Introduction to Mathematical Philosophy

WE have now carried our analysis of the series of natural numbers to the point where we have obtained logical definitions of the members of this series, of the whole class of its members, and of the relation of a number to its immediate successor. We must now consider the serial character of the natural numbers in the order 0, 1, 2, 3,.... We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series" in logical terms. The notion of order is one which has enormous importance in mathematics. Not only the integers, but also rational fractions and all real numbers have an order of magnitude, and this is essential to most of their mathematical properties. The order of points on a line is essential to geometry; so is the slightly more complicated order of lines through a point in a plane, or of planes through a line. Dimensions, in geometry, are a development of order. The conception of a limit, which underlies all higher mathematics, is a serial conception. There are parts of mathematics which do not depend upon the notion of order, but they are very few in comparison with the parts in which this notion is involved. In seeking a definition of order, the first thing to realise is that no set of terms has just one order to the exclusion of others. A set of terms has all the orders of which it is capable. Sometimes one order is so much more familiar and natural to our [Pg 29] thoughts that we are inclined to regard it as the order of that set of terms; but this is a mistake. The natural numbers—or the "inductive" numbers, as we shall also call them—occur to us most readily in order of magnitude; but they are capable of an infinite number of other arrangements. We might, for example, consider first all the odd numbers and then all the even numbers; or first 1, then all the even numbers, then all the odd multiples of 3, then all the multiples of 5 but not of 2 or 3, then all the multiples of 7 but not of 2 or 3 or 5, and so on through the whole series of primes. When we say that we "arrange" the numbers in these various orders, that is an inaccurate expression: what we really do is to turn our attention to certain relations between the natural numbers, which themselves generate such-and-such an arrangement. We can no more "arrange" the natural numbers than we can the starry heavens; but just as we may notice among the fixed stars either their order of brightness or their distribution in the sky, so there are various relations among numbers which may be observed, and which give rise to various different orders among numbers, all equally legitimate. And what is true of numbers is equally true of points on a line or of the moments of time: one order is more familiar, but others are equally valid. We might, for example, take first, on a line, all the points that have integral co-ordinates, then all those that have non-integral rational co-ordinates, then all those that have algebraic non-rational co-ordinates, and so on, through any set of complications we please. The resulting order will be one which the points of the line certainly have, whether we choose to notice it or not; the only thing that is arbitrary about the various orders of a set of terms is our attention, for the terms themselves have always all the orders of which they are capable. One important result of this consideration is that we must not look for the definition of order in the nature of the set of terms to be ordered, since one set of terms has many orders. The order lies, not in the class of terms, but in a relation among [Pg 30] the members of the class, in respect of which some appear as earlier and some as later. The fact that a class may have many orders is due to the fact that there can be many relations holding among the members of one single class. What properties must a relation have in order to give rise to an order? The essential characteristics of a relation which is to give rise to order may be discovered by considering that in respect of such a relation we must be able to say, of any two terms in the class which is to be ordered, that one "precedes" and the other "follows." Now, in order that we may be able to use these words in the way in which we should naturally understand them, we require that the ordering relation should have three properties:— (1) If precedes , must not also precede . This is an obvious characteristic of the kind of relations that lead to series. If is less than , is not also less than . If is earlier in time than , is not also earlier than . If is to the left of , is not to the left of . On the other hand, relations which do not give rise to series often do not have this property. If is a brother or sister of , is a brother or sister of . If is of the same height as , is of the same height as . If is of a different height from , is of a different height from . In all these cases, when the relation holds between and , it also holds between and . But with serial relations such a thing cannot happen. A relation having this first property is called asymmetrical. (2) If precedes and precedes , must precede . This may be illustrated by the same instances as before: less, earlier, left of. But as instances of relations which do not have this property only two of our previous three instances will serve. If is brother or sister of , and of , may not be brother or sister of , since and may be the same person. The same applies to difference of height, but not to sameness of height, which has our second property