Some Half-Baked Thoughts on Exchangeability and Identity
by David L Duffy
Exchangeability is a concept that is closely bound up with ideas of individuality and identity. In probability and statistics, it is a crucial formalized idea that underlies whole areas like Bayesianism and subjective probability.  In the mathematical context, exchangeable events or variables can be permuted without altering the properties of the system in which you are interested. A classic example is the number of heads out a series of coin tosses. You are not interested in what order the successes appear, just the number, so any pair of events could have been swapped in the sequence without altering the relevant outcome.
In the statistical mechanics of quantum physics (QFT), the exchangeability of fundamental particles is more than just a matter of the interests of the observer, it is an essential property of matter.  As you might guess, exchangeability is a type of symmetry relation in physics and mathematics.
The concept of a sortal in philosophy has a similar flavour, in that there is numerosity, with the things being counted identical to each other in respect of the properties used to define the sortal, e.g. “three brown dogs”. 
So, can two entities be exchangeable without being identical? That is, is it a weaker concept than being perfectly alike? Obviously, one would have to say “yes,” but then again it is hard to think of any pairs of things that are perfectly alike, as opposed to identity being purely self-reflexive. I have recently been enjoying Rodin’s Axiomatic Method and Category Theory, where in Chapter 5 he runs over some of the complications of defining and using the identity relation. He comments:
We see that Plato, Frege and Geach propose three different views of identity in mathematics. Plato notes that the sense of the “same” as applied to mathematical objects and to the ideas is different: properly speaking, sameness (identity) applies only to ideas while in mathematics sameness means equality or some other equivalence relation. Although Plato certainly recognizes essential links between mathematical objects and Ideas (recall the ideal numbers) he keeps the two domains apart. Unlike Plato, Frege supposes that identity is a purely logical and domain-independent notion, which mathematicians must rely upon in order to talk about the sameness or difference of mathematical objects, or any other kind at all. Geach’s [Theory of Relative Identity] has the opposite aim: to provide a logical justification for the way of thinking about the (relativized) notions of sameness and difference which he takes to be usual in mathematical contexts and then extend it to contexts outside mathematics…”Any equivalence relation … can be used to specify a criterion of relative identity.” 
Rodin then introduces what he variously describes as a constructivist (in that one specifies a procedure) or substantialist interpretation of the identity x = y: that there is an invertible transformation from x to y and y to x. His discussion now segues into categorical logic and category theory (which some folks might be impressed to see is explicitly Hegelian), but I will consider an analogous process in a simpler non-mathematical way for exchangeability of x and y in a broader context. 
Specifically, I am thinking of x and y as separate objects or events each sitting within their relationships with everything around them, and we perform symmetrical transformations of x to y and y to x. Let’s say these transformations are simple translations (movements) in space or time. So I might swap one coin for the other coin in two separate sequences of coin throws. In the traditional statistical setup for subjective probability, I might swap coin tosses from different times in a single sequence of throws. And if I were Max Black, I might have swapped the two identical spheres that are all a particular universe contains. 
So at the practical level, no one involved may notice there has been an exchange, or possibly there were obvious physical changes in the object of attention, or subtle changes in behavior detectable by prolonged observation, e.g. the statistical properties of the sequence of coin tosses in a change point model.  I might consider how much work is required to detect my substitution in an informational sense. The Leibnitzian idea of identity of indiscernibles implies that there is an absolute true state of nature (I won’t go off into quantum mechanics again), and the observer doing the discerning can spend an infinite amount of time and energy getting to the bottom of things while taking no time at all (like one of those hyper-computers). 
Is there a point to this? I see it as a nice way to think about various hoary thought experiments regarding identity:
- In the case of Black’s two spheres we can’t tell apart, I think it is safe to say they are exchangeable. This is actually tricky because of the underlying idea of the setup is that there are no reference coordinates by which we might specify which sphere is which, but I will merely say that a swap is to be performed.
- Those two versions of Theseus’s ship would be pretty hard to tell apart unless you looked at the ages of the different components. So they are exchangeable up to a certain level of discernment.
- How about I swap those one of those two versions of Theseus’s ship backwards in time with the original. Again component age would be the argument against exchangeability.
- I am put into the Star Trek transporter and accidentally doubled. If the two copies are immediately then swapped around by a further transporter malfunction, not even I would know what had happened.
- Swampman – exchangeable. 
- If I was swapped with myself 20 years ago, I think many observers would be able to work out that an exchange had been performed, and not just me myself. But over a few seconds, I and everyone else might just diagnose absent-mindedness or deja vu. More so for situations where I have retrograde amnesia, say following a head injury where there was no discernible physical brain damage.
- Mental uploading: this is essentially the same as Swampman. That is, if it were possible to live a simulated life that parallels the life of the physical individual, then exchangeability is one test of the adequacy of the simulation procedure.
So is exchangeability just another term for identity or equivalence in these examples? I find it useful because of its constructivist quality. That is, it specifies what action is to be taken to demonstrate that two entities are exchangeable. Time travel might be a little hard to realize – in the statistical settings the permutation is epistemic (and counterfactual) rather than ontological. More generally, one could object that many entities can be transformed into one another given enough work, without being even close to identical – consider a blob of clay and the resulting statue. Recall, though, that most of my examples were translations affecting each part of the object in the same way. More abstractly, mathematicians do often think of a mathematical object as that which is invariant under all its different possible representations. 
That the testing of equivalence is by way of checking the effects of both sides of the swap is an interesting feature that again speaks to a pragmatic definition rather than just a purely a priori attempt to specify identity or equality, for certain values of pragmatic. It seems to lend itself to interpretations of these various thought experiments that strike me as commonsensical. Where it can’t actually answer the question that two entities are actually exchangeable, it can suggest forms of test one might use in the future. Finally, in the literary sphere, there are all those stories starting with Twain’s The Prince and the Pauper (The Prisoner of Zenda, Double Star, Trading Places, Freaky Friday) that use this thought experiment to demonstrate human equality.
3) Richard E. Grandy (2016). Sortals. In Zalta EN The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/sortals/
4) Rodin A (2012). Axiomatic Method and Category Theory https://arxiv.org/abs/1210.1478
Also in hardback 2014, Springer Synthese Library.
6) Black M (1952). The Identity of Indiscernibles. Mind 61:153-164. https://www.jstor.org/stable/2252291
8) Forrest P (2016). The Identity of Indiscernibles. In Zalta EN The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/identity-indiscernible/
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