Eli5: Why Euclid’s fifth postulate cannot be proven?
In: 4
None of the postulates can be proven. To prove or disprove something, you have to relate it to something that you already know. But you have to start from somewhere, so you need some “obvious truth”, from which all others are derived. You cannot prove or disprove something if you know nothing at all.
Euclid’s “postulates” or “axioms” are such obvious truths. They are facts taken from observations of a real world, so they have no logical proof. All other statements are proven from them.
Postulates 1-4 are really obvious, but ancient mathematicians thought, that 5th postulate should follow from them. So they tried to prove 5th postulate using the first 4 as a base.
But they were wrong – 5th postulate does not follow from the first 4. It was proven, when one guy replaced the 5th postulate with its opposite and managed to build a system of logically valid proofs out of that. This new system is now called [Hyperbolic or Lobachevsky geometry](https://en.wikipedia.org/wiki/Hyperbolic_geometry).
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