Год выпуска: 2011 Автор: Wataru Asanuma Издательство: LAP Lambert Academic Publishing Страниц: 176 ISBN: 9783844323665
Описание
The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. The controversy over the Axiom of Choice (AC) is a case in point. Due to its non- constructive nature, the AC has seemingly unpleasant consequences. It leads to the existence of non- Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the latter is so called in the sense that it is a counter-intuitive theorem. To see that mathematical truths are of non- constructive nature, I draw upon Godel's Incompleteness Theorems. The Lowenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam's arguments assume a clear meaning. According to them, the AC depends for its truth-value upon the model in which it is placed. In my view, however, this shows a limitation of formal methods. In response to Benacerraf's challenge to Platonism, the book concludes that in mathematics, as distinct from natural...
