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.

## Important example.

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.

## To infinity

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.