Victor's Lines

[Katy]

OK so this makes little sense for genius other than it being in Vic's podcast, but, well, it makes less sense at the other board he posts at.

OK so am I on the right track here? (I fail miserably at trying to look this up online too much technical words so I ask you)

Paralell lines are lines that never intersect one another. In Euclidian space there can only be two through a point and a line because every other point would have to be equidistant from the original line an only one line can be drawn between any two points.

Eliptical space is essentially folding in on itself so that a line on it winds up being round like a line on the inside of a ball. It's possible to make a line where they appear to be equidistant, but since one has to be bigger than the other to do so (ie lines of lattitude). Since lines are no longer infinite (sorta) if there's gotta be the same number of points on a line... what?

we redefine line to mean an equator or great circle? Which would mean they all have to intersect at some point? But why isn't it a line that goes around a piece of it? Can you draw a line between any two points or only specific points now? Or are lines still 180degrees and having smaller angles inside changes things so lines automatically go around an equator?

Hyperbolic space works because if there's more than 360degrees of circle but a line is still 180, then you can move the line through the point to different angles and make multiple lines that won't intersect? They can touch because they can both be 180degrees and go off in different directions so that they haven't intersected.

Beh, that's pretty much pulled out thin air (along with a couple wiki pages that I couldn't understand a word of but had pictures that probably helped and a "math for non-math people" class I took some years ago that didn't do much aside from define hyperbolic space...) so I'm probably completely off? Yes? No?

[Katy]

oh - define a line as a great circle because like those things that you drop a coin in and watch it spin around where the coin has to be upright the whole time to keep spinning or else it falls? So it goes where it wants to go? (I'm not sure that sentence made sense)

[vicdan]

Parallel lines in elliptic space will intersect because they run longtitudinally, like meridians, not latitudinally like the equator and the tropic lines. In fact, on a globe, all latitudinal lines except for the equator are actually not straight in the elliptic geometry itself -- a straight line beginning on the tropic line would actually cross the equator quarter of a way around the globe, touch the corresponding opposite tropic half way around, touch the equator again three quarters of the way around, and return to its starting point after a full globe circumnavigation.

In an elliptic 2D space like the surface of a globe, any straight line you draw, no matter where, will eventually cross the equator twice. Any straight line will eventually cross any other straight line; but latitudinal lines aren't straight.

[Katy]

In fact, on a globe, all latitudinal lines except for the equator are actually not straight in the elliptic geometry itself -- a straight line beginning on the tropic line would actually cross the equator quarter of a way around the globe, touch the corresponding opposite tropic half way around, touch the equator again three quarters of the way around, and return to its starting point after a full globe circumnavigation.

Not straight because they are not naturally what would happen if something fell from there? Are they curving "upwards"

Am I on the right track with the hyperbolic space?

[DHodges]

"Latitudinal lines" in a spherical geometry are actually circles, not lines. This is most obvious if you look at one near a pole.

Are they curving "upwards"

Yes, they have a constant curvature and are equadistant from a central point - i.e., they are circles.

[vicdan]

Not straight because they are not naturally what would happen if something fell from there? Are they curving "upwards"

If the northern hemisphere, yes. If you follow a longtitudinal line in the northern hemisphere westward, from the point of view of the elliptic geometry, it will be curving to the right all the time.

Am I on the right track with the hyperbolic space?

Yup.

If you visualize elliptic space as a spheroid, you can visualize hyperbolic space as a saddle.

[Katy]

"Latitudinal lines" in a spherical geometry are actually circles, not lines. This is most obvious if you look at one near a pole.

Are they curving "upwards"

Yes, they have a constant curvature and are equadistant from a central point - i.e., they are circles.

Is a longitudanal line a circle and a line?

[DHodges]

Is a longitudanal line a circle and a line?

Yes. On a sphere, a line is a limiting case of a circle , having zero curvature.

It is still a circle, being equadistant from a point - actually, from two points, on opposite sides of the sphere from each other. E.g., the equator is a circle around the north pole and also a circle around the south pole.

[Katy]

So a line is still defined as 180degress?

[David Quinn]

The fact still remains that parallel lines on a 2-D plane can never meet, which is what Euclid originally proved. Thus, Victor's point on the show that, 2500 years later, we have somehow disproved this is false. All that has been proved is that parallel lines on something other than a 2-D plane can meet.

This might be an interesting fact, but it doesn't falsify Euclid's original proposition in any way.

-

[vicdan]

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.

