Relative identity
Out of the different approaches introduced above, the one which presents the fewest flaws (and no logical red flag I can think of), and matches most closely my general outlook, is that of a generous mereological universalism coupled with relative identity. With that in mind, let me add some remarks to further support that contention and flesh out my interpretation of this idea.
Absolute identity
is not denied — in other words I’m subscribing to a weak version of the relative identity thesis (unlike its creator, Peter Geach, who argued that absolute identity does not even exist). I share to an extent Quine’s concerns that without absolute identity, set theory would find itself irreparably damaged, or alternately that conserving absolute identity in set theory would create a wide rift with a physical discourse solely based on relative identity. One way to conceive of absolute identity is quite naturally at the most fundamental level. The identity between a cat and its tail-less self might be relative, but the identity of a wavefunction with itself is not.
As a caveat (and a fairly disturbing one at that), the quantum nature of reality has the potential to wreak havoc with the reassuring assumption that deep down, absolute identity persists. With each quantization we "lose" properties we would had assumed were integral to absolute identity. In 1905 we could conceive of objects as made up of a defined set of elementary particles, uniquely identifiable and with well-defined positions. The first quantization, epitomized by Heinsenberg’s uncertainty principle, makes position indeterminate, which in turn renders a particle’s "individual identity" much more of a torturous notion: if I have a non-zero probability of observing both particles A and B in a given position, how am I to know which one it is I’m detecting? So, after the first quantization we needed to accept that the "actual" physical object with absolute identity is the wavefunction, not the particle. But right when we could have started to feel somewhat comfortable comes the second quantization: particle fields become quantized operators themselves (like position and momentum had in the first quantization), with the practical consequence that "particle occupation states" are now eigenstates of these operators. Concretely, the number of particles of each type is now also indeterminate and subject to probabilistic laws. Again, we can simply state that the field operators are the real fundamental objects to which absolute identity statements can apply, rather than n-particles wavefunctions...
Going forward, we could expect this same reasoning to stay valid ad infinitum: as long as a fundamental theory can be expressed in mathematical language, we can find ourselves committed to the actuality of its mathematical objects (or rather, what those purportedly represent in the physical world), and make that our ontological ur-level. But as we get more and more removed from macroscopic reality, the relevance of absolute identity relations outside of fundamental physics (in which context it amounts to a mere tautology) becomes vanishingly small.
Ordinary objects are equivalence classes
: Given a given region of spacetime and its physical content (let’s say: my room \(R\)), one can build the set of physical objects (all possible mereological fusions) from that region: \(\mathcal{M}(R)\). Each relative identity relation acts upon pairs from that set. As explained above, if \(x\) and \(y\) are some mereological fusions, then \(x=_Sy\) ("\(x\) and \(y\) are the same \(S\)") can only be true if \(x\) and \(y\) are both examples of \(S\), where we had described \(S\) as a "type of thing". So, to state that "there is a table in the room", I need (1) a definition of the type "table", (2) a definition of the relation \(=_{table}\) and (3) at least one physical \(t\) such that \(t\) is a table. From (3) we know that \(\mathcal{M}(R)_{table}=\) \(\{x \in\mathcal{M}(R) | x \textrm{ is a table}\}\neq \emptyset\), and therefore that \(=_{table}\) can be well-defined as an equivalence relation over \(\mathcal{M}(R)_{table}\). An ordinary table \([t]\) is then the equivalence class over \((\mathcal{M}(R)_{table},=_{table})\) of which \(t\) is a member. The set of tables in the room is thus the quotient set \(\mathcal{M}(R)_{table}/=_{table}\).
Ordinary objects are not physical
: this is a direct consequence of what precedes — equivalence classes are not physical things. That seems to endanger the whole conceit: we want ordinary objects to have material properties, and for ordinary statements like "my dog weighs 40 pounds" or "I am touching this dog" to make sense. The only way out is to just accept that these ordinary properties don’t apply to physical objects but rather to ordinary objects: just like ordinary objects exist within the quotient set, these properties are functions over the quotient set rather than over the initial set.
