Absolutism vs Comparativism AboutQuantity

Shamik Dasgupta

We naturally think that material bodies have weights, sizes, masses, den-sities, volumes, and charges; that there are spatial distances between themtemporal durations between events involving them. These are all featuresthat fall under the category of quantity.

In this paper I discuss a question that arises for all quantities but whichis best illustrated by the case of mass. The property of having mass is adeterminable that appears to have two kinds of determinates. On the onehand, we naturally think that something with mass has a determinateintrinsic property, a property it has independently of its relationships withother material bodies. But we also think that things with mass stand invarious determinate mass relationships with one another, such as x beingmore massive than y or x being twice as massive as y. My question is: ofthe intrinsic masses and the mass relationships, which are fundamental?

According to a view I will call absolutism, the intrinsic masses are fun-damental. Loosely speaking, the view is that the most fundamental factsabout material bodies vis a vis their mass include facts about which intrin-sic mass they possess. The absolutist does not deny that things with massstand in determinate mass relationships, and she might even agree thatthose relationships are fundamental too. But more likely she will thinkthat those relationships hold in virtue of the intrinsic masses: if my laptopis twice as massive as my cup, the idea is that this is (at least partly) invirtue of the particular intrinsic mass that they each possess.1 In contrast,comparativism is the view that the most fundamental facts about materialbodies vis a vis their mass just concern how they are related in mass, andall other facts about their mass hold in virtue of those relationships.2

1I say at least partly because an absolutist might say that it is in virtue of their intrinsicmasses and various higher-order relations between those intrinsic masses (see for exampleMundy [24]). But this issue will not concern us here.

2Along, perhaps, with higher-order facts about those mass relationships, though againthis issue does not concern us here. Thanks to David Baker for helpful suggestions abouthow to formulate these views.

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I will describe both these views more precisely in Section 1, and aswe will see an analogous issue arises for all quantities. Given the centralrole that quantities play in our understanding of the natural world, thequestion of absolutism vs comparativism is central to any inquiry intowhat the natural world most fundamentally consists in. Moreover thereis more at stake in the issue than just our understanding of quantities,for comparativism brings with it a commitment to fundamental externalrelations and these have historically been treated with some suspicion. It istherefore surprising that the contemporary metaphysics literature containsrelatively little discussion of the issue.

In this paper I motivate and defend comparativism. But I am less inter-ested in pressing that particular view as I am in formulating the issue andbroadly surveying what I take to be the more important lines of argumentfor each position.

The main consideration in favor of comparativism is developed at theend of the paper in Section 8 and is based on the idea that we can onlyever observe mass relationships. More fully, the idea is that even if mate-rial bodies possessed the intrinsic masses posited by the absolutist, thoseintrinsic masses would be undetectable in a very strong sense: the struc-ture of the physical laws governing our world would guarantee that theycould never have an effect on our senses. As is well known, absolute ve-locity and absolute simultaneity are undetectable in this very same sense,and for this reason most contemporary metaphysicians and physicists callthem redundant or superfluous and dispense with them on Occamistgrounds. My thought is that the same reasons should compel us to dis-pense with the intrinsic masses posited by the absolutist. More precisely,the Occamist principle I appeal to is that positing undetectable structureis a vice, in the sense that if one theory of the material world posits un-detectable structure that another does not then all else being equal weshould prefer the latter. All else being equal, then, we should prefer com-parativism. Now, whether all else is indeed equal depends on whetherthere are stronger countervailing reasons to reject comparativism. So insections 27, the bulk of the paper, I consider a number of potential ob-jections to comparativism and argue that none are convincing; hence mypreference for comparativism.

The discussion will mostly be limited to the case of mass, and I willmake the simplifying assumption that the correct physical theory of massis a classical, Newtonian theory. The assumption is of course false, butmuch of the paper discusses rather general, philosophical considerationsthat do not depend on the physical details, and in those discussions thissimplifying assumption is harmless. Considerations from physics are dis-

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cussed towards the end of the paper, and there the assumption may beless harmless. But I continue to assume it because the considerations turnout to be conceptually rather delicate and so it is worth clarifying themin the context of this simplifying assumption before asking whether theygeneralize to more realistic physical theories. I think the considerationsstand a good chance of generalizing to at least some other quantities andphysical theories, but whether this so is a question for another time.

1 More on Absolutism and Comparativism

The absolutism vs comparativism issue per se has not received much dis-cussion. To be sure, views that count as absolutist have been defended byArmstrong [2] and [3], Mundy [24], and Lewis [22]; and views that countas comparativist have been defended by Ellis [13], Bigelow and Pargetter[5] and [6], and Field [14] and [15]. But those discussions are often inter-twined with other issues about quantites, so it will help to clarify the issueI have in mind before considering arguments either way.

I understand both views in terms of the idiom of one kind of factholding in virtue of another. As I use the phrase this is an explanatoryidiom: to say that a fact holds in virtue of another is to say that the lat-ter explains the former in a distinctively metaphysical sense. To illustrate,imagine asking what explains Europes being at war in 1939. A causalexplanation might describe events during the preceding 50 years that led,say, Chamberlain to declare war. But there is another kind of explanationthat would try to say what goings on in Europe at the time made it countas a continent at war in the first place. Regardless of what caused the con-flict, someone in search of this second explanation recognizes that war isnot a sui generis state and that there must therefore have been somethingabout the continent that made it count as being a continent at war ratherthan (say) one at peace. A plausible answer is that it was at war in virtueof how its citizens were acting, for example that some were committingpolitically motivated acts of violence against others. As I use the phrase,an explanation of this second kind is a statement of that in virtue of whichEurope was at war. I take this kind of explanation to be reasonably intu-itive: regardless of the truth of this explanation of why Europe was at war,we seem to understand the claim reasonably well.3

Earlier I said that the absolutist thinks that the most fundamental facts3The term ground has been used by Kit Fine and Gideon Rosen to describe this sense

of metaphysical explanation (see Fine [16] and [17], and Rosen [27]). But since that termhas also been used in other ways in the recent literature I have used the phrase in virtueof to try to minimize confusion.

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about material bodies vis a vis their mass include facts about which intrin-sic mass they possess. We can put this more precisely in terms of the aboveidiom: the view is that there are facts about which intrinsic mass a givenmaterial body possesses that are not explained (in the above metaphysicalsense) in other termsin particular, that are not explained in terms of itsmass relationships with other bodies.4 As I said earlier, the absolutist neednot deny that things with mass stand in various mass relationships, butshe will likely think that when they do this is at least partly explained (inthis metaphysical sense) by the particular intrinsic mass that each bodyhas.5 In contrast, the comparativist thinks that all facts about materialbodies vis a vis their mass are explained (in this metaphysical sense) interms of the mass relationships between bodies.6 This leaves open whatkinds of mass relations are most explanatorily basic: they might be massratios such as an object being twice as massive as another, orderings suchas an object being more massive than another, or even just linear structuressuch an object lying between two others in mass. But this in-house disputewill not matter for our purposes. For simplicity I will often assume thatthe comparativist under discussion is of the first type, but nothing willhang on this.7 As stated, both views are claims about what actually holdsin virtue of what and so are neutral on the relation between intrinsic massand mass relationships in other possible worlds. One might of course ar-gue that comparativism is necessarily true if true at all (mutatis mutandisfor absolutism) but I will not take a stand on this issue here.8

I have not yet mentioned facts about mass in a particular scale, suchas my laptops being 2 kgs, and one might consider this omission strangesince it is this sort of fact that we most often express when talking aboutmass. But it is not immediately obvious whether this is ultimately a fact

4This might be because there are facts about which intrinsic mass a material bodypossesses that are unexplained. Or, if there are infinite descending chains of explanations,it might be because any such fact is explained by another fact of the same kind; that is,another fact about which intrinsic mass something has.

5Though this is not enforced: the absolutist might conceivably think that there are alsounexplained facts about mass relationships along with unexplained facts about intrinsicmasses. But I do not know of anyone who holds this view or of any reasons in its favor soI will largely ignore it in what follows.

6Along, perhaps, with higher-order facts about the relations themselves; see footnote 2.7Baker [4] discusses which kinds of relations the comparativist should and should not

think are explanatorily basic. He also discusses (without endorsing) the idea that thecomparativist might introduce what he calls mixed relations. These are inter-quantityrelations, for example my mass being twice your length. I will not discuss these mixedrelations here.

8Both views, I should say, are neutral as to the status of facts about the nomic relationbetween mass and other quantities. The issue just concerns the status of catergoricalfacts concerning the masses of things.

