in advance the conditions under which the verifiably external can be known as such,—it is this, I say, which is the basis of Renouvier's principle of determinateness.
And now, what follows as to the external reality of infinite numerical aggregates? Even non-mathematical readers, like myself, have been struck of late by the very interesting efforts of men such as Moritz Cantor in Germany, of Mr. Charles Peirce in this country, to define infinite magnitudes and aggregates in a fashion such as introduces a new sort of determination of their nature, a determination not dependent upon counting. An infinite aggregation, according to this view, is one that can be demonstrably coördinated, element for element, to one of its own parts, as a straight line can be coordinated, by projection, point for point, with a line of any other length, however small. From this point of view the infinite aggregate cannot be counted, but it can be identified by any observer of its properties.
Problems in the modern Theory of Functions have led to this definition, which, of course, I have neither the right nor the time to judge here. I may say that the criterion of social verifiability, as I have now defined it, seems to me to permit of the acceptance of the objective existence, in our external world, of really countless magnitudes and aggregates, and so of infinite aggregates, in case you can so define them that in theory at least they could possibly be identified by any observer you please, and distinguished from all other objects. Now Cantor's or Mr. Charles Peirce's definition of the infinite aggregate enables you, in theory, not only to identify a given aggregate as conforming to this definition, but also (potentially, at least) to distinguish any one infinite aggregate from any other, through Cantor's remarkable symbolic device of the überunendliche Zahlen. From this point of view, counting would not become the only theoretically possible device for describing aggregates to your neighbor, for him to verify. As the countless aggregate is already a subjectively familiar experience, and as Cantor's devices deal with the general definition and description, in objectively verifiable terms, of infinite
