IS PHILOSOPHY 'AN IDLENESS IN MATHEMATICS'?
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IS PHILOSOPHY 'AN IDLENESS IN MATHEMATICS'? book
The dispute between logicists and intu~tionistsconcerning the applicability of the Law of Excluded Middle to propositions whose expression requires quantifiers ranging over an infinite domain is intimately connected with differing views about the nature ofthe entities in the domain. Intuitionists challenged the accepted use of the law in mathematics, and thereby raised the general question about what constitutes proof. Along with the question as to the validity of certain sorts of proof, questions have arisen concerning the legitimacy of certain mathematical operations (e.g. set formation in accordance with the axiom ofinclusion) , and the legitimacy of any use of impredicative definitions. At the core of disagreements over principles lay a number of ill-defined problems about the conception of the infinite. These problems arise in part over the conception ofan infinite totality (the 'consummated infinite'), and in part over the mere fact that individuals in an ordered series having no last member cannot all be examined for possession of a property. It might be supposed that what look to be philosophical questions about the individuals of a domain could be eliminated by the minimal requirement that the individuals shall constitute a well-defined, nonempty class.! But it may be that the notion of being well-defined cannot be made clear - for example, when the individuals are real numbers. As is known, the work of Russell in logic, as well as that of Frege, is an attempt to deduce arithmetic from logic. On Bernays's account ofFrege, logic is to be viewed as 'the general theory of the universe ofmathematical objects'. 2 These objects are held to exist independently of our constructions, whether they be points, sets of points, numbers, sets of numbers, functions, etc. Godel asserted that we have as good a ground for believing in the existence ofsets, namely, our perception ofthem, as is given by our perceptions of physical bodies.3