In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules.
For example, Euclidean geometry can be seen as a game whose play consists in moving around certain strings of symbols called axioms according to a set of rules called "rules of inference" to generate new strings. In playing this game one can "prove" that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules.
According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics).
Formalism is associated with rigorous method. In common use, a formalism means the out-turn of the effort towards formalisation of a given limited area. In other words, matters can be formally discussed once captured in a formal system, or commonly enough within something formalisable with claims to be one. Complete formalisation is in the domain of computer science.
Formalism stresses axiomatic proofs using theorems, specifically associated with David Hilbert. A formalist is a person who belongs to the school of formalism, which is a certain mathematical-philosophical doctrine descending from Hilbert.