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Euclid's Fifth Postulate
Philosophical and Logical Problems Contained in Euclid's Fifth Postulate
Euclid gave the world much of the information it has on planar geometry in his five postulates. While the first four are relatively easy to understand, the fifth one is very difficult in relation to the others. It is this fifth postulate that many people feel can never be proven. There are those that say it is simply incorrect, those that say it's both true and false, and others that say there is no possible way to prove it, and Euclid himself may have realized that the task was impossible. His fifth postulate states:
If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles."
There are numerous problems with the fifth postulate, not the least of which is that is it independent from the other postulates. One cannot be used to prove the other, and one cannot be said to be a side-effect of the other. The fact that the fifth postulate is independent of all of the other four postulates makes proving it completely impossible, no matter how many attempts are made (Bogomolny, 2002).
There is a school of thought that says Euclid knew when he created the fifth postulate that it could not be proven, and it troubled him so he didn't use it or put in it any of his works for quite some time. After he did start using it, many people challenged it. Seven hundred years later Procus, who studied Euclid's works, said that it was not even really a postulate, but more of a theorem, and that it should be removed from Euclid's postulates.
For whatever reason, it was not removed. It is still included in the postulates today, and no one has been able to prove yet whether Euclid was correct or not. Whether or not Euclid knew all this will never be known, but it makes the whole discussion of the postulate and its workings very difficult to comprehend.
Science and mathematics like to have things presented to them that are clear cut. It becomes difficult to work with a postulate or a theorem and use it to arrive at an answer to a problem, when the method used can't even be proven. It renders the whole problem and solution suspect because there is no way to know if the answer is actually right. Euclid, when he created this fifth postulate, certainly gave mathematicians something to ponder long after he went to his grave.
Philosophically, it is possible that Euclid did know how to prove the postulate and just never left any record of it. It is also possible he knew that it wasn't correct, and left it there to worry others. We don't know, and history doesn't say. Either way, the postulate continues to be a point of contention between those that say it is just plain wrong and those that say it simply cannot be proven, which is not the same thing. Just because something isn't provable, doesn't mean that it's not accurate.
Another logical problem was dealt with by Procus. His work and desire to prove the fifth postulate was either right or wrong continued most of his life. He thought that the postulate should be removed, and if he could prove it wrong, perhaps that could be done. He left some information behind that he felt was proof, but upon further inspect it is discovered that he hadn't really proven anything.
The reason that he hadn't proven anything is because his proof rests on an assumption that parallel lines are "always a bounded distance apart" (Bogomolny). This assumption is actually equivalent to the fifth postulate, not proof of it. So far, no one has been able to find a way to prove the fifth postulate as correct or incorrect. It seems that with all of the technology available today that wasn't there is the times of Euclid or Procus, mathematicians would be able to say something definitive about the fifth postulate, but it remains a mystery.
There are other problems with the fifth postulate as well. According to some, Euclid's fifth postulate means that there is an upper limit to the area of a triangle, because the two sides that extend out and slowly come toward each eventually have to touch at some finite point in space.
If this is the case, than many others who have studied mathematics have been wrong in assuming that if the two sides of the triangle are slowly heading toward one another, that doesn't mean that they will ever meet at a common point in space. They can just go on. They wouldn't have to meet. But Euclid says yes, they do. They have to meet, and it doesn't make logical sense to many mathematicians and scholars.
There are philosophical problems as well as logical problems with Euclid's fifth postulate. If the two lines that are heading on a seeming collision course at some point in space never meet, what happens to them? There is no way to rationalize the non-meeting of the lines, since they are not parallel to each other, and they are not moving away from one another. The triangle is getting smaller at the end that they are heading toward, so they must be getting closer to one another.
Logic says that they must meet and cross at some point, and philosophy, and apparently Euclid, wonders if there is some way that they don't have to. By discussing both logical problems and philosophical problems with Euclid's fifth postulate, it becomes even more confusing. It seems to make no sense either way, and it is amazing that no one has been able to prove it after all of this time, but it is still unproven.
Theoretically it would be possible for the lines to move toward one another so slowly, because of the low degree of angle, that they take a huge amount of space to come together at the end. But is it possible to have such a slight angle that the lines are almost parallel? They would be so close to parallel at that point that the impression that they are drawing closer together wouldn't be noticed unless they were looked at over miles at one time. That must be possible, but they still must meet somewhere in infinity.
Perhaps Euclid was right and the lines do meet somewhere, but the angles can be so minute that the lines go on almost to infinity, and we don't have the capabilities to calculate just how far that is yet. Perhaps Euclid is wrong and lines will go on into infinity still never touching, but only being a hair's width apart. Mathematicians may never know, since they haven't discovered any way to prove Euclid's fifth postulate by now. It is looking like there will never be any logical way to prove it.
Philosophically, it's a bit like faith in something. One might believe in something, or believe in someone, without actually knowing, physically, that that something or someone exists. That's what faith and belief are all about -- the assumption that something is without the knowledge or the proof to show that it must be.
Another problem with the fifth postulate is that even though mathematicians cannot prove that it is either true or false, they can prove that it is both true and false (Parallel, 2002). In doing this, they used the simplified version of Euclid's fifth postulate, that is much easier to understand, and says "Through a given point only one line may be drawn parallel to a given line" (Parallel, 2002).
While it seems ridiculous that one only parallel line could be drawn, it must be taken into account that words are not an exact science. The definitions and meanings of words like 'line' can be argued, as can the space that one is working in. If one is working with a curved surface, then lines must be drawn much differently so as to be parallel than they do on a flat surface. Some of the pickiness over terms and ideas of space is why the fifth postulate is said to be both true and false. It depends on where one is and what kind of space one is working with whether the postulate holds true.
Later scholars tried to fix Euclid's 'flaw' by finding a different postulate to replace the fifth one, but Euclid really didn't have a flaw in any of his postulates (Bennett, 2000). Just because one of them could not be proven by conventional methods did not mean it was flawed. Geometry is a very difficult and complex subject, and people would be ridiculous to think that everything about it made perfect sense and was easy to understand.
One of the reasons that geometry is so complicated…[continue]
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