If the M-R account is going to be able to do all I’ve promised it can, it needs to give definitions of the ordering, summation, and metrical ratio relations that capture the idea that they reflect something intrinsic to their relata. This is easy in the case of length ordering, since (2) is a natural reading of (1) and is also an intrinsic relation.

It’s less obvious how summation or metrical relations should be defined on this account. A common expression of length summation relations involves talking about length properties rather than lengthy objects. We say “x’s length is the sum of y’s and z’s lengths”. The natural expression of the relation between lengthy objects doesn’t use terms like ‘sum’ at all. Rather, it says

(3) has, if anything, more of a mereological ring to it than (1). Indeed, on a literal reading of ‘put together’, we can gloss (3) as: “x has the same length as an object, o, composed out of y and z put together, would”. However, while this reading is a mereological relation, it will not do as an analysis of (3). This is because it requires appeal to o, and in particular o’s length. But o might not be lengthy; that is, y and z might not be put together in the right way for o to have length (if, e.g., they make a ‘T’ shape). Or o might be lengthy but not have the right length (if y and z have some lengthy overlap, o’s length will not be the “sum” of their lengths).

The M-R account defines summation structure in a different way. It analyzes (3) in terms of y and z’s relations to x’s parts, together with the requirement that those parts are put together “in the right way” (a condition which will vary between different quantities):⁹

Here, the M-R account’s analysis goes beyond the intuitive mereological upshot of the summation relation. The M-R account understands both the ordering and the summation relations as specifying (among other things) something about the physical makeup of one of their relata. To say that b is shorter than a, or to say that a is as long as b and c put together, is to say something about a’s internal structure—specifically, whether a has any parts, what the configuration of those parts is relative to each other, and whether they share length properties with b or c. These relations, so defined, are intrinsic to the system composed by their relata. Indeed, they satisfy a stronger condition: since they depend only on the intrinsic properties of each of their relata—i.e. on how each relatum is in itself —they are not just intrinsic but internal relations.

The M-R account, similarly, defines ratio relations like “twice as long as” or “4.6-times the volume of” in terms of mereological relations and the sharing of intrinsic properties. Though our expressions of them appeal to numerical ratios like 2 : 1 or 4.6 : 1, the physical ratio relations should be understood as relations between concrete physical objects, not as relating objects to numbers. If that’s right, then there are an infinitude of distinct, two-place, ratio relations; i.e. “2-times the volume of” and “4.6-times the volume of” are distinct relations between voluminous objects. The M-R account gives a reductive definition, in terms of mereology and the sharing of intrinsic volume properties, for each such relation, by way of a procedure which pairs ratio relations with their mereological analyses.

The “ratio procedure”, performed on an ordered pair of voluminous objects, specifies the M-R account’s definition of the ratio relation they stand in. Let me give an example of how this works.¹⁰ Suppose we want to determine the volume ratio of b to a (how much more voluminous b is than a). We perform the ratio procedure on a and b.

First, we “take a out of b”, where this just means that we partition b into as many non-overlapping copies of a (i.e. parts with the same volume as a) as we can. In this case, that number is 3. Then there is a part of b which is our remainder, r₁, which is smaller than a. For step two, we do the same thing but taking r₁ out of a, which yields 2 non-overlapping copies and another remainder r₂. The third step follows this pattern, taking r₂ out of r₁. r₁ is composed of 2 non-overlapping copies of r₂ with no remainder. Since there’s no remainder, we stop.

This procedure determines the complex mereological property which will be the M-R account’s analysis of the “volume ratio” of b to a (let’s say that b “partitions into” some class of its parts iff no two of the members of that class overlap and b is their fusion):

How does this give us the volume ratio between a and b? Taking a out of b tells us approximately how much bigger b is than a. Taking r₁ out of a tells us approximately how much bigger a is than r₁. This, in turn, gives us a better approximation of how much bigger b is than a. Each time we repeat this procedure, we get a better and better approximation. If the procedure terminates, we have a perfect approximation. Indeed, r₂ goes evenly into a and b. From the procedure we can deduce that a is composed of 5 non-overlapping copies of r₂, and b 17 copies. So the ratio of b to a is 17/5, i.e. b=175*a=3.4*a. Indeed, “b partitions into 17 parts, all with the same volume as r₂, while a partitions into 5 such parts” amounts to the same thing as the definition given above. Why not just use that as the definition for volume ratios, then?

Here’s why: The ratio procedure is not guaranteed to terminate, and if it does not terminate, it cannot output a final remainder (like our r₂). However, there’s another way to determine the ratio between a and b from this procedure that doesn’t require appeal to the final remainder. Recall that the numbers, 〈3, 2, 2〉, output by the procedure, count up certain non-overlapping parts of b, a, and r₁. We can use these to construct what is called a “simple continued fraction”: 175=3+12+12

The list of integers output by this procedure is what is sometimes called an “anthyphairetic ratio”.¹¹ Continued fractions are one way to express this sort of ratio. Even when the ratio procedure does not terminate, it will still output a list of integers that count up the relevant sets of non-overlapping parts of a and b and the various non-final remainders. The only difference is that, when the procedure does not terminate, we get an infinitely long list. This is okay because continued fractions can, in fact, be continued indefinitely, and infinite simple continued fractions also pick out unique real numbers! It’s this formal feature which allows the mereological relations generated by this procedure to serve as the definitions for volume ratio relations.

