Foundations for Computable Topology

Chapter
Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 76)

Abstract

Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this using the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines.

References

  1. Appel, A. (1992) Compiling with Continuations, Cambridge: Cambridge University Press.Google Scholar
  2. Barr, M. and Wells, C. (1985) Toposes, Triples and Theories, New York: Springer.MATHGoogle Scholar
  3. Bauer, A. (2008) Efficient Computation with Dedekind Reals, in Brattka, V., Dillhage, R., Grubba, T. and Klutch, A., eds. Fifth International Conference on Computability and Complexity in Analysis, in volume 221 of Electronic Notes in Theoretical Computer Science, Amsterdam: Elsevier.Google Scholar
  4. Bauer, A., Birkedal, L. and Scott, D. (2004) Equilogical Spaces, Theoretical Computer Science 315, 35–59.MathSciNetMATHCrossRefGoogle Scholar
  5. Bishop, E. and Bridges, D. (1985) Constructive Analysis, Number 279 in Grundlehren der mathematischen Wissenschaften, Heidelberg, Berlin, New York: Springer.MATHCrossRefGoogle Scholar
  6. Bourbaki, N. (1966) Topologie Générale, Paris: Hermann, English translation, “General Topology” Berlin: Springer (1989).Google Scholar
  7. Bridges, D. and Richman, F. (1987) Varieties of Constructive Mathematics, Number 97 in London Mathematical Society Lecture Notes, Cambridge: Cambridge University Press.MATHCrossRefGoogle Scholar
  8. Brown, R. (1964) Function Spaces and Product Topologies, Quarterly Journal of Mathematics 15(1), 238–250.MATHCrossRefGoogle Scholar
  9. Carboni, A., Lack, S. and Walters, R. (1993) Introduction to Extensive and Distributive Categories, Journal of Pure and Applied Algebra 84, 145–158.MathSciNetMATHCrossRefGoogle Scholar
  10. Cardone, F. and Hindley, R. (2006) History of Lambda-Calculus and Combinatory Logic, Handbook of the History of Logic 5.Google Scholar
  11. Church, A. and Rosser, B. (1936) Some Properties of Conversion, Transactions of the American Mathematical Society 39(3), 472–482.MathSciNetCrossRefGoogle Scholar
  12. Cleary, J. (1987) Logical Arithmetic, Future Computing Systems 2, 125–149.Google Scholar
  13. Cockett, R. (1993) Introduction to Distributive Categories, Mathematical Structures in Computer Science 3, 277–307.MathSciNetMATHCrossRefGoogle Scholar
  14. Corfield, D. (2003) Towards a Philosophy of Real Mathematics, Cambridge: Cambridge University Press.MATHCrossRefGoogle Scholar
  15. Eilenberg, S. and Moore, J. (1965) Adjoint Functors and Triples, Illinois Journal of Mathematics 9, 381–98.MathSciNetMATHGoogle Scholar
  16. Fauvel, J. and Gray, J. (1987) The History of Mathematics, a Reader, London: Macmillan, Open University.Google Scholar
  17. Fourman, M. and Hyland, M. (1979) Sheaf Models for Analysis, in Fourman, M., Mulvey, C. and Scott, D., eds. Applications of Sheaves, number 753 in Lecture Notes in Mathematics, Berlin, New York: Springer, pp. 280–301CrossRefGoogle Scholar
  18. Fox, R. (1945) On Topologies for Function-Spaces, Bulletin of the American Mathematical Society 51.Google Scholar
  19. Gentzen, G. (1935) Untersuchungen über das logische Schliessen, Mathematische Zeitschrift 39, 176–210, 405–431, English translation, pages 68–131 in The Collected Papers of Gerhard Gentzen, edited by Manfred E. Szabo, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland.MathSciNetCrossRefGoogle Scholar
  20. Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M. and Scott, D. (1980) A Compendium of Continuous Lattices, Berlin, Heidelberg, New York: Springer, second edition, Continuous Lattices and Domains, Cambridge: Cambridge University Press (2003).MATHCrossRefGoogle Scholar
  21. Girard, J.-Y. (1987) Linear Logic, Theoretical Computer Science 50, 1–102.MathSciNetMATHCrossRefGoogle Scholar
  22. Hartshorne, R. (1977) Algebraic Geometry, Number 52 in Graduate Texts in Mathematics, Berlin, New York: Springer.MATHGoogle Scholar
  23. Hofmann, K. and Mislove, M. (1981) Local Compactness and Continuous Lattices, in Banaschewski, B. and Hoffmann, R.-E., eds. Continuous Lattices, number 871 in Lecture Notes in Mathematics, Berlin: Springer, pp. 209–248.CrossRefGoogle Scholar
  24. Howard, W. (1980) The Formulae-as-Types Notion of Construction, in Curry, H., Seldin, J. and Hindley, R., eds. To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, New York: Academic Press, pp. 479–490.Google Scholar
