The Nature of Infinity

[Cyperium]

It's just a thought but if the radius is infinite (which was what I meant, but it could also be the circumference that is infinite) then the circle becomes a straight line which meets itself at the end of infinity (on both sides of infinity). I believe that this is true, because at any point of a infinite line it is shown that infinity meets itself at that point coming from both sides of infinity.

A circle is defined as something that has equal distance from the center to all of its points (or all of it's points to the center), in a infinite circle all the distances to the center would be infinite so they would be equal and it would still be a circle if we go by definition.

 

[quantum_wave]

Motor Daddy answered your question. You cannot have an infinite distance between two points, so your radius originating at the center point, and extending to a point on the circumference, must have two points to define it, and therefore the radius cannot be infinite.

 

[Cyperium]

Motor Daddy answered your question. You cannot have an infinite distance between two points, so your radius originating at the center point, and extending to a point on the circumference, must have two points to define it, and therefore the radius cannot be infinite.

I think I see what you mean now, each point on the circumference would also be seperated by infinity (as a infinite distance from the center must always deviate in angle, even if ever so slightly).

I don't know if this is worth thinking about but I'll say it anyway; what if we have a number of infinite lines all originating from one point in angles, the distance between those lines would eventually be infinite, but since they do have a originating angle that is fixed then we still have a fixed distance between them, but since there is a infinity of distance between them, then perhaps infinite lines are also impossible? (at least infinite lines that has a beginning)

Point being that a fixed distance can't at the same time be a infinite distance.

Also, this would mean that two infinite lines can never have a angle between them, as at some point they would have a fixed distance, but in infinity have a infinity of distance between them. So all infinite lines are parallell to eachother at either a fixed distance or a infinite distance, but never both. This also means that two infinite lines can never cross.

 

[quantum_wave]

I think I see what you mean now, each point on the circumference would also be seperated by infinity (as a infinite distance from the center must always deviate in angle, even if ever so slightly).

You don't have a circle until you have a circumference and a radius. Also, you have to decide if your circle is on a plane, meaning that the subset of interior points are all on the same plane. Then the circle is the set of points in a plane that form a closed curve so that every point of the curve is the same distance from the center of the circle which is a single point.

If your circle complies with those postulates it does not have an infinte radius or an infinte circumference.

Now let's forget those constraints and try to imagine circle with an infinite circumference. If you are at the center of the circle and want to travel to the circumference you will never be able to get there because it is infinitely far away.

If you were to start out at the circumference and wanted to travel to the center, you would never be able to get there either for the same reason.

You imaginary circle cannot exist in incidence geometry.

Both of those statements about this imaginary circle are true because if you are at the center point, in order for the circumference to be an infinitely long closed curve it would have to have an infinitely long radius, and we have already claimed that no line with a start point and an end point, like a radius of the circle, can be infinite. Any straight line between two points in a plane is of finite length.

I don't know if this is worth thinking about but I'll say it anyway; what if we have a number of infinite lines all originating from one point in angles, the distance between those lines would eventually be infinite, but since they do have a originating angle that is fixed then we still have a fixed distance between them, but since there is a infinity of distance between them, then perhaps infinite lines are also impossible? (at least infinite lines that has a beginning)

You can have any number of infinite lines all originating from one point, even an infinite number of such lines, and any two of those lines will have a finite angle between them. If you never attempt to measure the distance between any two of them, you can say that the distance between each of them increases toward infinity. However, even if those lines are all infinitely long, the distance between any two of them will never reach infinity. You cannot have an infinite distance between any two of them, because that distance implies a line that connects a point on one of the lines to a point on another of those lines in order to make the measurement. You see, your lines can extend infinitely, but by establishing a distance between them you have to also establish a point on each of them, and those two points are going to be a finite distance from the center point, and the line between them will always be finite. The "infinity" in that case is simply that the distance between each line increases infinitely.

Point being that a fixed distance can't at the same time be a infinite distance.

This is correct, and though you can define a line that is infinitely long, and start that line at a point, once you establish a point somewhere else along the infinite line, the distance between the points can never be infinite.

Also, this would mean that two infinite lines can never have a angle between them, as at some point they would have a fixed distance, but in infinity have a infinity of distance between them. So all infinite lines are parallel to each other at either a fixed distance or a infinite distance, but never both. This also means that two infinite lines can never cross.

No, now you are not thinking this through. You can have two infinite lines with the same start point that have an angle between them. You just cannot have an infinite distance between the far end of those lines because infinite lines do not have two ends.

 

[Cyperium]

You don't have a circle until you have a circumference and a radius. Also, you have to decide if your circle is on a plane, meaning that the subset of interior points are all on the same plane. Then the circle is the set of points in a plane that form a closed curve so that every point of the curve is the same distance from the center of the circle which is a single point.

If your circle complies with those postulates it does not have an infinte radius or an infinte circumference.

Now let's forget those constraints and try to imagine circle with an infinite circumference. If you are at the center of the circle and want to travel to the circumference you will never be able to get there because it is infinitely far away.

