Kant and the Necessary 3-Dimensionality of Space

Here’s a thought.  Kant took it to be necessary that space was 3-dimensional.  Bracketing the possibility that space is transcendentally ideal for the moment—of course, you might think I’m bracketing the most interesting thing here—most people who are paid to think about these things now reject the necessary 3-dimensionality of space on the grounds that spaces with more dimensions are possible, and the standard argument for this is the following.  A 3-dimensional space can be modelled as $\mathbb{R}^3$ = $\mathbb{R} \times \mathbb{R} \times \mathbb{R} $ with each n-tuple representing a point in 3-dimensional space.  Methods of this sort allow for higher-dimensional spaces to be represented, since extending or generalizing the model to represent higher-dimensional spaces is quite straightforward. 4 dimensional space is represented as $\mathbb{R}^4$, 5-dimensional space as $\mathbb{R}^5$, and so on.

But why think a thing like this constitutes grounds for taking higher-dimensional spaces to be metaphysically possible?  Why think that because a model of 3-dimensional space can be extended in this sort of way (and remain coherent), higher-dimensional spaces themselves are possible, or even coherent?  Why think that because (i) there is a space which can be represented using $\mathbb{R}^3$, and (ii) there is nothing in consistent about $\mathbb{R}^4$, that (iii) there could be a space that is represented by $\mathbb{R}^4$?  Now, there may be other good reasons to think that (iii) is true, but the standard argument looks to be enthymematic at best.

[Cross-posted at Kant and Laws]