Deeper than primes

[The Man]

The circle -- as a one-dimensional object -- is not created the way multidimensional objects are by definition. If you drag a 0-dimensional point in one direction, you create a (0+1)-dimensional object other then the circle --the result of the dragging has to be the straight line (the first co-ordinate). In other words We hold these truths to be self-evident, that all 1-D objects are not created equal . . .

It depends on (or in) what space you do your dragging. If that space represents cycles (another representation of angle) then dragging from 0 to 1 gives you a complete circle (or cycle). Again the important thing is that the angle (or one dimensionality of a location on a circle) does not distinguish between circles. Radius or differing centers are required for that distinction, just as location in an orthogonal dimension distinguishes between parallel lines on a plane.


ETA:

Remember a sine wave is just a circle where the starting point is separated by two diameters from the 360 degree point along a line.


ETA2:
(When peak amplitude equals ¼ λ)

 

[epix]

It depends on (or in) what space you do your dragging. If that space represents cycles (another representation of angle) then dragging from 0 to 1 gives you a complete circle (or cycle). Again the important thing is that the angle (or one dimensionality of a location on a circle) does not distinguish between circles. Radius or differing centers are required for that distinction, just as location in an orthogonal dimension distinguishes between parallel lines on a plane.

That's the free style that delivers confusion into the works, unless conditions are exactly specified. See, dragging points and drawing objects on a n-D manifold is not the same thing.

Suppose that you want to draw a circle on a piece of paper - I mean a fairly precise circle. That requires a very good eye/hand coordination. But if you are not sure, you can use a co-ordinate that shows you exactly the required direction you move your hand, and the name of the co-ordinate is "template."

Any figure that requires one template to draw is a 1-dimensional object. But what if you want to draw a point? If you go to the store and ask for a template to draw a point, your request won't be quite understood. The creation of a point doesn't involve "dragging" your pencil, so there are no templates made for points. You don't need a eye/hand coordination for that task; you don't need a co-ordinate to guide you; no template, and therefore a point is a 0-dimensional object. But . . .

Suppose that you want to draw a precise line so it would appear on the screen of your graphing calculator. A template as an aid is not a possibility; the required co-ordination must be achieved by other means.

Suppose further that you have a simple graphing calculator equipped with two "templates" to guide your drawing attempts called "Cartesian co-ordinates." These co-ordinates look like a '+'; they comprise one horizontal line and one vertical line that intersect each other. (We all know that, but this may come handy to Doron who got a long distance to go to the basics, as he lives on the fringes of mathematics and needs to take the 1:30 flight from Organic Mathematics.) Both co-ordinates must be defined and a convention would do: the horizontal line/co-ordinate is called " x-axis" and the vertical line/co-ordinate is called the "y-axis.

Now you need to ask the calculator to draw the object of your choice. The calculator anticipate you request by a prompt, such as

y1 =

Does the 'y' have something to do with the y-axis?

It does, except the y-coordinate has been renamed on "dependent variable." It stands to reason that the empty space after 'y1=' should be completed with something that relates to the 'x-axis' and to renaming the x to stand for "independent variable." So let's try this to see what happens . . .

y1 = x

Once you complete it, a new prompt may appear:

y1 = x
y2 =

If there is a second entry, lets use 2 and complete the prompt:

y1 = x
y2 = 2x

Now you notice a button that says "Graph." So you hit it and observe. The intuitive approach to complete the prompt was right, as you see two straight line, which intersect each other where the x and y axis do, to appear.

So you are done -- unless someone shows up being curious about the dimension of both lines, not to be confused with the length of the lines. (Say what?)

That person can help to answer the question marveling over the precision the lines were drawn. "Better than done with the help of a template."

How many "templates" were needed to draw one of the straight lines? And how was it drawn?

