Hyper Dimensional Philosophizing

[Fredrik]

You again

(This belongs in the post I made before this one. It's annoying that we can't edit our posts after only a few hours).

I haven't really thought about it until now, but I see now that the definition that you offered in #141 (and other places) is different from all of the three definitions that I presented in the post preceding this one:

“dimension” is a dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.

This definition isn't bad. We could argue about the exact choice of words, but you've made it clear that what you have in mind is the definition that's used in the context of dimensional analysis.

The relationship between this definition and the one you presented in #79, is that the "dimensions" defined in #79 are what you call "base quantities" #141.

List of definitions

So far, at least five definitions of "dimension" have been explained in this thread:

1. Algebraic definition (the dimension of a vector space).
2. Geometric definition (the dimension of a manifold).
3. Your #79 definition (an abstract "direction").
4. Your #141 definition (dimensional analysis).
5. GodMark2's #128 definition (roughly meaning "size")

Now can we please agree that these definitions are different?

 

[Oppressed]

Fredrik,

Are you implying that I don't understand what space you're talking about? Of course I do.

I made the statement I did because I am trying to narrow the topic down to something smaller and more specific where most of it we can agree on and hopefully only a small part of which we disagree on in order to make the discussion more focused, understandable and maybe even work out the difference in understanding.

You went on to expanding the subject beyond this.

Both you and GodMark2 have made some good comments to respond to. I don’t have time to respond to either now. When I get back, I will respond to GodMark2 first because it is a simpler part of the issue to discuss and the point is to narrow the subject down to simple examples we can more easily understand our arguments over.

 

[illogical]

It doesn't matter what someone may or may not have said to you, because you habitually disregard the context in which it was said. What matters is what the mathematics says.

Grothendieck has a question: what is a meter?

 

[Jimbo07]

Grothendieck has a question: what is a meter?

"rhythm as given by division into parts of equal time"

 

[Oppressed]

Illogical,
Pulling from historical references:

A meter, as in a meter of length, is a unit of quantity of measurement in the dimension of length. It was an arbitrarily defined unit quantity of length defined in France as being one ten-millionth of a quadrant of the Earth. A prototype meter called Mètre des Archives was constructed in 1799 and used as a reference. This original meter reference was questioned as to whether or not it was the one they should use.

In 1872 a decision taken to make prototype metres, with the original metre held in the Archives de France serving as the reference. In 1875 the Convention of the Metre signed.

Between 1878 and 1889, preparation and measurement of thirty metre prototypes. This required matching, with an unprecedented precision, the new "X" cross-section metre line-standards to one another and to the 1799 metre (Mètre des Archives), which was an end standard. This entailed the development of some unique measuring equipment and of a reproducible, definable temperature scale. Selection of Metre prototype which became the international prototypes. Distribution of the national prototypes. On 28 September 1889 the International Prototypes were deposited at the BIPM, where they remain today.

In 1887, Michelson proposed the use of optical interferometers for the measurement of length. He subsequently received the 1907 Nobel Prize for physics for, among other things, his metrological work. Then, in 1892–1893, the Michelson interferometer was used at the BIPM (by Michelson and Benoît) to determine the length of the metre in terms of the wavelength of the red line of cadmium.

In 1906 the above measurement was confirmed by Benoît, Fabry and Perot using the interferometer made by Perot and Fabry.

In 1920 Nobel Prize for physics for Ch.-Ed. Guillaume, then Director of BIPM, for his invention of invar.
In 1921–36 the first verification of the national prototypes by intercomparisons among themselves and by comparisons with the International Prototype. This included new and improved determinations of the thermal expansion of the metre bars.

In 1927 The international accord, using the above 1893 and 1906 determinations of the wavelength of the red line of cadmium, defining the ångström; the ångström thus defined was henceforth used as the spectroscopic unit of length until abandoned in 1960.

In 1952 the CIPM decided to investigate the possibility of redefining the metre in terms of a wavelength of light, and established the Comité Consultatif pour la Définition du Mètre (now called the Consultative Committee for Length) for this purpose.

