The answer is that the higher dimensions are more real than our experience isolated to the third dimension. But to understand why this is the case is difficult. In order to understand it you have to rethink your view of the world somewhat. In our worldview Logos/Physus is the most basic duality. But between them is the Nomos (Order, Math) which allows theory to connect to experimental results. All good scientific theory mimics math in operationalized words that reflect how phenomena works. Without all three you could not have science. But Nomos being nondual comes before (is a priori to) the split between Logos/Physus. How do we know this? because if the universe did not have anything in spacetime there would still be the math. And the math has some very odd characteristics and those actually drive our world and constrain it in specific ways that show up in physical phenomena. The simple answer is that Logos and Physus describe Being and the Nomos describes existence as it looks from the Physus/Logos split. Being is in fact a unique Indo-European linguistic kink and thus is not universal in any stretch of the imagination, except by fiat of killing off many of the other peoples and worlds, and languages that did not have Being. So that means that nomos is nondual and prior to the rest of nature that appears when there is something in spacetime but it has structure even if nothing ever articulates that structure. Part of that structure is N-dimensions. But N-dimensions though infinite are not all the same. In fact the fourth dimension which is the dimension that we really live in has some pretty unique properties compared with the other dimensions. But dimensions are different from each other. And we see this best in the size of hyperspheres which get bigger then tail off into the infinitely small. The peak of size of hypersperes is around the 5th to the 7th dimensions. So although they are infinite this does not mean that the structures that are interesting go on forever, but rather oddly the interesting structures all exist at the lower dimensions, near where we are. The oddities like the hypercomplex algebras drive a lot of structure in the low dimensions. Also the fact that the fourth dimension uniquely has no set topology is important. So since Math constrains the physics it must be real, in fact in some sense more real than the physics, because those constraints would be there even if the matter were missing.
There is more to this answer than what I have mentioned but this is a good place to start to answer the question of why mathematical dimensions are real, more real than the phenomena that fills spacetime that the math constrains. The math has to constrain it otherwise science could not rely on mathematics to guide its theorizing about phenomena.
How do we know that mathematics (Nomos) is prior to phusis (Nature, Physics)?
(responding to a comment from Chase Quinton)
What you have to grasp to understand this idea is basically the essence of Kantian Philosophy. What we need to see really how deep the Kantian picture is in order to accept this idea. Basically he worked out an answer to Hume’s Skepticism. But more than that what has come out of Kant’s approach is the very basis of our scientific endeavors since his time. In effect no one has figured out how to surpass the limits he set on reason, nor the way that he has tied reason to experience to produce understanding, or the way he has managed to rescue causality from the mire of mere sequences of events. In other words Kant’s view has become our predominant view in our worldview concerning the efficacy of Science. And that view demands Transcendental Idealism as the route to Transcendental Realism. Everything else has gaps that cannot be explained. Kant got rid of the gaps, by saying first we project spacetime which we understand through math, and then we project the categories by which we get objects within spacetime and causality that may be objective, and the only way to know what is real is through these projections that we make unconsciously. What it does is by basing everything on projections is to make sure there is no gaps between us and what is real. Those projections we make are the reality, but that does not mean we are constructing that reality we see, but it means that what comes through the medium of the projections IS real, as real as anything gets. All other views have unexplainable gaps between reality and our attempts to be objective as subjects, and cannot explain things like causality on which Science is based.
Now since Kant philosophers have tried every possible way out of this box that Kant put us in. But they all fail one way or another and if you look at both Continental and Analytical philosophers they all accept and harken back to Kant one way or another. Because without his Copernican Turn toward critical philosophy by weakening reason and tying it to experience to produce understanding there is no way to explain how science actually works. All the other opinions are just tweaks to this basic positions or reactions that merely entrench it further. In other words we know that nomos comes before physus because logically spacetime has to come before anything can be in it. And Nomos is how we understand spacetime. And we can understand it become it comes out of us a a projection prior to our experience. Then we know that objects that are physical come next because things in space logically have to come after the place that they are in, and we know them because we are projecting the categories after spacetime. And we know these things because of our judgments that are aligned to the projections, and the judgments are in language. We project what we can then understand and interpret as judgments. Essentially this means that objects have to adapt to us and our projections rather than us adapting to them. We can only know what we are prepared in advance to know, we cannot know anything that we are not prepared to know prior to the knowing by our projections. Everything I am telling you is explained by Bernstein in his lectures of Kants at http://bersteintapes.com. So if you want to know more then listen to those tapes. We are projecting synthetic a priories, because that is the only way we can know anything. Whatever we know is already part of us from the beginning. And because of that what we know via our prior projections is the only thing that is Real to us.
It is like a current author (of Is that a Fish in your Ear) on translation said at the LA Times Bookfair recently. If we sent people off into space and they met aliens and learned their language and spoke to them, and they came back we would ask them what the Aliens said. If they said we would like to tell you but their language is untranslatable, we would doubt their veracity. By its very nature if they learned the language of the aliens and spoke to them then somehow what the aliens said would be translatable into something. This I think is another version of Kant’s argument. The very nature of language is that it is translatable into any other language somehow, to some degree, with some fidelity, perhaps not accurate or correct but to be language at all it must be translatable. Kant is merely saying the same sort of things about nature. Nature has to be experienceable. To be experienceable there are certain rules that it must adhere to and if the phenomena does not adhere to those rules governing our experience, we cannot experience it, and those rules come from inside us, not from outside. We may not be able to experience everything, what is real to us is what we are capable of experiencing and understanding. We use reason to aid in understanding. Left to its own devices reason makes up fantasies about the world, but with respect to interpreting phenomena by the rules of judgment to produce understandable experience reason does a good job and giving us access to the realities we can handle. Elliot’s quip about this is that Human beings cannot take too much reality, but the reality we can take is as real as anything gets.
I don’t want you to get confused about whose idea this is. It is Kant’s idea. Please argue with him if you must. People have been arguing with him ever since he writes the First Critique, but no one has found a way around this argument. You cannot accept it, in which case Science as we know it breaks down for you, or you can accept it in which case Science sorta works, and just enough to discover amazing things, but not enough to tell you how real the things are that they have discovered. So you have to kind of decide if you want Science to appear to work at least half way OK, or not at all. For Kant there are no noumena. He only posited noumena so he could show that this is an empty concept. Noumena do not exist because everything we experience via our projections is itself real, as real as anything can be. if there were nounena, i.e. realities that we cannot know, we would not know them, and so speaking about them would be a moot point.
This is not Platonic. For Plato and other metaphysicians before Kant there are unknowables, and the metaphysicians from their privileged position will tell you about them. But Kant is going to say every time to these in his critical philosophy, if it did not come from me ultimately then I cannot know it, and so it is irrelevant. I would rather than a science that told me novel things about the world that are interesting that I can know, that myriad unknowable things no matter how fascinating. If you want to know a little that is real about the universe then Kant is the only show in town. If you want to know a lot that is not knowable about it then there are other metaphysical paths to follow. There are a lot of problems with Kant’s thinking but on the essential point of the lack of value of ungrounded metaphysics there is unanimity. Un-grounded here means not grounded in our nature as human beings, and tied down to our experience, such that it produces understanding of the world. Another way to say this is what is real is what we can understand, and things e cannot understand ultimately because they go against our reason are not real, and are worthless to us.
I hope this helps.