A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms by a set of inference rules.[1]
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics.[2]
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.