Thursday, August 3, 2017

Connected and scattered objects

Intuitively, some physical objects, like a typical organism, are connected, while other physical objects, like a typical chess set spilled on a table, are disconnected or scattered.

What does it mean for an object O that occupies some region R of space to be connected? There is a standard topological definition of a region R being connected (there are no open sets U and V whose intersections with R are non-empty such that R ⊆ U ∪ V), and so we could say that O is connected if and only if the region R occupied by it is connected.

But this definition doesn’t work well if space is discrete. The most natural topology on a discrete space would make every region containing two or more points be disconnected. But it seems that even if space were discrete, it would make sense to talk of a typical organism as connected.

If the space is a regular rectangular grid, then we can try to give a non-topological definition of connectedness: a region is connected provided that any two points in it can be joined by a sequence of points such that any two successive points are neighbors. But then we need to make a decision as to what points count as neighbors. For instance, while it seems obvious that (0,0,0) and (0,0,1) are neighbors (assuming the points have integer Cartesian coordinates), it is less clear whether diagonal pairs like (0,0,0) and (1,1,1) are neighbors. But we’re doing metaphysics, not mathematics. We shouldn’t just stipulate the neighbor relation. So there has to be some objective fact about the space that decides which pairs are neighbors. And things just get more complicated if the space is not a regular rectangular grid.

Perhaps we should suppose that a physical discrete space would have to come along with a physical “neighbor” structure, which would specify which (unordered, let’s suppose for now) pairs of points are neighbors. Mathematically speaking, this would turn the space into a graph: a mathematical object with vertices (points) and edges (the neighbor-pairs). So perhaps there could be at least two kinds of regular rectangular grid spaces, one in which an object that occupies precisely (0,0,0) and (1,1,1) is connected and another in which such an object is scattered.

But we can’t use this graph-theoretic solution in continuous spaces. For here is something very intuitive about Euclidean space: if there is a third point c on the line segment between the two points a and b, then a and b are not neighbors, because c is a better candidate for being a’s neighbor than b. But in Euclidean space, there is always such a third point, so no two points are neighbors. Fortunately, in Euclidean space we can use the topological notion.

But now we have a bit of a puzzle. We have a topological notion of a physical object being connected for objects in a continuous space and a graph theoretic notion for objects in a discrete space. Neither notion reduces to the other. In fact, we can apply the topological one to objects in a discrete space, and conclude that all objects that occupy more than one point are scattered, and the graph theoretic one to objects in Euclidean space, and also conclude that all objects that occupy more than one points are scattered.

Maybe we should have a disjunctive notion: an object is connected if and only if it is graph-theoretically connected in a space with a neighbor-relation or topologically connected in a space with a topological structure.

That’s not too bad, but it makes the notion of the connectedness of a physical object be a rather unnatural and gerrymandered notion. Maybe that’s how it has to be.

Or maybe only one of the two kinds of spaces is actually a possible physical space. Perhaps physical space must have a topological structure. Or maybe it must have a graph-theoretic structure.

Here’s a different suggestion. Given a region of space R, we can define a binary relation cR where cR(a, b) if and only if the laws of nature allow for a causal influence to propagate from a to b without leaving R. Then say that a region of space R is connected provided that any two distinct points can be joined by a sequence of points such that successive points are cR-related in one order or the other (i.e., if di and di+1 are successive points then cR(di, di+1) or cR(di+1, di)).

On this story, if we have a universe with pervasive immediate action at a distance, like in the case of Newtonian gravity, all physical objects end up connected. If we have a discrete universe with a neighbor structure and causal influences can propagate between neighbors and only between them, we recover the graph-theoretic notion.


William said...

If objects were set to discrete and space to continuous, discrepancy theory (the study of arrangements of a limited set of points within continuous geometric spaces) might be useful.

Alexander R Pruss said...

That's true. Good point.