  25. Hudak, P. (1989) Conception, Evolution and Application of Functional Programming Languages, ACM Computing Surveys 21, 355–411.CrossRefGoogle Scholar
  26. Hudak, P., Hughes, J., Peyton-Jones, S. and Wadler, P. (2007) A History of Haskell: Being Lazy with Class, in History of Programming Languages, New York: ACM Press, pp. 12–55.Google Scholar
  27. Hyland, M. (1991) First Steps in Synthetic Domain Theory, in Carboni, A., Pedicchio, M.-C. and Rosolini, G., eds. Proceedings of the 1990 Como Category Conference, number 1488 in Lecture Notes in Mathematics, Berlin, Heidelberg, New York: Springer, pp. 131–156.Google Scholar
  28. Isbell, J. (1986) General Function Spaces, Products and Continuous Lattices, Mathematical Proceedings of the Cambridge Philosophical Society 100, 193–205.MathSciNetMATHCrossRefGoogle Scholar
  29. Johnstone, P.(1982) Stone Spaces, Number 3 in Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge University Press.MATHGoogle Scholar
  30. Johnstone, P. (1984) Open Locales and Exponentiation, Contemporary Mathematics 30, 84–116.MathSciNetCrossRefGoogle Scholar
  31. Joyal, A. and Tierney, M. (1984) An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society 51(309).Google Scholar
  32. Jung, A., Kegelmann, M. and Moshier, A. (2001) Stably Compact Spaces and Closed Relations, Electronic Notes in Theoretical Computer Science 45.Google Scholar
  33. Kock, A. (1981) Synthetic Differential Geometry, Number 51 in London Mathematical Society Lecture Notes, Cambridge: Cambridge University Press, second edition, number 333 (2006).MATHGoogle Scholar
  34. Kuhn, T. (1962) The Structure of Scientific Revolutions, Chicago: University of Chicago Press.Google Scholar
  35. Lakatos, I. (1963) Proofs and Refutations: The Logic of Mathematical Discovery, British Journal for the Philosophy of Science 14, 1–25, re-published by Cambridge University Press (1976), edited by John Worrall and Elie Zahar.MathSciNetCrossRefGoogle Scholar
  36. Lawvere, B. (1969) Adjointness in Foundations, Dialectica 23, 281–296, Reprinted with commentary in Theory and Applications of Categories Reprints 16 (2006), 1–16.MATHCrossRefGoogle Scholar
  37. Linton, F. (1969) An Outline of Functorial Semantics, in Eckmann, B., ed. Seminar on Triples and Categorical Homology Theory, number 80 in Lecture Notes in Mathematics, Berlin, Heidelberg, New York: Springer, pp. 7–52.CrossRefGoogle Scholar
  38. Mac Lane, S. (1963) Natural Associativity and Commutativity, Rice University Studies 49, 28–46.MathSciNetGoogle Scholar
  39. Mac Lane, S. (1971) Categories for the Working Mathematician, Berlin, Heidelberg and New York: Springer.MATHCrossRefGoogle Scholar
  40. Mac Lane, S. (1988) Categories and Concepts in Perspective, in Duren, P., Askey, R. and Merzbach, U., eds. A Century of Mathematics in America, vol. 1, pp. 323–365, Providence, RI: American Mathematical Society. Addendum in vol. 3, pp. 439–441.Google Scholar
  41. Manes, E. (1976) Algebraic Theories, Number 26 in Graduate Texts in Mathematics, Berlin, Heidelberg, New York: Springer.MATHCrossRefGoogle Scholar
  42. McLarty, C. (1990) The Uses and Abuses of the History of Topos Theory, British Journal for the Philosophy of Science 41, 351–375.MathSciNetMATHCrossRefGoogle Scholar
  43. Mikkelsen, C. (1976) Lattice-Theoretic and Logical Aspects of Elementary Topoi, PhD thesis, Århus: Århus Universitet, Various publications, number 25.Google Scholar
  44. Moore, R. (1966) Interval Analysis, Automatic Computation, Englewood Cliff, NJ: Prentice Hall, second edition, Introduction to Interval Analysis, with Baker Kearfott and Michael Cloud, Society for Industrial and Applied Mathematics (2009).MATHGoogle Scholar
  45. Paré, R. (1974) Colimits in Topoi, Bulletin of the American Mathematical Society 80(3), 556–561.MathSciNetMATHCrossRefGoogle Scholar
  46. Phoa, W. (1990) Domain Theory in Realizability Toposes, PhD thesis, Cambridge: University of Cambridge. University of Edinburgh Dept. of Computer Science report CST-82-91 and ECS-LFCS-91-171.Google Scholar
  47. Pitts, A. (2001) Categorical Logic, in Handbook of Logic in Computer Science, vol. 5, Oxford: Oxford University Press.Google Scholar
  48. Rosolini, G. (1986) Continuity and Effectiveness in Topoi, D. Phil. thesis, Oxford: University of Oxford.Google Scholar
