The constitution question
The structure of a physical ontology is not exhausted by its mereology. In discussing the problem of composition, I’m clearly leaning towards mereological universalism. But I’ve performed a big sidestep to avoid considering constitution and identity right away.
What is constitution?
Mereology makes stipulations about what objects we can legally combine to make more objects, but is not too concerned about the nature of the result. That’s what constitution is about.
It seems reasonable to argue that if I have pieces of iron, then welding and bolting a bunch of them together would result into a bigger piece of iron. It so happens that in Paris, there is a specific assemblage of iron girders that most people consider to be a definite object named "the Eiffel tower". Yet the mereological sum of these same iron girders and iron bolts also makes up a big piece of iron. That doesn’t seem to be a problem since most people would agree the Eiffel tower is indeed a big piece of iron. But are we so sure about the relationship between "big piece of iron" and "Eiffel tower"? This is what we’ll explore here.
The main question about constitution is whether it is the same thing as identity...But what is identity? It is most often characterized by these two properties:
(Reflexivity): \(\forall a (a=a)\)
(Substitution): \(\forall (a,b) \forall F (a=b \rightarrow F(a)=F(b))\)
In the above, \(F\) represents any sort of expression. In a philosophical context, the substitution property is often called the indiscernability of identicals (thereafter I.I.), or Leibniz’s law. One corollary is that two identical objects share all of the same properties: if \(a=b\) and a given property is true of \(a\), then it is true of \(b\) as well.
From this characterization we can deduce more corollaries. Let’s call \(T(\cdot)\) the truth-value function, and define \(F(\cdot)=T(b=\cdot)\). It follows that:
(Symmetry): \(\forall (a,b) (a=b \rightarrow F(a)=F(b) \rightarrow T(b=a)=T(b=b)=\text{True} \rightarrow b=a)\)
(Transitivity): \(\forall (a,b,c) (a=c \text{ and } c=b \rightarrow F(a)=F(c) \rightarrow T(a=c)=T(b=c)=\text{True} \rightarrow a=c)\)
Reflexivity, symmetry and transitivity, mean that identity is by definition an equivalence relation.
Venus and the clay
Since ancient times, the problem of constitution has been discussed through a clichéd example: that of a statue; philosophers soon remarked that a statue differs in its properties from the brute matter it is made of.
Let’s set up our puzzle. On Monday, a sculptor bought a big lump of clay — we’ll call it "Clay". He brings it to his studio, and on Tuesday sculpts the clay into a representation of the goddess Venus — let’s just call the resulting statue "Venus". It’s pretty easy to find ways in which these two differ. First there’s a difference in kind: Clay is a lump, while Venus is a statue (of course this statement only makes sense if one wants to believe in kinds; if not, it’s fine to ignore this one). Another difference is in persistence conditions: Venus would stop existing if it was squashed multiple times with a hammer, while Clay would perdure. We’ll focus on another difference in particular: Venus did not exist on Monday, but Clay did.
We can then propose the following argument:
(VC1) Venus did not exist on Monday.
(VC2) Clay did exist on Monday.
(VC3) If (VC1) and (VC2), then Venus \(\neq\) Clay.
(VC4) So, Venus \(\neq\) Clay.
Let’s consider the possible reactions one by one:
Accept the conclusion: I’ve been careful to call this story a puzzle and not a paradox. There is no logical inconsistency in accepting the premises, the argument, and therefore the conclusion that Venus and Clay are not identical objects. The question that springs to mind is, what differentiates them, that they differ so much in properties while having the same weight, shape, components, etc. One ever-popular response is what Aristotle called hylomorphism: the Stagiryte argued that every object had material parts, and one immaterial part: its form.
Now, there are three main cases.
Venus and Clay both exist on Tuesday. In that case we have two different objects, sharing all material properties and with the same parts, occupying the same region of spacetime: coincident objects. On Tuesday, Venus and Clay share material and temporal parts, but have different forms.
The main issue with this view is profligacy rearing its head again. It’s one thing to be a mereological universalist and having to accept "high-level" infinities of objects. It’s another running the gamut of all possible forms, leading to scores of coincident objects all over the place. Where a cat is sitting, there would coincidentally be a lump of organic matter, a set of living cells, a pet, and many more.
