Euclid's Parallel Postulate
Date: Fall 2013
I was wondering is Euclid's Fifth Parallel Postulate of parallel lines never intersecting, undecidable or essentially undecidable?
The basic premise of a mathematical postulate is that it cannot be tested to full extent. It cannot be proven. In theory, it is possible that the distance between lines and direction in real space are somehow curved at extreme distances from where we are located. One definition of parallel lines is that they have the same orientation, the same direction over an infinite distance in both directions. We do not really know whether such a structure can exist. In some models of curved space, two straight lines actually do meet. An example is two lines running perfectly northward. Both have the same ?direction?, but they will meet at the North Pole.
In math and science, the most basic premises cannot be proven. Until these basic premises are assumed, nothing else exists in the subject. We need these few postulates in order to get things started.
Dr. Ken Mellendorf
Illinois Central College
What your question refers to is whether the geometric space is convex, plane, or concave. The choice of these ?spaces? is up to the choice of which geometry you choice as the ?space? you want to use. In a ?concave? space, two lines will meet or even cross, in a ?convex? space two lines will not cross (unless the curvature of the space) is sufficiently convex that the space is closed. This would happen if the space is a sphere. Parallel lines on a plane space will not intercept. What you have to keep in mind is that none of these conditions is ?sacred?. They only depend upon what type of space you select. One is no or less ?true? than the other, it depends upon your assumptions you choose.
As an example: Consider a triangle. On a plane (Euclidean) space the sum of the angles is exactly 180 degrees. On a concave space (for example a sphere) the sum of the angles is greater than 180 degrees. The ?same? triangle on a convex sphere the sum of the angles of the same triangle will be less than 180 degrees.
None of these is ?right? or ?wrong?. It only depends upon which geometry you choose!!
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