John,
Thank you for the comments!
> "I would like to offer a few counter arguments in defense of space."
I don't deny space, but I am interested in your arguments anyway.
> "is physics possibly [overlooking] space?"
I am not aware of physics overlooking space. I am aware of theories which claim that space is emergent, but for the moment I don't have any reason to endorse this position (and no definite reasons to refute it).
> "Supposedly space arises from geometry, but could it be in fact that geometry is mapping properties of space?"
Here I take the position of mathematicians, that geometry is space, or at least that it is about space.
> "For instance, the dimensionless point is the essential geometric concept, yet it is explicitly a multiple of zero, being dimensionless and consequently, mathematically doesn't exist."
I think that statements like "dimensionless point [...] is explicitly a multiple of zero" and that from this follows that it doesn't exist mathematically are uncommon among mathematicians. Mathematicians tend to believe that if something is a multiple of zero, then it is zero, and that zero (and its multiples) exist mathematically. Maybe you want to say that the measure of a point is zero, but even this is not true for all measures. And even if it is true, it doesn't follow from this that the point doesn't exist mathematically. This would mean that no region of space can exist, being collections of points. Or it would mean at most that all subsets of space are empty sets.
> "Would a dimensionless apple have any existence?"
You are mixing apples and oranges. First you talked about mathematical existence, then you illustrate that it doesn't make sense by giving as example dimensionless apples. In mathematics you can only talk about things you defined, or at least you defined them implicitly by axioms. Explicit definitions are of the form "genus-differentia", or by construction. Implicit definitions are based only on properties, on relations between things. For example, Modern formulations Euclidean geometry define points and lines by their relations (although Euclidus said things like "A point is that which has no part"). The question is, how do you define a "dimensionless apple"? It seems that the genus is "apple", and the differentia is "dimensionless". But the "differentia" reffers to how you distinguish an object from a particular genus, from the others from the same genus. The "differentia" has to be a property that some of the objects of the genus have. If none of them has that "differentia", the set of objects satisfying your definition is empty. So it doesn't make sense. It is as if you define "strictly positive negative numbers", there is no such thing. Then you say you proved a contradiction, but your proof is based on nonexisting (logically inconsistent) concepts.
> "Consider that three dimensions are really just the xyz coordinate system, which is located by the 0,0,0 center point. As a subjective location, couldn't more than one center point exist in the same space? Much as billions of people, the originators of this system, are all the center points of their own space, but exist in the same space as all others. "
Right, I agree. "Think how much political conflict is about applying different coordinate systems to the same space" this is true :)))
> "How about longitude, latitude and altitude? Are they really a perfect foundational framework for the surface of this planet, or just a very useful mapping device?"
They are a mapping device. Here's how mathematicians think about differentiable manifolds. They define them using charts. Charts are mappings of pieces of the manifold on some n-dimensional space R^n. These mappings are one-to-one between a region of the manifold and an open subset of R^n. The first rule is that the manifold is covered by these charts. The second rule is that on a region covered by two charts, the composition of one of them with the inverse of the other (which is therefore a mapping of an open region in R^n to another open region of R^n) are differentiable. We call such a collection of charts an atlas of our manifold. Then, we add all such charts which satisfy the rules to the atlas, and the result is called a complete atlas. Now, think at a region of the manifold, and all the charts on that region. Mathematicians say these charts are just mappings, we used them to define the manifold, but we don't take them as the region of the manifold. They consider as "real" what is invariant, so charts are just perspectives, not reality. What is real is the result. The composed charts which give a mapping between open sets of R^n are called transition functions, and they allow us to change the perspective. And they define isomorphisms between those open regions of R^n. So what is real to mathematician is the equivalence class defined by these isomorphisms. Mathematicians use charts to define the manifold, then they throw away the charts as being just some props to obtain the definition. They continue to use the charts for example to make calculations, they translate the properties of the manifold into statements about R^n, hence about numbers.
So we see that the mathematicians agree that the charts are just mapping devices, but they disagree that the resulting manifold is a mapping device. An applied mathematician may use such a manifold to model some phenomena from another are of knowledge, and in this case the manifold is a tool too. But to geometers who don't apply geometry to something else, the manifold is not a tool, it is the ultimate object of their interest. So while even the hardcore mathematical Platonists will agree with you that charts are mapping devices, they would say that the manifold itself is real. They wouldn't say that they can prove it to you, but also you can't disprove them. So they are free to consider it real, and you are free to not consider it real.
> "Often it is assumed math exists as some platonic realm of order, but if it isn't and all order arose with the physical reality it describes and defines, would it be any different? If not, then wouldn't it be against Occam's razor to assume it does pre-exist the manifestation? Does the void need mathematical order, lacking any structure?"
Ockham's razor has two edges. The side you pick is that there is a physical reality, and there is a mathematical description of it. And if somebody believes both to be real, he has to explain a lot of things. Like why the behave the same, and then if they do the same things, why maintaining that the mathematical description is real and not a mere mapping device. (the same problem face Cartesian dualists with the mind-body problem) So you say let's use Ockham's razor to cut and separate the physical reality from the mathematical description, then discard the mathematical description as being something more than a mapping device. Someone who considers that the mathematical description is in fact the reality will agree with you to cut them exactly in the same place with Ockham's razor. The only difference is that she will discard the "physical reality". Your choice is motivated by your experience that physical reality is always there, while mathematics is in books. And you don't have to explain why the mathematics describing the physical world is isomorphic to the physical world, it is so by design, duh! But the "platonist" (this is not in fact what platonist means, but let's use it this way) would say that we have to objects doing the same thing, the physical reality and the mathematical structure. She knows very well what is a mathematical structure, but she doesn't know a definition of physical reality. So why not discarding physical reality and keeping the mathematical structure? Now the physical realist may say "but it is obvious that there is a physical world, here it is, I can experience it, I can taste an apple, qed". The mathematical realist may say "we only know about physical reality by our interactions, so what we know are not the physical objects as things-in-themselves, but our relations with them. So all there is experienced is relation. And a mathematical structure is nothing more than a collection of relations (see Universal algebra). We can't probe anything beyond the relations, hence the world is a mathematical structure. One may say though that we can keep as real the physical world and discard the mathematical structure as being a mere representation. But this is not a representation as the usual mathematical models. If you want to makea mathematical model of an apple, you will have to do a lot of complicate mathematics, and the result will be approximate. But a mathematical description of the physical laws makes our knowledge redundant. All phenomena are contained in the principles. Hence, the mathematical formulation of principles is not a mere model, because it is much more compact than the thing it models. This is why I choose to discard anything outside that mathematical structure." She may believe or not that the other mathematical structures exist physically too, but if we apply again Ockham's razor to keep the simplest option, which one you think is the simplest, the principle "all mathematical structures have physical existence", or "only the mathematical structure defined by [insert the mathematical formulation of the final theory of physics] has physical existence"?
> "So if we are looking for what's fundamental, wouldn't space be worth considering?"
Sure, and this is what I do (more precisely spacetime). If you look at my research papers you will see that they are about spacetime. Understanding singularities in general relativity, researching the possibility to make collapse consistent with the symmetries of spacetime, these were my main themes, and the other ones where I published less are also about spacetime. Yes, I know I said everything, including spacetime, is contained in the germ. So is this denying spacetime? I would say not. Holomorphic functions don't deny the domain on which they are defined. The domain is encoded in the germ at any of its points, but this doesn't mean that that domain doesn't exist. It only means that one of them is as real as the other.
> "Just a thought, but remember zero is between 1 and -1"
I can't. Zero is between -1 and 1.
Best regards,
Cristi