Section 4.1 Neutral Geometry
¶Euclid, along with many mathematicians who followed, believed that the parallel postulate was a necessary property of any (realistic) geometry. According to this view, the parallel postulate should follow from the other postulates. In other words, one should be able to derive the parallel postulate as a theorem, assuming the other postulates hold, rather than needing to separately assume that the parallel postulate also holds.
With the wisdom of hindsight, we now know that this position is incorrect, and that the parallel postulate is, in fact, independent of the other Euclidean postulates. Before considering alternatives to the parallel postulate, we first consider neutral geometry, in which all of the other Euclidean postulates hold, but in which there is no parallel postulate.