Overview

Constructive mathematics is based on the thesis that the meaning of a mathematical formula is given, not by its truth-conditions, but in terms of what constructions count as a proof of it. However, the meaning of the terms ""construction"" and ""proof"" has not been adequately explained (although Kriesel, Goodman and Martin have attempted axiomatizations). This monograph develops precise (though not wholly formal) definitions of construction and proof, and describes the algorithmic substructure underlying intuitionistic logic. Interpretations of Heyting Arithmetic and constructive analysis are given. The philosophical basis of constructivism is explored thoroughly in Part I. The author seeks to answer objections from Platonists and to reconcile his position with the central insights of Hilbert`s formalism and logic. The text should be of interest to philosophers of mathematics and logicians, both academic and graduate student, particularly those interested in Brouwer and Hilbert. Also, theoretical computer scientists interested in the foundations of functional programming languages and program correctness calculi.

Full Product Details

Author:   P. Fletcher
Publisher:   Kluwer Academic Publishers
Imprint:   Kluwer Academic Publishers
Edition:   1998 ed.
Volume:   276
Dimensions:   Width: 15.60cm , Height: 2.60cm , Length: 23.40cm
Weight:   1.850kg
ISBN:  

9780792352624

ISBN 10:   0792352629
Pages:   470
Publication Date:   31 October 1998
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Table of Contents

1. Introduction and Statement of the Problem.- 2. What’s Wrong with Set Theory?.- 3. What’s Wrong with Infinite Quantifiers?.- 4. Abstraction and Idealisation.- 5. What are Constructions?.- 6. Truth and Proof of Logical Formulae.- 7. The Need for a Theory of Constructions.- 8. Theories of Constructions.- 9. Hilbert’s Formalism.- 10. Open-endedness.- 11. Analysis.- 12. Introduction to Part II.- 13. Design of the Term Language.- 14. The Term Language.- 15. From the Term Language to the Expanded Term Language.- 16. The Expanded Term Language.- 17. The Protological Sequent Calculus.- 18. Commentary on the Protological Axioms and Rules.- 19. From Protologic to Expanded Protologic.- 20. Expanded Protologic.- 21. From Expanded Protologic to the Coding of Trees.- 22. The Coding of Trees.- 23. The Expanded Term Language as a Functional Programming Language.- 24. Introduction to Part III.- 25. From the Coding of Trees to Logic.- 26. Logic.- 27. From Logic to the Calculus of Proof Functions.- 28. Calculus of Proof Functions.- 29. From Calculus of Proof Functions to the Logic of Partial Terms.- 30. Logic of Partial Terms.- 31. From Logic of Partial Terms to Heyting Arithmetic.- 32. Heyting Arithmetic.- 33. From Heyting Arithmetic to Peano Arithmetic.- 34. Peano Arithmetic.- 35. Conclusions on Arithmetic.- 36. Introduction to Part IV.- 37. From Expanded Protologic to the Second-Order Coding of Trees.- 38. The Second-Order Coding of Trees.- 39. From the Second-Order Coding of Trees to Second-Order Logic.- 40. Second-Order Logic.- 41. From Second-Order Logic to Second-Order Calculus of Proof Functions.- 42. Second-Order Calculus of Proof Functions.- 43. From Second-Order Calculus of Proof Functions to Second-Order Logic of Partial Terms.- 44. Second-Order Logic of PartialTerms.- 45. From Second-Order Logic of Partial Terms to Second-Order Heyting Arithmetic.- 46. Second-Order Heyting Arithmetic.- 47. From Second-Order Heyting Arithmetic to Second-Order Peano Arithmetic.- 48. Second-Order Peano Arithmetic.- 49. Conclusions on Analysis.- References.- Index of symbols.- Index of axioms, theorems and rules of inference.- Index of names.- Index of topics.

Reviews

Author Information

Tab Content 6

Author Website:  

Countries Available

All regions