Axiomatic method

A way of organizing a theory by deriving results from explicit axioms and rules of inference. Its conclusions are theorems whose validity depends on those stated foundations and rules.

Axiom: A statement accepted as a starting assumption within a formal theory. Axioms supply the initial statements from which the method derives its results.
Formal language: A system of symbols and formation rules used to express statements precisely. Axiomatic theories need a language whose well-formed statements can be identified.
Euclid's Elements: A Greek mathematical treatise that presents geometry through definitions, postulates, and propositions. Its systematic derivation of propositions became the model case for axiomatic organization.
Empirical science: Knowledge-building inquiry that tests claims against observation and experiment. Unlike empirical testing, deduction within an axiomatic theory establishes only consequences of its assumptions.
Hilbert's program: David Hilbert’s project to secure mathematics through formalization and consistency proofs. It sought to justify mathematical theories by studying their axiom systems formally.
Rule of inference: A formal rule specifying how conclusions may be derived from premises. Inference rules determine which steps from axioms to theorems count as valid.
Model theory: The study of mathematical structures that satisfy formal theories. Models give semantic interpretations to axioms beyond their written form.
Euclid: An ancient Greek mathematician traditionally associated with the foundational geometry text Elements. Euclid’s presentation of geometry gave the method its most influential early exemplar.
Intuitionism: A philosophy of mathematics that treats mathematical objects as mental constructions and restricts acceptable proofs. Its standards for proof challenge classical assumptions often used in formal axiom systems.
Gödel's incompleteness theorems: Theorems showing limits on provability and consistency for sufficiently expressive formal systems. They established that axiomatization cannot provide every desired guarantee for arithmetic.

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