The x-independent variable refers to the points placed ON the x-coordinate; the y-dependent variable refers to the points placed ON the y-coordinate. In the case of the y2 line, the points were placed like this.

x: -5, -4, -3, -2,- 1, 0, 1, 2, 3, 4, 5
y2: -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10

The different placement of the points on the y-coordinate was affected by the modification of the x-variable from x to 2x. The Cartesian co-ordinate system requires that the points drawn on the x-coordinate were equidistant. The spacing of the points drawn on the y-coordinate depends (hence y is the dependent variable) on the modification of the x-independent variable. In other words, the spacing of points on the y-coordinate is a function of the independent variable x. This is expressed as

y = f(x)

and so

y2 = 2x
f(x) = 2x

are equivalent expressions.

All what is needed for the intended straight line to appear is to draw two perpendicular lines from the x and y-coordinates where the placed points given by the function are. Where both perpendicular lines intersect each other is one of the points that the straight line y2=2x is made of.

But the question still remains unanswered. What is the dimension of these lines?

Since we used two "templates" -- a system of two co-ordinates were required by the way the calculator is defined to draw such two lines, each line is 2-dimensional object.

Now, the person starts to object by clicking on Wikipedia. No, a line is a 1-dimensional object. That's true when you take a ruler (one coordinate) and draw the line on the piece of paper. There is no mentioning in the Wikipedia of an geometric figure generated by a function that consists of one independent variable x that places points on the x-coordinate and one independent variable that places points on the y-coordinate.

Does it mean that the system of Cartesian coordinates prevents to draw a 1-dimensional line? That doesn't seem to be right, because

y1 = 2x (two variables x and y --> two co-ordinates x and y -- two-dimensional line)
y2 = 2 ( one variable y --> one co-ordinate y --> one dimensional line.)

The red line is drawn by function y2 = 2 (where 2 is a just a constant) and so that line is 1-dimensional. But there is a slight problem: The y-coordinate is the ruler -- it makes sure that the line stays horizontal, but

x: -5, -4, -3, -2,- 1, 0, 1, 2, 3, 4, 5
y: 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Something had to move the pencil for the red line to appear, and that's the job the x-coordinate gets, otherwise there would be no intersection of [-5, 2] or [0, 2], for example.

But the implication dismisses the idea that a straight line drawn in the system of Cartesian co-ordinates is 2-dimensional when y=f(x)! That's because the hand that holds the pencil and moves the directon according to the ruler (the single dimension) is not considered a dimension. The hand happens to be the x co-ordinate.

But if there is the x-variable missing from the function y=2, then the hand is missing as well so the red line cannot be drawn, but we see it drawn -- the applet wouldn't draw it just like that.

Now what, Doron? You are the trickster, the OM-hardened analytic mind . . .

(y = 2) == ( y = 2 * x^0)

Well, my local-only reasoning couldn't figure out the non-local (something that I can't see and therefore consider) option. Now the hand is is present and the red line can be drawn.

The idea of dragging a point to create the first dimension is only good when it doesn't encounter an obstacle given the environment where the dragging is happening, like anything else which plagues definitions. The factor x^0 is a cop-out, but it may not be, coz

25 = 2*n^1 + 5*n^0 \where n = 10

and that's the concept of the decimal system.

The polar system of coordinates is the same as the Cartesian system except that one of the axis is replaced by a unit circle. So we draw the angular coordinate first

r1 = 1

and then we draw a tangent line to the coordinate/circle

r2 = tan(phi)/sin(phi)

Basically, you can set up any system of co-ordinates by dragging 0-dimensional point in any direction you wish. The problem is to figure the constant spacing of the-coordinate that happens to be a fancy curve. That's why there is no other curve coordinate used apart from the circle.

As far as the dimensionality of the "flower" is concerned . . .
Doron said it is 1-dimensional. I go by the number of different variables needed to write down the equation that defines and draws a geometric figure (and that includes the "hand".) If I'm wrong and see a result that looks different from what I intended, and the mistake lies in the dimensionality, then I adjust my definition of it.

 

[epix]

(y = 2) == ( y = 2 * x^0)

Errata: (y = 2) == ( y = 2 * |x^0|)

 

[doronshadmi]

No Doron it simply shows that your ridiculous K-Y assertions are just, well, ridiculous and has nothing to do with the dimensionality of an object or space, since you don’t get the correct answer (and deliberately so).

Your misunderstanding of a simple thing like k_X,k_Y is indeed ridiculous.

