bigideas

I came across the following quasi-logical puzzle the other day while looking for something else. It’s about identity and, specifically, the rule about the “identity of indiscernibles” known as Leibniz’s Law. (The example comes from a paper by Max Black, which isn’t available free online. A paraphrase can be found here).

Leibniz’s Law

Leibniz’s Law is very simple: if there are no differences between two things then they’re the same thing. Less cryptically, say I own a dog, Rufus, and you own a dog, Fido. Say everything that’s true of Rufus is true of Fido, and vice versa. Then Fido and Rufus are actually the same dog.

Most people think Leibniz’s Law is intuitively obvious. If Rufus and Fido were different, they’d be different in some respect. For instance, they’d occupy different regions of space relative to some fixed point. I could say, “Rufus is three metres to my left, whereas Fido isn’t”. If I can’t say something like that than I’ve only got one thing, not two.

(Note that being in different physical locations isn’t necessary for non-identity, though. Consider a statue made of clay; the statue and the clay are, on one account, not identical. For instance, the clay can be reformed into a cube without destroying it, but the statue can’t. Leibniz’s Law says that any property that enables you to “discern” between two things means the two things aren’t identical; the absence of all such properties means they’re the same thing. For more on the clay/statue question see our previous post on relative identity.)

The Two-Sphere Universe

Max Black asks you to imagine a universe in which there are two spheres. The spheres are identical in every way; it just happens that there are two of them. They’re made of the same kind of material, look, smell feel the same, and so on.

In particular, there’s nothing else in the universe to use as a spacial reference-point, so it’s hard to see how to describe their spacial positions, except by saying “the spheres are (say) 1 metre apart”. But that doesn’t translate into a property that’s true of one and false of the other.

It’s important that there’s no observer floating around in this universe. If there were, they could say “well, this sphere is on my left, while that one is on my right” and that would be a real difference between them. But there’s no such observer, or anything else, in this universe; just the two spheres.

Actually there is something else, which Black assumes without mentioning it; a 3-dimensional “ambient space” for the two spheres to live in. If you’ve been following our topology mini-series you’ll probably appreciate the difference between a space composed of only two unconnected spheres and a 3-dimensional space, part of which is occupied by the two spheres.

Some Unsuccessful Solutions

Nobody (I think) would want to accept Leibniz’s Law if it said that, contrary to all our intuitions, there was actually only one sphere in the two-sphere universe. It’s very tempting indeed to think that the spacial locations of the spheres can be used to show that’s not the case.

But without an observer, or some other reference-point, it’s quite challenging to find a property that one sphere would have that the other doesn’t in respect of their physical locations. You could point at one and say “that sphere is there, the other isn’t”, but in doing so you’ve fixed a reference-point (yourself) and used a vector (your pointing finger) to indicate a displacement from that reference-point to the sphere. The problem is, you don’t exist in this universe, so you can’t do that.

Here’s another trick you could try. Say each sphere has an identical blemish on its surface. Now, under certain circumstances that could be used to differentiate them. Imagine you do exist for a moment, and you’re looking at the two spheres side-by-side. The one on the right has a blemish on the right side as you look at it. Therefore, by hypothesis, so does the one on the left. But now one sphere has a blemish on the hemisphere that faces the other sphere. The other sphere, though, does not. That’s a genuine difference that doesn’t require an observer to be present. But if there are no blemishes, however microscopic, then this trick won’t work.

If the spheres have parts then there’s a prospect of a solution if you could pick out one of those parts. But since both spheres are identical, one would assume it follows that their parts are all identical too. It’s still possible that we could repeat the trick with the blemishes, though. If we assume the spheres don’t have “interesting” parts, though, that avenue seems blocked too.

I’ll admit that calling this example “artificial” would be quite an understatement. But it remains an interesting puzzle. The difficulty arises because you can’t talk about just one of the spheres, because to do that requires an observer who can say “that one”. Whatever you do to tell the difference between them, you have to do the same to each and describe what you’re doing in terms that make sense in the two-sphere universe.

