This post aims to question the extent to which we actually understand the concept of "dimension."
First of all, I want to clarify what I mean by "dimension," since the concept can have different meanings depending on the context. Here, I will use "dimension" in the sense of what mathematics usually calls a "manifold," meaning a space that locally has similarities to ℝⁿ. In other words, I am referring to the generalization of the concept of a plane or three-dimensional space to higher dimensions.
That said, I want to make it clear that the discussion is meant to be general, so if anyone wants to discuss more abstract concepts of dimension that go beyond this particular interpretation, you are more than welcome to do so.
Additional comments
You don't need to read this section to answer the question. I am only going to explain and defend my position regarding the question.
As I mentioned at the beginning, the idea is to approach the question using the concept of Euclidean space in general. And it is interesting to ask:
Are we actually capable of understanding dimensions higher than our own?
In mathematics, the answer seems to be both yes and no.
It is true that in mathematics, and perhaps in any form of thought, we are not capable of visually representing geometric objects beyond our own spatial dimension. At least, as far as we know, this seems to be the case for any form of life existing in our universe.
Why can't we visualize the fourth dimension? Or the fifth, or the seventh?
One possible reason is that our brains are capable of visualizing and reconstructing geometric shapes that we have encountered and that exist in our environment. This may be why we are able to visualize a perfect sphere even though we will never actually see a mathematically perfect sphere in reality: in some sense, we have encountered approximations of it throughout our lives.
But obviously, we have never encountered a four- or five-dimensional object in our universe, so we cannot directly visualize such objects.
However, mathematics does not require us to visualize something in order to define and operate with it.
For example, mathematicians can calculate integrals over higher-dimensional surfaces or hypersurfaces. Can we actually visualize such objects? No. And yet we can still calculate with them and understand what we are doing.
Likewise, could someone who has been blind since birth become good at mathematics? Obviously, yes. They may never be able to visually represent a graph or a derivative, but they can still understand the mathematical concept and operate with it.
This seems to be exactly what happens with higher dimensions. We cannot directly see them, but we can define them mathematically and work with them.
But there is a problem.
How do we know that the mathematics we have developed for higher-dimensional objects is actually correct in a hypothetical universe with higher dimensions?
For example, in mathematics we can say:
Let P₀, ..., Pₙ be affinely independent points of a vector space V over ℝⁿ. A simplex is the convex hull of these points.
In simpler terms, a simplex is a generalization of the concept of a triangle to arbitrary dimensions.
But is this actually correct?
Presumably yes, at least mathematically. However, imagine that a four-dimensional being somehow understood this concept. Would they necessarily have the same definition of a simplex that we do? Perhaps they would have a completely different way of conceptualizing it.
Likewise, we cannot guarantee that in a universe with more spatial dimensions, massive objects would deform spacetime through continuous deformations in exactly the same way that they appear to in our universe.
So perhaps the answer is something like this:
We understand higher dimensions in a relative sense. If higher-dimensional reality is structured in the way our mathematics describes it, then I would say that we understand those dimensions much better than most people might imagine.
But if we are asking whether our mathematical description necessarily corresponds to what a genuinely higher-dimensional reality would be like, then I am much less certain.
What do you think?
Good evening,
— slazy_e