What is an axiom and can an axioms be proofen?
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Axioms, by definition, are statements that are assumed to be true without proof that they are. If you can prove the statement, it’s not an axiom (it’s a theorem).
Axioms are necessary because you can’t deduce in a vacuum. In order to prove one statement is true, you need to show that it follows logically from other statements that are *already known to be true*. If you don’t assume anything, you can’t deduce anything.
Axioms are complete assumptions that you make, and follow, upon which all other mathematical statements are proven. Some axioms you might know are “objects that are both equal to the same object, are also equal to each other” or “let straight lines exist”
If you can prove an axiom from the other axioms you already have, then that’s great. It’s always good to minimize the number of assumed statements you have, but you can never prove *all* axioms.
An axiom is a concept in logic. It is a statement which is assumed to be true without question, and which does not require proof. It is also known as a postulate (as in the parallel postulate). The axiom is to be used as the premise or starting point for further reasoning or arguments, usually in logic or in mathematics.
This means it cannot be proved within the discussion of a problem. So inside some discussion, it is thought to be true. There are many reasons why it has no proof. For example,
1. The statement might be obvious. This means most people think it is clearly true. An example of an obvious axiom is the principle of contradiction. It says that a statement and its opposite cannot both be true at the same time and place.
2. The statement is based on physical laws and can easily be observed. An example is Newton’s laws of motion. They are easily observed in the physical world.
3. The statement is a proposition. Here, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. This means the emphasis is on what happens if the axiom is true. Whether the proposition is empirically true is not the goal of logic. This is a more modern definition of an axiom.
Logic can be used to find theorems from the axioms. Then those theorems can be used to make more theorems. This is often how math works. Axioms are important because logical arguments start with them.
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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