In 1960 the CGPM adopted a definition of the metre in terms of the wavelength in vacuum of the radiation corresponding to a transition between specified energy levels of the krypton 86 atom. At the BIPM, measurement of linescales in terms of this wavelength replaced comparisons of linescales between themselves; new equipment was installed for doing this by optical interferometry.

In 1975 the CGPM recommended a value for the speed of light in vacuum as a result of measurements of the wavelength and the frequency of laser radiation.

In 1983 the CGPM redefined the metre as the length of the path travelled by light in vacuum during a specific fraction of a second. It invited the CIPM to draw up instructions for the practical realization of the new definition. The CIPM, having anticipated the above invitation, outlined general ways in which lengths can be directly related to the metre as newly defined. These included the wavelengths of five recommended laser radiations as well as those of spectral lamps. The wavelengths, frequencies and associated uncertainties were specified in the instructions for the practical realization of the definition. At the BIPM, comparison of laser frequencies by beat-frequency techniques supplemented the measurement of linescales in terms of wavelengths of the same lasers.

In 1987 in order to check the accuracy of practical realizations of the metre based upon the new definition, a new round of international comparisons of laser wavelengths by optical interferometry and frequency by beat-frequency techniques was begun at the BIPM. Such international comparisons comprised comparisons of individual components of the laser, in particular the absorption cells containing the atoms or molecules upon which the laser is stabilized, as well as comparisons of whole laser systems (optics, gas cells and electronics).

In 1992 the CIPM decided, on the basis of new work in national laboratories and at the BIPM, to reduce significantly the uncertainties associated with the laser radiations recommended in 1983 and to increase their number from five to eight.

In 1997 the CIPM modified the 1992 instructions for the practical realization of the definition by further reducing the uncertainties and increasing the number of recommended radiations from eight to twelve. Work continues at the BIPM and elsewhere to identify those factors that at present limit the reproducibility of lasers as wavelength and frequency standards.

 

[GodMark2]

List of definitions

So far, at least five definitions of "dimension" have been explained in this thread:

1. Algebraic definition (the dimension of a vector space).
2. Geometric definition (the dimension of a manifold).
3. Your #79 definition (an abstract "direction").
4. Your #141 definition (dimensional analysis).
5. GodMark2's #128 definition (roughly meaning "size")

I think #1 and #2 are either identical, or one is a subset of the other. I remember having to prove that #2 is a subset of #1 in grad school, but my memory of those years is always suspect, due to copious alcohol consumption.

#3 seems to be a subset of #2, but is not well defined.

#4 is a unique meaning. I've been using the word "Unit", instead of dimension, to avoid confusion.

#5 is a unique meaning. I've been using the word "Measure", instead of dimension, to avoid confusion. It's a bit more complicated than simple size, as it can be all the measures necessary to replicate an object. (like the dimensions of a room being 10 feet by 12 feet by 8 feet plus an alcove at the meeting of the corner of the 10 foot and 12 foot sides, infringing three feet into the 10 foot side, extending the 12 foot side to 14 feet, for the full height of 8 feet)

And I think there were a few more, but these are definitely the most common.

Now can we please agree that these definitions are different?

That would be the hard part. I've suggested using synonyms for each of the definitions, rather than the easily confused word "dimension", to hopefully avoid accidentally using one meaning when another is proper for the context.

 

[Oppressed]

GodMark2,

Okay, keep in mind the narrowed subject is real physical flat Euclidian space. It is a well known simple model for space. This space has 3 dimensions and I believe we agree on this.

If we narrow these three dimensions to flat 2 dimensional space within the flat 3 dimensional space, we have a plane. I think we can agree on this as well.

This flat two dimensional space is a real space that we can observe, make measurements of, perform experiments on and so on, just as we can the three dimensional space it is a subset of.

So, when we are talking about this flat 2 dimensional space, we should at this point be able to agree what space we are talking about.

We can define a coordinate system to help us measure this space, such as the XY two axis Cartesian coordinate system. But, does the coordinate system determine the dimensions of the space?

Wish I could put pictures and drawings in this post.

When you use polar coordinates for defining points in a two dimensional flat plane, you do not change the dimensions of the plane, but how you are referencing them. Angle is a derived dimension, it is not one of the base dimensions of the plane and it is not an axis of the plane even when using polar coordinates.

