The (un)detectability of absolute Newtonian masses


Absolutism about mass claims that mass ratios obtain in virtue of absolute masses. Comparativism denies this. Dasgupta (in: Bennett, Zimmerman (eds), Oxford studies in metaphysics, Oxford University Press, Oxford, 2013) argues for comparativism about mass, in the context of Newtonian Gravity. Such an argument requires proving that comparativism is empirically adequate. Dasgupta equates this to showing that absolute masses are undetectable, and attempts to do so. This paper develops an argument by Baker to the contrary: absolute masses are in fact empirically meaningful, that is detectable (in some weak sense). Additionally, it is argued that the requirement of empirical adequacy should not be cashed out in terms of undetectability in the first place. The paper closes by sketching the possible strategies that remain for the comparativist. Along the way a framework is developed that is useful for thinking about these issues: Ozma games—how could one explain to an alien civilisation what an absolute mass is?

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  1. 1.

    Here I am using ‘ontology’ and ‘ideology’ in the Quinean sense (Quine 1951). Roughly, ontology refers to the (primitive) objects, and ideology to their (primitive) properties.

  2. 2.

    What laws the textbook equations exactly represent will be discussed in §6.2.

  3. 3.

    In the context of NG, inertial and gravitational mass are usually merely empirically identified, but still theoretically or conceptually distinguished. It would then be interesting to consider whether one could be an absolutist—defined below—about inertial mass but a comparativist—defined below—about gravitational mass, and vice versa. In this paper we will however follow the current literature in bracketing out such a possibility, by considering only a single mass determinable.

  4. 4.

    Although Dasgupta takes it to be an essential feature of absolute masses that they are intrinsic, I side with Roberts (2016) (and Sider, manuscript) in taking the crucial distinction between absolute masses and mass relations to be that the former are monadic properties, and the latter dyadic properties (i.e. binary relations). After all, regularity comparativism (Martens 2017b) grounds absolute masses in a complete four-dimensional mosaic of spatiotemporal relations and fundamental mass ratios, making them as extrinsic as they could possibly be, whilst leaving their monadicity untouched. The Higgs mechanism for generating inertial masses in the Standard Model of Particle Physics seems to similarly generate monadic masses that are nevertheless extrinsic, as the mass emerges from an interaction with, or immersion in the Higgs field (Bauer 2011).

  5. 5.

    It is! You should check it out online.

  6. 6.

    But see Hall (manuscript), Loewer (1996), Esfeld (2014), Esfeld et al. (2017, 2015), Esfeld and Deckert (2017), Vassallo et al. (2017), Hubert (2016), Martens (2017b).

  7. 7.

    Strictly speaking, in first order logic, there is just one mass relation, and several instantiations of that single relation. In this paper, as in most of the literature, when we talk about mass relationships, this is short for several instantiations of this single mass relation. Why this makes sense will become more clear in §3, where I will advocate defining absolute mass(es) and the mass ‘relation(s)’ using group theory.

  8. 8.

    By this I do not mean to rule out nominalistic versions of comparativism—such as Field’s approach which uses primitive congruence predicates as mass relations (Field 1980)—in favour of quantitative approaches—such as the numerical mass ratios used by, for instance, Dasgupta. I merely intend to exclude relations such as, in quantitative terms, ‘x is 2 kg more massive than y’.

  9. 9.

    See also his 1985 paper (Field 1984).

  10. 10.

    On the 20th of May 2019 the Bureau International des Poids et Mesures will redefine the kg in terms of natural constants.

  11. 11.

    Although I end up arguing against this view, that is not because it is not a coherent comparativist view.

  12. 12.

    That is if we assume the existence of mass ratios. It is an interesting question whether we truly need mass ratios (within NG), or whether they are merely an artefact of our representation in terms of quantities. As far as I am aware all of the comparativism literature assumes the existence of mass ratios, so we will do so as well (for now).

  13. 13.

