Scientific realism about mass

While I’ve grown up as a scientific realist, and been trained as one as a philosophy graduate student, and I suppose I still identify as one, I’ve been finding it more difficult to say what scientific realism claims.

For instance, what does it mean to be a realist about mass in a Newtonian context? A naive thought is that for each physical object, there is a positive real number, the mass of the object, which mathematically enters into the laws of nature such as F = ma and F = Gm₁m₂/r². But that seems to commit one to there being some odd objective facts, such as to which objects have the property that the square of their masses is less than their mass—a property that barely seems to make any sense, since normally in physics, we don’t compare masses with squares of masses, as they are measured in different units.

A more sophisticated thought is that there is a determinable mass, and a family of determinates, with various mathematical relations between them, with the family isomorphic with the positive real numbers with respect to the relations, but without necessarily a single isomorphism being privileged. But this more sophisticated thought is much more philosophy than physics: physicists hypothesize entities like forces and particles and the like, but not such entities like determinables and determinates. Indeed, this approach commits one to the denial of nominalism, and surely realism about mass in a Newtonian context shouldn’t commit one to such a controversial metaphysical thesis.

Is there some alternative? Maybe, but I don’t know.

Correction to "Goodman and Quine's nominalism and infinity"

In an old post, I said that Goodman and Quine can’t define the concept of an infinite number of objects using their logical resources. Allen Hazen corrected me in a comment in the specific context of defining infinite sentences. But it turns out that I wasn’t just wrong about the specific context of defining infinite sentences: I was almost entirely wrong.

To see this, let’s restrict ourselves to non-gunky worlds, where all objects are made of simples. Suppose, further, that we have a predicate F(x) that says that an object x is finite. This is nominalistically and physicalistically acceptable by Goodman and Quine’s standards: it states a physical feature of a physical object, namely its size qua made of simples. (If the simples all have some finite amount of energy with some positive minimum, F(x) will be equivalent to saying x has a finite energy.)

Now, this doesn’t solve the problem by itself. To say that an object x is finite is not the same as saying that the number of objects with some property is finite. But I came across a cute little trick to go from one to the other in the proof of Proposition 7 of this paper. The trick transposed to the non-gunky mereological setting is this. Then following two statements are equivalent in non-gunky worlds satisfying appropriate mereological axioms:

1. The number of objects x satisfying G(x) is finite.

2. There is a finite object z such that for any objects x and y with G(x) and G(y), if x ≠ y, then x and y differ inside z (i.e., there is a part of z that is a part of one object but not of the other).

To see the equivalence, suppose (2) is true. Then if z has n simples, and if x is any object satisfying G(x), then all objects y satisfying G(x) differ from x within these n simples, so there are at most 2ⁿ objects satisfying G(x). Conversely, if there are finitely many satisfiers of G, there will be a finite object z that contains a simple of difference between x and y for every pair of satisfiers x and y of G (where a simple of difference is a simple that is a part of one but not the other), and any two distinct satisfiers of G will differ inside z.

I said initially that I was almost entirely wrong. In thoroughly gunky worlds, all objects are infinite in the sense of having infinitely many parts, so a mereologically-based finiteness predicate won’t help. Nor will a volume or energy-based one, because we can suppose a gunky world with finite total volume and finite total energy. So Goodman and Quine had better hope that the world isn’t thoroughly gunky.

Goodman and Quine and shared bits

Goodman and Quine have a clever way of saying that there are more cats than dogs without invoking sets, numbers or other abstracta. The trick is to say that x is a bit of y if x is a part of y and x is the same size as the smallest of the dogs and cats. Then you’re supposed to say:

1. Every object that has a bit of every cat is bigger than some object that has a bit of every dog.

This doesn’t work if there is overlap between cats. Imagine there are three cats, one of them a tiny embryonic cat independent of the other two cats, and the other two are full-grown twins sharing a chunk larger than the embryonic cat, while there are two full-grown dogs that are not conjoined. Then a bit is a part the size of the embryonic cat. But (assuming mereological universalism along with Goodman and Quine) there is an object that has a bit of every cat that is no bigger than any object has a bit of every dog. For imagine an object that is made out of the embryonic cat together with a bit that the other two cats have in common. This object is no bigger than any object that has a bit of each of the dogs.