I mentioned before that the M-R account applies only to quantities that put the right constraints on the possible parthood structure of their instances. Here’s what that means: If the M-R definitions of an ordering relation, “LESS-Q”, or summation relation, “Q-SUM”, are to be any good, at the very least the definiens and definiendum must be necessarily coextensive. That is, a quantity, Q, is amenable to the M-R account only if it satisfies:

As well as the analogous necessary biconditionals for the ratio relations.

This means that for many (indeed most) quantities, the account cannot get off the ground. An M-R account of temperature, for instance, would get the quantitative relations almost entirely wrong. The ice in the freezer, at 30° Fahrenheit, is less warm than 212°F water boiling on the stove. But this fact about temperature ordering clearly doesn’t mean that the ice in the freezer is as warm as some proper part of the water on the stove!

What about quantities that, unlike temperature, put significant constraints on the mereology of their instances? Additive (also called extensive¹²) quantities are ones where, intuitively, wholes inherit their Q-properties from the Q-properties of their parts. More precisely, Q is additive just in case: whenever x and y instantiate Q-properties, and are “put together in the right way”, the mereological fusion of x and y instantiates the “sum” of their Q-properties. Being additive is necessary for a quantity to admit of an M-R account of its structure—a quantity is additive just in case it satisfies the right-to-left direction of both (5) and (6)—but it is not sufficient.

Here’s why: Consider the additive quantity, mass. On the standard model of particle physics, there are fundamental particles with different masses, like the electron (approx. 9.19 × 10⁻³¹kg), and the muon (approx. 1.88 × 10⁻²⁸kg). On a straightforward interpretation of this theory, Ellen the electron and Miriam the muon are mereological simples. This is inconsistent with both (5) and (6), since Ellen does not have a part with the same mass as Miriam, yet the standard model is not (and should not be) taken to be inconsistent with mass’s additivity. So, while additive quantities have a very close connection to mereology, a quantity’s being additive is not sufficient to support an M-R account of its structure.

In Perry (2015), I argue that some quantities put stronger constraints on the mereology of their instances than what additivity requires. These quantities I call “properly extensive”¹³ (recall that the unmodified term ‘extensive’ is equivalent to ‘additive’). The properly extensive quantities comprise a sub-class of the extensive quantities (quantities which are extensive but not properly so I call “merely additive”). Some quantities we classify as additive are, I claim, also properly extensive—specifically length, area, volume, temporal duration, and the invariant relativistic interval. Properly extensive quantities put stronger constraints on the relationship between quantitative structure and mereology than merely additive ones, like charge or mass, do.¹⁴ Most importantly, properly extensive quantities, intuitively, satisfy (5) and (6).

The connection that properly extensive quantities have to the parthood structure of their instances is what makes them amenable to the M-R account’s definitions of the quantitative relations. This amounts to more than just a restriction on the range of applicability of the account. It tells us how and why the M-R definitions work when they do. That is, the M-R account, on its own, only tells us that our representations of Q are faithful insofar as the structure of the mathematical entities we appeal to mirrors the mereological structure of that quantity’s instances, and the distribution of intrinsic Q-properties over that structure. The proper extensiveness of Q tells us why there’s a necessary correspondence of this sort between the mathematical and the mereological. This is what it means to say that the success of our mathematical representations of these quantities is explained by their connection to parthood.

Saying volume is properly extensive commits us to significant constraints on the possible physical structure of voluminous entities. Properly extensive quantities are unique in that these mereological constraints can require that a voluminous object have parts of a certain sort. One might wonder whether this puts a claim that length or volume is properly extensive in conflict with certain theses about mereology. In particular, the view that spatially extended mereological simples are possible.¹⁵ Extended simples are mereologically simple, which just means that they do not have any parts except themselves. However, unlike the mereologically simple massive point particles I mentioned earlier, they are also spatially extended, which is just to say that their length, area, and/or volume is non-zero.

The most commonly discussed kind of extended simple is a mereologically simple physical body that is spatially extended, yet occupies a region composed of smaller sub-regions (“simple body/composite region”). For instance, a 2m spherical hunk of matter, with no proper parts, occupying a 2m spherical region of Euclidean 3-space. Another sort of extended simple¹⁶ would be one which fully occupies a simple region—i.e. a region with no sub-regions (of non-zero volume). If space turns out to be composed of tiny, discrete “cells”, then a physical particle occupying the smallest possible region will not be a point-sized simple but an extended one (“simple body/simple region”).

