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Well, the math works. Volumes of hyperspheres are used to calculate otherwise incalculable integrals in statistical mechanics, particularly with Bose Einstein condensates. Whether that indicates the fundamental structure of these condensates is higher dimensional, or whether it's just a math trick, is a question for philosophers. Regardless of the answer, what we can say is that at low temperatures and pressures, some objects behave unlike anything we've ever seen, e.g. superfluid helium which seems to elevate itself to exit containers because of zero viscosity. This doesn't make sense in 3 spatial dimensions. Does it make sense in higher dimensions? Idk, maybe. Superfluidity is a bit outside my area of expertise.
