Formalism (philosophy of mathematics)

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."

According to formalism, mathematical statements are not "about" numbers, sets, triangles, or any other mathematical objects in the way that physical statements are about material objects. Instead, they are purely syntactic expressions—formal strings of symbols manipulated according to explicit rules without inherent meaning. These symbolic expressions only acquire interpretation (or semantics) when we choose to assign it, similar to how chess pieces follow movement rules without representing real-world entities. This view stands in stark contrast to mathematical realism, which holds that mathematical objects genuinely exist in some abstract realm.

Formalism emerged as a response to foundational crises in mathematics during the late nineteenth and early twentieth centuries, particularly concerns about paradoxes in set theory and questions about the consistency of mathematical systems. It represents one of the three major philosophical approaches to mathematics developed during this period, alongside logicism and intuitionism, though formalism encompasses a broader spectrum of positions than these more narrowly defined views. Among formalists, the German mathematician David Hilbert was the most influential advocate, developing what became known as Hilbert's program to establish the consistency of mathematics through purely formal methods.