A Sort Of Solution

I don’t have a direct solution, but I think can offer an indirect one. Below I give a procedure by means of which, without ever specifying one sphere separately from the other, one can construct a property that one has that the other does not, and vice versa.

The idea of the solution is to identify five points on each sphere, without ever relying on being able to differentiate between them, and then show that these five points give the spheres different orientations. It’s important to be pedantic about some of this because we want to be sure that we never rely on being able to choose one sphere and do something with it on its own, or do anything else that relies on an observer. For instance, we can’t refer to the “tops” of the spheres for just that reason.

One thing we can do is identify the points at which each sphere is closest to the other. I hope you’ll agree that these points don’t change depending on how you observe them; they’re “objective” geometric facts. We’ll label those points “A”:

Now pick the antipodal point — the one on the far side of each sphere — and label it “B”. You can do this by projecting the line from A to the centre of the sphere, which is the point that’s equidistant from every point on the surface.

Next, choose any line that passes through the centre of each sphere at right angles to the first line AB. Keep projecting it outwards until it meets the sphere, again at two antipodal points. We’ll call these the “polar points” of the spheres.

For the solution to work we need to label the polar points in a specific way. Intuitively, we want the “north poles” labelled “C” and the “south poles” labelled “D” (or vice versa):

We label two points C if (a) the points are on different spheres, and (b) the pionts are as close together as possible given condition (a). We then label the other points D. Which two points we start with is immaterial.

The simplest suggestion might be to randomly choose pairs of points until (a) and (b) were satisfied. In any event, this step is likely to involve a very weak version of the “axiom of choice“, which entitles me to choose any two of the four points. In most situations we’d take that for granted, but here it makes me uneasy because it seems to involve choosing one point, then another, and that seems like it might indirectly identify one sphere separately from the other, which breaks my rules. Well, I have no way around it, so let’s live with it for now.

Now we create one more point, E. To do this we use exactly the same idea. We start with the line through the centre of each sphere at right angles to the lines AB and CD. Unlike CD, this line is unique.

As these lines pass through the surfaces of the spheres they again pick out four points. Using the same labelling strategy as before, pick pairs of points until conditions (a) and (b) are met. Label the resulting points E.

We’ve now identified five identical points on each sphere, without ever considering one sphere separately from the other. And this is enough to enable us to find a difference between them.

Consider the arcs EA and imagine sweeping those arc around the spheres until they meet the points labelled C. On one sphere the arc will move clockwise, and on the other it moves anticlockwise. We’ve induced an orientation on each sphere that’s different from the other. We can therefore say that one is clockwise-oriented and the other is not.

This is not an observer-dependent property; or rather, in a sense, we’ve actually created an observer using the point E.

Summing Up

This is sufficient to make a logical difference between the two, although since it was a purely imaginary construction is makes no “real” difference. What it does show is that one can find a predicate on which the spheres disagree without arbitrarily inserting another object into the universe to act as a reference-point or treating the spheres differently before we’re entitled to do so. The spheres themselves, in the way they’re arranged, admit of the construction given without anything additional being introduced.

The solution has some weaknesses. If I come back later and perform the same construction I may end up with the orientations of the spheres swapped over, depending on which hemisphere E ends up on. So “being clockwise-oriented” is only true of a sphere x as long as the construction is in place.

The construction doesn’t rely on the perspective of an observer, but it still, in a sense, isn’t “objective”. The only “objective” property we’ve established is that, for each sphere, its orientation comes out different from the other when this construction is done. But each sphere has this property, and so it doesn’t really give us a way to discern between the spheres using a property. This goes fairly deep into what it means for a thing to have a property, and what “objective” facts are.

I should mention that Joseph Long has put forward an alternative solution that’s stronger, but it requires a modal language with world-indexical predicates, which is a bit more machinery that we might always expect to have to hand.

Any other ideas for solutions to this puzzle? I’m sure there must be a few ways to rescue Leibnitz’s Law from the two-sphere universe.