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The Instigator
Pro (for)
Winning
23 Points
The Contender
Con (against)
Losing
9 Points

# Two parallel lines can intersect.

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Post Voting Period
The voting period for this debate has ended.
after 8 votes the winner is...
JonathanCrane
 Voting Style: Open Point System: 7 Point Started: 5/29/2013 Category: Science Updated: 5 years ago Status: Post Voting Period Viewed: 7,856 times Debate No: 34337
Debate Rounds (4)

Pro

Ave.

Topic: Two parallel lines can intersect.
Position: Pro/For
Category: Science
Rounds: 4
Voting Period: 1 Week

Time to Argue: 72 Hours
Argument Max: 8,000 Characters

I am taking the Pro position. It is my burden to demonstrate that two parallel lines can intersect each other. It is the burden of Con to demonstrate that the arguments I use cannot affirm the resolution. The winner of the argument will be the side that demonstrates their case beyond a preponderance of the evidence. In other words, if 51% of the evidence favors one side, that side should win arguments.

There is not a category for this debate that I find suitable. I chose the closest thing to the subject, science. I prefer this debate to be in a mathematics section, but such a section does not exist.

There are four rounds in this debate. The first round is for acceptance of the rules and framework. I will present my opening statement in the subsequent round, and the next rounds will be devoted to rebuttals.

The voting period will be one week. Each side will have seventy-two hours to post each round.

The maximum number of characters is eight thousand. Using pictures in order to demonstrate geometric concepts is allowed in this debate.

Vote comments are allowed. Two lines that are parallel is defined as two lines with the same slope. This is an example of such a situation.

Intersection is defined as a point that is common to lines or surfaces that intersect.

This debate is not specific to any geometry.

Vale.

Con

Since Pro failed to provide definitions I would like to supply them.

Parallel- extending in the same direction, everywhere equidistant, and not meeting. [1]

Parallel Lines- Two lines on a plane that never meet. They are always the same distance apart. [2]

Geometry- a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids. [3]

Good luck in the up-incoming debate PRO.

[1] http://www.merriam-webster.com...
[2] http://www.mathsisfun.com...
[3] http://www.merriam-webster.com...

Pro

Ave.

Con asserts that I have not provided any definitions for parallel lines. This is not a factual assertion, and anyone who reads the first round would know that I provided definitions for the resolution. Two parallel lines was defined as two lines that have the same slope. Intersection was defined as a point that is common to two lines or surfaces that intersect. Therefore, my burden of proof is to demonstrate that two lines with the same slope can have a point that is common between them. I have provided all of the relevant definitions for this debate. Con's definitions are superfluous and unnecessary, and they should be ignored because all relevant terms have been defined already.

Con is really just trying to define his way to a victory. You can't enter a debate about parallel lines intersecting and then define parallel lines as lines that never touch. It's like entering a debate called 'Public health care is better than private health care' and defining public health care as 'the system of health care that is inferior to private health care'. It's a semantic trick called the 'definist fallacy' [1]. I'm not suggesting that Con did this on purpose, as I don't think there is any evidence of that, but it's a good reason to ignore the definitions he offered.

Now that the unpleasant business is out of the way, we can talk about geometry. There's a problem with trying to define the word geometry. It's rather like talking about sports. Saying 'I play sports' doesn't tell us much about what you're actually doing. There are sports like football, soccer, swimming, etc. Which sport do you play? Geometry is similar to this. Saying 'geometry' is like saying 'sports'. There are types of geometry in the same way that there are different types of sports. There's Euclidean geometry, hyperbolic geometry, sphreical geometry, absolute geometry, and others [2]. In the first round, I indicated that we wouldn't be focusing on any specfic geometry. So, taking an example from one geometry is good enough for my case.

Most people learn in high school geometry that two parallel lines never touch. It's so intuitively obvious that nobody who learns it ever questions it. You just imagine those two lines going on forever, never touching. However, geometry seems to be a lot more imaginative than we are. The idea that two parallel lines will never intersect is an aspect of Euclidean geometry. It's called the fifth postulate. The problem is that there are many geometries that reject this postulate. Spherical and hyperbolic, geometries I named earlier, are examples of this. My third reference talks about the implications of hyperbolic geometry, or 'saddle geometry'. (All pictures and text are from the website, I claim no credit.)

 In hyperbolic geometry, the sum of the angles of a triangle is less than 180°. In hyperbolic geometry, triangles with the same angles have the same areas.

 There are no similar triangles in hyperbolic geometry. In hyperbolic space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right. Lines can be drawn in hyperbolic space that are parallel (do not intersect). Actually, many lines can be drawn parallel to a given line through a given point.

