Mathematical Epistemology and Psychology

ISBN-13: 9789048183289 / Angielski / Miękka / 2010 / 326 str.

E. W. Beth; J. Piaget; W. Mays

Opis

One of the controversial philosophical issues of recent years has been the question of the nature of logical and mathematical entities. Platonist or linguistic modes of explanation have become fashionable, whilst abstrac tionist and constructionist theories have ceased to be so. Beth and Piaget approach this problem in their book from two somewhat different points of view. Beth's approach is largely historico-critical, although he discusses the nature of heuristic thinking in mathematics, whilst that of Piaget is psycho-genetic. The major purpose of this introduction is to summarise some of the main points of their respective arguments. In the first part of this book Beth makes a detailed study of the history of philosophical thinking about mathematics, and draws our attention to the important role played by the Aristotelian methodology of the demon strative sciences. This, he tells us, is characterised by three postulates: (a) deductivity, (b) self-evidence, and (c) reality. The last postulate asserts that the primitive notions of a demonstrative science must have reference to a domain of real entities in order to have significance. On the Aristote lian view discursive reasoning plays a major role in mathematics, whilst pure intuition plays a somewhat subordinate one.

Spis treści

I. Mathematical Reasoning Cannot be Analysed by Traditional Syllogistics.- II. The Psychological Interpretation of Mathematical Reasoning.- III. The Logicist Tradition.- IV. Strict Demonstration and Heuristic Procedures.- V. Intuitive Structures and Formalised Mathematics.- VI. “Thinking Machines” and Mathematical Thought.- VII. Lessons of the:History of the Relations Between Logic and Psychology.- VIII. General Psychological Problems of Logico-Mathematical Thought.- IX. General Psychological Problems of Logico-Mathematical Thought (Continued).- X. The Psychological Problems of “Pure” Thought.- XI. Some Convergences Between Formal and Genetic Analyses.- XII. Epistemological Problems with Logical and Psychological Relevance.- General Conclusions.- Name Index.