I would like to begin with an issue that I recently came across when I took a class on Logic. It started with Euclid, a Greek mathematician who, in “The Elements,” developed both Geometry and Logic.
Euclid built the study of Geometry on just a few very simple a priori assumptions, or postulates. The first four postulates are:
- A line segment connects any two points by a straight line;
- A straight line segment may be extended indefinitely in either direction;
- A circle can be created with a center point and a radius; and
- All right angles are equal.
According to every source I have found, Euclid’s Fifth Postulate cannot be proven. Euclid’s Fifth Postulate has to do with parallel lines–it assumes that two lines that are not parallel will eventually meet on the side where the two angles total less than 180 degrees. It is reported that the greatest minds have been trying to prove Euclid’s Fifth Postulate for 2300 years without success. Due to the work of Gauss, Beltrami, and other great mathematicians in the 1800s, it was reportedly shown that the idea of parallel lines cannot be proven. Therefore, we must accept the concept of parallel lines that are curved. This is developed in a book entitled “The Fifth Postulate,” by Jason Socrates Bardi.
Why is this important?
A geometrical proof may not seem that relevant to issues in the real world. After all, geometry is supposed to be abstract. But this is important because of the subsequent history of thought regarding the Fifth Postulate.
The failure of attempts to prove Euclid’s Fifth Postulate led to acceptance of the idea that parallel lines can be curved. Three mathematicians, Carl Friedrich Gauss, Niolai Lobachevsky, and Janos Bolyai, independently developed what is called Non-Euclidean Geometry. They all relied upon the failure of attempts to prove Euclid’s Fifth Postulate to build a geometry in which multiple “parallel” lines can run through a single point.
This, in turn, is important because it led to other scientific ideas that built the case for relativism. A letter to the editor published in Nature in 1920 (Euclid, Newton and Einstein, by W.G., Nature, Vol. 104, No. 2624, Feb. 12, 1920, p. 627) stated that acceptance of Non-Euclidean Geometry prepares the mind for reception of Einstein’s theories of relativity. Of course, relativity is not the same as relativism. But once a person accepts the ideas that time and length are variable based upon speed and that space is curved, one is mentally prepared to also accept relativism.
The proof of Euclid’s Fifth Postulate does not necessarily undermine relativity theory. However, it does provide an impetus for questioning one’s core beliefs.
The tale of Euclid’s Fifth Postulate is one of the clearest examples I have ever run across of the fact that information provided for public consumption is not always accurate. I find it difficult to believe that in 2300 years nobody could come up with a proof. Parallel lines are intuitive; they are not conceptually difficult.
Furthermore, there seems to be fallacious thinking in the way the failure to prove Euclid’s Fifth Postulate has been viewed.
First, Euclid’s “Elements” serves two functions. It develops both geometry and logic. The value for purposes of logic stems from the fact that Euclid began with very simple assumptions. He went so far as to apparently disallow a ruler, instead relying only upon an unmarked straightedge and a compass. Every proposition logically builds upon the few initial assumptions.
But geometry need not rely solely upon logic. Even if it were logically impossible to prove the existence of parallel lines based upon Euclid’s postulates, that would not mean parallel lines do not exist! Suppose an owner asked a contractor to build the Empire State Building with only rock, scissors, and paper. The contractor would likely say it could be built, but better tools and materials would be needed. Likewise, geometry can be built upon foundations other than the postulates Euclid chose.
Another issue is the idea that somehow abstract parallel lines can bend after they reach the edge of the paper. This is one of the assumptions that led to Non-Euclidean Geometry. Gauss, Lobachevsky, and Boyai apparently believed that “while the cat’s away, the mice are at play.” That conclusion from the mere failure of attempts to prove parallel lines from Euclid’s assumptions strikes me as twisted thinking.
From Euclid we get the idea of the Cartesian coordinate system. The Cartesian system is built upon orthogonal perpendicular axes in three dimensions. It is the familiar “XYZ” system in which points are located based upon their distance from the origin in three dimensions. For example, x=5, y=2, and z=1 means start from the origin (0) and move 5 across, 2 up, and 1 over. Based upon the proof on this site, there is no reason to believe that abstract parallel lines will not remain parallel indefinitely.
Relationship between Euclidean and Non-Euclidean Geometry
If Euclid’s Fifth Postulate can be proven, it would no longer possible to accept that parallel lines are non-Euclidean. Whenever parallel lines are extended, equal perpendiculars can also be constructed that keep them equal lengths apart using the above method.
Non-Euclidian Geometry does not change Euclidean Geometry. For example, a “triangle” in spherical geometry does not meet the Euclidian definition. Its angles are not all in the same plane and its connecting lines are not straight. Any Non-Euclidean shape can also be described in a Euclidean reference frame.
Consequently, Non-Euclidean Geometry is merely a special case of Euclidian Geometry. It is a way of describing points and relationship according to different principles. It could be said that Non-Euclidean Geometry reverts back to the Babylon-Egyptian geometry–Euclid’s geometry is usually understood to be a step forward because it was built on abstract concepts–because it takes geometry out of the abstract and binds it to some real (or conceptually real) shape, such as a sphere.