Many debates on the basis of mathematics and philosophy involve the concept of meta-mathematics, which cannot be regarded as what we usually call "problems". The basic assumption of metamathematics is that the content of mathematics can be obtained through a formal system, such as an order theory or an axiomatic set theory.

Metamathematics and mathematical logic are closely related, so their development is similar. The origin of metamathematics probably dates back to Frege's book: Conceptual Text. David hilbert first introduced the term "regular meta-mathematics" (see Hilbert Plan). This is now known as the theory of evidence. Another important modern branch is model theory. Other important figures in this field are: Bertrand Russell, Suraf Colm, Emile Post, Qiu Qi, Kleene, Quine, Paul Benacerraf, Putnam, Gregory Chapin, and the most famous Taskey and Godel. In particular, Godel proved that any finite axiom of a given Piano arithmetic has some correct propositions, which cannot be proved by the given axiom, that is, the so-called Godel's incomplete theorem. In a sense, this result is the highest achievement of metamathematics and mathematical philosophy so far.