It’s quite a strange thing to consider but the question of whether or not higher dimensions exist seems to me to be precisely the wrong kind of inquiry to be making of this deeply mysterious human experience. Given the implicit linguistic (as neurocognitive) orchestration of experience, we should always acknowledge that those things we might ever say with or understand through language are always and already bound by the vacillating uncertainties of a language that gains more benefit from ambiguity than closure and through which we can hardly expect directly encounter or circumscribe any kind of ultimate meaning. If we were to ask any kind of thing that might aspire to substantive existential utility, I suspect it might more closely resemble something like this:
If we finds ourselves inhabiting a multidimensional world, in what ways might the plausible incomprehensibility and unattainable rational closure of such a concept from within language and thought or any other form of intelligible symbolic algebra actually underlie the core generative discontinuities of language and thought themselves?
In other words – might we reach out beyond the endemic discontinuities of language and thought precisely by means of these endemic discontinuities of language and thought? The strangest thing about finding one’s self in a multidimensional tesseract is that there would not and never could be anything outside that system of definition, reference and experience.
So, if this is a multidimensional world, I dare say that our experience and descriptions of it is precisely what we might expect it to be. We just can not see all of it at once and that is, I think, an experience that we are already all so familiar with that it really should be no surprise that the lower-dimensional experience of a higher-dimensional world is pretty much exactly what this human life resembles.