Venus does not exist on Tuesday. That’s the nihilist view: if Venus doesn’t exist, then it seems hard to argue the lump kind is somehow privileged compared to the statue kind; all we have is a pile of mereological simples arranged lump-wise, then statue-wise.
Clay does not exist on Tuesday. In that case, Clay’s material parts persist, but the Clay-form has not survived the act of sculpture and has been replaced by a Venus-form. That is close to Aristotle’s view and is fine on paper, but carries unpalatable consequences. We’d like to say that Venus is a statue and a lump of clay at the same time, but that fails as some properties are incompatible (statues don’t survive being squashed, lumps of clay do). So, sculpting truly destroyed the Clay-form, even though sculpting does not appear to be something that could do that as obviously as, say, throwing it into the sun. Conversely, squashing Venus would somehow make the Clay-form pop back up into existence.
All of this becomes even trickier if you imagine the more subtle case of an artist who takes a rock and presents it to an audience as an artwork. Aesthetic conventionalists would agree the rock is now of the kind "art piece", but hylomorphists amongst them would need to agree the rock has disappeared. It doesn’t seem right that a speech can make a rock disappear; logical, perhaps, but unpractical as a model of object constitution that puts a premium on intuitiveness.
One way out of this is a theory called dominant kind. In that view, Venus is a statue and a lump of clay at the same time, so the Clay-form remains; the puzzle of inconsistent properties is solved by saying that "statue" is the dominant kind, therefore whenever there can be a conflict, Venus has statue properties rather than lump properties. This is quite attractive and more akin to how we think of objects, but difficult to implement for one basic reason: it’s not always as easy to define the dominant kind as in this textbook case.
Deny VC2: Again, this is the mereologically nihilist point of view that lumps of clay don’t exist (only mereological simples arranged lump-wise). Any constitution puzzle thus dissolves, since they are about objects, and objects don’t exist.
Deny VC1: How can Venus exist on Monday if that statue was only sculpted on Tuesday? Only proponents of mereological constancy have argued as much, assuming that Venus existed already insofar as all of its parts existed too. This turns out, in my opinion, to be a pure verbal game: if Venus existed on Monday then it doesn’t have any of the properties we associate with statues: that they need to be sculpted, that they cannot survive being squashed, that they are recognised as art pieces or representations, etc. Therefore Venus is totally identified with Clay.
This is another way of saying that statues can’t exist, and of circling back toward mereological nihilism (or to a constitution-less universalism that only admits raw compositions as objects — as we’ve already pointed out, nihilism and universalism are often two sides of the same coin).
Deny VC3: VC3 is a direct corollary from the inderscinability of identicals (I.I.). Denying it forces us to adopt an alternate concept of identity. It is precisely what Peter Geach proposed with relative identity.
From that perspective, it doesn’t make sense to make such statements as \(x=y\) or \(x\neq y\), instead we may be able to say that \(x=_Sy\), read as "\(x\) is the same \(S\) as \(y\)", where \(S\) is some type of thing. The relative identity relation \(=_S\) does not follow (I.I.), but it is (by definition) an equivalence relation. Furthermore, to say that \(x=_Sy\), one must first verify that "\(x\) and \(y\) are both \(S\)".
How does this help with our conundrum? Let’s start back from the beginning, where Clay is the object we start with on Monday, and Venus is the object we get on Tuesday after sculpting. Clay is a lump of clay, Venus is a lump of clay too. We can state "Clay and Venus are the same lump of clay". However Clay is not a statue, so it cannot be "the same statue as Venus".
We don’t have any issue with coincident objects here: on Tuesday there is only one object, Venus. However we also don’t have to deal with as much counter-intuiveness as before: sculpting the clay didn’t destroy Clay — Clay persists on Tuesday through the existence of Venus, they are one and the same lump of clay. We use a different proper name because Venus is a statue, so "not the same statue" as Clay. Throughout the week there’s been exactly one lump of clay, one statue, and no coincident objects at all. Difficulties with persistence properties also seem to dissolve; the property "can survive squashing" doesn’t make sense by itself since it appeals to a general concept of identity. One may say that Venus can survive squashing as a lump, but cannot survive squashing as a statue, so that if the result of Venus’s squashing is Clay-2, Clay-2 is the same lump as Clay and Venus, but not a statue at all.
If we imagine a scenario where a new statue is sculpted and then squashed ten times over, we will have had one lump and ten different statues in that room, which aligns well with intuition.
