Positive Topology

Informationen zum Autor 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). Klappentext This book contains the first systematic exposition of a new discipline called positive topology, which brings many conceptual and mathematical innovations and many benefits for mathematical practice and applications, for the first time extending Darwin's evolutionary vision to the practice of mathematics. Zusammenfassung 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. Inhaltsverzeichnis Preface About This Book 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 References Index ...

Auteur
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).

Texte du rabat

This book contains the first systematic exposition of a new discipline called positive topology, which brings many conceptual and mathematical innovations and many benefits for mathematical practice and applications, for the first time extending Darwin's evolutionary vision to the practice of mathematics.

Résumé
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.

Contenu

• Preface

• About This Book

• 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

• References

• Index