To infinity and beyond

[lemit]

I'm really confused now, because I had heard or read the infinity line (as it were) so many times that I thought it was common. I now think it may have been a way to get me to take my childish questions and go elsewhere. Such as:

 

"No, really, what happens to parallel lines? Do they meet somewhere?"

 

"Sure, of course, they meet in . . . uh . . . let's see . . . Infinity! Yeah, that's it! They meet in Infinity! All right? Now, please let me get back to my work."

 

So now I guess I wonder if Infinity is used as a Geometrical trashcan or just a Pedagogical trashcan. Of course, I would welcome an answer of neither or both, if either of those answers is defensible.

 

Thanks for not continuing my supposition that my question is childish and meaningless, even if it is one or the other--although those qualities might constitute parallel lines.

 

--lemit

(even more confused)

[C1ay]

Statements such as "intersect at infinity" must always be taken with a grain of salt. Similarly for saying that "one over zero is infinity" which is handy in computing limits but in terms of real numbers is really just a shorthand for "the reciprocal of an infinitesimal term is an infinite term".

 

And in effect that's all I'm really saying. Parallel lines are always parallel even when some math construct fails and indicates otherwise. You can show algebraically that 1=2 but in reality 1 never really equals 2, the math just appears to make it look that way.

 

In the case of parallelism, this cannot be done; there seems no way to "define" parallel lines that merely extends the axioms. So no, there is no such definition, parallelism must be grafted on as an additional postulate.

 

Sure it can. We are the creators of language and the definitions of terms it relies on. We can write our definitions however we like and even make them conditional. I am perfectly well satisfied with the definitions of parallel from one reference to another and see no real conflict arising from them. As a corollary though we must realize that just because we have defined what parallel lines are does not necessarily mean that they exist. If in fact lines that appear parallel ultimately converge at some point then they were never really parallel to begin with, according to the definition we have given that term.

[Michael Mooney]

Cool thread. Where were you guys when I was arguing against the non-Euclidean trashing of Euclid's fifth.

I'm still left wondering what kind of "reality" the non-Euclidean paradigm represents when, in that realm, parallel lines can intersect.

 

Same "reality" I guess as where there are no "straight lines", space "itself" being curved and all.

 

I especially appreciated C1ay's statements:

-

And in effect that's all I'm really saying. Parallel lines are always parallel even when some math construct fails and indicates otherwise. You can show algebraically that 1=2 but in reality 1 never really equals 2, the math just appears to make it look that way.

 

(You don't seem to understand the dogma here C1ay, that math is always superior to mere reason and common sense.)

...

and lemit's quote:

"No, really, what happens to parallel lines? Do they meet somewhere?"

 

"Sure, of course, they meet in . . . uh . . . let's see . . . Infinity! Yeah, that's it! They meet in Infinity! All right? Now, please let me get back to my work."

... and statements:

So now I guess I wonder if Infinity is used as a Geometrical trashcan or just a Pedagogical trashcan. Of course, I would welcome an answer of neither or both, if either of those answers is defensible.

 

Thanks.

Michael

[lemit]

So infinity can be a sort of science fiction construct in which the rules we live by don't necessarily apply? A mathematical dumpster for troublesome theories? The place where parallel lines are finally joined and live happily ever, for infinity?

 

That sounds good to me.

 

I like stories with happy endings, including mathematical ones--stories or endings or both.

 

--lemit

[Michael Mooney]

So infinity can be a sort of science fiction construct in which the rules we live by don't necessarily apply? A mathematical dumpster for troublesome theories? The place where parallel lines are finally joined and live happily ever, for infinity?

 

That sounds good to me.

 

I like stories with happy endings, including mathematical ones--stories or endings or both.

 

--lemit

I appreciate your sense of humor about it.

But seriously (more or less)... seems to me the thread title is an oxymoron. Infinity means without end in my lexicon of meaningful words.

Same thing applied to space being infinite. What kind of end could anyone propose... and, hey, what would be on "the other side" of this "end?" Could it be more space... infinite space? Ooopse!

 

I think that the problem here is that the finite mind can not comprehend infinity, yet it can not reasonably impose an "end of space" or a la la land in which parallel lines do meet, in violation of how they are geometrically defined... points well made above in this thread.

