# Philosophy of Local Time

Let us hypothesize on the notion of local time.

Existence of temporal order: For each concrete basic thing x ∈ Θ, there exist a single ordering relation between their states ≤.

We now give a name to this ordering relation:

Denotation of temporal order: The set of lawful states of x is temporally ordered by the ≤ relation.

The above is a partial order relation: there are pairs of states that are not ordered by ≤; e.g. given an initial condition (x0,v0) for a moving particle, there are states (x1,v1) that are not visited by the particle.

Proper history: A totally order set of states of x is called a proper history of x.

The axiomatics do not guarantee the existence of a single proper history: they allow many of them, as in “The garden of forking paths”. The following axiom forbids such possibility.

Unicity of proper history: Each thing has one and only one proper history.

Arrow of time: The axiomatics describe a kind of “arrow of time”, although it is not related to irreversibility.

A proper history is also an ontological history. The parameter t ∈ M has not to be continuous. The following axiom, a very strong version of Heraclitus’ hypothesis Panta rhei, states that every thing is changing continuously:

Continuity: If the entire set of states of an ontological history is divided in two subsets hp and hf such that every state in hp temporally precedes any state in hf, then there exists one and only one state s0 such that s1 ≤ s0 ≤ s2, where s1 ∈ hp and s2 ∈ hf.

The axiom of continuity is stated in the Dedekind form.

Continuity in quantum mechanics: Although quantum mechanical “changes of state” are usually considered “instantaneous”, theory shows that probabilities change in a continuous way. The finite width of spectral lines also shows a continuous change in time.

Real representation: Given a unit change (s0, s1) there exists a bijection T : h ↔ R such that

h1 = {s(τ)|τ ∈ R} —– (1)
s0 = s(0) —– (2)
s1 = s(1) —– (3)

Local time: The function T is called local time. The unit change (s0, s1) is arbitary. It defines an arbitrary “unit of local time”.

The above theory of local time has an important philosophical consequence: becoming, which is usually conceived as evolution in time, is here more fundamental than time. The latter is constructed as an emergent property of a changing (i.e. a becoming) thing.