What does this concretely mean about my dog’s weight? Physical objects \(p\) have a mass \(m(p)\) (let’s forget about quantum uncertainty, and suppose this is perfectly determinate at a given point in time). For most ordinary objects, their mass is intuitively thought out in maximal terms: it may be that my dog minus his hair or minus his tail would still be the same dog, but when I want to weigh him, he needs to step on a scale: everything that’s attached to him counts. Similarly, a table minus one of its leg may still be the same table, but the table’s mass is conventionally that of the maximal object in the equivalence class.
The "ordinary mass" of a table could then be defined as: \(\widetilde{m}([t])=sup\{m(t), t \in [t]\}\). Again, the relative identity concept allows for great flexibility: this maximal version of ordinary mass \(\widetilde{m}\) is close enough to the way humans speak to be acceptable for most ordinary objects, but unlike \(m\) it need not be universal.
Take the merino sheep: its typical weight ranges from 60 to 80kg, but its wool alone can grow to be more than 25kg. If the sheep got weighed by a veterinarian before and after shearing, would the vet be alarmed that the animal lost 25kg in a matter of days, a third of its weight? He likely would if he made the same observation on other animals, but not on a merino; it then stands to reason that vets may use a different function \(\widetilde{m}_{sheep}\), the maximal mass over all sheep-without-wool in the equivalence class. With ordinary properties we are not beholden to one-size-fits all definitions, which would be a poor match for our experience of ordinary objects.
The second example may seem trickier: how can I see or touch a dog if it’s not physical? The answer is that I can’t — I only see or touch mereological simples assembled in certain way. Right now I’m touching the table with my finger. I’m in physical contact (however I define this) with a set of mereological simples. For any mereological fusion that includes at least one of those simples, one may say I’m touching it. If one of those belongs to the equivalence class "table", then I’m "touching" the table. The relation "touching" is thus formally distinct whether it applies to ordinary objects or to physical objects. As regards the puzzle of coincident objects, we wondered what exactly I’m touching while caressing a sheep: an animal? some wool? Here we recognize that this puzzle, like others, stem from a conflation between terms relative to the physical world, and terms relative to the quotient set of ordinary objects. I am physically touching a well-defined set of mereological simples, but I’m "ordinarily" touching a sheep and wool.
Ordinary objects are arbitrary
: This construction of ordinary objects solves two major problems we had identified with mereological conservatism: Sorites puzzles and object coincidence. There’s a third one, however, that won’t go away that easily: arbitrariness can hardly be avoided.
The nice mathematical veneer of equivalence classes and quotient sets will not have fooled an astute reader: in that model, the actual definition of an object type, its identity conditions, its specific properties, do very heavy lifting, and they are largely arbitrary. This means that trouturkeys and incars have as much of a claim to "ordinary objecthood" as dogs and islands. Is this all for nothing then, should we have just stuck with base universalism or nihilism? Not so: the framework I’ve chosen admittedly doesn’t help distinguish "our" ordinary objects (let’s say: those with an entry in dictionaries) from the rest, but it formalizes the manner in which we identify them and explain their properties — something that naïve identification with physical particulars didn’t allow.
Ordinary objects are abstract
: This arbitrary nature, the type of examples we’ve picked, and the term itself may suggest that ordinary objects strictly belong to the lived-in Umwelt of dogs, trees and cars. This could lead to a form of object psychologism: perhaps there are only things arranged dog-wise, and the dogs only exist in our heads. The social- and mind-dependent nature of so many ordinary objects can only encourage us to explore that option.
Coming back to the most famous example of all: you need humans to sculpt a statue, but you also need humans to recognise it’s a statue. That there’s only a pile of things arranged statue-wise, while the statue is all in our head, seems only an inelegant way of saying that "beauty is in the eye of the beholder".
