Thursday, May 9, 2013

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.

In particular, 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.


  1. 1. It seems like location in time is at least as important for causal interaction. That's consistent with your story insofar as time and space location are fungible in Minkowski space, but it seems like you'd then have to take account of the asymmetry of causation and that time is usually taken as a separate variable in the wavefunction. Neither of these necessarily undermine your point, but I'm not fully envisioning how you'd mean to include them, either.

    2. Two objects can also fail to interact because they have insufficient charge or are insufficiently massive (neutrinos, say) despite being really close. Is location special because it cuts across all modes of interaction? Is there then a place for your definition of location in a world where we have a unified theory of all forces?

    3. What does location mean in the entangled case? It seems strange to give an account of location largely in quantum terms without taking account of this central quantum phenomenon. It seems like either solution to the Bell dilemma is problematic for your story because it undercuts our intuitions about either location or causality.

    None of this is to say that I have a better theory; just questions that occurred to me as I read your post.

  2. Hi Alexander,

    That sounds good for the case of location in physical space, but what about location in abstract spaces?
    Take the real one-dimensional space R with its usual topology. Here it is not even clear what the capability of any real number to interact with any other real number would be. I assume that an interaction between a real numbers is any operation definable over the real field. But then it is not clear that the "capability" (I use scare-quotes because I'm assuming that it is a term with modal load; and such load may be difficult to conceive of when talking about necessary beings, as numbers are assumed to be) is correlated with distance, or that such a tendency might exist.

    I have the intuition that perhaps we should not try to determine what location is "from the outside" (i.e. by its effects) but "from the inside": perhaps by abstracting them from the abstract notion of space or of structure.


  3. The standard mathematical explanation of location in abstract space is that it's just set membership: 7 is in R just in case 7 is a member of R.

    In any case, the "from the inside" approach in the classical won't distinguish between momentum and position, since classically both have the same mathematical structure: they have values whose space has the structure of R^3. But I want to say what makes position position, what makes it different from momentum, or charge, etc.

  4. Interesting. From the perspective of quantum mechanics, there's nothing particularly special about "spatial location" as opposed to any other measurable property. For example, there is a momentum wavefunction representation -- and indeed, a wavefunction representation for every self-adjoint operator. So, if quantum theory justifies talk about "partial location," then it also justifies talk about "partial properties" for every measurable property of a physical system.

  5. Right, but I have a stronger intuition in the case of position that there could be position in classical worlds. Intuitively, position is less tightly law-bound than, say, charge.

    So I am suspecting multiple-realizability about position (and maybe momentum, as its conjugate) but not, say, about charge. In worlds where classical electrodynamic laws hold, then, nothing has any charge, but there is instead charge*, which behaves like in our world charge does in the classical limit. But the classical worlds really do have position, though it is realized differently from how it is realized in our world.

    One could, I think, try to hold that both charge and position are on par. I doubt that one could come up with a very plausible functional characterization of charge, so that would require saying that both charge and position are law-bound, and in classical worlds nothing has any charge and nothing has location or shape. Moreover, what goes for space probably goes for time, so in classical worlds there would probably be no time or even change. This is counterintuitive.

  6. I worry that what people call 'location in abstract space' is just the use of a spatial metaphor (or analogy) to talk about the relation of any entity whatsoever to any whole whatsoever. So, 'location'—as x's relation to the yys in whatever whole of parts you can imagine—can be described, by analogy, as x's 'location' in the whole. As such the idea that 'location' is multiply realised is not surprising.