Infinity: Limitless or Limited? Exploring the Concept of Infinity in Mathematics

[MeJennifer]

So what *are* your definitions for length, area, and volume?

Sorry but to understand that a point has no length, area or volume, or that a line has no area or volume, or that a triangle has no volume is mathemathics 101.
I suggest you start with Euclid, he is good!

 

[shmoe]

Sorry but to understand that a point has no length, area or volume, or that a line has no area or volume, or that a triangle has no volume is mathemathics 101.
I suggest you start with Euclid, he is good!

Should I assume that you in fact don't have a definition for length, area, or volume and yet you feel you can declare which objects they are or isn't defined for? Does that not seem strange?

 

[MeJennifer]

Should I assume that you in fact don't have a definition for length, area, or volume and yet you feel you can declare which objects they are or isn't defined for? Does that not seem strange?

The only thing strange here is that some people wish to assert that objects of dimensionality n have properties that apply only to dimensions higher than n.

 

[shmoe]

The only thing strange here is that some people wish to assert that objects of dimensionality n have properties that apply only to dimensions higher than n.

Huh? So 'area' is a 2 dimensional property that should only apply to things of dimension 2 or higher (I guess you are actually saying strictly greater than 2)? That's what you are saying? I'd first ask what's the area of a cube. I'd then ask again what your definition of "area" is that you are willing to conclude what objects have this property?

It's not that difficult a concept. If you want to discuss "area" or whatever thing and you don't even have a definition for this thing, then you are just talking nonsense about nothing. These concepts don't have intrinsic definitions or properties, they have whatever properties that follow from the definitions that we give them, before that they are meaningless.

 

[shmoe]

Huh? So 'area' is a 2 dimensional property that should only apply to things of dimension 2 or higher (I guess you are actually saying strictly greater than 2)? That's what you are saying? I'd first ask what's the area of a cube. .

Ahh, nevermind, ignore this bit. I see what you're saying. The rest still applies.

 

[Hurkyl]

The only thing strange here is that some people wish to assert that objects of dimensionality n have properties that apply only to dimensions higher than n.

What's strange is that some people think we're talking about a property that applies only to "dimensions higher than n". :tongue:

There really is no room for debate here -- if you have a definition of "length", then we simply appeal to the definition to see whether or not a point has length. And if you don't have a definition of length, then (mathematically speaking) you cannot say whether or not it applies to a point.

When you check the definitions of "length" usually used in mathematics, you find they apply to points. There is no way around this fact.

We can debate metamathematically about the merits of different ways we could define the word "length", and whether or not is desirable to choose a definition that allows us to measure the length of a point, but none of this has any bearing on the fact that "length", as used in mathematics, applies to points.

(e.g. sometimes you might want to measure the length of something when you don't know its dimensionality. And if you find that its length is finite and nonzero, that's a proof that it is one-dimensional)

 

[CRGreathouse]

(e.g. sometimes you might want to measure the length of something when you don't know its dimensionality. And if you find that its length is finite and nonzero, that's a proof that it is one-dimensional)

Quite. An example would be a recursvely defined object that may be a fractal. If it is in fact a fractal it is in some sense 'between' dimensions, having (for example) finite area but an infinite perimeter. The perimeter is more than 1 dimensional.

 

[MeJennifer]

There really is no room for debate here --

That is fine, you folks keep thinking about the length of points and the volume of triangles as if there is nothing wrong with that and leave me in error thinking that a 1 dimensional geometric object does not have a length and that a 2 dimensional geometric object has no volume.
Silly me, I must be far gone.

An example would be a recursvely defined object that may be a fractal. If it is in fact a fractal it is in some sense 'between' dimensions, having (for example) finite area but an infinite perimeter. The perimeter is more than 1 dimensional.

Utter nonsense, a fractal has no area since it is a set of disconnected points.

 

[CRGreathouse]

Utter nonsense, a fractal has no area since it is a set of disconnected points.

...!

Look up Hausdorff dimension and tell me if you still don't believe in infinite length and finite area, or fractonal dimensions for that matter.

 

[matt grime]

Think of a point as emebedded in the real line (or higher dimensions), and its n-dimensional measure (which is it's n-volume) is zero. Same with lines in 3-d and higher. Sorry, MeJennifer, but you are mistaken.

You are also mistaken to say that fractrals have bound no area, and that they are a set of disconnected points.