It’s easy to fix this:

2. Every object that has an unshared bit of every cat is bigger than some object that has an unshared bit of every dog,

where an unshared bit is a bit x not shared between distinct cats or distinct dogs.

But this fix doesn’t work in general. Suppose the following atomistic thesis is true: all material objects are made of equally-sized individisible particles. And suppose I have two cubes on my desk, A and B, with B having double the number of particles as A. Consider this fact:

4. There are more pairs of particles in A than particles in B.

(Again, Goodman and Quine have to allow for objects that are pairs of particles by their mereological universalism.) But how do we make sense of this? The trick behind (1) and (2) was to divide up our objects into equally-sized pieces, and compare the sizes. But any object made of the parts of all the particles in B will be the same size as B, since it will be made of the same particles as B, and hence will be bigger than any object made of parts of A.

A privation theory of evil without lacks of entities

Taking the privation theory literally, evil is constituted by the non-existence of something that should exist. This leads to a lot of puzzling questions of what that “something” is in cases such as error and pain.

But I am now wondering whether one couldn’t have a privation theory of evil on which evil is a lack of something, but not of an entity. What do I mean? Well, imagine you’re a thoroughgoing nominalist, believing in neither tropes nor universals. Then you think that there is no such thing as red, but of course you can say that sometimes a red sign fades to gray. It is natural to say that the faded sign is lacking the due color red, and the nominalist should be able to say this, too.

Suppose that in addition to being a thoroughgoing nominalist, you are a classical theist. Then you will want to say this: the sign used to participate in God by being red, but now it no longer thusly participates in God (though it still otherwise participates in God). Even though you can’t be a literal privation theorist, and hold that some entity has perished from the sign, you can be a privation theorist of sorts, by saying that the sign has in one respect stopped participating in God.

A lot of what I said in the previous two paragraphs is fishy. The “thusly” seems to refer to redness, and “one respect” seems to involve a quantification over respects. But presumably nominalists say stuff like that in contexts other than God and evil. So they probably think they have a story to tell about such statements. Why not here, then?

Furthermore, imagine that instead of a nominalist we have a Platonist who does not believe in tropes (not even the trope of participating). Then the problems of the “thusly” and “one respect” and the like can be solved. But it is still the case that there is no entity missing from the sign. Yet we still recognizably have a privation theory.

This makes me wonder: could it be that a privation theory that wasn’t committed to missing entities solve some of the problems that more literal privation theories face?

An argument for nominalism

Assume theism. Then, there is nothing in existence that is intrinsically bad. For everything that exists is either God or created by God, and neither God nor anything created by God is intrinsically bad.

On radical nominalism, all that exist are substances: there are no relations, properties, tropes, accidents, essences, etc. And it is very plausible that no substance is intrinsically bad. The most plausible candidates for intrinsically bad things are non-substances, like properties (being in pain) or relations (being mistaken about something). Thus, radical nominalism has a neat and elegant way of preserving the theistic commitment to there not being anything in existence that is intrinsically bad.

This seems to me to be a significant advantage of radical nominalism over other theories.

Of course, this is not a decisive argument for radical nominalism: there are other ways of preserving the commitment to there not existing intrinsically bad entities, such as Augustine’s privation theory.

Properties, relations and functions

Many philosophical discussions presuppose a picture of reality on which, fundamentally, there are objects which have properties and stand in relations. But if we look to how science describes the world, it might be more natural to bring (partial) functions in at the ground level.

