## The metaphysics of quantities

I've long been puzzled by the nature of quantities, but I've never really followed the literature. Now I've read Jo Wolff's splendid monograph on the topic. I'm still puzzled, but at least my puzzlement is a little better informed.

The basic puzzle is simple and probably familiar. On the one hand, being 2m high or having a mass of 2kg appear to be paradigm examples of simple, intrinsic properties. On the other hand, these properties seem to stand in mysterious relationships to other properties of the same kind. First, there's an exclusion relationship: nothing can have a mass of both 2kg and 3kg. Second, there are non-arbitrary orderings and numerical comparisons: one thing may be four times as massive as another; the mass difference between x and y may be twice that between z and w. If 2kg and 8kg are primitive properties, why couldn't an object have both, and where does their quasi-numerical order and structure come from?

It's tempting to think that the answer is connected to procedures or conventions of measurement. To simplify, let's imagine that we've chosen a particular chunk of iron as our "standard kilogram", and that we determine the mass of other objects with the help of a pan balance, so that something has a mass of 2kg iff it balances against two objects each of which balances against the standard kilogram, etc. If that's how '1kg', '2kg', etc. are defined, then it is conceptually impossible that something has multiple masses. It is also clear what we mean when we say that x is four times as massive as y: it means that four copies of y balance against x.

OK. But we should distinguish our concept of mass from mass itself. Intuitively, there's a difference between having a mass of 2kg and being disposed to balance against two copies of the standard kilogram in a pan balance. The former explains the latter. More generally, if two objects are disposed to balance on a scale, then that's because they have the same mass. So mass shouldn't be identified with a certain (brute?) measurement disposition.

Or so it seems to me. Neo-Aristotelian dispositionalists might hold that having a mass of 2kg really is nothing but a bundle of brute dispositions, including measurement dispositions. This might help with our initial puzzle, at the cost of raising other issues. Let's set it aside. (Interestingly, the dispositionalist option is never mentioned in Jo's book. It doesn't seem to be popular among quantities people.)

Here's another, more promising option. On this view, the measurement dispositions are explained not in terms of intrinsic mass properties, but in terms of non-dispositional relations such as being more massive than, which are taken to be revealed by our measurement process. These relations are metaphysically fundamental; monadic properties like having a mass of 2kg are somehow derived.

We know from mathematical measurement theory that if certain relations satisfy certain structural conditions ("axioms"), then they can be represented by a non-trivial assignment of numerical values to the relata. Maybe that's how having a mass of 2kg is grounded in fundamental mass relations.

This would also solve the puzzle with which I began. But it has some downsides.

One is that a sufficiently determinate numerical representation requires infinitely many relata that cover the entire range of the relevant quantities. To get around this, Field 1980 uses spacetime points as his relata. This is obviously a hack. According to Jo, others construe the relata as Platonic universals or as points in more general quality spaces. Being more massive than is then a kind of second-order relation, linking first-order properties.

On the resulting view, we can't really reduce individual mass properties to mass relations, because the properties are needed as the relata. Rather, we have fundamental monadic properties as well as fundamental (higher-order) relations. But then it is unclear to me whether we can still solve the exclusion puzzle: if 2kg and 4kg are fundamental properties, why can't something have both? "Because they are related by an asymmetric fundamental relation" isn't much of an answer. Why couldn't they not be related in this way? And what makes the relation asymmetric to begin with?

Jo herself defends a version of this view. On her account, we also have fundamental relations and fundamental properties that serve as relata. However, the properties aren't traditional quiddities that account for intrinsic similarities and differences. Instead, they are mere "hooks" for the relations. Informally, if you wanted to give a complete description of fundamental reality, you wouldn't have to say which of the mass properties are instantiated by which objects; you could existentially quantify over them: 'there are properties p1,p2... such that this thing has p1 and p1 stands in R to p2, etc.' As a consequence, we don't have to allow for distinct possibilities in which the mass properties have traded places, so that, for example, the property we know as 2kg stands in the more massive than relation to the property we know as 4kg.

I find it hard to wrap my head around this view: what it takes for an object to have a mass of 2kg is that the object has some fundamental property p that stands in certain fundamental relations to other fundamental properties p1,p2,..., and there is no further fact about which property this p is. Hmm.

Another possible solution to the problem of missing relata might be to let the relations hold between merely possible objects. Jo doesn't mention this option, so perhaps it has serious problems that I don't see. I don't think cross-world quantity comparisons are generally problematic. We can easily make sense of statements like: 'there could have been a sphere of gold more massive than any actual sphere of gold.' It would be odd to assume fundamental facts involving merely possible spheres of gold. But that's not the idea. Rather, the idea would be that the character of the relevant relations (more massive than, etc.) is not fully revealed by their this-worldly extensions. And that seems independently plausible.