 

[MeJennifer]

You are also mistaken to say that fractrals have bound no area, and that they are a set of disconnected points.

Where did I mention "bound no area"?
Or do you imply that the area of something and the bound area are identical things?
If they are different things, then perhaps you could explain why you attempt to suggest that I am mistaken about something I did not write?

 

[matt grime]

Sorry, I changed my choice of words half way through. Yes, now I see what you're saying, and I would hate to put words in your mouth.

However, fractals are not, necessarily, disconnected sets. The cantor set is totally disconnected (and is the unique blah blay with this property). However, given a fractal F it is not possible, in general to write to find two open non-intersecting sets A and B with AnF and BnF non-empty. (Of course, the real line is a fractal*, and that certainly is not disconnected.)

* for me fractal means something with self similarity. Perhaps some definitions would expressly exclude the real line as a possible fractal, but it is a matter of convention.

 

[Hurkyl]

That is fine, you folks keep thinking about the length of points and the volume of triangles as if there is nothing wrong with that and leave me in error

It's not my fault that you don't want to listen.

 

[shmoe]

That is fine, you folks keep thinking about the length of points and the volume of triangles as if there is nothing wrong with that and leave me in error thinking that a 1 dimensional geometric object does not have a length and that a 2 dimensional geometric object has no volume.
Silly me, I must be far gone.

Very far gone indeed. Don't worry though, I'm sure burying your head in the sand will make it all go away.

Utter nonsense, a fractal has no area since it is a set of disconnected points.

In case the real line wasn't a satisfying example for a fractal that isn't a bunch of disconnected points,

http://mathworld.wolfram.com/KochSnowflake.html

 

[MeJennifer]

Yes, some fractals are not a disconnected set of points like the Koch snowflake.
The Koch snowflake is a 2 dimensional object, independent of how many iterations you go through.
A Hausdorf dimension is just a different way to define a dimension.

 

[shmoe]

Yes, some fractals are not a disconnected set of points like the Koch snowflake.
The Koch snowflake is a 2 dimensional object, independent of how many iterations you go through.

Since you call it 2 dimensional does it have an area? If not, why not? How are you deciding which 2 dimensional things do or don't have an area?

 

[Hurkyl]

I'm still curious why she calls the Koch snowflake a two-dimensional object. The only reason I can possibly imagine is because we usually draw it in R²... but we can draw points in R² too, and I can't imagine her calling them two-dimensional.

Maybe it's this... see the image here:

http://en.wikipedia.org/wiki/Koch_snowflake

The Koch snowflake is the black part -- it is not the purple part.


And, incidentally, the Koch snowflake does not consist of "iterations" -- the drawings you see on that page are a sequence of approximations to the Koch snowflake. None of them are the snowflake itself.

A Hausdorf dimension is just a different way to define a dimension.

Yes. What definition of dimension would you prefer to use? We working with things that are generally not manifolds, so we can't use that...

 

[MeJennifer]

Since you call it 2 dimensional does it have an area? If not, why not? How are you deciding which 2 dimensional things do or don't have an area?

The Koch snowflake certainly has an area. At each stage of the iteration one can determine the area.

All two dimensional objects have an area.

Yes. What definition of dimension would you prefer to use? We working with things that are generally not manifolds, so we can't use that...

The matter we were discussing was not fractals it was if for instance the volume of a circle exists.

And, incidentally, the Koch snowflake does not consist of "iterations" -- the drawings you see on that page are a sequence of approximations to the Koch snowflake. None of them are the snowflake itself.

Feel free do demonstrate that you can define the Koch snowflake non iteratively.

Out of curiosity do the people here think that there is a difference between something that does not exist and 0, or is that the same for you?

 

[shmoe]

I'm still curious why she calls the Koch snowflake a two-dimensional object. The only reason I can possibly imagine is because we usually draw it in R²... but we can draw points in R² too, and I can't imagine her calling them two-dimensional.

I was curious too, but was afraid to ask.

 

[shmoe]

The Koch snowflake certainly has an area. At each stage of the iteration one can determine the area.

All two dimensional objects have an area.

Okie, what is a two dimensional object then? Something that can be drawn in R^2? Why not a point then? Or a straight line?

Maybe you want 2-dimensional to mean something that can be drawn in R^2 but not R^1? If so, would two straight lines that meet at a kink have an area? Does a circle have an area (the boundary of a disc, not the interior)?