Neither of these cases is inconsistent with length, area, or volume’s proper extensiveness, though there are a few steps required to see exactly why that is. Volume, fundamentally, is a property of regions of spacetime and, to the extent that a material object can be said to have a volume, they only have volume derivatively. Material bodies, that is, have the volumes they do in virtue of occupying a region of that volume. The constraints that properly extensive quantities put on the mereology of their instances only apply to the entities which possess volume non-derivatively. If we accept this, then an extended simple body occupying a composite region of non-zero volume is entirely consistent with volume being a properly extensive quantity. So volume’s proper extensiveness is not a threat to the possibility of the “simple body/composite region” type of extended simple.¹⁷

On the face of it, nothing about the second kind of extended simple (“simple body/simple region”) is inconsistent with the proper extensiveness of volume. I’ve already claimed that volume’s proper extensiveness is compatible with space or spacetime being discrete. However, we can get a problem if we add the requirement that some of these extended simple bodies differ in size (i.e. one simple is more voluminous than the other). This does conflict with volume’s proper extensiveness, since the differences in size between the simples would be grounded in the different volumes of the regions they occupy. But if volume is properly extensive, then no two simple regions could differ in volume! Whenever there’s a simple extended region, R (i.e. a region of non-zero volume with no proper sub-regions), there can exist no regions that are less voluminous than R—since, if there were, then R would have to have a proper part of that volume, and so wouldn’t be simple.

So volume’s proper extensiveness rules out there being two simple regions that differ with respect to their volumes. Extended simple bodies of the “simple body/simple region” type are, therefore, only possible if they are all equivoluminous, since extended simple regions are only possible if they are all equivoluminous. Luckily, it’s generally only the first sort of simple that most fans of extended simples want to allow.¹⁸ And, of the version of the second type of simple which are discussed at all, it’s largely the version of that simple that’s consistent with length’s proper extensiveness, primarily by those interested in the possibility of certain kinds of discrete spaces.¹⁹

Sections 3 and 4 make good on the promises made in this section. There, I present a formal M-R account of volume which takes the necessary constraints obeyed by properly extensive quantities as axioms.²⁰ From the assumption of volume’s proper extensiveness, and very little else, we can show that the volume ordering and summation relations, as defined by the M-R account, and the volume ratio relations, whose M-R definitions are generated by the “ratio procedure” (formally defined in section 4.2), are faithfully represented by the arithmetical ordering, summation, and ratio relations on the real numbers. Section 5 concludes and clarifies some issues set aside in the previous sections. Before presenting the formal M-R account of volume, it will be useful to understand how and why its competitors end up committed to quantitative structure, in particular metric structure, being radically extrinsic.

Some, like Mundy (1987), have thought that the problem with these existence and richness assumptions about the domain comes from their contingency. Representation and uniqueness theorems rely on global structural assumptions that are not guaranteed to be satisfied by a given subdomain. But, so the worry goes, its possible that a given subdomain have been all that there is. It seems to me, however, that this contingency is only a symptom of the broader problem of extrinsicality. We think the metric relations between elements of a given sub-domain (of the lengthy objects, say) would have still obtained had that subdomain been all that there is because we think that length metric relations do not depend, for their instantiation, on anything outside their relata. This is an important result, if correct, since most accounts of quantitative structure which successfully avoid the contingency objection still render metric relations radically extrinsic.

For instance, Mundy (1987) posits primitive second-order relations of “ordering” and “summation”, which relate mass²⁴ properties. He accepts a Platonism about properties according to which these universals, and the primitive second-order quantitative structural relations they stand in, are necessary existents. The first-order comparative mass relations between objects are all grounded in higher-order relations between their properties, which allows Mundy to avoid the contingency objection.

Consider, for instance, his definition of “less massive than”:

No problem there. An instance of the primitive [<] relation doesn’t depend on anything, so its obtaining doesn’t depend on things extrinsic to U₁ and U₂ – or x and y for that matter. If x being less massive than y depends on their intrinsic properties standing in a primitive two-place relations, then “less massive than” is not an extrinsic relation. However, when we move from the ordering relations to metrical relations, things don’t look so good.

U₁(x) and U₂(y) and: (1) U₁ and U₂ are part of a domain of mass universals M such that the distribution of the primitive second-order ordering and summation relations over this domain satisfies axioms A₁, A₂, A₃, … ; (2) U₁ and U₂ are such that there’s a function φ, from M to ℝ – where for any universals a, b, c ∈ M, a[<]b iff φ(a) < φ(b), and ab[*]c iff φ(a) + φ(b) = φ(c), and φ(U1)φ(U2)=n).

Where ‘[*]’ is the three-place second-order summation relation over the mass universals. Here this definition, again, just depends on universals and the fundamental ordering and summation relations they stand in. If universals are necessary existents, and if the axioms governing the primitive second-order relations over them are necessary, then this definition avoids any contingency worry we might have. However, this doesn’t help at all with the problem of extrinsicality. The obtaining of a given metric relation between a and b will (in part) depend on universals neither a, b, nor any of their parts instantiate, and on the primitive relations those universals stand in.