It demonstrates that two parallel lines can touch each other quite nicely, which is due to the positive and negative curvature of the 'saddle'. Here's a picture. [4]

As the above diagram demonstrates, hyperbolic geometry allows two parallel lines to eventually diverge and intersect. The fifth postulate isn't a necessary truth, and it's only for true Euclidean geometry. Once you step outside of Euclidean geometry, two parallel lines can clearly touch. (It's also interesting that this diagram shows a triangle with an angle sum that is less than 180, which shows that another Euclidean postulate doesn't always have to be true.)

I'll turn this debate over to Con. I thank him for his wishes of good luck, and hope the best for him as well.

Vale.

References
1: http://www.logicallyfallacious.com...
2: http://soler7.com...
3: http://regentsprep.org...
4: http://www.debate.org...

Con

Why do you think I'm trying to define my way to victory? I decided to post clear definitions, I always use the 1st round of an argument for acceptance and Definitions, I also tend to add to the definitions as debates continue for clarity, there are a lot of youths who use Debate.Org.

Hyperbolic Geometry is a non-Euclidean geometry that rejects the validity of Euclid"s fifth, the parallel postulate, because of this it differs from many common types of geometry. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In hyperbolic geometry, two parallel lines are taken to converge in one direction and diverge in the other. In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles. In Euclidean, polygons of differing areas can be similar; and in hyperbolic, similar polygons of differing areas do not exist.

The axiom in Euclidian Geometry is that parallel lines never intersect.

The axiom in non-Euclidian Geometry is that parallel lines intersect at infinity, this is more like a Illusion though, and if you don't believe me consider this.

Consider an isosceles triangle (where 2 side are equal in length) - if you increases the length of the two equal sides the angle subtended by these lines decreases; as the length of the lines tend to infinity, so the angle tends to zero and the lines are therefore parallel. But as it is a triangle they must meet at infinity, though infinity itself is not a point.

Equally, if you and a friend drove 2 cars both 7 feet apart at a 180 degree angle (Straight Line) through an infinite field of snow the tire tracks would never meet.

1/9 = 0.1111111111111111111111111111...
2/9 = 0.2222222222222222222222222222...
3/9 = 0.3333333333333333333333333333...
4/9 = 0.4444444444444444444444444444...
5/9 = 0.5555555555555555555555555555...
6/9 = 0.6666666666666666666666666666...
7/9 = 0.7777777777777777777777777777...
8/9 = 0.8888888888888888888888888888...
9/9 = 0.9999999999999999999999999999...

But 9/9 is quite obviously =1 therefore
0.9999999999999999999999999999... = 1

So you might argue that at infinity there is still a small distance between the lines of the triangle, but the above equation (that is both right and wrong at the same time depending on the form of Geometry) this isn't the case.

Kudos on your argument Pro, it was well written and I wish you luck in the 3rd round.

Pro

Ave.

Con asked me why I think he's trying to define his way to victory. This is an odd request from my viewpoint, because I articulated my reasoning in an entire paragraph. Firstly, I don't think Con, when typing out his argument, was actually thinking of using definitions to win the debate. I just think that his definitions can't be accepted because they're not neutral. Remember my analogy to a debate about public v.s. private health care, and one side tries to define public health care as 'a health care system that is inferior to private health care'. It defines public health care in a way that assumes the very thing we're debating. On the same token, defining parallel lines as 'not meeting' is just assuming what we're supposed to be debating.The definitions should be rejected for that reason.

I understand that it's important to have clear definitions for a debate's resolution. However, all of the terms in the resolution were defined in the first round. There was no need for Con to post any new definitions because all of the relevant definitions had been given.

Now, let's return to geometry. I think it's peculiar to claim that parallel lines can't intersect. It has very strange implications. Is projective geometry, where 'parallel lines may intersect', completely wrong? [1] Of course it isn't. They intersect just fine. How, then, can one argue that parallel lines cannot intersect?

Con states that an axiom of the geometries I'm using is that parallel lines intersect at an infinity. This might be true for some non-Euclidean geometries, but the geometry I used, hyperbolic, has no such axioms. Con doesn't provide any sources in this entire round, so there's no reason to believe that this is true other than his word. His word is wrong, and I even consulted my math teacher to see if this claim was true. She said it wasn't. There is no reason to believe that this an axiom of the geometries I'm using.

