Reducing binary distance relations to unary properties

Some philosophers say that space is fundamentally constituted by points. Others that it is fundamentally constituted by regions, and points are logical constructions out of regions. Here is an interesting advantage of an approach base on regions. Relations are more mysterious that properties. A point-based account is likely to involve distance relations: x and y are α units apart.

But a region-based account need not suppose a distance relation, but a diameter property. Intuitively, the diameter of a region is the largest distance between two points in the region, and hence is defined in terms of a distance relation (to account for regions that are not compact, we need to say that the diameter is the supremum of the distances between points in the region). But we could also suppose that the diameter property is more fundamental than distance, and just as we might define points as constructions out of a region-based ontology, we might define distances as constructions out of diameters plus region mereology.

How this would work depends on the details of the point construction. One kind of point construction identifies points with (equivalence classes of) sequences of regions that get smaller and smaller. Some have done this with special concentric regions like balls, but one can also do it with more general regions making use of the diameter D(A) of a region A. Specifically, we can let a point be (an equivalence class of) a sequence of A₁, A₂, ... of regions, where we requires that later regions in the sequence always being subregions of the earlier ones, and that the limit of D(A_n) is zero.(The equivalence relation can be defined by stipulating that the sequences A₁, A₂, ... and B₁, B₂, ... are equivalent just in case D(A_n+B_n) converges to zero where A_n + B_n is the fusion of A_n and B_n.) We can then stipulate the distance between the points defined by the sequences A₁, A₂, ... and B₁, B₂, ... is equal to the limit of D(A_n+B_n). We’re going to need some axioms concerning diameters and regions for all this to be well-defined and for the distance to be a metric.

Or we can take a version of Lewis’s construction where points are just identified with balls of a specific diameter δ₀, with the intuition that we identify a point with the ball of diameter δ₀ "centered on it". And we can again define distances in terms of diameters: d(A,B) = D(A+B) − δ₀.

This does not rid us of all relations. After all, we are supposing the mereological parthood relation (in its "subregion" special case). However, one might think that parthood is more of a fundamental binary predicate than a relation. And at least it’s not a determinable relation, in the way that distance is.

I am not myself fond of mereology. So the above is not something I am going to push. But it would be fun to work out the needed axioms if nobody’s done it (quite likely someone has—maybe Lewis, as I haven’t actually read his stuff on this, but am going on hearsay). It would make a nice paper for a grad student who likes technical stuff.

Spacetime and Aristotelianism

For a long time I’ve been inclining towards relationalism about space (or more generally spacetime), but lately my intuitions have been shifting. And here is an argument that seems to move me pretty far from it.

Given general relativity, the most plausible relationalism is about spacetime, not about space.

Given Aristotelianism, relations must be grounded in substances.

Here is one possibility for this grounding:

1. All spatiotemporal relations are symmetrically grounded: if x and y are spatiotemporally related, then there is an x-to-y token relation inherent in x and a y-to-x token relation inherent in y.

But this has the implausible consequence that there is routine backwards causation, because if I walk a step to the right, then that causes different tokens of Napoleon-to-me spatiotemporal relations to be found in Napoleon than would have been found in him had I walked a step to the left.

So, we need to suppose:

2. Properly timelike spatiotemporal relations are grounded only in the later substance.

But what about spacelike spatiotemporal relations? Presumably, they are symmetrically or asymmetrically grounded.

If they are symmetrically grounded, then we have routine faster-than-light causation, because if I walk a step to the right, then that causes different tokens of x-to-me spatiotemporal relations to be found in distant objects throughout the universe.

Moreover, on the symmetric grounding, we get the odd consequence that it is only the goodness of God that guarantees that you are the same distance from me as I am from you.

If they are asymmetrically grounded, then we have arbitrariness as to which side they are grounded on, and it is a regulative ideal to avoid arbitrariness. And we still have routine faster-than-light causation. For presumably it often happens that I make a voluntary movement and someone on the other side of the earth makes a voluntary movement spacelike related to my movement (because there are so many people!), and now wherever the spatiotemporal relations is grounded, it will have to be affected by the other’s movement.