    Dasgupta defines absolutism as “the view that the most fundamental facts about material bodies vis-à-vis their mass include facts about which intrinsic mass they posses” (Dasgupta 2013, p. 105), c as “the view that the most fundamental facts about material bodies vis-à-vis their mass just concern how they are related in mass, and all other facts about their mass hold in virtue of those relationships” (Dasgupta 2013, pp. 105–6). As they stand, these statements could be interpreted either in the weak sense or in, what I call below, the strong sense, depending on whether ‘most fundamental’ is taken to be a statement about absolute fundamentality, or whether ‘most fundamental vis-à-vis their mass’ is read as being about relative fundamentality. Since Dasgupta also claims that the absolutist believes that “intrinsic masses are fundamental” (Dasgupta 2013, p. 105), he is best interpreted as explicitly discussing the strong versions of absolutism and comparativism. However, nothing in the rest of his paper turns out to hinge on considering the strong rather than the weak versions. In particular, none of the arguments gives any reason to go beyond the weak versions. Hence, as explained in the text, I believe it best in general to focus on the weak versions, especially since only they are mutually exhaustive.

  14. 14.

    Dees calls this Quantity Primitivism, but in the current paper ‘quantity’ is a highly technical notion (§3), distinct from my equally technical notion of ‘magnitude’, which is presumably what Dees had in mind (Dees 2018). I am choosing for Primitivism rather than Magnitude Primitivism though, because in the case of strong comparativism the primitivism refers not to the magnitudes but the ratios.

  15. 15.

    An analogous distinction in the space debate is made elsewhere (Martens, forthcoming).

  16. 16.

    Perhaps distance (or time) as it features in theories of discrete spacetime such as causal set theory (Dowker 2008) would be a more interesting example. In this case, spatial (or temporal) extension boils down to counting the spacetime atoms. However, in such a case it is questionable whether distance (or time) is still truly a dimensionful determinable.

  17. 17.

    Arntzenius and Dorr echo this thought (Arntzenius 2012, Ch. 8).

  18. 18.

    One might argue that the comparativist faces that same objection: a relation such as ‘x is twice as massive as y’ refers indirectly to the platonic number two. This is true for the quantitative approach to comparativism of, for instance, Bigelow et al. (1988) and Dasgupta (2013), but not for congruence-based comparativism, as advocated famously by Field himself (Field 1980). In the next section we will see that the same type of solution exists for the absolutist.

  19. 19.

    Perhaps this failure to distinguish magnitudes and quantities is sufficient to undermine what Dasgupta calls ‘The Objection from Kilograms’ (Dasgupta 2013, §5) (which is, in contrast to the worries discussed in the previous section of this paper, seen as an objection against comparativism).

  20. 20.

    Hall similarly points out that physical determinates, such as absolute mass, are not linguistic entities but (monadic or polyadic) magnitudes (Hall, manuscript).

  21. 21.

    One may wonder whether we also need multiplicative structure, as the gravitational force depends on the product of two masses (and in order to define the Active Leibniz Mass Scaling defined below). One response would be to argue that this structure is part of the laws and not the masses, although it could be retorted that we could make the same move for all the structure that we do attribute to masses here (cf. Dees 2018). Fortunately, in standard NG we need not worry about products of masses, since we have assumed an equivalence principle between empirical and inertial mass, such that the final equation that governs the dynamics, \(a = Gm/r^2\), does not contain any product of masses.

  22. 22.

    The abovementioned constraints on mass relations required to prove the representation theorem are exactly the constraints that ensure that the mass relations can be represented by monadic properties with these two structures.

  23. 23.

    It is for this reason that kinematic comparativism, as defined in the next section, is true.

  24. 24.

    Insofar as determinables are grounded in determinates, it would only be a small leap to argue from determinate quidditism to transworld identities of determinables.

  25. 25.

    These structures correspond nicely to Weyl’s operational definition of (inertial) mass (Weyl 1949; Jammer 2000).

  26. 26.