Make a plot with two axes, one for “r” and one for “θ”. You don’t get the same physical two dimensional plane by doing this.

Since I can’t draw a picture, lets define three points and if you want, you can draw them. I’ll use the same convention of “C” for Cartesian coordinates and “P” for polar coordinates.
The Origin will be C0 = (+0,+0) = P0 (+0,NA), such that NA means not applicable or doesn’t matter.
C1 = (+1,+0) = P1 = (+1,+0).
C2 = (+1,+1) = P2 = (+sqr(2),+1/4*pi).
C3 = (+0,+1) = P3 = (+1,+1/2*pi).
Cn = (+xn,+yn) = Pn = (+rn,+θn), such that n is some number indicating a point.

r is the directed distance from P0 to Pn.
θ is the directed angle, counterclockwise from the polar axis to the line segment P0 to Pn.
The polar axis is the line extending from P0 thru P1.
The axis for θ does not lie in the two dimensional plane.

θ is a derived dimension.

Hmm. I had two different books that explained this pretty well and I can’t find either. I am organizationally handicapped.

Okay, consider this. You can use the polar coordinate system to measure an area on the two dimensional plane. What are the dimensions you come up with in the measurement of that area?

If you measure the distance between P0 and P1, you have a distance that lies in a flat line and it has a single dimension in the plane being measured. This is a distance which is measured and it involves defining a unit of quantity. But that unit of quantity lies in a dimension. The distance between P0 and P1 lies in the dimension of length between P0 and P1.

If you measure the distance between P0 and P2, you have a distance that lies in a flat line and it has a single dimension in the plane being measured. But, this not in the same direction as the polar axis.

When you go to measure an area in the plane, using polar dimensions, the result is in a derived dimension of area that is length^2. Dimension is not a unit of measure. Dimension is not a quantity of measure. Dimension is what the quantity being measured exists in.

When you calculate an area in the two dimensional plane using polar coordinates, the result is in the dimension of length*length, not in length*angle.

I’ve must have those text books that do a good job talking about this somewhere around here. I think I have too many books.

Surely there are some others here that understand that when you use polar coordinates in a real physical flat two dimensional Euclidian space, a plane, that the dimension of angle does not lie in the plane, that it is a derived dimension and that it does not alter what dimensions the two dimensional plane consists of.

 

[Fredrik]

I think #1 and #2 are either identical, or one is a subset of the other. I remember having to prove that #2 is a subset of #1 in grad school, but my memory of those years is always suspect, due to copious alcohol consumption.

They are definitely not identical. This is how they're connected:

1. There's always an n-dimensional vector space T_pM associated with each point p of an n-dimensional manifold M. It's called the tangent space of M at p.

2. Any n-dimensional inner product space (a vector space with an inner product) can be given a manifold structure in a very natural way. The manifold constructed this way is n-dimensional. (A somewhat sloppy way to express this would be to say that most useful vector spaces are manifolds).

#3 seems to be a subset of #2, but is not well defined.

His #3 covers "dimensions" such as "luminous intensity" and "apples" as well as distances along any direction in space (basically anything that can be measured). It's not as well defined as #1 and #2, but definitions that aren't made within the framework of mathematics never are. I think it's good enough.

Even when #3 is applied specifically to space, it's not equivalent to #1 or #2 There's nothing in #3 that says that the total number of independent directions, or the total number of coordinates needed to represent a point, has any relevance whatsoever to the concept of a "dimension". Because of that, #3 implies that the actual physical space that we live in has an infinite number of dimensions. (You can measure distances in any direction and that makes every direction a dimension). And that implies that #3 is not equivalent to #1 or #2.

 

[Fredrik]

Surely there are some others here that understand that when you use polar coordinates in a real physical flat two dimensional[1 or 2] Euclidian space, a plane, that the dimension[3 or 4] of angle does not lie in the plane, that it is a derived dimension[4] and that it does not alter what dimensions[probably 3, maybe 1 or 2] the two dimensional[1 or 2] plane consists of.

You seem to be using at least three different definitions of "dimension" in this paragraph. (The numbers are referring to the list of definitions I posted earlier).

 

[Fredrik]

Grothendieck has a question: what is a meter?

