"Dimensions" = just a mathematical game?

[mox]

Hi, I often wonder about "dimensions"? You could say we live in a "3D" world - but in reality, isn't that just a "volume"? Does it even make any "realistic" sense to talk about anything less than "3D", or anything more? Theoretical physicists & mathematicians talk about "zero-dimensional points", but something without any "size" cannot even be said to be a "thing", surely? Likewise for a "one-dimensional" line, or a "two-dimensional" plane - there is no "breadth", so it can't really exist outside of a mathematician's imagination? Likewise for higher dimensions, say 5, or 11, etc - isn't this nonsensical? Is discussion of "dimensions" simply playful mathematics? In reality, isn't there simply just our familiar 3-dimensional "volume"? For something to exist, it needs a volume to exist in.

(Related)... Did Einstein actually conflate "space" with "time"? Did he actually claim that time was just another spatial dimension? Or did he simply say space and time were connected at a fundamental level, albeit different entities? Wouldn't it be better to say there is a "volume" moving through "time"?

[Phi for All]

10 minutes ago, mox said:

Hi, I often wonder about "dimensions"? You could say we live in a "3D" world - but in reality, isn't that just a "volume"? Does it even make any "realistic" sense to talk about anything less than "3D", or anything more? Theoretical physicists & mathematicians talk about "zero-dimensional points", but something without any "size" cannot even be said to be a "thing", surely?

A black hole is most definitely a thing.

 

10 minutes ago, mox said:

Did Einstein actually conflate "space" with "time"? Did he actually claim that time was just another spatial dimension? Or did he simply say space and time were connected at a fundamental level, albeit different entities? Wouldn't it be better to say there is a "volume" moving through "time"?

Spacetime is a continuum. Three spatial dimensions and a temporal one, to designate any place/time in the universe. I can give you 2 coordinates for longitude and latitude for a place on Earth, and together with an altitude and a time, we can meet for lunch on the 40th floor of the Empire State Building.

Looking at dimensions this way helps show how mass/energy curves spacetime in a way we feel as gravity.

[Strange]

10 minutes ago, mox said:

Hi, I often wonder about "dimensions"? You could say we live in a "3D" world - but in reality, isn't that just a "volume"?

Yes, 3D space is a volume. We live in 3D space

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Does it even make any "realistic" sense to talk about anything less than "3D", or anything more? Theoretical physicists & mathematicians talk about "zero-dimensional points", but something without any "size" cannot even be said to be a "thing", surely? Likewise for a "one-dimensional" line, or a "two-dimensional" plane - there is no "breadth", so it can't really exist outside of a mathematician's imagination?

Things like points and lines are mathematical concepts. They can be useful in physics (and other areas). For example, the route between London and New York is 1D line. That is not a "thing"; but it is mathematically useful.

Similarly, electrons (which may or may not be "things" depending how you define "thing") are modelled as 0D points. That is useful because it corresponds to the way they behave.

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Likewise for higher dimensions, say 5, or 11, etc - isn't this nonsensical? Is discussion of "dimensions" simply playful mathematics? In reality, isn't there simply just our familiar 3-dimensional "volume"? For something to exist, it needs a volume to exist in.

Many areas of science use multi-dimensional abstract spaces. (Even "soft" sciences like social science.) Because they are useful ways of describing the world.

Science, generally, isn't concerned with "reality" but about what we can measure and describe.

If you want to discuss "reality" then you probably want philosophy (or maybe religion) rather than science.

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(Related)... Did Einstein actually conflate "space" with "time"? Did he actually claim that time was just another spatial dimension? Or did he simply say space and time were connected at a fundamental level, albeit different entities?

He did actually say that we can describe the universe as a 4D construct that combines space and time (called, not suprisingly, spacetime). All four dimensions have equal status of being "real" whatever that means. And all four are affected by the presence of mass (resulting in effects like gravity or time dilation).

