see that the absolutely universal proposition, “Every whole has its single and separate correspondent member in any one of the various ordered series of selected whole numbers aforesaid,” is not only free from contradiction, but is easily demonstrable, and is a mere expression of the actual nature of the number-series, taken as an object of exact thought.
Highly important it is, however, to observe, that the property of the number-series here in question is most sharply conceived, not when one wearily tries, as Mr. Bosanquet has it, to count “without having anything in particular to count,”[1] but when one rather tries to reflect, and then observes that the single feature about the number-system upon which all this conceivable complexity depends, is the simple and positive demand that is determined by the thought which conceives any order whatever. For order, as we shall soon more generally see, is comprehensible most of all in cases of self-representative systems of the present type. The numbers are simply a formally ordered collection of ideal objects. Whoever anywhere orders his own thoughts, either defines just such a self-representative system, or sets in order some empirically selected portion of a world that, in its totality, is such a system. And any system once self-representative, in this particular way, is infinitely self-representative. And if you will count its elements, you shall, then, always find that you can never finish the task.
Yet we are not yet done with showing, in this abstractly simple case of the numbers, what this type of self-representation implies. The numbers, namely, form a system not only self-representative in infinitely numerous ways, but also self-representative according to each of these ways, in a manner that can be doubly brought under our notice. Take, namely, the collection of series thus represented:—
- ↑ Logic, Vol. I, p. 175. In the Theory of Numbers, the properties of the whole numbers are indeed interesting for themselves “without anything in particular to count,” just because they form an ordered series, whose properties are the properties of all ordered systems.
