Friederike Moltmann

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather 'aspects' of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted

Terms such as 'wisdom' or 'happiness' are commonly held to refer to abstract objects that are properties. On the basis of a greater range of linguistic data and with the support of some ancient and medieval philosophical views, I argue that such terms do not stand for objects, but rather for kinds of tropes, entities that do not have the status of objects, but only play a role as semantic values of terms and as arguments of predicates. Such ‘non-objects’ crucially differ from objects in that they are not potential bearers of properties.