Consider a bucket that can hold a certain amount of water. This is a property of the bucket. But if we were to break the bucket into pieces this property will disappear. If we continue breaking up the pieces, smaller and smaller, we will reach the level of atoms, yet not have found this property.
However, zooming back out, we see how atoms interact to create molecules. How molecules interact to become materials. How materials can be used to make parts. How parts are put together become things. Where every layer is contributing new properties and interactions to the higher layers so that the bucket as a whole can hold a certain amount of water.
Now consider the game tic-tac-toe, a game played on a three by three grid, with two symbols, a turn based placement rule and a win condition. Analyzing the possible moves we see the game has losing moves, defensive moves and winning moves. And that a winning move can only be played after the opponent has played a losing move. What kind of rule is that? How can yet another rule appear that limits when winning moves can be played?
The bucket with its capacity to hold water, and the game with its extra rule, these two effects are the same. Consequences of rule-like interactions is that often higher level effects appear which are themselves rule-like. This can be seen in the laws of physics, or in the rules of games. And in bottom-up systems, where we see small parts, simple rules, local interactions, causing endless patterns filtered by entropy. And in the penultimate example: math, best described as the rule based manipulation of symbols and rules.
This emergence of new rules can feel like magic. Especially when repeated a few layers upwards. Why is math so unreasonably effective? How can something be more than the sum of its parts? How can complex things come about from simple things? How do crystals grow? How can a process like evolution design anything? Why are markets efficient? Where does moral right and wrong come from? What is a culture?
This bottom-up view highlights it’s not magic, but natural. If we can describe the system as interactions between entities, we often see why the patterns and higher level interactions emerge as consequences.