Treccani History of Science
The second Scientific Revolution
I1 – Mathematics and logic
Part A - Between the Nineteenth and Twentieth centuries
Edited by Umberto Bottazzini, Jeremy Grey
Philosophy and mathematical practice
• 1. Mathematics or mathematics?
• 2. Groups and geometries
• 3. Sets
• 4. Bodies and ideals
• 5. Foundations of mathematics
• 6. Problems and discussions
• 7. Logics and formal systems
• 8. "A network of hidden structures"
Places and institutions
• 1. Mathematical societies
• 2. From Berlin to Göttingen
• 3. International congresses
• 4. Between the two World wars
Point set topology
• 1. Limits of continuous functions
• 2. The earliest construction of point-set topology
• 3. Topological category
• 4. The beginnings of general topology
Problems of complex analysis
• 1. The general complex function theory.
• 2. The dominance of complex function theory.
• 3. The Riemann-Hilbert problem.
• 4. Nonlinear differential equations
Foundations of geometry
• 1. Pasch’s ‘modern geometry’ and Peano’s principles
• 2. The geometry of hyperspaces
• 3. Segre’s ‘young students’
• 4. Hilbert’s Grundlagen der Geometrie
• 5. Poincaré and Enriques’s answers
Geometric calculus
• 1. Grassmannians and Quaternionists
• 2. The geometric calculus according to Peano and his school
• 3. Discussions and controversies
• 4. Epilogue
The Italian school of algebraic geometry
• 1. The beginnings: Luigi Cremona and his school
• 2. The influence of the German school: Clebsch, Brill and Noether
• 3. The legacy of the German school: Veronese, Bertini and Segre
• 4. Castelnuovo, Enriques and Severi
• 5. The first half of the twentieth century
• 6. The critique of foundations: crisis and rebirth
Hilbert’s problems and the mathematics of the new Century
• 1. Mathematical problems and theory formation
• 2. Hilbert’s problems: a research program
• 3. The limits of prophecy