Introduction to Mathematical Philosophy

logical terms. Although a transitive asymmetrical connected relation always exists wherever there is a series, it is not always the relation which would most naturally be regarded as generating the series. The natural-number series may serve as an illustration. The relation we assumed in considering the natural numbers was the relation of immediate succession, i.e. the relation between consecutive integers. This relation is asymmetrical, but not transitive or connected. We can, however, derive from it, by the method of mathematical induction, the "ancestral" relation which we considered in the preceding chapter. This relation will be the same as "less than or equal to" among inductive integers. For purposes of generating the series of natural numbers, we want the relation "less than," excluding "equal to." This is the relation of to when is an ancestor of but not identical with , or (what comes to the same thing) when the successor of is an ancestor of in the sense in which a number is its own ancestor. That is to say, we shall lay down the following definition:— An inductive number is said to be less than another number when possesses every hereditary property possessed by the successor of . It is easy to see, and not difficult to prove, that the relation "less than," so defined, is asymmetrical, transitive, and connected, and has the inductive numbers for its field. Thus by means of this relation the inductive numbers acquire an order in the sense in which we defined the term "order," and this order is the so-called "natural" order, or order of magnitude. The generation of series by means of relations more or less resembling that of to is very common. The series of the Kings of England, for example, is generated by relations of each to his successor. This is probably the easiest way, where it is applicable, of conceiving the generation of a series. In this method we pass on from each term to the next, as long as there [Pg 35] is a next, or back to the one before, as long as there is one before. This method always requires the generalised form of mathematical induction in order to enable us to define "earlier" and "later" in a series so generated. On the analogy of "proper fractions," let us give the name "proper posterity of with respect to " to the class of those terms that belong to the -posterity of some term to which has the relation , in the sense which we gave before to "posterity," which includes a term in its own posterity. Reverting to the fundamental definitions, we find that the "proper posterity" may be defined as follows:— The "proper posterity" of with respect to consists of all terms that possess every -hereditary property possessed by every term to which has the relation . It is to be observed that this definition has to be so framed as to be applicable not only when there is only one term to which has the relation , but also in cases (as e.g. that of father and child) where there may be many terms to which has the relation . We define further: A term is a "proper ancestor" of with respect to if belongs to the proper posterity of with respect to . We shall speak for short of "-posterity" and "-ancestors" when these terms seem more convenient. Reverting now to the generation of series by the relation between consecutive terms, we see that, if this method is to be possible, the relation "proper -ancestor" must be an aliorelative, transitive, and connected. Under what circumstances will this occur? It will always be transitive: no matter what sort of relation may be, "-ancestor" and "proper -ancestor" are always both transitive. But it is only under certain circumstances that it will be an aliorelative or connected. Consider, for example, the relation to one's left-hand neighbour at a round dinner-table at which there are twelve people. If we call this relation , the proper -posterity of a person consists of all who can be reached by going round the table from right to left. This includes everybody at the table, including the person himself, since [Pg 36] twelve steps bring us back to our starting-point. Thus in such a case, though the relation "proper -ancestor" is connected, and though itself is an aliorelative, we do not get a series because "proper -ancestor" is not an aliorelative. It is for this reason that we cannot say that one person comes before another with respect to the relation "right of" or to its ancestral derivative. The above was an instance in which the ancestral relation was connected but not contained in diversity. An instance where it is contained in diversity but not connected is derived from the ordinary sense of the word "ancestor." If is a proper ancestor of , and cannot be the same person; but it is not true that of any two persons one must be an ancestor of the other. The question of the circumstances under which series can be generated by ancestral relations derived from relations of consecutiveness is often important. Some of the most important cases are the following: Let be a many-one relation, and let us confine our attention to the posterity of some term . When so confined, the relation "proper -ancestor" must be connected; therefore all that remains to ensure its being serial is that it shall be contained in diversity. This is a generalisation of the instance of the dinner-table. Another generalisation consists in taking to be a one-one relation, and including the ancestry of as well as the posterity. Here again, the one condition required to secure the generation of a series is that the relation "proper -ancestor" shall be contained in diversity. The generation of order by means of relations of consecutiveness, though important in its own sphere, is less general than the method which uses a transitive relation to define the order. It often happens