What is location?

I will first argue that location is a multiply-realizable—i.e., functional—determinable. Then I will offer a sketch of what defines it.

A multiply-realizable determinable is one such that attributions of its determinates are grounded in different ways in different situations. For instance, running a computer program is multiply realizable: that something is running some algorithm A could be at least partly made true by electrical facts about doped silicon, or by mechanical facts about gears, or by electrochemical facts about neurons. Moreover, computer programs can run in worlds with very different laws from ours.

As a result, a multiply-realizable determinable is not fundamental. But location seems fundamental, so what I am arguing for seems to be a non-starter. Bear with me.

Consider a quantum system with a single particle z. What does it mean to say that z is located in region A at time t?[note 1] It seems that the quantum answer is: The wavefunction (in position space) ψ(x,t) is zero for almost all x outside A. And more generally, quantum mechanics gives us a notion of partial location: x is in A to degree p provided that p=∫_A|ψ(x,t)|²dx, assuming ψ is normalized. On these answers, being located in A is not fundamental: it is grounded in facts about the wavefunction.

But it is also plausible that objects that do not have wavefunction can have location. For instance, there may be a world governed by classical Newtonian mechanics, and objects in that world have locations but no wavefunctions. (And even in a world with the same laws as ours, it is possible that some non-quantum entity, like an angel, might have a location, alongside the quantum entities.) Thus, location is multiply-realizable.

Very well. But what is the functional characterization of location? What makes a determinable be a location determinable? A quantum particle is located in A provided that ψ vanishes outside A. But a quantum particle also has a momentum-space wavefunction, and we do not want to say that it is located in A provided that the momentum-space wavefunction vanishes outside A? Why is the "position-space" wavefunction the right one for defining location? Why in a classical world is it the "position" vector that defines location, rather than, say, the momentum vector or an axis of spin or even the electric charge (a one-dimensional position)?

I want to suggest a simple answer. Two objects can have very similar electric charges, very similar spins or very similar momenta, and yet hardly be capable of interacting because they are too far apart. In our world, distance affects the ability of objects to interact with one another. Suppose we say that this is the fundamental function of distance. Then we can say that a determinable L is a location-determinable to the extent that L is natural and the capability of objects to interact with one another tends to be correlated with the closeness of values of L. This requires that L have values where one can talk about closeness, e.g., values lying in a metric space. In a quantum world without too much entanglement and with forces like those in our world, the wavefunction story gives such a determinable. In a classical world, the position gives such a determinable.

(One could also have an obvious relationalist variant, where we try to define the notion of being spatially related instead. The same points should go through.)

Notice that on this story, it may be vague whether in a world some determinable is location. That seems right.

I think this story fits well with common-sense thought about distance and location, and helps explain why we maintained these concepts across radical changes in physical theory.

This story also reminds me a little of what Aquinas says about the location of angels: An angel is at location x if and only if the angel is causally interacting with something at x. But there is a difference. While Aquinas defines location of an angel in terms of actual interaction, I define location at two removes from interaction: I only talk of the capability for interaction and I do not define location in terms of that, but in terms of a fairly natural determinable closeness in respect of which tends to correlate with capability for interaction.

We have two non-philosophical tests for a theory of location. One is whether it coheres with science and the other is whether it coheres with theology, and especially with transsubstantiation. It is no surprise that this theory coheres with science, since it was designed to. What about transsubstantiation (John Heil referred to it as a supercollider for metaphysical theories in medieval times)?

I think it coheres with transsubstantiation quite well. The Catholic tradition tends not to talk about physical but sacramental presence. This is a real presence of course. There are multiple ways of being located for Aquinas: physically, by power (as angels are where they act and as God is everywhere), as well as this sacramental presence that he has an elaborate metaphysical theory of (see my paper here). Our above theory allows for a multiplicity of location determinables, all taking values in a common space, and so there might be some determinable that would give an account of sacramental presence.

But one could also defend the physical presence of Christ's body in the Eucharist coherently with my view. Christ's body could have multiple wavefunctions, each defining a different location. Or maybe space could curve in on itself as I suggest (but do not endorse as my preferred view) in the above-cited paper so that the location of the consecrated host and the location of heaven are literally the same.

So while I didn't design the view initially with the transsubstantiationon in mind, I think it passes the transsubstantiation test.