Product details:
| ISBN13: | 9780199232888 |
| ISBN10: | 0199232881 |
| Binding: | Hardback |
| No. of pages: | 624 pages |
| Size: | 234x156 mm |
| Language: | English |
| 700 |
Category:
Positive Topology
A New Practice in Constructive Mathematics
Series:
Oxford Logic Guides;
Publisher: OUP Oxford
Date of Publication: 6 March 2025
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Short description:
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.
Long description:
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.
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.
Table of Contents:
Preface
About This Book
Dynamic Constructivism: A New Conception of Mathematics
The Minimalist Foundation: Basic Notions and Tools
Basic Pairs: Symmetry and Duality in Topology
Relation-Pairs: Continuity is a Commutative Square
Concrete Spaces: Benefits of Keeping a Base
Convergent Relation-Pairs: Towards Topology With Points
Basic Topologies: Pointfree Topology Without Convergence
Continuous Relations: Respecting Covers and Positivities
Positive Topologies and Formal Maps: Pointfree Topology
Ideal Spaces and Maps: Ideal Aspects Over Real Topology
Topological Systems: a Place for all Topological Notions
Overlap Algebras: the Power of a Set as an Algebra
Overlap Topologies: Putting Topology in Algebraic Terms
Generating Positivity By Co-induction
References
Index
About This Book
Dynamic Constructivism: A New Conception of Mathematics
The Minimalist Foundation: Basic Notions and Tools
Basic Pairs: Symmetry and Duality in Topology
Relation-Pairs: Continuity is a Commutative Square
Concrete Spaces: Benefits of Keeping a Base
Convergent Relation-Pairs: Towards Topology With Points
Basic Topologies: Pointfree Topology Without Convergence
Continuous Relations: Respecting Covers and Positivities
Positive Topologies and Formal Maps: Pointfree Topology
Ideal Spaces and Maps: Ideal Aspects Over Real Topology
Topological Systems: a Place for all Topological Notions
Overlap Algebras: the Power of a Set as an Algebra
Overlap Topologies: Putting Topology in Algebraic Terms
Generating Positivity By Co-induction
References
Index