Formalism

Formalism is a philosophy of mathematics that treats mathematical theories as formal systems: symbols are manipulated according to explicit rules, without requiring them to describe an external reality.

Formal system: A collection of formal symbols, formation rules, axioms, and inference rules for deriving expressions. Formalism interprets mathematical theories through systems of symbols and rules.
David Hilbert: A German mathematician whose work shaped geometry, mathematical physics, and the foundations of mathematics. Hilbert made the systematic formalization of mathematics a central foundational program.
Gödel's incompleteness theorems: Two theorems showing limits on what sufficiently expressive, consistent formal systems can prove about arithmetic. They showed that Hilbert's consistency ambitions could not be achieved by the intended finitary means.
Structuralism in mathematics: A philosophy that treats mathematical objects as positions in abstract structures rather than as independently existing entities. It shares formalism's emphasis on relations and structures while giving mathematical objects a different account.
Formal language: A set of finite strings built from an alphabet according to specified formation rules. Its syntax determines which expressions count as legitimate mathematical statements.
Hilbert's program: A project to formalize mathematics and prove the consistency of its formal systems by finitary methods. It is the best-known foundational program associated with mathematical formalism.
Consistency proof: A mathematical proof that a formal system cannot derive a contradiction. Hilbert's program made consistency proofs a central goal for formalized mathematics.
Model theory: The study of mathematical structures that satisfy formal languages and theories. It connects formal syntax to interpretations in which mathematical statements acquire truth values.
Axiom: A statement accepted as a starting point within a deductive system. Formal mathematical derivations begin from stipulated axioms rather than semantic truths.
Foundations of mathematics: The study of the basic concepts, methods, and logical basis of mathematics. Formalism emerged as one answer to questions about what mathematics rests on.

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