What is an action?
Well, they do things. What does that mean? It means something starts out some way and changes. Old states become new ones.
Given a transition map
T: a x s -> s, we can rewrite it as
a -> (s -> s). This means actions can be seen as maps between states, so we say
act: state → state. Because they’re maps, we can compose them. The actions of
mix batter and
put in oven can be composed into the action
mix batter, THEN put in oven.
Dots and arrows. The arrows are the actions, carrying dots to new dots. Chaining arrows is the same as composing actions. Much can be represented this way.
The setup is currently discrete: each action is basically a sudden jump.
Let’s use the hyperdiscrete idea: hyperfinitely many infinitesimal, discrete things is a continuous object (at least after taking a standard part).
Imagine taking lots of actions, but each one only has a tiny influence, so each successor state is really close to its source state. Then hyperfinitely many of these actions are a flow.
For a visual of how discrete actions are transformed into a flow, think of an equilateral triangle, oriented counterclockwise. The vertices are the dots, and the edges are the actions, carrying one vertex to another. If we perform a walk, then we will see a 3-cycle.
Now go to infinitely many sides while keeping the perimeter fixed. We get a polygon that’s (infinitely close to) a circle, and the walk is now a flow.
Following arrows gives a smooth path, such as going from the top to the bottom of the circle along the left or right half of the perimeter, which is now a circumference.
Most important example of a flow: motion through space.
Discrete and Continuous
This is just another example of the link between discrete and continuous, specifically realizing the continuous as the hyperdiscrete.