Yet we can draw from similar discussions around natural kinds and wonder if there aren’t natural ways to define ordinary objects. If "dog" is a natural kind, it could well be that "being the same dog" is a natural relation. Staying in the natural realm and going further from our daily experience, there are the objects of our scientific theorizing (this is where the terminology could mislead: we don’t think of quasars or baryons as ordinary in any way). The relative identity concept is a very good fit for the kind of model-based approach used in every science. Take these two objects: "solid Earth", the core, mantle and crust of our planet, and "blue Earth", which includes those parts as well as the hydrosphere and atmosphere. Are those the same planet? Geologists would give a resoundingly negative answer. Most of their observations would be inexplicable without a hydrosphere and an atmosphere. Planets without them, like Mercury, are not geologically close to Earth at all. Astronomers, though, would say it is essentially "the same planet". At Solar System level, astral bodies can be represented as homogeneous spheres, or even as concentrated mass at their center. At such scales, Mercury’s lack of water and air is irrelevant. If Earth’s atmosphere disappeared and its water was absorbed into the crust, it wouldn’t change its trajectory around the Sun one iota. To be precise we should thus introduce two relations: "to be the same geological planet" and "to be the same astronomical body". In the latter, equivalence classes each contain a special representative that is just a pointlike mass.
This example is a bit of a caricature. I’m well aware of the fifty shades of grey here: many astronomers work at small enough scales that talking only of pointlike masses makes no sense; rigid distinctions between disciplines are artifical, etc. But the key point is the capacity of the ordinary object concept to accommodate a scientific viewpoint: science aims to encompass the whole richness of the physical world, but scientific theories only present a schematic view of it. Reversing the argument we could also say that our daily experience of the "ordinary world" is itself a form of folk theorization. The scientist saying "the Solar System is made up of 8 smaller spheres in orbit around one larger sphere" is performing the same mental operation I do while entering a room and thinking "there are three objects here: a ball, a chair and a table").
Does this suffice to prove that ordinary objects are not just mental representations? It doesn’t in itself, but going further in that direction would anticipate to much: we shall later discuss abstract objects. We’ll then argue that ordinary objects are not physical, nor mental, and are therefore abstract.
Ordinary objects are a bridge between the physical world and abstracta
. The Solar System example is a good illustration of this idea. Imagine a spectrum with, at one extreme, the full richness of the physical world: a grain of sand on Mars, crystals on an asteroid, spacetime curvature around the Sun, quarks within solar hydrogen... it’s all there. At the other extreme let’s put our astronomical model with 9 point-like masses governed by Newton’s laws— a bare mathematical object with staggeringly little information in comparison.
I contend that ordinary objects are a natural bridge between these contrasting descriptions of the world. The spectrum starts with a gap, since ordinary objects, as we’ve seen, are abstracta and are thus different in kind from the physical view. At that extreme, the asymptotic ordinary earth carries all the information the physical earth does, enough to explain any phenomenon we can experience at its surface. Moving on the spectrum, we encounter more stylized versions of ordinary Earth: for instance, a geophysicist’s would be a relatively simple ball in comparison, with some major layers governed by fluid mechanics. Along the line, we see the distinct features and staggering complexity of the physical world recede into the ligne claire of mathematics, until we reach the astronomer’s pointlike mass.
This approach opens the door to more unusual ordinary objects. Even though we’ve seen there can be many "ordinary Earths", the idea of Earth as an object is not bound to surprise us. But consider objects tackled in other scientific theories: for instance, social institutions like Parliament. It is quite straightforward to note that there can be no strict description of Parliament in terms of wavefunctions or spacetime lattice. Social institutions are abstract concepts; being thought of and talked about by humans, they end up having an enormous impact (at human scale) on the physical world. They also have a more direct footprint in that they can be instantiated: for Parliament, that could mean a building (the Halls of Parliament) and all human beings bearing the title of MPs (note that this is entirely conventional — we could have picked the building only, or the humans only, or something else altogether). We can use that convention to turn this instantiation into an ordinary object. This would allow to make statements in such a way that the fusion of the old palace of Westminster as it stood in 1714 and of all Members from that year "is the same Parliament as" the current building plus the set of MPs from 1979. This allows to cut a maximal slice of spacetime (the contour of this ordinary object) that, according to our arbitrary convention, represents the physical imprint of Parliament in the universe. Whether that is useful is an open topic...