Michael

[Ben]

seems to me the thread title is an oxymoron.

Well, I chose that title a) to grab attention and :shrug: to be slightly light-hearted. Sorry if it confused you.

Infinity means without end in my lexicon of meaningful words.

I am sorry to say you cannot impose your own lexicon on math, or any branch of science for that matter.

 

Have you ever taken any courses in science or math? If so, you will recall that a very large part of learning any rigourous subject is learning its own lexicon. In math, infinity has at least two different (but very precise) meanings. If you had read th OP you would have gathered that; in fact, that is what this whole thread was supposed to about

a la la land in which parallel lines do meet, in violation of how they are geometrically defined

You don't get it do you? Even in Euclidean geometry, "not meeting" is not part of the definition of parallelism - open any high school text.

 

And, hey! Euclidean geometry is not only not the only geometry, it is, for reasons that you, lemit and C1ay seem not to apprciate, not the most natural.

[Qfwfq]

So now I guess I wonder if Infinity is used as a Geometrical trashcan or just a Pedagogical trashcan. Of course, I would welcome an answer of neither or both, if either of those answers is defensible.

Neither.

 

We were being somewhat jocular, especially cuz it doesn't quite fall within the cardinality issues and the work of Cantor, but one possible reply to that question came to my mind: the generalized intersection is at infinity in both ways along the pair of lines and one could view it in terms of infinite period, so they "continue on" toward here from behind us. The meaning of the word beyond in the thread title could become clearer if folks would let Ben get on with it. It may seem an oxymoron because it actually is a bit lax, but at least it's short enough to fit into the title.:D

 

You can show algebraically that 1=2 but in reality 1 never really equals 2, the math just appears to make it look that way.

Actually, you can't truly show that. The math that appears to make it look like 1=2 is just plain bad math, whereas [imath]\frac10=\infty[/imath] is shorthand, and handy in a way although crass, for something that actually is meaningful.

 

Ben's point is very subtle and did not imply any impossibility of coherence. He more or less meant that Euclid's fifth is necessary for the notion of parallelism to make sense, at that stage of construction, and it's also sufficient to enable saying OK, so that one and only line we say is parallel to the first. Only later in the process it becomes possible to formulate other statements, as being equivalent and folks often arrived at some, when trying to figure if that axiom is redundant. Possible revisions of the axiomatic structure were proposed and from the whole endeavour came the discovery that other coherent geometries are possible. All this to say, beware of taking defs to be an absolute decree when it comes to the foundations of mathematics.

 

An alternative to my previous example in terms of the limit for the angle approaching zero would be to do it in the geometry of the spherical surface but with the angle fixed to zero and consider the limit in the arbitrary radius approaching infinity. This of course brings in another one of those irksome things (topological pathology but see my above reply to Lemit): the plane is a spherical surface of infinite radius and the same goes for straight line vs. circumference. Straight means zero curvature = infinite radius of curvature.

[C1ay]

Even in Euclidean geometry, "not meeting" is not part of the definition of parallelism - open any high school text.

 

Visit parallel at Wolfram. Not meeting is the same as not intersecting. Please cite a reference that shows Wolfram's definition of parallel to be incorrect.

 

Actually, you can't truly show that. The math that appears to make it look like 1=2 is just plain bad math, whereas [imath]\frac10=\infty[/imath] is shorthand, and handy in a way although crass, for something that actually is meaningful.

 

That's just more bad math since division by 0 is undefined. Zero is nothing and you cannot divide something into a nothing number of parts.

[Ben]

the plane is a spherical surface of infinite radius and the same goes for straight line vs. circumference.

Brilliant expalnation!! Thanks for that, I mean it. Now where's that beer you promised?

 

Anyway, I just thought an equivalent way to explain this to those that don't understand math (though what they are doing in this subforum is for them to answer.....).

 

Take a glass ball and place it in a sheet of paper. Now make the smallest possible mark at the North Pole - in fact let it be a zero-dimensional mark (howls form the non-mathematicians?).

 

Now place light source directly above the North Pole. Its shadow is, of course in in our paper.

 

As we gradually move the light source down some meridian, and we will find the shadow remains on our paper provided only that we keep enlarging our paper.

 

But when the light source is aligned with the equator, the trajectory of the shadow of our point on the North Pole is parallel to our paper, and the only way to keep the image of the North Pole on our paper is to have the paper infinitely large.