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that holds in virtue of my laptops intrinsic nature or in virtue of its massrelationships, so to avoid begging questions it is best to state the ab-solutism/comparativism issue without mentioning these facts and leavetheir status as a further question (I discuss this question in Section 5).

I have so far assumed that holding in virtue of is a relation betweenfacts and I will continue to do so for ease of prose. But those wary of factsmay express claims about what holds in virtue of what with the sententialoperator because. Thus, when I previously said that Europes being atwar holds in virtue of facts about the actions of its citizens at that time,one could re-state this without reference to facts as

Europe was at war in 1939 because a battalion of troops marchedto Normandie in 1939 and. . . 9

so long as because is understood in the metaphysical rather than causalsense. On this way of talking, the absolutist will assert things like

My laptop is more massive than my cup because my laptop hasthe intrinsic mass M and my cup has the intrinsic mass M*.

whereas the comparativist will deny such a claim. Moreover, I have sofar made free reference to such things as intrinsic masses and mass re-lations. For example, the natural interpretation of the indented sentenceabove takes M to be a term referring to a property and the expressionhas the instrinsic mass to be a relational predicate holding of my lap-top and the property M. But nominalists wary of properties and relationscan also make sense of the absolutism vs comparativism issue by readingthe indented sentence in another way. For example, they might take hasthe intrinsic mass M to be a primitive monadic predicate containing noreferential devices at all. Still, in what follows I will continue to refer toproperties and relations for ease of prose.10

Although I have focused on the case of mass, it should be clear that thesame issue arises for other quantities too. For example, consider the case ofspatial distance. When material bodies X and Y stand in the determinable

9This way of expressing claims about what holds in virtue of what is suggested byFine (see his [16] and [17]). Strictly speaking the right hand side will consist of a list ofsentences rather than a conjunction, but this complication does not matter to us here.

10This is not to say that the absolutism/comparativism issue is entirely independent ofthe issue of realism about properties. One might argue, for example, that the nominalisthas a hard time being an absolutist since her vocabulary would then be required to in-clude an infinite number of primitive predicates, one for each determinate intrinsic mass.Still, logically speaking the issues are orthogonal and in what follows I will not be con-cerned with considerations that depend on a resolution to the question of realism aboutproperties.

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relation of being spatially related, there are two kinds of determinate re-lationships that they enter into. On the one hand, it is natural to thinkthat X stands in a determinate distance relation to Y. But it is also naturalto think they stand in various comparative spatial relations to other bod-ies, for example the relation of X being twice as far from Y as Z is fromW, or even just of X being further from Y than Z is from W. But of theformer fact about the distance between X and Y, and the latter fact aboutthe comparative distance between X, Y, Z and W, which is fundamental?The absolutist says that the most fundamental facts about distance includefacts of the former kind, while the comparativist says that all facts aboutdistance are ultimately explained (in the metaphysical sense) in terms offacts of the latter kind.11 This is why I used the terminology absolutismand comparativism in the case of mass rather than intrinsicalism andrelationalism: when we generalize to other quantities such as spatialdistance, the facts that the absolutist takes to be most fundamental arethemselves facts about relationships.

As I said, my main reason for endorsing comparativism (developed inSection 8) is that the intrinsic masses posited by the absolutist would beundetectable in a very strong sense. The Occamist principle described ear-lier deems this to be a significant mark against absolutism. However, thisis not a decisive mark against the view and if there were stronger coun-tervailing considerations against comparativism then absolutism wouldremain our best all-things-considered theory. So in Sections 27 I con-sider six kinds of objection to comparativismobjections from intuition,from modality, from semantics, from kilograms, from Humeanism, and fromphysicsand argue that none are compelling. All things considered, then,I believe that comparativism is the preferable view.

2 Objections from Intuition

Let us start with objections to comparativism from intuition. When first in-troduced to the issue, absolutism strikes many as being the more intuitiveand plausible view. If my laptop is more massive than my cup, it ini-tially seems that this is (at least in part) because of their intrinsic masses.Moreover, comparativism conflicts with an intuitive locality principleconcerning mass, namely that given a connected region of spacetime R

11The absolutism/comparativism issue about distance should not be confused with thesubstantivalism/relationalism issue. The latter issue concerns the relata of spatial relationsand asks whether they are, fundamentally speaking, material bodies or regions of space.Whichever way that dispute is resolved, we may then raise the absolutism/comparativismdebate by asking whether, at the fundamental level, those relata stand in 2-place absolutedistance relations or 4-place comparative relations.

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composed of two sub-regions R1 and R2, the fundamental facts about masswithin R1 and the fundamental facts about mass within R2 together deter-mine the fundamental facts about mass within R.12 This principle strikesus as intuitively correct, but the comparativist must deny it since the massrelationships within R1 and those within R2 do not determine the massrelationships between a body in the one region and a body in the other.

So let us agree that absolutism is initially the more plausible view. Onemight then argue that this is a reason to think that it is true. But the lastsentence conflates a number of arguments. One unconvincing argumentis that just as we are endowed with a reliable faculty of perception, weare also endowed with a reliable faculty of intuition which delivers theverdict that absolutism is true. To this we might well object that thereis good reason to doubt that we have a faculty of this sort (for one thing,anatomists and neurologists have yet to find anything corresponding to it).Here we need not deny that we have a faculty of intuition that is a reliableguide to math and logic; all we need insist is that we do not have one thatdelivers reliable verdicts about what the fundamental physical propertiesand relations are, for it is this that absolutists and comparativists disagreeabout. The denial of such a faculty should therefore be uncontroversial.

Still, one might argue that the intuitive plausibility of absolutism car-ries epistemic weight without appealing to a faculty of intuition. Forexample, one might say that absolutism is a Moorean truth, a propo-sition in whose truth we are more certain than any premise used in anargument to the contrary. Or one might point out that absolutism is ourstarting point in the inquiry, and then argue for a principle of epistemicconservatism according to which our starting point is (defeasibly) justifiedmerely by virtue of being our starting point. Either way, the upshot wouldbe that our initial absolutist inclinations are epistemically significant.

In response, I do not object to the principle of epistemic conservativismor to Moorean approaches to some questions in philosophy. In particular,I concede that absolutisms initial plausibility is at least some reason tobelieve it. But any such reason is defeated by my Occamist argumentto the contrary. To see this, consider the case of absolute simultaneity.While it is initially plausible that there is such a thing as simultaneity,most would agree that considerations from special relativity are enough todefeat any consideration from Mooreanism or epistemic conservativenessin simultaneitys favor.13 Now, I take the initial plausibility of simultaneity

12Thanks to Eliot Michaelson for elucidating this locality principle for me.13To the Moorean, this shows that our belief in simulteneity is not, after all, more certain

than the premises of any argument against it. To the epistemic conservative, it shows thatwhile our belief in simultaneity may have been epistemically privileged by virtue of beingour starting point, its privilege was not enough to ward off arguments to the contrary.

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to be at least as strong as the initial plausibility of absolutism. Therefore,since my reason to reject absolutism is the same as our reason to dispensewith simultaneity (I leave it until Section 8 to make good on this claim),it will be be strong enough to defeat considerations from Mooreanism orepistemic conservativeness in absolutisms favor.

3 Objections from Modality

Another class of objections to comparativism appeal to modal considera-tions. The idea behind all the objections is that the comparativist cannotmake sense of certain situations that seem, at least on the face of it, to bepossible. I believe that those with absolutist inclinations are often movedby these kinds of considerations, so I will consider four such objections insome detail.

The first objection can be put aside reasonably quickly. The objectionis that while it seems possible for everythings mass to double tonightat midnight, the comparativist cannot make sense of this since the massrelationships between things tomorrow would be exactly the same as theywere today and so by the comparativists own lights there would havebeen no change. But this objection is not compelling, for the comparativistmay claim that some of the fundamental facts about mass concern howsomething at one time is related in mass to something at another time. Ifso, she can make sense of the possibility after all.

A second, more compelling objection along these lines appeals to theintuition that it seems possible for everythings mass to always have beendouble what it actually is. The possibility of uniform doubling underconsideration here is not one in which everythings mass doubles at aparticular time, but rather one in which the entire history of the universeis just like ours with the one exception that at any given time, everythingsmass is double what it actually is at that time. The objection is then thatwhile this intuitively seems to be possible, the comparativist can make nosense of it.14

But why think that the comparativist cannot make sense of the pos-sibility of uniform doubling? At this point the objector needs to choosewhether to express her objection with modal operators or in the frame-work of possible worlds. In the latter framework, the possibility of uni-form doubling is represented by a possible world just like ours but withthe one exception that everythings mass is double what it actually is. Toargue that the comparativist can make no sense of such a world, the ob-

14Eddon [12] objects to comparativism in this way. Hawthorne [18] outlines the objectionand seems to take it to have some force, but he does not explicitly endorse it.