  49. Russell, B. and Whitehead, A.N. (1910–1913) Principia Mathematica, Cambridge: Cambridge University Press.MATHGoogle Scholar
  50. Scott, D. (1972) Continuous Lattices, in Lawvere, F.W., ed. Toposes, Algebraic Geometry and Logic, number 274 in Lecture Notes in Mathematics, Berlin, Heidelberg, New York: Springer, pp. 97–136.CrossRefGoogle Scholar
  51. Scott, D. (1993) A Type-Theoretical Alternative to ISWIM, CUCH, OWHY, Theoretical Computer Science 121, 422–440. Written in 1969.CrossRefGoogle Scholar
  52. Seldin, J. (2002) Curry’s Anticipation of the Types Used in Programming Languages, Proceedings of the Canadian Society for History and Philosophy of Mathematics 15, 148–163.Google Scholar
  53. Simmons, H. (1978) The Lattice-Theoretic Part of Topological Separation Properties, Proceedings of the Edinburgh Mathematical Society (2) 21, 41–48.MathSciNetMATHCrossRefGoogle Scholar
  54. Smolin, L. (2006) The Trouble with Physics, New York: Houghton Mifflin, republished by Penguin (2008).MATHGoogle Scholar
  55. Spitters, B. (2010) Located and overt sublocales, to appear in Annals of Pure and Applied Logic, 162, 36–54.MathSciNetMATHCrossRefGoogle Scholar
  56. Steenrod, N. (1967) A Convenient Category of Topological Spaces, Michigan Mathematics Journal 14, 133–152.MathSciNetMATHCrossRefGoogle Scholar
  57. Stone, M. (1937) Applications of the Theory of Boolean Rings to General Topology, Transactions of the American Mathematical Society 41, 375–481.MathSciNetCrossRefGoogle Scholar
  58. Stone, M. (1938) The Representation of Boolean Algebras, Bulletin of the American Mathematical Society 44, 807–816.MathSciNetCrossRefGoogle Scholar
  59. Tarski, A. (1935) Zur Grundlegung der Boole’schen Algebra, Fundamenta Mathematicae 24, 177–198.Google Scholar
  60. Taylor, P. (1999) Practical Foundations of Mathematics, Number 59 in Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge University Press.MATHGoogle Scholar
  61. Thielecke, H. (1997) Categorical Structure of Continuation Passing Style, PhD thesis, Edinburgh: University of Edinburgh.Google Scholar
  62. Vermeulen, J. (1994) Proper Maps of Locales, Journal of Pure and Applied Algebra 92, 79–107.MathSciNetMATHCrossRefGoogle Scholar
  63. Vickers, S. (1988) Topology Via Logic, volume 5 of Cambridge Tracts in Theoretical Computer Science, Cambridge: Cambridge University Press.Google Scholar
  64. Wilker, P. (1970) Adjoint Product and Hom Functors in General Topology, Pacific Journal of Mathematics 34, 269–283.MathSciNetMATHCrossRefGoogle Scholar
  65. Paul Taylor’s papers on the Abstract Stone Duality programme, including the extended version of this one, are available from www.paultaylor.eu/asd/
  66. Taylor, P. (2000a) Geometric and Higher-Order Logic in Terms of Abstract Stone Duality, Theory and Applications of Categories 7, 284–338.MathSciNetMATHGoogle Scholar
  67. Taylor, P. (2000b) Non-Artin Gluing in Recursion Theory and Lifting in Abstract Stone Duality.Google Scholar
  68. Taylor, P. (2002a) Sober Spaces and Continuations, Theory and Applications of Categories 10, 248–299.MathSciNetMATHGoogle Scholar
  69. Taylor, P. (2002b) Subspaces in Abstract Stone Duality, Theory and Applications of Categories 10, 300–366.Google Scholar
  70. Taylor, P. (2002c) Scott Domains in Abstract Stone Duality, http://www.paultaylor.eu/asd/pcfasd.pdf
  71. Taylor, P. (2004a) An Elementary Theory of Various Categories of Spaces and Locales.Google Scholar
  72. Taylor, P. (2004b) Tychonov’s Theorem in Abstract Stone Duality.Google Scholar
  73. Taylor, P. (2005) Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe, Electronic Notes in Theoretical Computer Science 122, 247–296.CrossRefGoogle Scholar
  74. Taylor, P. (2006a) Computably Based Locally Compact Spaces, Logical Methods in Computer Science 2, 1–70.CrossRefGoogle Scholar
  75. Taylor, P. (2006b) Interval Analysis Without Intervals, Real Numbers and Computers.Google Scholar
  76. Taylor, P. (2009a) With Andrej Bauer, The Dedekind Reals in Abstract Stone Duality, Mathematical Structures in Computer Science 19, 757–838.MATHCrossRefGoogle Scholar
  77. Taylor, P. (2009b) Equideductive Categories and Their Logic.Google Scholar
  78. Taylor, P. (2010a) A lambda Calculus for Real Analysis, Journal of Logic and Analysis 2(5), 1–115.Google Scholar
  79. Taylor, P. (2010b) Computability for Locally Compact Spaces.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.

Personalised recommendations