    The easiest way to see this is by realising that there is no qualitative distinction between a first set of binary relations that obey these structures, and a second set obtained from the first by squaring each relation—call this a Leibniz Mass Ratio Squaring. (I would like to thank Zee Perry for pointing this out to me.) Note that it is for this same reason that the mass ratio structure, as defined in the text, does not fix a sum over mass ratios—as this would not be invariant under a Leibniz Squaring. Although a structure that defines such a sum is usually implicitly assumed for the rational numbers, i.e. quantities, it is not yet fixed for the much sparser concept of mass ratios defined in the text. Such a sum over mass ratios would require extra structure (Martens 2019). Here we see yet another reason why it is dangerous to confuse numerical quantities with the absolute magnitudes or mass ratios that they represent. Moreover, focusing on magnitudes and mass ratios with their sparse structure rather than numerical quantities with their rich structure is in line with the desideratum of metaphysical parsimony that is central to the debate between absolutism and comparativism (cf. §5; Martens 2019).

  27. 27.

    cf. Sider (manuscript), §3.2.

  28. 28.

    See, for instance, Roberts (2016), §5.

  29. 29.

    I adapt this terminology (as well as ‘dynamic relationalism’, see main text) from Huggett (1999), who uses the term ‘kinematic relativity’. However, (1) the notion of relativity is easily confused—in the context of the philosophy of space—with Einstein’s notion of the relativity of simultaneity, and (2) kinematic relativity is the epistemological or kinematic version of Leibnizian (metaphysical) relationalism, making ‘kinematic relationalism’ the more obvious choice.

  30. 30.

    The distinction bewteen kinematic comparativism and dynamic comparativism (see main text) is not only analogous to Huggett’s distinction (cf. fn.29), but also closely related to Van Cleve’s ‘existentially relational’ vs. ‘referentially relational’ properties (van Cleve 1987, p. 209) (which echoes Brentano’s distinction between the Relativ and the Relativlich (Brentano 1974, p. 272; van Cleve 1987, p. 209)) and Bennett’s ‘reference by telling’ vs. ‘reference by showing’ (which is relevant for the Ozma games discussed in the main text) (Bennett 1970; Earman 1991, p. 132; Van Cleve 1991, p. 15).

  31. 31.

    It seems that Mach had something like this in mind (Martens 2018), and that Roberts is invoking something like this (Roberts 2016, pp. 4, 12).

  32. 32.

    Zanstra (1924) suggests that Berkeley, Aristotle and Copernicus seem to make the invalid argument of inferring (metaphysical) relationalism from kinematic relationalism. At the very least they are confusing the two notions.

  33. 33.

    In Newton’s Scholium to the Definintions in his Principia (Newton 1686/7).

  34. 34.

    Unless one combines comparativism—specifically a version that is anti-realist about absolute masses (see Sect. 9)—with a verification theory of meaning.

  35. 35.

    ‘Quantities’ is used more colloquially by Saunders than the technical definition in the current paper.

  36. 36.

    Does it make sense to multiply magnitudes (rather than the quantities that represent them) by a scalar (Cian Dorr, personal communication)? We could perhaps respond to this question by adding extra (multiplicative) structure to the absolute masses, but it seems that that structure is not needed within Newtonian Gravity (cf. fn. 21), and it would reduce the metaphysical parsimony. Alternatively, we could piggy-back on the structure of the quantities: apply the scalar multiplication at the representational level and find out which new magnitudes are now represented by the quantity that represented the old magnitudes. Dasgupta develops a mass-counterpart theory that might be used for this purpose (Dasgupta 2013) (although this basically boils down to a passive Leibniz Scaling, as defined below). If one feels that either of these options would be cheating, it also suffices for the purposes of this paper to consider any automorphism that is not the identity.

  37. 37.

    Except, in a sense, in the case of regularity comparativism (Martens 2017b).

  38. 38.

    Despite, ironically, Clarke having been the one to introduce the Static Shift (Alexander 1956; Maudlin 1993).

  39. 39.

    This is closely related to Dasgupta’s mass-counterpart theory (Dasgupta 2013).

  40. 40.

    See also (Dasgupta 2013, pp. 133–134).

  41. 41.

