GENIUS FORUMS

Victor wrote:Parallel lines on 2D plane can meet, or they may be more than two parallel lines going though a point -- it all depends on curvature.

Victor forgets the difference between 'space' and 'plane' in his eager to prove other people wrong.

Wikipediais our friend:

"In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitely thin sheet oriented in some space."

So David was correct is stating that: "The fact still remains that parallel lines on a 2-D plane can never meet".

David Quinn wrote: If two lines can meet, by whatever means, then by definition they are not parallel.

In non-Euclidean geometry the definitions are more murky but many still do use definition above. Some call it 'ultra parallel' though and refer to "geodesics" instead of lines.

Hahaha. David, you totally miss the point. Parallel lines on 2D plane can meet, or they may be more than two parallel lines going though a point -- it all depends on curvature. Just like in 3D space.

If two lines can meet, by whatever means, then by definition they are not parallel.

Victor should have stated this as "two lines that cross a third line at right angles" - in a Euclidean geometry, such lines will be parallel (i.e., always equadistant from each other). In a non-Euclidean geometry, they will converge or diverge.

So if curvature causes two initially parallel lines to meet, then all that's happened is that the initial parallel lines have been eliminated and a new pair of non-parallel lines have taken their place.

I would rather say that in non-Euclidean geometries, the idea of unique parallel lines is not valid. In an elliptic space, there are no parallel lines, while in a hyperbolic space, there are hyper-parallel and ultra-parallel lines.

David Quinn wrote:If two lines can meet, by whatever means, then by definition they are not parallel.

Hence the point that given a line and a point off that line, you may be able to draw no parallel lines through that point, only one line, or more than one line, depending on the type of geometry you work with.

So if curvature causes two initially parallel lines to meet, then all that's happened is that the initial parallel lines have been eliminated and a new pair of non-parallel lines have taken their place.

In other words, Euclid's point still holds.

No, you bloody idiot, Euclid's point is that given a line and a separate point, you can draw only one and exactly one straight line through that point which is parallel to the first line. And no, it doesn't hold.

No, it doesn't break down. You're doing the very thing you have accused me of doing - namely, in order to make your point you are extending the definition of "parallel lines" to include lines which do in fact meet.

I used a conceptual shorthand. The actual Euclid's 5th axiom is as I have described it above.

You're engaging in self-contradiction here. If it is true that a person "can have certainty or relevance to the world, but not both", then it is equally true that a person who thinks he can have both is deluded. Thus, instantly, a piece of certainty which is relevant to the world arises.

You lack understanding, grasshoppa. :)

In short, your position on this matter (and Quine's) is invalid.

A position cannot be invalid. An argument can be invalid, if it employs invalid syntactic manipulations. An argument can be unsound, if it is either invalid or relies on false premises. This is an important distinction -- you can make a totally valid argument which is profoundly unsound.

Victor,

Euclid's point is that given a line and a separate point, you can draw only one and exactly one straight line through that point which is parallel to the first line. And no, it doesn't hold.

It does hold for lines on a flat plane, which was the context of Euclid's point. Whatever happens on curved planes is irrelevant to this.

DQ: You're engaging in self-contradiction here. If it is true that a person "can have certainty or relevance to the world, but not both", then it is equally true that a person who thinks he can have both is deluded. Thus, instantly, a piece of certainty which is relevant to the world arises.

V: You lack understanding, grasshoppa. :)

No, I think you've been found out.

DQ: In short, your position on this matter (and Quine's) is invalid.

V: A position cannot be invalid. An argument can be invalid, if it employs invalid syntactic manipulations. An argument can be unsound, if it is either invalid or relies on false premises. This is an important distinction -- you can make a totally valid argument which is profoundly unsound.

You're not dealing with the issue at hand. You're employing a common trick used by lawyers and politicians to deflect attention.

I will take it as read that you have no answer to my point.

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