Hyper Dimensional Philosophizing

[PixyMisa]

I have stated that physical flat Euclidian two dimensional plane has two dimensions of length which are perpendicular to each other.

There's no such thing.

The meaning of dimension which I am clearly trying to focus on is the meaning of dimension as it is used when referring to the dimension of length in this space

That's your problem, right there.

GodMark2 has stated that the dimensions of this space can be length and angle

Where did he say that?

You seem to be unwilling to disagree with this. Yet from the way you write, you should be disagreeing. Why?

Why what?

I have stated that in the two dimensional plane we are talking about, you can create a coordinate system and how you place that coordinate system does not affect or change the dimensions of the two dimensional space. It does not matter whether or not you translate the origin of the coordinate system. It does not matter whether or not you rotate the direction of the coordinate system. It does not matter whether you use the Cartesian coordinate system or the polar coordinate system. The two dimensions of the plane remain two dimensions of length perpendicular to each other.

If you take a Euclidean space and assign the property of length to its dimensions, then that property remains regardless of the co-ordinate system you use.

 

[Oppressed]

Fredrik,

I agree that it makes some sense to say that the subset of the plane that's bounded by the circle "lies in the same two dimensions[1 or 2]" as the plane, but I would never say that myself because it's a strange thing to say. If someone says that to me, I'll translate it to "the interior of this circle is a subset of the plane" and think "why is he telling me something completely obvious in a very strange way?".

The reason I am stating the obvious is because GodMark2 has stated that by shifting to polar coordinates the dimensions of the real physical flat Euclidian two dimensional plane changes those dimensions to length and angle. Then he states angle is a unit and not a dimension.

You do not question this?

I have stated several times, trying in different ways, to make it clear I am talking about dimension as it is referenced/used in the two dimensions of this flat two dimensional plane.

I have stated the dimensions of this flat plane are two perpendicular lengths. Do you disagree with this?

I have stated that regardless of the coordinate system you use on this two dimensional plane, it does not change the two dimensions of the plane. Do you disagree with this?

GodMark2 responds with stating the dimensions for the two dimensional plane can be length and angle. I give the example of defining a circular subset of the two dimensional space to show that the dimensions of the space do not change. You say that is obvious, but apparently not to GodMark2. Why don’t you comment about that?

 

[Oppressed]

If you were talking about a physical object like a clock, yes, the axis would be perpendicular. But there's no such thing as a "real physical flat Euclidean two dimensional plane" or a "real physical flat Euclidian three dimensional space", nor can you construct one from the subset of the other. Planes, spaces, and sets are mathematical concepts. So none of this makes any sense.

You can define a polar co-ordinate system for a two-dimensional space. You don't need a third dimension for the axis to stick up into, because the axis isn't real.

I’m sorry, but what you say does not make sense to me.

Begin with basics.

You open your eyes and look around you. In addition to looking you listen and feel your surroundings. What is it you exist in?

We give a term to it, “SPACE”.

What is this “SPACE”?

We begin a process of trying to define it. We develop a model for this “SPACE” which has three dimensions, three lengths, perpendicular to each other. We can measure this space and do experiments on this space.

The simple model used as a base for this space is the Euclidian model.

This space is real.

There is a meaning to dimension when we speak of the three dimensions of space.

This is the meaning I am trying to focus us down onto.

Other meanings or uses of the meaning for dimension we disagree on.

But, we should be able to work our way towards an understanding of what dimension means when talking about the three dimensions of real space.



You don’t need to plot angle when using the polar coordinate system for a two dimensional space. But, if you are going to use angle as an axis of measure, then the axis for angle would be plotted perpendicular to the two dimensional plane.

 

[PixyMisa]

Space - in the sense of our Universe - is real.

It isn't Euclidean, though. That is, a mathematical model that accurately represents the properties of the space of our Universe is non-Euclidean.

What's more, that mathematical model is a mathematical model. As such, you don't need to define new dimensions for the axes of your co-ordinate systems; they are, after all, nothing but mathematical functions.

You can't take rules about the real world and apply them blindly to mathematical models, and you can't take rules about mathematical models and apply them blindly to the real world, or you end up talking nonsense.

 

[Oppressed]

PixyMisa,

By far, the most used model to represent the real physical space we live in is the flat Euclidian model. It is the simpler model developed and like other models of our real physical space, it is based on real observation, measurement and testing of our real world. It is the simpler and easier model to discuss the basic concept of what a dimension means without complicating it by going into something like the theory of relativity.