What is a 2 dimensional object to you? Try not to avoid the question, tell me how you can determine if an object is two dimensional or not. I want clear and concise rules so there is no confusion.

Feel free do demonstrate that you can define the Koch snowflake non iteratively.

He's just saying the Koch snowflake is not equal to any of the iterations.

Out of curiosity do the people here think that there is a difference between something that does not exist and 0, or is that the same for you?

Yes, there's a difference. length, area, volume, etc are all functions from some collection of sets to the non-negative real line. Saying this function is undefined on a set is very different than saying this function has the value 0 on a set.

 

[MeJennifer]

Does a circle have an area (the boundary of a disc, not the interior)?

A circle definately has an area. The boundary of a two dimensional object is a one dimensional object, and as you know a one dimensional object has no area in my view.
However, if I am not mistaken you think that the boundary of a disk does have an area am I correct? The area of the boundary of a disk in your view is 0, correct?

Yes, there's a difference. length, area, volume, etc are all functions from some collection of sets to the non-negative real line.

A function from a set?
The non-negative real line?
What are you talking about?

Length, area and volume are properties of certain geometric objects.

A two dimensional object is an object composed of one or more lines that form a closed curve. For instance a triangle and a circle are two dimensional objects.

 

[shmoe]

A circle definately has an area. The boundary of a two dimensional object is a one dimensional object, and as you know a one dimensional object has no area in my view.
However, if I am not mistaken you think that the boundary of a disk does have an area am I correct? The area of the boundary of a disk in your view is 0, correct?

a circle of radius 1 is the set of points in R^2 {x^2+y^2=1}, it is not the interior bits. yes, it has an area, this area is 0.

Circle:

http://mathworld.wolfram.com/Circle.html

Disc:

http://mathworld.wolfram.com/Disk.html

A function from a set?
The non-negative real line?
What are you talking about?

Length is a function defined on sets. matt's definition of 'length' he gave earlier takes an interval on the real line and maps it to a non-negative real number.

Likewise for area and volume.

Length, area and volume are properties of geometric objects.

No real difference I can see, at least not when you are trying to assign some numerical value to them.

 

[shmoe]

A two dimensional object is an object composed of one or more lines that form a closed curve. For instance a triangle and a circle are two dimensional objects.

So what about the surface of a sphere in R^3? So there's no confusion let's look at the set of points in R^3 satisfying x^2+y^2+z^2=1. Is this not 2-dimensional?

 

[Hurkyl]

Out of curiosity do the people here think that there is a difference between something that does not exist and 0, or is that the same for you?

Of course -- I was the one who brought it up, I thought. In the technical usage, saying something has "zero volume" means the volume exists, and it is zero. Saying something has "no volume" means that the volume does not exist.

Why would mathematicians have made such a definition? Well, if the volume doesn't exist, how can it be zero? :tongue:


P.S. do you realize that a point is a closed curve? And that there are curves that pass through every point on the inside of a cube?

 

[MeJennifer]

P.S. do you realize that a point is a closed curve?

I suppose I fail to realize that.
What is the shape, is it a circular curve?
So the tangent of any points on this curve exists as well I suppose. How about the radius, the radius of a point exists as well?
So perhaps I also fail to realize that a point is not just a close curve but also a closed surface or hypersurface right?
Perhaps I should extend my views and stop calling a point a zero dimensional object, it has really an infinite number of dimensions right, all of them zero but they do exist right?

And that there are curves that pass through every point on the inside of a cube?

Yes, there is no limit to the amount of things that can pass a point, but I suppose I fail (again) to realize how that is in any way relevant.

 

[matt grime]

MeJennifer, you seem to be using lots of mathematical terms, but just not in a rigorous sense, whilst attempting to draw rigorous conclusions (such as telling us we don't know what we're talking about). For example you're using dimension in the sense of measurement and no one else here is. A point is zero dimensional. That use of the word dimension is stictly different from referring to the dimensions of a box as 1m by 2m by 2m, say.

It would also be best to fix notation. When talking about a polygon or circle, we are referring to the boundary only. By abuse of language, referring to the area of a circle commonly means the are bound by te circle, but we should really refer to it as the area of the disc.