Objects have attributes like mass, momentum, charge, DNA sequence, size and shape. These attributes associate values, like 3.4kg, 15~kg m/s north-east, 5C, TTCGAAAAG, 5m and sphericity, to the objects. The usual philosophical way of modeling such attributes is through the mechanism of determinables and determinates. Thus, an object may have the determinable property of having mass and its determinate having mass 3.4kg. We then have a metaphysical law that prohibits objects from having multiple same-level determinates of the same determinable.

A special challenge arises from the numerical or vector structure of many of the values of the attributes. I suppose what we would say is that the set of lowest-level determinates of a determinable “naturally” has the mathematical structure of a subset of a complete ordered field (i.e., of something isomorphic to the set of real numbers) or of a vector space over such a field, so that momenta can be added, masses can be multiplied, etc. There is a lot of duplication here, however: there is one addition operator on the space of lowest-level momentum determinates and another addition operator on the space of lowest-level position determinates in the Newtonian picture. Moreover, for science to work, we need to be able to combine the values of various attributes: we need to be able to divide products of masses by squares of distances to make sense of Newton’s laws of gravitation. But it doesn’t seem to make sense to divide mass properties, or their products, by distance properties, or their squares. The operations themselves would have to be modeled as higher level relations, so that momentum addition would be modeled as a ternary relation between momenta, and there would be parallel algebraic laws for momentum addition and position addition. All this can be done, one operation at a time, but it’s not very elegant.

Wouldn’t it be more elegant if instead we thought of the attributes as partial functions? Thus, mass would be a partial function from objects to the positive real numbers (using a natural unit system) and both Newtonian position and momentum will be partial functions from objects to Euclidean three-dimensional space. One doesn’t need separate operations for the addition of positions and of momenta any more. Moreover, one doesn’t need to model addition as a ternary relation but as a function of two arguments.

There is a second reason to admit functions as first-class citizens into our metaphysics, and this reason comes from intuition. Properties make intuitive sense. But I think there is something intuitively metaphysically puzzling about relations that are not merely to be analyzed into a property of a plurality (such as being arranged in a ball, or having a total mass of 5kg), but where the order of the relata matters. I think we can make sense of binary non-symmetric relations in terms of the analogy of agents and patients: x does something to y (e.g. causes it). But ternary relations that don’t reduce to a property of a plurality, but where order matters, seem puzzling. There are two main technical ways to solve this. One is to reduce such relations to properties of tuples, where tuples are special abstract objects formed from concrete objects. The other is Josh Rasmussen’s introduction of structured mereological wholes. Both are clever, but they do complicate the ontology.

But unary partial functions—i.e., unary attributes—are all we need to reduce both properties and relations of arbitrary finate arity. And unary attributes like mass and velocity make perfect intuitive sense.

First, properties can simply be reduced to partial functions to some set with only one object (say, the number “1” or the truth-value “true” or the empty partial function): the property is had by an object provided that the object is in the domain of the partial function.

Second, n-ary relations can be reduced to n-ary partial functions in exactly the same way: x₁, ..., x_n stand in the relation if and only if the n-tuple (x₁, ..., x_n) lies in the domain of the partial function.

Third, n-ary partial functions for finite n > 1 can be reduced to unary partial functions by currying. For instance, a binary partial function f can be modeled as a unary function g that assigns to each object x (or, better, each object x such that f(x, y) is defined for some y) a unary function g(x) such that (g(x))(y)=f(x, y) precisely whenever the latter is defined. Generalizing this lets one reduce n-ary partial functions to (n − 1)-ary ones, and so on down to unary ones.

There is, however, an important possible hitch. It could turn out that a property/relation ontology is more easily amenable to nominalist reduction than a function ontology. If so, then for those of us like me who are suspicious of Platonism, this could be a decisive consideration in favor of the more traditional approach.

Moreover, some people might be suspicious of the idea that purely mathematical objects, like numbers, are so intimately involved in the real world. After all, such involvement does bring up the Benacerraf problem. But maybe we should say: It solves it! What are the genuine real numbers? It's the values that charge and mass can take. And the genuine natural numbers are then the naturals amongst the genuine reals.