I have stated over and over again that the point of choosing this space and the simple model we have for this space is for the purpose of simplifying and narrowing down the topic under discussion, rather than having things become obscured by needless complications.

Unless you desire to cloud the issue, you should be able to help focus on the very specific example I have repeatedly referred to.


Have you tried creating the 3D plot and examining it yet?


Why are the two dimensions of a plane perpendicular to each other?

Well, consider the following. Think about vectors.

You have a one dimensional vector length “X1”. We define its direction in two dimensions as X1 = (1*a, 0*b) such that a is a unit vector lying in the first dimension of length which we will call D1 and that b is a unit vector lying in the second dimension of length which we will call D2.

Being as we are talking about a real space, we can observe and measure this.

Now, we define a second vector Y1 = (0*a, 1*b).

X1 has a quantity measurement of 1 in the D1 dimension but no (or zero) quantity measurement in the D2 dimension.

Similarly, Y1 has a quantity measurement of 1 in the D2 dimension but no quantity measurement in the D1 dimension.

You are familiar with how to determine if two vectors are perpendicular to each other.

Clearly X1 has only quantity in the first dimension and likewise Y1 has only quantity in the second dimension.

If third vector is defined, vector XY1 = (1*a, 1*b), this vector does not define a new or different dimension of the two dimensional plane, but it has components of measured quantity in both the first and the second dimension.

The two dimensions of length are perpendicular.

 

[PixyMisa]

You have a one dimensional vector length “X1”. We define its direction in two dimensions as X1 = (1*a, 0*b) such that a is a unit vector lying in the first dimension of length which we will call D1 and that b is a unit vector lying in the second dimension of length which we will call D2.

Being as we are talking about a real space, we can observe and measure this.

No you can't. Vectors are mathematical concepts. If you want to talk about a piece of string, talk about a piece of string.

I have stated over and over again that the point of choosing this space and the simple model we have for this space is for the purpose of simplifying and narrowing down the topic under discussion, rather than having things become obscured by needless complications.

Yes, you've said that, and then you invariably make a hash of various meanings of the word "dimension" and intermingle mathematical and physical concepts, so obviously it's not working so well.

I'm not sure what the point of the rest of that post was.

You have two pieces of string at right angles and a third at 45 degrees. Okay.

 

[Fredrik]

The reason I am stating the obvious is because GodMark2 has stated that by shifting to polar coordinates the dimensions[3 or 4] of the real physical flat Euclidian two dimensional[1 or 2] plane changes those dimensions[3 or 4] to length and angle. Then he states angle is a unit and not a dimension.

You do not question this?

First of all, stop using the word "dimension" in several places in a paragraph where it means different things in different places. (Note the numbers that I'm now routinely inserting when I'm quoting you. Those are the numbers of the definitions that make sense right there).

I haven't read every detail in the posts GodMark2 wrote to you, so I don't know exactly what's been said, but I do know that he has several times proposed that we use the word "unit" instead of "dimension" when we mean the 4th kind of dimension in my list. When you say that "he states angle is a unit and not a dimension" you're making it very clear that you have failed to notice this, or just chosen to ignore it.

This is what he must have been trying to tell you: "Angle" is a dimension[3 or 4] but it's not a dimension[1] and it's not a dimension[2].

Personally, I don't care if you use the word "unit" or "dimension" as long as you make it clear which definition you're using (and you almost never do).

GodMark2 is definitely aware that a change of coordinate systems doesn't change the dimensions[1 or 2] of a plane. If you think he isn't, it's just because you keep confusing different definitions of one word.

I have stated several times, trying in different ways, to make it clear I am talking about dimension as it is referenced/used in the two dimensions of this flat two dimensional plane.

I think that you may actually believe that, but you keep writing paragraphs where the word just can't mean the same thing at every place where it occurs. (This particular sentence is not an example of this).

I have stated the dimensions of this flat plane are two perpendicular lengths. Do you disagree with this?

I have stated that regardless of the coordinate system you use on this two dimensional plane, it does not change the two dimensions of the plane. Do you disagree with this?

Yes and no. I answered those questions thoroughly in my previous post. I feel like I've answered them in almost every post I've written in this thread, even before you asked.

GodMark2 responds with stating the dimensions[4] for the two dimensional[1 or 2] plane can be length and angle. I give the example of defining a circular subset of the two dimensional space to show that the dimensions[???] of the space do not change. You say that is obvious, but apparently not to GodMark2. Why don’t you comment about that?

