How does the axiomatic method build a scientific or mathematical theory?

The axiomatic method builds a scientific or mathematical theory by establishing a formal system through logical deduction from a foundational set of propositions. This process involves several key steps:12

• Primitive Terms: The system begins with a set of undefined terms, which serve as the basic building blocks of the theory.12
• Axioms: Certain basic propositions, known as axioms or postulates, are adopted as the starting points. These are either constructed arbitrarily or chosen based on intuitive warrant.13
• Formation Rules: A formal language is established using rules that specify how primitive symbols can be combined into well-formed formulas.23
• Rules of Inference: The system defines specific rules for deriving new formulas from existing ones. A theorem is then defined as any formula that is either an axiom or the result of a finite sequence of valid inferences.23

When developing such a system, scholars prioritize criteria like consistency—ensuring no statement and its negation can both be derived—and richness, ensuring the system captures the desired scope of the theory. This method allows for the rigorous analysis of deductive structures independent of specific interpretations.42