Originally Posted by **benit13** No. There's no distinction between mathematical dimensions and physical dimensions. Dimensions are just names for the orthogonal bases in coordinate systems.
They're not irrelevant. The object can still be described using the 3D Cartesian coordinate system you adopted.
There are physical phenomena that we can model using the laws of physics. Those laws of physics are framed using well known coordinate systems, like Cartesian, cylindrical and spherical polar coordinates, but there's nothing stopping you from defining your own coordinate systems and using those. For example, solid state physics makes use of reciprocal space, relativity can be framed using time as a dimension. Nuclear physicists sometimes use complex forms for things that are not complex, which can sort of be seen as a new 'dimension'.
Yes, most of nature can be described by using three spatial dimensions in your coordinate system, but that's not a fundamental property of the Universe. You could, for example, use a 1D or 2D system of coordinates if you want to describe a certain situation, which might be fine. However, there are other situations where one or two dimensions are not good enough. Similarly, some scientists find it easier to describe certain phenomena using many dimensions, but that doesn't mean that they are 'real'.
I'm not very good at relativity, but from what I understand, I think the answer is no. Einstein developed his theories by investigating the properties of objects as they approach the speed of light and never assumed or treated space as a material that can be warped and bent. |

With respect, since you have written some very good posts here,

This question was posted at University level, in the General Relativity section.

Surely we should not offer the simplifications used in school?

The difficulty (which can by illustrated using an x, y frame) with cartesian systems is this.

Consider a point, (x, y).

This needs two pieces of information to specify it doesn't it, x and y ?

But waitup, does it?

Suppose I measure distance along some arbitrary curve to reach this point.

Then I only need one piece of information, not two, viz the distance.

The curve or line I measure along is a one dimensional manifold that we have embedded in our two dimensional x y frame.

But we don't need to if the one dimensional manifold is all there is.

And that is the concept behind General Relativity that is so difficult to get across, especially as we have a three or four dimensional (Space or Spacetime) manifold in play there.

Mathematically we can 'embed' our manifold in as many higher dimensions as we like and calculate as though they exist.

Mathematically it gets worse, because a Peano curve can reach every point in the plane (or even a solid cube) in this way, at the expense of the calculus, since we loose the concept of a neighborhood in doing this.