by David L Duffy

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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. [1] 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. [2] 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”. [3]

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.” [4]

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. [5]

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. [6]

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*. [7] 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). [8]

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. [9]- 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*. [10]

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.

Notes

1) https://en.wikipedia.org/wiki/Exchangeable_random_variables

2) https://en.wikipedia.org/wiki/Identical_particles

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.

5) https://ncatlab.org/nlab/show/Science+of+Logic

6) Black M (1952). The Identity of Indiscernibles. Mind 61:153-164. https://www.jstor.org/stable/2252291

7) https://en.wikipedia.org/wiki/Change_detection

8) Forrest P (2016). The Identity of Indiscernibles. In Zalta EN The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/identity-indiscernible/

9) Neander K (2018). Teleological Theories of Mental Content. In Zalta EN The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/archives/fall2008/entries/content-teleological/#4.2

10) Avigad J, Morris R. Character and object. Review of Symbolic Logic 9 (2016) 480-510 https://arxiv.org/abs/1404.4832

## Comments

## 6 responses to “Some Half-Baked Thoughts on Exchangeability and Identity”

Maybe I don’t understand this correctly, but I feel your approach is ignoring half of the problem. In general we say that x is interchangeable with y when the following logical relationship holds:

R(x) if and only if R(y).

In other words: if x has relationship R with something, then y has this relationship too (and the other way around).

When invertible transformations f(x) are introduced, the question becomes: if R(x), can we say that R(f(x)) holds too?

This is certainly true for some R. In Newtonian mechanics, under a constant, not time-dependent force, the equation of motion x = f(t) has the same relationship with the force F as the equation x=f(t+T). In this sense, t and t+T are interchangeable. Newton’s second law is invariant under translations of time.

But is the moment t the same moment as t+T? Are they interchangeable?

Everything depends on the choice of R. We could take another R, say: the distance in time to moment 0. Then t and t+T are clearly different and not interchangeable.

So it’s not really – or not only – the transformation f(t) = t+T that says something about interchangeability. It’s also the choice of the relationships R.

My favorite example is a pizza. If you cut it in two, you can eat half a pizza. If you cut it in four and take two pieces, you can eat half a pizza too. If R means “how much pizza can I eat?”, both situations are identical. But as a physical object, one half of a pizza is not the same as two quarters.

Hi Couvent. you are right about R(), as exemplified in that first example of total number of heads arising from a sequence of tosses, or in the question of what level of discernment defines indiscernible. To take pizza cutting, if the cuts are fine enough and the pizza not moved during the procedure, it is unchanged under certain types of test (and if the topping is still hot, that will glue itself back together ;))

There is a lot of information to lose yourself in. I need to revisit it later. Thank you.

David.

at the beginning of the Greater Science of Logic, Hegel argues that Nothing and Absolute Being (the totality of existence) are logically equivalent – they are both equally useful – or, rather, they are both equally useless. This should give pause to both Hegelians and anti-Hegelians, but the only philosopher to wrestle with it was Heidegger..

In the Modern era,Analytic Ontology is really as useless (but nowhere near as interesting) as Idealist or Phenomenological Ontology.

The solution to the problem of Theseus’s ship for instance – the question is, what use has it in what context? If you’re engaged in archeology, you want the original ship. If you’re setting a diorama or you want to see what a ship of that type could do, then you build a model – what’s the problem?

Metaphysical Ontology – whether classical or Analytic – presumes that the distance between the individual and the universal is of some profound importance. Actually, it’s of little importance, except insofar as the individual works according to the proposed model.

When I bought a lawn mower recently, the ‘owner’s manual’ promised that if I fit ‘bolt A’ into ‘Slot 2’ my lawn-mower would be ready for use. This didn’t happen. Consulting a mechanic, it was discovered that access to Slot 2 required loosening of Screw iii. What a bummer! Fortunately I wasn’t charged anything for this. .

