Identity, Quantum Vagueness and E.J. Lowe
Everything is identical to itself, and nothing is identical to anything except itself. No two things are ever identical. If A and B are identical then "they" are one, not two.
These are platitudes about identity, or rather about a somewhat technical use of "identity" common in mathematics and philosophy.
No doubt there are other uses. For instance, "identity" and its cognates are often used to express sameness of kind, as in "this record is the same Jones bought last week". Sometimes, "identity" is used as a singular term for a thing's characteristc properties or individual essence, as in "the festival has lost its identity". The conceptual platitudes above do not apply to these other uses.
In a number of publications, E.J. Lowe has attacked the conceptual platitudes, arguing that they only hold for a restricted class of things. (Lowe uses "thing" for things in that class; I use "thing" as Lowe uses "entity", as a blanket term). They do not hold, he says, for "non-objects" like tropes and "quasi-objects" like electrons. I'll focus on the latter here.
If A and B are electrons then according to Lowe it is sometimes neither true that A and B are identical nor that A and B are not identical, even though it is true that A and B are two. He offers two examples (see, e.g., The Possibility of Metaphysics, sec. 3.3), one dealing with diachronic and one with synchronic identity:
Suppose at time t0 we detect an electron, which we call "A". Then we put A into contact with a Helium ion having only a single electron, so that we get a neutral Helium atom with two electrons. This is time t1. Later, at time t2, we take away again one of the atom's electrons, detect it and call it "B".
Now Lowe says it is neither true nor false that A is the same electron as B. Rather, it is indeterminate whether it is. That's the diachronic example.
I find this example rather unconvincing. Let's agree for the sake of the argument that it is indeterminate whether A is B. Unfortunately, for this very reason it is also indeterminate whether A and B are two electrons or one. So if Lowe wants an example of (determinately) two things of which it is nevertheless indeterminate whether they are identical, this is simply not an example of this kind. At least not unless Lowe offers us a reason to believe that A and B are two. Which he doesn't.
Still, one might think the example does show that something is wrong with the naive account of identity. For this account leaves no room for vague identity: Once it is made clear exactly what thing "A" refers to and what thing "B" refers to, there can be no uncertainty about the truth of "A = B". On the naive account it is therefore impossible that we should find objects A and B such that their identity is indeterminate. But electrons, and quasi-objects and non-objects in general, are such objects.
Well, this "naive account of identity" is just the rejection of ontic vagueness. The most obvious alternative to ontic vagueness is not to deny that there is vagueness at all -- in this case, to deny that "A = B" is indeterminate --, but to locate the vagueness in our language. And this works pretty well in the current example: It is simply not true that the reference of "A" and "B" has been precisely determined. In particular, when I said that "A" is to denote the electron detected at t0, I failed to specify A's future and past extension. Once we make up our minds which spacetime-worm "A" denotes it will no longer be indeterminate whether this worm contains the electron we detected at t2. (Question: How do endurantists deal with this case?)
In sum, Lowe's diachronic example doesn't even begin to challenge any platitudes about identity, nor does it provide any reason to believe in the ontic account of vagueness and the existence of quasi- or non-objects.
For the synchronic example we focus on time t1:
The single electron shell of a neutral helium atom contains precisely two electrons: and yet, apparently, there is no determinate fact of the matter as to the identity of those electrons. [...] Our inability to say which electron is which is not merely due to our ignorance [...]: not even God could say which electron was which, because there is simply no fact of the matter about this. [...] What this means, then, is that an identity statement of the form 'x is the same electron as y' may simply not be either determinately true or determinately false. (Lowe, ibid., p.62)
This time we have at least a prima facie candidate for a challenge to the platitudes: The helium atom definitely contains two atoms, but their identity is indeterminate.
It seems to me, however, that Lowe confuses several different uses of "identity". So there is no fact of the matter as to which electron is which. What does that mean? Perhaps it means that the electrons are in some sense qualitatively indistinguishable because they share certain physically relevant properties. This would resemble the "same kind" usage of "identity", with kinds being understood very narrowly. (Note however that the two electrons differ in spin; so they are not qualitatively identitical.)