More puzzles
The debtor’s puzzle: One of Epicharmus’s comedies tells us a joke. Achilles owes Atreus a debt and is approached by him at a banquet. He tells him: "One man is growing, another is diminishing, and all are constantly in the process of change. You and I are different today from who we were yesterday, and by the same argument we will be different again and never the same in the future." Atreus agrees. Achilles then concludes that he is not the same man who contracted the debt yesterday; in that case he can hardly be held responsible for the debt. Atreus, exasperated, strikes Achilles a blow. Achilles protests, but Atreus points out that by now he is already quite different from the man who struck the blow a minute ago.
The type of puzzle time goes back a long way, arguably all the way to Heraclitus’s remark that one cannot bath twice in the same river. The simplest way to expose it in the context of absolute identity is as such:
(DP1) In 1986, Alice was 2 foot tall.
(DP2) In 2021, Alice is 6 foot tall.
(DP3) So, 1986-Alice \(\neq\) 2021-Alice.
Relative identity has no problem sidestepping this puzzle. 1986-Alice is not the same physical body as 2021-Alice (it may not have single cell in common), but it is the same person. Whatever the relation of "person-identity" is, none of the properties it preserves are material ones such as height or weight.
Tibs and Tail: Take a cat, Tibbles, and call Tibs the portion of that cat that excludes the tail. Tibbles is a cat with a tail, Tibs is a cat without a tail. Then amputate the tail. We end up with the now familiar set of uncomfortable options:
Two perfectly coincident cats exist at the same time and place,
Only Tibs exists— then Tibbles didn’t survive the loss of a tail,
Only Tibbles exists — then Tibs disappeared when the tail was cut, although it wasn’t his tail.
Again, relative identity seems able to save the day: Tibbles and Tibs are and have always been the same cat, but Tibbles and Tibs are different physical bodies (that is, because of their different history; they can be said to be "the same physical body at time t" if one wishes to introduce this concept).
The ship of Theseus: The ancient historian Plutarch recounts the story of the famous ship of Theseus, which was displayed in Athens for many centuries. "Over time, the ship’s planks wore down and were gradually replaced. The ship became a standing example among philosophers, for the logical question as to things that grow; one side holding that the ship remained the same, and the other contending it was not the same".
Interest in this story has persisted, owing in part to an additional twist imagined by Hobbes. Suppose that the original, worn-down planks were collected and stored in a warehouse, and much later on, were put back together by naval historians. Where, then, is the famous Ship of Theseus (SoT)?
Some would say that seafaring vessel is the one displayed in the museum ("the reconstructed ship"); they would argue that ships can survive the replacement of their parts, provided that the change is sufficiently gradual. Others would say it is in the warehouse ("the mended ship"), since ships can survive being dismantled and reassembled.
The complete range of options seems to be limited to those:
(1) The SoT is the reconstructed ship.
(2) The SoT is the mended ship.
(3) Both ships are the SoT.
(4) Neither ship is the SoT.
As it turns out, all of those are defendable options! Using relative identity helps highlight how, and leads us to conclude that the original puzzle is only a verbal game that stems from disagreements about what "being the same ship" means, and can be dispelled through rephrasing.
Studies at the Oak Ridge Atomic Research Center have revealed that "about 98 percent of all the atoms in a human body are replaced every year.... Experts have further concluded that there is a complete, 100 percent turnover of atoms in the body at least every five years". Yet "Alice" is clearly not identical to the mereological fusion of her atoms from 10 years ago (which are sprinkled within the ground, in the atmosphere, in other living beings’ bodies, etc.) — this takes us back to the notion of person-identity we evoked when discussing the debtor’s puzzle.
Intuitively, it seems possible to follow (1) in saying that the reconstructed ship is the Ship of Theseus if certain conditions are fulfilled, such that (a) the rate of replacement of parts be slow enough (if 9 planks out of 10 were replaced on the same day, the claim would be harder to support), and (b) there is some social recognition by humans that the identity of the object is unchanged (if the museum curator one day changed the label to say it is now "a replica of Theseus’s ship", claims that this is the SoT would be weakened). So using relative identity, we are able to define a relation of "being the same socially-recognized ship". But the criterias outlined should appear for what they are: fuzzy, imprecise and often mind-dependent. Relative identity can salvage identity statements about ordinary objects from paradoxes, but it is at the price of logical rigor.