Spacetime: Beyond substantivalism and relationalism

According to substantivalism, spacetime or its points or regions is a substance, and location is a relation between material things and spacetime or its points or regions. According to relationalism, location is constituted by relations between material things. Often, the two views are treated as an exhaustive division of the territory.

But they're not. Lately, I've found myself attracted to a tertium quid which I know is not original (it's a story other people, too, have come to by thinking about the analogy between location and physical qualities like charge or mass). On a simplified version of this view, being located is a determinable unary property. Locations are simply determinates of being located. This picture is natural for other physical qualities like charge. Having charge of 7 coulombs is not a matter of being related to some other substances--whether other charged substances or some kind of substantial "chargespace" or its points or regions. It's just a determinate of the determinable having charge.

This determinate-property view is more like the absolutism of substantivalism, but differs from substantivalism by not positing any "spacetime substance", or by making the locations into substances. Locations are determinates of a property, and hence are properties rather than substances. If nominalism is tenable for things like charge or mass, the theory won't even require realism about locations.

A nominalist reduction

Suppose that there were only four possible properties: heat, cold, dryness and moistness. Then the Platonic-sounding sentences that trouble nominalists could have their Platonic commitments reduced away. For instance, van Inwagen set the challenge of how to get rid of the commitment to properties (or features) in:

1. Spiders and insects have a feature in common.

On our hypothesis of four properties, this is easy. We just replace the existential quantification by a disjunction over the four properties:

2. Spiders and insects are both hot, or spiders and insects are both cold, or spiders and insects are both dry, or spiders and insects are both moist.

And other sentences are handled similarly. Some, of course, turn into a mess. For instance,

3. All but one property are instantiated

becomes:

4. Something is hot and something is cold and something is dry but nothing is moist, or something is hot and something is cold and something is moist but nothing is dry, or something is hot and something is dry and something is moist but nothing is cold, or something is cold and something is dry and something is moist but nothing is hot.

Of course, this wouldn't satisfy Deep Platonists in the sense of this post, but that post gives reason not to be a Deep Platonist.

And of course there are more than four properties. But as long as there is a finite list of all the possible properties, the above solution works. But in fact the solution works even if the list is infinite, as long as (a) we can form infinite conjunctions (or infinite disjunctions—they are interdefinable by de Morgan) and (b) the list of properties does not vary between possible worlds. Fortunately in regard to (b), the default view among Platonists seems to be that properties are necessary beings.

Goodman and Quine's nominalism and infinity

[This post is largely wrong. -- Note added in December, 2024]

The argument in this post is highly compressed. It's even more a note to self than other posts are.

Goodman and Quine have a very clever nominalist metalanguage that lets them handle first-order logic. There seems, however, to be a serious problem in their system. As it stands, the system will be inconsistent in certain infinite worlds, if it allows excluded middle (which it does, being classical). And they do not have the resources for specifying that the worlds they're using are finite.

The problem stems from the fact that Goodman and Quine nowhere specify that the sentences of their target language are finite. Because they fail to specify that, they cannot rule out infinite sentences corresponding to something like "~~~......~~~p". Think of the front part of it as two infinite sequences of smaller and smaller tildes, with the infinite tails touching. (Goodman and Quine work with Sheffer strokes, and it's a touch harder to explain how such an infinite sentence is done with Sheffer strokes, but I think it can be done, too. I will work with ordinary FOL.)

Now, why would you want to rule out such sentences? Well, write N for the doubly infinite sequence of negations, as above. Then by excluded middle, we have Np or ~Np. But ~Np is the same sentence as Np. Hence, we have Np or Np. Hence we have Np. But again ~Np is the same sentence, so we have both Np and ~Np. And that's pretty bad, because everything now follows by explosion, again because we have a classical logic.