The only account that comes close to avoiding extrinsicality is Field’s. The part of his account which fares best is the theory of spatial (or spatiotemporal) distance. Intrinsicality is a bit different for a relational quantity like distance. We shouldn’t think of facts about the distance from a to b as needing to be intrinsic to a and b. Rather, we should think of them as a matter of being intrinsic to (the shortest) straight path from a to b. If this is right, then Field’s definition of “the distance from x to y is twice that from z to w” (which we’ll express as ‘xyR₂zw’) in terms of betweenness and congruence does satisfy the intrinsicality condition:

This relation between x, y, z and w is intrinsic to the straight lines xy and zw. It holds in virtue of the existence of a part, u, of xy, and the fundamental congruence relation, ‘Cong’, between zw and some parts of xy. This definition of “R_n” is only available for rational n; irrational metric relations (like “the distance from x to y is π-times the distance from z to w”) are more difficult to define on this account. However, there’s at least some reason to believe that these will be either intrinsic or, at least, not radically extrinsic.²⁵

Unfortunately, Field’s success does not extend to monadic quantities like length, mass, volume, or temporal duration.²⁶ Field (1980) describes how to extend his account to apply to scalar quantities: replace the spatiotemporal “betweenness” and “congruence” relations with “SC-betweenness” and “SC-congruence”²⁷ (‘SC’ for scalar). However, the analogue of Field’s definition schema using these relations does not avoid the extrinsicality problem. That is, the scalar analogue of (11),

is not an intrinsic relation. This is because, while the spatiotemporal relation “between(yxz)” entails that x is a part of physical straight line yz, its scalar analogue “SC-between(yxz)” merely indicates that y ≼ x and x ≼ z where ≼ is that quantity’s ordering relation. With no guarantee that x is part of either y or z, the relation V₂(xy) according to the scalar version of definition 11 will not be intrinsic to x, y, or their fusion. The same will go for an application of this definition schema to other scalar quantities, like mass, temperature, length, or area.²⁸

One might wonder if the M-R account faces a version of the extrinsicality worry, insofar as it is taken to apply to macroscopic “solid” voluminous material bodies like apples, chairs, people, and automobiles.

Here’s the concern: we ascribe approximate volumes to medium-sized dry goods like tables or swarms of bees based not on the combining the volumes of their miniscule parts (which, on an atomic theory of matter, would be extremely small), but based on the volume of some salient contiguous, simple, table-shaped region (or, rather, a class of very similar regions) throughout which those parts are distributed. But, if the table is composed of only those material parts, then this volume attribution appeals to a region that the table does not occupy.

I will take it as uncontroversial that the volume of a spatial or spatiotemporal region is intrinsic to that region.²⁹ Whether the volume of a particular hunk of matter occupying that region is intrinsic to that hunk of matter will depend not just on what you think matter is, but also on your preferred definition of intrinsicality.³⁰

However, I think we can safely set the differences between specific definitions of intrinsicality aside for two reasons: First, the intrinsicality intuition is the intuition that the things playing into our physical explanations be features of, or entities involved in the physical phenomena to be explained. Insofar as the volumes of macroscopic material bodies make a difference in the physical world, it is the approximate volumes we ascribe to them based on the regions they trace out (and not the sum of its microscopic voluminous parts, supposing it has any). And so the M-R account, on which the volume ratio relations between voluminous regions are genuinely intrinsic to those regions, satisfies the spirit of the intrinsicality intuition for medium-sized dry goods insofar as their macroscopically-relevant volumes depend on those regions. Again, this is because whenever the volume of a macroscopic material body is “involved in” a particular physical phenomenon, the volume of the contiguous bounding region it (partly) occupies is thereby also so involved.

Second, the intrinsicality intuition is dialectically important because it distinguishes the M-R account from its competitors: The M-R account satisfies the intuition; its competitors don’t. But, as we’ve now seen, the extant competitors to the M-R account (with the partial exception of Field) don’t merely fall short of satisfying the intrinsicality intuition, they’re views on which volume ratio relations are radically extrinsic. In most cases, volume ratio relations were grounded in holistic structural facts about the global domain of actual voluminous entities (or of all volume properties, in the second-order case). The extrinsicality of these theories isn’t anywhere close a borderline case.³¹ So, even if what the M-R account achieves falls short of genuine intrinsicality according to the demands of a specific definition, the M-R account still does much much better, in this regard, than any available alternative. This is sufficient for my purposes.

This system assumes the axioms of classical extensional mereology (CEM).³³ (P1)Pxx(P2)(Pxy∧Pyz)→Pxz(P3)¬Pyx→∃z(Pzy∧z≠y∧¬Ozx)(Sum)∃z(z∈S)→∃x(∀w(w∈S→Pwx)∧∀w(Pwx→∃y(y∈S∧Owy)))Oxy=df ∃z(Pzx∧Pzy)Cxyz=df Pxz∧Pyz∧∀w(Pwz→(Owx∨Owy)) ‘Pxy’ reads “x is a part of y”. According to this system, parthood is reflexive (P1) and transitive (P2). I also assume the principle of strong supplementation (P3),³⁴ and unrestricted composition: (Sum) says that for any (non-empty) set of objects, there exists an object, x, which is their mereological sum. The predicates ‘Oxy’ and ‘Cxyz’ are to be read, respectively, as “x overlaps y” and “x and y compose z”.

