The Principles of Mathematics (1903)

§ 145

The only point which it is important, for our present purpose, to observe as regards the above definition of ratios is, that they are one-one relations between finite integers, which are with one exception asymmetrical, which are such that one and only one holds between any specified pair of finite integers, which are definable in terms of consecutiveness, and which themselves form a series having no first or last term and having a term, and therefore an infinite number of terms, between any two specified terms. From the fact that ratios are relations it results that no ratios are to be identified with integers: the ratio of 2 to 1, for example, is a wholly different entity from 2. When, therefore, we speak of the series of ratios as containing integers, the integers said t obe contained are not cardinal numbers, but relations which have a certain one-one correspondence with cardinal numbers. The same remark applies to positive and negative numbers. The nth power of the relations of consecutiveness is the positive number +n, which is plainly a wholly different concept from the cardinal number n. The confusion of entities with others to which they have some important one-one relation is an error to which mathematicians are very liable, and one which has produced the greatest havoc in the philosophy of mathematics. We shall find hereafter innumerable other instances of the same error, and it is well to realize, as early as possible, that any failure in subtlety of distinctions is sure, in this subject at least, to cause the most disastrous consequences.(§ 145 ¶ 1)

There is no difficulty in connecting the above theory of ratio with the usual theory derived from multiplication and division. But the usual theory does not show, as the present theory does, why the infinite integers do not have ratios strictly analogous to those of finite integers. The fact is, that ratio depends upon consecutiveness, and consecutiveness as defined above does not exist among infinite integers, since these are unchanged by the addition of 1.(§ 145 ¶ 2)

It should be observed that what is called addition of ratios demands a new set of relations among ratios, relations which may be called positive and negative ratios, just as certain relations among integers are called positive and negative integers. This subject, however, need not be further developed.(§ 145 ¶ 3)