Anyway, the problem of missing relata is not the only problem for reducing quantity properties to relations. Another problem is the choice of relations. Length, for example, can be characterised in terms of ordering and concatenation, but also in terms of betweenness and congruence, and in countless other ways. It's implausible that all of these relations are fundamental. And if they are, some of our initial puzzles return. For example, why does the instantiation of ordering fix the instantiation of betweenness? (Jo mentions this problem, but she doesn't explain how her own account gets around it. Or maybe she does, and I missed it.)

I also worry that the familiar laws of physics involve particular masses, not relations like more massive than. Since these laws relate, say, masses to velocities, it is not obvious to me that they could be rewritten in terms of mass relations and independently specified velocity relations. Wouldn't we also need relations linking different types of quantities?

Finally, and most simply, it just seems counter-intuitive that properties like mass and length are grounded in relations like more massive than and longer than. Intuitively, these are "internal" relations: they hold in virtue of the properties of their relata.

All in all, I'm not convinced that this is the way to go. I'm more inclined to return to the original view on which individual mass properties are fundamental, and relations like more massive than derivative.

We're then still left with the two puzzles from the beginning. Why can't an object have several mass properties, and how do these properties get their numerical structure?

One hypothesis is that these are metaphysically contingent facts about our world. In this world, you need to put four objects with property p1 on one side of a pan balance in order to balance out one object with property p2. Not so in other worlds. It's not guaranteed by the intrinsic nature of p1 and p2. On this view, it is contingent a priori that objects with a mass of 4kg are twice as massive as objects with a mass of 2kg. Perhaps it is similarly contingent a priori that nothing can have both a mass of 2kg and a mass of 4kg.

The main downside of this view might be that it sees an astounding regularity where intuitively there's nothing astoundingly regular at all. In our world, the continuum many properties we know as masses or lengths just happen to be related in a simple and systematic fashion, so that we can conceptualise them as values of a single quantity.

It would be nice if we could avoid this assumption. On some days, I think it's a prejudice to think that fundamental properties are always binary, simply applying or not applying to objects. Why not allow for properties that fundamentally apply to an object to a certain degree? More generally, why not allow for properties (or predicables) with "values" that form a certain structure? These values would, of course, not be numbers, but their structure would sometimes be homomorphic to the structure of certain numbers. The structural relations between the derived binary properties (like having a mass of 2kg) would then be genuinely internal.

I'm not sure if this view is coherent. As I said, some days I think it is, some days I don't.

Overall, I'm still puzzled.

# on 26 February 2021, 11:08

Hi Wo,

Thanks for this---I didn't know this book existed, but it is next on my reading list now.

I wondered about this: "I also worry that the familiar laws of physics involve particular masses, not relations like more massive than. Since these laws relate, say, masses to velocities, it is not obvious to me that they could be rewritten in terms of mass relations and independently specified velocity relations."

Maybe I'm misunderstanding, or I'm thinking of a bad example, but in any case I'm not yet seeing what the issue here is. Consider p = m/v. Can I not understand this as: "for any (possible) objects x and y: if y has n times x's mass and r times x's volume then y has n/r times x's density, and if x has n/r times y's density then for some a/b = n/r, y has a times x's mass and b times x's volume." We then spell out what "times" means in the usual way involving relations and operations over concatenations.

Maybe the problem here is with the example, especially as I'm talking about definition of a derivative quantity. But I'm not seeing why things would be importantly different for, say, Newton's law of gravitation. If I'm missing something, then clarification would be appreciated!

Also on this: "Another possible solution to the problem of missing relata might be to let the relations hold between merely possible objects. Jo doesn't mention this option, so perhaps it has serious problems that I don't see."

I hope there's no serious problems with it, as that's how I've been thinking about it.

# on 27 February 2021, 09:53

Thanks Ed,

my worry is that something physically real would be lost by the relational translation, in cases where a law links separate fundamental quantities. The law of gravitation says that F = G m1m2/r^2. So the absolute value of the force F depends on the ratio between the absolute values of the masses and the absolute distance. If we say that there is no such thing as an absolute mass value, but only relations like "being n times as massive", then how is the value of F determined? It seems that it isn't determined at all: the supposed mass facts are invariant under rescaling m1 and m2, and any such rescaling yields a different value of F. So we're committed to the idea that F also doesn't have an absolute value. Moreover, we're committed to the force facts being invariant under a particular transformation: the one induced by the transformations on m1 and m2. Since F is similarly related to momentum and thereby to distance and time etc., we're also committed to those quantities being invariant under certain transformations. And maybe they are not.

I might well be confused here.

Good to hear that you also like the possibilist response to the problem of missing relata. Would be interesting to see that spelled out in more detail (probably using some higher-order metaphysics).