# Formal system

A **formal system** is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the **logical calculus** of the formal system.
A formal system is essentially an "axiomatic system".[1]

In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics.[2] A formal system may represent a well-defined system of abstract thought.

The term *formalism* is sometimes a rough synonym for *formal system*, but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation.