Mathematics and Logic

Defines the scope, nature, and fundamental relationship between mathematics and logic, exploring major philosophical schools of thought regarding their truth and applicability.

Traces the parallel and intersecting evolution of mathematical and logical thought from ancient civilizations through the foundational crises to the modern era.

Examines the fundamental logical machinery underpinning mathematics: axioms, definitions, theorems, and the methods of proof (deductive, inductive, constructive).

A detailed breakdown of major logical systems, including propositional logic, predicate logic, modal logic, and their syntax, semantics, and proof theories.

Surveys the principal classifications within pure and applied mathematics, exploring their unique objects of study and connections to logic.

Analyzes the critical role of mathematical and logical frameworks in computer science, linguistics, physics, economics, cryptography, and artificial intelligence.

Discusses major controversies, such as the foundational crisis, Gödel's incompleteness theorems, and the philosophical limits of formal systems.

Explores current research in areas like homotopy type theory as a foundation, automated theorem proving, computational complexity theory, and the logic of machine learning.