Handbook of Constructive Mathematics
Handbook of Constructive Mathematics
Hajime Ishihara, Japan Advanced Institute of Science and Technology
Michael Rathjen, University of Leeds
Helmut Schwichtenberg, Ludwig-Maximilians-Universität Munchen
Constructive mathematics – mathematics in which 'there exists' always means 'we can construct' – is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches.
- Gives a complete overview of modern constructive mathematics, including pure constructive mathematics, foundational issues, and applications to computing and probability
- Each chapter is a commissioned survey from leading specialists of a particular sub-topic within constructive mathematics
- The book begins with introductory chapters giving graduate students and other newcomers the foundations they need to engage with the rest
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Table of Contents
- Preface Douglas Bridges, Hajime Ishihara, Michael Rathjen and Helmut Schwichtenberg
- Part I. Introductory:
- 1. Introduction to intuitionistic logic Michael Rathjen
- 2. Introduction to CZF: an appetizer Michael Rathjen
- 3. Bishop's mathematics: a philosophical perspective Laura Crosilla
- Part II. Algebra and Geometry:
- 4. Algebra in Bishop's style: a course in constructive algebra Henri Lombardi
- 5. Constructive algebra: the Quillen-Suslin theorem Ihsen Yengui
- 6. Constructive algebra and point-free topology Thierry Coquand
- 7. Constructive projective geometry Mark Mandelkern
- Part III. Analysis:
- 8. Elements of constructive analysis Hajime Ishihara
- 9. Constructive functional analysis Hajime Ishihara
- 10. Constructive Banach algebra theory Robin Havea and Douglas Bridges
- 11. Constructive convex optimization Josef Berger and Gregor Svindland
- 12. Constructive mathematical economics Matthew Hendtlass and Douglas Bridges
- 13. Constructive stochastic processes Yuen-Kwok Chan
- Part IV. Topology:
- 14. Bases of pseudocompact Bishop spaces Iosif Petrakis
- 15. Bishop metric spaces in formal topology Tatsuji Kawai
- 16. Subspaces in point free topology and measure theory Francesco Ciraulo
- 17. Synthetic topology Davorin Lešnik
- 18. Apartness on lattices and between sets Douglas Bridges
- Part V. Logic and Foundations:
- 19. Countable choice Fred Richman
- 20. The Minimalist Foundation and Bishop's constructive mathematics Maria Maietti, Giovanni Sambin
- 21. Identity, equality, and extensionality in explicit mathematics Gerhard Jäger
- 22. Inner and outer models for constructive set theories Robert Lubarsky
- 23. An introduction to constructive reverse mathematics Hajime Ishihara
- 24. Systems for constructive reverse mathematics Takako Nemoto
- 25. Brouwer's fan theorem Josef Berger
- Part VI. Aspects of Computation:
- 26. Computational aspects of Bishop's constructive mathematics Helmut Schwichtenberg
- 27. Application of constructive analysis in exact real arithmetic Kenji Miyamoto
- 28. Efficient algorithms from proofs in constructive analysis Mark Bickford
- 29. On the computational content of choice principles Ulrich Berger and Monika Seisenberger
- Index.