Sure, you can define a geometry which confirms to all five of Euclid's postulates, and where for any line and a point off that line, one and only one parallel line exists; but it would be so by definition -- you try to establish a correspondence between this definitional geometry and real space,and it breaks down. Just as I had spoken about. You can certainty, or you can have relevance to the world, but not both.

[Katy]

David - this is still 2 dimentional space. Because you're still stuck in a plane and can't move around in space. that plane just happens to be curved.

[DHodges]

Victor, was your point that , in Euclid's day, geometry was considered to be absolute truth, rather than an axiomatic system separate from reality?

[vicdan]

More or less. My point, specifically as related to Dan's and David's argument, was that in Euclid's day, his formulation of the 5th postulate was considered true as indubitable; yet as we see, being unable to doubt something coherently does not in fact establish that this something is true.

[DHodges]

being unable to doubt something coherently does not in fact establish that this something is true.

This seems closely related to (the logical fallacy) the Argument from Ignorance: I don't see how it could be otherwise, therefore it must be so.

[Dan Rowden]

You can't doubt something that is defined. It's nonsensical. Definitions aren't true or false. The duality of certain/uncertain doesn't meaningfully apply to definitions. They just are what they are. The formal content of geometry doesn't exist in the world outside the abstract. There's no such thing as planes and lines and squares and so forth. There are only nominal empirical correlates (this fact doesn't detract from their utility, of course). This has no relation to the arguments we make about things and existence as there is nothing but the content of those ideas in the world.

[Matt Gregory]

More or less. My point, specifically as related to Dan's and David's argument, was that in Euclid's day, his formulation of the 5th postulate was considered true as indubitable; yet as we see, being unable to doubt something coherently does not in fact establish that this something is true.

That may be true in certain circumstances, but Euclid's 5th postulate is not one of them, at least not in the way you are arguing. It is indubitable within the parameters that he defined it in (a flat curvature), and simply positing the existence of a different type of curvature doesn't change the logic of the flat curvature that he was speaking with respect to.

[ExpectantlyIronic]

You can't doubt something that is defined.

It is indubitable within the parameters that he defined it in

So you can't doubt an assertion? Okay...

In that case I shall define definitions as doubtable. There. Problem solved. Definitions are now doubtable by definition.

Dan,

There's no such thing as planes and lines and squares and so forth.

There's no such thing as apples or the color red either. Such things are just terms we use to describe the world. Exactly like the terms "planes" and "lines".

This has no relation to the arguments we make about things and existence as there is nothing but the content of those ideas in the world.

Yet you can only assert that they are in the world. Whereas I can show you a line, or a plane, or an apple. Oh, but the plane isn't an actual plane, and the line isn't an actual line. Kind of like an apple isn't the perfect apple form I have in my head. Without such a form, how should I know that different lines, or planes, or apples are all the same? What then is this form? Does it exist in some odd Platonic realm? Obviously not. It is simply our idea of something that is formed upon observing it.

[Matt Gregory]

Dan wrote:

You can't doubt something that is defined. It's nonsensical. Definitions aren't true or false. The duality of certain/uncertain doesn't meaningfully apply to definitions. They just are what they are.

You can doubt them in relation to one another. You can doubt whether or not your definition is the same as someone else's that you may have learned it from or may have taught it to. You can doubt your ability to remember things and thus question the fact of defining something as you defined it a moment ago, or even the fact that you ever defined it at all if you really wanted to. But yeah, without relating a definition to another, what could it possibly mean to doubt it?


EI wrote:

You can't doubt something that is defined.


It is indubitable within the parameters that he defined it in

So you can't doubt an assertion? Okay...

In that case I shall define definitions as doubtable. There. Problem solved. Definitions are now doubtable by definition.

How would you define "doubt" then?

[ExpectantlyIronic]

Matt,

How would you define "doubt" then?

You know I was just making a point. Nevertheless, you came up with a reasonable list of things that could be doubted about definitions. If someone were to say "a fish is a flying bird", we should assume them to be jesting.

But yeah, without relating a definition to another, what could it possibly mean to doubt it?

You can doubt whether a term is being used in its conventional manner. Even if one had a language consisting of one sign, that sign could be interpreted as used in a way we would say is incorrect. For instance, if the term "slab" is always used to tell someone that they are to move a slab, and one day it is used in the presence of no slabs, it would be reasonable to doubt that the individual saying the word is using it correctly. Language is a social thing, and it depends upon certain conventions to remain useful and coherent. Given that the term "fish" isn't normally used to indicate a flying bird, it is correct to say that such a usage is wrong.