Accepting (2) seems pretty straightforward in terms of relative identity: consensually, the mended ship is "the same physical ship as" the SoT. Again, by leaving aside absolute identity (out of necessity), we’re not in a perfectly rigorous domain. The old planks in the warehouse have gained and lost atoms, via rot and erosion. Radioactive decay has turned some atoms into others. "Same physical ship" is thus far from equated with "same physical body at a fundamental level"; instead we would need to agree on some reasonable criteria defining the relation "same plank" and say that the "same physical ship" is made from the same planks, arranged in the same shape.
(3) is compatible with (1) and (2) as we’ve just discussed them. The SoT is relatively identical to both the mended ship and the reconstructed ship, based on distinct identity relations, so transitivity doesn’t come into play; indeed we don’t think of the mended ship and the reconstructed ship, which both exist at the same time in different physical locations, as identical (note that would be possible with some very loose criteria, where "being the same ship" would only mean "having the same shape and materials, and being named the same by its owners").
Conversely, (4) is compatible with a very strict relative identity relation. We could demand that "being the same ship" mean: 90% of the planks are identical and there is no history of dismantling and re-assembling. In that case neither the mended ship nor the reconstructed ship qualify.
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...
Metaphysics
Substance: are particulars just bundles of properties?
Position: Substance holism
It is quite common for philosophical jargon to take on different meanings across eras or schools of thought. I won’t attempt a historiography of the term substance — that would be a major encyclopaedia entry in itself. Instead I will sum up the mainstream view: we think of particulars (individual objects) as instantiating universals (properties, relations...). If the object I’m considering is a rock, it can instantiate many properties: being grey, being hard, being on Earth, etc. Now, if that rock ceased to instantiate each of its properties, one by one, what would remain? The answer, not very surprising, is either something, or nothing.
The former view posits that we would be left with something named a substratum, a thin particular which Aristotle called the primary substance of the object: an ontological primitive that is property-less, independent and durable. In that case, a particular is more than the universals it instantiates: at its core is a sort of pincushion on which universals can be poked. That is substance.
Conversely, one can consider that taking off all universals just leaves us with nothing. In that view, objects are simply bundles of universals. From a physical perspective, the distinction seems scholastic and perhaps ineffectual. Most of classical physics takes the substratum view implicitly: "let \(A\) be an atom, with mass \(m\), momentum \(\mathbf{p}\), position \(\mathbf{x}\) at time \(t\), etc.". Through a change of perspective, one could easily take a bundle view by calling \(A\) the bundle of "being an atom", "having mass \(m\)", "having momentum \(\mathbf{p}\) at time \(t\)", "having position \(\mathbf{x}\) at time \(t\)".
From a metaphysical perspective however, the question is quite momentous. It really boils down to the topic of objecthood. At first glance, it seems we’re going backwards. Haven’t we been using "individual objects" and in particular concrete ones since we started setting the stage of this formalized ontology? Sure we have, but the question of their ontological priority is still up in the air. Most statements we’ve made about objects were in the mold of \(\exists x Px\) : we summon an object based on some property it instantiates. But would \(\exists x\) alone make any sense?
As I’ve mentioned, the existence of individual susbtances is still taken as somewhat of a default stance, more or less consciously. Yet, I consider we should rather defend a form of bundle relationism coupled with monist substratum theory which I’ll call substance holism. That choice is not borne out of some ideological or aesthetic preference; it is an inevitable consequence of taking modern physics seriously. The combined actions of quantum theory and of relativity have completely revised the metaphysical underpinning that had been taken for granted by "serious" (as in: acquainted with the natural sciences) philosophers. Of course, our "best theories" will eventually be replaced, but they still imply some very real constraints on what their successors will be like. Bell’s theorems, to take a famous example, preclude the principle of locality from poking its head again, even via thus-far-hidden variables.
