There are a number of paradoxes that come up with commonly used definitions of indiscerniblity and identity. I think these can likely be resolved if we stop assuming a "God's eye," view of the world, where observations happen from nowhere/everywhere and see everything, and instead allow our metaphysics to be informed by the empirical evidence against this "view from nowhere," or "view of things in themselves, as they are absolutely."

My argument is that this view necessarily muddies the waters and leads to paradox because it posits a magical viewpoint without an observer, which is arguably something inconceivable to us and leads to nonsense phrases in our languages. It's also a view that contradicts our knowledge from the natural sciences.

We would like to say that identity is "the relation everything has to itself and to nothing else. But this is circular, since “nothing else” just means “no numerically non-identical thing”. It can be defined, equally circularly (because quantifying over all equivalence relations including itself), as the smallest equivalence relation (an equivalence relation being one which is reflexive, symmetric and transitive, for example, having the same shape)."

In order to improve on this circularity, the common criteria for identity are as follows:
1. Reflexivity: x = x
2. Liebnitz' Law: if, for every property F, object x has F if and only if object y has F, then x is identical to y. Or in the notation of symbolic logic: ∀F(Fx ↔ Fy) → x=y.
3. The Necessity of Identity: That these relationships are necessary. If y = x, this is true by necessity
4. Necessary distinctness: if Y is not x then this is true by necessity.

Identity is normally taken in two ways, as existing in one frozen moment in time, and as existing throughout time.

An initial paradox the above definition runs into is that of change over time. Take a pet dog. We would like to say it is the same dog over time. However, the old dog has many properties that the puppy does not. A common way around this is to assume that properties are related to a specific time. So the identity of dog D has certain properties at time T, when it is a puppy, and time T' when it is an adult, but the identity is all the properties the dog will have. Easy enough.

But the problems become more acute in other examples. Take for example a lump of clay that is formed into a statue. On the following day, the statue is destroyed as the clay is rolled back into a ball. The clay is then used to make a second statue.

Clearly the clay has different properties at different times. The statues presumably have significantly less density than the ball for example. We can say our clay has the property of being the first statue and then the second statue at times T and T', but what about the statues? They appear to have an identity that is created and destroyed. But if the statues have an identity, and the clay has its own identity, then you have two objects occupying the same area of space at the same time. If the statues are really just the clay, and lack their own identity, then are we saying what things are made of determines identity? This seems like a problem because humans replace the vast majority of individual atoms in their bodies several times over their life cycle, which would mean a person's identity is always changing. It would also mean that your car is not the same car after a part is replaced.

Parts of objects also cause a problem. It seems intuitive to say that a statue of a woman missing an arm is the same statue, it has just been broken. However, if we bisect the statue, it is now totally unclear which half is the original statue. Have we created two new identities? And if we have, what if we roll the clay from the two halves back together? Has our original clay's identity ceased to be when the statues were broken in half, but returned when we rejoined them?

These examples are all on the SEP article on relative identity. I just wanted to summarize them as typical examples.

Here is my own, which gets at my point more:

Imagine a loved one has been abducted by aliens. The aliens set about reorganizing their brain. They do not add new materials to it, so your loved one's body continues to have the same constitution, it's just that some structures have been rearranged. These rearrangements were done in such a way that your loved one now has an entirely different set of memories, entirely different preferences, and an entirely different personality. However, they still look the same, and the molecules making up their body haven't changed any more over the week they've been gone than they would have had they been living on Earth.

Is the person who comes back your loved one? I think most people would say no.

But let's look at a more common occurrence. Let's say your computer gets a particularly brutal virus. It can't boot up, all your information is being compromised, the only step that will fix it is a factory reset. So you bring it in to a repair shop. They make changes to your computer, but it is largely made up of the exact materials it was made up of when you dropped it off. All that has changed is that some work was expended to erase stored information. Your computer now boots up, but with none of the old files. Is it the same computer? Generally, I think most people would say yes.

Likewise, we tend to think that if our dog has his tail chopped off, our dog still exists and the chopped off tail exists, but is not a dog. When we see our cat on a mat, we suppose there is one cat, not a cat for every hair the cat has, 10,000 cats in the same place. These wacky solutions, thousands of cats in one, tails as dogs, etc. are only dreamed up by philosophers who realize that our common conception of identity is a mess to put into logic. Indeed, identity can't even be defined in first order logic, although Quine had argued that in a first order language indiscerniblity is identity.

