Geometry's Foundations
May 9, 2025
Starting with propositional and predicate logic, including formulas and theories, A Course in Old and New Geometry II: Basic Euclidean Geometry clearly and systematically explains the logical underpinnings of mathematics. It precisely defines key concepts such as statements, negation, equivalence, and implication, using truth tables and logical equivalences effectively, making it particularly useful for those seeking a fundamental understanding of mathematical logic.
The book is a valuable asset for understanding the impact of numbers and formulas on mathematics. It successfully conveys the rigor and depth of thought of influential mathematicians, highlighting its historical and philosophical significance. Its critical analysis of limitations and discussion of subsequent advancements demonstrate the lasting relevance of these mathematical ideas. It's a commendable effort to encapsulate key contributions to logic, geometry, and the formalist program.
Moreover, the book doesn't just summarize mathematical work but also critically examines its limitations and potential for further development. For example, it notes the initial absence of circle-related axioms and theorems in Hilbert's foundations and discusses later work by Greenberg and Hartshorne, adding analytical depth to the review.
The book also assesses mathematical models, offering a perceptive distinction between their use in natural sciences and mathematics. It emphasizes the dual-level thinking required in mathematical modeling and the importance of relative consistency proofs, providing valuable insight into the abstract nature of modern mathematics.
In conclusion, A Course in Old and New Geometry II: Basic Euclidean Geometry offers a profound exploration of mathematics best suited for readers with a strong mathematical background interested in its theoretical and philosophical aspects. Expanding its accessibility could broaden its impact.
