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Carnegie Mellon University, Department of Philosophy,
Spring 2001
The Nature of Mathematical Reasoning
Dirk Schlimm
Handouts
History of mathematics
-
Stanislas Dehaene: The Number Sense, 1997.
- Chapter 4: The Language of Numbers, pages 92-106.
-
Robert Osserman: Poetry of the Universe, 1995.
- Chapter I: Measuring the Unmeasurable, pages 1-19.
-
John Barrow: Pi in the Sky, 1992.
- Chapter 1: From Mystery to History, pages 1-25.
-
John Barrow: Pi in the Sky, 1992.
- Counting with base 2, 5, 60, pages 51-67.
Kinds of reasoning
- Dirk Schlimm: Terminology for proofs and arguments.
-
Anthony Weston: A Rulebook for Arguments, 1992.
- Chapter IV: Deductive Arguments, pages 46-59.
-
David Hume: An Enquiry Concerning Human Understanding, 1748.
- Section IV: Sceptical doubts concerning the operations of the understanding. Excerpts from: Cahn, Classics of Western Philosophy.
-
Paolo Mancosu: Philosophy of mathematics and mathematical practice in
the seventeenth century. 1996.
- Section 4.3: Proofs by contradiction from Kant to the Present,
pages 105-117.
- Gottlob Frege: Letter to Russell, 22.6.1902.
From: Michael Beaney, The Frege Reader, Blackwell Publishers, 1997; pages 252-4.
-
Clark Glymour: Thinking Things Through, 1992.
- Chapter 1: Proofs, pages 3-15.
The structure of mathematical theories
-
Berlinghoff, Grant, Skrien: A Mathematics Sampler - Topics for the liberal arts, 1996.
- Sections 3.3-3.5: Axiom systems. Models. Consistency and Independence., pages 85-103.
-
David Hilbert: Foundations of Geometry, 1899.
- Chapter I: The five groups of axioms, pages 1-5.
-
John Byrnie Shaw: Lectures in the Philosophy of Mathematics, 1918.
- Chapter III: Space and the Geometrization of Mathematics,
pages 31-46.
-
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.
- Chapter I: Natural numbers, pages 1-13.
-
Bertrand Russell: Introduction to Mathematical Philosophy, 1919.
- Chapter I: The series of natural numbers, pages 1-10.
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J.M. Bochenski: The Method of Contemporary Thought, 1964.
- 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.
-
Dirk Schlimm: Gödel's Incompleteness Theorems.
- 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
-
Robert Audi (ed.): The Cambridge Dictionary of Philosophy, 1995.
- Probability by E.Eels, pages 649-651.
Philosophical digression: Category Mistakes
-
Gilbert Ryle: Descartes's Myth, 1949.
- From: Bratman and Perry, Introduction to Philosophy, pages 305-310.
Logic & Formal proofs
-
Clark Glymour: Thinking Things Through, 1992.
- Chapter 2: Aristotle's logic, pages 44-57.
-
Dirk Schlimm: Natural deduction rules.
-
Dirk Schlimm: Mathematical induction problems.
Theory of the infinite
-
Dirk Schlimm: Important definitions: Sets & Functions.
-
William Dunham: Journey through Genius, 1990.
- Chapter 11: The Non-Denumerability of the Continuum, pages
245-266.
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© Dirk Schlimm, Last modified: Tue May 1 10:32:47 EDT 2001