What I said is obvious is just that a subset of a plane is a subset of a plane, nothing more. And again, you're using several different definitions of the same word in one paragraph without making it clear which ones you're using. I don't think the problem is just that you aren't expressing yourself clearly enough. I think you just don't pay any attention to what definitions the rest of us are using, possibly because you think it doesn't matter.

 

[Fredrik]

Oppressed, I strongly recommend that you read my posts #160 and #161 over and over again until you understand them. Everything you need to know is in there.

Also read the last paragraph of #168. It explains an important part of #160 in a better way.

 

[Oppressed]

Okay, let us go back to the basics.

At what point were we able to agree on anything.

(1) We have an observable phenomenon, the space that we exist in.

(2) A model for this space is the Euclidian model which states the space has three (type a) dimensions which are perpendicular to each other.

(3) To try and avoid misunderstanding, I will specify the meaning of dimension I am using by writing it as (type a) dimension. The (type a) dimension is the meaning of dimension when we speak about space, (the space identified in (2) above), as being in three dimensions. Therefore I am saying this space is in three (type a) dimensions.

Before I move to (4), who can agree with the above points? Note, at this point I am making no reference to any other meaning of dimension other than the (type a) dimension.

 

[Fredrik]

I agree with all of this (post #189) except for the inclusion of the word "perpendicular".

The next step is to define exactly what kind of mathematical structure we're going to use to represent space. The obvious choice is the vector space R³ (because that's the model used in Newtonian mechanics).

The next step after that is to give your (type a) dimension a precise meaning. I did that in #160 (in the algebra section).

 

[GodMark2]

GodMark2 has stated that the dimensions of this space can be length and angle, but then also stated that angle is a unit not a dimension.

No, I did not. I have not used the term dimension when describing anything for several pages. You have substituted the term dimension here, in an attempt to make it appear that I am being inconsistent.

I stated that the axes of the space describing a Euclidean plane cane be angle and distance.

I stated that radian is a unit, with which "angle" may be measured.

You keep insisting on using dimension, and not the other term that can adequately describe the concepts you are using. The only reason I can now consider for this, especially considering that you have now taken what I have stated using the separate terms and restated it using the single term, is that you only wish to be able to confuse the terms.

 

[GodMark2]

I haven't read every detail in the posts GodMark2 wrote to you, so I don't know exactly what's been said, but I do know that he has several times proposed that we use the word "unit" instead of "dimension" when we mean the 4th kind of dimension in my list. When you say that "he states angle is a unit and not a dimension" you're making it very clear that you have failed to notice this, or just chosen to ignore it.

Aw. I'm hurt. Not, you know, a lot or anything

But you and I have really been saying almost the exact same things, you with technically accurate mathematical language, and me with less jargon-driver engineering/physics language.

I proposed he use "unit" when referring to a unit (dimension [4]), and to "axis" when referring to an axis(dimension[1 or 2]).

 

[Oppressed]

I agree with all of this (post #189) except for the inclusion of the word "perpendicular".

(1) We have an observable phenomenon, the space that we exist in.
We agree.

(2) A model for this space is the Euclidian model which states the space has three (type a) dimensions which are perpendicular to each other.
We agree on the three dimensions but not the requirement for the dimensions to be mutually perpendicular.

(3) To try and avoid misunderstanding, I will specify the meaning of dimension I am using by writing it as (type a) dimension. The (type a) dimension is the meaning of dimension when we speak about space, (the space identified in (2) above), as being in three dimensions. Therefore I am saying this space is in three (type a) dimensions.
We agree that there is one meaning of dimension being spoken of here.

Okay, why are the 3 dimensions perpendicular or not perpendicular?

Let us begin with a point. We can mark a point in space and use it as a reference point. We can call that point the origin.

From that point we can place another point some distance away. The shortest distance between those two points defines a line segment. This line segment lies in one dimension of the three dimensional space. We can move a point back and forth along the line segment without moving in either of the other two dimensions.

We define a unit vector in that dimension and call it D1.

We then move a distance away from the origin and the line defined by the line segment so that the point does not lie in the same dimension D1. When moving the distance away from the origin point in this new dimension, we do not change or move in the first dimension. This movement in the new dimension is independent of D1 and that is why we can move in that dimension while not changing position in the first dimension.