Could Goodman and Quine cleverly exclude such infinite sentences? It seems that they can't do it using their present metalanguage primitives and axioms, nor by means of any straightforward extension of them. For their metalanguage is also classical and first-order, and hence unable to express sentences that "logically imply" that there are finitely many Fs (say, letters in s)—i.e., sentences that are true in all and only all interpretations on which finitely many things satisfy F (this is easy to prove by compactness, and I think does not even require the Axiom of Choice).

That looks pretty much fatal. Except that Goodman and Quine might be able to help themselves if they could use some heavy duty metaphysics to establish that our world either has only finitely many objects or is a single continuous plenum. For given that the world is a single continuous plenum, we might be able to express the idea that a sentence is finite by saying that for every mereological sum of curvy arrows in the world (arrows being certain arrow-shaped parts of the plenum—we need mereological universalism for the system to work) such that every letter of the sentence is at the tail end of exactly one arrow, and every arrow points to a letter of the sentence, and no two arrows point to the same one, every letter of the sentence is pointed to. But this only works given a continuous plenum where there is enough stuff to make enough arrows to ensure this isn't spuriously satisfied. And I doubt there is a good way to express the fact that we have a continuous plenum in the Goodman and Quine system. So the system can only be made to work on a quasi-empirical assumption that the system, apparently, cannot state. And it is bad that whether a logical system is consistent depends on how matter is arranged in the world—if it is arranged in a plenum or there are only finitely many objects, it's consistent, otherwise possibly not.

Another move would be to require that all the letters be exact copies of each other and that they be all in a straight line. There is no way to form "~~~......~~~p" in an Archimedean universe where all the letters are in a straight line. But, again, their logical system will depend for its consistency on the assumption that our world is Archimedean. And that's weird.

Goodman and Quine mention something related to the finiteness issue in footnote 14, in the context of the alternative framed ingredients method. I think the alternative framed ingredients method also requires an assumption of finitude.

Plato might have been a "nominalist"

I was reading the SEP entry on nominalism by Rodriguez-Pereyra. Rodriguez-Pereyra sees nominalism as basically the rejection of causally inert non-spatiotemporal entities. If so, then Plato might have been a nominalist. It seems that Plato did not think the Form of the Good was causally inert--it caused the good arrangement of things in the universe. I don't know if Plato generalized from that case, but he might well have--he might have taken all of the Forms to be capable of causing things to be like them. So, for all I know, Plato was a nominalist.

And Leibniz might have been was a nominalist despite going on and on about abstract objects, because he thought of them as ideas guiding God's deliberation, and hence perhaps we should say that on his view they had a causal role in creation.

This isn't a big deal. Rodriguez-Pereyra's account nicely captures a rejection of modern forms of Platonist.

I wonder, too, whether a belief in Newtonian space is compatible with nominalism by this definition. Newtonian space seems to be causally inert (perhaps unlike the Riemannian manifold of General Relativity). And it may be a category mistake to say that space is spatiotemporal. Though maybe it's fine to say that space is spatiotemporal in some trivial sense.

Pluralist theories of predication

According to anti-pluralist theories of predication, there is only a small handful of fundamental predicates and they are all of a highly general and abstract nature.  Sometimes there is only one.  For instance, strong Platonism has as fundamental only the multigrade predicate Instantiates.  All other predications should be analyzed in terms of it.  Resemblance nominalism has the fundamental predicate ResemblesInRespect and then needs some story about respects (which story may involve one or two more fundamental predicate).  Bundle theory will have the fundamental predicate CobundledWith, plus perhaps the predicates of set theory (∈ and IsASet) or of some other highly general theory for constructing objects out of bundles.

According to pluralist theories of predication, there are many fundamental predicates and many of them are of a very concrete nature.  For instance, the pluralist is likely to have predicates like Horse, Daphnia and NegativelyCharged.  She may also have highly abstract predicates like ∈ as well.

Ostrich nominalists are pluralists.  But one can also be a weak Platonist and a pluralist.  I am inclined to think that the solution to the problem of the unity of form and matter given in Metaphysics H.6 commits Aristotle to pluralism.

