Eli5: Why Euclid’s fifth postulate cannot be proven?
In: 4
Lets use a toy version with less complexity. I postulate:
1. There are three kinds of things: Rables, Gerbles and Blurbs. Only Rables are able to feel hatred.
2. Every Gerble is green.
3. There is at least one Blurb that happens to be blue.
4. Rables hate red things with a passion, but not each other and nothing of another color.
5. All three can coexist together: in a group of one of each, none will be hateful towards another.
So, is the fifth one unnecessary to postulate, in other words: can we conclude that the fifth one already follows from the other 4 statements? We are only allowed to use logical reasoning, as we otherwise know nothing about this weird culture.
Playing around with the given first four statements we can observe a few conclusions, for example:
* A group of only Rables and Gerbles does not cause issues.
* Neither does a group of only Gerbles and Blurbs.
* Rables are fine with some of the Blurbs.
Now lets look at a sample population: we have exactly one of each, lets denote them by R(able), G(erble) and B(lurb). What do we know in that case?:
* G and B do not hate each other, themselves or R by (1).
* G is green by (2).
* As there is no other Blurb, the one, B, we have must be the blue one from (3).
* By (4), we conclude that R does not hate G nor B.
* In total, that group of three is fine, (5) holds!
But what if we instead have a slightly larger population, say R, G as above but now two Blurbs Blu and Blo, which are blue and blood-colored. Note that (1)-(4) are all satisfied again. But this time, (5) is not true, because R hates Blo, i.e. in the group R, G, Blo there will be conflict!
We conclude that there are two settings satisfying each of (1)-(4) perfectly, but only one of them also has (5) as true. Hence there cannot ever be a logical argument that (5) does follow from the others.
Important tidbit: it is not enough to only look at the second setup where (5) is wrong. Without knowing the first example, it could be that there is absolutely no viable world at all. Or reversely stated: every possible (i.e.: no!) world satisfies (5).
Now with geometry, that’s the same, the rules are only way more complex, or rather emergent. Showing that two example exist where (1)-(4) hold but only one of them has (5) as correct becomes more difficult, but it has been done. The standard examples are geometry of the plane (satisfying the 5th) and hyperbolic geometry (violating it).
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