What is the Meaning of Dimension

[radou]

I do not see the point of this thread, and I certainly do not see how it fits into General Math. This is only my humble opinion.

 

[matt grime]

Your's is not the only humble opinion that leans that way (mathwonk, me, any mathematician who's looked at it...)

 

[JOPearcy]

Physics Help and Math Help - Physics Forums > Mathematics > General Math

So general math as it applies to mathematics as it applies to physics forums under the category of physics help and math help has nothing to do with dimensions or the meaning of dimension?

Given we are talking about one dimensional length in real physical space, one of three accepted dimensions of length for real physical space.

You reject my above definition for how to define the meaning of dimension as it applies to this one dimension of length. So give your definition for this very specific example.

How do you define the meaning of dimension as it applies to this one dimension of length?

 

[matt grime]

You're missing the point - you're talking abuot Minkowski space time as 4 dimensional - that is it lives what mathematics has decreed dimension means - we're talking about sweeping out manifolds in space time etc. Then you're spouting hand wavy stuff about what for clarity of argument we will now call UNITS.

So, for the rest of this post, with whomever you're talking, let's agree that UNITS is the word you should use to avoid any clashes of convention.

I am side stepping your question. It isn't about maths, and I don't care to discuss the semantics of physics.

 

[JOPearcy]

Mind boggling.

Matt Grime,

No I am not talking about Minkowski spacetime which is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated.

I thought I made that very clear. The point is to seek a simple common ground, so that one can not cloud the issue with other complexities which are not pertinent to the real question.

Simplicity.

I’m talking about flat Euclidian Space, which is the simple basic view of the real physical space we live in. This simplifies the subject matter.

I have repeatedly stated the point of seeking a simple common ground that is well known. This simple flat 3D model of space and its definitions are well known.

“Three-dimensional space is the physical universe we live in. The three dimensions are commonly called length, width, and breadth; although any three mutually perpendicular directions can serve as the three dimensions.”

Going into the more complicated model of Einstein's theory of special relativity complicates the question which is a basic question and thus can be answered using the simple model instead of the complex model.

To further simplify the example, we can focus only on 1 of the 3 mutually perpendicular directions that can serve as the three dimension of this flat simple model of space which is well known and well understood. This leaves us with a single dimension to focus on.

Now, we should have a single simple well understood dimension to discuss.

You imply that you understand this subject better than I and that, I am wrong and you are right.

Okay, here is a single simple well understood dimension to discuss. Show me how you understand this better than I do.

How do you define the meaning of dimension as it applies to this one dimension of length?

 

[K.J.Healey]

How about : A parameter of the space in question.

And Minkowski really is the simple model. 3D+1 right? Pretty easy.

 

[rcgldr]

I prefer to keep the definition of dimension simple, and not to dilute it to include all qualites of an universe and objects in that universe.

So my concept is bascially the same as wikipedia's:

A dimension is a direction within a universe (real or abstract). If that universe is mapped by more than one dimension, then each dimension of that universe is oriented so that it is pependicular to / orthoganal to / independent of (pick your favorite here) of all the other dimensions that map that universe.

The term dimension should not be diluted to include other qualities of a universe and objects, like time, temperature, size, mass, velocity, or ...

 

[JOPearcy]

K.J.Healey,

How about : A parameter of the space in question

I would say yes, you are correct, relatively. I would accept that as a starting point.

So we are looking at a one dimensional length. What is the parameter? Is the parameter the dimension of length?

What does parameter mean in relation to this one dimensional length? Could parameter here mean a quantity that defines certain relatively constant characteristic of the system?

If that is the case, then does it really fit the meaning of dimension as it is used in the phrase “one dimensional length”?

Does not quiet fit to me. This definition of dimension seems more appropriate for what dimension means in the phrase “the dimension of the line segment is …”.

Considering that, it does make it more confusing if you are thinking of this meaning of dimension as opposed to the meaning of dimension as it relates to the phrase “one dimensional length”.