[Matt Gregory]

Matt,

How would you define "doubt" then?

You know I was just making a point.

You were missing the point.

Nevertheless, you came up with a reasonable list of things that could be doubted about definitions. If someone were to say "a fish is a flying bird", we should assume them to be jesting.

Why assume anything about a situation that has not even occurred? Maybe there is a legitimate reason to define "fish" in that way that we haven't thought of. We don't need to cling to intellectual biases like this.

But yeah, without relating a definition to another, what could it possibly mean to doubt it?

You can doubt whether a term is being used in its conventional manner.

I covered this in my last post. That's doubting the comparison between two definitions. How would you doubt a definition itself, though? It's nonsensical.

Language is a social thing, and it depends upon certain conventions to remain useful and coherent.

Language is also useful for thinking and communicating with oneself. You could create your own language and keep a journal with it and it would be totally useful, yet not social.

Given that the term "fish" isn't normally used to indicate a flying bird, it is correct to say that such a usage is wrong.

Only if you choose to follow the dictionary's definition. It's each person's intellectual decision to follow the dictionary on a particular definition. Thinkers routinely override dictionary definitions for the sake of greater clarity, and that's a very important skill. George Orwell made some good demonstrations of what happens when people have no mastery of language.

[Jamesh]

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.

Surely a curved plane must automatically be considered as a 3D environment. If not then why?

[ExpectantlyIronic]

Matt,

Why assume anything about a situation that has not even occurred? Maybe there is a legitimate reason to define "fish" in that way that we haven't thought of. We don't need to cling to intellectual biases like this.

If you are not interpreting what someone is saying to you, then you are not listening to them. An interpretation depends on a number of assumptions. Mind you, to assume something tentatively isn't to ignore evidence that suggests your assumption was wrong. For instance, I assume that this post will be read by someone. I wouldn't see any reason to post it otherwise. I imagine that you assumed that I intended to use the term "assume" to mean something much stronger then I did (or at least assumed that I meant something by what I said).

I covered this in my last post. That's doubting the comparison between two definitions. How would you doubt a definition itself, though? It's nonsensical.

What is a definition beyond a statement explaining how a word is used? To doubt a definition is to doubt that such a statement is an accurate portrayal of how the word in question is typically used. I do agree, though, that there is nothing to doubt about someones explanation of how they use a term.

Language is also useful for thinking and communicating with oneself. You could create your own language and keep a journal with it and it would be totally useful, yet not social.

Language is useful for thinking only if one exists in a society where it is used. This is to say that there is no survival benefit to knowing a private language in complete and life-long social isolation. There is, admittedly, one possible exception to this that I can think of. It could be useful to keep written reminders of various things, but I have trouble visualizing what sort of reminders those might be for someone with obligations only to themselves. Perhaps one might want to draw a map to where they hid some food from other animals. Although they would have to remember what the signs that comprised the map meant in such a case, so I'm not so certain that even that would be useful.

Only if you choose to follow the dictionary's definition. It's each person's intellectual decision to follow the dictionary on a particular definition. Thinkers routinely override dictionary definitions for the sake of greater clarity, and that's a very important skill.

Nevermind the dictionary. We don't typically learn how words are used from it, so I don't see what it has to do with what I'm saying. In most cases we learn how a word is used in the situation in which it is used.

[ExpectantlyIronic]

Surely a curved plane must automatically be considered as a 3D environment. If not then why?

Imagine that you were to draw something on a sheet of paper. If the paper were bent slightly, would you then say that the drawing was 3D? I imagine not. For something to be 2D it doesn't need to be flat, it just can't have any depth to it.

[David Quinn]

Victor,

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.

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.

Sure, you can define a geometry which confirms to all five of Euclid's postulates, and where for any line and a point off that line, one and only one parallel line exists; but it would be so by definition -- you try to establish a correspondence between this definitional geometry and real space,and it breaks down.

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. But unlike how I operate, this really is a sham piece of reasoning.

Just as I had spoken about. You can certainty, or you can have relevance to the world, but not both.

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.

In other words, the certainty of the initial truth automatically extends into a certainty about something within the empirical world - namely, that the living, breathing person residing in the empirical world is in a state of delusion. But this is precisely the sort of certainty which the initial truth claims is impossible. Thus, the initial truth breaks down from its own momentum and becomes incoherent.

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

-