Summary:
Any entity E which is constituted by extrinsically indiscernible parts A and B remains extrinsically the same, in all stages of the change, even if A changes to B and B to A (concurrently).
More generally, an entity E which is constitutable by a set of parts {A, B} as well as another set of parts {C, D} remains extrinsically indiscernible even if A changes to C and B to D and vice versa.
Accordingly, (McTaggart's) B-series may be described as consisting of multiple sequences with different beginnings (sets of parts) which are alternating and therefore are non-simultaneous but all constituting/causing the same whole (or equivalently simultaneous and indiscernible wholes).
If causal is the flow of constitution (parts-to-whole direction), there are multiple causal routes each from any of the sets of parts to the whole. Thus, it seems appropriate to suggest that temporal is the flow of de-constitution (whole-to-parts) -- time as the transition from a whole which is constitutable by each of many sets of parts to the set of parts that generates the shortest path.
SEP article on "Change and Inconsistency" discusses the notion that change involves some sort of Inconsistency. There seems to be a sense in which such an inconsistency is minimal: the entities (say E) that are constituted by parts undergoing change are to remain extrinsically unchanged during and after the change. It means that A changes to B if and only if
- B changes to A concurrently,
- A and B are extrinsically indiscernible (have identical constitutional roles)
- and for all entities E if A is a part of E, then B is a part of E.
If we take into account the stage during the change as well not just before and after the change, A changes to B if and only if
- there are C and D such that A changes to C before changing to B and B changes to D before changing to A, and that the whole constituted by A and B is extrinsically indiscernible from the whole constituted by C and D,
- and for all entities E if A is a part of E, then B is part of E and if C is a part of E, then D is a part of E.
The entity E before the change (constituted or caused by A and B) is extrinsically indiscernible from E during the change (constituted or caused by C and D) and after the change (constituted or caused by B and A).
If we translate this structure into the language of something like McTaggart's B-series, both pairs (A, B) and (C, D) are in the block "before" the block containing the entity E (since the pairs both cause E independently) but do NOT occur at the same time. However, the entity E caused by the first pair and that by the second pair are extrinsically indiscernible as if both effects occur at the same time regardless of which pair is at work at any given time. Thus, B-series may be described as consisting of multiple sequences with different non-simultaneous alternating beginnings (sets of parts) all converging to the same end (or equivalently, to simultaneous and indiscernible wholes).
if causal is the flow of constitution (parts-to-whole direction), there are multiple causal routes from each of the sets of parts to the whole. This may be described as the source of inconsistency. It seems that some consistency is gained if there is a shortest temporal path from the end of the b-series to its beginning (more precisely, one of its beginnings). That is to suggest that temporal is the flow of de-constitution (whole-to-parts) -- the transition from a whole which is constitutable by each of many sets of parts to the set of parts that generates the shortest path. This also seem to implicate that the temporal flow must deploy a different causal mechanism than the one operating in the causal routes.
(it is worth noting that in that (a, b) and (c, d) are "before" e in the b-series, "before" does not mean "earlier than").
In the context of the Laplacian universe (initial conditions + deterministic laws) this formulation is equivalent to the case in which the initial conditions are only critically determined at the beginning of time, enough to set off the temporal process, and are further determined (crystallized) towards the end of time, as causal routes are excluded from the set of candidates for the shortest path.