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Against Laplacian Reduction of Newtonian Mass to Spatiotemporal Quantities

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Abstract

Laplace wondered about the minimal choice of initial variables and parameters corresponding to a well-posed initial value problem. Discussions of Laplace’s problem in the literature have focused on choosing between spatiotemporal variables relative to absolute space (i.e. substantivalism) or merely relative to other material bodies (i.e. relationalism) and between absolute masses (i.e. absolutism) or merely mass ratios (i.e. comparativism). This paper extends these discussions of Laplace’s problem, in the context of Newtonian Gravity, by asking whether mass needs to be included in the initial state at all, or whether a purely spatiotemporal initial state suffices. It is argued that mass indeed needs to be included; removing mass from the initial state drastically reduces the predictive and explanatory power of Newtonian Gravity.

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Notes

  1. One might respond that we could simply hold a massive object in each hand, and feel the potentially different weights, despite there being no spatiotemporal difference. However, the experience of the strength of that force in one’s brain is correlated with and reducible to the depth to which the massive object protrudes into our skin, that is our somatosensory nervous system. Even weighing an object in one’s hand is consistent with the claim that all observation is spatiotemporal.

  2. Similar sentiments are echoed by Bell (“[I]n physics the only observations we must consider are position observations...” [10, p. 166]), Holland (“[A]ll experiments in the real world ultimately reduce to the determination of position. As far as we are aware this [assumption] is completely in accord with actual laboratory practice” [11, p. 350]) and Jammer (“[I]n the last analysis all measurements in physics are kinematic [i.e. spatiotemporal] in nature...” [12, p. 6]).

  3. Except perhaps for negative mass values, but these can be made irrelevant if we take it as a fundamental feature of the gravitational law that it is attractive and thus only cares about the magnitudes of the masses. (See Jammer [12, Ch. 4] for a historical overview of the search for negative (gravitational) masses.)

  4. In the regime of applicability of the theory.

  5. See footnote 6.

  6. Correct in (at least) the sense of being in agreement with folk science—“If I had dropped this cup, it would have fallen to the ground”; “Two celestial bodies cannot execute the Argentine tango”; “If the two particles described in Sect. 3.1 had (in)sufficient initial velocity, they would escape (collide)”. Of course we have access to counterfactuals only indirectly: we first choose the best theory to represent and explain the actual world (within the relevant regime), and only then infer counterfactual claims by inspecting what evolution this theory would generate for counterfactual initial conditions. In a sense it thus seems circular to claim that Newtonian Gravity generates the correct set of empirically possible worlds (within the Newtonian regime), beyond the non-trivial claim that Newtonian Gravity is indeed the best theory in light of the empirical data in the actual world (within the Newtonian regime). It is logically possible and it would be interesting to consider a totally different Newtonian theory, without mass, that manages to represent and explain the actual world as well as or even better than Newtonian Gravity with primitive masses, whilst nevertheless having different modal consequences. I criticise a similarly revisionist option elsewhere [7, Ch. 3 & 4.4.2]. In this paper I am in the first instance interested in the less radical project of trying to remove a primitive notion of mass from standard Newtonian Gravity without revising that theory—that is, while maintaining the same modal consequences. We are considering reducing mass to other primitives within the same theory, not coming up with a distinct theory. Once this fails, a distinct project could then be to search for such a revisionary theory as just described, but that is outside the scope of this paper.

  7. Zanstra [13] adopts a similar ‘substitution approach’ in the analogous debate on relationalism about space.

  8. Whether a massive particle is less or more massive than another particle with a different mass determinate.

  9. The ratio between the masses of two massive particles.

  10. How the mass of one massive particle compares to the combined mass of two other massive particles.

  11. Dees [15] advocates a position like this. Perry [16] considers a more varied range of reductionist projects, including the Machian project.

  12. See also Esfeld & Deckert’s project [17] and my paper on regularity comparativism and regularity eliminativism about mass [14].