If you were to start out at the circumference and wanted to travel to the center, you would never be able to get there either for the same reason.

You imaginary circle cannot exist in incidence geometry.

Both of those statements about this imaginary circle are true because if you are at the center point, in order for the circumference to be an infinitely long closed curve it would have to have an infinitely long radius, and we have already claimed that no line with a start point and an end point, like a radius of the circle, can be infinite. Any straight line between two points in a plane is of finite length.
You can have any number of infinite lines all originating from one point, even an infinite number of such lines, and any two of those lines will have a finite angle between them. If you never attempt to measure the distance between any two of them, you can say that the distance between each of them increases toward infinity. However, even if those lines are all infinitely long, the distance between any two of them will never reach infinity. You cannot have an infinite distance between any two of them, because that distance implies a line that connects a point on one of the lines to a point on another of those lines in order to make the measurement. You see, your lines can extend infinitely, but by establishing a distance between them you have to also establish a point on each of them, and those two points are going to be a finite distance from the center point, and the line between them will always be finite. The "infinity" in that case is simply that the distance between each line increases infinitely.
This is correct, and though you can define a line that is infinitely long, and start that line at a point, once you establish a point somewhere else along the infinite line, the distance between the points can never be infinite.
No, now you are not thinking this through. You can have two infinite lines with the same start point that have an angle between them. You just cannot have an infinite distance between the far end of those lines because infinite lines do not have two ends.

I thought it through and I believe that you were wrong about the infinite line not being a circle.

The circumference of a perfect circle is infinitely defined, and this already is a infinite line (if you zoom in on the circumference to infinity then it would never bend). Because of this there is no reason to assume that a infinite line would not be a infinite circle.

So the definition itself of the circumference of a circle is proof that it is a infinite line. I can see no way around this (pun not intended).

So if you would see a perfectly defined circle then you would actually see a infinite line which crosses infinity and meets itself. I guess that infinity itself has a boundary somehow where it "wraps" around. Either that or it might just be that a infinite line just can't exist.

 

[quantum_wave]

Do you have a link to a web page that will explain it to me? I'm not familiar with the geometry that you are describing.

 

[brokenpower]

Here is something that has always bothered me. If infinity does exist (or can be explained) wouldn't it need a starting point? What would that starting point be?

 

[Cyperium]

I thought it through and I believe that you were wrong about the infinite line not being a circle.

The circumference of a perfect circle is infinitely defined, and this already is a infinite line (if you zoom in on the circumference to infinity then it would never bend). Because of this there is no reason to assume that a infinite line would not be a infinite circle.

So the definition itself of the circumference of a circle is proof that it is a infinite line. I can see no way around this (pun not intended).

So if you would see a perfectly defined circle then you would actually see a infinite line which crosses infinity and meets itself. I guess that infinity itself has a boundary somehow where it "wraps" around. Either that or it might just be that a infinite line just can't exist.

No special geometry is needed so I don't have a link. If there is a proof then I'm afraid you have to find it yourself, it's not something I have looked up.

I'll explain it again;

You said that a infinite circle couldn't exist, but exactly the same thing happens with a circle that is infinitely defined (since it actually is a infinite circle, but in the other direction). Therefor the same reasoning that would be applied to a circle of infinite radius, can be applied to a circle that has a infinite definition.

Since the circumference is defined by the diameter times Pi, it has to be infinitely precise, as Pi has infinite precision. There is no special geometry needed and the circle doesn't need to be of infinite size.

Is my reasoning correct on this or is there some kind of fault I'm making to draw the wrong conclusion?

 

[Cyperium]

Here is something that has always bothered me. If infinity does exist (or can be explained) wouldn't it need a starting point? What would that starting point be?

I'm not sure about this, but I think infinity works both ways. If there is a infinity with a starting point, wouldn't it, if looking at it the other way around, have described a infinity which has now come to an end at the starting point? The trouble would then be that it isn't sure if everything actually works both ways. The thing about infinity is that there are many kinds of it, I think that a infinite line (if it would exist) can't have a starting point, however, infinite time is easier to conceive that it has a starting point but no end (if you see what I mean?).

So I really can't tell, but I think that some infinities require a start, while some might not.

 

[river]

To understand infinity is to understand the opposite , finite

Then dwell upon the mathematical

Then understand the essence of the mathematical , shape(s)

Then dwell upon the essence of shape(s)

Objects

Then dwell upon the existence of objects , in the infinite and finite

Infinite , never ends

Finite , objects end

Now dwell again

 

[arfa brane]

A finite length, straight line can have a slope which is infinite.

When you make a strip of paper into a Mobius strip, the twist is projected to infinity when you look at local sections (since none of them are twisted).
Both forms of "infinity" are mathematical, but the second requires the notion of a local frame of reference.

 

[river]

A finite length, straight line can have a slope which is infinite.

When you make a strip of paper into a Mobius strip, the twist is projected to infinity when you look at local sections (since none of them are twisted).
Both forms of "infinity" are mathematical, but the second requires the notion of a local frame of reference.