Since you don’t get it let us use baby steps.

n=1 to ∞
k=0 to n-1

X = "Dimensional space"

Y = "Dimensional element"

k_X - k_Y = 0 amount of coordinates that are used to determine k_Y location w.r.t k_X, no matter of k > or = 0.

For example, there is no difference between the magnitudes of 1_X and 1_Y and as a result there is 0 amount of values that determine their location w.r.t each other (they are not “local” AND not “non-local” w.r.t each other).

This lack of value has nothing to do with the fact that 1_X or 1_Y are existing dimensions.

Exactly the same reasoning holds in the case of 0_X and 0_Y w.r.t each other (they are not “local” AND not “non-local” w.r.t each other).

This lack of value has nothing to do with the fact that 0_X or 0_Y are existing dimensions, and a point is an exact manifestation of such existing dimension.

Since 0_dimension is an existing dimension (and a point is a concrete example of such existing dimension), than 0_dimension, 1_dimension,2_dimension, ... etc. Obey the rule of inclusion\exclusion, and indeed 1_dimension is included NXOR excluded w.r.t 0_dimension, where 0_dimension is included XOR excluded w.r.t 1_dimension.

The Man, your ridiculous reasoning does not hold water.

You have no ability to understand the difference between “How much?” (Magnitude) and “How many?” (Multitude) and as a result you are not able to get the notion of Point as an existing dimension.

The notion of existence is not limited here to physical existence, and a point is an existing abstract element, whether there is an ordinate or a coordinate that is related to it, or not.

 

[doronshadmi]

The idea of dragging a point to create the first dimension

A point is 0 dimensional element, and 0 dimensional element can't be dragged, wiggled, stretched or be changed by scale factor, and still be considered as 0 dimensional element.

Only k>0_dimensional elements can be dragged, wiggled, stretched or be changed by scale factor.

k>0_dimensional element can't be reduced into 0_dimensional element and still be considered as k>0_dimensional element.

These essential differences between existing 0_dimensional and existing k>0_dimensional prevents form infinitely many existing 0_dimensional elements to fully cover a given existing k>0_dimensional element, and enables an existing k>0_dimensional element to be included NXOR excluded w.r.t any given existing 0_dimensional element along it.

 

[Little 10 Toes]

Here are your expanded definitions of local and non-local:

1) If the result of one domain (labeled A) sharing a given domain NXOR not sharing a given domain with regard to a second domain (labeled B) is TRUE, then domain A is non-local with regard to domain B.

2) If the result of one domain (labeled A) sharing a given domain XOR not is sharing a given domain with regard to a second domain (labeled B) is TRUE, then domain A is local with regard to domain B.

Because you like to make your own definitions of words, I wanted your definition of domain. Here's the following messages:

Please define your usage of the word domain.

The researched.

The researched what?

Domain is "that is researched".

I'll repeat my question:

What is researched?

Define researched.

 

[zooterkin]

A point is 0 dimensional element, and 0 dimensional element can't be dragged, wiggled, stretched or be changed by scale factor, and still be considered as 0 dimensional element.

No-one is saying it is. By dragging the point, you are defining, for example, a line, which is 1-dimensional.

 

[The Man]

Your misunderstanding of a simple thing like k_X,k_Y is indeed ridiculous.

So your ridiculous calculation did not give you the wrong (and deliberately so) answer for the number of coordinates of some “>0” dimensional space or element? Again it is just you that is misunderstanding, well, yourself.

Since you don’t get it let us use baby steps.

Use all the “baby steps” you want, but until you start using an accurate representation of dimension for an object or space your still just jerking yourself around.

n=1 to ∞
k=0 to n-1

X = "Dimensional space"

Y = "Dimensional element"

k_X - k_Y = 0 amount of coordinates that are used to determine k_Y location w.r.t k_X, no matter of k > or = 0.

Once again your ridiculous calculation fails, and that is the fault of no one but you.

For example, there is no difference between the magnitudes of 1_X and 1_Y and as a result there is 0 amount of values that determine their location w.r.t each other (they are not “local” AND not “non-local” w.r.t each other).

This lack of value has nothing to do with the fact that 1_X or 1_Y are existing dimensions.

