The Finite Space Constant.

[throng]

Oh the relevence of relativity.

 

If we have an infinite space any portion would be infinitesmal by proportion.

 

We have an infinite space and a proportionate point.

 

If there were no point no space could 'arise' and without space there is no point, so these are simultaneous.

 

The space is everywhere and the point is anywhere, but it does exist as fundamentally relative.

 

Only anywhere and everywhere are possible for no relative portion exists, and this can be seen as two 'places' without a possible variant.

 

So, there is a constant finite because the 'places' are reliant.

 

Thus the assertion of space derives a finite value.

 

Thank you, I hope you 'see the point.' Hehehehehe.

[Sisyphus]

Thank you, I hope you 'see the point.' Hehehehehe.

 

I'm afraid I don't. Let's see:

 

Oh the relevence of relativity.

 

If we have an infinite space any portion would be infinitesmal by proportion.

 

Alright. In other words, there is no ratio between the finite and the infinite. True enough. Thus, if space is infinite, then any finite portion (like the visible universe) is no fraction of the total.

 

We have an infinite space

 

Is that a supposition or an assertion?

 

and a proportionate point.

 

I guess. The analogy you're making, if I understand you correctly, is finite is to infinite as infinitessimal is to finite. That makes sense in that neither composes any fraction of its counterpart, but it's not exactly the same thing. No ratio is no ratio.

 

And after that, I think you lost me...

[throng]

I'm afraid I don't. Let's see:

 

 

 

Alright. In other words, there is no ratio between the finite and the infinite. True enough. Thus, if space is infinite, then any finite portion (like the visible universe) is no fraction of the total.

 

Yes - an infinite proportion necessitates a fractal measure be an infinitismal ratio.

 

 

 

Is that a supposition or an assertion?

 

Assertion, yes apparently based.

 

 

 

I guess. The analogy you're making, if I understand you correctly, is finite is to infinite as infinitessimal is to finite. That makes sense in that neither composes any fraction of its counterpart, but it's not exactly the same thing. No ratio is no ratio.

 

And after that, I think you lost me...

 

The assertion of space is entirely reliant on the comparison, so the relative portions can only be simultaneous, and can only be one ratio. Infinitismal.

 

All or nothing as it were.

 

We can say by this fundamental that two states of space are possible, two 'places'.

 

Therefore a definite constant value is fundamental to space, because the only possibilities are reliant.

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We have infinite space and any part is proportionately infinitismal, therefore two possible states exist as relatives.

 

The states define eachother and are co-dependent.

 

In this way space is two simultaneous places.

 

'Two' is the only possibile, by relative proportion, hence a constant value.

 

They are completely inter-reliant, hence the value is finite.

 

Hope that's understandable.

[the tree]

We have infinite space

This is the type of statement that either requires evidence - or should be made as a supposition.

 

 

[something]

 

Hope that's understandable.

It's really not.

[throng]

This is the type of statement that either requires evidence - or should be made as a supposition.

 

Yes it is supposition, yet any space is infinite in relative proportion to a point.

 

 

It's really not.

 

First a space is proportionately infinite to the point required as spacial justification.

 

Space can't exist without a reference point for comparison and the proportional relationship is identical in any space.

 

So there is no space without an infinitismal relative.

 

The relationship is requisite.

 

There are only two states. A unit seperates and defines 'TWO' as the exact quantity. (like distance is a unit defining exactly two points.)

 

The value is definite.

 

No other relative is possible so the value is constant.

 

The value is not quantifiable - but it is definite and constant.

 

I hope you see the point, it is undefiable and space itself defines a definite constant.

[Sisyphus]

Yes it is supposition, yet any space is infinite in relative proportion to a point.

 

I still don't understand the bulk of what you're trying to say, but this I understand and must disagree with. There is no proportion between a mathematical point and a finite volume. Nor between a line and a finite volume, nor an area.

[throng]

I still don't understand the bulk of what you're trying to say, but this I understand and must disagree with. There is no proportion between a mathematical point and a finite volume. Nor between a line and a finite volume, nor an area.

 

A space can't exist without a point for comparison.

 

The relationship is point to space.

 

The value of seperation defines 'Two' states, exactly two. It is definite.

 

The value is only discernable as itself. It is constant.

 

Empty space provides a constant definite value.

 

I see this as reasonable and can't find a flaw.

