Existence: Coming Into & Going Out Of

How does a part of the world come into being or disappear?

In phenomenological terms, this is an event -- in state s, we find that a particular part of the world is "not present"; at a next state T(s) it is. But setting aside the way it seems, this is is not a real event. Real events are transitions from the state of a part of the World to a different part state (the state is the only aspect of a part which can change). Coming into or going out of existence is not a change of state, hence is not an event in this sense. However, we can define something analogous to the phenomenological concept, using the concept of epochs. These are sets of states interpreted as "stretches of time", and "coming into being" or "going out of existence" can then be given a technical meaning in terms of such epochs.

What is a stretch in this context? In conventional theories of time, time is modelled by the "real line" or continuum R and a stretch is an interval subset of the continuum. But within the framework of mathematical holism, all temporal relations are constituted by state transitions, so we need to find an analogue to intervals within the framework of transitions.

We define an epoch as a sequence of states weakly connected by the transition relation. The set of states in the future of a state s is an epoch.

Now, one way to think of a part of the World as "going out of existence" in a particularl state is to say that in the future of that state, the part never changes -- so restricted to the epoch of all the future states, the part is reduced to the null part O.

"Coming into existence" is the dual of "going out of existence" -- a part which does not change in the past of a World state s and hence is O on the past of s exhibits the possibility of change by having on some larger epoch at least two states. The part then does not reduce to O in such an epoch, and thus exists over that epoch. It can be said to "come into being" at the latest state in whose past the part reduces to O.

We note that if T is not a function so that transitions are non-deterministic, then a part may have more than one state at which it begins or ceases to exist -- "earliest" and "latest" must be interpreted in the sense of "minimal" or "maximal" rather than "least"or "greatest" time value.

For a given part a, there exists a particular epoch

E^a = { s : T(s) contained in a(s) };

Note that if a part is null in a particular epoch, then (since it is always in the same state) it transmits no information to any other part nor causally controls any other part of the World in that epoch.

Thus, we have three temporal levels at which we can talk of being and non-being of parts of the world. (At least some philosophical problems of existence arise from a confusion of these levels.)

• At the universal level of the World states, all parts exist "forever"; that is, they are defined for all states.

• At the level of future and past epochs, every part has a (possibly empty) past in which it does not "yet" exist, and a (possibly empty) future in which it "no longer" exists.

  At the level of arbitrary epochs, it is logically possible that a part comes into and goes out of existence repeatedly.