Positive Topology A New Practice in Constructive Mathematics

Constructive topology is an important new branch of constructive mathematics and logic and this monograph is the first account of constructive topology in book form. Aimed at researchers in mathematics, philosophy, and logic, the author provides discussion of the technical development of the field and also outlines the philosophical and methodological motivations underlying the evolution of the discipline.

Aimed at researchers in mathematics, philosophy and logic, this book provides the first organic exposition of dynamic constructivism and the mathematics ensuing in practice, including discussion of the technical development of the field and outlining the philosophical and methodological motivations underlying the evolution of the discipline. In dynamic constructivism, mathematics is seen as the result of a dynamic process of interaction between the construction of mathematical entities, by abstraction and by idealization, and their selection according to their efficiency in applications to reality and in the organisation of mathematics itself. The crucial benefit of this vision is its independence from dogmas and external authorities. A practical consequence is full respect for the diverse areas of mathematics - mainly computation, spatial intuition, deduction, and abstract axiomatic method - without reducing one to another. As a second consequence, a dynamic interaction between different 'epistemological levels' is always active and present, in the development of mathematics in practice, the study of its foundations and its formalisation in a computer language.

1 - Dynamic Constructivism: A New Conception of Mathematics
2 - The Minimalist Foundation: Basic Notions and Tools
3 - Basic Pairs: Symmetry and Duality in Topology
4 - Relation-Pairs: Continuity is a Commutative Square
5 - Concrete Spaces: Benefits of Keeping a Base
6 - Convergent Relation-Pairs: Towards Topology With Points
7 - Basic Topologies: Pointfree Topology Without Convergence
8 - Continuous Relations: Respecting Covers and Positivities
9 - Positive Topologies and Formal Maps: Pointfree Topology
10 - Ideal Spaces and Maps: Ideal Aspects Over Real Topology
11 - Topological Systems: a Place for all Topological Notions
12 - Overlap Algebras: the Power of a Set as an Algebra
13 - Overlap Topologies: Putting Topology in Algebraic Terms
Appendix A - Generating Positivity By Co-induction

Giovanni Sambin has been doing research in logic and the foundations of mathematics since the 1970s. He is among the founders of provability logic and has been developing the field of topology since the 1990s, adopting a dynamic view in mathematical practice, leading to positive topology, a new, richer approach to constructive topology based on a dynamic vision of the nature of mathematics. He was the first president of the Italian Association of Logic and its Applications (AILA).