The big insight of the pluralist is that the puzzle of predication is no less of a puzzle when that puzzle concerns a small handful of fundamental predicates.  There may be theoretical simplicity grounds to prefer particular anti-pluralist theories of predication over particular pluralist theories, but I suspect these will result in a stalemate.  And then the pluralist will win, as her fundamental predicates fit better with our intuitions, I think.

Goodman and Quine's nominalism

Goodman and Quine's "Steps Toward a Constructive Nominalism" is an ingenious attempt to show how one might begin to give nominalist analyses of claims that prima facie involve abstracta like functions, numbers or types. For instance, the analysis of "There are more cats than dogs" would be that there is no one-to-one function pairing every cat with a different dog. Here is the clever analysis. Say that an object is a bit iff it is exactly as big as the smallest animal among the cats and dogs. Say that an object is a bit of z iff it is a bit and it is a part of z. Now, there are more cats than dogs iff every object that has a bit of every cat is bigger than some object that has a bit of every dog.

The first issue here is that this—like many of Goodman and Quine's definitions—only works given mereological universalism. We need to be able to form an object that has a bit of every dog. But while the theory requires mereological universalism, it is not clear that one can state the relevant mereological universalism without adverting to sets or properties. For instance, it seems that the account of "There are more cats than dogs" only works if from the fact that every dog has a bit we can infer that there is a minimal object among the objects that have a bit of every dog. The relevant axiom is something like this: given non-overlapping Fs and an object x no bigger than any F, there is a minimal object that has an x-sized portion of every F. But this axiom seems to quantify universally over kinds F. Without such quantification, we will simply have a separate axiom for each kind, and then we cannot state the fact that the counting method works "in general".

The second problem is technical. There might not actually be a smallest cat-or-dog if there is an infinite chain of smaller and smaller cat-or-dogs. Perhaps the definition only works if there are only finitely many cat-or-dogs. But it is not clear how one can say that there are only finitely many cat-or-dogs on the theory. A natural suggestion is that the fusion of all the cat-or-dogs has finite size, i.e., no proper part of it is the same size as the whole. But that won't do, because it could be that the fusion of the cat-or-dogs is of finite size, while there are infinitely many cat-or-dogs. In fact, this point suggests that in general the notion of finitude is going to be difficult for Goodman and Quine to express.

The third issue is that the crucial basic concept here is that of being "bigger than". One sense of "y is bigger than x" is that a rigid motion could bring x wholly within the space occupied by y. This sense won't do here, however. For there need not be a smallest cat-or-dog in this sense, as the shapes might not nest. The relevant sense of "y is bigger than x" is that y has greater mass or volume than x. But, now, how do we understand that in a nominalist way? Here is one problem: mass and volume are relative to a reference frame. Maybe, though, we fix some particle r and then understand mass or volume relative to r. This would require a ternary relation BiggerThan(y,x,r). I suppose this is doable. But what about massless and volumeless objects, like photons? Well, maybe then we need the notion of energy comparison instead. However, now we have the oddity that the concept of counting depends on the concept of energy. But surely there could be more As than Bs in a world whose laws were so different from our world's laws that there could be no concept of energy there.

It is hard to be a thorough-going nominalist.

Two problems for conspecificity as primitive

Here is something growing out of last night's neo-Aristotelian metaphysics class with Rob Koons. Suppose we take the relation of conspecificity as a primitive, in order to be a nominalist about species. (The context here is Aristotelian, so "species" may include "Northern leopard frog", but it may also include "electron".) Then we will have a hard time making sense of claims like:

1. Possibly, none of the actual members of x's species exist (in the timeless sense), but there is some member of x's species.

Suppose for instance x is an electron. Then, surely, there is a possible world where there are electrons, but none of the actual world's electrons exist. But to make sense of (1) on an account that takes conspecificity to be primitive would require a conspecificity relation between an electron in the actual world and an electron in the possible world. But how can there be a relation one of whose relata does not exist? (Intentional relations are like that, but I don't think we want conspecificity to be like that.) The realist about species doesn't have this particular problem. She just explains (1) by saying that if s is the species of x, then possibly none of the actual members of s exist but s nonetheless has a member. Also, if one takes conspecificity as primitive but allows the existence of non-actual individuals, the problem disappears, since then we can unproblematically relate a non-actual individual with an actual one.

