10 CONTAINER SPACE AND RELATIONAL SPACE Leibniz's attack on Newton's conception of space and Einstein's special theory of relativity are the two important starting points for modern philosophical discussions of space. The theories of Leibniz and Einstein contradict Newton's ideas but in different ways. Newtonian space is a complex notion, it is possible to criticize it in some respects and defend it in others. In order to capture the category of space its different constituent aspects must be carefully distinguished. In post-Newtonian discussions they have all too often been run together in an unfortunate manner. Leibniz's criticism is mainly that Newtonian space is not relational. The specifically Einsteinian criticism has been that space has a relativistic metric. There is also a third kind of criticism, associated with Ernst Mach, which should be kept clearly apart from the other two. It claims that what is wrong with Newtonian space is that it is not relative. Newtonian space is neither relational, nor relative, nor relativistic. It is absolute, but 'absolute' means one thing when contrasted with 'relational', something else when contrasted with 'relative' and something else again when contrasted with 'relativistic'. I shall therefore avoid this ambiguous term as far as possible and instead use 'container space', 'non-relative space', and 'non-relativistic space' respectively. I shall argue that these distinctions are related as in Figure 10.1.1 The important thing about this figure is that it shows that the category 'container space' subsumes the different concepts of space which are or have been, proposed within physical science. I shall not take into account what might be called post-Einsteinian conceptions of space, but the arguments put forward below can, I 145 ONTOLOGICAL INVESTIGATIONS non-relativistic (Newton) non-relative relativistic non-relativistic (Mach) relative space relativistic (Einstein) relational (Leibniz) Figure 10.1 think, be developed to take into account such conceptions.2 My aim is only to stress the inevitability of a container conception of space. If the table has something to say to the physicist, it might be that a non-relative relativistic conception of space is a consistent one; Figure 10.1 also shows the real relationship between the theories of Einstein and Mach. I shall explicate the table from left to right. 10.1 CONTAINER SPACE The container view of space claims that actual space is in one sense literally a container. Space contains things in exactly the same sense in which an ordinary container can contain things. But space cannot have ordinary limits and so in this sense it is not a container. The similarity justifies the concept 'container space', the difference indicates that a container space is categorially different from a state of affairs. States of affairs exist in container space, but space does not exist in anything else. The difference between these categories becomes obvious when one realizes how they behave with regard to aggregation. Things (= states of affairs) can be aggregated into larger things; i.e. things of second order (cf. section 6.1). An aggregate does not exist in and through itself, but only in and through its parts. The parts (the things) are primary and the totality (the aggregate) secondary. With a container space matters 146 CONTAINER SPACE AND RELATIONAL SPACE are quite different. The subvolumes or regions of space exist only as parts of space.3 Here, the totality is primary and the parts secondary. It makes no sense to speak of aggregation of the subvolumes. By 'aggregation' I refer not only to a thought operation in the sense in which one considers a number of things as if they were a single thing, but to real aggregation as when things are actually placed in space in such a way that they take on the character of a single entity. Things may in principle be moved around in space without losing their identity, but the parts of space can not be so moved. That is why container space cannot be an aggregate. That space is a container space means also that it, as a category, might be instantiated without it being the case that the category of state of affairs is instantiated. In other words, there is nothing in the category itself which implies that space could not be empty. The category of state of affairs, however, presupposes the category of space. States of affairs can only exist in a space. At this point a reminder of the rarefied level of abstraction of an ontological system is in order. Nothing we have said so far is in conflict with a theory of physics implying that the container space which we live in is, as a matter of fact or physical theory, not empty. Relational space, on the other hand, cannot even in principle be empty. Container space is a particular, even if it is a special kind of particular. Particulars like instances of states of affairs and quality- instances are characterized by the fact that they only can be at one place at a time. Of course, the same cannot be said of space. Places only exist in space and so space cannot be in a place. Nevertheless, container space has to be a particular. Otherwise it could not contain other particulars. A mere universality cannot contain what is particular. If one takes a nominalist position, the opposition between particulars and universale is clearcut; universals do not exist and so what is particular cannot be universal. From a nominalist point of view a container space does not bring in universality at all. But on the view adopted in the present work, immanent realism, everything that exists has to have both a particularity aspect and a universality aspect. 'Universals are nothing without particulars. Particulars are nothing without universale', to quote Armstrong again (cf. page 11). This means that container space, too, has to have a kind of universality. 147 ONTOLOGICAL INVESTIGATIONS One might think that the universal aspect of space is nothing more than a possibility (or even actuality!) that there may be several numerically distinct container spaces. But this is wrong. There can be one and only one container space. My argument for this claim goes as follows. The concept of 'container space' entails a distinction between 'no-thing' (= 'void') and 'nothing'. If there is a container space there may be a void, but a void is a something not a nothing. Assume there are two container spaces. Then, either there is void between them or nothing between them. If there is a void, this void connects them into one single container space, and if there is nothing they are already connected and make up one space. We have a reductio ad absurdum argument to the effect that there can only exist one container space. This, it must be noted, does not at all imply that space cannot have many different qualities (e.g. geometrical qualities), even qualities which may vary over space and time. But how, it may be objected, can something which necessarily has only one instance have a universal aspect? This brings us back to the last chapter and the relations of existential dependence. One relation defined there was that of just existential dependence (D9.1 and D9.1') and another that of existential exclusion (D9.5). The relation of plural self-dependence was defined (D9.1.1) as a special kind of existential dependence. Ά is plurally self-dependent if and only if for every instance of A it is logically impossible for A to exist if another instance of A does not also exist.' Examples given were 'competitor' and 'sibling'. In the same way as D9.1.1 was obtained from D9.1 we can now create a new definition out of D9.5. We thereby obtain: D10.1: Л is plurally self-excluding if and only if for every instance of Л it is logically impossible for A to exist if another A also exists. A universal conforming to this definition is a 'singularity-universal' (cf. 'plurality-universal', page 139). The argument above shows that container space has a universal aspect, because that is what the argument is about. And the upshot of it may now be expressed by saying that container space is a singularity-universal. Like everything else really existing, container space is both particular and universal, albeit a very special kind of universal. But that is what should be expected. 148 CONTAINER SPACE AND RELATIONAL SPACE The concept of 'container space' implies both containment and necessary singularity, but also a third thing: independence. Container space may exist without anything else at all existing. 'Independence' is here meant exactly in the sense defined in D9.2 of the last chapter. Moreover, container space is in this theory of categories the only absolutely independent category. It is independent not only with regard to one of several kinds of universal, it is existentially independent of all universals. This fact together with the fact that it is a singularity-universal gives container space the unique role it has.4 10.2 RELATIONAL SPACE Relational space may be defined by denying that the attributes of containment, necessary singularity, and independence apply to space. Relational spaces are dependent upon other entities, either ordinary things or Leibnizian monads, and are some kind of relation between these entities. They are grounded in some universal aspect of the entities, and are therefore themselves ordinary universals (i.e. not 'singularity-universals'). Being grounded universals, these spaces cannot possibly be empty and cannot contain states of affairs in the sense that container space does this. In order to clarify the view that space is merely a system of relations between things, more has to be said than was needed to introduce the container view of space.