An axiom is a statement accepted as a starting premise without proof, from which a mathematical system's other results are derived by logical deduction. Euclid's Elements, with its five postulates and five common notions, carries the most famous ancient use; modern mathematics treats axioms as formal stipulations defining a system rather than as self-evident truths, a shift the discovery of hyperbolic geometry forced. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Partially Attested
Origin YearPlaced at circa 300 BCE for Euclid's Elements, the earliest surviving systematic use of axioms and postulates; the underlying practice of assuming unproved first principles is older and not datable to a single year. Connections
Associated With
In Branch
Source Wikipedia: Axiom
Sources
1. Wikipedia: Axiom
Wikimedia FoundationEtymology; Historical development
The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience.
Early Greeks
The classical approach is well-illustrated by Euclid's Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).
In Category: Concepts, lead paragraph, opening definition
An axiom, postulate, or assumption, is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
In Branch: Logic and Foundations, Mathematical logic section
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical.
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