A formal axiomatic system is like a parlour board game. It has playing pieces and rules defining possible moves, as well as a game space within which the action takes place. Transformation rules are the algorithms by which expressions may be handled in order to derive theorems.
Theorems are the output of the system.
Unlike the elements of a causal system, those of a formal system are intentional and unambiguous. They exist exactly and only as what they have been declared to be. Like the rules of a game, they are well-defined and true because we agree to them. For example, even though perfect right angles and dimensionless points cannot actually be drawn, their idealizations can be manipulated in thought with complete precision. Formal systems contain explicitly only what has been defined for them and, implicitly, the logical consequences of these definitions.