Abstract
Since doing mathematics is concerned with investigating generalizations and convincing both yourself and others that the generalizations are valid, it would be nice to be able to say definitely what constitutes a valid and totally convincing argument. Unfortunately, this is rather difficult. In order to be mathematically precise, it would be necessary to provide minute details justifying every little step. Even then there might be some assumption hidden in the implicit meanings of the words. To be absolutely certain, the words have to be converted into symbols which are manipulated mechanically according to formal rules. All sense of meaning is abandoned. Such a proof of anything interesting would be so cumbersome as to be uncheckable, and certainly it would be uninformative as to why the proof was valid and the statement true.
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Further Reading
Imre Lakatos, Proof and Refutations, Cambridge University Press (1976). A famous mathematical theorem is gradually clarified by means of a conversation in which arguments and objections are put forward by various characters linked to the historical development of the ideas.
George Polya, Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving, Wiley, Combined Edition (1981). An expensive reprint, but by far the most important and useful book on mathematical problem solving. It investigates specializing and generalizing in depth with hundreds of examples.
John Mason, Leone Burton and Kaye Stacey, Thinking Mathematically, Addison Wesley (1982). Starting with specializing and generalizing, a framework is developed for improving mathematical thinking by learning from experience—complete with a psychological theory of how to learn from experience, and over a hundred problems to think about.
Philip J. Davis and Reuben Hersh, The Mathematical Experience, Harvester (1981). An excellent discussion that all students of mathematics should read.
Susan Pirie, Mathematics Investigations in the Classroom, Macmillan Education, 1987. Discusses techniques for getting pupils working investigatively.
Alan Graham, Investigations in Everyday Maths, Edward Arnold, 1987. Intended for use with GCSE lower-grade pupils, putting mathematics in everyday context.
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© 1988 The Open University
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Mason, J.H. (1988). When is an Argument Valid?. In: Learning and Doing Mathematics. Palgrave, London. https://doi.org/10.1007/978-1-349-09782-1_5
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DOI: https://doi.org/10.1007/978-1-349-09782-1_5
Publisher Name: Palgrave, London
Print ISBN: 978-0-333-44942-4
Online ISBN: 978-1-349-09782-1
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