Metamathics is the branch of logic in which mathematical assumptions are treated as mathematical objects. Metamathics is dedicated to finding connections between mathematical disciplines. Metamathics is based on supposition rather than assumption.
First Order Metamathics[]
Though metamathics deals with all possible logical structures in which mathematical assumptions can be treated as objects, one of the most popular and illustrative suppositions is that mathematical assumptions follow the group theoretic axioms - closure, associativity, identity, and invertibility. In this case, the identity element is most often taken to be zeroth order logic, though the sub-field of Elementary Metamathics Algebra explores results based on different identity elements such as elementary arithmetic and elementary functional arithmetic.
Higher Order Metamathics[]
Quickly after the foundation of metamathics, magisters began using metamathical structures as elements in common mathematical logic systems. The proofs that result are considered second order metamathics.
Third order metamathics is the application of metamathic suppositions to metamathic structures. Resulting proofs expressed in the mathematical language of the elemental logical structures have been shown to be both impossible to prove and impossible to disprove. Hieronymus Findorff famously proved that such results are still valid in the case where identity elements exist and the same identity element is taken at every order. Such structures are called the identity element of that axiomatic logic, ie the "group theory identity", or the "elementary arithmetic identity".