The problem here is that of interworld conspecificity. What makes an individual a₁ in a world w₁ conspecific to an individual a₂ in w₂? If there is an individual a₂ in w₁ conspecific to a₁ who also exists in w₂ and is conspecific to a₂, by transitivity of (Aristotelian) conspecificity this is not a problem. We can generalize this solution by saying that a₁ in w₁ is conspecific to a₂ in w₂ provided that there are chains of worlds W₁,...,W_n and entities A₁,...,A_n such that

• W₁=w₁, W_n=w₂, A₁=a₁, and A_n=a₂
• b_i is in both W_i and in W_i+1 for i=1,...,n−1
• b_i and b_i+1 are conspecifics in W_i+1 for i=1,...,n−1.

For this approach to give a good account of interworld conspecificity it has to be the case that conspecificity is transitive and that species membership is essential. (But the approach can also work if species membership is not essential, as long as we have individual forms, and the membership of an individual form in a species is essential. For then we can give the story not in terms of chains of particulars, but chains of individual forms.)

The above account does, however, entail the following metaphysical principle:

2. Whenever worlds w₁ and w₂ contain individuals a₁ and a₂ who are members of species s (understood nominalistically), then there is a finite chain of possible worlds, starting at w₁ and ending at w₂, such that every pair of successive members of the chain has a common member of s.

Is (2) true? Well, it seems hard to come up with counterexamples to it, at least. If we could imagine a species whose possible members could be divided into two classes, A and B, such that no member of A could exist in a world that contains a member of B, then we would have a violation of (2). But I am not sure we have much reason to think such species exist.

But now consider a different problem for the account. Two photons can collide and produce an electron-positron pair. Suppose we are in a world where there are lots of photons, but only one collision has occurred, producing electron e (and a positron that I don't care about). We now want to be able to say this:

3. A pair of photons p₁ and p₂ jointly have the power of producing an electron.

Presumably this should reduce to some claim about how they have the power of producing a conspecific to e. But that is an extrinsic characterization of the power of the photons. Yet it is an intrinsic feature of the joint power of p₁ and p₂ that it is a power to produce an electron (and a positron). Moreover, supposing that no collisions occurred, and hence there was no e in sight, we would still want to be able to say this:

4. A pair of photons p₁ and p₂ jointly have the power of producing a conspecific to something that photons p₃ and p₄ jointly have the power of producing.

Tricky, tricky. Here is a suggestion. We slice powers, considered as particulars ("x's power to do A") finely enough that we can talk of a particular power that p₁ and p₂ jointly have (or maybe one has the power to operate on the other in some Aristotelian way), the power of producing an electron (this power can only be exercised together with a power to produce a positron). Now, we can talk of the primitive conspecificity not just of particles, but of productive powers, and we can characterize the conspecificity of two entities disjunctively:

5. e₁ and e₂ are conspecific (non-primitively) if and only if either e₁ and e₂ are primitively conspecific or e₁ results from the exercise of a power primitively conspecific to a power the exercise of which results in e₂ or e₁ results from the exercise of a power which results from the exercise of a power primitively conspecific to a power the exercise of which results in a power the exercise of which results in e₂ or ....

Assuming that powers are characterized by what they produce, any disjunct further down in the disjunction entails all the disjuncts further up in the disjunction. Now we can make sense of (3) and (4) in an intrinsic way, in terms of the conspecificity of the powers of producing electrons. Moreover, we can make the chain-of-worlds move as needed for non-primitive conspecificity. This will yield a very complicated analogue of (2), but that analogue will, if anything, be even more plausible than (2).

This is all too messy, but maybe mess is unavoidable.