of using certain familiar and easily observed collections (our fingers, for instance) as means for defining the nature of less familiar and more complex collections. The number-names, derived from these elementary processes of finger-counting, come to our aid in the further development of our thought about numbers. The decadic system makes possible, through a simple system of notation, the expression of numbers of any magnitude. And so the number-concept in its generality is born.
This usual summary view of the origin of the numbers has its obvious measure of historical and psychological truth. It leaves wholly unanswered, however, the most interesting problems as to the nature of the number-concept. For numbers have two characters. They are cardinal numbers, in so far as they give us an idea of how many constituents a given collection of objects contains. But they have also an ordinal character; for by using numbers, as the makers of watches, and bicycles, or as the printers of a series of banknotes, or of tickets, use them, we can give to any one object its place in a determinate series, as the first, the tenth, or the ten thousandth member of that series. Such ordinal use of numbers is a familiar device for identifying objects that, for any reason, we wish to view as individuals. Now, a very little consideration shows that the ordinal value of the numbers is of very fundamental importance for their use in giving us a notion of the cardinal numbers of multitudes of objects. For when we count objects by using either the fingers or the number-names, we always employ an already familiar ordered series of objects as the basis of our work. We put the members of this series in a “one-to-one” relation to the members of the collection of objects which we wish to count. We deal out our numbers, so to speak, in serial order, to the various objects to be counted. We thereby label the various objects as they are numbered, just as the makers of the banknotes stamp an ordinal number on each note of a given issue. Only when this process is completed do we recognize the cardinal number which tells us how many objects there are in the collection of the objects counted.
