A Defence of Arbitrary Objects

There is the following view. In addition to individual objects, there are arbitrary objects: in addition to individual numbers, arbitrary numbers; in addition to individual men, arbitrary men. With each arbitrary object is associated an appropriate range of individual objects, its values: with each arbitrary number, the range of individual numbers; with each arbitrary man, the range of individual men. An arbitrary object has those properties common to the individual objects in its range. So an arbitrary number is odd or even, an arbitrary man is mortal, since each individual number is odd or even, each individual man is mortal. On the other hand, an arbitrary number fails to be prime, an arbitrary man fails to be a philosopher, since some individual number is not prime, some individual man is not a philosopher. Such a view used to be quite common, but has now fallen into complete disrepute. As with so many things, Frege led the way. Given his own theory of quantification, it was unnecessary to interpret the variables of mathematics as designating variable numbers; and given the absurdities in the notion of a variable number, it was also unwise. It was with characteristic irony that he wrote: 'Perhaps there is a seminal idea here which we could also find of value outside mathematics' ([5], p. 160). Where Frege led, others have been glad to follow. Among the many subsequent philosophers who have spoken against arbitrary objects, we might mention Russell ([12], pp. 90-91), Lesniewski ([8], pp. 22-3, 27), Tarski ([13], p. 4), Church ([3], p. 13), Quine ([10], pp. 127-8), Rescher ([11], pp. 134-7) and Lewis ([7]) p. 203. If more philosophers of the present day have not added their voices to the protest, it is probably because they have not thought it worth the bother. As Menger says ([9], p. 144), the thesis that there are no variable numbers is 'today one of the few propositions about which logicians as well as mathematicians are in general agreement'. In the face of such united opposition, it might appear rash to