The Identity of Indiscernibles

This post distinguishes between substantivalism and relationalism, and how the principle of the identity of indiscernibles threatens substantivalism. Then we’ll evaluate the principle, showing under what conditions it is plausible.

We will assume Leibniz’s view of relationalism. Broadly, this is the view that there are no substances such as space and time. Rather space and time are real only insofar as they are basic relations and properties between material objects – they do not exist independently of material objects. Facts about space and time are derived from material bodies, and without material bodies or events, there simply are no facts about space, time, or their relations. So, what makes it the case that the basketball is five feet away from the hoop is nothing over and the fact that there is a basketball and a hoop that are such-and-such apart.

Substantivalism is the denial of relationalism: space and time are substances which exist independently of any material objects. Consequently the substantivalist can imagine an empty universe, or an empty space/time container, where only facts about space and time obtain. So, what makes it the case that the basketball is five feet from the hoop is that the basketball occupies one region of space and the hoop occupies another, and those two regions are some equivalent of five feet apart. This way we can explain spatial relations without taking them as basic in the sense that relationalism does.

To explain the problem of a Leibniz shifted world, we need to explain the principle of the identity of indiscernibles. This principle states that if any two objects are indiscernible, then those two objects are identical. Two objects are indiscernible if they share all their properties and there does not exist a property belonging to one which does not belong to other (or vice versa) by which you could tell them apart. Jackson and Geoffrey are discernible in that one wears glasses and the other doesn’t. So Jackson and Geoffrey are not identical. Mark Twain and Samuel Clemens, however, are indiscernible. You’ll find they have an identical genetic structure, both occupy the actual world, and both are named both “Mark Twain” and “Samuel Clemens”.

Substantivalism holds absolute time as an ordered set of instants, and the universe must have come into existence in (an arbitrary) one of these instants. These two facts entail that there could be a possible world which would have come into existence at different instant, but otherwise contains all and exactly the same events as our actual world with all the spatial and temporal relations between material bodies being the same. This is one example of a Leibniz shifted world, a world which is identical to ours, except differing in some “absolute” property (e.g. time of instantiation, where that entire world is drifting to the “left” at 5m/s).

Because their relative spatial and temporal relations are identical, the only way they could possibly differ is in an absolute spatial or absolute temporal property. But their arbitrary absolute differences are weak grounds for discernibility. Why should “God” choose to instantiate one world at one time and another identical world at another? Intuitively, there is no reason that God should choose to instantiate two identical worlds at two arbitrary points in time, with some particular duration of absolute time between them. In the spirit of parsimony, the worlds are indiscernible, and so, by the principle of the identity of indiscernibles, they must be identical. But their identity contradicts the claim that there could be a relatively identical world to ours that comes into existence at some other instant of absolute time. If you subscribe to absolute space or time, then your only option is to abandon the principle of identity of indiscernibles.

But is problematic, for this entails the possibility of Leibniz shifted worlds, and this entails an unnacceptable proliferation of relatively identical possible worlds – indeed, we could have an infinite set of these, each differing only in, say, absolute time instantiation. Prima facie, we have better reason to maintain the principle of identity of indiscernibles than to admit the possibility of this proliferation.

The substantivalist might reply by rejecting the principle of indiscernibles. It is logically possible that the universe should have contained nothing but two exactly similar spheres. Every quality and relational characteristic of one holds also of the other, in virtue of the symmetric relations between the two spheres. You might think that the one sphere has the property of being not the other sphere. But this property is also true of the other sphere.

If we introduced an observer, he might name one sphere “A”, and the other sphere “B”. Then A has the property of being -B, a property which B cannot have, so they are discernible and not identical. But there are two considerations here. The first is that by naming them different things, an observer is merely stipulating that they have the property of being different, which is uninformative. The second is that this difference in property is dependent on there being a “namer” in the universe, so this purported difference in property is really more a feature of how observers apply names than a feature of genuine property difference between the two spheres.

It is tempting to think that the spheres differ in virtue of the fact that one is located in a place that the other is not. But their spatial relations are identical, so the only way we could point to this difference in property is if we say that A occupies location R and B occupies location S. But this move can only be made if we admit that regions of space are independently existing things, not merely a consequence of brute spatial relations between objects. So the relationalist would have to concede absolute space to the substantivalist.