Edited June 10, 2009 by throng

[Sisyphus]

A space can't exist without a point for comparison.

 

I don't know what this means. I should think it would take at least 4 points to define a 3D volume. Or are we talking about something else entirely? What do you mean by "comparison?" You have to explain yourself.

 

The relationship is point to space.

 

What relationship?

 

The value of seperation defines 'Two' states, exactly two. It is definite.

 

Define "value of separation" and "states" as used in this sentence.

 

The value is only discernable as itself. It is constant.

 

Empty space provides a constant definite value.

 

I don't understand what this means. Don't just repeat yourself - explain. In everyday language.

[Sayonara]

I think "Throng... so what?" is an apt question at this point.

[throng]

If there is no point of reference then space is not apparent.

 

Hence point and space are dependant relatives.

 

Any portion of a space still requires a point of reference.

 

Point and space are entirely co-dependant.

 

Any space has (and must have) this dual relationship.

 

Point is one state and space is the other.

 

A value seperates the states. There are two states. The value is singular and defines 'two'. It is finite.

 

The value is only discernable as itself so is constant (effectively).

 

So what? I derive a space constant from emptiness... has that ever been done before?

 

If emptiness is given a finite value... well that's quite relevent isn't it?

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I should think it would take at least 4 points to define a 3D volume. .

 

I hope the above clarifies some questions, I think there is substantial reasoning there and if a defined value can be got from empty space then - hey, something new.

 

Ever considered c as prior to motion? A fundamental primary?

 

You wouldn't believe the trouble I have had explaining that four locations are required for 3D spacial volume. So simple!

 

In this case space is four points infinitely spaced (but infinity makes no discernable shape so I didn't need that.)

 

However if you prefer, the point I refer to is a point related to four infinitely spaced points, but it is more complex that way, though still, it would justify my position above.

Edited June 10, 2009 by throng
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[Sayonara]

So what? I derive a space constant from emptiness... has that ever been done before?

What value does this add to the sum of human knowledge?

[J'Dona]

Hi throng. As you've restated yourself a number of times I hope you don't mind if I use a contracted version of one (post #5) to frame my question:

First a space is proportionately infinite to the point required as spacial justification.

...

The relationship is requisite.

...

The value is definite.

No other relative is possible so the value is constant.

The value is not quantifiable - but it is definite and constant.

Possibly I'm being very stupid but it sounds to me like you're asserting that there exists a "finite constant" [math]n[/math] such that

 

[math]\infty \cdot n = 1' date='[/math']

 

where [math]n[/math] is definite, i.e. not undefined, but not quantifiable, i.e. undefined.

 

[i have assumed for this that relative to the Universe in any dimension the size of a point may be normalised to unity, as I assume you did not mean to suggest that

 

[math]\infty \cdot n = 0[/math]

 

for definite values of [math]n[/math]. This is, of course, an assumption which should be justified.]

 

Unless I've interpreted your use of these words incorrectly, I'm afraid this does not appear to be a significant contribution to science.

[throng]

What value does this add to the sum of human knowledge?

 

It is a value derived from the relationship required for existant empty space.

 

It is a way of defining two spacial properties by removing change.

 

A value free from change - It's just ... new.

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Hi throng. As you've restated yourself a number of times I hope you don't mind if I use a contracted version of one (post #5) to frame my question:

 

Possibly I'm being very stupid but it sounds to me like you're asserting that there exists a "finite constant" [math]n[/math] such that

 

[math]\infty \cdot n = 1,[/math]

 

where [math]n[/math] is definite, i.e. not undefined, but not quantifiable, i.e. undefined.

 

[i have assumed for this that relative to the Universe in any dimension the size of a point may be normalised to unity, as I assume you did not mean to suggest that

 

[math]\infty \cdot n = 0[/math]

 

for definite values of [math]n[/math]. This is, of course, an assumption which should be justified.]

 

Unless I've interpreted your use of these words incorrectly, I'm afraid this does not appear to be a significant contribution to science.

 

Ok, we have the point. It has no location so can be condidered to be 'anywhere'. (point A)

 

We have a space which is 'everywhere'. (point E)

 

These are reduced to two possibilities, inevitibilities rather.

 

Ok - A and E are the simultaneous states, the only two, for no change is apparent.

 

The value (V) is the seperation of A and E. And that seperation is the same for any empty space.

 

So A and E are reliant, or simultaneuosly required.

