Eli5: Why Euclid’s fifth postulate cannot be proven?
In: 4
A postulate is an assumption. It’s something we assume to be true to build other ideas off of. You can’t prove them because they are your starting point. None of Euclid’s postulates can be proven, because they are the starting points of euclidean geometry. So maybe the better question is why did people try so hard to prove the fifth postulate?
Let’s look at the postulates:
1. A straight line can be drawn joining any two points.
2. Any line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (Basically defines what a parallel line is)
You can probably tell one of these postulates is not like the others. Postulate 5 is far longer and more complicated than the the others. It really feels like postulate 5 should be a theorem not a postulate. In order to make it a theorem you would need to use the first 4 postulates to show that postulate 5 is always true.
If you could prove it then any system where the first 4 postulates are true the 5th would also need to be true. This turns out not to be the case. People have developed self consistent geometrical systems where the first 4 postulates are true and the fifth is not true (google “non-Euclidean geometries”) so it can’t be a result of the other 4 postulates.
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