The Principles of Mathematics (1903)

§ 147

There is no doubt that the notion of half a league, or half a day, is a legitimate notion. It is therefore necessary to find some sense for fractions in which they do not essentially depend upon number. For, if a given period of twenty-four hours is to be divided into two continuous portions, each of which is to be half of the whole period, there is only one way of doing this: but Cantor has shown that every possible way of dividing the period into two continuous portions divides it into two portions having the same number of terms. There must be, therefore, some other respect in which two periods of twelve hours are equal, while a period of one hour and another of twenty-three hours are unequal. I shall have more to say upon this subject in Part III: for the present I will point out that what we want is of the nature of a magnitude, and that it must be essentially a property of ordered wholes. I shall call this property magnitude of divisibility. To say now that A is one-half of B means: B is a whole, and if B be divided into two similar parts which have both the same magnitude of divisibility as each other, then A has the same magnitude of divisibility as each of these parts. We may interpret the fraction ½ somewhat more simply, by regarding it as a relation (analogous to ratio so long as finite wholes are concerned) between two magnitudes of divisibility. Thus finite integral fractions (such as n/1) will measure the relation of the divisibility of an aggregate of n terms to the divisibility of a single term; the converse relation will be 1/n. Thus here again we have a new class of entities which is in danger of being confused with finite cardinal integers, though in reality quite distinct. Fractions, as now interpreted, have the advantage (upon which all metrical geometry depends) that they introduce a discrimination of greater and smaller among infinite aggregates having the same number of terms. We shall see more and more, as the logical inadequacy of the usual accounts of measurement is brought to light, how absolutely essential the notion of magnitude of divisibility really is. Fractions, then, in the sense in which they may express relations of infinite aggregates--and this is the sense which they usually have in daily life--are really of the nature of relations between magnitudes of divisibility; and magnitudes of divisibility are only measured by number of parts where the aggregates concerned are finite. It may also be observed (though this remark is anticipatory) that, whereas ratios, as above defined, are essentially rational, fractions, in the sense here given to them, are also capable of irrational values. But the development of this topic must be left for Part V.(§ 147 ¶ 1)