Notices
Results 1 to 25 of 25
Like Tree1Likes
  • 1 Post By Liongold

Thread: Proving All Five of Euclid's Postulates

  1. #1 Proving All Five of Euclid's Postulates 
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    After a lot of time spent trying to prove all five of Euclid's postulates, I now believe I have succeeded. I reproduce here my 'proofs' asking anyone to find a mistake in my use of rigour or logic, as I wish to verify my proofs.

    As some of these proofs use other postulates, the proofs I am writing down here will be out of order. I start with the fifth postulate, as this is in most need of verification. I use Euclid's original version before going on to show that it is equally valid for Playfair's equivalent version.

    Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

    Proof: This means to say that the two lines meet on the side whose sum is less than 180 degrees.

    From the diagram, we see that if we extend the lines indefinitely, we eventually get a triangle.

    It has been proved that the sum of the angles of a triangle sum to 180 degrees.

    This implies that the two angles formed by the third line which goes through the other two lines) cannot be equal to 180 degrees, as it would then violate the angle sum property of that triangle.

    The triangle is only possible if the two angles are not equal to 180 degrees or more. This implies that the lines may only meet on those sides where the angles together sum up to less than 180 degrees.

    Hence, Euclid's version is proved.

    For Playfair's version, which states that given a line l and a point P, there is one and only one line through P parallel to l,

    We take the line and the point. We must then construct a line from the point to the line perepndicular to line l. Now, apart from the special case where the line through P is perpendicular to to l, we see that the question has now reverted to the Euclid's original version.

    Now, any line whose angle with the perpendicular produces a sum of the interior angles on the same side which is less than 180 degrees must go through l.
    Only those lines whose angle when taken with the angle formed by l with the perpendicular produce a sum of 180 degrees may not meet the line l.

    An obtuse angle implies that the line meets on the other side, where the angle formed is acute, as in order to obey the rule that the angles on a line must be equal to 180 degrees, the other angle must always be acute. An acute angle implies that the line meets on that side, as an acute angle added to a right angle is always less than 180 degrees.

    Therefore, this implies that no line which produces an obtuse or acute angle may be parallel to l. Therefore, the only other angle possible is a right angle, which taken in conjunction with the other interior angle forms 180 degrees, implying that any line which produces 90 degrees with the perpendicular is always parallel to line l.

    As only one such line is possible, this proves Playfair's equivalent version.

    I apologise if the language may make the text a little hard to understand.

    Postulate 4: All right angles are equal to each other.

    Proof: Let us assume that this is not true, and all right angles are not equal to each other.

    This instantly leads to a contradiction, as it implies that a triangle may have more than one right angle.

    Therefore, by reducio ad absurdum, we see that all right angles must be equal to each other.

    Another proof is by looking at the definition of a right angle. A right angle is any angle equal to 90 degrees.

    We know that 90 = 90 = 90 ...

    We see therefore that all right angles are equal to 90 degrees and as 90 degrees is equal to 90 degrees,

    this implies that all right angles are equal to each other.

    Hence proved.

    Postulate 2: A finite straight line may be extended indefinitely.

    Proof: There are an infinite number of points in a region.

    This implies that there are an infinite number of collinear points, as any operation with infinity that does not involve another infinity results in infinity. By collinear, I mean points between which a straight line may be drawn. ( I clarify this in order to prevent accusations of using a circular argument with the first postulate)

    This implies that a line may be extended infinitely.

    Postulate 3: A circle may be drawn with any center and any radius.

    Note: By the term "collinear", I mean that it is possible to draw straight line from it another specific point.

    Proof: This is a little trickier to prove, so I divided the problem down into two parts. I will first prove that a circle may have any radius.

    Taking point A as centre, we may look at the radius as a line. By Postulate 2, we know that line may be extended indefinitely.

    Therefore, the radius may be extended indefinitely.

    This implies that a circle may have any radius.

    The second part is to prove that a circle may have any center.

    Taking any collinear point, we see that it is possible to draw a straight line between this and any other straight line.

    By rotating the line by 360 degrees, we obtain a circle.

    This implies that any collinear point may be the center of a circle, as the straight line that can be drawn may be considered a radius, and rotating the radius produces a circle.

    Our next challenge is to show that all points are collinear, in order to fully prove this postulate. Fortunately, that is also the next postulate.

    Postulate 1: A straight line may be drawn from any point to any other point.

    Proof: Finally proved only yesterday, we must refer to the third and second postulate in order to fully prove this one. In order to prevent accusations of lack of rigor, I will use the still incomplete third postulate only in those cases where it may be applied.