David;

Very interesting post on a difficult and often confusing subject. I think I’ll go follow every link and reference down whatever rabbit hole it goes. I’m particularly thrilled about Rodin. I have some previous knowledge of category theory, from a mathematics and computer science perspective, and I was thinking about ways to maybe implement that way of thinking into philosophy, so that just hit the spot.

Re: Black’s spheres; I don’t think you are actually permitted to do any transformation. We are invited to consider two spheres, but we are not permitted any means of distinguishing between them. How, then, may we say that there are two? Or that they are spheres? How may sphericity be predicated of an indiscernible? From a categorial perspective, they fail to even satisfy the criterion for objecthood. There may of course be some other sense in which they are objects, but until that sense is clearly explained to me, all I can do is shrug, and move on to something else.

An object a in a category is given by its identity relation 1a ➡️a (“the identity on a is a”); that is, the raw identity of an object as itself is considered not as an abstract universal, but as the feature that identifies it as an object. Note that there may be all kinds of mappings from an object to itself that are not the identity relation. For instance, a symmetry group may be given as a monoid – a category of one object, and mappings the group transformations, only one of which will be the identity on the object. Beyond that, identity may only ever be given up to isomorphism. A and B (that may be objects, or categories, or categories of categories…) are isomorphic iff a mapping f from A to B, and a mapping g from B to A exist, such that f following g is the identity on B, and g following f is the identity on A.

This seems to be the stricter version of what you’re after. But it also seems to say that we must always follow the transformation both ways, to see that we have the identity on A and the identity on B. As you noted, the clay and the statue seems problematic, but let’s see; turning a lump of clay into a statue, and then back into a lump of clay: I would accept that as the identity on the lump of clay. But turning a statue into a lump of clay, and then back into a statue? I don’t think that counts as the identity on the statue; which statue? The first, or the second? Oh, but what if they are indiscernible? But they are not indiscernible, or, rather, only indiscernible in so far as we are allowed ignorance, or allowed to forget what happened in between. It may be that there’s a way of modeling this as pair of what is called forgetful functors, but that’s a can of worms I’d rather not open today.

Categorial reasoning is not a cure for all seasons. Category theory is first and last about the composability of transformations, and its primitive objects a kind of mathematical fiction, completely without intrinsic properties.

You may have noticed that I use ‘categorial’ in the sense of ‘as pertains to categories’ reserving ‘categorical’ for its sense of ‘without exception’.

I would have liked to have posted sooner, so we might have had a better chance of an exchange, but sometimes it’s just one damn thing after another. All the same, though, keep on baking those thoughts!

Hi EJ. Both Rodin’s book and especially Screiber, Corfield and others at

https://ncatlab.org/nlab/show/Science+of+Logic#FormalizationDictionary

spend a fair bit of space discussing how categorical logic (or categorial in Jesper’s vocabulary) maps onto Hegelian objective logic eg

eg “being-for-one” = “infinitesimal flat modality” 😉

Theseus’s ship of course maps onto provenance – van Meegeren had some interesting thoughts about identity and indiscernibility in these cases. In theology, the idea of demons copying goodness sufficiently well comes up, I believe. It pops up in a few favourite novels.

Hi Jesper.

The example of clay statue presumably can be mapped appropriately by some abstractions. Thinking concretely again in terms of information and energy, the appropriate pairwise swap would leave the “total system” evolving in the same way as it would have in the absence of the two transformations, but requires a lot of (actual) work by the morphism. This doesn’t seem quite the same concept as a permutation, which I thinks implies the objects involved somehow retain some kind of integrity. My rule in the examples was that they retain spatial (and functional) integrity, but that may be begging the question about identity. In Couvent’s way of specifying the problem, we carry out the swap, and all the existing relations of x are successfully taken up by y, and vice-versa. The clay-statue successfully passes this latter. So we need to go up one level of abstraction to the “2-morphisms”, to specify which ones represent bilateral relationship preserving for least amount of effort. Otherwise, we end up with a universal oneness that requires maximal effort (I have exceeded my smiley quota).