On the M-R account, volume is a determinable quantity associated with a class of fully determinate magnitudes, i.e. intrinsic volume properties. Since each such property is a fully determinate way of having volume, an object can instantiate at most one volume property. Let’s introduce the two-place predicate ‘≈’. x ≈ y just in case x instantiates the same determinate volume property as y. It can be pronounced more simply as: “x has the same volume as y”, or “x is as voluminous as y”. Those who are uncomfortable embracing a realist conception of properties, or who are sympathetic to comparativism³⁵ about quantities, may accept a variant of my account on which ‘≈’ is not a derived relation, but an unanalyzed primitive two-place predicate.³⁶ The rest of my presentation will be consistent with either approach (that is, I will not need to make any appeals to volume properties outside of my use of ‘≈’).

Since the determinate volume properties exclude one another, if x instantiates a different volume determinate from y, it follows that x doesn’t instantiate the same volume determinate as y. From this, and the symmetry and transitivity of identity, we can derive: (≈ Sym)∀x∀y(x≈y→y≈x)(≈ Trans)∀x∀y∀z((x≈y∧y≈z)→x≈z) I will pronounce ‘x ≈ x’ as “x is voluminous” (since x ≈ x just in case x instantiates a determinate volume property). From (≈ Sym) and (≈ Trans) we can derive a limited form of reflexivity: if x bears ≈ to anything, then x is voluminous, i.e.

Let me introduce the three-place predicate ‘xy ○ z’, which stands for “x and y are put together in the right way and compose z”, or “x and y concatenate to make z”.³⁷ The definition of ‘○’ differs between different quantities. For voluminous objects, all that is required for a and b to be put together in the right way is for them not to overlap: ab○c=df¬Oab∧Cabc The defined-up ‘○’ predicate can be used to formulate axioms that apply to quantities like length or temporal duration just as well as they apply to volume. That is, while different properly extensive quantities will disagree about what’s required to be “put together in the right way”, they will agree about the overall structure of the axioms. Hence, one could straightforwardly adapt this system to apply to a quantity like length, temporal duration, or the invariant relativistic interval, simply by introducing a different definition for ‘○’.³⁸

In general, a combination principle encodes the role of ‘○’ or ‘put together in the right way’, however it’s defined, in a broader account of that quantity’s structure. When it comes to the Volume Combination Principle, or (V-Comb), we encode the role of ‘○’ as it’s defined for volume (above) in our account of volume’s quantitative structure. That is (filling in ‘○’s definition): if a and b are voluminous, don’t overlap, and compose c, then c is voluminous.³⁹ (V-Comb)a≈a∧b≈b∧¬Oab∧Cabc→c≈c

The M-R account defines “a is less voluminous than b”, or ‘a ≺ b’, and “a is at least as voluminous as b”, or ‘a ≼ b’ as follows:

That is, a is less voluminous than b just in case they differ in volume and a has the same volume as one of b’s parts. The M-R account defines “c is as voluminous as a and b put together” (or “c’s volume is the sum of a and b’s volumes”) as “there exists some x ≈ a and y ≈ b such that xy ○ c”.

One reason to choose spatial volume as our example is that the ordering relation on voluminous objects is, plausibly, a total order. That is, if a and b are voluminous but don’t have the same volume, then either a’s volume is greater than b’s or vice versa.  (Totality*)a≈a∧b≈b→(a≼b∨b≼a) Put another way: for any voluminous a and b, a is either less voluminous than, more voluminous than, or of the same volume as b. (‘a ≼ b ∨ b ≼ a’ is equivalent to ‘a ≺ b ∨ b ≺ a ∨ a ≈ b’). Not all properly extensive quantities satisfy unrestricted totality. While all of them satisfy some form of a totality axiom, for some quantities their ordering is only total within certain sub-domains.⁴⁰ The M-R account should be, and, indeed, is, applicable to those properly extensive quantities as well.

Expressed in the fundamental, mereological terms of the M-R account, the totality axiom satisfied by volume says:  (Totality)a≈a∧b≈b→∃x((Pxb∧a≈x)∨(Pxa∧b≈x)) In prose: If a and b are voluminous, then either a has the same volume as some part of b or vice versa.

The axioms (Additivity) and (Prop. Extended) jointly characterize volume’s proper extensiveness. I’ll discuss them in turn. (Additivity)a≈a∧b≈b∧ab∘c→∀x∀y∀z(x≈c∧yz∘x→(y≈a→z≈b))

(Additivity), takes a bit of unpacking. All properly extensive quantities are additive: if a and b concatenate to make c, then c’s volume is the “sum” of a’s and b’s volumes. Importantly, if a and b’s volumes sum to c’s, then c’s volume cannot be the sum of a’s volume and some volume other than b’s (just as 6 + 9 = 15 means that 15 cannot be the sum of 6 and some other number ≠ 9). Here’s how this feature is encoded in the axiom (Additivity): If z is a voluminous object composed of voluminous x and y, put together in the right way, then either x ≈ a and y ≈ b , or vice versa (since a and b’s volumes sum to c’s volume), or neither x nor y share their volumes with a or b.

