what is Euclidean’s fifth postulate, and how was it refuted?

Question

*Euclid dammit

In: 6

Answer 1

The 5 postulates of Euclidean geometry are based on plausible properties of lines, circles, and angles in an “ideal” infinite plane.

If you look at the 5 postulates, the 5th one is more complicated than the rest. It gives conditions under which two lines in a plane eventually meet if they are extended far enough. This postulate turns out (assuming the other 4 postulates) to be equivalent to saying for each point P not on a line, exactly one line passes through it that doesn’t cross the first line. We’d recognize that as the line through P that is parallel to the original line, so the 5th postulate is called the parallel postulate.

Because the 5th postulate did not seem as basic or fundamental as the other 4, everyone felt for centuries that it ought to be provable from the other 4. What they tried to do was assume the first 4 postulates of Euclidean geometry and also that the 5th postulate is *false* and then derive a contradiction. That would then show that in the presence of the first 4 postulates, the negation of the 5th postulate is impossible, and hence the 5th postulate must be true when you assume the other 4 postulates.

Numerous people explored such “exotic” plane geometries where the first 4 postulates hold plus the negation of the 5th one and said they found a contradction, but their reasoning always turned out to have a mistake. Finally, in the 1800s, several people (Gauss, Lobachevsky, and Bolyai) realized independently that these exotic geometries are *not* contradictory, but in fact are as consistent in their own way as Euclidean plane geometry.

Two examples of these “new” geometries are the hyperbolic plane and the sphere. In them, the term “line” is not what ordinary experience would suggest should be regarded as a line, but it genuinely fits all the necessary axioms. On a sphere, the word point has its normal meaning but the word “line” means a great circle (equators in any position. On a sphere, all great circles cross each other, or in the Euclidean language, “all lines cross”. So there are no parallel lines. On the hyperbolic plane, it turns out for each point P off a line, there are infinitely many lines through P that don’t meet the original line. So geometry on a sphere and on the hyperbolic plane negate Euclid’s 5th postulate in different ways.