Round squares

Most people think that roundness and squareness are incompatible. Circles have curved sides and no corners, while squares have straight sides and four corners. Meinong is sometimes interpreted as having thought there was a round square, but he thought it was a non-existent object, and may even have thought that its having incompatible properties explained its non-existence. But I’ve recently started wondering whether round squares could have existed after all.

If I understood rightly, I was told on Friday that if space was curved in an odd enough way then things in it could be round and square. It sounds like the sort of thing that might be true. That’s not what I’ve been wondering about though; I’m suspecting that round squares might even fit into both Euclidean space and ordinary space.

The problem comes from multiple location. If things can be wholly in two places at once, then I don’t see how round squares could be impossible. The simplest way multiple location could allow round squares is if absolutely anything could be absolutely anywhere and in any number of places. Then an object could be in a round region and a square region, thereby being a round square, since things are the same shapes as their locations.

Prima facie, one can plausibly deny that the possibilities for multiple location are this permissive. However, if multiple location is possible at all I’d have definitely thought that a point-sized object could be at two points at once. (Maybe there really is only one of each kind of point particle and each is in a lot of places. Feynman talked about something like this in his Nobel lecture.) Now take continuum-many point-sized objects, and let them each have two locations. The first locations could be arranged into a square and the second locations could be arranged into a circle. The square region and circular region wouldn’t even have to be the same size, although they could be. The object made out of these point particles would thereby have a circular location and a square location, so it would be a round square.

If this argument works, I suppose it could be used to argue that shape wasn’t an intrinsic property of objects, but rather a property they had in virtue of where they were. That’s weird of course, but multiple location is weird. I’m actually quite attracted to the view that how things are and where they are have a fairly loose relationship, but now I’m starting to worry that if it’s true then analogous arguments to the one about shape will show that almost no properties of located things are intrinsic, and that way lies madness. Or supersubstantivalism.