Actually your ridiculous calculation has nothing to do with the fact that you have already specified both your “X = "Dimensional space"” and “Y = "Dimensional element"” to be, well, 1 dimensional. Subtracting some one dimensional object (like a line segment) from a one dimensional space (like a line or a line segment) can still leave you with a one dimensional object or multiple one dimensional objects. However, it does not change the dimensionality of the space being considered or that of the object subtracted. In the case of the space being the interval [4,5] and the object being the interval [4,5), your resulting difference is the zero dimensional element referred to as point 5. However, the space it is being considered in is still one dimensional as its unique location is still defined by the singular ordinate of 5.

Exactly the same reasoning holds in the case of 0_X and 0_Y w.r.t each other (they are not “local” AND not “non-local” w.r.t each other).

The same reasoning does hold, and as usual it just isn’t yours, as you have again specified the dimensional space and dimensional object to already be zero dimensional. Your ridiculous reasoning still has no standing nor does your simply self contradictory “not “local” AND not “non-local”” assertion.

This lack of value has nothing to do with the fact that 0_X or 0_Y are existing dimensions, and a point is an exact manifestation of such existing dimension.

“existing dimensions”? How? You have specified no dimensions. Do you claim 0 apples are existing apples? Again a point specificly lacks dimension.

Since 0_dimension is an existing dimension (and a point is a concrete example of such existing dimension), than 0_dimension, 1_dimension,2_dimension, ... etc. Obey the rule of inclusion\exclusion, and indeed 1_dimension is included NXOR excluded w.r.t 0_dimension, where 0_dimension is included XOR excluded w.r.t 1_dimension.

Since “0_dimension” as “an existing dimension” is just your fantasy, it is of concern to no one but you. What “rule of inclusion\exclusion” are you referring to? Your “included NXOR excluded w.r.t 0_dimension” that is always false, except in your fantasies? Doron your fantasies “rule” no one and nothing other than simply you.

The Man, your ridiculous reasoning does not hold water.

Doron your fantasy "dimension 0" doesn’t hold anything, including a dimension. What you are apparently missing, and perhaps deliberately, is that a 0 dimensional element is used to define dimension (in this consideration) and by that (and its) definition it has no dimension.

You have no ability to understand the difference between “How much?” (Magnitude) and “How many?” (Multitude) and as a result you are not able to get the notion of Point as an existing dimension.

You have no ability (meaning zero ability) to understand the difference between your self contradictory fantasies and self consistent definitions and usages whether about ““How much?” (Magnitude) and “How many” (Multitude)” and as a result you ascribe some fantasy "dimension" for just a point yet can not locate that zero dimensional object (a point) in a zero dimensional space (a point). So once again it is simply you who, again evidently deliberately, can not understand that both the ““How much?” (Magnitude) and “How many?” (Multitude)” of the dimensionality for just a point is just zero.

The notion of existence is not limited here to physical existence, and a point is an existing abstract element, whether there is an ordinate or a coordinate that is related to it, or not.

The notion of dimension is “an existing abstract” concept, that evidently you simply want to ignore. That concept (in this consideration) is based on and defined by the also “existing abstract” concept of a point (an element without dimension), evidently that too you simply want to ignore, both of which define the point as having no dimension. Again you seem to be confusing the concept of a point defining dimension (in this consideration) with that point having dimension. Unfortunately, that would simply make the existence of some dimension part of the definition of a point (as opposed to the specific lack of dimension) and thus the existence of a point the basis of the definition of a point. So the definition of a point and of dimension (as in this consideration) simply becomes circular. Also making the minimum number of dimensions > 0 dimension, thus no dimension as well as 0 dimensional is simply self contradictory as both must be > 0 dimension. Fortunately most of us here are already well aware that circular concepts are as critical to your OM as are self-contradictory concepts.

 

[epix]

A point is 0 dimensional element, and 0 dimensional element can't be dragged, wiggled, stretched or be changed by scale factor, and still be considered as 0 dimensional element.

That's right. The verb "drag" is just a loose verbal approximation that describes the way dimensions d>0 are created, but it is a wrong choice of description.

