A laymans question

osmethod

Please correct me:-

The following premises

An equilateral triangle has 3 sides of equal length, where each side intersects at a common point.
Intersection of two lines is defined as the intersection of a common point between the two lines.
A point is defined as that which has position but no space or time.
A line has position, space and time

Problem:

Take 3 lines and see them as a collection of points of equal number. e.g. each line is 6 points long.

If I follow the rules above I can't actually construct the equilateral triangle.

Why?

Because in order for the lines to remain the same length after constructing the equilateral triangle I'd have to have 2 points occupy the same position!

ecksor

osmethod wrote:

The following premises

An equilateral triangle has 3 sides of equal length, where each side intersects at a common point.
Intersection of two lines is defined as the intersection of a common point between the two lines.
A point is defined as that which has position but no space or time.
A line has position, space and time

Are you offering these as interpretations of the standard definitions in euclidean (school) geometry or are you trying to redefine these terms to invent your own geometry?

Take 3 lines and see them as a collection of points of equal number. e.g. each line is 6 points long.

I've never seen any definition of a line in any geometry that regards it as a finite set of points.

Because in order for the lines to remain the same length after constructing the equilateral triangle I'd have to have 2 points occupy the same position!

I don't see what you're getting at. Points can belong to more than one set (in this case a line being a set of points).

Capt'n Midnight

Try using the word "meet" instead of "intersect"

Also if you have 18 points you can make a triangle whose sides are 6 units long - 7 dots along each side.

If you are at the Equator on 0 degrees and head North till you reach the North pole. Then turn 90 degrees and head South. At the equator you take another 90 Degree turn and arrive back at your starting point. You are now 90 degrees away from your starting point. 90+90+90=270 The triangle has 270 degrees

osmethod

I am not trying to redefine anything... as I said "A laymans Question". I'm not qualified as a Mathematician but I respect the discipline.

"Also if you have 18 points you can make a triangle whose sides are 6 units long - 7 dots along each side"

Question: Then in a countable sense is that not 3 * 7 = 21 points when you actually started with 18?

In order for 18 points to form the three sides strictly of equal length and maintain the number of dots; 3 sets of 2 points have to occupy the same position? This being the case, if I disassemble the 18 points triangle it now becomes a triangle of 15 points.

Please comment further and I will offer an alternative conjecture (not sure if that is the correct terminology).

osmethod

Is it not the case that a line is an infinite set of points by the divisibility theorm. However, when you construct a geometrical object you have an implied finiteness (length)?

osmethod

If you are at the Equator on 0 degrees and head North till you reach the North pole. Then turn 90 degrees and head South. At the equator you take another 90 Degree turn and arrive back at your starting point. You are now 90 degrees away from your starting point. 90+90+90=270 The triangle has 270 degrees


Have you moved to a 4 dimensional geometry in this case. e.g. a Rheiman Sphere?

I wanted to keep it simple initially to see if what i was seeing was worth while, thats all.

ecksor

Is it not the case that a line is an infinite set of points by the divisibility theorm. However, when you construct a geometrical object you have an implied finiteness (length)?

Don't see what you mean by the divisibility theorem here but the finiteness in length doesn't imply a finite number of points. Any line segment has an infinite number of points.

This is getting tied in knots. I'm assuming that by points you mean points on a line segment spaced out by some standard unit beginning and ending on either end of the segment.

osmethod wrote:

Question: Then in a countable sense is that not 3 * 7 = 21 points when you actually started with 18?

Firstly, with the line segments you described I don't see where Capt'n Midnight is getting his 7 points. Those 3 line segments with 6 points each corresponds to a triangle with 15 points (because each line segment has two points that it shares with two other line segments).

In order for 18 points to form the three sides strictly of equal length and maintain the number of dots; 3 sets of 2 points have to occupy the same position? This being the case, if I disassemble the 18 points triangle it now becomes a triangle of 15 points.

If 3 sets of 2 points occupy the same position in the way you describe, then you only need 3 points to do that.

If those three points are a,b,c then the line segments are ab, bc, ac

In other words, line segment 1 contains a and b, line segment 2 contains b and c and line segment 3 contains a and c.

Have you moved to a 4 dimensional geometry in this case. e.g. a Rheiman Sphere?

The Riemann sphere is a 3 dimensional object, but the geometry on the surface is isomorphic under stereographic projection to the extended complex plane which is 2 dimensional as I understand it. It's also irrelevant to your original question.

osmethod

If 3 sets of 2 points occupy the same position in the way you describe, then you only need 3 points to do that.

If those three points are a,b,c then the line segments are ab, bc, ac

In other words, line segment 1 contains a and b, line segment 2 contains b and c and line segment 3 contains a and c.


