The word formalism has several meanings:

• A certain school in the philosophy of mathematics, stressing axiomatic proofs through theorems specifically associated with David Hilbert.
• A certain rigorous mathematical method: see formal system.
• A set of notations and rules for manipulating them which yield results in agreement with experiment or other techniques of calculation. These rules and notations may or may not have a corresponding mathematical semantics. In the case no mathematical semantics exists, the calculations are often said to be purely formal. See for example scientific formalism.
• In the study of the arts and literature, formalism refers to the style of criticism that focuses on artistic or literary techniques in themselves, in separation from the work's social and historical context. See formalism (art), formalism (literature).

See also: formalist.