This second line segment allows us to proceed as we did with the first line segment and define a unit vector in that dimension which we can call D2.

D1 and D2 are perpendicular to each other.

In vector terminology we have;

D1 = (1 in the first dimension, 0 in the second dimension) = (1,0)
D2 = (0 in the first dimension, 1 in the second dimension) = (0,1)

You understand vector, see if these vectors are perpendicular or not.

We can then perform this process one more time. Again beginning at the origin point, we can move away from the point such that we make no movement in either the first dimension or the second dimension and select another point. This wil result in a line segment lying in the third dimension relative to the first and second dimensions. We can then call this dimension D3.

Now we have three dimensions;

D1 = (1 in the 1st dim, 0 in the 2nd dim, 0 in the 3rd dim) = (1,0,0)
D2 = (0 in the 1st dim, 1 in the 2nd dim, 0 in the 3rd dim) = (0,1,0)
D3 = (0 in the 1st dim, 0 in the 2nd dim, 1 in the 3rd dim) = (0,0,1)

Do the math, the three vectors are perpendicular to each other.

 

[GodMark2]

(2) A model for this space is the Euclidian model which states the space has three (type a) dimensions which are perpendicular to each other.
We agree on the three dimensions but not the requirement for the dimensions to be mutually perpendicular.

(3) To try and avoid misunderstanding, I will specify the meaning of dimension I am using by writing it as (type a) dimension. The (type a) dimension is the meaning of dimension when we speak about space, (the space identified in (2) above), as being in three dimensions. Therefore I am saying this space is in three (type a) dimensions.
We agree that there is one meaning of dimension being spoken of here.

Okay, why are the 3 dimensions perpendicular or not perpendicular?

Let us begin with a point. We can mark a point in space and use it as a reference point. We can call that point the origin.

From that point we can place another point some distance away. The shortest distance between those two points defines a line segment. This line segment lies in one dimension of the three dimensional space. We can move a point back and forth along the line segment without moving in either of the other two dimensions.

If you restrict "dimensions (type a)" to length, then they must be mutually perpendicular.

But you don't need to have three lengths to define a space. You just need three "independent axes". "Independent axes" are spatial axes such that a change in the measure along one axis causes no change in any of the remaining axes.

Perpendicular linear axes are independent, but not all independent axes are perpendicular, nor are all axes linear.

 

[Oppressed]

GodMark2,

Given I am talking about (type a) dimension as the meaning of dimension when speaking about the 3 dimensions of length in the space we exist in, do you have 3 (type a) dimensions which are not three lengths to describe the three dimensions of space?

I’m only talking about one meaning for dimension here. I believe the meaning of this (type a) dimension can apply to more than just the (type a) dimension of length.

 

[GodMark2]

GodMark2,

Given I am talking about (type a) dimension as the meaning of dimension when speaking about the 3 dimensions of length in the space we exist in, do you have 3 (type a) dimensions which are not three lengths to describe the three dimensions of space?

Two angles and one length.

or

One angle and two lengths.

Each of those will describe any space that three lengths could.

I’m only talking about one meaning for dimension here. I believe the meaning of this (type a) dimension can apply to more than just the (type a) dimension of length.

I think you may still be trying to cram more than one meaning into this "Dimension (type a)".

 

[Fredrik]

Okay, why are the 3 dimensions perpendicular or not perpendicular?

I explained this in #160 (under "algebra"). You really should read it over and over again until you understand it. You may have to work through some of the details with a pen and paper to fully understand linear independence, but it's absolutely necessary that you do.

Now we have three dimensions;

D1 = (1 in the 1st dim, 0 in the 2nd dim, 0 in the 3rd dim) = (1,0,0)
D2 = (0 in the 1st dim, 1 in the 2nd dim, 0 in the 3rd dim) = (0,1,0)
D3 = (0 in the 1st dim, 0 in the 2nd dim, 1 in the 3rd dim) = (0,0,1)

Do the math, the three vectors are perpendicular to each other.

These three basis vectors will do just as well:

E1=(1,0,0)
E2=(1,1,0)
E3=(1,1,1)

Now you do the math.

You're doing one thing right here though. You want to define the word "dimension" in a particular context, and it would have been crazy to try to do that without first choosing the context, so I'm pleased to see that you have chosen to focus on the vector space structure of R³.

 

[Fredrik]

If you restrict "dimensions (type a)" to length, then they must be mutually perpendicular.

