Handbook of Constructive Mathematics

Gives a complete overview of modern constructive mathematics and its applications through surveys by leading experts.

Douglas Bridges (Edited by), Hajime Ishihara (Edited by), Michael Rathjen (Edited by), Helmut Schwichtenberg (Edited by)

9781316510865, Cambridge University Press

Hardback, published 11 May 2023

800 pages
25 x 17.6 x 4.9 cm, 1.63 kg

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.

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.

Subject Areas: Stochastics [PBWL], Game theory [PBUD], Topology [PBP], Geometry [PBM], Algebra [PBF], Set theory [PBCH], Mathematical logic [PBCD], Mathematical foundations [PBC], Philosophy of mathematics [PBB], Mathematics [PB]