What is an axiom and can an axioms be proofen?
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I will give an answer from the perspective of modern mathematics.
A set of axioms is a set of statement where you can use in proof without further justification. Why can you use it without justification? There are 3 main reasons:
– You believe that those mathematical object (mentioned in those axioms) truly exist in some abstract world and this is their properties.
– You consider the axioms as a definition for certain kind of objects, e.g. the object is only called X because it satisfied the axioms for X.
– You find the consequence of them interesting enough and want to explore more.
The first kind are usually the foundational axioms, things that are supposed to underlie the entire mathematical world, or things that people have a very precise intuitive idea what they are. Things like set theory axioms (ZFC or some other things), or Peano’s arithmetic.
The second kind is everywhere, they are axioms that underlie a specific field of math. In this case, they serve to narrow down the scope of a field. Note that these axioms are really *definitions*, but they’re not called “definitions” because they generally consist of many interacting parts instead of a simple “X is just Y that has property Z”. For example, Euclidean axioms define the concepts of “line”, “point”, “angle” and “circle” *simultaneously*.
The third kind are very frequently used by working mathematician to explore further certain concepts. If they pick up traction, it can turns into either of the previous kind.
Note that the question of what kind of axioms is what is just a matter of judgment, not objective reality. It depends on what you believe the axioms is supposed to do. Because of this, there isn’t a singular idea of what is an axiom. Depends on your philosophy toward mathematics, you can consider certain kind of axioms as more important than others, or certain kind of axioms to be non-existence. From a pure abstract math point of view, every math proof start with a list of assumptions and end with a conclusion, it doesn’t matter why you put up those list of assumptions.
Generally, not all things that people find to be obviously true and get used in proof are called “axioms”, though they play very similar. For example, in computer complexity theory and cryptography, it’s very common to make an assumption of the form “problem X is too hard that it can’t be solved within limitation Y”, and with enough works using the same assumptions, it plays a role very much like an axiom. The reason why such assumptions exist because if it’s true, they’re very difficult to prove, but a lot of interesting things can be proved if you assume they’re true.
Can it be proven? Yes and no.
Foundational axioms is generally considered to be not provable within the context of working mathematics. After all, the math have to start somewhere. However, that doesn’t stop people (who do research in foundation) from developing new foundational axioms, then prove that the old axioms follow from it, in order to show that all the old math still carry over. However, it is sometimes also possible that people who write the axioms produced redundancy: some axioms can be proved from others, but this is generally avoided.
Definitional axioms are, well, definitions. You can prove that certain objects actually satisfy the axioms, and hence is an object the axiom refers to.
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