I suppose routine faster-than-light causation isn’t too terrible if it can’t be used to send signals, but it still does seem implausible. It seems to me to be another regulative ideal to avoid nonlocality in our theories.

What are the alternatives to relationalism? Substantivalism is one. We can think of spacetime as a substance with an accident corresponding to every point. And then we have relationships to these accidents. There is a lot of technical detail to work out here as to how the causal relationships between objects and spacetime points and the geometry of spacetime work out, and whether it fits with an Aristotelian view. I am mildly optimistic.

Another approach I like is a view on which spacetime position is a nonrelational position determinable accident. Determinable accidents have determinates which one can represent as values. These values may be numerical (e.g., mass or charge), but they may be more complex than that. It’s easiest in a flat spacetime: spacetime position is then a determinable whose determinates can be represented as quadruples of real numbers. In a non-flat spacetime, it’s more complicated. One option for the values of determinate positions is that they are “pointed spacetime manifold portions”, i.e., intersections of a Lorentzian manifold with a backwards lightcone (with the intended interpretation that the position of the object is at the tip of the lightcone). (What we don’t want is for the positions to be points in a single fixed manifold, because then we have backwards causation problems, since as I walk around, the shifting of my mass affects which spacetime manifold Napoleon lived in.)

Measuring rods

In his popular book on relativity theory, Einstein says that distance is just what measuring rods measure. I am having a hard time making sense of this in Einstein’s operationalist setting.

Either Einstein is talking of real measuring rods or idealized ones. If real ones, then it’s false. If I move a measuring rod from one location to another, its length changes, not for relativistic reasons, but simply because the acceleration causes some shock to it, resulting in a distortion in its shape and dimensions, or because of chemical changes as the rod ages. But if he’s talking about idealized rods, then I think we cannot specify the relevant kind of idealization without making circular use of dimensions—relevantly idealized rods are ones that don’t change their dimensions in the relevant circumstances.

If one drops Einstein’s operationalism, one can make perfect sense of what he says. We can say that distance is the most natural of the quantities that are reliably and to a high degree of approximation measured by measuring rods. But this depends on a metaphysics of naturalness: it’s not a purely operational definition.

Some fun distinctions

Isn’t it funny how very similar gestures can signal respect and disrespect? Under ordinary circumstances, crossing to the other side of the street to avoid near someone is a form of disrespect. But in a pandemic it signals a respectful desire not to make the other nervous. Though I suppose even apart from a pandemic, one would have moved out of the way of dignitaries.

We have another neat little thing here. There is a difference between going out of one’s way to ensure that one isn’t in another’s personal space and going out of one’s way to ensure that the other isn’t in one’s personal space, even though in an egalitarian society, x is in y’s space if and only if y is in x’s space.

And notice how hard it is to formulate that point without reifying “personal space”, just by using distance. I can hear a difference between avoiding my being within a certain distance of another and avoiding the other being within a certain distance of me, but I can’t tell which is which! Maybe, though, we can distinguish (a) avoiding imposing on another the bad-for-them of us being within a certain distance and (b) avoiding imposing on me the bad-for-me of us being within that distance. In other words, the reasons for the two actions are grounded in the same state of affairs but considered as bad for different individuals.

I suppose similar things can happen entirely in third person contexts. I can work for a friendship between x and y considered as a good for x, considered as a good for y, or considered as a good for both. And these are all three different actions.

Real Presence and primitive locational relations

According to relationalism, space is constituted by the network of spatial relations, such as metric distance relations (e.g., being seven meters apart). If these relations are primitive, then there is a very easy way for God to ensure the Real Presence of Christ: he can simply make there be additional spatial relations between Christ and other material entities, spatial relations that are exactly like the relations that the bread and wine stood in to other material entities.