Yes! However, in a countable sense, if 3 sets of 2 points occupy the same position and you say you then only need 3 points have you not then lost 3 points. I was of the opinion that 2 points can't occupy the same position. Also, consider the disassembly of the triangle into line segments, after reducing 3 sets of 2 points to 3 individual points:

Line segment [ab] is 6 points long

Line segment [bc] is 5 points long (because you used the point at b in[ab] you can't use it again - it's gone!)

Line segment [ac] is 4 points long (because you used the point at b in[ab] and the point at c in[bc] you can't use them again - there gone!)

This is in a countable sense. I would appreciate any further criticism/comment etc

ecksor

Your language is very unclear. Points don't have length so "5 points long", "6 points long" etc are painful to read.

Yes! However, in a countable sense, if 3 sets of 2 points occupy the same position and you say you then only need 3 points have you not then lost 3 points.

No, you only had 3 points to begin with, just that they all belonged to more than one set.

I was of the opinion that 2 points can't occupy the same position.

That's correct because it's just one point. Your confusion is in thinking that a point can't be on more than one line.

What does countability have to do with this?

elivsvonchiaing

osmethod wrote:

Please correct me:-
...
1. A point is defined as that which has position but no space or time.
2. A line has position, space and time

It seems to me you are trying to relate mathematics to the real world - which it was initially meant to do, but only in an abstract sense. You need to understand this as a concept. E.g.

1.1 a point. can be just a point (zero dimension) e.g. universe before big-bang singularity.

1.2 a point can be one dimensional this can have a value of -infinity - 0 - +infinity. The number of distinct values between 1 and 1m is infinite. The number of values between 1 and 2 is not one millionth of these it is also infinite. In mathematics there are no "atoms". You can describe a line here with a, b.

1.3 a point can be two dimensional this can have a value (a, b) both a and b subscribing to the rules of 1.2. You can describe a line here with (a,b), (c, d). Or even with (a, b), (a, c) a vertical line. No points are double-parking here. Here the values b and c or d (i.e the second dimension) could represent time. In which case if you carefully watch the dimension and are infinitely perceptive you will see a point appear and disappear immediately. Try to understand the concept is all I'm saying.

2. See above.

osmethod

Yes this is the kind of understanding I was looking for - thank you for the courtesy of offering this knowledge.

I'm constructing a "countable model interpretation" of building an object wrt to time. By layman I mean I can visualise it but I'm not a mathematician to be able to translate it. I read logical/mathematical books only as a hobbyist.

Your point:
1.2 a point can be one dimensional this can have a value of -infinity - 0 - +infinity. The number of distinct values between 1 and 1m is infinite. The number of values between 1 and 2 is not one millionth of these it is also infinite. In mathematics there are no "atoms". You can describe a line here with a, b.

If the point is 1 dimensional is it not a line? i.e. not a point in the strictest sense of linearity? Initially, my construct needs to be in the same dimension.

Accepting your concept thus: the one dimensional nature can be bound wrt time because one can imply a direction.

Consider the following: (i'm not sure if this is a good illustration but anyhow)

Take 2 points and name them a, b respectively. A ply (implication) operaton can bring you all the way from a to b but not touch b. There is an infitesimally small space left. Vice versa from b to a.

Now I have 2 infitesimally small spaces. If i then proceed to use standard operations and say add the 2 infitesimally small spaces together I have 1 infitesimally small space that is actually smaller in size than when they were independent wrt time. This is because I have moved further on in time and the original 2 independent infitesimally small spaces were getting infitesimally smaller because of the ply operation continuing in time.

I'd appreciate further comment....

ecksor

osmethod wrote:

If the point is 1 dimensional is it not a line? i.e. not a point in the strictest sense of linearity? Initially, my construct needs to be in the same dimension.

A point has zero dimension. I think what he means is that a point can exist in a one dimensional space, such as a line.

Take 2 points and name them a, b respectively. A ply (implication) operaton can bring you all the way from a to b but not touch b. There is an infitesimally small space left. Vice versa from b to a.

Now I have 2 infitesimally small spaces. If i then proceed to use standard operations and say add the 2 infitesimally small spaces together I have 1 infitesimally small space that is actually smaller in size than when they were independent wrt time. This is because I have moved further on in time and the original 2 independent infitesimally small spaces were getting infitesimally smaller because of the ply operation continuing in time.

I'd appreciate further comment....

You can't really add infinitesimally small spaces in a meaningful way. To the best of my knowledge they don't even let mathematicians talk about infinitesimals nowadays until they're postgraduates, but that's slightly beside the point (unless physicists use this language a bit more freely).