The distance that light travels in 1/299792458 seconds (in a vacuum).

 

[Oppressed]

Fredrik,

You seem to be using at least three different definitions of "dimension" in this paragraph. (The numbers are referring to the list of definitions I posted earlier).

Probably the biggest disagreement between you and me here lies in what I consider to be a base dimension in the two dimensional plane and what I consider to be a derived dimension of area and angle.

Trying to stick towards finding common ground first, do you agree that the two dimensional space I’ve narrowed the topic to is comprised of two dimensions of length which are perpendicular to each other? This is a physical reality which can be observed, measured, tested and analyzed.

If I have a circular quantity defined in this two dimensional plane, I can use Cartesian coordinates to help me measure and calculate the area of that circle.

That circular area is a defined quantity of the two dimensional plane. Do you understand and agree that this area, this quantity, lies in the same two dimensions that the plane lies in?

If I change my coordinate system to polar coordinates and use those polar coordinates to measure the same circular area, do you understand that this are still lies in the same two dimensional plane and thus lies in the same two dimensions, even though the coordinate system has changed?

 

[GodMark2]

Fredrik,

Probably the biggest disagreement between you and me here lies in what I consider to be a base dimension in the two dimensional plane and what I consider to be a derived dimension of area and angle.

AAAHHHGH! Two different dimensions in the opening paragraph!

I'll try to rewrite it to avoid confusion

Probably the biggest disagreement between you and me here lies in what I consider to be an axis in the two dimensional plane and what I consider to be a derived unit of area and angle.

continuing

Trying to stick towards finding common ground first, do you agree that the two dimensional space I’ve narrowed the topic to is comprised of two dimensions of length which are perpendicular to each other? This is a physical reality which can be observed, measured, tested and analyzed.

I don't know why you insist on starting with a volume, which is then discarded except for a single plane, when we could have started with a single plane. You could have just said: We have a Euclidean plane.

so:
We have a Euclidean plane.

If I have a circular quantity defined in this two dimensional plane, I can use Cartesian coordinates to help me measure and calculate the area of that circle.

That circular area is a defined quantity of the two dimensional plane. Do you understand and agree that this area, this quantity, lies in the same two dimensions that the plane lies in?

If I change my coordinate system to polar coordinates and use those polar coordinates to measure the same circular area, do you understand that this are still lies in the same two dimensional plane and thus lies in the same two dimensions, even though the coordinate system has changed?

Yes, any measurement will be completely unaffected by coordinate system. The result will be the same, as it's the same measurement. Coordinate systems do not make the measurement.

GodMark2,Okay, keep in mind the narrowed subject is real physical flat Euclidian space. It is a well known simple model for space. This space has 3 dimensions and I believe we agree on this.

If we narrow these three dimensions to flat 2 dimensional space within the flat 3 dimensional space, we have a plane. I think we can agree on this as well.

This flat two dimensional space is a real space that we can observe, make measurements of, perform experiments on and so on, just as we can the three dimensional space it is a subset of.

So, when we are talking about this flat 2 dimensional space, we should at this point be able to agree what space we are talking about.

Again, why start with a volume, when all you need is a plane? It just adds unnecessary complexity

We can define a coordinate system to help us measure this space, such as the XY two axis Cartesian coordinate system. But, does the coordinate system determine the dimensions of the space?

No. The number of ordinals required to create a coordinate space is a result of the number of axes in the space.

Wish I could put pictures and drawings in this post.

When you use polar coordinates for defining points in a two dimensional flat plane, you do not change the dimensions of the plane, but how you are referencing them.

You are choosing a different set of axes with which to reference points on the plane.

Angle is a derived dimension, ...

Angle is a derived unit.

... it is not one of the base dimensions of the plane and it is not an axis of the plane even when using polar coordinates.

The highlighted section is currently gibberish. Which word instead of 'dimension' could be applied and leave the meaning intact?

And "angle from reference' IS an axis of the plane when using polar coordinates. For the definition of axis of "one of a set of independent vectors in a manifold".

Make a plot with two axes, one for “r” and one for “θ”. You don’t get the same physical two dimensional plane by doing this.

