Treccani History of Science
The second Scientific Revolution
I2 – Mathematics and logic
Part B - The problem of the foundations and the new logics
Edited by Umberto Bottazzini, Jeremy Grey
Schools of philosophy of mathematics
• 1. The general reduction process: the contributions of Dedekind, Peano and Cantor
• 2. Frege’s logicist program
• 3. Russell’s logicist program
• 4. Predicativism: Poincaré and Weyl
• 5. Zermelo’s axioms for set theory
• 6. Brouwer’s intuitionism
• 7. Hilbert’s finitist program
Brouwer’s intuitionism
• 1. Weak counterexamples and creative subject
• 2. Brouwer’s program
• 3. Intuitionistic logic and arithmetic
Set theory
• 1. The axiom of choice
• 2. The axiomatization of Zermelo
• 3. The theory of Zermelo-Fraenkel
• 4. The constructible sets
• 5. The forcing
• 6. New axioms
Gödel incompleteness theorems
• 1. The Hilbert program
• 2. The incompleteness of arithmetic
• 3. The incompleteness and mathematical practice
The Leopoli-Warsaw school
• 1. The beginnings
• 2. Stanislaw Les´niewski
• 3. 1930
• 4. Tarskian semantics
Modal logic
• 1. The syntactic phase
• 2. The possible worlds semantics
• 3. The algebraic semantics
• 4. Further developments
The recursion theory
• 1. The precursors
• 2. Consolidation
• 3. The new Prometheus
• 4. Development