Con proceeds to give an argument that is, in my opinion, bizarre. Remember, I gave an argument in favor of the resolution based on hyperbolic geometry. This is the geometry of what is called a "saddle", which can be seen in the last picture. Hyperbolic geometry does not deal with flat surfaces, but surfaces with curvature. Con's argument, which talks about you and your friend driving cars, deals with flat surfaces. There is no way to demonstrate my argument to be false using flat surfaces, because you're dealing with different geometries. [2] Besides, the entire argument is predicated on an unsupported claim, which is that my geometries have the axiom of 'two parallel lines intersect at infinity'.

This debate isn't about one specific geometry. It is about any form of geometry. [3] I only need to show that parallel lines intersect in one, such as hyperbolic or spherical. It could be the case that parallel lines do not intersect when you're dealing with flat surfaces. This does not mean Con has won the debate, because all I need to do is show that they intersect when dealing with something else, like saddles. The argument from saddles hasn't been answered to, because there is no way to move from 'parallel lines do not intersect when dealing with flat surfaces' to 'parallel lines do not intersect when dealing with saddles'. It is a non-sequitur.

In conclusion, there are no reasons to think that parallel lines cannot intersect. I have given a good example of a case where parallel lines do intersect. I have also provided more sources than Con has, so my factual assertions have more weight than Con's. Therefore, the preponderance of evidence favors a Pro vote.

Vale.

References
1: http://robotics.stanford.edu...
2: http://en.wikipedia.org...
3; Consult the first round of the debate

Con

1. Non-Euclidean Geometry has been used historically as a cover for bad math. Its needless complexity, and its inherent lack of rigor have opened it up to broad and one might say universal misuse. [1]

2. Infinity itself isn't a point proving that most non-Euclidean Parallel Lines doesn't intersect, no source is needed for this obvious statement.

3. [2]
-Euclidean geometry is a mathematical well-known system attributed to the Greek mathematician Euclid of Alexandria.
Euclid's text Elements was the first systematic discussion of geometry.
-It has been one of the most influential books in history, as much for its method as for its mathematical content.
-Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fitted together into a comprehensive deductive and logical system.
- Euclidean has been validated and acknowledged by almost all Mathematicians, and is in constant use in the field of Computers. While Non-Euclidean has bee ignored by many great Mathematicians, often hard to prove, and hard to understand.

http://milesmathis.com... [1]
http://www.sciencedaily.com... [2]

Pro

Ave.

Con cites Miles Mathis to argue that non-Euclidean geometry has been abused as a cover for bad mathematics. I'm not sure what the application of non-Euclidean geometry has to do with its actual truth. People like Deepak Chopra abuse quantum mechanics regularly, but this obviously doesn't demonstrate that quantum mechanics is false. It is simply a non-sequitur to suggest that, because an idea has been misapplied and abused, it is therefore not true.

I also have to question the validity of citing Miles Mathis. Who is Miles Mathis? I've done extensive research on the name and haven't found anything that shows he's qualified to discuss mathematics.I understand that he's published articles and two books on mathematics, but there is nothing which indicates that his ideas are taken seriously by mathematicians. Personally, I can't take anyone who claims that pi is equal to four as a serious person. Yes, you read that correctly. Miles Mathis actually thinks that pi is equal to four. [1] This guy is the basis of Con's argument?

I didn't ask Con to prove that infinity is not a point. I asked Con to show that 'an axiom of the geometries I'm using is that parallel lines intersect at an infinity.' Con misses the point of what I said, so he doesn't actually give a source to prove the earlier claim. Therefore, it's a failed argument.

Con then makes three factual claims.

1: Euclidean geometry is a mathematical well-known system attributed to the Greek mathematician Euclid of Alexandria.
2: Euclid's text Elements was the first systematic discussion of geometry.
3: It has been one of the most influential books in history, as much for its method as for its mathematical content.
4: Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fitted together into a comprehensive deductive and logical system.

All four of these statements are completely irrelevant to the resolution. We are debating whether two parallel lines can intersect. We are not debating whether Euclid made Euclidean geometry, what the first systematic discussion of geometry was, how influental his work is, or why his work was original. Stating these four facts means nothing.

Con makes a fifth factual assertion, which I seperated from the other four because it's actually relevant.

Euclidean has been validated and acknowledged by almost all Mathematicians, and is in constant use in the field of Computers. While Non-Euclidean has bee ignored by many great Mathematicians, often hard to prove, and hard to understand.

According to Con, this statement can be found by clicking on his second reference. I clicked on his second reference, did some investigating, and the above statement appears nowhere. I copied the text of the statement to see if I could find some results. I didn't get any. I even used control + f just to make sure. The above statement is not in his source, so it has no factual basis. (It's also important to point out that the word 'been' is spelled incorrectly. Supposedly, Con copy and pasted this from his reference, yet they spelled the word been wrong. I would not expect this from a serious website like ScienceDaily. It seems clear to be that this spelling error was due to Con typing the statement himself rather than getting it from his source.)

