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 substances 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.