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jector would appeal to a plausible necessitation principle: that if a factY holds in virtue of a fact X, then every world in which X obtains is also aworld in which Y obtains.15 This necessitation principle is standard in theliterature on the in virtue of relation and is extremely plausible. For exam-ple, if the fact that Europe was at war in 1939 holds in virtue of facts aboutthe actions of its citizens, then those actions are what accounts for andmakesit the case that that Europe was in a state of war rather than (say) a state ofpeace. Consequently, it seems that any world in which the citizens of Eu-rope act in that way must also be a world in which the continent is at war,just as the principle implies. Now, along with the necessitation principlecomparativism implies that worlds agreeing on all the mass relationshipsagree on all facts about mass. But the so-called doubled world agreeswith the actual world on all mass relationships; hence according to thecomparativist it is not a world that differs from the actual world regardingany fact about mass and is therefore not a world in which everythingsmass is doubled.

That is the objection stated in the framework of possible worlds. Itcan also be stated using modal operators by formulating the necessitationprinciple in terms of modal operators instead. In what follows I will onlydiscuss the possible worlds version of the objection just outlined. Thedetails of the modal operator version differ subtly, but there is no space todiscuss those details and in any case the main morals of our discussion interms of possible worlds will carry over.

How might the comparativist respond to the objection? One responseis to become a modal realist in Lewis sense and say that the fundamentalfacts of the world are really facts concerning a plurality of concrete worlds.The comparativist may then think that the fundamental facts concerningmass relationships include how objects in different worlds relate to oneanother in mass. A comparativist of this sort will point out that the actualworld and the doubled world disagree on their inter-world mass relations,and therefore the necessitation principle does not imply that they agree onall facts about mass. But I do not wish to rest my case on this response.Putting aside the unpopularity of modal realism, a more important worryis whether the modal realist can legitimately allow fundamental relation-ships between objects in different worlds. For example, the generalization

15I am simplifying here. As I describe in Section 5, I believe that the in virtue of relationis irreducibly plural in the sense that some facts Y can hold in virtue of some facts X eventhough no Y taken on its own holds in virtue of anything. If that is right, the necessitationprinciple should be rephrased with plural variables as follows: if some facts Y hold invirtue of some facts X then every world in which the X all obtain is also a world in whichthe Y all obtain. But this complication is not relevant to the current objection so I willignore it here.

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of this approach in the case of spatial distance is that fundamental spatialrelations hold between bodies in different possible worlds, and one mightargue that this conflicts with Lewis account of a possible world as themereological sum of spatio-temporally related things.16

So it is worth asking whether the comparativist can respond to the ob-jection without appealing to fundamental inter-world mass relationships.I believe she can. In the first place, she may argue that her failure to makesense of the possibility of uniform doubling is no real vice; and in the sec-ond place she may argue that she can, perhaps surprisingly, make senseof the possibility without fundamental inter-world mass relations after all.Both responses seem reasonable to me, so I will discuss each one.

Start with the first. The objection rests on the intuition that a uniformdoubling of mass really is possible, but is this right? I find that my in-clinations here depend on my theoretical convictions: when absolutismstrikes me as attractive it seems possible, but when I am in the grip ofcomparativism I feel that the possibility is a silly philosophical mistake.This should not be surprising, since the absolutist and the comparativistare both likely to agree that if absolutism is true then uniform doubling inmass is possible. Now, if our intuition that doubled worlds are possiblerests on a prior belief that absolutism is true, the current argument wouldat best collapse into the argument from intuition just considered or, atworst, beg the question. So the question is whether we have an inclinationto think that uniform doubling is possible that is independent of any priorbelief in absolutism, and if so how strong that inclination is. I am not surehow one might go about answering this question, but an answer is crucialto the current argument. For now, then, it seems reasonable to take theissue of uniform doubling to be a case of spoils to the victor.

The second response is that the comparativist can, perhaps surpris-ingly, make sense of the possibility of uniformly doubling without appeal-ing to fundamental inter-world mass relationships. To see how let us turnto the third modal objection to comparativism, for our response to it willprovide the materials needed to make sense of uniform doublings. So,putting aside for a moment the possibility of everythings mass being dou-bled, note that it is surely possible for just my laptop to have been twiceas massive as it actually is. The third modal objection is that the compar-ativist cannot even make sense of this. On the face of it, this objection isfar more powerful than the last. For while we might reasonably deny thatit is possible for everythings mass to have been doubled we must surelyagree that my laptop could have been more massive than it is. If the com-

16Lewis himself thought that there can be no perfectly natural relationships betweenobjects in distinct possible worlds. See his [21], p. XX.

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parativist cannot even make sense of this latter possibility, that is a viceindeed. I am therefore surprised not to have seen or heard this argumentexpressed by those with absolutist inclinations.

But why think that the comparativist can make no sense of the possi-bility? Again, the objection can be developed in the framework of possibleor with modal operators, but let us stick to the former. The idea, then, isshe can perfectly well make sense of a world W just like ours except thatthe mass-ratio between my laptop and all other things is double what itactually is. But without inter-world mass relationships, there is no factof the matter as to whether W is a world in which my laptop is twice asmassive as it actually is, or one in which my laptop is the same mass andeverything else is half as massive as they actually are. Notice that if theactual bodies and those in W had intrinsic masses then the problem wouldnot arise, for those intrinsic masses would determine mass relationshipsbetween bodies in the one world and bodies in the other. But withoutfundamental inter-world mass relationships, the comparativist has no re-sources to make a similar inter-world comparison.17

How might the comparativist respond? I believe she can accuse theobjection of resting on an incorrect model of how a possible world rep-resents my laptops mass and introduce a better model that allows her tomake sense of the possibility in question. To see this, it will help to workwith a specific model of how a possible world represents something de reof my laptop in the first place. I will work with Lewis famous proposalthat it does so not by containing my laptop but by containing one of itscounterparts, though nothing hangs on this choice. Given this assumption,the world W introduced above can be re-described as a world containing acounterpart of every actual material body such that if my laptop is r timesas massive as another body x, my laptops counterpart in W is 2r times asmassive as xs counterpart in W.

Now, consider an object other than my laptop, such as my cup. Noticethat the mass relationships it enters into are extremely similar to thoseentered into by its counterpart in W. The only difference is that while mycup is (say) half as massive as my laptop, my cups counterpart in W isone fourth as massive as my laptops counterpart in W. Other than that,my cups mass relationships to all other bodies are exactly the same as itscounterparts relationships to theirs. But this is not so with my laptop: themass relationships it enters into differ systematically from those that itscounterpart in W enters into by a factor of 2. So the comparativist may

17The situation is not improved by noting that comparativism is a contingent claim andallowing the material bodies in W to have intrinsic masses. For so long as the actualmaterial bodies lack intrinsic masses, there remains no fact of the matter as to whether Wis a world in which my laptop is twice as massive as it actually is.

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say that it is in virtue of this asymmetry that W represents my laptop asbeing twice as massive as it actually is and everything else as being thesame mass as they actually are.

In effect, the comparativist just introduced a mass-counterpart re-lation in addition to the ordinary, Lewisian counterpart relation. Sincemy cup and its counterpart in W resemble one another with respect totheir mass role, we call them mass-counterparts. And because my cupscounterpart is also its mass-counterpart, W represents my cup as beingthe same mass as it actually is. Here the mass-counterpart relation is do-ing analogous work to Lewis counterpart relation: just as the latter is notidentity but stands in for it when determining what a world representsde re, the mass-counterpart relation is not the same-mass-as relation butstands in for it when determining what a world represents about mass.

Indeed, the comparativist can introduce a slew of mass-counterpartrelations, one for each real number. My cups mass role resembles itscounterparts mass role, so we call them mass1-counterparts. My laptopsmass role does not resemble its counterparts mass role in the same way,but since the mass ratios my laptop stands in are uniformly half thoseof its counterpart, their mass roles resemble each other modulo a factor of2. As a result, the comparativist can call them mass2-counterparts. Moregenerally, she can say that x and y are massr-counterparts just in case xsmass role resembles ys mass role modulo a factor of r. And she may thenpropose the general principle that, relative to a counterpart relation anda set of mass-counterpart relations, W represents an actual object x asbeing r times as massive as it actually is just in case x has a counterpartin W that is also xs massr-counterpart. Relative to the mass-counterpartrelations just described, this delivers the desired result that W representsmy laptop as being twice as massive as it actually is.