  13. Note that in general \(\frac{a_{21}}{a_{12}}\) depends on the reference frame, although it will be constant across inertial reference frames. Mach’s definition therefore also depends on the first law [12, p. 15], which provides the notion of an inertial frame. However, operationalising the inertial frames brings with it its own problems. Pendse [19] proves that there exists an infinite set of special non-inertial frames such that observers at rest with respect to those frames obtain positive and constant values for the mass ratios via Mach’s operational definition, which nevertheless differ from the corresponding values found in the inertial frames. Importantly, those observers will not be able to tell that they are not in an isolated, inertial frame. Note that these problems are irrelevant to the project in this paper: we take the initial spatiotemporal quantities with respect to some inertial frame to be given, and use them to attempt to calculate the emergent masses and the evolution of the system. The Machian project—“the heuristic aspect” in Pendse’s terminology [19, p. 55]—on the other hand starts out with observed trajectories only, and needs to somehow reconstruct the inertial spatiotemporal quantities before one may derive the mass (ratios).

  14. Here I sympathise with Barbour’s warning not to be “misled by the special circumstances of our existence... Take a billion of particles and let them swarm in confusion—that is the reality of ‘home’ almost everywhere in the universe. The stars do seem to swarm... We must master celestial [determination of mass ratios] and not be content with the short cuts that can be taken on the Earth, for they hide the essence of the problem” [2, pp. 137–138].

  15. For comparativism about mass—defined below—an analogous issue of consistency or transitivity arises [21] [7, Ch.5].

  16. Mass relations, more generally. But I will follow Baker [6] in focusing on mass ratios.

  17. In the case of non-zero angular momentum, the set of solutions that features coinciding particles is of measure zero. In that case we may need to turn to the evolution of shapes/angles to empirically distinguish the models, rather than coincidence.

  18. Modulo some well-known exotic counter-examples [22, 23].

  19. This inequailty governs the special case where the mass ratio is one, but this could easily be generalised.

  20. We may call this a (spatiotemporally) local operational definition. In Ref. [14] I discuss an attempt at a global definition, namely a reduction of the absolute masses to the full 4D mosaic of particle trajectories (and perhaps their mass ratios). Dasgupta provides an example of an alternative global definition. He introduces a notion of plural grounding, and argues that the totality of kilogram facts is plurally grounded in the totality of mass ratios [24].

  21. Pendse [27] objects that we do not have independent empirical access to the Gravitational Law. However, in the context of the project in this paper we simply take the laws as given. In fact, Mach and Pendse’s own projects take Newton’s Laws as given, so why could Narlikar not add the Gravitational Law to this?

  22. This awkward arbitrariness stems from the slightly surprising attempt to reduce the n degrees of freedom of a scalar, mass, to the 3n degrees of freedom of a vector, acceleration [7, §6.6]. This arbitrary choice of an x-axis seems especially unjustified in the homogeneous Euclidean space in which Newtonian Gravity lives (pace Knox [28])—we seem to have implicitly chosen a preferred axis, in a space which has no structure to ground such a notion. Of course, if mass is a primitive notion it is easy to show that nothing depends on the choice of axis; it is not clear why the same would necessarily be true in the reductionist framework. As argued elsewhere, removing this arbitrariness by using all components of acceleration instead forms no improvement [7, §6.6].

  23. Although these equations may be linearly independent in general, presumably not all specific instances will be so. What to do with those deviant cases? Perhaps it will turn out that these specific systems are of measure zero in the space of solutions, and that that gives us some reason to ignore them. Or perhaps these cases result in infinitely many solutions which are all empirically equivalent. Or perhaps choosing a different set of instants to measure the distances suffices to restore linear independence. All of this remains to be shown though.