    Although I have taken the qualifier ‘absolute’ in ‘absolute masses’ to mean ‘monadic’ (cf. fn.4), for an absolutist about mass these masses are also intrinsic, even though they will not necessarily be for a comparativist who is a realist about absolute masses (eg. regularity comparativism (Martens 2017b)). In the case of acceleration magnitudes, however, I take it that even an absolutist about those magnitudes will not (necessarily) hold that they are intrinsic, since, according to the “at-at” theory, acceleration is reducible to the limit of velocity differences (which in turn are reducible to a limit of differences in position), cf. §8.3.

  42. 42.

    See fn.36.

  43. 43.

    It is not clear though that either this reinterpretation of the gravitational law, or L2, really are laws. They are not in the form of differential equations, and thereby not suitable for solving initial value problems. Moreover, it can be shown that highly deviant evolutions satisfy these two ‘laws’ (Martens 2017a, §4.4.1); solutions that we would expect not to be physically allowed by Newtonian Gravity. Machian comparativism overcomes both these problems (Martens 2017a, §4.2.3; Martens 2019).

  44. 44.

    As opposed to the comparative acceleration, rather than the relative acceleration.

  45. 45.

    I believe Baker’s Friction World and Shell World arguments to suffer from the same mistake (Baker 2013a, b).

  46. 46.

    Unless it is being reduced to, say, the curvature of spacetime, of course.

  47. 47.

    I believe that both these problems also (fatally) plague Baker’s Friction World argument (Baker 2013a, b).

  48. 48.

    Technically L2 could not even be applied here, since there are no non-zero force relations. This could be remedied by introducing another particle with a force acting on it.

  49. 49.

    See also Schlick’s ‘method of coincidences’ (Schlick 1922, 1925, 1949; Stevens 2014).

  50. 50.

    In fact, if we place particle detectors infinitely far to the ‘left’ and ‘right’, we have in effect constructed the absolute mass detector of which Dasgupta argued it could not exist (Dasgupta 2013, §8.3).

  51. 51.

    As opposed to relational.

  52. 52.

    An even closer analogy, using particles rather than a bucket, would be Skow’s two-particle scenarios (Skow 2007) (discussed by Pooley, manuscript) or Barbour’s three-particle scenarios (Barbour 2000, Fig. 13). As all these scenarios are used to make the same point, and Newton’s bucket is the most famous of the three, I have chosen to adopt the name ‘comparativist’s bucket’.

  53. 53.

    However, in the general multi-dimensional, multi-particle case, presumably only a subset of the space of solutions of measure zero will have colliding particles at all. Perhaps this is even a reason to believe that we can and should ignore that subset. In the general scenario it will then be necessary to make our criterion of empirical equivalence of worlds more strict by considering not only coincidence of trajectories but also angles (i.e. shapes) or ratios of distance over time. The absolutist and comparativist should still be able to agree on this criterion.

  54. 54.

    This idea originates with Aristotle and was developed in the Middle Ages (Arntzenius 2000; Grant 1981; Dijksterhuis 1986). In more modern times it is associated, amongst others, with Russell (1903).

  55. 55.

    It may be suggested that scaling t could still change whether the inequality is satisfied, but it is not clear, on the at-at theory, that t remains as an independent variable alongside distances, once one has reductively defined velocity and acceleration in terms of distances. As Baker puts it: “[w]hether the inequality holds or not is a scale-independent fact about t’s neighborhood” (Baker 2013b, p. 20).

  56. 56.

    Strictly speaking this is only true for possible worlds with only two particles, call these the scarce worlds. If there are more particles, for simplicity say several (approximately) isolated two-particle systems, these systems might stand in the correct mass relation to the alpha-two-particle system that is the counterpart of the system in the scarce worlds to allow for a different evolution. The question then remains however how to determine which of the two-particle systems is the alfa-system. These issues are discussed elsewhere (Martens 2019).

  57. 57.

    A Machian rather than this Laplacian (Martens 2018) form of reducing mass to acceleration is discussed elsewhere (Martens 2017b) under the name of ‘regularity eliminativism’, but leads to a form of weak absolutism, rather than a defence of comparativism.

  58. 58.

    One may, for instance, have expected this mistake from the title of Pooley’s forthcoming book on the substantivalism–relationalism debate, The Reality of Spacetime, although this does not in fact occur in the book (Pooley, manuscript).