The ordinary counterpart relation is context sensitive, in the sense thatthe features of individuals relevant to determining whether they resembleone another, and therefore whether they are counterparts, are sensitive tothe context in which the modal claim is made. Similarly, the comparativistcan allow that those aspects of a bodys mass role relevant to determin-ing what its massr-counterparts are depend on the conversational context.For example, a context in which mass relationships to my laptop are par-ticularly salient might be one in which my laptops counterpart in W isalso my laptops mass1-counterpart (since the latter agrees with my lap-top on all mass relationships to itself). Relative to this mass-counterpartrelation, W represents my laptop as being the same mass as it actually isand everything else as being half as massive as they actually are!

Although the discussion so far assumed Lewis own theory of de re

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modality, the mass-counterpart theory just introduced is consistent withmany other theories including ersatz ones. There is of course much moreto say about it, but much of the ensuing discussion will resemble the lit-erature on Lewis own counterpart theory so instead let me return to thesecond modal argument we left earlier and explain how the comparativistcan use the mass-counterpart theory just introduced to make sense of thepossibility of uniform doubling. The problem, remember, was that a uni-formly doubled world agrees with our world on all mass relationshipsand so, according to comparativism and the necessitation principle out-lined earlier, also agrees with our world on all facts about mass whatsoeverand is therefore not a doubled world after all. But with mass-counterparttheory in hand, I believe the comparativist can accuse the objection of ig-noring the distinction between worlds and possibilities. That is, she canrespect the necessitation principle and concede that she can make no senseof a uniformly doubled world, but insist that she can nonetheless makeperfectly good sense of the possibility of uniform doubling.

To see how, note that it is a familiar fact that worlds and possibili-ties come apart in ordinary counterpart theory. To use Lewis example, Imight have been either one of a pair of twins: I might have been the firstborn, and I might have been the second born.18 Here we have one possibleworld with twins but two possibilities, one in which my counterpart is thefirst born and the other in which my counterpart is the second born. In-deed, we have already seen an analogous distinction between worlds andpossibilities in mass-counterpart theory: the worldW discussed above rep-resented two possibilities depending on which mass-counterpart relationwe focused on, one in which my laptop is twice as massive as it is and onein which everything else is half as massive as they are. In his discussionof worlds and possibilities, Lewis also says that our counterparts need notalways be in other worlds. When I consider the unhappy possibility ofbeing my neighbor Fred, Fred himself (my worldmate) is acting as mycounterpart and represents me as having all his properties.19 In this casethe actual world, along with a certain counterpart relation, is representinga non-actual possibility for me.

With mass-counterpart theory, the comparativist can model the pos-sibility of uniform doubling analogously to how Lewis models the pos-sibility of my being Fred, namely by using the actual world along witha suitable mass-counterpart relation. To see how, note that the possibil-ity of uniform doubling is a possibility for all the material bodies takentogether. If we let S be an ordered set of all those bodies, then accord-

18See Lewis [21], p. 231.19See Lewis [21], p. 232.

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ing to Lewisian counterpart theory a world represents a possibility forthose bodies by containing a counterpart of S. Well, surely S can be itsown counterpart in normal contexts. Moreover, S can also be its ownmass2-counterpart in normal contexts. This is because Ss mass rolethepattern of mass-relations entered into by the members of Sresemblesits mass role perfectly modulo a factor of 2: put intuitively, the pattern ofmass-relations are exactly as they would be were everything doubled inmass! And relative to these counterpart and mass-counterpart relations,our mass-counterpart theorysuitably generalized to apply to orderedsetsimplies that the actual world itself represents the possibility of uni-form doubling. So in this sense, uniform doublings of mass are possibleeven for the comparativist.

This is not to deny that there is another sense in which uniform dou-blings of mass are impossible for the comparativist. What we just did,effectively, is distinguish between two modal notions. One is the notionthat quantifies over what I called possible worlds above and might becalled strict possibility. This is the notion on which the above necessita-tion principle is true, that if a fact Y holds in virtue of a fact X then everypossible world in which X obtains is also a possible world in which Y ob-tains. The comparativist should concede that she can make no sense ofthe strict possibility of uniform doublingsindeed as the third modal ob-jection rightly points out the comparativist cannot even make sense of thestrict possibility of my laptop being more massive than it actually is! Butthere is another modal notion that quantifies over what I called possibil-ities above, where a possibility can be thought of as a triple of a possibleworld W, an assignment of objects to a counterpart in W, and an assign-ment of objects to a massr-counterpart in W.20 We might call this loosepossibility. As argued above, the comparativist can then make perfectlygood sense of the loose possibility of my laptops being more massive thanit actually is, and indeed the loose possibility of uniform doublings.

What the modal objections to comparativism need to establish, then, isnot just that the envisaged situations (uniform doublings of mass, or justmy laptop being more massive than it actually is) are possible in somegeneric sense but rather that they are strictly possible, for it is only thislatter claim that the comparativist must deny. I have seen no attempt byabsolutists to establish that they are strictly possible and I am skepticalthat they can do so. At the very least, the burden is now on the absolutistto establish the required premise. For this reason I remain unmoved by

20If one is attached to the idea that it is possible that P iff there is a world in which P istrue, one should feel free to call the possibilities possible worlds instead when using theloose notion. The nomenclature is not what matters here.

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these modal objections to comparativism.There is a more general lesson here, namely that when engaging in

modal reasoning (as these objections do) it is crucial to take great careto clarify the modal notion in use. For there may be many notions ofpossibility related to but distinct from strict possibility. If so, then whenfaced with the objection that she cannot make sense of this or that pos-sibility the comparativist can always explore the response of saying thatthe envisaged situation is possible in one of these other senses even if itis impossible in the strict sense. Indeed I have no particular allegianceto the counterpart-theoretic response developed above and am open toresting my response on another alternative to strict possibility, if one canbe developed. The counterpart-theoretic response is dialectically useful,though, since it nicely illustrates the existence of at least one of these othermodal notions.21

It is worth mentioning a fourth kind of modal argument against com-parativism, namely that the comparativist cannot make sense of a possibleworld containing just one massive body.22 In particular, one argument isthat she cannot make sense of it having the determinable property of hav-ing mass, while a second argument is that she cannot make sense of ithaving any particular determinate mass (say, the mass of an electron). Butin response to the first argument, the comparativist can say that havingthe determinable mass consists in standing in determinate mass relation-ships, and then point out that the lone particle stands in such a relationwith itself, namely the same-mass-as relation. And in response to the sec-ond objection, the comparativist can appeal to mass-counterpart theoryand say that whether the particle counts as having the mass of an electrondepends on the mass-counterpart relations allowed by the conversation inwhich the world is being discussed.

I conclude, then, that objections to comparativism based on modal con-siderations are not convincing.

21David Chalmers distinguishes two notions of metaphysical possibility that he callsprior and posterior possibility (he mentions the idea briefly in footnote 5, p. 449 of his [7]and is developing the idea in other work). His notion of prior possibility is I think thesame as the notion of strict possibility in use here. It is less clear to me how his notionof posterior possibility is related to the notion of loose possibility discussed in the text,though his notion of posterior possibility is certainly more general since the notion ofloose possibility is (as defined above) restricted to the case of mass.

22Thanks to Michaela McSweeney for helping me to appreciate the force of this argu-ment. Similar arguments are given by Eddon [12].

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4 The Objection from Semantics

One might instead try to refute comparativism on semantic grounds. Tosee how, recall Kripkes famous claim that we use the term meter withthe stipulation that it is to refer to the length of the standard meter inParis.23 The analogous view in the case of mass is that we use kilogramswith the stipulation that it is to refer to the mass of that lump of platinum-iridium alloy in Paris that serves as our standard of measurement, knownas the International Prototype Kilogram (IPK). But the entity that the Krip-kean theory takes the referent of kilogram to be, namely the mass of IPK,sounds suspiciously like an intrinsic property of IPK. After all, if the fun-damental facts about mass were just facts about mass relationships, it isdifficult to see what the mass of IPK could possibly be. So, the argumentgoes, if comparativism were true then kilogram would fail to refer andsentences like My laptop is 2 kilograms would fail to be true.

