Axiom

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. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.^[1]^[2]

The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.^[3] In modern logic, an axiom is a premise or starting point for reasoning.^[4]

In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example a + 0 = a in integer arithmetic.

Non-logical axioms may also be called "postulates" or "assumptions". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.

Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.^[5]

^ Cf. axiom, n., etymology. Oxford English Dictionary, accessed 2012-04-28.
^ Stevenson, Angus; Lindberg, Christine A., eds. (2015). New Oxford American Dictionary (3rd ed.). Oxford University Press. doi:10.1093/acref/9780195392883.001.0001. ISBN 9780199891535. a statement or proposition that is regarded as being established, accepted, or self-evidently true
^ "A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. Oxford English Dictionary Online, accessed 2012-04-28. Cf. Aristotle, Posterior Analytics I.2.72a18-b4.
^ "A proposition (whether true or false)" axiom, n., definition 2. Oxford English Dictionary Online, accessed 2012-04-28.
^ See for example Maddy, Penelope (June 1988). "Believing the Axioms, I". Journal of Symbolic Logic. 53 (2): 481–511. doi:10.2307/2274520. JSTOR 2274520. for a realist view.

[1] Cf. axiom, n., etymology. Oxford English Dictionary, accessed 2012-04-28.

[2] Stevenson, Angus; Lindberg, Christine A., eds. (2015). New Oxford American Dictionary (3rd ed.). Oxford University Press. doi:10.1093/acref/9780195392883.001.0001. ISBN 9780199891535. a statement or proposition that is regarded as being established, accepted, or self-evidently true

[3] "A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. Oxford English Dictionary Online, accessed 2012-04-28. Cf. Aristotle, Posterior Analytics I.2.72a18-b4.

[4] "A proposition (whether true or false)" axiom, n., definition 2. Oxford English Dictionary Online, accessed 2012-04-28.

[5] See for example Maddy, Penelope (June 1988). "Believing the Axioms, I". Journal of Symbolic Logic. 53 (2): 481–511. doi:10.2307/2274520. JSTOR 2274520. for a realist view.

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