    Take any two collinear points A and B, where collinear means it is possible to draw a straight line between them. It is possible therefore to draw a straight line between them.

    Now any points on the line AB must also be collinear, for otherwise a straight line could not have been drawn. Hence, it is also possible to draw a line from A to any point upon the line.

    Now, let us rotate the line, such that the collinear point A is the centre of the circle so produced.

    Now, it is possible to draw a straight line from A to any point in the circle. This is because the radius of the circle is a straight line, and upon rotation, it covers all the points in the circle, implying that a straight line can be drawn from all the points covered by the radius to the center of the circle, which is A.

    Therefore all points in the circle are collinear to A i.e. they produce a straight line to A.

    It is easy to show that all points in the plane are collinear: merely extend the radius infinitely, so the resultant circle encompasses the entire region.

    Repeating the above for any point in the circle, we see that it is possible to draw a straight line from that point to any other point in its circle, and so on.

    From the information above, we can deduce that all points are collinear to each other, or

    It is possible to draw a straight line from any one point to any other point.

    Hence proved.


    I apologise to anyone who has had difficulty understanding me through my use of language, and I am willing to explain it again to anyone who has had difficulty understanding.

    Thank you for taking the time to read this.


    trfrm likes this.
    Reply With Quote  
     

  2.  
     

  3. #2  
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    But … aren’t Euclid’s postulates supposed to be the axioms of Euclidean geometry? They are what, taken together, defines Euclidean geometry. What you are doing is trying to prove a definition – which doesn’t make any sense at all. :?


    Reply With Quote  
     

  4. #3  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    Certianly true, yes, but then why did so many mathematicians expend their efforts trying to prove the fifth postulate?

    The axioms are not definitions, as such. They are claims e.g. you could claim that it is possible to draw a circle with any radius and centre, but you don't know that for sure. Once you prove it, you know it will hold steadfast whatever you do.

    What I am doing is verifying claims. Can you spot any mistake in my proofs?
    Reply With Quote  
     

  5. #4  
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    What mathematicians were doing was to try and derive the fifth postulate from the other four postulates in the mistaken assumption that the fifth postulate was redundant. In other words, they thought that Euclidean geometry could be based on just the first four postulates. Their efforts floundered until non-Euclidean geometry was discovered, which finally proved once and for all that the fifth postulate was not redundant – that is, it cannot be derived from the other postulates. In other words, if you leave out the fifth postulate, you won’t get Euclidean geometry any more.

    You need to understand that mathematics is not always about how to do things: often it is why you do things that matters. In the case of the fifth postulate, it’s not about how to prove it. Rather, it’s about why you don’t prove it. It’s a definition (or part of a definition system) and you don’t prove definitions. True, people did try once, long ago, to prove the fifth postulate because they thought it wasn’t definition – but non-Euclidean geometry has confirmed that it is a definition and so stands in no need of proof.

    Don’t worry, if you still don’t understand, DrRocket will hopefully return from his Christmas break and explain everything all over to you. :P
    Reply With Quote  
     

  6. #5  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    All right, so the fifth postualte is merely a definition. All I am trying to do is show that it is applicable anywhere in Euclidean geometry, that it holds true in Euclidean geometry.

    Likewise, I'm trying to show that the other four also hold true in Euclidean geometry, and that there can be no doubt about their correctness.

    You could say I'm trying to prove definitions, but the postulates themselves are not definitions. They do not define Euclidean geometry exclusively; the fourth postulate for example is true in every geometry. In such a case, would you say the postulate that all right angles are equal is a definition of a geometry if it holds for every single geometry? Likewise, the first, second and third all hold in different geometries. How can you say that they are definitions if they do not define any one single thing?
    Reply With Quote  
     

  7. #6  
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    When you wish to prove something, you need to base it on a definition or some definitions. Otherwise you’d just be going round in meaningless circles.

    Euclid’s postulates provide the definition from which all provable statements in Euclidean geometry are derived – statements such as “the interior angles of a triangle add up to 180°”, and so on. You want to “prove” Euclid’s postultes? Then what are you going to base your proof on? What definition are you going to use to show that Euclid postulates follow from that definition?

    All right, this is what you can do. Instead of the Euclidean postulates, choose your own postulates. Choose what statements you like to take for granted. These statements may be provable by Euclid’s postulates, but you may, for the present purpose, assume that they are true. Then, and only then, can you legitimately attempt to prove Euclid’s postulates from your own. You won’t be then be trying prove Euclid’s postulates in a vacuum.

