Introduction to Mathematical Philosophy

the field of . Thus the serial relation which is to be defined as the sum of and is not " or " simply, but " or or the relation of any member of the field of to any member of the field of ." Assuming that and do not overlap, this relation is serial, but " or " is not serial, being not connected, since it does not hold between a member of the field of and a member of the field of . Thus the sum of and , as above defined, is what we need in order [Pg 57] to define the sum of two relation-numbers. Similar modifications are needed for products and powers. The resulting arithmetic does not obey the commutative law: the sum or product which they are taken. But it obeys the associative law, one form of the distributive law, and two of the formal laws for powers, not only as applied to serial numbers, but as applied to relation-numbers generally. Relation-arithmetic, in fact, though recent, is a thoroughly respectable branch of mathematics. It must not be supposed, merely because series afford the most obvious application of the idea of likeness, that there are no other applications that are important. We have already mentioned maps, and we might extend our thoughts from this illustration to geometry generally. If the system of relations by which a geometry is applied to a certain set of terms can be brought fully into relations of likeness with a system applying to another set of terms, then the geometry of the two sets is indistinguishable from the mathematical point of view, i.e. all the propositions are the same, except for the fact that they are applied in one case to one set of terms and in the other to another. We may illustrate this by the relations of the sort that may be called "between," which we considered in Chapter IV. We there saw that, provided a three-term relation has certain formal logical properties, it will give rise to series, and may be called a "between-relation." Given any two points, we can use the between-relation to define the straight line determined by those two points; it consists of and together with all points , such that the between-relation holds between the three points , , in some order or other. It has been shown by O. Veblen that we may regard our whole space as the field of a three-term between-relation, and define our geometry by the properties we assign to our between-relation.[13] Now likeness is just as easily [Pg 58] definable between three-term relations as between two-term relations. If and ' are two between-relations, so that "" means " is between and with respect to ," we shall call a correlator of and ' if it has the field of ' for its converse domain, and is such that the relation holds between three terms when ' holds between their -correlates, and only then. And we shall say that is like ' when there is at least one correlator of with '. The reader can easily convince himself that, if is like ' in this sense, there can be no difference between the geometry generated by and that generated by '. [13]This does not apply to elliptic space, but only to spaces in which the straight line is an open series. Modern Mathematics, edited by J. W. A. Young, pp. 3-51 (monograph by O. Veblen on "The Foundations of Geometry"). It follows from this that the mathematician need not concern himself with the particular being or intrinsic nature of his points, lines, and planes, even when he is speculating as an applied mathematician. We may say that there is empirical evidence of the approximate truth of such parts of geometry as are not matters of definition. But there is no empirical evidence as to what a "point" is to be. It has to be something that as nearly as possible satisfies our axioms, but it does not have to be "very small" or "without parts." Whether or not it is those things is a matter of indifference, so long as it satisfies the axioms. If we can, out of empirical material, construct a logical structure, no matter how complicated, which will satisfy our geometrical axioms, that structure may legitimately be called a "point." We must not say that there is nothing else that could legitimately be called a "point"; we must only say: "This object we have constructed is sufficient for the geometer; it may be one of many objects, any of which would be sufficient, but that is no concern of ours, since this object is enough to vindicate the empirical truth of geometry, in so far as geometry is not a matter of definition." This is only an illustration of the general principle that what matters in mathematics, and to a very great extent in physical science, is not the intrinsic nature of our terms, but the logical nature of their interrelations. We may say, of two similar relations, that they have the same [Pg 59] "structure." For mathematical purposes (though not for those of pure philosophy) the only thing of importance about a relation is the cases in which it holds, not its intrinsic nature. Just as a class may be defined by various different but co-extensive concepts—e.g. "man" and "featherless biped,"—so two relations which are conceptually different may hold in the same set of instances. An "instance" in which a relation holds is to be conceived as a couple of terms, with an order, so that one of the terms comes first and the other second; the couple is to be, of course, such that its first term has the relation in question to its second. Take (say) the relation "father": we can define what we may call the "extension" of this relation as the class of all ordered couples which are such that is the father of . From the mathematical point of view, the only thing of importance about the relation "father" is that it defines this set of ordered couples. Speaking generally, we say: The "extension" of a relation is the class of those ordered couples which are such that has the relation in question to . We can now go a step further in the process of abstraction, and consider what we mean by "structure." Given any relation, we can, if it is a sufficiently simple one, construct a map of it. For the sake of definiteness, let us take a relation of which the extension is the following couples: , , , , , , , where , , , , are five terms, no matter what. We may make a "map" of this relation by taking five points on a plane and connecting them by arrows, as in the accompanying figure. What is revealed by the map is what we call the "structure" of the relation. It is clear that the "structure" of the relation does not depend upon the particular terms that make up the field of the relation. The field may be changed without changing the structure, and the structure may be changed without changing the field—for [Pg 60] example, if we were to add the couple in the above illustration we should alter the structure but not the field. Two relations have the same "structure," we shall say, when the same map will do for both—or, what comes to the same thing, when either can be a map for the other (since every relation can be its own map). And that, as a moment's reflection shows, is the very same thing as what we have called "likeness." That is to say, two relations have the same structure when they have likeness, i.e. when they have the same relation-number. Thus what we defined as the "relation-number" is the very same thing as is obscurely intended by the word "structure"—a word which, important as it is, is never (so far as we know) defined in precise terms by those who use it. There has been a great deal of speculation in traditional philosophy which might have been avoided if the importance of structure, and the difficulty of getting behind it, had been realised. For example, it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must have differences inter se corresponding with the differences in the phenomena to which they give rise. Where such hypotheses are made, it is generally supposed that we can know very little about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world, and allowing us to infer from phenomena the truth of all propositions that can be stated in abstract terms and are known to be true of phenomena. If the phenomenal world has three dimensions, so must the world behind phenomena; if the phenomenal world is Euclidean, so must the other be; and so on. In short, every proposition having a communicable significance must be true of both worlds or of neither: the only difference must lie in just that essence of individuality which always eludes words and baffles description, but which, for that very reason, is irrelevant to science. Now the only purpose that philosophers [Pg 61] have in view in condemning phenomena is in order to persuade themselves and others that the real world is very different from the world of appearance. We can all sympathise with their wish to prove such a very desirable proposition, but we cannot congratulate them on their success. It is true that many of them do not assert objective counterparts to phenomena, and these escape from the above argument. Those who do assert counterparts are, as a rule, very reticent on the subject, probably because they feel instinctively that, if pursued, it will bring about too much of a rapprochement between the real and the phenomenal world. If they were to pursue the topic, they could hardly avoid the conclusions which we have been suggesting. In such ways, as well as in many others, the notion of structure or relation-number is important. [Pg 62]