The Philosophical Status of Diagrams

by Greaves, Mark

• ISBN13: 9781575862941
• ISBN10: 1575862948
• Format: Paperback
• Copyright: 2001-03-01
• Publisher: Stanford Univ Center for the Study

Summary

The use of diagrams in logic and geometry has encountered resistance in recent years. For a proof to be valid in geometry, it must not rely on the graphical properties of a diagram. In logic, the teaching of proofs depends on sentenial representations, ideas formed as natural language sentences such as "If A is true and B is true...." No serious formal proof system is based on diagrams. This book explores the reasons why structured graphics have been largely ignored in contemporary formal theories of axiomatic systems. In particular, it elucidates the systematic forces in the intellectual history of mathematics which have driven the adoption of sentential representational styles over diagrammatic ones. In this book, the effects of historical forces on the evolution of diagrammatically-based systems of inference in logic and geometry are traced from antiquity to the early twentieth-century work of David Hilbert. From this exploration emerges an understanding that the present negative attitudestowards the use of diagrams in logic and geometry owe more to implicit appeals to their history and philosophical background than to any technical incompatibility with modern theories of logical systems.

Table of Contents

Acknowledgments	vii
Introduction	1	(12)
Part I: Geometry	13	(70)
Diagrams for Geometry	15	(4)
Euclidean Geometry	19	(16)
Descartes and the Rise of Analytic Geometry	35	(8)
Geometric Diagrams in the Nineteenth Century	43	(34)
Diagrams in the Geometry of Poncelet	43	(10)
Non-Euclidean Geometries and the Rejection of Kantianism	53	(12)
Pasch, Hilbert, and the Rise of Pure Geometry	65	(12)
Summary	77	(6)
Part II: Logic	83	(124)
Diagrams for Logic	85	(8)
The Logical Framework of the Syllogism	93	(14)
Diagrams for Syllogistic Logic	107	(32)
Introduction	107	(3)
The Linguistic Formulation of the Syllogism	110	(5)
Early Diagrams for Syllogistic Logic	115	(6)
Euler Diagrams and the Rise of Extensional Logic	121	(18)
Diagrams for Symbolic Logic	139	(54)
Introduction	139	(2)
Boole's Symbolic Logic	141	(5)
Boolean Logic and Venn Diagrams	146	(16)
Peirce's Extensional Graphs	162	(13)
Logic at the End of the Nineteenth Century	175	(18)
Summary	193	(8)
Conclusion	201	(6)
References	207	(4)
Index	211