Theories of Dimension from Ancient to Modern Times

This book provides a history and analysis of the ideas, problems, concepts, and theories of dimension from ancient to modern times. It focuses primarily on topological and, to a lesser extent, measure-theoretic approaches. It offers a detailed account of the contributions of key mathematicians who shaped dimension theory, including Bernard Bolzano, Bernhard Riemann, Georg Cantor, Richard Dedekind, Henri Poincaré, L.E.J. Brouwer, Paul Urysohn, Karl Menger, Witold Hurewicz, Paul Alexandroff, and Felix Hausdorff. The book examines the twists and turns in the development of the subject, including its controversies, and explores the philosophical aspects of its history. As the first in-depth historical study of dimension theory, it serves as a reference for historians and philosophers of mathematics. It also provides general historical background for students and researchers with broader mathematical interests.

Dale Johnson worked for many years at the University of Hertfordshire, UK, where he taught and did research in the history and philosophy of mathematics, publishing papers on Bolzano, Brouwer, and the history of dimension theory. In a second career he contributed research in cybersecurity at The MITRE Corporation, publishing several papers in the field. He is now retired.

1 Introduction to Theories of Dimension.- 2 Ideas Touching on Dimension in Ancient Greek Philosophy and Mathematics.- 3 Dimension from the Middle Ages to the End of the Eighteenth Century.- 4 Bolzano's Investigations into Geometry and Dimension.- 5 Nineteenth-Century Theories of Higher-Dimensional Spaces.- 6 Cantor and Dedekind: Discovery of a Counterintuitive Example concerning Dimension.- 7 Early Efforts to Prove Invariance of Dimension, 1878-1879.- 8 Early Development of Point-Set Theory and Point-Set Topology.- 9 Poincaré's Theories of Dimension.- 10 Riesz, Baire, Fréchet: New Approaches to Dimension.- 11 Brouwer: Development of Topology, Invariance of Dimension.- 12 New Ways of Proving Invariance of Dimension; Brouwer-Lebesgue Dispute.- 13 Problems of Curves and Dimension in the Early Twentieth Century.- 14 Urysohn's Theory of Curves and Dimension.- 15 Menger's Theory of Curves and Dimension.- 16 Development of Theories of Dimension and Curves.- 17 Historical, Mathematical, and Philosophical Perspectives.- Appendix A: Original Language Texts of Quotations.