 

[Mordred]

A simple way to look at dimensions under physics is any independent mathematical object such as a variable or group etc. With spacetime any coordinate can vary in value without changing any other coordinate value.

If the group etc has an infinite set then you can compactify that dimension as any infinite set contains finite a finite set.

Edited January 24, 2020 by Mordred

[mox]

Thank you, this may well be where my lack of formal mathematical training is clouding my reasoning eh!

I appreciate the responses, nice forum here! 🙂

[studiot]

1 hour ago, mox said:

Wouldn't it be better to say there is a "volume" moving through "time"?

It came as a great and hard suprise to the world when Benoit Mandelbrot showed the world that nature is even more peculiar than that.

You should get hold of his book; it is quite accessible to amateurs.

The Fractal Geometry of Nature where he discusses the easily observable fact that dimensions in Nature are not whole numbers.

Mandelbrot coined the term Fractal to describe this.

https://en.wikipedia.org/wiki/The_Fractal_Geometry_of_Nature

 

Essentially he explores the question

What do we mean by length, area and volume ?

What happens when we look more and more closely at greater and greater magnifications ?

His famous question was "What is the true length of a coastline?"

[J.C.MacSwell]

27 minutes ago, studiot said:

It came as a great and hard suprise to the world when Benoit Mandelbrot showed the world that nature is even more peculiar than that.

You should get hold of his book; it is quite accessible to amateurs.

The Fractal Geometry of Nature where he discusses the easily observable fact that dimensions in Nature are not whole numbers.

Mandelbrot coined the term Fractal to describe this.

https://en.wikipedia.org/wiki/The_Fractal_Geometry_of_Nature

 

Essentially he explores the question

What do we mean by length, area and volume ?

What happens when we look more and more closely at greater and greater magnifications ?

His famous question was "What is the true length of a coastline?"

"How long is the coast of Britain?" is the line I associate with him (no doubt he has said both)

[studiot]

3 minutes ago, J.C.MacSwell said:

"How long is the coast of Britain?" is the line I associate with him (no doubt he has said both)

You could be right about the exact wording.

If so, thank you for the correction.

[swansont]

2 hours ago, mox said:

Hi, I often wonder about "dimensions"? You could say we live in a "3D" world - but in reality, isn't that just a "volume"?

No, it means you need three coordinates to locate a point, using some coordinate system. Corner of 5th and Elm, ninth floor. Not inherently a volume.

[Strange]

41 minutes ago, swansont said:

No, it means you need three coordinates to locate a point, using some coordinate system. Corner of 5th and Elm, ninth floor. Not inherently a volume.

Succinctly put. And, as Phi for All said, you need the 4th coordinate (“when”) if you want to successfully meet up with someone there. 

You can’t get away with omitting any of these, and you can’t replace one with some combination of the others, so they meet the “independent” requirement in Mordred’s definition. 

[MigL]

Definitely not a mathematical game, but certainly a mathematical tool.

You could specify a particular volume, but that wouldn't be unique.
That is why we specify length, width, and height if we need better precision, as two equal volumes may be shaped differently.
And to uniquely identify a particular volume, we specify the (x,y,z) co-ordinates ( in cartesian, or radius and angles in spherical polar ) that designate its volume  about the arbitrary origin.
We similarly use 2dimensional co-ordinates for a specific area, and an 1dimensional co-ordinate for a distance from the origin

[studiot]

On 1/24/2020 at 10:32 PM, studiot said:

Essentially he explores the question

What do we mean by length, area and volume ?

What happens when we look more and more closely at greater and greater magnifications ?

His famous question was "What is the true length of a coastline?"

 

There is of course mathematically more than this.

One alternative view of geometry in general developed during the early and mid 20th century has come to the fore in more recent years.

The idea of instead of patterns of points in space forming the basics of geometry, the idea of simplexes as unit building blocks for these patterns.

So complete geometries have been buit up using a line segment, a triangle, a unit cuboid and so on.
Whilst there were earlier workers, the current heading is Finslerian Geometry and Johnson Polyhedra.