How to tell the Axioms from the Assumptions

Question

Are there criteria, that allow one to decide, whether something, which isn't a consquence of a theorie's axioms, but is exploitet in a proof, is a further axiom and not an assumption?

A concrete example is the statement, that in euclidean geometry, to every line and a point, that is not on that line, there exist a unique parallel which contains the point.
What is it, that qualifies the statement as an axiom?

Another example would be the assumption, that the points, that are fed into a computational-geometry algorithm, are in "general position"; why isn't that an axiom.

Of course, I don't doubt that every mathematician knows the difference, but what if you were asked to explain it to math "newbies"?

Answer 1

I am not completely sure I understand the distinction you are making between "axioms" and "assumptions", so this response may be missing the mark, but here is my interpretation of the question.

Consider the following example of a theorem in, say, ring theory:

Let $u$ be an invertible element in a ring $R$. Then... (something something something)

Now the assertion that $u$ is an invertible element is an assumption of the theorem; it is the hypothesis that logically entails a conclusion. Not all elements in a ring have multiplicative inverses, but for the sake of this theorem (whatever it is) we are thinking about an element that does have a multiplicative inverse.

On the other hand, every element of a ring has an additive inverse; that is built into the axioms of a ring, and as soon as we say we are thinking about a ring those axioms are implicitly invoked. So the fact that $-u$ exists is not an assumption, it is an axiom.

If this is the kind of distinction you have in mind, then I would say that the difference between axioms and assumptions has to do with whether the assumptions are local and temporary -- the conditions under which a certain conclusion holds -- or whether they are global and stable -- part of the conditions that define the sort of object we are thinking about.

There is of course some gray area in between. If we are studying general rings, then asserting that $R$ is a commutative ring might be a (local) assumption for a particular theorem or a particular set of theorems. But if we say in that specialized theory for a while, it might eventually become part of the background context and appear more like an axiom of the sub-theory.

Answer 2

Of course, I don't doubt that every mathematician knows the difference, but what if you were asked to explain it to math "newbies"?

There are some ways to approach this for math newbies, one more 'to the bone' than the other. Which explanation you choose depends on the expected level of math proficiency (or tolerance).

• Axioms are atomic, indivisible building blocks. They are the basic rules with which we build a system. Because axioms create the system, the system itself cannot meaningfully prove them. That would be like proving blue is prettier than red by assuming blue is prettier than red.

• Axioms are the what-if genesis of a story. What if there's only one line parallel to another through a given point? What kind of story (mathematical system) does that imply? Like in a story, there is no benefit in trying to prove the genesis: the Harry Potter series starts with "there are wizards;" it's axiomatic to the story.

• Axioms are like types of Lego blocks: all of the tall 2x2 blocks are an axiom, and all of the flat 1x4 are an axiom, and so on. With these types of blocks, you can build structures (theorems). A proof of such a theorem is like the blueprint of a structure.

Are there criteria, that allow one to decide, whether something, which isn't a consquence of a theorie's axioms, but is exploitet in a proof, is a further axiom and not an assumption?

The difference between an axiom and an assumption is contextual and fudgeable, and not really set in stone. Assumptions that are considered axiomatic in one field of study may be contestable in other fields.

Loosely speaking, an assumption is a statement you take to be true with no proof that it is so, so it could be false. An axiom is the same, but it is essential to your system that it not be false.

Answer 3

Axioms are general assumptions, they are taken to hold always.

The specific case you mention, "points in general position" is more making explicit what wasn't said: Unless I'm told that the points, say, lie on a circle, I have to take them to lie anywhere.