Well, I get the same plane, I don't know how you're not. In fact, I can even put both axes on the same plane simultaneously

Since I can’t draw a picture, lets define three points and if you want, you can draw them. I’ll use the same convention of “C” for Cartesian coordinates and “P” for polar coordinates.
The Origin will be C0 = (+0,+0) = P0 (+0,NA), such that NA means not applicable or doesn’t matter.
C1 = (+1,+0) = P1 = (+1,+0).
C2 = (+1,+1) = P2 = (+sqr(2),+1/4*pi).
C3 = (+0,+1) = P3 = (+1,+1/2*pi).
Cn = (+xn,+yn) = Pn = (+rn,+θn), such that n is some number indicating a point.

And you manage to apply the axes to the same physical plane simultaneously, so I guess you got over that "They can" bit pretty fast.

r is the directed distance from P0 to Pn.
θ is the directed angle, counterclockwise from the polar axis to the line segment P0 to Pn.
The polar axis is the line extending from P0 thru P1.
The axis for θ does not lie in the two dimensional plane.

Incorrect. Every angle used for θ lies in the plane, and every possible angle for θ lies in the plane. The axis θ, lies in the plane.

θ is a derived dimension.

θ has derived units

Hmm. I had two different books that explained this pretty well and I can’t find either. I am organizationally handicapped.

Okay, consider this. You can use the polar coordinate system to measure an area on the two dimensional plane. What are the dimensions you come up with in the measurement of that area?

What are the units you come up with in the measurement of that area?
Area. Area is measured in units of area.

If you measure the distance between P0 and P1, you have a distance that lies in a flat line and it has a single dimension in the plane being measured. This is a distance which is measured and it involves defining a unit of quantity. But that unit of quantity lies in a dimension. The distance between P0 and P1 lies in the dimension of length between P0 and P1.

If you measure the distance between P0 and P2, you have a distance that lies in a flat line and it has a single dimension in the plane being measured. But, this not in the same direction as the polar axis.

You freely use dimension, but everything said here has no bearing on the questions you're asking.

When you go to measure an area in the plane, using polar dimensions, the result is in a derived dimension of area that is length^2.

Now we're going to get some good stuff.

Dimension is not a unit of measure.

Then why do you keep using it when you mean unit? One of the possible meanings for 'dimension' is 'unit'. Such as, "What is the dimension of 'area'?" No other definition of 'dimension' applies to that example.

Dimension is not a quantity of measure.

One of the definitions listed is contrary to this. One of the possible definitions of 'dimension' is 'a measure'. As in "What's the dimension of that opening again? 3 feet?"

Dimension is what the quantity being measured exists in.

When you calculate an area in the two dimensional plane using polar coordinates, the result is in the dimension of length*length, not in length*angle.

And no one has ever said otherwise. Area must have units of "square length". This has no effect on, or from, the axes of the manifold in which the area lies. Axes are simply a tool used to reference points.

I’ve must have those text books that do a good job talking about this somewhere around here. I think I have too many books.

Surely there are some others here that understand that when you use polar coordinates in a real physical flat two dimensional Euclidian space, a plane, that the dimension of angle does not lie in the plane, that it is a derived dimension and that it does not alter what dimensions the two dimensional plane consists of.

Would you please stop using "dimension"? The word is overloaded with meanings, which can all be stated using other, less misunderstandable, words.

Unit. Measure. Axis of a manifold.

Substituting synonyms will not render an argument invalid. But, when your ideas are restated, using these words, you quickly run into situations where the word needed is indeterminate, which indicates a conflict.

 

[PixyMisa]

A Euclidean plane is two-dimensional because it takes two numbers to specify a point in that plane. Never less, never more. You can define various co-ordinate systems, but they can always be reduced to two numbers.

You can create a co-ordinate system that takes three numbers and maps onto a Euclidean plane. That's known as a projection. It doesn't change the number of dimensions in the plane; that's always two.

In the sense that the Euclidean plane is two dimensional, there's no such thing as "derived dimensions".

 

[Oppressed]

I am beginning with something that is real that is most commonly associated with having dimensions, real physical space, and working towards isolating a single dimension of this real space.