It might seem contradictory for Christ to stand in two distance relations: for instance, being one mile from me (in one church) and three miles from me (in another). But I doubt this is a contradiction. New York and London are both 5600 km and 34500 km apart, depending on which direction you go.

According to substantivalism, on the other hand, points or regions are real, and objects are in a location by standing in a relation to a point or region. If relations are primitive, again there should be no problem about God instituting additional such relations to make it be that Christ is present where the bread and wine were.

In other words, if location is constituted by a primitive relation—whether to other objects or to space—there is apt to be no difficulty in accounting for the Real Presence. The reason is that we expect, barring strong reason to the contrary, primitive relations to be arbitrarily recombinable.

If location, however, is constituted by a non-primitive relation, there might be more difficulties. For instance, as a toy theory, consider the variant of relationalism on which spatial relations are constituted by gravitational force relations (two objects have distance r if and only if they have masses m₁ and m₂ and there is a gravitational force Gm₁m₂/r² between them). In that case, for God to make Christ present in Waco would require God to make Christ stand in gravitational force relations of the sort that I stand in by virtue of being in Waco. For instance, the earth’s gravitational force on Christ would have to point from Waco to the center of the earth—but since the Eucharist is also in Rome, it would have to point from Rome to the center of the earth as well. And that might be thought impossible. But perhaps there could be two terrestrial gravitational forces on Christ: one along the Waco-geocenter vector and the other along the Rome-geocenter vector. This would require some sort of a realism about component forces, but that’s probably necessary for the gravitational toy theory. And then God would have to miraculously ensure that despite the forces, Christ is not affected in the way he would normally be by these forces. All this may be possible, but it’s less clear than if we have primitive relations.

A reduction of spatial relations to an outdated physics

Consider a Newtonian physics with gravity and point particles with non-zero mass. Take component forces and masses as primitive quantities. Then we can reduce the distance at time t between distinct particles a and b as (m_am_b/F_ab)^1/2, where F_ab is the magnitude of the gravitational force of a on b at t, and m_a and m_b are the masses at t of a and b respectively (I am taking the units to be ones where the gravitational constant is 1); we can define the distance between a and a to be zero. For every t, we may suppose that by law that the forces are such as to define a metric structure on the point particles.

If we want to extend this to a spatiotemporal structure, rather than just a momentary temporal structure, we need to stitch the metric structure into a whole. One way to do that is to abstract a little further. Let S be a three-dimensional Euclidean space. Let P be the set of all particles. Let T be the real line. For each object a in P, let T_a be the set of times at which a exists, and let m_a(t) be the mass of a at t. For any pair of objects a and b and time t in both T_a and T_b, let F_ab(t) be the magnitude of the gravitational force of a on b at t. Let Q be the set of all pairs (a,t) such that t is a member of T_a. Say that a function f from Q to S is an admissible position function provided that:

1. If t is a member of both T_a and T_b, then F_ab(t)=m_a(t)m_b(t)/|f(b,t)−f(a,t)|².
2. f''(a,t) is equal to the sum over all particles b distinct from a of (f(b,t)−f(a,t))F_ba(t)/(m_a(t)|f(b,t)−f(a,t)|).

The laws can then be taken to say that the world is such that there is an admissible position function. We can then relativize talk of location to an admissible position function, which plays the role of a reference frame: the location of a relative to f at t is just f(a,t).

The above account generalizes to allow for other forces in the equations.

So, instead of taking spatial structure to be primitive, we can derive it from component forces, masses and objects, taking the latter trio as primitive.

I don't know how to generalize this to work in terms of a spatiotemporal position function instead of just a spatial position function.

Of course, component forces are hairy.

Perhaps the method generalizes to less out-of-date physics. Perhaps not. But at least it's a nice illustration of how spatial relations might be non-fundamental, as in Leibniz (though Leibniz wouldn't like this particular proposal).