Classical Arithmetic is Part of Intuitionistic Arithmetic
Michael D. Potter, Editions Rodopi · Grazer Philosophische Studien · 1998
mathematics is an argument to show that the correct logic to apply in mathematical reasoning is not classical but intuitionistic. In this article I wish to cast doubt on Dummett’s conclusion by outlining an alternative, motivated by consideration of a well-known result of Kurt Gödel, to the standard view of the relationship between classical and intuitionistic arithmetic. I shall suggest that it is hard to find a perspective from which to arbitrate between the competing views. Let me start, then, by stating the standard view of the relationship, with which the account I shall be canvassing is to be contrasted. 1 The standard view Although some of what I shall be saying can be applied to areas of mathematics other than the arithmetic of the natural numbers, much of it depends on a feature of arithmetic that is not readily generalizable, namely that its atomic sentences are decidable. It will therefore simplify matters greatly to restrict our attention solely to the arithmetical case. The relationship between classical and intuitionistic mathematics in general is complicated by the fact that the intuitionistic