  59. 59.

    For instance, but not necessarily, as a ‘real pattern’, a phrase coined by Dennett (1991).

  60. 60.

    In fact, Dasgupta seems to have some sympathy with realism about absolute masses, insofar as he finds it important that comparativism must be able to explain what he calls the kilogram facts (e.g. the fact that my laptop is 2 kgs), which seem closely related to absolute masses. He even develops a novel, plural notion of grounding (Dasgupta 2013, 2014) in order to be able to explain these facts. It is not directly clear to me how he takes this sympathy with realism about absolute masses to be compatible with denying the detectability of absolute masses.

  61. 61.

    Arguably, Roberts’ rough definition of comparativism could be interpreted as such: “A comparativist about a quantity-type—about mass, for instance—says that what has significance is relations among the values of quantities of that type, rather than values of particular quantities of that type” (italics in original) (Roberts 2016, p. 3) (see also his fn.3).

  62. 62.

    If one had chosen to redefine absolutism (comparativism) as realism (anti-realism) about absolute masses, this loophole would not be an option. Regularity comparativism would be off the table, ab initio.

  63. 63.

    Since comparativist mass ratios are generally also quidditistic (§3)—Machian comparativism being the exception (Martens 2019)—a similar problem seems to occur for comparativist mass ratios.

  64. 64.

    In fact, absolute masses are underdetermined in a sense that goes beyond even their inexpressibility (Martens 2017a, §2.7). (Moreover, since comparativist mass ratios are generally also quidditistic—Machian comparativism being the exception (Martens 2019)—this further underdetermination seems to apply to comparativist mass ratios as well.)

  65. 65.

    §8.3 of Dasgupta’s 2013 paper (Dasgupta 2013) suggests that Dasgupta is sympathetic to this way of thinking. §8 of his 2015 paper (Dasgupta 2015b) suggests otherwise.

  66. 66.

    Of course we could just describe a specific elementary particle to them and use its mass as the unit of mass, but here we consider NG only.

  67. 67.

    See also Maxwell’s mass unit (Maxwell 1873, vol.1,§5, 3–4).

  68. 68.

    That is, yet another alternative besides the ideology proposed by Baker (i.e. distances, velocities and accelerations).


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I would like to thank David Baker, Harvey Brown, Adam Caulton, Eddy Keming Chen, Erik Curiel, Shamik Dasgupta, Neil Dewar, Patrick Dürr, Sam Fletcher, Dennis Lehmkuhl, Niels Linnemann, Tushar Menon, Thomas Møller-Nielsen, Zee Perry, Oliver Pooley, Carina Prunkl, John Roberts, Simon Saunders, Syman Stevens, Reinier van Straten, Chris Timpson, Teru Tomas, David Wallace and Alastair Wilson for useful discussions, comments on earlier drafts of this essay, and for their generosity with their time. I am grateful for questions and comments from the audiences at the Ockham Society, the Socrates Society, the DPhil seminar, the Philosophy of Physics Research Seminar and the Philosophy of Physics Graduate Lunch Seminar at the University of Oxford, as well as SOPhiA 2014 in Salzburg, the 2014 Bucharest Graduate Conference in Early Modern Philosophy, the 2014 and 2015 Tübingen Summer Schools in HPS, the 2015 Metaphysics of Quantities Conference at NYU, the 2015 Graduate Workshop in Mathematical Philosophy at the MCMP in Munich, and the DPG2015 in Berlin. This material is based on work supported by the Arts and Humanities Research Council of the UK, a Scatcherd European Scholarship, and in part by the DFG Research Unit “The Epistemology of the Large Hadron Collider” (Grant FOR 2063). The major part of this essay was written while I was at Magdalen College and Department of Philosophy, University of Oxford, including a two-month research visit to Princeton University (supported by the AHRC Research Training Support Scheme and a Santander Academic Travel Award).

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Martens, N.C.M. The (un)detectability of absolute Newtonian masses. Synthese 198, 2511–2550 (2021).

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  • Physical quantities
  • Comparativism
  • Absolutism
  • Mass
  • Newtonian gravity
  • Detectability