What is an Axiom?

In mathematics, we often come upon terms such as axiom, theorems, postulates, etc., but what are the differences between them, and how are they used in mathematics? After taking a look into Euclidean Geometry again, I thought it would probably be a nice refresher for myself to look more closely into the definitions. 

Axioms and postulates are two words I hear interchangeably used in mathematics, but I have always been confused whether they are the same thing. Sometimes a statement is an axiom and sometimes the same statement can be referred to as a postulate, depending on who you are talking to. While looking further into this, I found that axioms are often referred to as postulates because like axioms, postulates are statements that do not need to be proven as true because it is self-evident that they just are true. For example, since we recently have looked at Euclidean Geometry, Euclid's Fifth Postulate is "If a straight like falling on two straight lines make the interior angles on the same side less that two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than." While reading, we can see that this is obviously a true statement because if we have to lines that intersect another line, and the interior angles created are less than 90 degrees, the two lines will have to intersect eventually. We don't need a proof to show that this is true, so it can be considered an axiom or postulate by definition.

Axioms and postulates are different from theorems because of the fact that they don't need to be proven, some even say that they cannot be proven because they are so self-evident there is no need for them to be proven because it is just logical. Theorems tend to be challenged more often then axioms since they are subject to more interpretations and because there are many various methods of deriving theorems. Axioms and postulates are often used to help prove theorems.

Axioms and postulates are evident throughout all mathematics, but one area where there are many famous ones are in Euclidean Geometry.  The historic mathematician Euclid came up with many postulates and theorems in Euclidean Geometry, that we still use today. Euclid's most famous postulates are the first five postulates:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All Right Angles are congruent.

5. If a straight like falling on two straight lines make the interior angles on the same side less that two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than.

His Fifth Postulate, which I mentioned above, is one of the most famous postulates in mathematics. It caused a lot of controversy among mathematicians for many years that resulted in many proofs trying to prove and disprove the postulate. 

Euclid came up with his axioms and postulates to establish initial assumptions to help him prove other theorems throughout his studies. This, to me, is why axioms and postulates are so important to mathematics. Without axioms, we wouldn't have any set assumptions about mathematics that we can all agree are true without any controversy. Axioms make proving theorems so much easier. They give us a foundation of true assumptions, which then are used to prove theorems. Without them, we would have try to prove these self-evident statements, but with them we just have to acknowledge that it is a true assumption. 

Now that I have clarified the differences between axioms, postulates, and theorems, and I have refreshed my understanding of how helpful axioms and postulates really are, I have realized that I don't think I would want to try proving much without them!