but not our first. The relation "father," on the other hand, has our first property but not [Pg 31] our second. A relation having our second property is called transitive. (3) Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. For example, of any two integers, or fractions, or real numbers, one is smaller and the other greater; but of any two complex numbers this is not true. Of any two moments in time, one must be earlier than the other; but of events, which may be simultaneous, this cannot be said. Of two points on a line, one must be to the left of the other. A relation having this third property is called connected. When a relation possesses these three properties, it is of the sort to give rise to an order among the terms between which it holds; and wherever an order exists, some relation having these three properties can be found generating it. Before illustrating this thesis, we will introduce a few definitions. (1) A relation is said to be an aliorelative,[10] or to be contained in or imply diversity, if no term has this relation to itself. Thus, for example, "greater," "different in size," "brother," "husband," "father" are aliorelatives; but "equal," "born of the same parents," "dear friend" are not. [10]This term is due to C. S. Peirce. (2) The square of a relation is that relation which holds between two terms and when there is an intermediate term such that the given relation holds between and and between and . Thus "paternal grandfather" is the square of "father," "greater by 2" is the square of "greater by 1," and so on. (3) The domain of a relation consists of all those terms that have the relation to something or other, and the converse domain consists of all those terms to which something or other has the relation. These words have been already defined, but are recalled here for the sake of the following definition:— (4) The field of a relation consists of its domain and converse domain together. [Pg 32] (5) One relation is said to contain or be implied by another if it holds whenever the other holds. It will be seen that an asymmetrical relation is the same thing as a relation whose square is an aliorelative. It often happens that a relation is an aliorelative without being asymmetrical, though an asymmetrical relation is always an aliorelative. For example, "spouse" is an aliorelative, but is symmetrical, since if is the spouse of , is the spouse of . But among transitive relations, all aliorelatives are asymmetrical as well as vice versa. From the definitions it will be seen that a transitive relation is one which is implied by its square, or, as we also say, "contains" its square. Thus "ancestor" is transitive, because an ancestor's ancestor is an ancestor; but "father" is not transitive, because a father's father is not a father. A transitive aliorelative is one which contains its square and is contained in diversity; or, what comes to the same thing, one whose square implies both it and diversity—because, when a relation is transitive, asymmetry is equivalent to being an aliorelative. A relation is connected when, given any two different terms of its field, the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical). It will be seen that the relation "ancestor," for example, is an aliorelative and transitive, but not connected; it is because it is not connected that it does not suffice to arrange the human race in a series. The relation "less than or equal to," among numbers, is transitive and connected, but not asymmetrical or an aliorelative. The relation "greater or less" among numbers is an aliorelative and is connected, but is not transitive, for if is greater or less than , and is greater or less than , it may happen that and are the same number. Thus the three properties of being (1) an aliorelative, (2) transitive, [Pg 33] and (3) connected, are mutually independent, since a relation may have any two without having the third. We now lay down the following definition:— A relation is serial when it is an aliorelative, transitive, and connected; or, what is equivalent, when it is asymmetrical, transitive, and connected. A series is the same thing as a serial relation. It might have been thought that a series should be the field of a serial relation, not the serial relation itself. But this would be an error. For example, are six different series which all have the same field. If the field were the series, there could only be one series with a given field. What distinguishes the above six series is simply the different ordering relations in the six cases. Given the ordering relation, the field and the order are both determinate. Thus the ordering relation may be taken to be the series, but the field cannot be so taken. Given any serial relation, say , we shall say that, in respect of this relation, "precedes" if has the relation to , which we shall write "" for short. The three characteristics which must have in order to be serial are: (1) We must never have , i.e. no term must precede itself. (2) must imply , i.e. if precedes and precedes , must precede . (3) If and are two different terms in the field of , we shall have or , i.e. one of the two must precede the other. The reader can easily convince himself that, where these three properties are found in an ordering relation, the characteristics we expect of series will also be found, and vice versa. We are therefore justified in taking the above as a definition of order [Pg 34] or series. And it will be observed that the definition is effected in purely