(Prop. Extended) has two jobs (indeed, my original temptation was to break it into two distinct axioms). If we take the M-R definitions on board, then (Prop. Extended) is equivalent to saying that (1) the ≼ ordering on voluminous objects is transitive, and (2) whenever a is less voluminous than b, b is as voluminous as a and something else put together. If a is one of b’s voluminous parts, then anything with the same volume as b, call it d must have a part with the same volume as a. If a ≈ b, then this part is d itself; otherwise d is composed of a pair of voluminous parts, one of which has the same volume as a.

Finally, I’ll introduce an assumption that, while not strictly necessary to obtain all the results we want from this system, greatly simplifies our presentations of the definitions in the next section. It amounts to the stipulation that there can be no voluminous entity which is infinitely more voluminous than some other one. More technically, it says that, if b is some voluminous entity, then b cannot be composed of an infinite set of non-overlapping parts, all with the same volume.

One interesting consequence of this Archimedean assumption is that there cannot be a “zero magnitude” of volume. If by “b has zero volume” we mean that ∀a∀c(ab ○ c → c ≈ a), then the Within-Object Archimedean assumption entails that any such b must not be voluminous (i.e. b ≉ b).⁴¹ This assumption, therefore, implies that points of space, one-dimensional lines, or two-dimensional planes in space are quite literally volume-less—they do not instantiate a volume magnitude. This doesn’t mean that we deny that these entities exist; it just means we deny that such entities are voluminous (and so aren’t picked out by phrases like “so-and-so’s voluminous parts”). I could have accepted a weakened Archimedean assumption that allows for a zero magnitude of volume, but it would add unnecessary complexity while making no difference to what the system can prove.⁴² I discuss a more substantive way we might weaken (W-O Archimedean) in section 5.1.

The first step will be to define a different procedure, which I call “taking x out of y” for some voluminous x and y, which tells you how many “copies” of x can “fit inside” y, and whether there’s some remainder. The ratio procedure, we shall see, is defined in terms of repeated applications of this procedure.

To “take x out of y” is to determine the maximum number of non-overlapping proper parts y can be partitioned into such that all (except, perhaps, one) of those parts bear ≈ to x. That is, whenever we take x out of y, for x ≉ y, the procedure outputs a pair of entities: A part, r, of y such that r ≼ x, which we’ll call the “remainder”. The second is an integer (we’ll call it the “count”), which is the cardinality of a particular set, S, such that (1) every member of S bears ≈ to x, (2) no member of S overlaps any other member, (3) y is the mereological sum of all the members of S ∪ {r}.

We take a out of b, where a ≈ a and b ≈ b, as follows: If b ≺ a, then there are no parts of b which bear ≈ to a. The output of this procedure is the integer 0, and the remainder is b. If a ≈ b, then the output of this procedure is the integer 1 and there is no remainder. b ≈ a so b is the fusion of 1 copy of a without remainder. The third case, a ≺ b, is the more interesting one:

Continue this procedure for every d_n arrived at in this way. I.e.:

From this definition, it’s easy to see that taking a out of b has a defined output for any voluminous a and b.⁴³ This procedure is unique up to the volume of the remainder, and taking x out of y, where x ≈ a and y ≈ b, has the same output (up to the volume of the remainder) as taking a out of b.

When d_n ≈ a, then a “goes evenly into” b—i.e. b can be partitioned into n + 1-many non-overlapping parts, all ≈ a. This brings us to our first Lemma, which says that whenever there is a minimal element,⁴⁴ u, every voluminous entity is the fusion of k non-overlapping parts all ≈ u, where k is some integer.

Lemma 1. If there exists a minimal element, call it u, – that is, if ∃u∀x(x ≈ x → u ≼ x) – then, for all voluminous b, the remainder d_n left after we take u out of b bears ≈ to u.

Proof. Since d_n is part of the output of taking u out of b, it must be that this procedure terminated with d_n. Hence, by the definition of the procedure, d_n ≼ u. But, by the minimality of u, u ≼ d_n. Hence, by the definition of ≼, u ≈ d_n. □

We use the procedure for “taking x out of y” to define the volume ratio relations—i.e. those designated by statements like “b is n-times the volume of a” for some n∈ℝ and voluminous pair a, b. We will define a “ratio procedure” which, as I mentioned before, will consist of repeated application of the taking out procedure: first taking a out of b and then, if there’s a remainder, taking that remainder out of a, and so on. Each application of the taking out procedure gets us a better and better approximation of the ratio of a to b.

