Abstract
The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive nature of the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel’s Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam’s arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to Benacerraf’s challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality.
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