Of course, the comparativist might bite the bullet and concede thatkilogram sentences are not true. According to this error theory response,the term kilogram is similar to phlogiston: both were used with the stip-ulation that they are to refer to whatever entity satisfies some description,but since nothing answers to the description they both fail to refer. Themain difference between the two cases is that there is a pragmatic reasonto continue using the term kilogram that is lacking in case of phlogis-ton. For as long as our use of kilograms is governed by the inferencerule

a is r kilograms

b is s kilograms

Therefore, a is r/s times as massive as b

we can use kilogram as a convenient way of storing and communicat-ing information about mass ratios even if sentences containing it are nottrue.24 If this error theory sounds radical and unwarranted, compare it tothe case of absolute simultaneity. If semantic investigation revealed thatthe truth of our ordinary talk requires there to be such a thing as absolutesimultaneity but it subsequently turned out for reasons of physics andmetaphysics that there is no such thing, we would have no qualms con-cluding that our ordinary talk is in error. Similarly, if the above semanticargument showed that the truth of kilogram sentences requires the truthof absolutism but it subsequently turned out for reasons of physics and

23See Kripke [19].24I discuss the role of this form of inference in more detail in Dasgupta [10].

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metaphysics that absolutism is false, we should have no qualms acceptingthe resulting error theory.

But while this error theory is defendable, there is no need for the com-parativist to adopt it since the semantic objection fails to establish thatthe truth of kilogram sentences requires the truth of absolutism in thefirst place. In fact, it fails for two reasons. First, the Kripkean theory ofreference-fixing it presupposes is false. To see this, imagine reading in theTimes that the French have been subjecting us to an illusion that makesIPK appear twice as massive as it actually is. Imagine that the article ex-plains that the illusion has been systematic, so that whenever we used IPKto calibrate our measuring instruments, the calibration succeeded eventhough we were misled about the properties of the lump. So, if we wereto put IPK on one of the many calibrated measuring instruments aroundthe world, it would read 500 grams rather than 1 kilgoram.

How would we report this discovery? Intuitively, by saying that wediscovered the surprising fact that IPK is 500 grams! But the Kripkeantheory predicts otherwise. Since the theory is that 1 kilogram is stipu-lated to refer to the mass of IPK whatever that mass is, it implies that thearticle should instead be reported as telling us that while the standardobject is (of course) still 1 kg, all other material bodies are actually halfthe mass in kilograms that we previously thought they were. And this, Iclaim, is not how we would intuitively report it.

But there is a second and perhaps more decisive reason why the se-mantic objection fails to establish that the truth of kilogram sentencesrequires absolutism. Even if we granted the Kripkean theory of reference,the argument is supposed to be that the entity to which kilogram is stip-ulated to refer, namely the mass of IPK, is not identical to IPKs massrelationships, and it concludes that the comparativist must say that thereis no such thing. But this last step is a non sequitur. All the comparativistclaims is that the most fundamental facts about mass are facts about massrelationships; it is perfectly consistent with this that there is such a thingas the mass of IPK which is not identical to any mass relationships, so longas any fact of the matter concerning it holds in virtue of facts about IPKsmass relationships. So the comparativist is free to agree that there is sucha thing as the mass of IPK to which the term kilogram refers after all.

The non sequitur exhibited by the semantic objection is vividly exem-plified in the following case. Consider a physicalist who claims that allfacts hold in virtue of facts concerning physical entities, and imagine anobjector who says The term stock market refers to the stock market, butthe stock market is not a physical entity; therefore your physicalism isfalse. In response, our physicalist will surely point out that the argument

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misses its mark entirely: her view was never that everything is a physi-cal entity but rather that all facts about the world hold in virtue of factsconcerning physical entities. The comparativist can say exactly the sameabout the semantic objection to comparativism.

In sum, I do not believe that there is semantic evidence that the truthof kilogram sentences requires absolutism to be true. But even if therewere, I believe that the appropriate response for the comparativist wouldbe to adopt an error theory about kilograms.

5 The Objection from Kilograms

Since comparativism holds that all facts about mass hold in virtue of massrelationships, one might naturally try to refute the view by finding a coun-terexample, i.e. a fact about mass that does not hold in virtue of massrelationships. For example, consider the fact that my laptop is 2 kgs. Ifone can argue that there are no mass relationships in virtue of which thisobtains, one would naturally take oneself to have refuted comparativism.

Absolutists should find this strategy promising, for I believe that thereare good arguments to the effect that my laptops being 2 kgs does nothold in virtue of any mass relationships. Unfortunately there is no room todiscuss these arguments in full detail, but let me say something to motivatethe idea.25 First, note than an absolutist has no problem accounting for mylaptops being 2 kgs: she can say that it is either identical to, or else holdsin virtue of, my laptops having a certain intrinsic mass. If absolutismwere true, this would be an extremely plausible view. For if materialbodies really did have intrinsic masses, it would be natural to think thatterms of the form r kilograms would refer to those properties (even if theKripkean view about what fixes the referents of the terms is incorrect). Ifso, then it is almost irresistible to say that my laptops being 2 kgs is eitheridentical to, or else holds in virtue of, its having a certain intrinsic mass;namely the one that is the referent of 2 kgs.

But the fact that this account of my laptops being 2 kgs is so naturaland satisfying shows that, at least intuitively, the mass relationships that itenters into are entirely irrelevant when it comes to explaining what makesit 2 kgs. My laptop stands in all sorts of mass relationships to standardobjects in Paris and measuring instruments in Paruguay and electrons onPluto, but the fact that the absolutists explanation is so satisfying showsthat, intuitively, all these relationships are irrelevant to an explanation ofits being 2 kgs. Therefore, the argument goes, whichever mass relation-ships the comparativist picks in order to explain its being 2 kgs, she will

25These arguments are developed in more detail in Dasgupta [10].

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violate our intuitions as to what is relevant to explaining that fact. Ofcourse, the comparativist might concede this and reply that revising ouropinions about what is explanatorily relevant is a natural consequence oftheoretical inquiry. To an extent, this reply is well taken. But all handsshould agree that this would be a significant revision of pre-theoretic be-lief and therefore counts as at least a point against her view.

As I said, there is no space to develop this kind of argument in detail.But let us give the absolutist the benefit of the doubt and suppose thatthere are no mass relationships in virtue of which my laptop is 2 kgs.Where does this leave the comparativist? Of course, one option would beto adopt the error theory described earlier, according to which there areno facts about mass-in-kilograms in the first place. But I believe that thereis a better option. The key is to recognize that the in virtue of relation isirreducibly plural, in the sense that a plurality of facts Y can sometimeshold in virtue of another plurality of facts X even though no Y when takenon its own holds in virtue of anything. Given this pluralistic conceptionof the in virtue of relation, the comparativist may take the set K of allkilogram facts and the set R of all facts about mass relationships, andpropose that the members of K (plurally) hold in virtue of the membersof R even though no kilogram fact taken on its own holds in virtue ofanything. This view neatly sidesteps the problem of relevance discussedabove, for R does not contain irrelevant information when it comes toexplaining the members of K. To be sure, R contains irrelevant informationwhen explaining my laptops being 2 kgs, such as information about itsmass relationships to electrons on Pluto, but since K contains facts abouthow massive those electrons are in kilograms the relationships betweenthem seem perfectly relevant when explaining Ks members all together.By adopting this position, the comparativist can then agree that there is afact of my laptops being 2 kgs (contra error theory), concede that there areno mass relationships in virtue of which it obtains, and yet nonethelessinsist that this is perfectly consistent with comparativism since it remainsthe case that all the facts about mass in kilograms when taken together as aplurality hold in virtue of the underlying mass relationships.

There are many virtues of this view. One is that the comparativist canrespect the intuition described earlier that facts about my laptops massrelationships are not part of what explains my its being 2 kgs. They arenot part of the explanation of this fact, on this view, because the fact on itsown has no explanation in the first place! Another virtue is that it neatlyexplains why absolutism is initially the more intuitive and attractive view.For as we have seen, we have a strong intuition that my laptops being 2kgs does not hold in virtue of its mass relationships to other bodies. Ac-

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cording to the current approach, the absolutists mistake is just to take thisto imply that its being 2 kgs must be explained in terms of its intrinsicnature, when instead the correct conclusion is that we can only explainfacts about mass in kilograms when they are taken all together as a plu-rality. The absolutists mistake is therefore understandable, but a mistakenonetheless.

Now I have not argued that this pluralist explanation of kilogram factsis satisfactory, and unfortunately there is no space to do so here.26 But atthe very least, it is clear that if the in virtue of relation is irreducibly pluralthen refuting comparativism is significantly more difficult than one mighthave thought. For it is then not enough for the absolutist to argue thatmy laptops being 2 kgs fails to hold in virtue of its mass relationships.In addition, she would need to argue that the plurality of kilogram factstaken together do not hold in virtue of the totality of mass relationships.Until she shows this, comparativism remains a live option.