I’ve been so focused on the meaning of dimension and trying to explain it that I overlooked the idea that you could be thinking of the other meaning for dimension. The two definitions are very similar and yet not the same.

I should have thought about that confusion, my mistake there.

Now I understand better Matt Grimes comment about calling dimensions “units” instead of dimensions, if he is thinking about dimension as being a quantity measured, like the dimension of a line segment.



Apply your definition to a one dimensional length and see if that really works for defining the meaning of dimension as it is used here.
A one dimensional length.
A one parameter length.
A single quantity that defines certain relatively constant characteristic.

Can you explain better how this definition applies to dimension as used in the phrase “one dimensional length”?

Can you understand how my definition of dimension applies to dimension as used in the phrase “one dimensional length”? Have you tried to work it out?

My definition of dimension will not allow what Matt Grimes suggests.

In the definition of dimension that I am using, the definition I say applies to dimension as it is used in the phrase “one dimensional length”:

• Dimension does NOT equal a quantity.
• Dimension does NOT equal a unit.
• Dimension does NOT equal a unit of quantity.

Dimension as it is used in the phrase “one dimensional length” means:
A dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.

So what does this mean with relationship to the phrase “one dimensional length”?

Well, you can arbitrarily begin with any quantity of length of this “one dimensional length”. Let’s say we have X quantity the base quantities.

To measure that quantity you define some base quantity of length of this “one dimensional length”. Let’s say we define a base quantity called a “hand of length”, like when measuring how tall a horse is.

The system of quantities is simple, because we have only one dimension, so the system is the system of quantities of the “one dimensional length”.

The representation by the product of the powers of the factors corresponding to the base quantities is also simply, simply being “hand of length”^1.

• The X quantity of “hands of length” exists in the dimension of the “one dimensional length”
• The base quantity of “hand of length” exists in the dimension of the “one dimensional length”

• You can not add dimension as it is meant in “one dimensional length”.
• You can not measure dimension as it is meant in “one dimensional length”.

• You can add quantities of length that exist within the dimension of the “one dimensional length”.

It all holds together defining what dimension means when using the phrase “one dimensional length”.

Have you tried to work out the understanding of the definition I gave?

 

[rcgldr]

What I've been trying to explain is that a dimension is a direction without any implied magnitude. It's just a direction. If you want to determine distances between objects, then magnitudes are required, but if you just want to know the number of dimensions occupied by an object (abstract or real), then the magnitudes (as long as they aren't zero) don't matter. In a N dimensional universe, if an object has N (or more) dimensions, then that object exists and occupies space within that universe, and the magnitudes (other than zero) don't matter.

So a "single dimension" is just a direction, or a line with unspecified magnitude, regardless of the total number of dimensions in an universe.

 

[JOPearcy]

Jeff Reid,

The dimension of mass is a dimension. It is not a direction. We are agreed that dimension in this reference has no magnitude. A quantity of mass can be measured relative to a defined unit quantity of mass. Both the quantity of mass being measured and the defined unit quantity of mass to relate scale too, exist with the dimension of mass.

If you dig around you will find authoritative texts directly referring to the dimension of mass and corresponding defined base units of mass.

The dimension of time is a dimension. It is not a direction. A quantity of time can be measured relative to a defined unit quantity of time. Both the quantity of time being measured and the defined unit quantity of time to relate scale too, exist with the dimension of time.

If you dig around you will find authoritative texts directly referring to the dimension of time and corresponding defined base units of time.

The dimension of luminous intensity is a dimension. It is not a direction. We are agreed that dimension in this reference has no magnitude. A quantity of luminous intensity can be measured relative to a defined unit quantity of luminous intensity. Both the quantity of luminous intensity being measured and the defined unit quantity of luminous intensity to relate scale too, exist with the dimension of luminous intensity.

If you dig around you will find authoritative texts directly referring to the dimension of luminous intensity and corresponding defined base units of luminous intensity.

And so on…

 

[Integral]

How has this thread continued for this long?

LOCKED!