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8.2 Extensiveness, Space and Time In the previous section, places in spacetime were defined as possibilities of a certain kind. The question to be asked next is how these places are `brought together' to form one world with continuous space and time. We want to know how the particular things in the world can be at different places, yet stand in definite relations to one another. We must also consider the question of the ontological status of space and time: whether they are absolute, relative, and whether space and time exist separately from a joint `space-time' which may be formulated. In the previous chapter, it was remarked that not all places could be identical, as then it would be impossible to have any separate existences or variety in the world. Neither should we go to the other extreme, and have all places distinct but with no real relation to each other.     Both Leibniz and Bergson tended to this extreme when they tried to deny the reality of extension.   We may observe, very generally, that a particular thing will tend to act on (and be changed by acts of) other particulars which are `nearer', rather than those which are `further away'. To explain this fact, some kind of extensiveness is necessary. There must be some kind of `real extensiveness' which is related to how particulars can interact with each other (and not merely some kind of `phenomenal space' that only describes how things appear to us).     Some philosophers, such as the pre-critical Kant8.2, or Bergson perhaps, have tried to derive extension from these mutual interactions. In Kant's doctrine of acting monads, extensive relations were brought into being by the acting of the monads on each other, because each monad `fills an assignable space' by its intrinsic activity.   The trouble with this approach, as Leclerc8.3 points out, is that the very fact of the mutual interacting of distinct substances presupposes that the participants are in some sense `other' to each other. Thus extension must be necessarily prior to any mutual interacting.   Whitehead points out that extension has the double role of allowing for the externality of distinct places while, at the same time, bringing them together by means of geometrical and chronological co-relations: ``extensive connection with its various characteristics is the fundamental organic relationship whereby the physical world is properly described as a community''8.4.     Extensiveness and the Extensive Continuum Extensiveness is therefore defined, in the present philosophy of nature, as a fundamental real relation between places. It is a relation between places that holds independently of whether or not the related places are filled. We could think of it specifying absolutely the metric distance between any two points in space-time. By means of extensiveness we intend to create a `dimensional order' for all places in space and time. Furthermore, since places are kinds of possibilities, and since (as we saw in chapter 6) possibilities can form a continuum, we have that extensiveness can naturally give rise to a continuous order of places. I am not arguing that extensiveness is necessarily continuous, only that continuity is not incompatible with the basic notions we have put together. We will discuss later how all places can be ordered according to their mutual extensive relations.       When this is done, they then form what Whitehead has termed an `extensive continuum'. This continuum is the manifold of all places apparent when they are juxtaposed with respect to their relations of extensiveness. The extensive continuum is the most general form of order in the world, as it is prior to the contingent things which actually happen. It is not however to be conceived as actuality, or as any kind of definite `container'. As the full definition of the extensive continuum is that it is an `ordered manifold of places', it is therefore the `continuous order of possibilities for actuality'. It would only be if we erroneously conflate the concepts of possibilities a we would mistake the extensive continuum for an actual entity. Such conflation may be permissible within mathematics, but has disastrous consequences elsewhere. In understanding the extensive continuum in terms of our usual space and time, we should remember the places, as possibilities for actuality, are `wheres' and `whens'. This means, as pointed out at the start of the chapter, that the extensive continuum has more of a spatiotemporal nature, than of a purely spatial or purely temporal nature. We can refer to the extensive continuum as `spacetime', but only provided we remember that such a spacetime is not the actual existent of many interpretations of Einstein's relativity theories. Rather, it is the extensive manifold of all possibilities for actuality. On this basis we can admit the `objective' nature of the four-dimensional spacetime of relativity theory, but not as an actual or fully-determinate thing.     