The possibilities are endless even though every move is restricted. It is a world made of rigid rules that empower us. It is a common misconception that the rules of mathematics are somehow bestowed upon us. Yes, they are in the sense that we have inherited them. However, the rules of mathematics have had to be built from scratch (or almost scratch) at some point in time. The rules are either built out of necessity or curiosity.
Necessity-driven rules are built to satisfy an end-goal. An end-goal usually places limitations on us. However, it has been proven, times and times again, that limitations push us to innovate and think creatively. We all know the saying, "think out of the box". However, it is only that for the existence of the box, we are able to think out of it. On the other hand, curiosity-driven rule building is self-serving. One makes a rule which will either give rise to interesting properties or other interesting rules to be built on top of it.
Both necessity-driven and curiosity-driven rules push us to explore new territories we would otherwise not explore. Almost always, necessity-driven rules are built on top of curiosity-driven rules and vice-versa. Thus, the resulting total rule structure of mathematics is strongly intertwined with both kinds of rules.
|Picture from last night's blackboard discussion.|