Beginners' References - Constructive Mathematics - Mathematics and Statistics - University of Canterbury - New Zealand

Constructive Mathematics

A bibliography of constructive mathematics

Compiled by: Douglas S. Bridges

The following is a brief list of references that might interest people who are new to constructive mathematics. First, we have two websites:

  • Constructive Mathematics—Stanford Encyclopedia of Philosophy

    Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct”. In order to work constructively, we need to re-interpret not only the existential quantifier but all the logical connectives and quantifiers as instructions on how to construct a proof of the statement involving these logical expressions....

  • Luitzen Egbertus Jan Brouwer—Stanford Encyclopedia of PhilosophyL.E.J. Brouwer

    Dutch mathematician and philosopher who lived from 1881 to 1966. He is traditionally referred to as ‘L.E.J. Brouwer’, with full initials, but was called ‘Bertus’ by his friends.

    In classical mathematics, he founded modern topology by establishing, for example, the topological invariance of dimension and the fixpoint theorem. He also gave the first correct definition of dimension.

    In philosophy, his brainchild is intuitionism, a revisionist foundation of mathematics. Intuitionism views mathematics as a free activity of the mind, independent of any language or Platonic realm of objects, and therefore bases mathematics on a philosophy of mind. The implications are twofold. First, it leads to a form of constructive mathematics, in which large parts of classical mathematics are rejected. Second, the reliance on a philosophy of mind introduces features that are absent from classical mathematics as well as from other forms of constructive mathematics: unlike those, intuitionistic mathematics is not a proper part of classical mathematics...

Next we have books and articles.

  1. M. J. Beeson, Foundations of Constructive Mathematics, Springer–Verlag, Heidelberg, 1985.
  2. E.A. Bishop, Foundations of Constructive Analysis, McGraw–Hill, New York, 1967.
  3. E.A. Bishop, Schizophrenia in Contemporary Mathematics, Amer. Math. Soc. Colloquium Lectures, Univ. of Montana, Missoula; reprinted in: Contemporary Mathematics 39, 1–32, Amer. Math. Soc., 1985.
  4. D.S. Bridges, “Constructive truth in practice”, in: Truth in Mathematics (Proceedings of the conference held at Mussomeli, Sicily, 13–21 September 1995, H.G. Dales and G. Oliveri, eds), 53–69, Oxford University Press, Oxford, 1997.
  5. D.S. Bridges, “Constructive Mathematics: a Foundation for Computable Analysis”, Theoretical Computer Science 219 (1–2), 95-109, 1999.
  6. D.S. Bridges and L.S. Vîta (Dediu), “Paradise lost or paradise regained?”, EATCS Bulletin 63, 141–152, October 1997.
  7. D.S. Bridges and L.S. Vîta, Apartness Spaces, Universitext, Springer–Verlag, New York, 2006.
  8. M.A.E. Dummett, Elements of Intuitionism, (2/e), Oxford Logic Guides 28, Clarendon Press, Oxford, 2000.
  9. A. Heyting, Intuitionism—An Introduction (Third Edition), North Holland, 1971.
  10. B.A. Kushner, Lectures in Constructive Mathematical Analysis, Amer. Math. Soc., Providence RI, 1985.
  11. P. Martin–Löf, ‘Constructive mathematics and computer programming’, in Proc. 6th. Int. Congress for Logic, Methodology and Philosophy of Science (L. Jonathan Cohen, ed.), North–Holland, Amsterdam, 1980.
  12. R. Mines, F. Richman and W. Ruitenburg, A Course in Constructive Algebra, Universitext, Springer–Verlag, Heidelberg, 1988.
  13. F. Richman, “Intuitionism as a Generalization” , Philosophia Math. 5, 124–128, 1990.
  14. F. Richman, “Interview with a Constructive Mathematician”, Modern Logic 6, 247–271, 1996.
  15. G. Stolzenberg, Review of “Foundations of Constructive Analysis” [2], Bull, Amer, Math. Soc. 76, 301–323, 1970.
  16. A.S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction (two volumes), North Holland, Amsterdam, 1988.
  17. Warschawski, “Errett Bishop—In Memoriam”, Contemporary Mathematics 39, 33–40, Amer. Math. Soc., 1985.
  18. K. Weihrauch, Computable Analysis, Springer–Verlag, Heidelberg, 2000.