After defining this procedure I show how it allows us to generate the M-R account’s definitions of volume ratio relations, and I’ll argue that the relations picked out by this procedure are ratio relations properly-so-called. The relations themselves will not require appeal to, or quantification over, numbers or other mathematical objects. The M-R definition will make use of nonnegative integers, but only in the case where they serve to count the members of some well specified, finite class of voluminous entities.

We can now determine the general definition schema for the volume ratio relations. Recall that, b “partitions into” some class of its parts iff they are all voluminous, none of the members of that class overlap, and b is their fusion. The schema is as follows:

The right side of this definition is precisely the sufficient condition for the ratio procedure on 〈a, b〉 to output the particular list of integers K(a, b) = 〈k₀, k₁, k₂, …〉. This definition only involves appeal to a, b and their parts, and, beyond the mereological relations, only appeals to ‘≈’, i.e. “instantiates the same determinate volume property as”. Therefore, the volume ratio relations are intrinsic. Since every positive real number can be uniquely picked out by the simple continued fractions generated from a list K(a, b), this definition allows us to associate ordered pairs of voluminous objects with a unique real number which characterizes their volume ratio. This schema is a formalized and mereologized version of the “anthyphairetic ratio” between some pair of objects, and the ratio procedure is closely related to the process of anthyphairesis.⁴⁶

I have argued that we can, simply by counting up the right sets of their parts, match each ordered pair of voluminous objects to a unique real number. I’ve also suggested that there’s good reason to think these numbers correctly characterize the physical volume ratio between that pair. The usual punch-line to an account of metric structure involves proving representation and uniqueness theorems. The M-R account of volume’s metric structure, however, does not need to appeal to result of such a theorem to establish that there are volume ratio relations. Representation and uniqueness theorems are not necessary to give an account of the quantitative relations we appealed to in the explanations at the beginning of this paper.

It would be possible to prove representation and uniqueness theorems about this system, but not as part of its account of volume’s metrical structure. Rather, they would show that the physical volume ratio relations I characterize above imply that these relations can be faithfully represented by the right mathematical ratios. As such, we will not need to prove the usual sort of theorem, that starts with only the ordering and summation relations over the domain of voluminous objects, and gets to the ratio relations by showing that functions from the domain to the real numbers which preserve ordering and summation all agree about certain metric facts.

We, on the other hand, can appeal directly to the volume ratio relations, whose physical definitions are fixed by the ratio procedure, and consider whether mappings from objects to numbers preserve volume’s ratio structure.⁴⁷ As such, the M-R account grounds metric structure in a thoroughly unit-free way. That is, we do not need to specify a particular function, φ, and some arbitrary voluminous object, u, to serve as the “unit” such that the image of any other voluminous object is defined in terms of the end result of the ratio procedure u and that object. Rather, we can express a simple rule which any function φ will satisfy just in case it faithfully represents/preserves volume’s metric structure:

What (RULE) does is show a correspondence between certain basic numerical relations and certain mereological ones. This is important because the definition of the ratio procedure for a given a and b is defined entirely in terms of repeated applications of the “taking out” procedure for various pairs of a’s and b’s parts. What this means is that the very ratio procedure defined in the previous section, combined with (RULE), will be able to provide a full specification of the numerical ratio between the numbers that a function must assign to a given voluminous pair, which is provably identical to the number which characterizes the volume ratio between that pair.

That is, this rule, while simple in expression, turns out to allow us to prove what I call the Direct Ratio Theorem for volume:

Direct Ratio Theorem. Every function φ:V↦ℝ+ satisfies (RULE) if and only if:

Moreover, for any pair of functions φ and φ′ which both satisfy (RULE), there exists some m∈ℝ+ such that, for all x ∈ V: φ(x)=m*φ′(x) Where m is such that, if there exists some u, v ∈ V where φ(v) = φ′(u), then Vrat:m (u, v).

Rather than bothering with ordering or summation structure, this theorem concerns ratio structure directly. The proof of this theorem requires no postulation of an arbitrary unit. It directly concerns the feature which all such functions must have if they are to preserve the ratio structure of the voluminous entities. I maintain (though there is not enough room to formally demonstrate it here) that the system presented in this section is rich enough to prove the Direct Ratio Theorem.⁴⁸

One might object that this account does rely on a contingent assumption about the structure of the domain after all, since I assume that the world is Archimedean. However, there are two reasons this assumption is acceptable. The first is that the “Within-object Archimedean property” is still an intrinsic assumption. It says, roughly, that for any two voluminous a and b, there is always a finite number of non-overlapping copies of a that “fit” in b, and vice-versa. This basically amounts to the assumption that there are no pairs of voluminous objects that stand in what we might describe as an “infinite ratio”. Since the focus of this paper is on metricality, it makes sense to simplify things for ourselves and rule this out.

However, this Archimedean assumption is not one that this system genuinely needs, even though it’s a reasonable assumption to make about the actual world. That is, if we were to drop this assumption, we could still recapture many of the results of the view. The sort of “ratio” relations we would be able to define in the non-Archimedean case would correspond to something over and above what we think of as ordinary metric structure. The right representational tool would likely be some sub-structure of the surreal numbers. I think the M-R account could be extended in this way, though I won’t argue for this in detail.

