Axiom
An axiom is a statement accepted without proof and used as a starting point for everything else in a subject. Nothing can be derived from an earlier statement forever, so each mathematical subject begins with a short list of assumptions and builds outward. Euclid opens the Elements with definitions, five postulates, and a set of common notions. Modern practice treats axioms less as self-evident truths than as the rules of a chosen system: change the rules and you get a different mathematics, which may be just as consistent.
Why it matters
Once you see axioms as choices rather than facts, the nineteenth-century discovery of alternative geometries stops looking like a paradox and starts looking inevitable.
Also written: axioms, postulate
Where it comes up
Sources
- Axiom Wikipedia
- Euclidean geometry Encyclopaedia BritannicaSets out Euclid's five postulates and what each of them assumes.
