Two attempts at deriving internal time from the causal order of modes

It would be nice to define the internal time of a substance in terms of the causal order of its accidents.

For each mode (i.e., accident or substantial form) α that a finite substance x has, there is the event c_α of α’s being caused. Causal priority provides a strict partial ordering on the events c_α.

Perhaps the simplest theory of the internal time of the substance x is that the moments of internal time just are the events c_α and their order just is the causal priority order.

This has the consequence that internal time need not be totally ordered, since one can have cases where α ≠ β but there is no priority relation between c_α and c_β. This consequence is welcome and unwelcome. It is welcome, as it allows one to give a nice account of bilocation involving the bifurcation of internal time. It is unwelcome, as intuitively time is linear. Let’s see if we can do something to reduce the unwelcome consequence.

Let’s suppose—as per causal finitism—that causal interactions are discrete. Then we can define a fundamental distance between the moments of internal time: d(c_α, c_β) is the length of the longest unidirectional causal priority chain between c_α and c_β. One might reasonably hypothesize that d(c_α, c_β) is something of the order of magnitude of the temporal distance between c_α and c_β in the rest frame of the substance in units of the order of Planck time. (Note that d is not a metric because of the unidirectionality constraint on the chains.)

This lets us have a second way of defining the internal time of a substance x. Let f be x’s substantial form. Then we can define “the start time” of a mode α as d(f, α): the length of the longest internal causal priority chain from c_f to c_α. Now likely some modes will have a simultaneous internal start time—they will have the same distance to c_f.

For this to define an intuitively plausible time sequence, we need the substance to have lots of interconnections between its accidents. Ordinary substances do seem to have that.

And perhaps some accidents won’t have an internal start time—if God turns me blue right now, my blueness won’t have an internal start time. But nonetheless that blueness can be “attached” to my internal temporal sequence by noting that it will be close according to d to some of my near-future accidents. For that miraculous blueness will interact with some of my other accidents to produce new accidents that are properly in my middle age. For instance, it will interact with my memories of observations of things not turning blue to generate the accident of surprise.