Wiki:

An inductive definition of dimension can be created as follows. Consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. In general one obtains an n+1-dimensional object by dragging an n dimensional object in a new direction.

You can easily demonstrate that dragging a 0-D point in one direction wouldn't create a 1-D (or 1_D by your syntax) straight line. Just put an apple on the table and drag it along from one side of the table to the other. When you are done, you see that there is still only one apple, not a collection of many apples that form an "apple line."

A straight line is created by the accumulation of points organized according to a coordinate, such as the ruler. Here is a simple example: Take a pen, a big roll of paper, a ruler and start drawing a line. There will be time ahead when the line will end due to the fact that there is no more ink in the pen. The pen is the dispenser of the points.

Doron:

Only k>0_dimensional elements can be dragged, wiggled, stretched or be changed by scale factor.

Absolutely incorrect by your first proposition.

1) Line is a (k>0)_D object and a collection of 0-D points the number of which approaches infinity.
2) Points cannot be dragged.
Therefore
3) "Only k>0 dimensional elements can be dragged" is false.


(You can cancel the false conclusion by re-defining the line not to be a collection of points. But that would make the line a "solid" 0-D element with the same property as the 0-D point has, and therefore you can't drag the line to create a 2-D surface area either. To be precise, I can't and that's because I am Non-Doron. You'll surely figure a way . . . )

 

[doronshadmi]

a point (an element without dimension)

In that case point is anything but dimension, and such an element can't be used for anything where Dimension is involved.

For example: a point can't be used, for example, as an origin for 1-dimension, 2-dimension, 3-dimension, ... n-dimension, etc ..., because a point is anything but dimension (according to No_dimension determination).

"No" and "0" are not the same thing, because 0 is a number exactly as 1,2,3, etc... are numbers, where "No" is a logical connective.

An element that has no dimension, does not have any degree of dimension , which is determined by a given number, including 0.

So, an element that has no dimension and an element that has 0 dimension are not the same element.

 

[doronshadmi]

By dragging the point, you are defining, for example, a line, which is 1-dimensional.

Line is a (k>0)_D object and a collection of 0-D points the number of which approaches infinity.

Nope.

A line is not defined by a point, or collection of points.

A line has non-local quality w.r.t a point and a point has local quality w.r.t a line.

The difference of these qualities is an invariant fact as long as a point and a line are strictly compared w.r.t each other.

If there is no strict distinction between qualities, then these qualities are actually vanished into the un-manifested, exactly as they are derive from the un-manifested:

Some natural law is non-local w.r.t any given space-time zones of several laboratories that discover it by well established scientific experiments.

Nevertheless the discovered natural law and the space-time zones of several laboratories that discover it, are vanished within a black-hole.

The big-bang is actually a white bulge, where the natural law and the given space-time zones of several laboratories that discover it, are derive from.

 

[doronshadmi]

What is researched?

Define researched.

A measurable realm.

 

[doronshadmi]

Again a point specificly lacks dimension.

In that case it specifically can't be used as an ordinate or coordinate, because it has nothing to do with the notion of Dimension, since it lacks this property, according to your reasoning.

You still get the existence of things by counting them (The Man: "Do you claim 0 apples are existing apples?") but this is not the case about dimension's magnitudes, for example:

|{0}| = |{1}| = |{2}| = ... = |{n}| = 1, where cardinality 1 means "existing element" and the content of each set is the magnitude of some dimension.

Your "no dimension" is |{}| = 0

 

[zooterkin]

In that case it specifically can't be used as an ordinate or coordinate, because it has nothing to do with the notion of Dimension, since it lacks this property, according to your reasoning.

Nonsense. You don't use the point as a co-ordinate, the co-ordinates tell you where the point is.

 

[Little 10 Toes]

What is researched?

Define researched.

A measurable realm.

What is "a measurable realm"?

 

[epix]

Nonsense. You don't use the point as a co-ordinate, the co-ordinates tell you where the point is.

Thanks for saving me a couple of sentences. Doron uses terms explained in junior high, but doesn't understand their function - or can't recall what the heck an orthogonal line called the "co-ordinate" is all about. Maybe he played hookie when the math teacher did the explaining; maybe he was present in the class, but no comprende.