    In mathematics, you do not prove something out of nothing. Mathematics is not politics.
    Reply With Quote  
     

  8. #7  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    I'll see what I can do.

    However, if I may, Euclid's postulates do not directly have anything to say about angles (apart from the fifth postulate, which is independent from Euclidean geometry), nor do they much affect the relam of angles. So could I use angles to prove them?

    Also, what about lines? Euclid's first postulate says that it is possible to draw a straight line from any point to any other. However, given that a triangle has been drawn, can I then say that it is possible to draw a straight line from the three points (where the arms meet) of a triangle? That it is true in this case?

    I'm still trying to understand why they provide the definition for Euclidean geometry, when they so clearly do not resemble definitions. For example, there is no reason to propose the first postulate when you can simply draw a line and say that for the two points, it is possible to draw a straight line. They resemble claims in their nature, and I have always thought that mathematics is driven by proofs, and not by philosphical considerations such as a definition.

    I understand why you cannot use a definition to prove a definition. Yet, when I am using methods that do not rely upon the first four postulates - can,indeed, be used without them - , I hardly think I'm leading to a contradiction.

    I'm sorry, but I still think I need further convincing.
    Reply With Quote  
     

  9. #8  
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    Well, I’m not familiar with all the finer details of the Euclidean geometry. I’m afraid you’ll have to wait for DrRocket to come back and help you out.

    Quote Originally Posted by Liongold
    I'm still trying to understand why they provide the definition for Euclidean geometry, when they so clearly do not resemble definitions.
    They are what are called postulates or axioms. No, they are not definitions individually – but taken together, they help to define Euclidean geometry. Suppose someone asks you, “What is Euclidean geometry?” You might answer by saying that Euclidean geometry is a system of points and lines in space that satisfy Euclid’s postulates – and you go on and list Euclid’s postulates. This is called an axiomatic definition. (To be absolutely rigorous, you will also have to explain what you mean by “point”, “line” and “space” and define their properties – again I leave it to DrRocket to sort out the details. )
    Reply With Quote  
     

  10. #9  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    I understand, now, what you mean by definitions in this case. Thanks for clearing that up.

    However, my other question is this: how exactly do I invaldiate myself if I'm using proofs that aren't based on Euclid's postulates? For example, looking at my proofs through your viewpoint, you could say my proofs for the second and fourth postulate do not base themselves on Euclidean geometry. So you could argue that they can be considered true. I then go on to show that my proof for the second postulate allows us to prove the third postulate (partially, but that is later resolved) before showing that these two taken together allow us to prove the first postulate.

    These four, when taken together, can provide a clear enough definition of enough portions of Euclidean geometry to allow us to use their results for the fifth postulate.
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  11. #10  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by JaneBennet
    What mathematicians were doing was to try and derive the fifth postulate from the other four postulates in the mistaken assumption that the fifth postulate was redundant. In other words, they thought that Euclidean geometry could be based on just the first four postulates. Their efforts floundered until non-Euclidean geometry was discovered, which finally proved once and for all that the fifth postulate was not redundant – that is, it cannot be derived from the other postulates. In other words, if you leave out the fifth postulate, you won’t get Euclidean geometry any more.

    You need to understand that mathematics is not always about how to do things: often it is why you do things that matters. In the case of the fifth postulate, it’s not about how to prove it. Rather, it’s about why you don’t prove it. It’s a definition (or part of a definition system) and you don’t prove definitions. True, people did try once, long ago, to prove the fifth postulate because they thought it wasn’t definition – but non-Euclidean geometry has confirmed that it is a definition and so stands in no need of proof.

    Don’t worry, if you still don’t understand, DrRocket will hopefully return from his Christmas break and explain everything all over to you. :P
    No need. You are doing fine. You have also shown that the fifth postulate cannot be proved, for if it could there would be no non-Euclidean geometry.
    Reply With Quote  
     

  12. #11  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    Actually, my proof for the fifth holds only in a geometry where the angles of a triangle are equal to 180 degrees. It shows that the fifth is correct and holds true only when this condition is satisfied.
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  13. #12  
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    Actually you didn’t properly prove the fifth postulate. The postulate states that if the interior angles of two lines with a third line add to less than 180°, the two lines when extended will meet at a point. You didn’t prove that, but the converse instead. What you proved was that if two lines meet at a point, then their interior angles with a third line add to less than 180°. That is not the same thing.
    Reply With Quote  
     

  14. #13  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    Thank you for pointing that out. :-D

    I'll try and see if there is any way to correct it.
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  15. #14  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    Alright. Will this one work?