Adding doesn't happen with respect to time. I think what you want to do is define a function of time such that the result can become arbitrarily close to a point as time increases. I'm not quite sure what you're trying to do though.

elivsvonchiaing

osmethod wrote:

Your point:
1.2 a point can be one dimensional this can have a value of -infinity - 0 - +infinity. The number of distinct values between 1 and 1m is infinite. The number of values between 1 and 2 is not one millionth of these it is also infinite. In mathematics there are no "atoms". You can describe a line here with a, b.

If the point is 1 dimensional is it not a line? i.e. not a point in the strictest sense of linearity? Initially, my construct needs to be in the same dimension.

No it is still a point on a line - the line being the 1 dimension
[EDIT] already answered above [/EDIT]

osmethod wrote:

{Previous in post} I'm constructing a "countable model interpretation" of building an object wrt to time.... Accepting your concept thus: the one dimensional nature can be bound wrt time because one can imply a direction.

Consider the following: (i'm not sure if this is a good illustration but anyhow)

Take 2 points and name them a, b respectively. A ply (implication) operaton can bring you all the way from a to b but not touch b. There is an infitesimally small space left. Vice versa from b to a.

Now I have 2 infitesimally small spaces. If i then proceed to use standard operations and say add the 2 infitesimally small spaces together I have 1 infitesimally small space that is actually smaller in size than when they were independent wrt time. This is because I have moved further on in time and the original 2 independent infitesimally small spaces were getting infitesimally smaller because of the ply operation continuing in time.

I'd appreciate further comment....

I'm really trying to visualise what you're trying to achieve and the only thing that comes to mind is Gantt diagrams for some reason.

Can you expand more on ""countable model interpretation" of building an object wrt to time"?

osmethod

"No it is still a point on a line - the line being the 1 dimension"

Trying to build a construct in time!

Consider (A)....(Using a static framework as measurement)

I start with a point. I then have space between this point and the next point. Consider the points separately. There not a line unless you make it so.

Not use a ply (unary operation) on the first point. Direction wise I can get to the next point but not touch it. So now I have a point and a ply operation still as a unit. (Picture it as a dot with an arrow coming from it!)

I can cover all space except an infitesimally small amount. (I still haven't introduced an extra dimension)

Consider (B).... (An adjusted framework)

A typical X Y axis has 0 as the common point. Remove 0 and all both axes to move but (by above) neither axis can reach -1 or +1, they'll always be infitesimally small from doing so.


By using this framework I can build a construct such that when I use the ply operaton from (Consider A) on the framework and the using the (Consider "A") construct, I have a situation where i can potentially reach the second point in time.

Thus the possibility of continuum and a finiteness in time.


My "countable model interpretation" counts the infitesimally small spaces to show how to build a contruct to measure behaviour similar to the above descriptions.

elivsvonchiaing

osmethod wrote:

...I can cover all space except an infitesimally small amount. (I still haven't introduced an extra dimension)
...Thus the possibility of continuum and a finiteness in time.


My "countable model interpretation" counts the infitesimally small spaces to show how to build a contruct to measure behaviour similar to the above descriptions.

Still don't get this. Are we talking a computer model/simulation here?

Are we talking quantum time here eg. Planck intervals (quantum physics) or clock-cycles (synchronous logic)? What area are you working in is what I mean by this

osmethod

Nothing Specific except my understanding of the following.

Stephen Hawking's thesis on the singularity theorm implied that all time rewound back to an instant (point). Hence, the laws of physics broke down at this instant, no further analysis could be performed.

In 1994 I believed this to be incorrect. I believed it rewound back to a unit ok but the unit was an infitesimally small space, therfore the laws of physics wouldn't break down.

I understand (laymanly) that he has revised his thesis on the singularity. He now conjectures that its not a singularity but there is AN amount of space/time.

I looked at Axiomatic set theory to build a model that could justify my initial interpretation of the "unit of space/time" being the instant/point (in standard interpretation). I decided against as there is too much pilosophical argument about the "Axiom of Choice". I looked at "Cantor's continuum" hypothesis but in the end Godel's work looked most promising using logical countable models and constructive lattices.

Hope thats a bit clearer. Bare in mind I am a layman.... thanks for your patience.

ecksor

You're trying to do physics here, not maths. Your ideas are phrased as physics constructs. Nothing wrong with that, but if you want to do maths then you need to know the areas you're working in and I suspect that the maths that Hawking uses aren't areas you're familiar with.

This book might be a useful resource for you: http://www.amazon.com/exec/obidos/tg/detail/-/0262510049/