I don't know what it would mean to restrict (type a) dimensions to length. Does it mean that we choose to look at the vector space structure of R³ instead of at the manifold structure? If that's what you meant, then what you said in the text I quoted above is wrong. I don't need to explain why because you did so yourself in the text I'm quoting below.

But you don't need to have three lengths to define a space. You just need three "independent axes". "Independent axes" are spatial axes such that a change in the measure along one axis causes no change in any of the remaining axes.

Perpendicular linear axes are independent, but not all independent axes are perpendicular, nor are all axes linear.

 

[Oppressed]

If you restrict "dimensions (type a)" to length, then they must be mutually perpendicular.

But you don't need to have three lengths to define a space. You just need three "independent axes". "Independent axes" are spatial axes such that a change in the measure along one axis causes no change in any of the remaining axes.

Perpendicular linear axes are independent, but not all independent axes are perpendicular, nor are all axes linear.

Two angles and one length.

or

One angle and two lengths.

Each of those will describe any space that three lengths could.

I think you may still be trying to cram more than one meaning into this "Dimension (type a)".

I’m sorry, but I was very, very specific here.

I am talking about the three (type a) dimensions of the physical space we live in. I have gone way overboard to make it clear, we have a physical space which is described as having (type A) three dimensions. I have spoke of no other types of dimensions, but spoken only of these three dimensions.

I have said that these three (type a) dimensions of the physical space we live in are three dimensions of length which are perpendicular.

But you have replied that I am wrong and that space can be described in
1) Not only in length, length, length such that all three lengths are perpendicular but
2) Two angles and one length
3) One angle and two lengths

Clearly I was talking about the three (type a) dimensions of physical space, thus you are clearly stating here that angle is a (type a) dimension. This is what you did earlier.

Then you stated that angle is not a (type a) dimension.

This is a logical inconsistency in what you are saying.

Fredrik and PixyMisa, why don’t you argue with GodMark2 when he says this?

Further, you state:

If you restrict "dimensions (type a)" to length, then they must be mutually perpendicular.

GodMark2, why don’t you argue with Fredrik over the fact he is stating that the three (type a) dimensions of length I am referring to for space do NOT have to be mutually perpendicular.

This is one of the things that make this forum seem hostile. Three of you gang up arguing with me but not arguing with each other, even when your arguments contradict each other.

GodMark2, you have indicated that you do understand the three (type a) dimensions of length I am referring to have to be mutually perpendicular. Why don’t you argue this with Fredrik? Why don’t you explain it to him?

E1=(1,0,0)
E2=(1,1,0)
E3=(1,1,1)

These three vectors do not lie in three independent dimensions. They are described in a three dimensional vector space. E1 lies in only 1 dimension of the vector space. E2 lies in 2 dimensions of the vector space. E3 lies in all three dimensions of the vector space.

To show that the three dimensions of the vector space are independent, you need three representative vectors, each lying only in one of the three dimensions. They don’t even have to have the same scalar value in each dimension, but they must lie exclusively in one and only one of the three dimensions of the three dimensional vector space.

V1=(a,0,0)
V2=(0,b,0)
V3=(0,0,c)

Each of these three vectors are perpendicular to each other.

 

[Fredrik]

I have said that these three (type a) dimensions of the physical space we live in are three dimensions of length which are perpendicular.

But you have replied that I am wrong and that space can be described in
1) Not only in length, length, length such that all three lengths are perpendicular but
2) Two angles and one length
3) One angle and two lengths
...
Fredrik and PixyMisa, why don’t you argue with GodMark2 when he says this?

You have a point here. As I said, it doesn't make sense to try to define what "dimension" means in a specific context without first stating what that context is, and we all seemed to agree to let that context be vector spaces, at least for now. So I was a bit surprised to see that GodMark2 decided to talk about dimensions in another context (manifolds). I don't think it's a good idea to do that right now. Also, he didn't explicitly say that he was switching to another context.

You're right that I would have pounced on you immediately if you had done the same, and that I didn't care that much when he did it. The reason is that I think that he understands what he did. He was a bit sloppy there, but he knows it. I haven't been getting the same impression from you.

This is one of the things that make this forum seem hostile. Three of you gang up arguing with me but not arguing with each other, even when your arguments contradict each other.

If you check the last post I wrote before this one, you will see that I was talking directly to GodMark2, and that I pointed out an inconsistency in what he said. And that's definitely not the first time I have criticized other people than you in this thread.