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Carnegie Mellon University, Department of Philosophy,
Summer 2000

The Nature of Mathematical Reasoning

Dirk Schlimm


Handouts

    History of mathematics

  1. Stanislas Dehaene: The Number Sense, 1997.

  2. - Chapter 4: The Language of Numbers, pages 92-106.

  3. Robert Osserman: Poetry of the Universe, 1995.

  4. - Chapter I: Measuring the Unmeasurable, pages 1-19.

  5. John Barrow: Pi in the Sky, 1992.

  6. - Chapter 1: From Mystery to History, pages 1-25.

  7. John Barrow: Pi in the Sky, 1992.

  8. - Counting with base 2, 5, 60, pages 51-67.

    Kinds of reasoning

  9. Dirk Schlimm: Terminology for proofs and arguments.


  10. Anthony Weston: A Rulebook for Arguments, 1992.

  11. - Chapter IV: Deductive Arguments, pages 46-59.

  12. David Hume: An Enquiry Concerning Human Understanding, 1748.

  13. - Section IV: Sceptical doubts concerning the operations of the understanding. Excerpts from: Cahn, Classics of Western Philosophy.

  14. Gottlob Frege: Letter to Russell, 22.6.1902.
    From: Michael Beaney, The Frege Reader, Blackwell Publishers, 1997; pages 252-4.


  15. Paolo Mancosu: Philosophy of mathematics and mathematical practice in the seventeenth century. 1996.

  16. - Section 4.3: Proofs by contradiction from Kant to the Present, pages 105-117.

  17. Clark Glymour: Thinking Things Through, 1992.

  18. - Chapter 1: Proofs, pages 3-15.

    The structure of mathematical theories

  19. Berlinghoff, Grant, Skrien: A Mathematics Sampler - Topics for the liberal arts, 1996.

  20. - Sections 3.3-3.5: Axiom systems. Models. Consistency and Independence., pages 85-103.

  21. David Hilbert: Foundations of Geometry, 1899.

  22. - Chapter I: The five groups of axioms, pages 1-5.

  23. John Byrnie Shaw: Lectures in the Philosophy of Mathematics, 1918.

  24. - Chapter III: Space and the Geometrization of Mathematics, pages 31-46.

  25. Edmund Landau: Foundations of Analysis - The Arithmetic of Whole, Rational, Irrational, and Complex Numbers. A Supplement to Text-Books on the Differential and Integral Calculus, 1926.

  26. - Chapter I: Natural numbers, pages 1-13.

  27. Bertrand Russell: Introduction to Mathematical Philosophy, 1919.

  28. - Chapter I: The series of natural numbers, pages 1-10.

  29. J.M. Bochenski: The Method of Contemporary Thought, 1964.

  30. - Chapter VI: The Axiomatic Method, Section 16: Example of the axiomatic method in practice - Axiomatization of the sentential logic of Hilbert and Ackermann, pages 87-90.

  31. Dirk Schlimm: Gödel's Incompleteness Theorems.

  32. - Containing quotes from Constance Reid, Hilbert (1970) and Kurt Gödel, On formally undecidable propositions of Principia Mathematica and related formal systems I (1931).

    Case Study I: Theory of Probability

  33. Robert Audi (ed.): The Cambridge Dictionary of Philosophy, 1995.

  34. - Probability by E.Eels, pages 649-651.

    Logic & Formal proofs

  35. Clark Glymour: Thinking Things Through, 1992.

  36. - Chapter 2: Aristotle's logic, pages 44-57.

  37. Dirk Schlimm: Natural deduction rules.

  38. Dirk Schlimm: Claims to prove by mathematical induction.

  39. Theory of the infinite

  40. Dirk Schlimm: Important definitions: Sets & Functions.

  41. William Dunham: Journey through Genius, 1990.

  42. - Chapter 11: The Non-Denumerability of the Continuum, pages 245-266.


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© Dirk Schlimm, Last modified: August 4, 2000