The use of diagrams as permissible parts of formal proofs in logic and geometry encountered stiff resistance in the 20th century. Contemporary definitions of validity in logic are stated purely in terms of relationships between strings of a formal language, and it is now commonplace to observe that a valid geometric proof must not be based on any pictorial properties of an accompanying diagram. Yet diagrams have been a constant part of informal proof practice in both logic and geometry since antiquity, and it is a rare mathematician who can execute a formal proof in plane geometry without one. What accounts for the exclusion of diagrammatic methods from modern proof techniques?
This book explores the reasons why structured graphics have been largely excluded from contemporary formal theories of axiomatic systems. In particular, it traces how several systematic forces in the intellectual history of mathematics and logic drove the adoption of sentential representational styles rather than diagrammatic ones. This book shows the progressive effects of these forces on the evolution of diagram-based systems of inference in logic and geometry, stretching from the Greeks to the early twentieth-century work of David Hilbert. This exploration makes clear that the familiar prejudice against diagrammatic inference in logic and geometry owes more to history and philosophical context than to any technical incompatibility with modern theories of axiomatic systems.
Please note that Johan van Benthem's name is misspelled on page vii of the Acknowledgments.
is Director of the Joint Logistics Technology Office for the Defense Advanced Research Projects Agency.
- 1 Introduction
- Part I: Geometry
- 2 Diagrams for Geometry
- 3 Euclidean Geometry
- 4 Desacates and the Rise of Analytic Geometry
- 5 Geometric Diagrams in the Nineteenth Century
- 5.1 Diagrams in the Geometry of Poncelet
- 5.2 Non-Euclidean Geometries and the Rejection of Kantianism
- 5.3 Pasch, Hilbert, and the RIse of Pure Geometry
- 6 Summary
- Part II: Logic
- 7 Diagrams for Logic
- 8 The Logical Framework for the Syllogism
- 9 Diagrams for Syllogistic Logic
- 9.1 Introduction
- 9.2 The Linguistic Formulation of the Syllogism
- 9.3 Early Diagrams for Syllogistic Logic
- 9.4 Euler Diagrams and the Rise of Extensional Logic
- 10 Diagrams for Symbolic Logic
- 10.1 Introduction
- 10.2 Boole's Symbolic Logic
- 10.3 Boolean Logic and Venn Diagrams
- 10.4 Peirce's Extensional Graphs
- 10.5 Logic at the End of the Nineteenth Century
- 11 Summary
- 12 Conclusion