We’ve noted above that the ontological priority of objects is at stake in this debate. If bare particulars exist, then they are prior, or at least on equal footing with universals they instantiate. If particulars are bundles of universals, then by definition the latter are prior. To put it another way: is the rock endowed with the property "being massive" by virtue of the "bare rock" having mass? Or does the rock have mass because it instantiates the universal "being massive"? That’s a matter where quantum field theory, in particular, flips the table. As Cassirer had it, the field "is a system of effects (Wirkungen), and from this system no individual element can be isolated and retained as permanent, as being ‘identical with itself’ through the course of time. The individual electron no longer has any substantiality in the sense that it per se est et per se concipitur; it ‘exists’ only in its relation to the field, as a ‘singular location’ in it." In a classical field theory, the field acts on particles that are either prior to it (e.g. the gravitational field, which exists only by virtue of particles having mass), or at most on a par with them. The field and particles’ interaction is predicated on fundamental physical properties that are intrinsic to particles: an electric charge denotes the magnitude to which a given particle reacts to a given electric field. In quantum field theory, "elementary particles" are defined as excited states of the field: their existence without the field is not conceivable, but a field can exist without any particle (in the "ground state"), and the fundamental equations of the theories are expressed in terms of fields only. Mass, spin, electric charge, or any other seemingly intrinsic property proceed directly from the field equations and the corresponding gauge symmetry group.
Behind this state of affairs looms the even more troubling question of individuality. The ability to distinguish between particulars has been a key role for substance. But even classical physics did easily dispense with substance for that purpose: since no two particles could share all of their relational, in particular spatio-temporal, properties anyway, they can be tracked and are always discernable. In a way, the combination of impenetrability, locality and determinism enforce discernability, and there is no haecceity or "primitive thisness" at the core of that individuality. With quantum mechanics, the issue becomes much thornier, as there are entangled quantum states of many particles that attribute exactly the same intrinsic and relational properties to each of them. In quantum field theory, elementary particles are excited states of a quantum field, and the question of discernability doesn’t even make any sense. Trackability is also conceptually impossible due to the uncertainty principle. Ultimately, the hard knock that quantum theory gives to metaphysics forces a clarification from classical ambiguity: either elementary particles are not individuals at all; or they are, but their individuation must refer only to some bare haecceity rather than the instantiation of any property. I lean towards the former, as it is not entirely clear how the bare particularity of quantum particles could gel with the theoretical framework of gauge-invariant QFT in a meaningful way (that is, in a fashion that would not just make it an aesthetic add-on devoid of explanatory power).
The next question seems to be whether the fields themselves are individuals, or, since field quantities are normally attributed to spacetime points or regions, whether those are the true indviduals, whilst fields are really their properties. That question is bound up with the substance debate within the foundations of general relativity... We really are caught between Scylla and Charybdis here! The main problem now is the general covariance of the field equations in general relativity. Any spacetime is entirely equivalent in all observable respects to its image under a diffeomorphism. Another way to say this is that spacetime points are entirely indiscernible: swapping their properties around should make no difference since the structure remains the same.
In conclusion, it appears that if there are individuals, they ought to be the universe-spanning objects governed by the field equations. In our current theories, those would be spacetime as a whole, and all quantum fields from the Standard Model of particle physics. Now, it’s also important to note that the unification process that has guided particle physics in the past half century tends to reduce the number of fundamental fields more and more. In quantum electrodynamics (\(U(1)\)), the electron and neutrino fields are separate. But in the Standard Model (\(SU(2)\times U(1)\)) they are part of a unique, first-generation leptonic field. The decomposition into electron and neutrino is a consequence of the theory’s symmetries, and thus is ontologically secondary. Modern physics has aimed to keep this process going: Grand Unified Theories would get leptons and quarks together; supsersymmetry would unify fermions (leptons and quarks) with bosons (interaction-carrying or mass-giving particles); some superstring models see spacetime itself as as a byproduct of fields’ excitation. It might well be then that there is not "several" but instead one individual.
That unique physical individual would be ontologically prior to the physical universals, since it is the interaction of that Ur-field with the fundamental symmetry group which generates conserved quantities and the corresponding properties. The colloquial objects of physics (elementary particles, spacetimes points...) would then be derived, being properties of the field. Metaphysically, I then subscribe to substratum theory since there is at least one object that’s not ontologically dependent on universals. Should there be one fundamental object, this would be a monist substratum theory. Particles and spacetime points are entirely dependent on universals being instantiated by the Ur-field(s), hence they follow a form of bundle relationism. I call the combination of the two substance holism: substance’s role is not to individuate particulars at all — yet the mathematical structure of our physical theory forces us to recognize there is at least one bare object, and thus, at least one substance.