The way to go about it, I think, would be to define an equivalence relation over the quantities, such that each equivalence class contains all and only quantities which bear finite ratios to one another. We could, for instance, do this by way of something like the taking out procedure: for every x ≈ x and y ≈ y, x and y are “finitely comparable” iff there exists an n∈ℝ+ such that either x ≼ y and y can be partitioned into, at most, n non-overlapping copies with the same volume as x, or y ≼ x and x can be partitioned into, at most, n non-overlapping copies with the same volume as y.

The Within-object Archimedean assumption will hold within each equivalence class carved out by this relation, and so the M-R account, unmodified, will apply to them. Ratios within equivalence classes, then, will be finite and defined in the normal way. Ratios between voluminous entities which are not finitely comparable would be infinite. We could just define “infinite ratio” to be the failure of finite comparability. Via the ordering we could define two kinds of infinite ratios (intuitively “infinitely-many-times more voluminous than” and “infinitely-many-times less voluminous than”). It’s not clear if much else would need to be done to accommodate the non-Archimedean case, but my guess is that the M-R account of volume’s quantitative structure has the resources to capture it.

Volume is a properly extensive quantity whose ordering satisfies an unrestricted totality condition. I mentioned above that there are quantities which do not satisfy totality. Consider, for instance, the case of the invariant relativistic interval, “I”, in special relativity. If we understand the interval as measuring something like the spatiotemporal “length” of a path through Minkowski space time, then the quantitative ordering relation is not total over the domain of all spatiotemporal paths. No space-like path, i.e. a path composed of events which are each at space-like separation from all of the others, is either shorter or longer than any time-like trajectory connecting two time-like separated events.⁴⁹ On the various ways I is represented, numerically, space-like paths are assigned negative (or imaginary) numbers, while time-like ones are assigned positive (or just real) numbers.

The ordering relation does apply, however, within each sub-domain (i.e. of all the time-like trajectories, or of all the space-like paths) and, indeed, the relation is total. So, in these cases, while I is, plausibly, a properly extensive quantity that does not satisfy (Totality) in general, there are analogues of the axiom which are satisfied by certain sub-domains. Within those sub-domains, ratio relations will be definable and faithfully representable with the right mathematical structure. These ratio relations will remain silent on the relationship between a time-like trajectory and a space-like path (since the ratio procedure for such a pair will be unperformable), but, in such a case, that’s exactly what we want.⁵⁰

From the start (see the discussion at the start of Section 1) I have made it clear that, on the whole, the alternate theories of quantity in the literature are more general than the M-R account, in that they apply to more quantities. The M-R account has many advantages, but these advantages only extend as far as the properly extensive quantities.

Moreover, there’s no prospect to tweak the M-R account to generalize it, since the view crucially depends on a mereological feature that quantities like mass or temperature do not have. One might read this as a (perhaps defeasible) disadvantage of my account. I think this would be a mistake. Generality is a good-making feature of a theory insofar as we want to give a unified account rather than a disjunctive one. But a unified account is valuable only to the extent that it does not paper over metaphysically important distinctions.

For the M-R account, the restriction to properly extensive quantities is not a handicap of the view; it’s an explanation of what it is about these quantities that grounds their physical quantitative structure, and of what about them makes it such that this structure is faithfully represented by a given bit of mathematics. Losing the restriction to properly extensive quantities means losing the explanatory strength of the M-R account. The M-R account tells us about how the metrical structure of certain quantities is grounded in the kinds of things they are. That it does not extend to other kinds of quantitative properties, whose relationship with the physical world is different, merely shows that the explanation the M-R account provides really does turn on the characteristics I identify (namely, the proper extensiveness of these quantities).

Moreover, I think the M-R account may help us make strides in the direction of an account of the structure of non–properly extensive quantities (i.e. merely additive or intensive quantities) as well. The reason for such hope is that physical quantities do not exist in isolation, they influence, interact with, and depend on other aspects of the physical world and other physical quantities. As such, there’s prima facie justification for a “hierarchical” theory of quantity, on which the metric structure of one quantity may be grounded in/defined in terms of the metric structure of another quantity (spatial length or volume, perhaps), which is already established via independent means (e.g., by the M-R account presented here).

Burgess (1984) and (1991) demonstrates that such a theory is formally viable, by developing an account of Newtonian mechanics which generates the ratio structure of mass (and others scalars) from the distance ratios between collinear points, which is defined independently. Indeed, we may take a cue from Mach (1893) and consider whether there are genuinely physical connections⁵¹ between the quantities, like dynamical behavior, which might give us a means to ground the quantitative structure of one in the structure of the other. That is, just as Mach attempted to define mass in terms of acceleration (or tendencies to acceleration), we might take mass ratios as determined by acceleration ratios (which, in turn, are grounded in the structure of properly extensive quantities like length or temporal duration). This would allow us to give a hierarchical account of mass’s metric structure in terms of the metric structure of properly extensive quantities.⁵²