Now for my favorite example, because it is so strange. Imagine a universe containing nothing but two large, completely identical glass spheres sitting two miles apart. Do we have one sphere or two? The two are identical and can only be defined as different by reference to the other identical sphere.

One proposed solution here is "that such a completely symmetrical situation of two spheres could be re-interpreted as one sphere in a non-Euclidean space. So what might be described as a journey from one sphere to a qualitatively identical one 2 units apart could be redescribed as a journey around space back to the very same sphere."

There are problems here too though. Imagine three spheres in a row. The same arguments could be made about it really just being one sphere. But now imagine the middle sphere is slightly different, it has a scratch. Now we have to have three spheres. But the scratch is something that is contingent, which leads to the strange case where three perfectly identical spheres are actually one sphere, until one is scratched, and then three spheres emerge.

My own addition here would be that objects in universes with flat or curved space are differently discernable because space can play a role in the discernment of objects. Imagine we are an astronaut plopped into this strange universe in the middle of the two spheres. We see two spheres, one to the left, and one two the right. These are different for us. We can fire up our jet pack and go to one of the other.

If space curves back on itself, our experience moving several miles to the right would be seeing a continuous number of identical spheres pass by. We would have the experience of seeing infinite spheres, as we loop back on our old position. Thus, the one sphere argument holds. If space is flat, after we pass one sphere we head out into an invite void. Upon turning around we would see two spheres at a distance, not one. Space thus defines objects. Space is not just a receptacle/container ala Newton, but a description of relationships ala Whitehead.

Now, let's go back to the universes with space that loops back on itself. Here we have an infinite series of the same single sphere. But say we drop a beacon above one of the spheres after traveling in this loop for several rounds. Now, if there were actually always two spheres, we discover that the spheres are different. We will still experience an infinite number of spheres if we get in one direction, but now we will see our beacon flashing above every other sphere that we pass. We can thus confirm that there are two spheres and that we are moving in a loop (or else that this universe has the weird property of duplicating things left around a sphere infinitely). What can be described as one sphere by an external observer, looking at the universe from the outside, can in turn be distinguished by an actor within the universe to be two spheres, but only if a relationship can exist that allows for differentiation. Identity depends on relations to other things, not a thing's relationship to itself.

The interesting thing is that our astronaut's experience of the universe is consistent with the single sphere hypothesis at first. However, as soon as he drops a beacon or scratches a sphere, there is now more than one sphere. Since we don't want to claim that the astronaut's action created a second sphere, we want to say there was always two spheres.

And indeed, the ability for our astronaut to interact with and identify the number of spheres in the universe is dependant on the possible relations the astronaut's actions can have relative to the spheres. Scratching a sphere in a one sphere universe is a different relationship than scratching a sphere when there are multiple spheres. If we posit just one sphere in a space that loops back on itself, dropping the beacon will produce different results, as our astronaut sees the beacon in the same place relative to the sphere on every pass through, instead of seeing the beacon stop one sphere in every two.

But, try the same thought experiment with a magical mote that can observe the spheres but cannot interact with them. Now there is no possible way to distinguish a one sphere and two sphere universe. Always and forever, everything that can be said about a one sphere scenerio can be said about the two sphere scenerio, for all observations occurring inside the universe. Saying the two would still be different requires positing some sort of absolute standpoint that "sees things as they are." But as we've seen in the objections before, the one and two sphere universe can also be described as the same universe from an external frame, so this absolute viewpoint doesn't seem to exist anywhere except in our heads. It is a viewpoint of pure brute facts we've dreamed up.

The points here are that the shape of space matters, as it defines relationships, and that relationships between things can alter identity and indiscerniblity.

The problem with the absolute standpoint is that:
1. There is no way to tell if you have reached it.
2. Physics suggest that this sort of viewpoint is impossible as only a non-physical entity could aquire a magical "view from nowhere."
3. It is unclear if talking about the existence of things that no observer can observe is coherent.

For an example of point three, imagine that we actually have two types of protons in our universe. Regular protons, and "notons." These are completely indistinguishable because of the way our laws of physics work. However, had the symmetry breaking that gave rise to our universe's physics gone ever so slightly differently, the "noton" would have a slightly different charge from the proton, because it is in fact different, we just can't observe that it is.

At the view from nowhere level, the noton and proton are different. They appear the same to all observers, always and forever, but this is merely contingent. They do in fact differ in a way that is unobservable.

