The Impossibility of Higher Dimensions in Gunky Space

Some people think that space is gunky. That is some people think that every region of space has proper subregions. There are two interesting questions that might be answered if space turns out to be gunky. First, are there higher spatial dimensions? Second, could there be higher spatial dimensions? I am going to suggest that the person who thinks that space is gunky should also think that there are no higher spatial dimensions. Moreover, the person who thinks that space is gunky should also think that there could be no higher spatial dimensions (or at least that there could be no higher gunky spatial dimensions).

Here is how the argument goes against higher spatial dimensions goes. If space is gunky, then there are no points, there are no edges and there are no surfaces. Moreover, (P1) if space is gunky and there are four spatial dimensions (or more) rather than three, then there are no volumes either. This is because volumes would be like the surfaces of hypervolumes and just as we should deny the existence of surfaces in a gunky space of at least three dimensions, so too we should deny the existence of volumes in a gunky space of at least four dimensions. But, (P2) there are volumes (for example, my tea cup encompasses a volume of space and you and I also encompass volumes of space). So, (C1) if space is gunky, then there are not four spatial dimensions (in fact there are no more than three spatial dimensions).

Here is how the argument against the possibility of higher spatial dimensions in gunky space goes. (P3) If there could be more than three gunky spatial dimensions, then there could be an exact duplicate of our spatial world within a four (or more) dimensional gunky space. Moreover, (p4) if there were an exact duplicate of our spatial world within a four (or more) dimensional gunky space, then there would be volumes in such a gunky space. But, (P5) necessarily, if there are four (or more) dimensions of gunky space, then there are no volumes in such a gunky space. So, (C2) there could not be a duplicate of our spatial world within a four (or more) dimensional gunky space. So, (C3) there could not be more than three gunky spatial dimensions.

Interestingly, similar reasoning should get us that there are not, nor could there be fewer gunky spatial dimensions than there in fact are. This is a rather radical and suprising result.

I am inclined to think the first argument is sound. I might worry about the second argument though. In particular, I might worry that (p3) is false. But, then if (P3) is false, then I worry that there are restrictions on plausible modal recombination principles. I also wonder if (P3) might be stronger than is needed. In any case, it seems to me that (P3) is the weakest part of the argument.