Eli5: Why Euclid’s fifth postulate cannot be proven?
In: 4
Because there are geometries where it is false. Something that is false cannot be proven true. Or, if it can, you better find a different framework to operate from.
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.
There are places where all four of the other postulates are true, but the fifth is false. [Hyperbolic Geometry](https://en.wikipedia.org/wiki/Hyperbolic_geometry) is an example, it’s the geometry of a “saddle-shaped” plane. Because there are legitimate geometries where all the other postulates are true, but the fifth is not, it follows that the fifth cannot depend on the other four.
If you accept only the four postulates and make no commitment to the fifth postulate either way, then you get [Absolute Geometry](https://en.wikipedia.org/wiki/Absolute_geometry) whose results are weaker (as they have no Pythagorean Theorem or alternatives), but the results apply to both Euclidean and Non-Euclidean geometry.
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).
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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