Extended Atoms
In my last post, I said that I do not believe that every extended thing must have parts. Sam disgrees, arguing that whenever something is extended over length h, we can restrict our attention to a part of it with length h/n for any n < h.
I do agree that all ordinary extended things have parts. And I do agree that extended things without parts are really very strange. I'm just not sure that they are impossible.
First, couldn't there be a quantized world where every thing's extension is a multiple of some positive length h? That doesn't sound contradictory to me. But if every thing's extension is a multiple of h then also every part's extension is a multiple of h. So there are no parts smaller than h. In such a world, there are no rulers with marks for h/2 on them, nor can we restrict our attention to parts of length h/2 because there are no such parts.
Another interesting possibility (which I mentioned last year) is that ordinary things are really microphysical collections, not of extended but of unextended things. That is, it is possible that every extended thing is ultimately composed of point-sized bits of matter with lots of empty space (or spacetime) in between. Something like this seems to be described by classical physics. It seems not entirely unreasonable to me not to count the bits of empty space as parts of the extended things. Certainly one shouldn't count them if one is a relationalist about spacetime, a matter about which I have no opinion. But if so, and we have a thing consisting of only two point-particles at a distance of h from each other, this thing will have only two parts, not infinitely many.
Finally, there is the issue of tropes and immanent universals (which I also mentioned last year in a posting where I suggest yet another possibility, but I wouldn't place to much weight on that one). If such things exist then presumably at least some of them are spatiotemporally extended but lack parts.
[Update (1 hour later): Lots of abstract things besides universals might also be spatiotemporally extended without having spatiotemporal parts. My singleton for instance is arguably located just where I am, but it doesn't have a (proper) part that is located at the region of my head. Something similar can be said about species, governments, increases, currencies, bank accounts, and even numbers if one holds that numbers exist at every time. However, for some of these it is important how precisely 'extension' is understood. If for instance being extended means being located at two different spacetime points, or to be located at every point between two spacetime points, then my singleton isn't spatially extended unless it is located at the top of my head, which perhaps it isn't. Perhaps my singleton is only located at the entire region where I am. Then it is still spatially extended in the sense of being located at an extended region of space. I'm not sure what exactly we mean by 'extension'. (So I'm also not exactly sure what I mean by 'persistance'.)]