in a series that there are an infinite number of intermediate terms between any two that may be selected, however near together these may be. Take, for instance, fractions in order of magnitude. Between any two fractions there are others—for example, the arithmetic mean of the two. Consequently there is no such thing as a pair of consecutive fractions. If we depended [Pg 37] upon consecutiveness for defining order, we should not be able to define the order of magnitude among fractions. But in fact the relations of greater and less among fractions do not demand generation from relations of consecutiveness, and the relations of greater and less among fractions have the three characteristics which we need for defining serial relations. In all such cases the order must be defined by means of a transitive relation, since only such a relation is able to leap over an infinite number of intermediate terms. The method of consecutiveness, like that of counting for discovering the number of a collection, is appropriate to the finite; it may even be extended to certain infinite series, namely, those in which, though the total number of terms is infinite, the number of terms between any two is always finite; but it must not be regarded as general. Not only so, but care must be taken to eradicate from the imagination all habits of thought resulting from supposing it general. If this is not done, series in which there are no consecutive terms will remain difficult and puzzling. And such series are of vital importance for the understanding of continuity, space, time, and motion. There are many ways in which series may be generated, but all depend upon the finding or construction of an asymmetrical transitive connected relation. Some of these ways have considerable importance. We may take as illustrative the generation of series by means of a three-term relation which we may call "between." This method is very useful in geometry, and may serve as an introduction to relations having more than two terms; it is best introduced in connection with elementary geometry. Given any three points on a straight line in ordinary space, there must be one of them which is between the other two. This will not be the case with the points on a circle or any other closed curve, because, given any three points on a circle, we can travel from any one to any other without passing through the third. In fact, the notion "between" is characteristic of open series—or series in the strict sense—as opposed to what may be called [Pg 38] "cyclic" series, where, as with people at the dinner-table, a sufficient journey brings us back to our starting-point. This notion of "between" may be chosen as the fundamental notion of ordinary geometry; but for the present we will only consider its application to a single straight line and to the ordering of the points on a straight line.[11] Taking any two points , , the line consists of three parts (besides and themselves): [11]Cf. Rivista di Matematica, IV. pp. 55 ff.; Principles of Mathematics, p. 394 (§ 375). (1) Points between and . (2) Points such that is between and . (3) Points such that is between and . Thus the line can be defined in terms of the relation "between." In order that this relation "between" may arrange the points of the line in an order from left to right, we need certain assumptions, namely, the following:— (1) If anything is between and , and are not identical. (2) Anything between and is also between and . (3) Anything between and is not identical with (nor, consequently, with , in virtue of (2)). (4) If is between and , anything between and is also between and . (5) If is between and , and is between and , then is between and . (6) If and are between and , then either and are identical, or is between and , or is between and . (7) If is between and and also between and , then either and are identical, or is between and , or is between and . These seven properties are obviously verified in the case of points on a straight line in ordinary space. Any three-term relation which verifies them gives rise to series, as may be seen from the following definitions. For the sake of definiteness, let us assume [Pg 39] that is to the left of . Then the points of the line are (1) those between which and , lies—these we will call to the left of ; (2) itself; (3) those between and ; (4) itself; (5) those between which and lies —these we will call to the right of . We may now define generally that of two points , , on the line , we shall say that is "to the left of" in any of the following cases:— (1) When and are both to the left of , and is between and ; (2) When is to the left of , and is or or between and or to the right of ; (3) When is , and is between and or is or is to the right of ; (4) When and are both between and , and is between and ; (5) When is between and , and is or to the right of ; (6) When is and is to the right of ; (7) When and are both to the right of and is between and . It will be found that, from the seven properties which we have assigned to the relation "between," it can be deduced that the relation "to the left of," as above defined, is a serial relation as we defined that term. It is important to notice that nothing in the definitions or the argument depends upon our meaning by "between" the actual relation of that name which occurs in empirical space: any three-term relation having the above seven purely formal properties will serve the purpose of the argument equally well. Cyclic order, such as that of the points on a circle, cannot be generated by means of three-term relations of "between." We need a relation of four terms, which may be called "separation of couples." The point may be illustrated by considering a journey round the world. One may go from England to New Zealand by way of Suez or by way