Foundations of Mathematics

Mathematical and Formal Logic

Does everyone agree with axioms?

No, not everyone agrees with all axioms, the self-evident truth upon which knowledge must rest and other knowledge is built. For example, not all epistemologists (philosophers who deal with the nature, origin, and scope of knowledge) agree that any true axioms exist. However, in mathematics axiomatic reasoning is widely accepted, where it means an assumption on which proofs are based.

The word axiom (or postulate) comes from the Greek word axioma, and means “that which is deemed worthy or fit,” or “considered self-evident.” Ancient Greek philosophers used the term axiom as a claim that was true without any need for proof. In modern mathematics, an axiom is not a proposition that is self-evident, but simply means a starting point in a logical system. For example, in some rings (for more about rings, see “Algebra”), the operation of multiplication is commutative (said to satisfy the “axiom of commutativity of multiplication”), and in some it is not.



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