6 The Objection from Humeanism

Humean Supervenience (HS), says Lewis, is the view that

all there is to the world is a vast mosaic of local matters of par-ticular fact, just one little thing and then another. . .We havegeometry: a system of external relations of spatio-temporaldistances between points. . . And at those points we have lo-cal qualities: perfectly natural intrinsic properties which neednothing bigger than a point at which to be instantiated. . . Andthat is all. There is no difference without a difference in thearrangement of qualities. All else supervenes on that.27

So stated, HS is inconsistent with comparativism since it asserts that thefundamental physical quantities like mass are intrinsic and are instanti-ated at single points of spacetime. But HS is supported in the literatureby wide range of arguments. If it is inconsistent with comparativism, theargument would be, so much the worse for the latter.

However, most arguments for HS are perfectly consistent with com-parativism. To see this, note that HS is the conjunction of two theses: onestating that everything supervenes on the categorical nature of the physicalworld, and a second describing what the categorical nature of the phys-ical world is like. The above quote focuses on the second and says that

26I motivate and defend this pluralist explanation at some length in Dasgupta [10].27Lewis [22], p. ix.

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the categorical nature of the world consists in the distribution of intrinsicproperties across spacetime. And that second thesis is indeed inconsistentwith comparativism.

But much of the literature on Humean Supervenience focuses on thefirst thesis, the view that everything elseincluding chances, causes, coun-terfactuals, minds, morals, etcsupervenes on the worlds categorical na-ture. And a brief glance at that literature reveals that none of the argu-ments depend on whether those underlying categorical facts consist in theinstantiation of intrinsic properties (as Lewis says) or in the instantiationof comparative relations (as the comparativist says). Even if one is movedby those arguments, one may still adopt comparativism.

To be sure, the second thesis does play a role in Lewis metaphysics.For example, he famously analyzes de re modals in terms of counterparts,and he says that objects are counterparts insofar as they resemble eachother, and he says that resemblance is ultimately a matter of sharing in-trinsic properties. But one can easily restate his view in terms friendlyto the comparativist by allowing resemblance to ultimately be a matterof participating in the same pattern of relations instead. Indeed, Lewisonly thought that resemblance was ultimately a matter of sharing intrin-sic properties because he thought that it was a matter of sharing perfectlynatural features, and the second thesis of HS states that those perfectlynatural features are all intrinsic (save for geometric relations). Thus, ifone gives up that second thesis in favor of comparativism and allows thatsome perfectly natural features are relations, it will follow from the restof his system that resemblance is sometimes a matter of participating inthe same pattern of relations after all. As a result, a large chunk of Lewissystem remains essentially unchanged even if we endorse comparativism.

This is not to say that all of Lewis views are easily recast in compar-ativist terms. Still, once one sees how many of them can be, the abovequote stating that the categorical world consists in the distribution of in-trinsic properties sounds less like an essential part of his view and morelike a convenient working assumption.

7 Objections from Physics

The objections to comparativism considered so far have been broadly apri-ori, but is there empirical evidence against the view? If we could see theintrinsic mass had by a given material body or detect it with the help ofmechanical devices, that would presumably count as empirical evidencefor absolutism. But I will argue in Section 8 that if material bodies had theintrinsic masses posited by the absolutist, those intrinsic masses would

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be invisible to the naked eye and undetectable by any physically possibledevice.

Still, if the absolutist could show that intrinsic mass is indispensible toour best confirmed scientific theories, one might then think that empiri-cal evidence confirming those theories would thereby count as empiricalevidence that each material body has an intrinsic mass even if we cannottell which particular one it is. To see how this idea might be developed,consider one simple law governing mass, f=ma, and let us pretend forsimplicity that our best confirmed physical theory states that it is the onlylaw governing the motions of material bodies. Now, consider a world Wexactly like ours with the one exception that everythings mass is doublewhat it actually is. One might argue that if the equation f=ma actually ob-tains then it does not obtain in W since doubling everythings mass whileleaving their forces and accelerations unchanged would break the equal-ity.28 Since W is just like ours in all mass relational respects, the argumentwould be that the truth of f=ma depends not just on the mass relationsbetween things but also on which intrinsic masses they have. Therefore,empirical evidence confirming f=ma is ipso facto empirical evidence con-firming absolutism.

But the argument does not convince, for it depends on a controver-sial interpretation of the equation f=ma. Taken at face-value, it states amathematical relationship between the numbers and vectors that representforce, mass and acceleration in a particular scale. But what does it stateabout the quantities themselves? There are at least two interpretations,one on which it states something about absolute quantities and anotheron which it states something about comparative quantities. On the firstinterpretation it states that the absolute masses, accelerations and forcesall line up in a specific way. For example, part of the content of the equa-tion on this interpretation is that anything with the determinate intrinsicmass M which is accelerating at a determinate rate A will have a particulardeterminate force F acting on it, and so on for other determinate masses,

28This is where it helps to ignore any other laws specifying the force acting on eachparticle and pretend that f=ma is the only law governing our world. For we therebysidestep the complication that with the identification of inertial mass and gravitationalmass, mass plays a unique role in classical mechanics: not only is it a brake on accel-eration as described in f=ma, it is also a determiner of the gravitational force betweenthings as described in the inverse-square gravitational force law. The fact that it plays thisdual role might tempt one to think that doubling everythings mass would preserve thetruth of the classical mechanical laws, since the increase in gravitational forces would becounter-balanced by the increased brake effect experienced by each body. But even ifthis line of reasoning were sound, it would not generalize to other quantities. I considerf=ma in isolation from whatever force laws it might couple with precisely because we arelooking for general considerations.

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accelerations and forces. And, in the other direction, part of the contentof the equation will be that anything with the particular determinate forceF acting on it will either have the determinate mass M and be accelerat-ing at the determinate rate A, or else will have the determinate mass 2Mand be accelerating at the determinate rate A/2 (where 2M is the intrinsicmass that is twice M and A/2 is the determinate rate of acceleration that ishalf A), and so on for other pairs of determinate masses and accelerations.More generally, we might express this interpretation as follows:

(L1) For any material thing x,

(a) For any reals r1 and r2, if x has mass r1M and acceleration r2A,then x has force r1r2F acting on it.

(b) For any real r3, if x has force r3F acting on it, then there arereals r4 and r5 whos product is r3, such that x has mass r4Mand acceleration r5A.

By contrast, the second interpretation of the equation takes it to state aconnection between mass relationships, force relationships and accelera-tion relationships. For example, part of the content of the equation on thisinterpretation is that if a body x is twice as massive as another body y andan equal force is applied to both, then y will accelerate at twice the rate asx. More generally:

(L2) For any material things x and y,

(a) For any reals r1 and r2, if x is r1 times as massive as y and isaccelerating r2 times the rate of y, then x has r1r2 times as muchforce acting on it than y.

(b) For any real r3, if x has r3 times as much force acting on it thany, then there are reals r4 and r5 such that r4r5 = r3, and such thatx is r4 times as massive as y and is accelerating r5 times the rateof y.29

The argument under consideration assumed that the actual physical lawsdo not obtain in the world W in which all masses are doubled. We cannow see that whether this is true depends on whether the fundamentallaw governing the quantities is (L1) or (L2). The assumption is true if the

29This is a very simplistic example of a comparativist rendering of a physical law. For amore realistic attempt to render physical theories in comparativist friendly terms, see Field[14]. I will discuss (L2) here because it is particularly easy to grasp but the discussion isintended to generalize to more realistic comparativist friendly laws such as those describedby Field.

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fundamental law is (L1), since in W the absolute masses, forces and accel-erations line up differently than they actually do. But the assumption isfalse if the fundamental law is (L2), for (L2) only talks of mass relation-ships and those are the same in W as they are in the actual world. Indeed,it is clear that if the fundamental law is (L2), then whether it obtains doesnot depend on material bodies having the intrinsic masses posited by theabsolutist, contra the argument under consideration.

So in order to run the indispensability argument the absolutist mustargue that the empirical evidence confirming f=ma is evidence that con-firms (L1) and disconfirms (L2). Only then would the intrinsic massesposited by the absolutist be indispensable to what is confirmed by theempirical evidence. But what evidence would favor (L1) over (L2)? Thedifference between the laws is this: (L1) implies that if the state of theworld at present differed only in that everythings mass were double whatit actually is, things would proceed to accelerate at half their actual rate.30

(L2) does not imply this because the mass ratios would be exactly the samein the doubled state and that is all that the law makes reference to. So thetwo laws issue these different predictions, but how could we test whichprediction is correct? The obvious idea is to construct two isolated labora-tories that are exactly alike at an initial time except for the fact that one isa doubled-mass version of the other. One might think that if the bodies inthe doubled-mass laboratory proceed to accelerate at half the rate as thebodies in the other, this would confirm (L1) and disconfirm (L2).