We’ve only talked of the physical in this section - what of the mental realm? In a very speculative move, I’ve posited that some theory of the physico-mental governed the concrete as a whole, with a "qualium field" as a purely phenomenal object whose interaction with the physical universe gave rise to mental states. This is another cosmic-wide object we can posit to have its own substance (it could conceivably be secondary to some hypothetical physico-mental field, but there’s of course no evidence of that whatsoever).
And what of abstracta? In the buildup of my ontology, I’ve provided a clear-cut interpretation: they are essences. They are fully individuated by what they encode, and thus are by nature durable and independent. Like concrete objects, they do exemplify properties, but there’s no ambiguity as to what remains if we could strip them all down: that bare identity is the essence, the very encoded definition of these objects. To use a slogan: in the abstract realm, substance is essence.
Essence: what makes ordinary objects what they are?
Position: Essence deflationism
Essence is substance’s sibling in traditional metaphysics. As Aristotle would have it, Alexander the Great had some individual substance (the "pincushion"), that happened to instantiate countless predicates. Essence was the sum total of Alexander’s essential properties, the ones that he "must have had". Alexander could have been a bad general, and he would still be Alexander. He could also have died as a ten-year old. None of these properties are essential to his core identity. However, Alexander could not have been a turtle, as "being human" is part of his essence... Since this account of essence is grounded in properties, bundle theorists can also incorporate it in their thinking, as it is quite distinct from, and not grounded in a notion of substance.
What will we do? As a consequence of the model I’ve been expounding so far, I will need to reject essence as a metaphysical concept. The question of what something we call "Alexander the Great" can or cannot be is definitely of value, but as a semantic problem. Questions of sense and reference, of de re and de dicto statements then pop up left and right, and not to worry, we’re not done yet with semantics in this book. But to consider essence, as Aristotle would, as some ontologically primitive part of an object, independently of how it is referred to hardly seems compatible with what precedes.
Aristotle’s worldview relied (implicitly) on a mereological conservatism where individuals, endowed with essence, formed a clearly defined class. This matching of object to essence had of course deep ties to Aristotelian physics and their natural kinds. Alexander is a human, your dog Buck is a dog, this rock is a rock, and so on... Mereological fusions or divisions are then accepted on a bespoke basis (a part of a rock is still a rock), again based on physical principles.
Nothing of the sort will do for us. First, mereological universalism introduces a strange bestiary of objects. The fusion of every trout head with Alexander’s left foot doesn’t seem to have much that’s "essential" about it. Or if we try to force it, we’d soon devolve to a situation where essences are individual, much like substance. But we’ve seen, precisely, that substance holism would prevent us from even trying that, since there are just a few (perhaps only one) objects that are durable, independent and "naturally" individuated. As a result, essence does not play a primitive metaphysical role in our model, and my approach on the matter of essence in the concrete realm could be deemed deflationist. That is not to say that Alexander, Buck, rocks, or other ordinary objects have nothing "essential" about them. But our next section will remind us that ordinary objects do not exactly belong to the concrete realm.
Constitution: how are ordinary objects individuated?
Position: Constitutional pluralism
Objecthood seems entirely elusive so far. In my model, there’s no individual substance save for universe-spanning behemoths, and as explained previously, concrete objects are accidents to which we cannot assign essential properties. Yet, I speak of dogs, tables, humans and myself with ease!
In the relevant part of the "physical world" section, I followed a route that promised to resolve such puzzles: although we shan’t deny the existence of absolute identity, we need to adopt relative identity as a way to individuate ordinary objects. In doing so, we give up the pretense of endowing the objects of discourse and thought with pure concreteness. Instead, they are abstract, but form a bridge between the concrete and abstract realms.
How do I, and all fellow humans, then manage to talk of ordinary men, dogs or tables? Let’s go back to Buck the dog and work step by step.
We can quickly remind ourselves why Buck cannot just be a concrete object, a member of \(\aleph^{\bigodot}\) (e.g. some mereological fusion of various atoms).
Sorites problems: if Buck had five fewer hairs on his head, he would still be Buck. But two concrete objects not sharing every property (such as the number of their hairs) cannot be equal...