Let's assume it was in his source, however. Maybe I missed something and Con will point it out in the next round. The first thing to realize is that mathematicians like Lobachevsky, Gauss, and Bolyai were involved in creating and developing non-Euclidean geometry. Suggesting that it hasn't been taken seriously by great mathematicians is absurd. The second thing to realize is that 'hard to understand' has nothing to do with truth. Nuclear physics is hard to understand. That doesn't mean it's false.

I'm dissapointed to see the massive number of points that Con dropped. I argued that his analogy to you and your friend riding a car was flawed because it dealed with flat surfaces. This was ignored. I argued that projective geometry has parallel lines which intersect. This was ignored. My diagram of two parallel lines diverging? Similarly ignored. Con's entire round was extremely derivative, and he took virtually every single thing he said from his sources, which caused him to drop several points and make many irrelevant assertions.

The sheer number of dropped arguments, and the apparent deception applied by Con that I demonstrated above, warrant argument and conduct votes. The preponderance of evidence favors my side. Vote Pro.

Vale.

References

1: http://milesmathis.com...

Con

Conduct
Pro must just be laughing after turning this into a troll debate.
" The sheer number of dropped arguments, and the apparent deception applied by Con that I demonstrated above, warrant argument and conduct votes."
If I recall, I had 2 rounds in which I made arguments ad backed them, in no way is this statement correct. As you can tell, I haven't dropped or denied any of my own arguments, I had ten points, I made 5 in the second round, and 3 in the third round.

Sources
Pro made a heavy effort to attack my sources
" It is simply a non-sequitur to suggest that, because an idea has been misapplied and abused, it is therefore not true."
I don't know if he even read what he said, i never said Non-Euclidean Geometry wasn't true, I said it was needlessly complex, i said it is constantly misused, and i said it was a poor way of doing Math, I never denied it altogether.

Pro even attacked miles Mathis, below you will find a Miles Mathis quote on what he was actually talking about.

"For all those going ballistic over my title, I repeat and stress that this paper applies to kinematic situations, not to static situations. I am analyzing an orbit, which is caused by motion and includes the time variable. In that situation, `0; becomes 4. When measuring your waistline, you are not creating an orbit, and you can keep `0; for that. So quit writing me nasty, uninformed letters." [1]

Pro eve suggested that was I said about infinity wasn't valid, although in truth Pro said.
"This debate is not specific to any geometry."

To anyone who has read the entire debate, I thank you and hope for a vote and/or comment.

Source: http://milesmathis.com... [1]
18 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Enji 5 years ago
Con's definition of parallel lines is contingent upon Euclidean geometry - since the debate was not limited to Euclidean geometry Pro's definition was better.
Posted by JonathanCrane 5 years ago
No, it's just an assumption.
Posted by ClassicRobert 5 years ago
I may be wrong, but I'm pretty sure Con's definition of a parallel line was correct. The equivalent slopes part is just a property of parallel lines.
Posted by JonathanCrane 5 years ago
I don't understand this line of argument at all. If you have two parallel lines on a flat surface, and then you curve the surface, causing the lines to diverge, how in any sense were they perpendicular the whole time?
Posted by uhex116 5 years ago
Lines are only parallel if they don't intersect, if you want to prove parallel lines intersect and you succeed then you've just proved that those lines aren't parallel and you get back to the start.
Posted by the_croftmeister 5 years ago
To be completely fair, since pro stated that the argument was not specific to any particular geometry then neither side could possibly have a winning argument since without an underlying geometry the statement is neither or both true and false.
Posted by Enji 5 years ago
I should clarify that I'm not giving conduct against Pro for Con's ignorance - rather the debate could only reasonably be expected to be taken by someone with Con's ignorance.

RyuuKyuzo seems to think that someone who is more knowledgeable about geometry could convincingly argue to the contrary - if anyone can find such an argument, I'll change my vote.
Posted by Enji 5 years ago
Con's decision to accept was based on ignorance which would be expected from anyone who has only learned about geometry in high school (and judging from the comments, this ignorance is somewhat common).
Posted by F-16_Fighting_Falcon 5 years ago
Enji, why does Pro lose conduct for a bad faith resolution? Wouldn't the fact that Con accepted the debate negate any inherent unfairness in the resolution?
Posted by Ragnar 5 years ago
Glancing over things, those look like curves instead of lines to me... However, this level of math is way over my head.
8 votes have been placed for this debate. Showing 1 through 8 records.