Mathematically speaking, what is an ‘Axiom’?
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Axioms are what mathematicians call “axioms”. Seriously. There are no strict mathematical definitions. Axioms is used as part of assumptions a proof can make, but assumptions can be classified differently depends on its purpose. The view that axioms are things that are self-evidently true is a very classical view traced back to Euclid, but it is not how axioms are treated in modern mathematics.
Mathematically, every proofs have assumptions, things that are assumed to be true for a conclusion of the proof to hold. Assumptions fall into a few kinds:
– Hypothesis. Something assumed to be true about a particular unknown object, because the proof is only meant to be applicable to that object. For example “If x is an odd number”.
– Axioms. Assumptions that you can use without mentioning it explicitly. People are sure that they should be true. There are at least 2 kinds of axioms, as mention below.
– Well-founded assumptions. Claim that can be conceivably false, but so far it was considered reasonable enough chance to be true to produce mathematics out of. Example, “assume integer factorization is hard”. Proof that use well-founded assumptions are considered “conditional” proof.
There are 2 kinds of axioms, depends on how mathematicians view them.
– Definitional axioms: axioms that define the limitation of what a field study. Essentially they are just definitions of terms. For example, group axioms define what a “group” is, and that is the object to be studied by “group theory”. There are no fundamental different between definitional axioms and a definition.
– Foundational axiom: axioms that just run in the background, and considered to be things that should be true. Most mathematics will use the same shared collection of foundational axioms. For example, “if something is true about number 1, and if it’s true for any natural number then it’s true for the next number, then it’s true for all natural numbers”.
Even that above is not a strict dividing line. What one mathematician considered to be foundational axiom could be definitional axiom by another mathematician.
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