Lowe also frequently talks about "the identity" of the electrons, refering for example to our alleged inability "to distinguish between the identities of those electrons". This looks like the "individual essence" usage of "identity". And apparently it is indeed a peculiar fact about entangled electrons that they lack characteristic individual properties which we could use to tell them apart or label them or track them through time. But this has nothing to do with identity in the strict and philosophical sense.
Lowe is not alone in denying that there are facts of the matter about the identity of particles in quantum mechanics. Thus Hesse (1970) claims that
[w]e are unable to identify individual electrons, hence it is meaningless to speak of the self-identity of electrons
Interestingly, Hesse uses "self-identity" instead of "identity". It is rather clear however that by this he means something like individuating properties or individual essences. Also interesting is the fact that unlike Lowe, Hesse does not say that the (self-)identity is indeterminate. Rather, he says that speaking of (self-)identity is here meaningless. This echoes Schrödinger (1952) who held with respect to entangled particles that
[i]t is beyond doubt that the question of 'sameness', or identity, really and truly has no meaning.
(I've taken both quotes from the article "Quantum Vagueness" by French and Krause in the July edition of Erkenntnis. See there for references.)
Again, it is not clear just what is supposed to be meaningless: the assumption that our two electrons are not identical in the strict and philosophical sense? or the assumption that they have characteristic individual essences? or the assumption that they differ qualitatively? To get anywhere near a challenge of the identity platitudes we have to look at the first possibility.
Unfortunately, the closer we get to a challenge of the platitudes, the more unintellitible the position. (That's why the platitudes are platitudes.) If we have two electrons, rather than one or three, then certainly those two electrons are not identical, as otherwise they wouldn't be two. So how on earth can it be meaningless to say that they are not identical? As far as I can see, the only answer is to give up the claim that they are really two.
I know nothing about quantum mechanics, but this might not be a completely unattractive position: When we describe a helium atom as containing two electrons, this is only loose speaking. Strictly, the electron is in a two-electron-state, where this state is characterized by certain properties, and typically results from a helium nucleus combining with two electrons (hence the name). But however it is properly described, it must not be said that atoms in this state really literally contain two electrons. They don't, and so it is meaningless to ask whether these two electrons are identical.
I think something like this is the view French and Krause defend in the second part of their paper, though they formulate it in a very different way -- in a way that seems to challenge the identity platitudes. Here is what I think has happened.
Almost always when in some discourse certain statements are not to be understood literally one can find (non-standard) formal systems that capture the inferential relations in that discourse. So it is for quantum mechanics with the not-to-be-understood-literally statements about "two electrons" in the helium atom. The formal system is some Schrödinger logic together with quasi-set theory.
In this system, we have terms (names and variables) for "electrons" in entangled states (even though really there are no such things). But we only allow these terms in certain contexts, e.g. in "the helium atom contains two electrons A and B". We want to rule out questions about the identity of A and B, so we stipulate that "=" is not applicable to the quasi-terms. Unfortunately, in classical set theory, the cardinality of {A,B} cannot be two unless not(A = B). So we introduce quasi-sets for which cardinality is interpreted in some way that does not depend on identity being applicable to the elements.
Then, as always, some people spend so much time speaking and writing in this formal system that they forget that they are using the quantifiers, "identical" and "cardinality" in a very unusual way. In the end, some of them say that quite literally there are entities -- quasi-objects, or "non-individuals", as French and Krause call them -- A and B such that A and B are two even though it is not true that they are not identical, because identity is not applicable to them (they are not even identical with themselves).
Do these people challenge my platitudes about identity? No. They simply speak a different language. In my language, what they literally say is unintelligible nonsense, mere noise. The only way that I can make any sense of what they mean is to reinterpret their words, and then they no longer contradict any platitudes about identity.