We can see, detect, measure, test and analyze this real space. This real space does not exist as a point, a line or a plane. The real physical space we live in does not exist as a two dimensional plane or a one dimensional line. That is why I am talking about three dimensional space as we narrow it down to discussing two dimensions and one dimension within the three dimensional space.

PixyMisa,

I am speaking about the dimensions of what should be a well defined space, the Euclidean two dimensional plane which is a subset of the Euclidean three dimensional space.

In the sense that the Euclidean plane is two dimensional, there's no such thing as "derived dimensions".

Are you stating that angle is a dimension of this plane and that it is not derived?

Angle is a derived unit.

GodMark2 states Angle is not a dimension but a derived unit.

I state the flat plane I am talking about has two dimensions of length which are perpendicular to each other.

GodMark2 is stating I am wrong and using the example cylindrical coordinates in 3D space and polar coordinates in 2D space as an example of proving me wrong.

Independent axes are axes in which movement along one causes no change in movement along the other, for the entire length of the axis (dx/dy = 0). Note: In general, they are not required to be perpendicular. In particular, the three axes chosen for Cartesian coordinates are not perpendicular, as at least one is a plane (or is indeterminate, residing in a plane).

When GodMark2 made the above, he later made the correction that he meant to be using Circular coordinates, which did not really make sense for a 3D space, but I assume he meant cylindrical and/or spherical coordinates.

I mistyped, and didn't proofread the suggested correction. I apologize. I meant to type "Circular coordinates".

Cartesian coordinates are the example of three perpendicular axes.

Circular coordinates are an example of non-perpendicular coordinates in Eucledian space.

I make a distinct difference in definition between coordinate system and dimension in the space we are talking about. What GodMark2 is saying seems mixed to me, but you do not saying anything about that?

GodMark2 has made the argument that angle can be one of the dimensions of the two dimensional plane and that angle can be one or two of the dimensions three dimensional space.

But then he has also stated that angle is a unit, not a dimension.

Why are you not arguing with him about that? I’m bringing this up because while you have stated you like the way I said something better than someone else before, it mostly feels like you are all ganging up on arguing with me while ignoring inconsistencies between each other.

What I am stating is consistent, even if you disagree with me.

When we are talking about the two dimensional flat plane which is a subset of the three dimensional real space we are talking about, do you consider angle to be a valid dimension of that space?

Do you consider angle to be a dimension?

 

[Oppressed]

Angle is a derived unit.QUOTE]

GodMark2,

I am talking about the two dimensions of the two dimensional plane which is a subset space of the three dimensional space we have been talking about.

I have stated the two dimensions of this space are two dimensions of length which are perpendicular to each others.

You have said I am wrong and used the example that the two dimensions of this plane can be the dimension of length and the dimension of angle.

But then you state that angle is not a dimension, it is a unit.

This is inconsistent.

The dimension of angle is not a unit.

What does 3.5 angle + 2.1 angle equal? I’m sorry, but that does not even make sense.

What does 3.5 degrees of angle + 2.1 degrees of angle equal? Aha! Now we have defined units of quantity of the dimension of angle. The answer is 5.6 degrees of angle.

What does 3.5 radians of angle + 2.1 radians of angle equal? Aha! Now we have defined units of quantity of the dimension of angle. The answer is 5.6 radians of angle.

Now, we may continue to disagree about whether or not the definition for “dimension” as it is used for angle is the same or different from the definition for “dimension” as it is used for length, but “dimension does not equal unit”.

 

[Fredrik]

Probably the biggest disagreement between you and me here lies in what I consider to be a base dimension[3 or 4] in the two dimensional[1 or 2] plane and what I consider to be a derived dimension[4] of area and angle.

Trying to stick towards finding common ground first, do you agree that the two dimensional[1 or 2] space I’ve narrowed the topic to is comprised of two dimensions[???] of length which are perpendicular to each other? This is a physical reality which can be observed, measured, tested and analyzed.

The disagreement doesn't seem to be so much about what should be considered a "base dimension"[3 or 4] and what should be considered a "derived dimension"[4] as about whether those concepts have any relevance at all in this context.

