# Why is the Euclid's Fifth Axiom/Parallel Postulate unnatural?

Why is the Euclid's Fifth Axiom/Parallel Postulate unnatural for many mathematicians such that they want to find a way to prove it from the other four axioms? Is there an example that show its unnaturalness?

• – Blue Dec 8 '14 at 13:05

## 2 Answers

The fifth axiom is interesting, in that he did not introduce it at the start of the book, but seemingly only added it after the 28th proposition, realising he would need it for the next one.

It is an interesting axiom for mathematicians, as it ONLY holds for Euclidean geometries.

Have a read about non-euclidean geometries:

http://en.wikipedia.org/wiki/Non-Euclidean_geometry

Mathematicians are always interested to know if the axioms of some system are or are not independent. So the desire to show that the fifth postulate is or not implied by the first four is as much about logical neatness as anything else.

Euclid's system was probably the first to come under such scrutiny. Whether or not the fifth axiom is different from others in some qualitative sense is subjective. I don't know of any mathematician in particular who has called it 'unnatural', do you?

That being said, such scrutiny was definitely worthwhile, because non-Euclidean geometries have turned out to be remarkably fruitful.