Positive Topology

Abstract This book provides a comprehensive exposition of dynamic constructivism—mathematics as a human cultural product. Looking at idealization as distinct from abstraction; sets as distinct from propositions, subsets, and collections; and operations as distinct from functions, it covers the coexistence of real and ideal mathematics, overcoming the clash between classical and constructive views. It provides a minimalist foundation to maximize information. Mathematics respects positive proofs, predicative definitions, and computational content. Sambin offers a look at the new mathematics emerging from dynamic constructivism. He explores duality and symmetry underlying topology, continuity as a commutative diagram, and the basic picture—the language of topology as a synthetic expression of applied logic. He describes the ‘dark side of the moon’, namely, mathematical treatment of existential quantifications: overlap, positivity relation, coinduction, and positive topology as an enrichment of formal topology. He investigates pointfree topology and explores a proof re: Grothendieck. He describes the new connection between discrete and continuous: ideal spaces as the limit of positive topologies closer to intuition. The book offers a roadmap towards a natural-dynamic paradigm in mathematics. It provides the view of bottom-up information management rather than top-down objective truth and modular and dynamic approach. It describes how formal systems are not strictly necessary and reassesses human understanding and intuition. Computational content and proof-assistant friendly, it shows how mathematics in the minimalist foundation is compatible with all others. It explains active acceptance of pluralism, construction of a common core, and how we need to develop mathematics in the natural-dynamic paradigm.