And we recognize this result of counting by the simple device of giving to the whole collection counted a cardinal number corresponding to the last member of the ordinal number-series that we have thus dealt out. If, for instance, the last object labelled is the tenth in the series of objects set in order by the ordinal process of labelling, then the counted collection is said to contain ten objects.
Unless the numbers were, then, in our minds, already somehow a well-ordered series, they would help us no whit in counting objects. Nor does counting consist in the mere collection of acts of synthesis by which we each time add one more, in mind, to the collection of objects so far counted. For these acts of synthesis, however carefully performed, soon give us, if left to themselves, only the confused sense, “There is another object, — and another, — and another.” In such cases we soon “lose count.” We can “keep tally” of our objects only if we combine the successive series of acts of observing another, and yet another, object, in our collection of objects with the constant use of the already ordered series of number-names, whose value depends upon the fact that one of them comes first, another second, etc., and that we well know what this order means.
The ordinal character of the number-series is therefore its most important and fundamental character. But upon what mental process does the conception of any well-ordered series depend? The account of the origin of the number-series by the mere use of fingers or of names, does not yet tell us what we mean by any ordered series at all.
To this question, whose central significance, for the whole understanding of the number-concept, all the later discussions and the modern text-books recognize, various answers have been given.[1] The order of a series of objects, presented or
- ↑ Couturat, in the work cited, gives an admirable summary of the present phases of the discussion; only that he fails, I think, to appreciate the importance and originality of Dedekind’s method of deducing the ordinal concept. The views of Helmholtz and Kronecker are discussed with especial care by Couturat. Veronese, in the introduction to his Principles of Geometry (known to me in the German translation, Grundzüge der Geometrie, übers v. Schepp, Leipzig, 1894) gives a very elaborate development of the number-concept upon the basis of the view that the order of a series of conceived objects is an ultimate fact or absolute datum for thought (op. cit., § 3, 14-28, 46-50). Amongst the recent text-books, Fine’s Number-System of Arithmetic and Algebra holds an important place. See also the opening chapter of Harkness and Morley’s Introduction to the Theory of Analytic Functions.