Also, you're definition of a 2-d object in your language ought to be: it is a shape whose bounday is composed of lines, not a shape composed of lines. A line is composed of lines, but I doubt you think that is 2-d. You even say that your 2-d object is a curve, and that cannot be true: a cruve is something that is generically (i.e. except for trivial degenerate cases), locally, 1-d. (Of course a 1-d complex curve is locally 2-d as a real manifold, just to annoy you some more.)

 

[CRGreathouse]

I'm still curious why she calls the Koch snowflake a two-dimensional object. The only reason I can possibly imagine is because we usually draw it in R²... but we can draw points in R² too, and I can't imagine her calling them two-dimensional.

Surely she's talking about the interior of the Koch snowflake, which has a well-defined area (3/4, I think, if the "triangle's" legs are of unit length).

The fact that the length of the Koch snowflake itself is infinite might be problematic for MeJennifer, I don't know.

 

[MeJennifer]

The fact that the length of the Koch snowflake itself is infinite might be problematic for MeJennifer, I don't know.

Not at all, given that this "object" can only be defined by applying an infinite number of operations.

 

[MeJennifer]

MeJennifer, you seem to be using lots of mathematical terms, but just not in a rigorous sense, whilst attempting to draw rigorous conclusions (such as telling us we don't know what we're talking about). For example you're using dimension in the sense of measurement and no one else here is. A point is zero dimensional. That use of the word dimension is stictly different from referring to the dimensions of a box as 1m by 2m by 2m, say.

It would also be best to fix notation. When talking about a polygon or circle, we are referring to the boundary only. By abuse of language, referring to the area of a circle commonly means the are bound by te circle, but we should really refer to it as the area of the disc.

Also, you're definition of a 2-d object in your language ought to be: it is a shape whose bounday is composed of lines, not a shape composed of lines. A line is composed of lines, but I doubt you think that is 2-d. You even say that your 2-d object is a curve, and that cannot be true: a cruve is something that is generically (i.e. except for trivial degenerate cases), locally, 1-d. (Of course a 1-d complex curve is locally 2-d as a real manifold, just to annoy you some more.)

You are correct, I should be more careful with definitions in the future.

But, to me nothing changes, a disk for instance, which is a 2 dimensional surface, does not have a volume. The property volume does not apply to surfaces. It is not even 0!

 

[shmoe]

You are correct, I should be more careful with definitions in the future.

You say this in one breath, then in the next you're back to:

But, to me nothing changes, a disk for instance, which is a 2 dimensional surface, does not have a volume. The property volume does not apply to surfaces. It is not even 0!

Where you are using your own personal definition for volume that is different from everyone else's. This is assumining you even have a definition for volume, there's no real evidence that you do.

 

[matt grime]

You are correct, I should be more careful with definitions in the future.

But, to me nothing changes, a disk for instance, which is a 2 dimensional surface, does not have a volume. The property volume does not apply to surfaces. It is not even 0!

listen to your own advice. Any disk is naturally embeddable in 3-space (or n-space for any n>3, where its measure is zero. Measure in 3-space is what you would term volume.

 

[MeJennifer]

listen to your own advice. Any disk is naturally embeddable in 3-space (or n-space for any n>3, where its measure is zero. Measure in 3-space is what you would term volume.

I do not disagree that a disk occupies a volume of zero in 3-space. But that is not what we are talking about!

Who is talking about embedding a disk in 3-space? Or measuring it in 3-space?

I am talking about the properties of a disk, a disk which is a 2 dimensional geometric object, not neccesarily some plot of some function in a multi-dimensional cartesian coordinate system or so.

 

[matt grime]

You said it is not possible to apply the word volume to it, now you admit it has a volume of zero when considered as an object in 3 space? That is contradictory, to say the least.

A (closed) disc (or disk if you are not English, I seem to interchange between the two without noticing these days) is a 2-manifold (with boundary). That is how you should say it. Not that 'the concept of volume does not apply to it'.

 

[MeJennifer]

You said it is not possible to apply the word volume to it, now you admit it has a volume of zero when considered as an object in 3 space? That is contradictory, to say the least.

Not at all, by analogy consider:

$$\sqrt -1$$

In R the result does not exist but it does exist as an imaginary or complex number.
Nothing contradictory here!

Same thing with volume, in R² volume does not exist but it does exist in R³ or higher.

 

[matt grime]

Erm? No, your analogy would be to assert that there was no such thing as the square root of -1 because there was no such element in R.