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According to the analyses of the last chapter, the number concepts have presented themselves as an indeterminate sequence of concepts, apparently continuing even to infinity, whose clarity and easy mutual distinguishability seemed to be beyond question and to render superfluous further investigations for the purpose of precise reciprocal delimitation. One and one is sharply distinct from one and one and one, and this in turn from one, one, one and one, etc. One sees, however, that the ease of differentiation noticeably diminishes the further we advance in the sequence of numbers. Nineteen is much less easy to distinguish from twenty than nine is from ten, and the latter less easy than three from four. That this circumstance does no harm, that in spite of it we consider number determinations and number distinctions to be the most rigorous in the domain of our knowledge, has its basis in certain instrumentalities [Hilfsmitteln]. Through these we are in position — in cases where direct intuition is either totally denied or could easily err — to attain the goal of rigorous differentiation indirectly and to confine error to a very narrow range. The instrumentalities are those of enumerating and calculating, i.e., certain "mechanical' operations, as it were, the true basis for which lies in the elemental relations between the numbers. The psychological analysis of these relations of more and less is the goal which we now set before us.

DOI: 10.1007/978-94-010-0060-4_6

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