The two-dimensional[1 or 2] plane consists of two dimensions[1 or 2], and it consists of infinitely many dimensions[3]. The number of dimensions depend on what kind of dimensions we're talking about, just as as the likelihood that a baseball player will score a home run when he swings his bat in a game depends on if it's the kind of bat that most players use, or the kind that lives in caves.

If I have a circular quantity defined in this two dimensional[1 or 2] plane, I can use Cartesian coordinates to help me measure and calculate the area of that circle.

That circular area is a defined quantity of the two dimensional plane[1 or 2]. Do you understand and agree that this area, this quantity, lies in the same two dimensions[1 or 2] that the plane lies in?

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?".

However, it would make no sense to say that the area is in those two dimensions. "Area" is a dimension of type 3 or 4, so it's fundamentally different than anything that can "lie in the same two dimensions[1 or 2]" as the plane.

If I change my coordinate system to polar coordinates and use those polar coordinates to measure the same circular area, do you understand that this are still lies in the same two dimensional plane and thus lies in the same two dimensions, even though the coordinate system has changed?

The circle along with its interior is just a set of points in the plane. A coordinate system is just a function that assign numbers (coordinates) to points, so a change of coordinate systems obviously can't change what set of points we're talking about. It just changes the numbers we associate with them.

 

[PixyMisa]

I am speaking about the dimensions of what should be a well defined space, the Euclidean two dimensional plane which is a subset of the Euclidean three dimensional space.

The Euclidean plane is a well-defined space in and of itself. You don't need to consider it as a subset of anything.

Are you stating that angle is a dimension of this plane and that it is not derived?

No. I'm saying that, if you are using the word "dimension" in the sense that the Euclidean plane is two-dimensional, the term "derived dimension" is meaningless. You can't derive a dimension in that sense.

The Euclidean plane has two dimensions.
If you apply a Cartesian co-ordinate system, it has two dimensions.
If you apply a Polar co-ordinate system, it has two dimensions.
If you apply a Mercator projection to map the surface of a sphere onto the plane, it has two dimensions.

GodMark2 states Angle is not a dimension but a derived unit.

Yep.

I state the flat plane I am talking about has two dimensions of length which are perpendicular to each other.

And that's where you start having problems.

A Euclidean plane has two dimensions. Notice the full stop at the end of that sentence? That's because that's where the definition of the Euclidean plane ends.

You can have a Euclidean plane with two dimensions of length. Here, you are assigning a property to the dimensions. GodMark2 calls this property the unit of the dimensions, which seems a reasonable way to avoid the confusion in which this thread has been mired from the beginning.

 

[Oppressed]

I don’t know how or if pictures can be added to a post. Do you have access to a program which will allow you to make a 3D plot? If so, try the following.

Let “r” = a constant value of 1.
Let “N” = a constant value of 720.
Let “θ_init” = - pi.
Let “θ_inc” = pi/180.
Let “n” = 0 to N+1, incrementing in steps of 1.

Select one axis and plot the angle “θ” on it for “θn = θ_init + θ_inc * n”.

Select another axis perpendicular to the first axis and plot “x” on it for “xn = r * cos(θn)”.

Select the third axis perpendicular to both the first and the second axis and plot “y” on it for “yn = r * sin(θn)”.

The point represented by the polar coordinate (r,θn) on the xy-plane is the projection onto the xy-plane of the three dimensional point (axis1=θn angle, axis2=r*cos(θn) length, axis3=r*sin(θn) length).

The axis for angle in the use of polar coordinates is perpendicular to the real physical flat Euclidian two dimensional plane which is a subset of space from real physical flat Euclidian three dimensional space.

Now, we might be disagreeing over what dimension means with relation to the dimension of length versus the dimension of angle, but using polar coordinates to specify a point in the real physical flat Euclidian two dimensional does not change the dimensions of that plane from being two perpendicular lengths to being a length and an angle.

 

[Oppressed]

PixyMisa,

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

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.

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.

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

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.

Do you disagree with this?

 

[PixyMisa]

I don’t know how or if pictures can be added to a post.

Yes, you can attach images. I think you can - you need a minimum number of posts for that.

The axis for angle in the use of polar coordinates is perpendicular to the real physical flat Euclidian two dimensional plane which is a subset of space from real physical flat Euclidian three dimensional space.

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.