 

Parallel lines meet at infinity

[Michael Mooney]

Ben,

Sorry to have butted in on your mathematical discourse on infinity.

You are quite right to say:

 

I am sorry to say you cannot impose your own lexicon on math, or any branch of science for that matter.

 

I was just wondering, with lemit, if parallel lines intersecting in some sense of infinity made any sense in "the real world," i.e., the realm of existing phenomena and scientific observation.

 

And, with C1ay, just wondered if the math had referents in said realm.

 

Have you ever taken any courses in science or math? If so, you will recall that a very large part of learning any rigourous subject is learning its own lexicon. In math, infinity has at least two different (but very precise) meanings. If you had read th OP you would have gathered that; in fact, that is what this whole thread was supposed to about

 

I took math including algebra, geometry and a distasteful taste of calculus in high school, but I am quite lame in these areas, save geometry. I did read the OP and it was, as you know, over my head. (But I'm really good at understanding *concepts that make sense.*)

You don't get it do you? Even in Euclidean geometry, "not meeting" is not part of the definition of parallelism - open any high school text.

I guess not. All definitions I know of parallel lines say that they don't intersect or share a point in common (aka, "meeting.")

 

I also know that non-Euclidean geometry posits "curved space"(of various shapes), but it does not address the ontology of what it is that curves. So, if it remains simply emptiness, volume without walls or limits in the known three spacial dimensions, then there is nothing to curve, and non-Euclidean geometry is rendered nonsense ('cept as a devised matrix or metric to make the math more graphic and clear.)

And, hey! Euclidean geometry is not only not the only geometry, it is, for reasons that you, lemit and C1ay seem not to apprciate, not the most natural.

 

Edit: Missed the "not." So how unnatural is it? Supernatural, or merely mental without external referents in the observable cosmos?

Michael

[C1ay]

Parallel lines meet at infinity

 

And if and when they meet then they are no longer parallel...

[Jay-qu]

And if and when they meet then they are no longer parallel...

the problem is that infinity is no where or when in the conventional sense..

[lemit]

the problem is that infinity is no where or when in the conventional sense..

 

Infinity is that magical place where parallel lines meet, fall in love, and live happily after after.

 

--lemit

[modest]

Wikipedia says something that I find a little odd without giving a source:

If the word "parallel" is defined as constant separation, the Euclid's fifth postulate can be proved from his first four postulates. However, if the definition is taken so that parallel lines are lines that do not intersect, Euclid's fifth postulate is independent to his first four postulates.

Parallel postulate - Wikipedia, the free encyclopedia

Then wolfram seems to say what I would expect:

Equidistance Postulate:

Parallel lines are everywhere equidistant. This postulate is equivalent to the

parallel axiom

.

Equidistance Postulate -- from Wolfram MathWorld

Is there a significant difference between the "equidistant postulate" and Euclid's 5th, and so much so that equidistant can be proved from the first 4 postulates?

 

~modest

[Jay-qu]

Infinity is that magical place where parallel lines meet, fall in love, and live happily after after.

 

--lemit

On this note I suggest that Ben be allowed to continue with the thread as he directed and further discussion of parallel lines be redirect into its own thread.

[Qfwfq]

Yes I think we should let Ben reach his goals and then go on with other related things (which can actually include the matter of parallel lines).:) Meantime just a quick couple of points:

 

That's just more bad math since division by 0 is undefined. Zero is nothing and you cannot divide something into a nothing number of parts.

Try reading Affinely Extended Real Numbers -- from Wolfram MathWorld and I doubt Ben meant to make a semantic distinction between meet and intersect. We should perhaps recall Euclid's fifth:

 

If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid's Postulates -- from Wolfram MathWorld

[C1ay]

This postulate is equivalent to what is known as the parallel postulate. [/indent]

Euclid's Postulates -- from Wolfram MathWorld

 

Do recall though that I distinctly made the point that I was not arguing in support of Euclid's 5th postulate, only the definition of "parallel". Definitions are just that, definitions, not axioms, postulates or theorems and they are meaningless as definitions if we throw out their meaning for convenience. What exactly would be the point in defining a term to mean something for the sake of language and communication and then saying, "well, it really only means that most of the time but not always"?