But on further reflection this experimental outcome is predicted by(L2) and would therefore not disconfirm it. After all, the bodies in thedoubled-mass laboratory are twice as massive as their counterparts in theother and are subjected to the same forces as those counterparts, and theexperimental outcome is that they accelerate at half the rate as their coun-terparts. But this is exactly what (L2) would predict! The trouble is thatthe two laws make different predictions about what would happen if theentire world were doubled in mass, but when attempting to test which pre-diction is correct we can do no better than to compare different parts of theworld (our two laboratories) and the laws make exactly the same predictionabout what would then occur.

How else might we obtain evidence that confirms (L1) and disconfirms(L2)? Suppose we could see or detect which particular intrinsic mass, forceand acceleration each material body has, and suppose it turned out thatthey were lined up in the way stated by (L1).31 Would this confirm (L1)

30Again, I am bracketing the effect that doubling everythings mass might have on theforces acting on things. See footnote 28.

31I will argue in Section 8 that we could never see or detect such a thing. But I do notwant to rest my argument here on the results of that section. So here I am supposing, per

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and disconfirm (L2)? No, because that observation is consistent with thehypothesis that material bodies have absolute quantities but are governedby (L2). For if that were the case then the absolute quantities are boundto line up in one way or another, it would just be a matter of accident (orperhaps, more specifically, a matter of initial conditions) that they line upas they do whereas according to (L1) they would line up like that as amatter of law.

It therefore appears that there is no possible evidence that would con-firm (L1) and disconfirm (L2). Under the pretense that f=ma is the onlylaw of our best confirmed physics, then, the intrinsic masses posited bythe absolutist are not indispensable to our best physics after all.32 Thepretense is of course false, but it seems reasonable to hope that similarreasoning will apply in the case of better confirmed physical theories. ButI leave the question of whether this is so for another time.33

8 In Favor of Comparativism

8.1 The Occamist Argument

Having surveyed a number of objections to comparativism, I find noneconvincing. But is there any positive reason to be a comparativist? Ibelieve there is. I will argue that if material bodies really did have theintrinsic masses posited by the absolutist, those intrinsic masses wouldbe undetectable. Our Occamist principle says that it is a mark against atheory if it posits undetectable structure, so this is reason to prefer com-parativism.

This epistemic argument is not the only possible argument for com-parativism. Some might favor comparativism purely on grounds of on-

impossible, that we can make such observations just for the sake of argument.32Baker [4] argues that the idea that the fundamental laws of our world are

comparativist-friendly laws like (L2) (or, more realistically, the kind of laws that Fieldexpresses in his [14]) has serious problems. There is no space here to discuss his argu-ments in the detail they deserve but I hope to do so in further work. Very briefly, theproblems he describes concern modal properties of the laws such as whether they are de-terministic. Now I argued in section 3 that when discussing comparativism it is importantto think carefully about how modal claims about quantities are to be evaluated. There Iargued in favor of a mass-counterpart interpretation of modal claims. I believe that ifthe comparativist endorses this mass-counterpart theory she has all the resources neededto respond to Bakers objections. But I hope to defend this idea in future work.

33Field [14] has famously made a good start at expressing a portion of physics in purelycomparativist terms. However, the current point does not depend on the success of Fieldsproject, which was to express physics without reference to numbers, sets or other abstracta.For example, (L2) serves the comparativists purposes but it freely quantifies over realnumbers.

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tological parsimony, the idea being that while everyone recognizes massrelations it is only the absolutist who goes further and posits extra intrin-sic masses that are not explicable in terms of those mass relationships.Others might favor comparativism because it leads to a simpler axioma-tization of a theory of mass.34 But my epistemic argument is particularlyimportant. For if we could see or otherwise detect which intrinsic mass agiven body hasif the intrinsic masses posited by the absolutist were re-ally part of our datathen a comparativist theory that dispensed withthose intrinsic masses would not be empirically adequate. And in that casecomparativism should obviously be rejected outright purely on groundsof empirical inadequacy, and the proposed benefits of ontological parsi-mony and theoretical simplicity would then be neither here nor there. Soa hidden assumption in these other arguments for comparativism must bethat the intrinsic masses posited by the absolutist are not detectable afterall, which is precisely what my epistemic argument tries to establish.

Much depends on what I mean by undetectable. If I used the termto include anything that we cannot see with the naked eye, our Occamistprinciple would recommend that we become radical scientific anti-realistsand dispense with so-called theoretical entities such as electrons. Butthat is not how I use the term here. Instead, something is undetectable inmy sense of the term if, roughly speaking, it follows from the structureof the laws of motion governing our world that it is physically impossiblefor it to have an impact upon our senses. Electrons are therefore per-fectly well detectable in this sense because there are physically possibleprocesses, such as those that occur in particle accelerators, by which thepresence of an electron can be made to have an impact on our senses viaits impact on (say) the movement of a dial or an image produced on acomputer screen. In contrast, features like absolute velocity and absolutesimultaneity are undetectable in my sense: even if they were real, it turnsout that the laws of motion governing our world are set up in such a wayas to guarantee that it would be impossible for them to ever have an im-pact on our senses. That is why they are considered to be redundant orsuperfluous in modern physics, and most physicists and metaphysicianstherefore believe on Occamist grounds that those features are not real afterall. I will argue here that the same goes for intrinsic mass.

The Occamist principle I use here is not verificationist: there is noclaim that talk of these undetectable features is meaningless. It is just aprinciple of theory choice, a principle stating that (all else being equal) weshould favor those theories without undetectable features. Nor does our

34This is an idea that Field emphasizes in his [15]. Thanks to a referee for encouragingme to think about this consideration.

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Occamist principle say that we should always dispense with undetectablefeatures. It just says that undetectable features are undesirable, so that allelse being equalor at least near enough equalwe should prefer the-ories like comparativism that dispense with them. But in the previoussections I have been arguing that there are no overwhelming reasons toreject comparativism and therefore that all is indeed near enough equalafter all.

Our crucial premise, then, is that if material bodies really possessedthe kind of intrinsic mass posited by the absolutist, those intrinsic masseswould be undetectable in our sense of the term. How should we argue forthis? There is a reasonably well known argument for the analogous claimin the case of absolute velocity, so let us rehearse it before applying it tothe case of intrinsic mass.

8.2 The Case of Velocity

What is absolute velocity? We often talk of a material bodys velocity rela-tive to another body: a car might have a velocity of 65 mph in a particulardirection relative to the highway and 10 mph in the same direction relativeto the train traveling alongside it. But how fast is it really going, indepen-dent of any material reference point? If there is an answer to this question,that is a statement of its absolute velocity.

Now if there were such a thing as absolute velocity, why would itbe undetectable? At least naively, one might think that the speedometerfound in an ordinary car is a device that detects the vehicles absolutevelocity. But we can argue that such a device at best measure relative ve-locity and that absolute velocity is undetectable after all.35 For in orderto detect absolute velocity, there would need to be some physically possi-ble process that, when initiated at t0 to measure the absolute velocity of agiven body, will generate a readingan image on a computer screen, say,or the position of a needlethat indicates what that bodys velocity wasat t0. Moreover, the outcome that would be produced if the body weretraveling at one velocity must be discernibly different from the outcomethat would be produced if it had a different velocity, on pain of our notbeing able to tell what velocity a given outcome indicates.36 So, if we sim-

35This argument that follows has received perhaps its clearest written expression inRoberts [26]. I heard similar arguments orally in seminars given by Tim Maudlin at Rut-gers and David Albert at Columbia. However, all these theorists run the argument inimportantly different ways. My presentation here is similar to my presentation in Das-gupta [8].

36At least, that is the ideal: in practice, we do not mind if the outcomes that wouldbe produced by velocities differing only by some tiny amount are indiscernible. Moreaccurately, then, what we require is that the outcomes would be discernible when the

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ply wanted to measure whether a given body was in a state of absoluterest or absolute motion, the process would need to produce one outcomeif the body was at rest at t0for example an inscription of At restand adiscernibly different outcome if the body was moving t0an inscription ofMoving, say. Finally, since the process is a physical process, the outcomeproduced will depend on the physical laws governing our world. Puttingthis all together, we can therefore say that absolute velocity is detectableonly if there is a physically possible device which at a given time t0 hastwo properties: first that, according to the laws, it will display At rest ona computer screen at a later time t1 iff it was presented with a body at restat t0; and second that (according to the laws) it will display Moving on acomputer screen at t1 iff it was presented with a body that was moving att0.37

But according to most of our best confirmed physical theories, it isphysically impossible for a device to have both properties. For suppose Itake a device with the first property and present it with a body at rest at t0,and it therefore displays At Rest at t1. We can show that it does not havethe second property by considering a world W just like ours with the oneexception that at all times the absolute velocity of all bodies is five mphgreater in a certain specified direction. Now, W is a world in which thedevice is presented with a moving body at t0, and yetsince the relativepositions of all bodies at all times are (by construction) the same in W asthey actually arethe device still displays At rest at t0. But it turns outthat according to our best physics, the laws of motion governing W are thesame as those governing ours. Therefore, the behavior of the device in Wrepresents how it behaves according to our laws of motion; hence it doesnot have the second property listed above. QED.