 

It's the same for any space: constant.

 

Only two possibilities: finite.

 

It is like Heads and Tails, a coin is required. We don't regard the monetary value, because any coin has this quality.

 

We could say coin = 1 or V = 1.

[J'Dona]

Ok, we have the point. It has no location so can be condidered to be 'anywhere'. (point A)

 

We have a space which is 'everywhere'. (point E)

Okay, I had thought this might be the case but assumed that you meant some ratio between a point (with an assumed infinitesimal but non-zero width in a given dimension) and the size of the Universe (in that same dimension). Of course, a point is that which has no part, and so has zero width, so we aren't really talking about points, but infinitesimal scales. I take it that when you imply point E may be everywhere, you mean that we can choose a value anywhere, so that a distance between it and a point A is still well defined.

The value (V) is the seperation of A and E. And that seperation is the same for any empty space.

 

So A and E are reliant, or simultaneuosly required.

 

It's the same for any space: constant.

 

You say that any finite distance as compared to an infinitesimal object (which you call a point), that is, the distance V between a point A and a point E is a "finite constant" which defines a relationship between them, so that even if the physical distance changes, relative to an infinite Universe the value of V is constant. Scaling the position of E up to infinity (as should be acceptable for the infinite space you assert in the first post), we would presumably have an object of finite, non-infinitesimal size at A.

 

In that case the "finite constant" V doesn't obey normal rules of algebra (as described my last post), sort of like dividing by zero or, in this case, multiplying by infinity. V is undefined because the infinity can absorb any definite factor (to become infinity again) so that halving the constant and doubling the infinity to leave the finite distance on the right-hand side fixed is the same as halving the constant and doing nothing to the infinity, while still equating to unity. So the "finite constant" is neither necessarily constant nor defined.

 

Not to belabour the point, but to be completely clear, let us define [math]V[/math] such that

 

[math]V = \frac{d_A}{d_E} \quad \implies\quad d_E \cdot V = d_A[/math],

 

where [math]d_A[/math] is the physical size (at A) and [math]d_E[/math] is the physical distance (to E). (If this is incorrect, do tell, but based on the minimal description provided this is the only case I can see which preserves a constant [math]V[/math] for finite physical distances and scales.) So extending to the case where [math]d_E \to \infty[/math] and [math]d_A \to 1[/math], which must be acceptable for a Universe of infinite size as you assert, we have

 

[math]\infty \cdot V = 1[/math] .

 

But infinity is one of those quantities (along with zero) that breaks normal algebraic rules, so that multiplying by it leads to contradictions, due to odd properties like [math]2 \cdot \infty = \infty[/math]. Specifically, we may obtain

 

[math]\infty \cdot 2 \cdot \frac{V}{2} = 1 = \infty \cdot \frac{V}{2} = \infty \cdot V[/math],

 

from which we can not just divide by infinity and claim that [math]V[/math] is equal to half of itself (as infinity divided by infinity is undefined, otherwise from above we'd have 2 = 1 and thus that I am the pope), but we can still see that the solution satisfying the condition is not unique, so that [math]V[/math] is not well defined. The resolution of this is to accept that there exists (at least not in this form) no definite constant which can reduce an infinity to a finite value, and so that your concept of a finite space constant V is not useful.

 

Since a lot of the confusion here seems to stem from your use of certain words (as the multiple requests for you to define certain terms suggests), you should be more careful in referring to "points", which have no geometrical dimension, when you really mean an infinitesimal quanity, which has dimension. It also changes the algebraic properties of the quantities you are talking about (e.g. an infinitesimal divided by an infinitesimal may be, say, a perfectly well-defined derivative, whereas zero divided by zero is undefined). It's extremely important that you use the corrent terms and definitions when describing something, particularly in mathematics, or you'll end up wasting a lot of people's time as they try to understand something other than what you meant to explain.

[ajb]

A space can't exist without a point for comparison.

 

The relationship is point to space.

 

A topological space is a set (collection of points) and a topology. So ok, points and spaces are in "relationship" with each other.

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Any portion of a space still requires a point of reference.

 

So, subspaces are subsets with the subspace topology. I.e. they consist of points.

 

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Like your other posts on points and spaces I really miss what you are trying to say.

Edited June 11, 2009 by ajb
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[Sayonara]

It is a value derived from the relationship required for existant empty space.

 

It is a way of defining two spacial properties by removing change.