  24. See Ref. [14] for a discussion of reconstructing absolute masses using this approach.

  25. Beyond of course the obvious errors in the quantum and relativistic regimes.

  26. Or perhaps an infinitesimally small initial period of time, see footnote 36.

  27. For similar reasons Schmidt’s reduction of mass [12, 29] is disqualified.

  28. This is true only because Newtonian Gravity contains both Newton’s third law and the principle of equivalence of gravitational and inertial mass—without for instance the latter Eq. 6 would not have been as simple. This seems to suggest that if we were to go beyond Newtonian Gravity by adding other forces which, for instance, do not obey a similar equivalence principle (such as the Coulomb Force), the argument against reductionism would collapse. This cannot be true however, since this would only introduce more unknowns (i.e. the electric charges) without extra ‘knowns’ to determine those unknowns (unless perhaps the additional force depended on velocity and we could measure the velocities to aid us).

  29. Or, that the gravitational law does not care about the sign of the masses.

  30. Pooley [33, §5.4] discusses analogous issues (concerning Sklar’s relationalist manoeuvre of adding primitive accelerations to the initial conditions) in the analogous substantivalism–relationalism debate about space. See also Arntzenius [34, §5.7] on piggy-back relationalism.

  31. It might be argued that one cannot compare which particle is left or right of the middle between different solutions. One could avoid this by adding an extra particle sufficiently far from these free particles to be dynamically isolated from them, in order to serve as a reference for, say, ‘left’. However, the two solutions are clearly not each other’s mirror image, so adding an extra reference particle is not really required.

  32. On reflection, it is quite strange that we are trying to reduce the n degrees of freedom of mass, a scalar, to acceleration, which—as a vector—has 3n degrees of freedom, in the first place. This asymmetry provides the basis for an additional argument against reductionism developed elsewhere [7, §6.6].

  33. Dasgputa [35] is developing an analogue of this project in response to the accusation that relationalism about handedness, space and mass are all indeterministic. That case seems much simpler though than the case considered in this paper.

  34. Although, for e.g. a system with four masses on the vertices of a square, all with accelerations of equal magnitude pointing outwards along the diagonals, there is one unique solution if we were to allow negative masses. (It consists of masses of equal magnitude (the exact value depending on the acceleration magnitude), but the masses on one diagonal have a negative sign, whereas the masses on the other diagonal have a positive sign.)

  35. I would like to thank David Wallace for pointing this out to me.

  36. Perhaps the following serves as a plausibility argument for an upper bound on the order k of initial spatiotemporal data that would guarantee removing the underdetermination of mass (although the overdetermination problem, resulting in conspiratorial (i.e. unexplained!) constraints, still remains). On one popular view, the “at-at” theory of motion, (initial) velocities are not in fact properties of an (initial) instant, but of an infinitesimal (initial) period of time. After all, velocity is usually defined as \(\lim \limits _{dt \rightarrow 0} \frac{r(t+dt) - r(t)}{dt}\), which is a property intrinsic to \([t, t+dt]\). In general, the initial \(k^\text {th}\)-order time derivative of r is a property of \([t_0, t_0 + kdt]\). In a slogan: ‘God was not done when he created the initial configuration and the laws, but he had to also specify the subsequent \(k-1\) configurations (depending on the order of initial spatiotemporal data that we are considering)’. Now, if \(k = n + 1\), this initial period (of \(n+1\) instants) effectively contains \(n-1\) independent sets of accelerations (and even more sets of distances). Narlikar’s method then guarantees that this initial data fixes the masses.

  37. See footnote 30.

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  1. Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen, Aachen, Germany

    Niels C. M. Martens

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Correspondence to Niels C. M. Martens.

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I would like to thank Tushar Menon, Tom Møller-Nielsen, Oliver Pooley, Carina Prunkl and David Wallace for useful discussions and comments on earlier drafts of this paper. I am grateful for questions and comments from the audiences at “Foundations 2016” at LSE, London, and the Ockham Society at the University of Oxford. This material is based upon work supported by the Arts & 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 paper was written while I was at Magdalen College and the 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. Against Laplacian Reduction of Newtonian Mass to Spatiotemporal Quantities. Found Phys 48, 591–609 (2018). https://doi.org/10.1007/s10701-018-0149-0

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