THE THEORY OF PLANE AREA AT THE CROSSROADS IBD

Sinopse

This book explores a cluster of philosophical, historical, and logical problems concerning the foundations of the theory of plane area in elementary geometry. The motivation of this study is a notable geometrical proposition known as De ZoltâÇÖs postulate, which asserts that a polygon cannot be equal in area to a proper polygonal part. The book is the first systematic investigation of the philosophical and foundational significance of this proposition, which can also be described as the 'fundamental theorem' of the theory of plane area.This volume provides a comparative study of EuclidâÇÖs development of the theory of area in theáElementsáand its modern reinterpretation in HilbertâÇÖs classical monographáFoundations of Geometry.áIt connects the historical reflections on De ZoltâÇÖs postulate with the nineteenth-century program of providing a purely geometrical foundation for Euclidean geometry, uncovering a rich array of intertwined conceptual problems. It also shifts the perspective and provides a logical analysis of this geometrical postulate within an original development of the abstract theory of magnitudes, called compatible magnitudes. Finally, it extends the previous formal treatment of De ZoltâÇÖs postulate to the case of three-dimensional geometry by producing a type system for polyhedral geometrical mereology. The innovative combination of philosophical, historical, and logical perspectives results in a novel discussion of a fascinating problem at the crossroads of (late) nineteenth-century geometry. áThis volume will interest readers in the fields of history and philosophy of mathematics, logic, and formal philosophy.