A Minimalist Foundation at Work

Chapter
Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 75)

Abstract

What is the nature of mathematics? The different answers given to this perennial question can be gathered from the different theories which have been put forward as foundations of mathematics. In the common contemporary vision, the role of foundations is to provide safe grounds a posteriori for the activity of mathematicians, which on the whole is taken as a matter of fact. This is usually achieved by reducing all concepts of mathematics to that of set, and by postulating sets to form a fixed universe, which exists by itself and is unaffected by any human activity. Thus it is a static vision.

Keywords

Open Subset Closed Subset Classical Logic Type Theory Intuitionistic Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. Battilotti, G. and Sambin, G. (2006). Pretopologies and a uniform presentation of sup-lattices, quantales and frames. Annals of Pure and Applied Logic, 137:30–61.CrossRefGoogle Scholar
  2. Bell, J. L. (1988). Toposes and Local Set Theories: An introduction, volume 14 of Oxford Logic Guides. Clarendon, Oxford.Google Scholar
  3. Bishop, E. (1967). Foundations of Constructive Analysis. McGraw-Hill, New York, NY.Google Scholar
  4. Bishop, E. (1970). Mathematics as a numerical language. In Kino, A., Myhill, J., and Vesley, R., editors, Intuitionism and Proof Theory, pages 53–71. North-Holland, Amsterdam.Google Scholar
  5. Bishop, E. and Bridges, D. (1985). Constructive Analysis. Springer, Berlin.Google Scholar
  6. Ciraulo, F. (2007). Constructive Satisfiability. PhD thesis, Università di Palermo.Google Scholar
  7. Coquand, T., Sambin, G., Smith, J., and Valentini, S. (2003). Inductively generated formal topologies. Annals of Pure and Applied Logic, 104:71–106.CrossRefGoogle Scholar
  8. Dummett, M. (1977). Elements of Intuitionism, volume 2 of Oxford Logic Guides. Clarendon, Oxford.Google Scholar
  9. Hancock, P. and Hyvernat, P. (2006). Programming interfaces and basic topology. Annals of Pure and Applied Logic, 137:189–239.CrossRefGoogle Scholar
  10. Hyland, J. M. E. (1982). The effective topos. In Troesltra, A. S. and van Dalen, D., editors, The L.E.J. Brouwer Centenary Symposium, Studies in Logic and the Foundations of Mathematics, pages 165–216. North-Holland, Amsterdam.CrossRefGoogle Scholar
  11. Maietti, M. E. (1999). About effective quotients in constructive type theory. In Altenkirch, T., Naraschewski, W., and Reus, B., editors, Types for Proofs and Programs. TYPES ’98, volume 1657 of Lecture Notes in Computer Science, pages 164–178. Springer, Berlin.Google Scholar
  12. Maietti, M. E. (2005). Modular correspondence between dependent type theories and categories including pretopoi and topoi. Mathematical Structures in Computer Science, 15:1089–1149.CrossRefGoogle Scholar
  13. Maietti, M. E. (2009). A minimalist two-level foundation for constructive mathematics. Annals of Pure and Applied Logic, 160:319–354.CrossRefGoogle Scholar
  14. Maietti, M. E. and Sambin, G. (2005). Toward a minimalist foundation for constructive mathematics. In Crosilla, L. and Schuster, P., editors, From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics, volume 48 of Oxford Logic Guides, pages 91–114. Clarendon, Oxford.Google Scholar
  15. Maietti, M. E. and Sambin, G. (201x). Choice sequences in a minimalist foundation. In preparation.Google Scholar
  16. Maietti, M. E. and Valentini, S. (1999). Can you add power-sets to Martin-Löf’s intuitionistic set theory? Mathematical Logic Quarterly, 45:521–532.CrossRefGoogle Scholar
  17. Martin-Löf, P. (1970). Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm.Google Scholar
  18. Martin-Löf, P. (1984). Intuitionistic Type Theory: Notes by G. Sambin of a Series of Lectures Given in Padua, June 1980. Bibliopolis, Napoli.Google Scholar
  19. Martin-Löf, P. (2006). 100 years of Zermelo’s axiom of choice: what was the problem with it? The Computer Journal, 49:10–37.Google Scholar
  20. Martin-Löf, P. and Sambin, G. (in press). Generating positivity by coinduction. In The Basic Picture: Structures for Constructive Topology. Oxford University Press, Oxford, to appear.Google Scholar
  21. Nordström, B. and Petersson, K. (1990). Programming in Martin-Löf’s Type Theory: An introduction. Oxford University Press, Oxford.Google Scholar
  22. Sambin, G. (1987). Intuitionistic formal spaces—a first communication. In Skordev, D., editor, Mathematical Logic and Its Applications, pages 187–204. Plenum, New York, NY.Google Scholar
  23. Sambin, G. (1995). Pretopologies and completeness proofs. Journal of Symbolic Logic, 60:861–878.CrossRefGoogle Scholar
  24. Sambin, G. (2002). Steps towards a dynamic constructivism. In Gärdenfors, P., Kijania-Placek, K., and Wolenski, J., editors, In the Scope of Logic: Methodology and Philosophy of Science, vol. 1, number 315 in Synthese Library, pages 263–286. Kluwer, Dordrecht.Google Scholar
  25. Sambin, G. (2003). Some points in formal topology. Theoretical Computer Science, 305:347–408.CrossRefGoogle Scholar
  26. Sambin, G. (201x). The Basic Picture: Structures for Constructive Topology. Oxford University Press, Oxford. to appear.Google Scholar
  27. Sambin, G. and Valentini, S. (1998). Building up a toolbox for Martin-Löf’s type theory: subset theory. In Sambin, G. and Smith, J., editors, Twenty-five Years of Constructive Type Theory, number 36 in Oxford Logic Guides, pages 221–244. Clarendon. Proceedings of a Congress held in Venice, October 1995.Google Scholar
  28. Troelstra, A. and van Dalen, D. (1988). Constructivism in Mathematics: An Introduction, volume 2 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam.Google Scholar
  29. Vickers, S. (2006). Compactness in locales and in formal topology. Annals of Pure and Applied Logic, 137:413–438.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.University of PaduaPaduaItaly

Personalised recommendations