    We know that a triangle may not have more than one right angle. It is not possible for a triangle to be formed if two angles are greater or equal to 90 degrees. However, if two angles are collectively less than two right angles (they are acute), then it is possible to form a triangle from their lines.

    If it is possible to form a triangle, then they may not be parallel, but instead must meet at a point, for otherwise a triangle could not be formed. As long as the angles do not add upto 180 degrees, then a triangle is always possible and may be formed by extension.

    Does this finally imply that the fifth postulate is correct for Euclidena geometry?
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  16. #15  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    No replies?
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  17. #16  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by Liongold
    No replies?
    There is nothing worth replying to. Your argumens are circular. You cannot prove Euclid's axioms without assuming things that are equivalent to those axioms. For instance your use of the fact that there are 180 degrees in a triangle in the proof of the parallel axiom, is circular.

    You don't seem to really understand the issues.
    Reply With Quote  
     

  18. #17  
    Forum Sophomore
    Join Date
    Jan 2008
    Posts
    130
    Hi there!

    To me, having the postulates in the first place seems a little pointless... It is logical that a circle can be formed by a center and a radius, same with any two points forming a straight line... Note I'm refering to the first four postulates though...

    Unless I'm missing something deeper here?
    Reply With Quote  
     

  19. #18  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    Your arguments are circular. You cannot prove Euclid's axioms without assuming things that are equivalent to those axioms. For instance your use of the fact that there are 180 degrees in a triangle in the proof of the parallel axiom, is circular.
    The proof of a triangle having 180 degrees was proved using the first four postulates as the basis, before proving it on the basis of the knowledge of angles. I know you will be familiar with this.

    My proof for the parallel postulate is built first by proving the postulates which first laid the basis for the proof of the fact that a triangle has 180 degrees. Once that is proved, I can then proceed to use this simple nugget of information for the last postulate.

    In fact, if you do look at my proofs, please consider the second and fourth postulates to be completely proved, as they rely on other mathematics not connected with geometry for proof. I then used the second to help aid my proof for the third, before using both to ultimately prove the fourth.

    I have not assumed things equivalent to Euclid's axioms, as my proofs take nothing for granted. At least, I hope they do. I ask only that you look at them and my revised version of the parallel postulate once more, as I am quite sure that the proof of the angle sum property of a triangle is independent of the parallel postulate and can be taken as solid fact. Thank you.

    To me, having the postulates in the first place seems a little pointless... It is logical that a circle can be formed by a center and a radius, same with any two points forming a straight line... Note I'm refering to the first four postulates though...

    Unless I'm missing something deeper here?
    Mathematics is built on proof and logic. By taking these as postulates, Euclid avoided the difficulty of having to prove all of the postulates, and enabled them to be taken simply for granted. They are "assumed" to be the truth, which I do not think befits the logical foundations of mathematics.

    That is why I am trying to prove all five of them.
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  20. #19  
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    Liongold and rgba:

    It seems that you (still) don’t fully understand what is meant by an axiomatic approach in mathematics. Have a look at this: http://en.wikipedia.org/wiki/Axiomatic_system.

    You may or may not find it useful. Read it anyway, in case you do.
    Reply With Quote  
     

  21. #20  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    Thank you, JaneBennet. I did read the article, and I do know what an axiomatic approach is, but I didn't see the relevance. We have no reason to suspect that any of Euclid's postulates are independent. Euclid;s geometry may be axiomatic, but because we do not know if any postulates are independent, I don't see much relevance for the issue here.
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  22. #21  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by Liongold
    The proof of a triangle having 180 degrees was proved using the first four postulates as the basis, before proving it on the basis of the knowledge of angles. I know you will be familiar with this.
    You cannot possibly prove that a triangle has 180 degrees from the first four postulates. There exist other geometries, non-Euclidean geometries, that satisfy the first four postulates but in which the sum of the angles in a triangle is not 180 degrees.

    http://en.wikipedia.org/wiki/Non-euclidean_geometry

    The bottom line is that none of your proofs are valid and you need to pay attention to what Jane Bennett told you and learn something. It ought to be a hint that many very fine mathematicians have considered these problems over hundreds of years. The postulates cannot be proved without assuming others things that are equivalent to the postulates themselves.
    Reply With Quote  
     

  23. #22  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    I realise what you're saying, DrRocket, but I assure you I have not assumed things equivalent to the postulates themselves. I took cases where the postulates worked, and used them to show that they would work for all cases.

