The Theory of Plane Area at the Crossroads : Philosophical, Historical, and Logical Perspectives

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 Zolt’s 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 Euclid’s development of the theory of area in the Elements and its modern reinterpretation in Hilbert’s classical monograph Foundations of Geometry. It connects the historical reflections on De Zolt’s 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 Zolt’s 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.  This volume will interest readers in the fields of history and philosophy of mathematics, logic, and formal philosophy.

Eduardo N. Giovannini is Associate Researcher at the National Scientific and Technological Research Council (CONICET, Argentina) and Assistant Professor at the National University of Litoral (Argentina). He also has held research positions at the Department of Philosophy of the University of Vienna and the University of California at Berkeley (USA). His research areas include the history and philosophy of mathematics, especially in the nineteenth and early twentieth centuries; the history of modern axiomatic geometry, with a particular emphasis on David Hilbert's metatheoretical investigations, and the history and philosophy of modern formal logic, particularly model theory. He has published extensively on Hilbert's foundational work in geometry, the history of the modern axiomatic method, the geometrical roots of model theory. He is the author of David Hilbert y los fundamentos de la geometría (College Publications, 2015).

Edward Hermann Haeusler is Associate Professor of the Department of Informatics at the Pontifical Catholic University of Rio de Janeiro (PUC-Rio, Brazil). His research areas are proof theory, logic, and theory of computation. Besides papers and book chapters, he is the author of Category Theory for Computer Science (Sagra Luzzato and UFRGS), co-editor of Celebration of Dag Prawitz's Work (Springer-Verlag, 2013), co-editor of A Question is More Illuminating than an Answer. A Festschrift for Paulo A. S. Veloso (College Publications, 2021), and co-editor of Why is this a Proof ? Festschrift for Luiz Carlos Pereira (College Publications, 2015). He is also co-editor of the special issue volume 1, number 18 of the Logic Journal of IGPL (Oxford University Press, 2009) and of the special issue of Computación y Sistemas 21(3) (2017).

Abel Lassalle-Casanave is Full Professor of the Department of Philosophy at the Federal University of Bahia (UFBA, Brazil) and Researcher at the National Council of Research and Technological Development (CNPq, Brazil). His area of research is philosophy of formal sciences, with emphasis in verbal, symbolic and diagrammatic proofs, modern philosophy of mathematics and Hilbertian formalism. Beside papers and chapters of books, he is the author of Por construção de conceitos: em torno da filosofia kantiana da matemática (Editora PUC-Rio, 2019), editor of Symbolic Knowledge from Leibniz to Husserl (College Publications, 2012), and co-editor of Visualização nas ciências formais (College Publications, 2012, with Frank Sautter) and of El árbol de los números: cognición, lógica y práctica matemática(Editorial Universidad de Sevilla, 2016, with José Ferreirós).