And the second therefore needs objects to frame the reference , in otherwords

The Nature of infinity is more completely understood within the understanding of objects or energy and matter than mathematical concepts of infinity

Since the exploration into the infinity of energy and matter have a more relevant or concrete consequence to thinking upon infinity

And therefore existence of everything

 

[river]

Therefore the real Nature of infinity , lies in exploring the infinity or the finite Nature of energy and matter , in depth

And the consequences thereof of each idea and the ramifications

 

[arfa brane]

The Nature of infinity is more completely understood within the understanding of objects or energy and matter than mathematical concepts of infinity

The kicker here is, our understanding of such 'physical' objects is mathematical.

Besides, energy is an abstraction--we can't measure it directly, only what happens in some system to abstract 'inputs and outputs of energy'.
Matter is a field, a field is fairly abstract. It isn't surprising then that we use mathematical abstractions to define physical objects, parameters, etc.

 

[arfa brane]

Some of the abstract objects in Einstein's theories are: events, inertial frame of reference, connection, geodesic, and perhaps most significantly, curvature in four dimensions.

Minkowski space is a visualisation of four-dimensional spacetime which is (locally) flat. It describes mathematically the space of simultaneous events, with a 'spherically-expanding' future plus a spherically-contracting past. Each horizontal section of the past or future part of an observer's lightcone is the boundary of a sphere. It represents a certain 'causal structure' which is the geometry of a surface of revolution. The lightcone is symmetric about the observer's past and future (of all possible events, which are just objects that have a coordinate in some basis, a 4-position, say). It separates events, or, the function that "geometrises" the space partitions the set of total events--the observer can only assign 'cause' to those coordinates lying on or inside their lightcone.

 

[river]

Originally Posted by river
The Nature of infinity is more completely understood within the understanding of objects or energy and matter than mathematical concepts of infinity

The kicker here is, our understanding of such 'physical' objects is mathematical.

Besides, energy is an abstraction--we can't measure it directly, only what happens in some system to abstract 'inputs and outputs of energy'.
Matter is a field, a field is fairly abstract. It isn't surprising then that we use mathematical abstractions to define physical objects, parameters, etc.

The understanding of an object can be further gained by mathematics , but mathematics doesn't make the object

The energy would be of plasma , protons and electrons

Anyway my point is that to understand infinity is not based on some mathematical based abstraction , that is pure mathematics

To really understand infinity means that you have to look at the energy and matter in the way of , if neither is around or disappears completely , never to return or be replaced

 

[river]

Some of the abstract objects in Einstein's theories are: events, inertial frame of reference, connection, geodesic, and perhaps most significantly, curvature in four dimensions.

Minkowski space is a visualisation of four-dimensional spacetime which is (locally) flat. It describes mathematically the space of simultaneous events, with a 'spherically-expanding' future plus a spherically-contracting past. Each horizontal section of the past or future part of an observer's lightcone is the boundary of a sphere. It represents a certain 'causal structure' which is the geometry of a surface of revolution. The lightcone is symmetric about the observer's past and future (of all possible events, which are just objects that have a coordinate in some basis, a 4-position, say). It separates events, or, the function that "geometrises" the space partitions the set of total events--the observer can only assign 'cause' to those coordinates lying on or inside their lightcone.

But what has any of this have to do with infinity ?

 

[arfa brane]

But what has any of this have to do with infinity ?

I'm just sayin', you know?

In special relativity you can consider the speed of light is infinite in certain circumstances.
And you seem to be saying that despite our understanding of physicality being based on mathematical abstractions, infinity can be understood without mathematics. How would that work, what would it tell us?

 

[river]

Originally Posted by river
But what has any of this have to do with infinity ?

I'm just sayin', you know?

sure

In special relativity you can consider the speed of light is infinite in certain circumstances.

In what circumstances ?

And what is the essence or what is fundamental reason that the speed of light could be infinite ?

And you seem to be saying that despite our understanding of physicality being based on mathematical abstractions, infinity can be understood without mathematics. How would that work, what would it tell us?

The understanding of the physicality by mathematical abstractions is important but misses the point of the infinity of the existence of..objects

Its not the understanding of the object that defines the infinite existence of the fundamental object to my mind

But rather whether if the fundamental energy and matter were not infinite and the consequences thereof

This is why the mathematics of the object becomes irrelevant

It works by comprehending that infinity is actually physically based

And it tells us that energy and matter are infinite and always will be, to be otherwise would be an extremely complicated scenario

 

[arfa brane]

The understanding of the physicality by mathematical abstractions is important but misses the point of the infinity of the existence of..objects

Mathematics does this with the existence of objects: "Let S be the set of all objects".
So now you can do: "Let S' be the set of all objects in an observer's worldline".

You don't have to worry about whether S or S' are infinite sets, instead it's some property that every member of the set has. But you still need to define what an object is, as well as what a worldline or observer is.

This is why the mathematics of the object becomes irrelevant

Well, you just defined a set containing at least one of "objects whose mathematical representation is irrelevant". A set is (a) mathematical (structure), so are you saying it's irrelevant if this object is in some set?