    For example, in postulate 2, I used the fact that there an infinity of points in a region to show that this must mean that there an infinite number of collinear points on a line, meaning that a line could be extended indefinitely. I then showed that it is possible to draw a cricle with any radius and with a collinear point as the centre. Finally, I took this case and showed that it implied that every point within a circle is collinear, and since a circle can be increased infinitely, an infinite number of points are collinear i.e. a straight line may be drawn between them, which in turn shows that a centre may be drawn with any center and radius.

    I am not cosidering the parallel postulate exclusively, and even then I am trying to prove it works only in Euclidean geometry which does hold that a triangle has 180 degrees. My proof does not work if a triangle cannot have 180 degrees, so the parallel postulate cannot be considered valid there.

    I am aware that mathematicians have considered these problems over several centuries, and I do not mean to imply they may be wrong. All I ask is evidence for any flaw in my proofs, as I sincerely believe that they can be proved, or an evaluation of my proofs. The parallel postulate is, I know, a special object, but I seek to prove its correctness in Euclidean geometry and not in any other. Thank you for your time in replying to me.
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  24. #23  
    Forum Masters Degree
    Join Date
    Dec 2008
    Posts
    627
    I see I have no replies. Best to let this thread die, then...
    In control lies inordinate freedom; in freedom lies inordinate control.
    Reply With Quote  
     

  25. #24  
    New Member
    Join Date
    Nov 2012
    Posts
    2
    Your "proofs" are full of mistakes and none of them are valid.
    I will go in the order of your "proofs"

    Postulate 5
    you start off with assuming that the lines will eventually meet and form a triangle, the point of a proof is to prove that something will happen, not assume; also, you cannot extend the lines at all because you haven't "prooved" the second postulate yet, which allows you to extend lines.
    Begrudging these mistakes, you later go on to state that the sum of the angles in a triangle is equal to 180 degrees, while this is true, it can only be proven with the 5th postulate itself

    Postulate 4
    Again you assume that a triangles angles add up to 180 degrees, and again this proof rests on several postulates, notably the fifth, o fwhich your proof is invalid. Next you state that a right angle is equal to 90 degrees, again technically true but you assume too much. A right angle is quite simply any angle such that when one line intersects another all four of the angles produced are the same. Centuries after this concept was developed an angle with 90 degress was stated to be a right angle, so your logic is again wrong. The only reason anyone could assign a degree value to a right angle at all is because all right angles are the same, not because they inherently equal 90 degrees (fun fact, for several months France decided that a right angle had 100 degrees, and consequently a circel would have 400, in it to simplify angles and trigonometry).

    Postulate 2
    Really this was an honest mistake that any mathematician could make. A point, at the time these postulates were first stated, was a line with no length. This does not imply however, as you assumed in your proof, that there exists any points in a given region. A point has to be made, declared if you will, none simply exist ni euclidean geometry until someone makes them, which is the basis for this postulate.

    Postulate 3 and 1
    These are somewhat more sound than your first two "proofs". However there exists a fundamental flaw in both proofs that completely destroys any and all attempts to prove them. Your proof of postulate 3 is based on the fact that postulate 1 is true, yet your proof for postulate 1 is based on the fact that postulate 3 is true. This is known as a loginc loop, one cannot be proven without the other, which i ntur nrequires the first to prove. The loop is a common problem facing people who try to prove Euclids Fifth.

    I am not trying to be mean here. I am tearing apart your proofs here to show you where you made mistakes so you will not make them again, after all that is the ultimate goal of learning.
    Reply With Quote  
     

  26. #25  
    New Member
    Join Date
    Nov 2012
    Posts
    2
    You made a valid attempt to prove the postulates, however you do not have a complete appreciation of the math, particularly geometry, that was available at the time these postualtes were derived by Euclid. All of the stuff about math and geometry especially can be traced back to these postulates and the subsequent proofs that came of them. As someone who has tried to prove Euclids Fifth postulate, I can assure you that you cannot use any of the math you are accustomed to. Within the limits of Euclidean Geometry, and even some Non-Euclidean Geometries, it is impossible to prove any of these postulates, my hope holding out for the fifth, of which I have submitted a proof to several people to review. Props for trying though. If this kind of stuff interests you, you might consider purchasing a copy of Euclids Elements and reading through the myriad of proofs that Euclid was able to derive from these simple postulates.
    Reply With Quote  
     

Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •