Dimensions and nothing

[Sorcerer]

Can a point can exist in 0 dimensions and if so, does it mean it's wrong to say 0 dimensions is nothing.

 

It seems logical to say nothing has no dimensions. But if a point can exist in 0D then is no dimensions the same thing.

 

If we look at the geometric progressions from 3D to 0D, solid, sheet, line, point. Could we then say nothing has -1 dimensions. Why isn't the empty set included?

 

Is the nomenclature of dimensions chosen for a mathematical reason. How would it alter maths if a point was said to exist in 1 dimension ie all dimensions were renamed as n+1?

 

Doesn't it make more sense to say a point exists in 1 dimension, the first being existence, the second length, the third width etc?

[ajb]

A point can be thought of as a zero dimensional space (or manifold even).

 

The dimension of a vector space (or a manifold) is really given in terms of how many numbers are needed to describe an arbitrary point in that space. So, an 'isolated' point takes no numbers to describe it: you just have a single point and that is it. A point in the real line takes 1 number, a point on the plane take 2 numbers etc...

[andrewcellini]

my mistake, i accidentally downvoted your post ajb.

 

hopefully someone can correct this.

[studiot]

my mistake, i accidentally downvoted your post ajb.

 

hopefully someone can correct this.

 

A pleasure, done.

[conway]

Ajb

 

Is it ever necessary to describe an "isolated point", that needs no numbers, that is a point not on the real's?(QM for example?)? If then we only ever describe a point or vector on the real number line, then has not value also been applied to that point, dimension, or space, or vector. That is the value of the "number" assigned in order to describe the point in the first place? Does this not show an interchange between space and value, and show them working separately in ideas if not in equations?

 

 

Sorcerer

 

I would think that by definition nothing does not exist. So as you point out, it does not posses a dimension. So then 0 can never and does not represent "nothing" of anything. Dimensions or otherwise. Integers are not non existence they are opposite of a given existence. So then integers can not represent nothing. If the set is empty, then there is not a set. But there is never "nothing".

Edited November 17, 2015 by conway

[ajb]

Is it ever necessary to describe an "isolated point", that needs no numbers, that is a point not on the real's?

I don't quite follow. A point can be considered as a topological space such that, as a set it contains just one element.

 

Such things are needed and useful.

 

 

 

(QM for example?)?

The phase space of quantum mechanics is not a set! We are now in the world of noncommutative geometry: this is another story and one to discuss later.

 

 

If then we only ever describe a point or vector on the real number line, then has not value also been applied to that point, dimension, or space, or vector. That is the value of the "number" assigned in order to describe the point in the first place? Does this not show an interchange between space and value, and show them working separately in ideas if not in equations?

Value seems to be to be similar to picking coordinates. For example, on the plane I can describe any point by two number (x,y), but to do that I have to pick some point to label (0,0). Once I have done that, and decided how I am going to label all the other points relative to (0,0), I have a coordinate system. That is a realisation of how to assign the two numbers. However, there is almost never a canonical choice of coordinate system. I can choice some other origin point and change the way I assign numbers to the other points.

 

Moreover, points on the plane exists independently of the coordinates.

[TheGeckomancer]

I am not sure I understand how this would work. A point is hypothetical in the first place. It kind of IS nothing. If you put a point IN nothing then you cannot mathematically relate it to anything. I think in that context is doesn't exist. I think you can say a 0d point can exist as long as we describe it relation-ally to the things around it.

[studiot]

A point can be thought of as a zero dimensional space (or manifold even).

 

The dimension of a vector space (or a manifold) is really given in terms of how many numbers are needed to describe an arbitrary point in that space. So, an 'isolated' point takes no numbers to describe it: you just have a single point and that is it. A point in the real line takes 1 number, a point on the plane take 2 numbers etc...

 

I think there is rather more to it than this.

 

We are again talking about the difference between the value zero and nothing (= no thing).

 

So let us consider density = mass /volume.

 

What sense does it make to state "the density at a point is"?

 

A point has zero volume so you are dividing by zero.

 

Yet the applied maths world happily uses density every day.

 

We get around this conundrum by taking a limit.

 

Of course the same issue applies to other properties besides density, pressure for instance.

Pressue is Force/Area and we need the use the same limiting process when we consider pressure at a point.

 

So should I consider a point as a line with zero length, a square of zero area or a cube of zero volume?

 

These have 1, 2 or 3 dimensions respectively.

 

 

To make matter worse we can also consider the reverse situation.

 

How many dimensions has a cube?

 

Am I bid 3?

 

What about (1 dimensional) Peano curves then?

 

At this point (pun intended) you need to ask the Hausdorf/Mandelbrot question

 

What is a dimension?

[ajb]

A point is hypothetical in the first place. It kind of IS nothing.

We are discussing mathematics, so a point is really no more than an element of a set.

What is a dimension?

As you know, there are more exotic things that are also called dimension, like Minkowski dimension and Hausdorff dimension (both are defined on metric spaces). You also have the Lebesgue covering dimension for topological spaces. And other constructions as well.

 

What I have been discussing is the dimension of a topological manifold, which is tied to the notion of dimension for a vector space. This is probabily, what most people would think of as 'dimension' when discussing points and lines.

[TheGeckomancer]

We are discussing mathematics, so a point is really no more than an element of a set.

 

As you know, there are more exotic things that are also called dimension, like Minkowski dimension and Hausdorff dimension (both are defined on metric spaces). You also have the Lebesgue covering dimension for topological spaces. And other constructions as well.

 

What I have been discussing is the dimension of a topological manifold, which is tied to the notion of dimension for a vector space. This is probabily, what most people would think of as 'dimension' when discussing points and lines.

 

 

I thought that was the point I was making. A point can exist relationally(mathematically). You can describe it's position on a line or in space. But since it itself is not a thing, if you put it in nothing, you lose the ability to describe it.

 

Ignore this. I see why. I never understood what he was asking. Got it. I totally misunderstood the initial question.

Edited November 27, 2015 by TheGeckomancer

[studiot]

 

We are discussing mathematics, so a point is really no more than an element of a set.

 

 

 

Exactly.

 

So it depnds upon which set, which was the essence of my post.

 

So by choosing an appropriate set you can endow your point with any number of dimensions you care to, according to any definition you choose to use.

[Sorcerer]

 

I think there is rather more to it than this.

 

We are again talking about the difference between the value zero and nothing (= no thing).

 

So let us consider density = mass /volume.

 

What sense does it make to state "the density at a point is"?

 

A point has zero volume so you are dividing by zero.

 

Yet the applied maths world happily uses density every day.

 

We get around this conundrum by taking a limit.

 

Of course the same issue applies to other properties besides density, pressure for instance.

Pressue is Force/Area and we need the use the same limiting process when we consider pressure at a point.

 

So should I consider a point as a line with zero length, a square of zero area or a cube of zero volume?

 

These have 1, 2 or 3 dimensions respectively.

 

 

To make matter worse we can also consider the reverse situation.

 

How many dimensions has a cube?

 

Am I bid 3?

 

What about (1 dimensional) Peano curves then?

 

At this point (pun intended) you need to ask the Hausdorf/Mandelbrot question

 

What is a dimension?

Interesting points, I'll look up the mandelbrot question myself later, but if you could elaborate it would be helpful.

 

Personally I think limits are a product of our perspective, we have written the rule in a seemingly 3D world, and need a way to simplify reality so it can be calculated. In reality there must be either an actual limit (ie something similar to the planck scale) or infinite divisibility.

 

The interesting paradox which arises here is that if reality is discrete, then the points which form it have dimension and limits surpass this. And if reality is infinitely divisible then at no scale can we find an area that consists of only 1 point, which limits rely on to measure anything. The math uses 0 and infinity as limits, but the reality is they don't exist. Things either get smaller to a point of finite size, or there isn't any limit, and even the concept of reaching it at infinity fails, because at every scale there are nested infinities, the math would need to be altered to account for infinite concentric sets. It's fine for the abstract, the mathematical surface is defined, but isn't truly the surface we observe.

 

Also my thanks to ajb you have been very helpful.

We are discussing mathematics, so a point is really no more than an element of a set.

 

As you know, there are more exotic things that are also called dimension, like Minkowski dimension and Hausdorff dimension (both are defined on metric spaces). You also have the Lebesgue covering dimension for topological spaces. And other constructions as well.

 

What I have been discussing is the dimension of a topological manifold, which is tied to the notion of dimension for a vector space. This is probabily, what most people would think of as 'dimension' when discussing points and lines.

Well actually he has a point, pun intended, that perhaps mathematical ideas are abstract and their existence isn't tangible as reality is. When we say they exist, it would seem to imply therefore they aren't nothing, but maths existing devoid of the physical world or possible worlds it describes isn't really anything at all.

 

Which gives rise to the interesting multiverse speculation , where every physical possibility described by every form of mathematics does exist. And nothingness never was because it was always just a field of possibilities which had to exist.

 

IE mathematics existence is physical existence. Mathematics is abstract. Abstract things are axiomatic. Physical existence is inevitable because of the possibilities created by the abstract, and an absence of time by which to place a relative moment of existence, the possibilities all just seemingly are.

A point can be thought of as a zero dimensional space (or manifold even).

 

The dimension of a vector space (or a manifold) is really given in terms of how many numbers are needed to describe an arbitrary point in that space. So, an 'isolated' point takes no numbers to describe it: you just have a single point and that is it. A point in the real line takes 1 number, a point on the plane take 2 numbers etc...

I may be uncessarily complicating this, I assume by "describe" you mean label its position. So for 1D, which requires atleast 1 more point, we can choose an origin arbitrarily at any point along the now line segment, the original point which is a boundary, needn't be labeled as such. Infact there is infinite choice due to the infinite number line of points between. I'm trying to resolve how additions or subtractions of dimensions to and from a coordinate system retain or lose their required similarities.

 

It just seems to me (not since last edit) that there's something special about a change from 0D where only 1 point exists to 1d where only infinite points can. Is there a link between 0 becoming 1 and 1 becoming infinite?

 

Edit: Thinking on that it seems perhaps these two values aren't part of the same labelling system, one labels the number of coordinates needed to define a point, the other the number of points. My brain likes to try to find patterns where there are none sometimes. The confusion was because I didn't recognize "dimension" as a unit. And "coordinate" as a way to define position on that dimensional unit.

 

It is possible to have an infinite series of numbered points between any 2 points. If we then describe dimensions with a finite number of points (ie dividing the infinite set evenly into a finite amount of sets), those points themselves must then have dimensions, beginning the problem again. And the infinity isn't resolved just shifted by our definitions.(Sorry this is physics, not math, but in quantum uncertainty points should appear to be smeared, or probabilistic. Perhaps uncertainty is rather just an artefact of macroscopic thought creating the math, the wave particle duality, might actually just be a observation we came to because of our mathematical failure to recognise only a continuum, because nothing occupies only 1 point and as soon as 2 points are occupied, so too are an infinite amount of infinite concentric sets of other points).

 

__________

 

It seems the numbers associated with dimensions should be whole and positive and consecutive. I guess this is required because, as ajb said, they are a count of the number of coordinates.

 

Does it have any mathematical meaning to have fractional dimensions? If we alter the definition of dimension to also include it's size, we could describe the number of coordinates and also show the amount. If all subsequent dimensions are 1:1 proportional in size to the initial dimension, the series would be noted 1,2,3 etc. So if the 1st dimension was bounded between 2 points but infinite in positions between a 2nd dimension labeled 2 would therefore have this same property.

 

Could we however have a 1.5 dimension, with the half dimension contained, relative to the first, only half the number of possible coordinates? Or would we still label that as 2. Why is it only the number of directions that is labeled, surely there is importance in amount of possible coordinates or size with respect to dimensions. Given that all dimensions, labeled as a number line are infinitely divisible, what significance is there between greater and lesser infinities anyway?

 

Is there any meaning in having dimensions like -1D or iD? I've seen i represented as a vector system. In that case is i the 3rd dimension, where the size of the dimension is dependent on the property of i? If i is a possible dimension, which shares properties of negative numbers, what prevents there being actual negative dimensions?

 

Assuming it makes sense, to be negative, to alter the labelling and include relative size as well as number of coordinates, if a space contained 1D and it's inverse -1D, wouldn't it cancel to appear as 0D, how do 2 infinities cancel out to leave the single point contained in 0D? Where does the remaining 1 point come from?

Edited November 27, 2015 by Sorcerer

[studiot]

 

Personally I think limits are a product of our perspective, we have written the rule in a seemingly 3D world, and need a way to simplify reality so it can be calculated. In reality there must be either an actual limit (ie something similar to the planck scale) or infinite divisibility.

 

 

 

Quickly since you are still online.

 

I think you have misunderstood my comment about limits (though ajb did not)

 

I mean the formal mathematical limit the we use

 

[math]\mathop {\lim }\limits_{\delta x \to 0} f(x)[/math]

 

[Sorcerer]

 

Quickly since you are still online.

 

I think you have misunderstood my comment about limits (though ajb did not)

 

I mean the formal mathematical limit the we use

 

[math]\mathop {\lim }\limits_{\delta x \to 0} f(x)[/math]

 

Maybe, it's been a while, I am confusing myself a bit.

 

That says that the equation is conditional on there being a measurable point even at infinity where the change in x approaches 0.

 

There never is this point, just more infinities and always change. Points are perfect concepts flawed by any deviation from that perfection.

https://en.m.wikipedia.org/wiki/Aleph_number

 

After a quick Google I'm reading this. I think this is what I was getting at lol.

https://en.m.wikipedia.org/wiki/Axiom_of_choice

[studiot]

 

it's been a while

 

Yes, I looked up the Mandelbrot reference for you and I didn't realise how long ago his famous book (in English) came out. 1982.

 

You really ought to read

 

The Fractal Geometry of Nature.

 

It is not too mathematical and I don't think there is even any calculus in it.

However the book is a masterpiece.

It answers many of your questions about fractional dimensions, the nature of dimensions, and even (if I remember correctly) discussues the Peano space filling curves I referred to.

It has lots of pictures and diagrams , which should keep Mike Smith Cosmos happy as well.

 

Be aware the the word point as used in set theory is just another name for an element or member of a particular set.

 

So in the set of all numbers, each number is called a point, even though there are are an infinite number of them in any interval.

 

Finally your thinking about points is perhaps closer to the ancient Greek thinking

 

Euclid definition:

A Point : That which hath no part.

A Line: A breadthless length.

 

Note that Euclid regards lines as starting and finishing so had at least two points (each end).

Modern maths regards a line as going on forever, without end points.

 

Further there was a religious dimension (aspect) to ancient Greek mathematics.

They regarded geometric figures, points, lines etc a somehow 'perfect' or embodiments of perfection as achieved in whatever heaven they believed in.

Our real physical world was regarded as just a pale shadow or copy of the 'real' thing.

[Sorcerer]

Thanks Studiot, I might check that book out, constantly learning.

 

Well yes I guess it's close to Euclid, simply because I am adding/subtracting essential points as I step up/down dimensions.

 

0 dimensions contains 1 point. 1 point must be added to transition to 1d, so the smallest possible line does have 2 points on either end, but it is also infinitely divisible. Because a point has no part, no part can be placed an infinite amount along this line. Any addition of a point to the line creates a smaller but also infinite set. There are infinite possible infinite lines (as defined by all containing unique coordinates) and infinite overlapping lines.

 

The entire set of infinite sub sets is bounded by a point on either end. So I guess my definition of a line is similar to both Euclid and modern math.

 

My view is that reality is the perfection and mathematics must adapt to describe it as best it can. Euclid geometry can be envisioned and attempted to be formed in reality, but scale magnifies the reality that it isn't any match.

 

I feel some of our deepest problems with modelling reality are due to this backwards view. Reality allows us the mind with which to create the math to describe it. The dependence is on wether we can perfect that description, do we have the mental faculty?

 

https://en.m.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

 

Someone once said "if the brain were so simple that we could understand it, we'd be so simple we couldn't." Similarly the perfection doesn't lie with our creation of axioms, but with the reality which we can create them in. Our logic is conditional on any limits imposed, and any limits don't necessarily need to be perceivable.

 

I tend to think there is an epistemological graph with knowledge increasing over time, towards an absolute limit of truth. It has assymptotes on each end. We all ways know something, but will never know everything. The funny thing is when people assume we're closer to the assymptote approaching truth. We don't know where we are, that's one of the things we never can.

 

I must add there is no reason that a line cannot be unbounded. To say there is a positive and negative point at increasing and decreasing infinity makes some sense if you consider them hypothetically, but when considered as reality, there can always be another point, this type of infinity really makes no sense to use as a limit because it isn't approached, all other points are paradoxically equidistant from both. Which also means each polar infinity is also equidistant from all other points including their counterpart.

 

Only does the paradox begin to resolve when we consider points as being on a line with aleph number coordinates.

Edited November 27, 2015 by Sorcerer

[ajb]

I assume by "describe" you mean label its position.

Yes.

 

So for 1D, which requires atleast 1 more point, we can choose an origin arbitrarily at any point along the now line segment, the original point which is a boundary, needn't be labeled as such. Infact there is infinite choice due to the infinite number line of points between.

Right. Just think of a line. You can associate to any point a number once you have picked a point to label as 0. There is no canonical way of deciding what point should be assigned zero.

 

 

 

I'm trying to resolve how additions or subtractions of dimensions to and from a coordinate system retain or lose their required similarities.

I do not follow what you are trying to do here.

 

 

 

It seems the numbers associated with dimensions should be whole and positive and consecutive. I guess this is required because, as ajb said, they are a count of the number of coordinates.

This is true for the dimension of a vector space and so the dimension of a topological manifold. We can also allow infinite dimensions also, but this usually involves more work to make proper sense of.

 

 

Does it have any mathematical meaning to have fractional dimensions?

Not in the context of vector spaces and topological manifolds. However, there are other kinds of dimension that are used, especially in the context of fractals, that can take on fractional values.

 

Is there any meaning in having dimensions like -1D or iD?

There are some notions of negative dimensional spaces in topology. Again, these cannot be topological manifolds as we usually understand them.