On standard substantivalist theories of spacetime, there is a special contingent concrete entity *spacetime* and locational properties of objects depend on the objects' relations to the parts or components of spacetime. There is more than one way of spelling this out, depending on which relations one takes to be primitive and which parts or components of spacetime the relations relation the objects to. For instance, you can have a point theory, on which the fundamental relation is something like *being at least partly located at*, or you can have a region theory, on which the fundamental relation is something like *being wholly located within*, *exactly occupying* or *spatially overlapping*.

Substantivalist theories have at least two costs: (1) they bloat the ontology and do so with a fairly mysterious object or objects; and (2) as Leibniz argues in his correspondence with Clarke, they make it mysterious why the contents of space (or spacetime) are where they are rather than all shifted in some direction.

I will offer, without in any way endorsing (in fact, I am strongly inclined to disendorse the theory due to some theological worries and as I am mildly inclined towards ifthenism in mathematics), a substantivalist story that helps with cost (1), and when combined with theism also helps with (2), and that may even have the bonus of helping with the epistemology of mathematics. I will do this in the context of a point theory where *L* will be the *being at least partly located at* relation, but it can be done equally well with a region theory.

The theory is simple. Physicists and mathematicians model spacetime with a mathematical object, say Euclidean four-dimensional space, Minkowski space or some curved Riemannian manifold. Don't take that to be a model: take it to be the reality. In other words, objects are literally, and not merely within a model, located at points of a mathematical object.

Put that way, it just struck me that this sounds mysterious. How could we be "inside" a mathematical object? Aren't mathematical objects just sets? But how can one be inside a set, except metaphorically, unless you yourself are a mathematical object? But really, there need be nothing much to this. To be at least partly located at a point *x* of a mathematical object is just to be *L*-related to *x*, and the relation *L* may just be a fundamental relation. There is nothing mysterious about being related to mathematical objects. If there are any mathematical objects, you're related to many of them by the *thinking about* relation. There is nothing particularly mysterious about supposing another relation that material (or enmattered) objects can have to mathematical objects. And why couldn't that be the location relation? Granted, it sounds odd to say we're inside mathematical objects, but I think the weirdness comes from not paying attention to the shift in the meaning of "inside" between ordinary cases where one material object is inside another and the idea of an object being inside a region of spacetime. It would be weird indeed if we were inside a mathematical object in the first sense but it isn't so weird to say we're inside a mathematical object in the second sense. In fact, we might take the second sense of being inside as somewhat metaphorical even on an ordinary substantivalist theory. One might worry that only mathematical objects can be inside a mathematical object, but there is little reason to think that. One might as well think that only regions can be in a region. Besides, a set can have concrete members in a set theory with ur-elements, and that's a way of being "inside".

Back to the theory. Well, actually, we're done with the fundamental layer. We just have a mathematical object *S* and *L* is a relation between objects and points of *S*. If we want to define other locational relations, such as being wholly located in, and so on, we need to do more work, but I will leave those details to the reader.

Now, there will be a question of what sort of an object *S* is and how it is related to its points. I think the right answer on the theory is: We don't know. That goes too far beyond the empirically observable. But we can speculate. For instance, in standard mathematical fashion, we might well take *S* to be a set with some structure, say a topological or a metric one, and "the points of *S*" will then be members of that set. For instance, in a Euclidean setting, we could just take *S* to be *R*^{4}, the set of all quadruples (*x*,*y*,*z*,*t*) where *x*, *y*, *z* and *t* are real numbers.

There is an arbitrariness worry here. Why *this* set and not another? There is actually a lot of arbitrariness here. For instance, in my Euclidean example, I talked of "real numbers". But what are real numbers? On a fairly standard view, they are objects in the universe of sets, constructed with some construction procedure, like Dedekind cuts or Cauchy sequences. Different construction procedures yield different but isomorphic real fields. Why is our spacetime based on the real field *R* rather than some isomorphic field *R**? That's a good question, but notice that (a) it is not clear that the question is insuperable, and (b) the question is no more problematic than ordinary substantivalism. It is not clear that the question is insuperable because some constructions are more elegant than others, and God could have simply chosen a particularly *elegant* denizen of the universe of sets to be our spacetime (e.g., I have a non-arbitrary preference for the Cauchy sequence method over the Dedekind cut method because it seems to me to have a greater generality). And it's no more problematic than ordinary substantivalism because presumably there are many possible spacetimes we could have inhabited, and we could have inhabited different portions of them.

Now as to the advantages of this theory. First, bloat. Many philosophers think we need mathematical entities, especially those of set theory, anyway, independently of questions of spacetime. If they're right (and I am not sure of this), then there is no additional ontological bloat. If spacetime is a set, and we're already committed to sets, then we've added no new and mysterious object to the ontology.

Second, the Leibniz problem. The problem is why would the contents of spacetime be where they are rather than shifted over by symmetry of the spacetime. Now, this problem is lesser in curved spacetimes where there may not be any symmetries of the right sort. So perhaps given general relativity we don't need to worry about it so much. But let's worry about it. To make the worry as big as we can, take Leibniz's setting where spacetime can be modeled with the Euclidean space *R*^{4}. Well, on the view at hand, spacetime is some mathematical object, and let's suppose it just is *R*^{4}. Then Leibniz's worry arose from the fact that all points in spacetime were exactly alike, and so there would be no reason for an object to be at one point rather than another. But it is false that all points in *R*^{4} are exactly alike. For the points in *R*^{4} are quadruples of numbers, and it is false that all numbers are exactly alike. On the contrary, 0 and 1 are very different indeed, both arithmetically—anything multiplied by 0 is 0, but anything multiplied by 1 is itself—and on the standard set-theoretic construction of arithmetic, 0 is empty and 1 has a member. So since not all points are exactly alike, Leibniz's argment fails. If God exists, then he can make a non-arbitrary choice of where to locate things. For instance, he might put some theologically important event, like the first object being created, at (0,0,0,0).

Finally, I said that this might help with epistemology of mathematics. The major problem with the epistemology of mathematics is that on standard views mathematical entities do not stand in causal relations, so it is hard to see how we can know about them. But on this view, while mathematical objects may still be causally inert, enmattered objects have causally relevant relations to mathematical entities. Which locations things are at *is* important to causal explanation in the sciences. We can, arguably, even *see* where an object is. So relations to those mathematical entities that make up spacetime become causally relevant and perhaps even observable, and that may help us get a foothold in the mathematical realm (interestingly, this would make something like analysis or geometry be the basic part of mathematics). I think that this is a stretch, and there are probably better ways of getting an epistemology of mathematics going (option 1: theism; option 2: observe that while mathematical entities are causally inert they are not explanatorily inert), but it does help somewhat.

## 3 comments:

Liebniz's idea that all positions in spacetime are alike is a reidentifiability difficulty. If points aren't reidentifiable then R4 positions won't map to any spacetime points.

Well, if this theory is plausible, that's good reason to deny Leibniz's intuition that all points are exactly alike.

In order for Liebniz to say that all points are exactly alike he would have to have a means of identifying them spatially.

So it cannot be the fact that all points are alike that Liebniz cannot identify space. It must be the fact that spatial points are not reidentifiable, in principle.

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