The Definition of Parallelism
The geometry inside a plane includes straight lies of infinite lengths falling inside a plane. It is regarded as a fact that two points inside a plane can define a unique straight line. By intuition we assume that one point and one line also can define a unique straight line if we demand the defined line to be parallel to the given line.
However, parallelism has to be defined. According to an about 2000 years old definition parallel lines have no common point. This definition is based on a common point that does not exist. A correct definition should be based only on concepts that really exist. Therefore, when the number of common points takes the value zero and the concept no longer exists we are not allowed to use the concept in a definition. We should base a definition on existing concepts and describe what these concepts really do and not on something that they do not do, because there are a lot of other things that they do not do.
The disappearance of common point is more like a consequence of parallelism and instead the concept constant separation (or equidistance) is a better representation for the essence of parallelism. We can therefore find a better definition based on constant separation. This means that any arbitrarily chosen point on one of the lines always is on the same distance to the other line. Based on this definition we can prove that one point and one line can define a unique line parallel to the given line. From a correct definition we find that our intuition was correct.
Why was parallelism explained in that way 2000 years ago and why has the definition survived for such a long time? Perhaps both questions have the same answer. The reason can be that we beleave too much in simplicity (Occam’s Razor) and therefore have a very high ambition to describe a complex relation in very few words. The older definition really is very short. The human desire to be very short may have fooled us to use a concept that does not exist in a definition for a long time.
We cannot define a concept by concepts that do not exist and not by actions that never are done.