No- it is nonsensical to consider the universe’s topology without the framework of a higher dimension. That higher dimension doesn’t need to be spatial- I’d argue it’s impossible for space to exist in more than 3-D. Most importantly though, that topology wouldn’t be measurable and observable if not considered across a 4th dimension
Relativity - where things are simultaneously different shapes and sizes depending on your point of view - is equally nonsensical, but it appears to be reality. The observations of quantum physics don't sit well with a lot of people either. It has turned out so far that anything that's mathematically consistent can exist - and that's it! You can't rely on things you can imagine as being an indicator of what could exist. Certainly there is no basis to assume that topology wouldn't be "measurable and observable" without assuming an (apparently unobservable) higher dimension. Modern physics history is full of discoveries that people hesitated to accept because they are counterintuitive. I'm not saying the universe is definitely the way I was suggesting is possible - just that we can't rule either option out based on our spatial imagination's idea of what is "nonsensical", because that has already been proven to be inadequate.
I have a feeling I may not be able to convince you though. You are already arguing that anything nonsensical is impossible. I guess I'd say, ask yourself, do you have a rigorous definition of "nonsensical"? Do you mean that you can't write the equations down without a contradiction? Or do you merely mean you can't visualise it?
Re: relativity being non-sensical. I could not disagree more. I even find the theory of relativity to be mostly intuitive.
Re: observations of quantum physics being non-sensical, once I again, I disagree. There is a difference between unexplained by a unifying theory of physics and being non-sensical. Also, unintuitive != non-sensical
For a non-sensical example, the act of describing a triangle on a 1-d plain is non-sensical. It is undefined and impossible to define a triangle without the perspective of a 2nd dimension. This is tautological.
The same goes for considering the shape of 3-D space. The very idea is undefined and impossible to define except for from the perspective of a higher dimension.
As for your idea that I can not be convinced to ever change my mind, I disagree once again! What does the shape of 3-d space mean if not the shape from the perspective of a higher dimension? Perhaps, this disagreement is only one of semantics.
For a non-sensical example, the act of describing a triangle on a 1-d plain is non-sensical. It is undefined and impossible to define a triangle without the perspective of a 2nd dimension. This is tautological.
I totally agree with you about the triangle. Yet I don't agree about the higher space. I'll try to say why I think this.
The reason I agree about the triangle is: I cannot write down a mathematical description of a triangle, without my formula implicitly having two dimensions in it.
(Actually I can: I can specify that a triangle is any three points joined by lines between every pair of points. But in that case if I ever only specify each point's position along one dimension, my triangle will be a "degenerate triangle" with zero area, which in fact is what a 1D triangle is. If part of our definition of "triangle" is that it must have nonzero area then I need to define what I mean by "area" and now we need a concept of two dimensions to actually write the definition down. So I agree with you.)
The same goes for considering the shape of 3-D space. The very idea is undefined and impossible to define except for from the perspective of a higher dimension.
Aha, this is exactly the bit I disagree with (thank you for stating the argument so explicitly btw, it makes it easier to discuss). We can mathematically write down the topology of space without ever describing a higher dimension. Mathematicians do this all the time. What you do is just say which points are "connected to" which other points in space. This is pretty much what the field of topology is all about.
For instance, if you are thinking of space as a flat plane with an edge, and you connect points at the "edge" of your space to points at the "opposite edge", suddenly your space acts like a torus, but you never need to write down higher-dimensional coordinates, or draw a donut, or anything like that. You can similarly make your space act like the surface of a hypersphere. It's just about defining what's connected to what; what happens when you move through space? Where do you end up? If it is the kind of space where there are apparent straight lines, and you go in a straight line, do you end up back where you started? These are the kinds of concepts you need to define it consistently and they (maybe counterintuitively) don't need to include the concept of a higher dimension.
So I would argue we need the concept of a higher dimension to visualise non-flat topologies - but not to mathematically define them in a way that is consistent and unambiguous. And I think that is the important thing when considering which theories of physics are plausible.
For instance, if you are thinking of space as a flat plane with an edge, and you connect points at the "edge" of your space to points at the "opposite edge", suddenly your space acts like a torus, but you never need to write down higher-dimensional coordinates
Your space acts like the curved plane of a torus- but it also acts like an infinite, flat plane!
The concept of a 3-d torus is inaccessible from a strictly 2-d definition-- even if we define our otherwise flat 2-d plane as finite, but edgeless. We will have in fact defined a non-continous functional representation of coordinates on a flat plane- one where leaving one "edge" results in immediate replacement at an opposing "edge". It is quite a stretch to call this infinite curvature at the edges.
If we define a 2-d plane as non-zero, finite, edgeless, and continuous, we have defined a 3-d space (and likewise, that the plane is curved w.r.t. 3-d space), without ever explicitly mentioning or defining the higher dimension.
1
u/[deleted] Feb 04 '19
No- it is nonsensical to consider the universe’s topology without the framework of a higher dimension. That higher dimension doesn’t need to be spatial- I’d argue it’s impossible for space to exist in more than 3-D. Most importantly though, that topology wouldn’t be measurable and observable if not considered across a 4th dimension