If we introduce an asymmetric observer, however, one sphere will be on the “right” and the other on the “left”. Then the spheres would have different properties. So it is logically possible that the spheres have different properties. The substantivalist admits this, but holds it is in virtue of newly acquired relational characteristics. But the relationalist might argue modal properties are real properties.

Here’s how that argument might run. The substantivalist holds that A and B have the same properties. The spheres are equivalent in that they both satisfy all the same propositions. Each sphere must bear a reflexive relation to itself and a symmetric relation to the other. But the substantivalist admits that if there were an asymmetric observer, then the two spheres might have different properties. But we can say something stronger. For an asymmetric observer must be an observer which is closer to the center of one sphere than the center of another, and so it is necessary that the spheres would have different properties (qua to the observer).

So the substantivalist has admitted that it is possible that it is necessary that A and B are not equivalent. This, together with A and B being symmetric, entail that it is not the case that A and B are equivalent, by the axiom of symmetry.1 But if A and B are not equivalent, then they cannot be identical, contrary to the substantivalist’s supposition.

But the substantivalist might reply as follows. In instantiating the asymmetric observer, the spheres are no longer symmetrical in all their spatial properties. Using the axiom of symmetry, then, is not valid.

At this point, it’s hard to see who is right. Because the relationalist might think that it is possible that there is an asymmetric observer before we move to instantiate one. And from that observer’s perspective, it would be necessary that the two spheres are distinguishable in virtue of a spatial-relational property. So before we have disrupted the purported symmetry, there is already a difference in (modal) properties between the two spheres.

The tension here is that discernibility seems to essentially involve the idea of some observer doing the discerning, whereas identity is an objectively necessary condition for this being this, regardless of observation. Two things are indiscernible iff it is logically impossible for an observer to tell them apart. The asymmetrical observer can discern between the spheres (so it is logically possible). But when left out, it is logically impossible to discern between them, so the relationalist cannot explain why they are not identical. (There’s a certain antimony here.)

The notion of a point of view is required for discernibility to be logically possible, if the two spheres are not in fact one and the same. I am not sure what resources the relationalist has to be able to make sense of “point-of-viewness” – without positing an observer – in this situation. But if he can make sense of this, then he will have a means of discerning between the two spheres and so can explain why they are not identical. But if he cannot, then we really can conceive of two spheres which are indistinguishable but not identical. That would falsify the principle of the identity of indiscernibles, and so defang Leibniz shifted worlds.

1 Axiom of symmetry: <>[]p → p

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2 thoughts on “The Identity of Indiscernibles

  1. Interesting work here. I never took symbolic logic, so I missed that segment of your paper. On this:

    “It is tempting to think that the spheres differ in virtue of the fact
    that one is located in a place that the other is not. But their spatial
    relations are identical, so the only way we could point to this difference
    in property is if we say that \alpha occupies location R and \beta occupies location S. But this move can only be made if we admit that regions of space are independently existing things, not merely a consequence of brute spatial relations between
    objects. But then the relationalist would simply be conceding to the
    substantivalist.”

    I’m not sure I’m following. When you say “their spatial relations are identical” do you mean sphere 1 is related to sphere 2 in the same way as sphere 2 is to sphere 1? And therefore they have properties which are identical?

    I guess the main question is, why do we have to admit that regions of space are independently existing things? How does time factor into this example?

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  2. When I say “their spatial relations are identical” I mean that any relational property you can predicate of one sphere, you can predicate of the other. The sphere are each related to each in the same way, since we are talking about a universe with just these two spheres, all their properties are symmetrical. If one sphere has the property, then the other sphere has the property.

    But the spheres are not identical, because they are two different spheres. To say this sphere is here and that sphere is there is to say that this sphere is in one region of space and the other sphere is in some other region of space. The idea is that the relationalist cannot use this fact to distinguish between the spheres, because that fact entails that space is more than just a relation between two material objects, namely space is the sort of thing that an object occupies (which smells like substantivalism).

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