the moon. But “one” is not a property of the moon itself, which may equally well be regarded as many molecules: it is a property of the general term “earth’s satellite.” Similarly, 0 is a property of the general term “satellite of Venus,” because Venus has no satellite. Here at last we have an intelligible theory of the number 0. This was impossible if numbers applied to physical objects, because obviously no physical object could have the number 0. Thus, in seeking our definition of number we have arrived so far at the result that numbers are properties of general terms or general descriptions, not of physical things or of mental occurrences.
Instead of speaking of a general term, such as “man,” as the subject of which a number can be asserted, we may, without making any serious change, take the subject as the class or collection of objects—i.e. “mankind” in the above instance—to which the general term in question is applicable. Two general terms, such as “man” and “featherless biped,” which are applicable to the same collection of objects, will obviously have the same number of instances; thus the number depends upon the class, not upon the selection of this or that general term to describe it, provided several general terms can be found to describe the same class. But some general term is always necessary in order to describe a class. Even when the terms are enumerated, as “this and that and the other,” the collection is constituted by the general property of being either this, or that, or the other, and only so acquires the unity which enables us to speak of it as one collection. And in the case of an infinite class, enumeration is impossible, so that description by a general characteristic common and peculiar to the members of the class is the only possible description. Here, as we see, the theory of number to which Frege was led by purely logical