 

[The Man]

In that case point is anything but dimension, and such an element can't be used for anything where Dimension is involved.

As that is just your ridiculous notion it is simply your problem.

For example: a point can't be used, for example, as an origin for 1-dimension, 2-dimension, 3-dimension, ... n-dimension, etc ..., because a point is anything but dimension (according to No_dimension determination).

Again, only your assertion, so only your problem.

"No" and "0" are not the same thing, because 0 is a number exactly as 1,2,3, etc... are numbers, where "No" is a logical connective.

So again, when you have no apples you don’t have 0 apples? How many apples do you have when you have no apples?

An element that has no dimension, does not have any degree of dimension , which is determined by a given number, including 0.

Acctualy, that “given number” 0 is determined by that fact that it has no dimensions. Just as 0 apples is determined by the fact that you have no apples.

So, an element that has no dimension and an element that has 0 dimension are not the same element.

Incorrect. So having no apples and having 0 apples is not the same thing in your OM?

In that case it specifically can't be used as an ordinate or coordinate, because it has nothing to do with the notion of Dimension, since it lacks this property, according to your reasoning.

A point is not “used as an ordinate or coordinate” an ordinate or coordinates are used to distinguish between points in some space. At least one dimension (of ordering) is required to distinguish one point from another. Just as second dimension was required to distinguish the same angular location on one circle from that on another or just one circle from another.

You still get the existence of things by counting them (The Man: "Do you claim 0 apples are existing apples?") but this is not the case about dimension's magnitudes, for example:

|{0}| = |{1}| = |{2}| = ... = |{n}| = 1, where cardinality 1 means "existing element" and the content of each set is the magnitude of some dimension.

Your "no dimension" is |{}| = 0

As I said before…

The origin of a zero dimensional space (a point) has no ordinate or coordinates (because it technically has no origin)

()

As a set it is, well, empty

{}

This set has a cardinality of 0.

You still simply do not, or do not want to get, the concept of dimension or of a point having no dimension. Again if you think just a point has dimension then give us a set of the ordinate or coordinates to locate that point in itself and give us the cardinality of that set (and thus the dimensionality of your "point" with dimension)

 

[epix]

Nope.

A line is not defined by a point, or collection of points.

It's a part of the definition that refers to the line -- the line that has been used and applied to idealize some of the physical properties of this universe. The reliance on the definition of the line contributed to our knowledge of the nature.

A line has non-local quality w.r.t a point and a point has local quality w.r.t a line.

The purpose of your line is to demonstrate total collapse of reason.

 

[epix]

Incorrect. So having no apples and having 0 apples is not the same thing in your OM?

The aliens have been using two distinct terms: zero-space and empty-space, with the following numerical example of the difference:

You have a number such as 123456 comprised of the digits. Suppose that digit 4 needs to go to the restroom. It leaves the collection of digits called Number. That absence may alter the "genetic makeup" - Number would evolve into two new species (123 and 56), coz the effect of the absence results in 123 56. But since 4 is expected to come back from the restroom, the empty space is marked by zero: 123056. It's similar to the occasion when you go to the restroom and put some article on your chair hoping that a new guest would understand the symbolism and assumes that the chair is still occupied.

If 4 leaves and is not expected to come back, the zero space is referred to as the empty space, and it depends on the decision of 3 and 5 how to react to it. The "decision" is given by the properties of digits 3 and 5, which can be some elements. 3 and 5 can also create "tools" that are instrumental to the creation of other classes than 12356 within the species called Number. For example, you see Number/Class 12156. You can form a theory of the origin of this particular class based on the hypothetical presence and consequent finite absence of 4:

123456 --> 123 56 --> 12[3 5]6 --> 12[3*5]6 --> 12156

 

[doronshadmi]

So having no apples and having 0 apples is not the same thing in your OM?

Since you are in the business of counting things, then you probably know that a point is an existing thing with cardinality 1 if our universe is based on the concept of Dimension, for example |{.}| = 1.

If our universe is based on the concept of Apples then 0 apples or no apples have cardinality |{}| = 0.

See the difference?