Foundations of mathematics

Foundations of mathematics studies the axioms, logic, and structures on which mathematical reasoning rests, and examines what mathematical objects and proofs are.

Mathematical logic: The study of formal languages, inference, proof, and mathematical theories. It provides the formal tools used to state and analyze foundational systems.
Logicism: The view that mathematics can be reduced to logic, usually with suitable definitions and principles. It seeks a logical basis for mathematics rather than treating mathematical axioms as independent starting points.
Euclid's Elements: An ancient Greek treatise that organized geometry and number theory into definitions, postulates, and propositions. Its axiomatic presentation became a lasting model for deductive mathematics.
Gödel's incompleteness theorems: Results showing that sufficiently expressive, consistent formal theories cannot prove every arithmetic truth or establish their own consistency. They sharply limited the ambitions of Hilbert's program.
Lean (proof assistant): An interactive theorem prover based on dependent type theory, used to formalize mathematics and verify proofs. It implements a type-theoretic foundation for machine-checked mathematical arguments.
Axiomatic system: A formal framework consisting of axioms and rules from which theorems are derived. Foundational programs assess which axioms can support mathematical theories.
Formalism: A philosophy of mathematics that emphasizes formal symbol systems and derivations under explicit rules. It locates mathematical rigor in formal manipulation, unlike programs centered on mathematical objects.
Non-Euclidean geometry: Geometries in which Euclid's parallel postulate is replaced by a different assumption. Its discovery showed that geometrical axioms need not describe one uniquely necessary system.
Consistency: A property of a formal theory in which no contradiction can be derived from its axioms and rules. Consistency is a central standard for evaluating foundational systems.
Coq: An interactive theorem prover based on the calculus of inductive constructions. It uses constructive type theory to certify proofs and programs.

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