To be clear, the argument here does not assume that absolute velocitycan only be detected by a device with these two properties. The assump-tion is rather that if absolute velocity is detectable then it is detectable witha device with these two properties (even if it is also detectable in otherways too). The argument then is that it is not detectable with a deviceof this kind, therefore it is not detectable. I will discuss the assumptionfurther in section 8.5.38

This is not to say that speedometers in cars are useless, for the argu-ment here is consistent with the thesis that they detect the cars velocity

velocities differ by more than some amount x, in which case we say that the processmeasures absolute velocity up to an accuracy of x.

37We use biconditionals here because we not only want each initial velocity to issue ina readable outcome, we also want each outcome to be uniquely associated with that initialvelocity so that we know what the outcome indicates.

38Thanks to a referee for urging me to clarify the assumption used here.

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relative to a given body such as the road. All the argument shows is thatthey do not detect the cars absolute velocity. And of course if absolutevelocity is undetectable our Occamist principle then recommends (all elsebeing equal) endorsing a theory of spacetime according to which there isno such thing as absolute velocity.39

8.3 Undetectable by Us

Returning to the case of intrinsic mass, let us suppose for reductio thatmaterial bodies have the intrinsic masses posited by the absolutist. Forsimilar reasons as just discussed in the case of velocity, I claim that whichparticular intrinsic mass each body has would be undetectable.40

The claim may initially sound implausible. After all, my laptop andmy cup feel different in mass when I pick them up, so am I not therebydetecting the intrinsic mass of each item? Similarly, one might naturallythink that an ordinary bathroom scale is a device that allows us to detectthe intrinsic mass of the object placed on it by displaying its mass in theposition of a needle. But it turns out that this is a mistake: just as aspeedometers at best allow us to detect the velocity of a car relative to

39Readers may recognize this as one line of reasoning that can be used to extract meta-physical conclusions (in this case about the structure of spacetime) from facts about thesymmetry or invariance groups of the fundamental laws of motion (in this case the relevantsymmetry group being uniform velocity boosts). The general idea of using symmetries tomotivate various metaphysical views is not new. See Earman [11] chapter 3, and North [25]for discussions of this idea with respect to the metaphysics of spacetime. It is also used(explicitly and implicitly) in contemporary discussions of diffeomorphism invariance inGeneral Relativity and permutation invariance in Quantum Mechanics to motivate vari-ous structuralist metaphysical conclusions (see Ladyman [20] for a discussion of boththese issues, and see Dasgupta [9] for a discussion of the former). Where I may differ fromothers is my belief that the best such arguments go by way of intermediary conclusionsabout what is undetectable. That is, the best arguments in my view first use facts aboutsymmetries to draw an intermediary conclusion to the effect that some putative featureis undetectable, and only then use the Occamist principle to draw the metaphysical con-clusion that those undetectable features are not real. The question of whether symmetryarguments must take this epistemic form is vexed and not one I can discuss here, thoughI hope to do so in future work. In any event there is no need to settle the matter here forwe can evaluate the argument in the text on its own merits regardless of whether there areother ways of drawing metaphysical conclusions from symmetries. Thanks to a referee forencouraging me to point out the connections between the discussion in the text and theseother debates.

40I should stress that the argument concerns the kind of intrinsic mass as posited by theabsolutist; that is, the kind of intrinsic mass the possession of which is not explicable interms of mass relationships. As I argued in section 5 the comparativist can agree thatthere are facts about my laptops mass in kilograms such as its being 2 kgs, and perhapsthis is a fact about its intrinsic mass in some sense of the term intrinsic. But she insiststhat these facts hold in virtue of facts about mass relationships so my argument is notdirected at these kinds of intrinsic masses.

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the highway, so too picking things up or putting them on bathroom scalesat best allows us to detect the mass relationships between things and nottheir intrinsic masses.

The argument is a little more complex than in the case of velocity, butlet us start by running an analogous argument and modify it as required.For convenience, let us suppose that the terms 1 kg and 2 kgs labelparticular intrinsic masses. Following the above discussion, we can thensay that intrinsic mass is detectable only if there is a physically possibledevice which at a given time t0 has two properties: first, that (accordingto the laws) it will display 1 kg on a computer screen at a later time t1iff it was presented with a 1 kg object at t0; and second, that (accordingto the laws) it will display 2 kgs on a computer screen at t1 iff it waspresented with a 2 kg object at t0. We now argue that it is physicallyimpossible for a device to have both properties. To this end, suppose Itake a device with the first property and present it with a 1 kg object at t0and it therefore displays 1 kg on a screen at t1. We show that it does nothave the second property by considering a possible world W just like ourswith the one exception that everythings mass is double what it actuallyis; that is, a world in which our device is presented with a 2 kg object at t0but in which it nonetheless displays 1 kg at t0. And we now make a keyassumption, namely that the laws of motion governing W are the same asthose governing ours. It then follows that the behavior of the device in Wrepresents how it behaves according to our laws of motion; hence it doesnot have the second property listed above. QED.

But is this key assumption correct? As before, let us suppose for sim-plicity that our best confirmed physical theory states that there is only onelaw of motion governing our world: f=ma. In the last section, we saw thatthis equation might express one of two things: a law governing absolutequantities like (L1), or a law governing comparative quantities like (L2).We also saw that if the actual law turns out to be of the second kind thenit would obtain in W too, while if the actual law turns out to be of theformer kind then it would not obtain in W (in the latter case W does notrepresent how our device behaves according our laws of motion and it re-mains open that it has the two properties after all). So the assumption thatthe law of W are the same as the actual laws amounts to the assumptionthat the laws are like (L2) and govern comparative quantities.

Can the comparativist assume this? The assumption does not presup-pose the falsity of absolutism, since the hypothesis that the laws governcomparative quantities is consistent with the idea that material bodies alsohave intrinsic quantities in virtue of which those comparative quantitiesobtain. But is it reasonable for the comparativist to believe that the as-

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sumption is true? Recall that in the last section we argued that there isno possible evidence that would confirm the hypothesis that law governsabsolute quantities but disconfirm the hypothesis that it governs compar-ative quantities. Importantly, the argument did not assume that intrinsicmass is undetectable, so we can appeal to the result of that argument with-out begging questions. So, if the evidence does not settle the matter eitherway, what is it reasonable to believe?

This is a delicate issue in epistemology concerning what it is reason-able to believe given certain evidence. An extremely permissive view ofrationality would say that if the evidence does not settle whether p or not-p, it is reasonable to believe either. On this view, the comparativist mayreasonably believe that the actual law governs comparative quantities, inwhich case she can run the above argument as written. But one mightworry that the argument would then be dialectically weak, since on thispermissive view it would also be reasonable for the absolutist to believethat the laws govern absolute quantities in which case she will remain un-moved by the argument. Moreover, other views in epistemology wouldinsist that if the evidence equally supports two views, we should remainagnostic between the two rather than believe either.

Luckily there is no need to settle the issue here. For what our centralargument above shows is that whether or not a device with two propertieslisted above is physically possible depends on which kind of law governsour world. More specifically, the argument shows that a given device hasthose two properties only if our laws govern absolute rather than com-parative quantities. Therefore, my evidence that a given device has thosetwo properties can be no higher than my evidence that the laws governabsolute quantities rather than comparative quantities. But the conclusionof the last section was that there is no possible evidence that the actuallaws govern absolute quantities and not comparative quantities. There-fore, regardless of the delicate epistemic question of what it is reasonableto believe given certain evidence, all hands should agree that I can have noevidence that a given device has the two properties listed above. And thisconclusion is strong enough for our purposes. For even if a given devicedoes in fact have the two properties and gives a reading of 1kg whenI present it with a material body, the fact remains that if I have no evi-dence that the device has those two properties then the reading gives meno evidence as to what the bodys intrinsic mass is. So our conclusion isthat even if absolute mass is in some sense detectable by the device it is notdetectable by us, which is the point we were trying to establish all along.

Th