NUMBER AS DETERMINING FORM OF GROUP 33
classical riddle : How many grains of wheat make a heap ? For since one, two, three, four grains do not do it, but a thousand certainly, there must accordingly be between these two numbers a boundary, at which the addition of a single grain makes those previously present into a " heap." If one, however, makes the trial of further enumeration, it appears that no one can announce discovery of this boundary. The logical ground of these diffi- culties is found in the fact that a quantitative series is given which, on account of the relative insignificance of each single element, seems to be a continuous and uniformly ascending series, and that this from a certain point on must permit the application of a qualitatively new idea, sharply set off from the idea previously applied. This is obviously a contradictory demand : by virtue of its very idea, the continuous cannot justify purely of itself a sudden break and change. The sociological difficulty has now a further complication aside from that in the ancient sophistry, for by the "heap" of grains we understand either a piling-up, and then one is logically justified in this use of terms so soon as only one layer appears above the lowest layer ; or, only a quantity is designated by the term. In this case it is quite unjustifiable to demand of an idea like "heap," which in its very essence is quite variable and undetermined, that it should lend itself to application to perfectly defined and unequivocally bounded realities. 1
In those sociological cases, however, specifically new aggre- gate phenomena appear, when quantity increases, which are not present pro rata in the case of smaller numbers. A political party has qualitatively another significance from that of a small clique. A few curious persons standing together betray different traits from those of a mob (Auflau/}, etc. The indeterminate- ness which attaches to these ideas from the impossibility of fixing numerically the corresponding quantities may perhaps be
'Still more evident is this mistake in the negative direction, in the case of the question: How many hairs must one lose before one may be called bald? If we take this latter idea seriously, it applies only to him who has no hair at all. If we apply it, however, to any case in which there is possession of hair, we thereby surrender the unequivocal severity of the idea, and we may not wonder that we possess no objectively precise criterion of its application, since we have put such application out of our power.
