Concepts
Axiom
Also Known As Postulate
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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.
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. Cross-Tradition Connections
Sources
1. Wikipedia: Axiom
Wikimedia FoundationEtymology; Historical developmentQuote, Etymology; 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.
1. Wikipedia: Axiom
Wikimedia FoundationHistory sectionQuote, History section
The classical approach is well-illustrated by Euclid's Elements, where a list of postulates is given (common-sensible geometric facts drawn from our experience), followed by a list of common notions (very basic, self-evident assertions).
1. Wikipedia: Axiom
Wikimedia FoundationIn Category: Concepts, lead paragraph, opening definitionQuote, 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.
1. Wikipedia: Axiom
Wikimedia FoundationIn Branch: Logic and Foundations, Mathematical logic sectionQuote, 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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