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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.
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.
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
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:
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:
In our case, however, the extensiveness s(p,q) is definite for pairs of places even if the particular coordinate allocation of is dependent on the choice of some observer.
All the information present in is also present in 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.
Next: 8.3 Past and Future Up: 8. A Theory of Previous: 8.1 Possibilities as Places Prof Ian Thompson