Metrics Extensiveness has been defined as a relation between places, but we have still to see how it may be more precisely formulated. We might expect, because extensiveness can be continuous, that places can be related a `continuous relation'. Mathematically this could be represented as a function s(p,q), as a continuous function of pairs of places p, q. Our general philosophy of nature cannot determine the exact form of this function, for to know its characteristics would be to know the number of dimensions of space and time in spacetime, and to know the precise metric of the spacetime manifold.   I see no way of deriving these a priori. The 3+1 dimensional nature of spacetime, for example, seems to be a `brute fact'. Both the Newtonian and Einsteinian accounts of spacetime should however be permitted by our general philosophical arguments. In the following, I will hence be discussing a plausible formulation of extensiveness which can accommodate both Newtonian and relativistic metrics. It is supposed that the extensiveness function is given as the function s(p,q) for all pairs of places p & q, such that 1. s2(p,q) is real-valued, 2. s(p,p)=0 for all places p, 3. s(p,q) = - s(q,p) for all p, q, and 4. $s(p,q) \neq 0 \Rightarrow p \neq q$(though $s(p,q) = 0 \mbox{ need not imply } p = q$) These are just the preliminary relations which are supposed to be satisfied. The more useful relations `p precedes q' and `p alternates with q' are now defined by $$\equiv s(p,q) \geq 0 ~~ \wedge ~~ \lnot ( p=q \vee plc(p,q) )$$ p  precedes  q (8.1) $$\equiv \lnot (p ~precedes~ q) ~~ \wedge ~~ \lnot (q ~precedes~ p)$$ p  alternates with  q (8.2) $$\equiv s(p,q) \mbox{ non-zero imaginary } \vee p=q$$ (8.3) $$\mbox{ where } plc(p,q) \equiv s(p,q) = 0 ~~ \wedge ~~ ( \exists r) s(p,r) > s(q,r) > 0$$ (8.4) The relation plc(p,q) is a special condition necessary with relativistic metrics to mark the case of p being on what is called the `past light cone' of q An inverse relation succeeds can also be defined: $$p ~succeeds~ q \equiv s(q,p) \geq 0 ~~ \wedge ~~ \lnot ( p=q \vee plc(q,p) )$$   In relativistic physics, precedence is known as the `time-like' or `topological past' relation, and alternation as the `space-like' or `topological present' relation between two places. This correspondence obtains because the form of the relation s(p,q) that is used above is based on the characteristics of a differential metric that could be either the relativistic ds 2 = c2 dt2 - dx2 - dy2 - dz2 , or that developed from Newtonian space and time: $$ds = \left\{ \begin{array}{ll} i (dx^{2} + dy^{2} + dz^{2}) ... ...=0$\space } \\ dt & \mbox{ otherwise. } \end{array} \right.$$ In our case, however, the extensiveness s(p,q) is definite for pairs of places even if the particular coordinate allocation of $( \vec{x} ,t) = (x,y,z,t)$ is dependent on the choice of some observer.     Momentum Space? From the mathematical point of view, an function $f(\vec{x})$ over the coordinates $\vec{x} \equiv (x,y,z)$ can be equivalently expressed as a function $\hat{f}$ over momentum space $\vec{k} \equiv (k_x ,k_y ,k_z )$ using the Fourier transformation $$\hat{f} (\vec{k}) = \int f (\vec{x}) e ^ {-i \vec{x} \cdot \vec{k} } d \vec{x} .$$   All the information present in $f(\vec{x})$ is also present in $\hat{f} (\vec{k}).$ Schrödinger's equation (among others) can be expressed equally as well in momentum space as in ordinary space, so it is not clear in quantum mechanics which space is the `more real'. In the philosophy of nature, the question of which space is the `real' space can be decided by means of the argument used above to introduce space in the first instance. The mutual interaction of distinct substances presupposes that they are in some sense `other' to each other, and hence there must be a real extension or a real space which is necessarily prior to any mutual interacting. A space is therefore real if that kind of spatial proximity is the prerequisite for interactions. On the basis of this criterion, ordinary space is real compared with momentum space. The `process logic' can be applied equally well to both spaces, but interactions tend to be local in ordinary (rather than momentum) space. Thus proximity in momentum space is not of itself a necessary condition for interactions occurring, and hence momentum space is not a `real space' for processes and propensities.

    

Author: I.J. Thompson (except as stated)

Email: IJT@generativescience.org