 

A value free from change - It's just ... new.

OK, what you have done here is taken the word "value" as I have used it, and changed the meaning to something else.

 

So I will rephrase:

 

What useful new explanatory and/or predictive power does this add to the sum of human knowledge?

[throng]

OK, what you have done here is taken the word "value" as I have used it, and changed the meaning to something else.

 

So I will rephrase:

 

What useful new explanatory and/or predictive power does this add to the sum of human knowledge?

 

It is new therefore it is 'added'.

 

If indeed there is a value of definition constant to any space, we need no inert location and geometric relationship starts prior to assertion of inertia.

 

Relevent? I think so. Some say observation is intergral to quantum behaviour so new geometric models entailing an observer's perception could well be relevent. I think so.

 

Not everything has been thought of and there is always a new idea.

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A topological space is a set (collection of points) and a topology. So ok, points and spaces are in "relationship" with each other.

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So, subspaces are subsets with the subspace topology. I.e. they consist of points.

 

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Like your other posts on points and spaces I really miss what you are trying to say.

 

Lets just say a space requires a point and a point requires a space.

 

I really want to include the observer, and speculate his 'experience' of change and time etc.

 

First I will try to explain why two points do not discern a distance.

 

If the observer was on a point A and was to travel to point B this would be his apparency:

 

While on point A he would observe one point B. The moment he left point A both points would be observed but there's no perceivable change until arrival at point B, which is identical to his origional observation.

 

Thus, No time expires (no change) then it's the same again.

 

So no velocity, distance or time are perceivable to the 'intergrated' observer.

 

In this model, we have an overall awareness of the relative observer's experience within a geometric system.

 

Distance is relative and not singular.

Edited June 13, 2009 by throng
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[Sayonara]

throng, I have highlighted the salient words in my question:

 

What useful new explanatory and/or predictive power does this add to the sum of human knowledge?

[ajb]

Lets just say a space requires a point and a point requires a space.

 

For (classical) topological spaces this is a tautology. Spaces consist of a set of elements of which are defined to be points and a topology.

 

As I have said before, you can have "spaces" without points.

 

Now as for the rest of your post, can you formulate your ideas using maths. It is the only way we are going to understand you.

[GutZ]

throng, I have highlighted the salient words in my question:

 

What useful new explanatory and/or predictive power does this add to the sum of human knowledge?

 

That's not going to change anything. You are talking to a wall....don't take my word for it...let's see how it plays out.

 

*cough* like i stated in my thread....*cough* *chokes on his own saliva and dies*

 

 

*watches with bowl of popcorn O_O*

[J'Dona]

throng, it should be clear that to convey an abstract, mathematical idea as you describe, you need to use mathematics. Your statements about a "finite space constant" are explicit and have derivable consequences, which are what I explored in my last post. However, you ignored my post and commented on the ones astride it, which frankly given the simplicity of the argument in it (and the ease with which you could destroy it if it makes false assumptions) doesn't speak well of your understanding of the mathematics.

 

Please address the points made in it—not just this post—and provide a mathematical description of what you're trying to say so we can understand you.

[the tree]

*watches with bowl of popcorn O_O*

See, now I think you're getting it.

provide a mathematical description of what you're trying to say so we can understand you.

I'm not all that hopeful, TBH.

As I have said before, you can have "spaces" without points.

Right. This thread is now about pointless topology, not least because it has the best name of any field of study ever. Urm... does anyone here know anything about it?

[ajb]

I don't know anything about pointless topology. I know a little about noncommutative schemes, locally ringed spaces and in particular the geometry of supermanifolds.

 

I am interested, but know very little about noncommutative geometries more generally.

[the tree]

Presumably, commutitive pointless geometries would have something going for them as well. It seems that really, it's just a step of abstraction above regular topologies, so ways that you want to define a regular topology could be translated into a pointless version.

[ajb]

The notion of a point can be far more involved in algebraic geometry than naive topological spaces.

 

For example consider schemes*. It is not true that all the "information" is given by the underlying set. However, lots of set-theoretical structures of schemes can be "recovered" using the functor of points. This soon gets very categorical.

 

From a physics angle, point-less geometries seem a must. For example as a semiclassical description of fermions or quantum mechanical phase spaces. Maybe even quantum gravity requires them.

 

 

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* This also includes supermanifolds where are more like schemes than manifolds