ABSTRACT
Infinite aggregates are often denied. Even Leibniz, favourable as he was to the actual infinite, maintained that, where infinite classes are concerned, it is possible to make valid statements about any term of the class, but not about all the terms, nor yet about the whole which (as he would say) they do not compose.† Kant, again, has been much criticized for maintaining that space is an infinite given whole. Many maintain that every aggregate must have a finite number of terms, and that where this condition is not fulfilled there is
no true whole. But I do not believe that this view can be successfully defended. Among those who deny that space is a given whole, not a few would admit that what they are pleased to call a finite space may be a given whole, for instance, the space in a room, a box, a bag or a book-case. But such a space is only finite in a psychological sense, i.e. in the sense that we can take it in at a glance: it is not finite in the sense that it is an aggregate of a finite number of terms, nor yet a unity of a finite number of constituents. Thus to admit that such a space can be a whole is to admit that there are wholes which are not finite. (This does not follow, it should be observed, from the admission of material objects apparently occupying finite spaces, for it is always possible to hold that such objects, though apparently continuous, consist really of a large but finite number of material points.) With respect to time, the same argument holds: to say, for example, that a certain length of time elapses between sunrise and sunset, is to admit an infinite whole, or at least a whole which is not finite. It is customary with philosophers to deny the reality of space and time, and to deny also that, if they were real, they would be aggregates. I shall endeavour to show, in Part VI, that these denials are supported by a faulty logic, and by the now resolved difficulties of infinity. Since science and common sense join in the opposite view, it will therefore be accepted; and thus, since no argument à priori can now be adduced against infinite aggregates, we derive from space and time an argument in their favour.