Coincidence problem: Buck has been Buck since his birth, so any reasonable definition of Buck as a concrete object must have temporal extension (roughly, spacetime worldlines). But we can define another object: the heap of specific atoms that constitute Buck at an instant t. This object is distinct from Buck (they don’t have the same origin), yet at some point it coincides with Buck. For ordinary objects to be of any practical use, we want to be able to count them — when I’m looking at Buck I need to somehow know I’m looking at one object, not two nor a million. Defining ordinary objects as concrete entities leads to infinitely many coinciding objects, thus making countability elusive.
The first thing we’ll need to properly build an ordinary object is a sortal. A sortal is a concept that can tell us about a certain kind of thing
if a given concrete object belongs to that kind.
how to count members of that kind.
In common discourse, the assignation of ordinary objects to kinds is often implicit: if I say "this is Buck" while pointing to a dog-shaped heap of atoms, everyone will agree that Buck was born within the past 30 years, and not billions of years ago in the Sun. That is because the ordinary object I’m referring to is understood by all in association with the kind "dog" rather than the kind "heap of atoms".
There remains the question of countability, which is linked to individuation and permanence. 10 years ago, Buck was small, while now he is big, yet unless we herald Heraclitus and assert that one never sees the same dog twice, it would be nice for our ordinary object concept to consider that past- and present-Buck add up to just one dog. Similary, actual-Buck and Buck-with-fewer-hairs are one and the same. We can see that by starting from sortals, we’re landing on something that looks a lot like a relative identity relation.
Let’s call \(P_{dog}\) the property of being a dog. \(P_{dog} \in (\mathbf{3}^{\bigodot})_\aleph\): it is a property of macro-objects from \(\aleph ^{\bigodot}\).
Then let’s call \(=_{dog}\) the corresponding relative identity relation ("to be the same dog as"). This is an equivalence relation between concrete objects that are dogs (that exemplify \(P_{dog}\)), which we can picture as enforcing that two terms share some characteristic properties.
For the sake of example, a reasonable characterization of an individual dog would be based on its DNA and exact time of birth, since no two distinct dogs on our Earth share those.
The next step is then to define ordinary objects as equivalence classes.
We call \(\aleph ^{\bigodot_{dog}}\) the set of "concrete dogs", i.e. concrete objects belonging to that kind: \[\aleph ^{\bigodot}_{dog}=\{x \in \aleph^{\bigodot} | P_{dog} x \}\]
What we call an ordinary dog belongs to the quotient space of the class of macro-objects by the "same dog" relative identity relation. \[\mathbf{Ord}_{dog}=(\aleph ^{\bigodot} _{dog}/=_{dog})\] The size of that set should be the number of dogs that have ever been or ever will be (with the appropriate restriction to avoid "multiverse dog twins" and the likes...we will come back to this point).
It’s important to remember that being equivalence classes, ordinary objects are technically sets of concrete objects. If \([d_0]\) is an ordinary dog (\([d_0] \in \mathbf{Ord}_{dog}\)), then \[[d_0]=\{ d \in \aleph ^{\bigodot}_{dog} | d = _{dog} d_0\}\]
This will allow us to define "ordinary properties". To achieve that, we could consider maximal properties.
If \(T_t\) is the concrete property "having a tail at time \(t\)" (with an extension in \(\aleph ^{\bigodot}\)), we can define its ordinary equivalent \([T_t]\) (also acting over \(\aleph ^{\bigodot}\)), as: \[[T_t]d_0 \hat{\longleftrightarrow} \exists d \in [d_0] (T_t d)\] In plain English: a concrete object that is a dog has a tail when one of the dogs that are the same has a tail — in other words: Buck-minus-his-tail may not exemplify the regular property "having a tail", but since it’s "the same dog as" maximal-Buck, it exemplifies the ordinary property "having a tail".
More generally, we can say that a property \([P]\) qualifies as an ordinary property adapted to a sortal \(s\) when \[\forall x \in \aleph ^{\bigodot}_{s} ([P]([s]) \in \{ \{\top\},\{\bot\}\})\] In other words: for a given ordinary object, the ordinary property is either exemplified by all its concrete members, or exemplified by none. This ensures that we are committing an acceptable sleight-of-hands when casually talking of the properties of ordinary objects: we’re actually talking of properties of their members, but any ambiguity is resolved by the property’s uniformity over a whole equivalence class.
This same construction can be easily expanded to relations.