Is the "noton" a coherent idea? Or does it make more sense to say "the identity of protons was contingent on circumstances early in the universe, and while a noton was possible, it never became actual."

What if, through some unknown mechanism, the laws of physics change. Suddenly, the difference between the noton and proton can be observed, but the difference is small enough that it doesn't cause radical changes to our world. Would we say that notons were created at the moment they became discernable from protons, or that they must have always been different? If you take the God's eye view, and assume the difference could have always been there, there are two different possible scenerios here, one where a new identity is created late in the history of the universe, another where a different identity only became observable later in the universe but was always there. But always and forever, for all observers, both scenerios are identical. This seems like a strange example, but some takes on the unification of fundemental forces at high enough energy levels say something like this has occurred.

My argument is simply that the plethora of paradoxes emerging from the concepts of indiscerniblity and identity, and the counterintuitive solutions to these paradoxes (which still fail to resolve them), shows there is something deeply flawed with how we are thinking of the concept. If the "noton" can exist as distinct from the proton because deep down, seen from some God's eye level, it truly is different, but for all observers it appears the same as a proton, then it seems we can posit an infinity of differences that don't make a difference. That the unique identity of identical subatomic particles in quantum mechanics, our best theory for how the world works, is generally denied, should be another strong indication that the concept is flawed.

It appears that the parts of which all physical objects are constructed lack concrete identity, any essential "thisness" or haecceity. So why do we claim "classical sized objects," have this property? It seems more likely to me that identity is the result of our perceptual system. Probabilistically, classical sized objects tend to exhibit something like identity. They are indeed constantly changing, but not enough to make a difference to us.

While an object may change substantially over time (e.g., a young dog and an old one), they do not tend to change that rapidly from our perspective. When things do change rapidly, like a statue being split in half, we end up with paradox and confusion. We have assumed some invention of logic will solve this issue, but it seems to me that it is much more likely that the concept of identity is tied to how evolution shaped our perceptions, not to anything real.

We have tried to remove relational/derived properties from identity because we don't want things' identity changing when they move around, but the sphere examples show identity is deeply tied to relations.

It seems to me that a better model would look at indiscerniblity as defined within given systems or for given observers. This has the deficit of being arbitrary, but the absolute conception has the deficit of giving rise to the possibility of infinite noton-type objects, or being incoherent, larger problems IMO.

Fears of unrestrained relativism need not hold us back. We can have a weaker definition of absolute indiscerniblity and identity by positing these as the sum total of all actual relationships that occur between a thing and other things. This still has the problem that "objects" and "systems" are defined in seemingly arbitrary ways, but that's a problem for another day.

Indiscerniblity can be defined using information theory as two messages that appear identical for a single recipient. For example, the two "ts" at the end of "that" and "that" are discernable because you see two copies of the word "that". But the fact that I erased the t at the end of the second "that" and retyped it four different times does not change that t's discerniblity for you, despite its different history. Indiscerniblity and thus identity would then depend on the channel of an individual message.

It is odd to think of physical interactions as messages, but let me show you how two things can be indiscernible in a simple physical interaction. Imagine a giant ball of cotton candy in space and define this as a system. Now imagine shooting balls at it. These are balls of identical volume and with identical energy. The balls vary slightly in chemical composition. This isn't discerned by our system. The balls blow a hole through the cotton candy all the same. All that can be discerned is their various trajectories and the paths they take through our system. Information about their paths shows up in our system, in the form of holes, but not information about differences in composition. And, if we shoot similar sized balls through holes we have already made, we actually don't have any information exchange. Our system is unchanged.

Meanwhile, if our system is made up of ballistics gel, the balls get stuck in the system, and information about their composition becomes information within the system. Likewise, if one ball has a composition that varies from the others and which causes it to explode on contact with cotton candy, then composition can change the information exchanged.

We can still formalize indiscerniblity and identity under a relative metric, we would just do it looking at the level of individual channels. My dog stays the same dog as he ages because the channel of my perceptual system carries enough information about his difference from all other dogs and objects that I can keep him in mind as a distinct entity. Difference over time, change, and relative similarity can be handled with the idea of morphisms from category theory. This post is already too long so I won't get into that here.

This form of relative identity seems like it would resolve the afformentioned paradoxes. Absolute identity would only hold for abstract objects. While it would impossible to know in full the absolute identity of any one thing when that absolute identity is defined as the sum total of all the information it exchanges, all its relationships, this is also true of the God's eye absolute definition. There is no way to know when you've hit rock bottom.