Abstract
Wilson [Dialectica 63(4):525–554, 2009], Moore [Int Stud Philos Sci 26(4):359–380, 2012], and Massin [Br J Philos Sci 68(3):805–846, 2017] identify an overdetermination problem arising from the principle of composition in Newtonian physics. I argue that the principle of composition is a red herring: what’s really at issue are contrasting metaphysical views about how to interpret the science. One of these views—that real forces are to be tied to physical interactions like pushes and pulls—is a superior guide to real forces than the alternative, which demands that real forces are tied to “realized” accelerations. Not only is the former view employed in the actual construction of Newtonian models, the latter is both unmotivated and inconsistent with the foundations and testing of the science.
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Notes
I’m adopting the language of “real” forces from the literature. In my mouth, at least, this means solely that the forces in question are causally efficacious, that they are—in Newton’s terms—“true and physical” (Newton 1726/1999, p. 408) as opposed to mathematical vectors that represent such forces. I have no intention of trying to determine whether mathematical objects such as vectors are “real” in some important sense and don’t deny that there will be cases in which the best explanation involves the total force acting on a body. My point, in declaring some elements of Newtonian physics real and others not, is to demarcate which of these elements are taken to have causal implications of their own and which represent or encode the effects of the former.
Wilson (2009) also gives an experiential or phenomenological argument in support of total forces. I take it that the same considerations apply.
It has been objected to me that this reading begs the metaphysical question (against resultant forces, in particular). But that’s wrong: the point is that we can transform any Newtonian models using the principle of composition, regardless of whether or not the real forces are usefully read off of their vectors. As such, the point is also independent of the issues investigated by Lange (2011), which have to do with how the principle is proven and its modal implications.
It’s worth noting that Cartwright’s interests and opinions are different from those tackled in the contemporary discussion. For a detailed discussion of both her complaint and her (changing) views, see Spurrett (2001); for an extremely worthwhile response to Cartwright’s original worries, see Forster (1988).
Can the same claim be made about the law of universal gravity and physical interactions? I doubt it, but I’ll accept that the two “types” of force are analogous in this manner for the sake of argument.
The argument from Smith’s work is drawn explicitly by Massin (2017); what follows is a reconstruction of what I take this argument to be, not what I take Smith’s original conclusion to be. I’ve transformed Smith’s harmonic oscillator problem into a gravitational problem. For his discussion, see Smith (2010, pp. 361–363); for a discussion of a similar problem, see Moore (2012, pp. 367–368).
I’m simplifying: we’d also need to account for the interaction between j and k and the effect of i on each. The point doesn’t depend on these details.
It also doesn’t appear to match Newton’s historical formulation of the second law in terms of motive forces, though things are much more complicated here because it isn’t clear what Newton’s version of the second law actually says—it certainly wasn’t interpreted by his contemporaries as equivalent to \(F=ma\). Euler, whose “Discovery of a New Principle of Mechanics” (1752) popularized \(F=ma\), clearly understood this “principle” as distinct from any of Newton’s laws; we probably owe the interpretation of the second law in terms of \(F=ma\) to Lagrange’s Analytical Mechanics, which argues that Euler can be understood as having generalized the second law (Lagrange 1811/1997, p. 172). There’s a substantial historical literature on the subject, which does not, or does not clearly, support the contention that the second law must be understood in terms of total forces. My favored interpretation—offered by Pourciau (2011)—indicates the opposite: Newton’s version of the second law ought to be understood in terms of each force acting on a body and only derivatively (in virtue of the principle of composition!) on their total effect.
My argument builds on a problem originally raised by Wilson (2009), discussed in the next paragraph. The rest of the section demonstrates that the other classes of accelerations that one could privilege fare no better.
Why total? Because any other acceleration will be partial, violating the motivating intuition.
There’s nothing wrong with frame-relativity in itself; what’s objectionable is the frame-relative interpretation of Newtonian forces. Besides the broad mismatch between this concept and how actual practitioners understood Newtonian physics, understanding forces in a frame-relative manner has the effect of undermining the understanding of inertial frames and true motion in Newtonian physics discussed below: if forces are frame-dependent, they can’t be used to define the true motions and thus to distinguish between Tychonic and Keplerian models of the solar system.
It is also dialectically important that Massin (2017, p. 819), at least, takes the fact that “component accelerations” are unobservable theoretical posits as one of the main reasons to doubt their existence—but the same motivation rules out privileging anything beyond relative accelerations as the “actual” or “realized” accelerations.
Note that similar considerations undercut Wilson (2009)’s experiential argument for total force: the total forces that we experience are total only relative to a particular choice of local inertial frame—we don’t experience the total force pulling us towards the sun in the appropriate sense, for example.
Notice that if we build gravity into the spacetime structure á la Newton–Cartan theory, we can define a frame-invariant notion of acceleration using the tangent field of the appropriate timelike curve and then define a notion of force on this basis (see Malament 2012, pp. 252–253). This result may appear to belie my claim that the real accelerations are defined in terms of the real forces, but it doesn’t: since gravity is encoded in the metric, the metric is playing the role that forces (the forces found in specific force laws!) play in a force-based interpretation, which (to my eyes) just serves to reinforce the point that the relevant notion of acceleration must be defined in terms of whatever is playing that role.
Technically, this requirement is met by any observer moving with a constant velocity relative to the center of force. For our purposes, this caveat can be ignored.
Contemporary scientists who employ Newtonian physics are still concerned with the latter of these projects, of course.
See Forster (1988) for a different way of employing much the same point against Cartwright.
Massin (2017) recognizes this fact and some of the problems that it poses to a view that posits forces where there is nothing for the acted-on body to interact with—i.e., any view that holds that the total forces are causally efficacious.
Since I’m primarily interested in the implications of the third law, I’m passing over the question of whether Newton actually provided a sufficient argument for his solution. But note: while this is a particularly thorny question for Newton scholarship, precisely the issue for Newton’s contemporaries—such as Cotes and Huygens—is that the mathematical models employed to demonstrate the inverse-square rule don’t discriminate between attractions and impulses, and this difference has physical consequences. For detailed discussion of both this issue and the derivation more broadly, see Stein (1990), Smith (2002), and Harper (2012, pp. 346–355).
The next two paragraphs follow Smith (2014).
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Acknowledgements
I would like to thank Nic Teh; Katherine Brading; James Nguyen; Evan Arnet; audiences at Bloomington, Leeds, and Notre Dame; and a number of anonymous reviewers for helpful comments on earlier drafts of this paper. I am additionally indebted to George E. Smith for his invaluable instruction on Newton and Newtonian mechanics, without which I would not have been able to write this essay.
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Dethier, C. Forces in a true and physical sense: from mathematical models to metaphysical conclusions. Synthese 198, 1109–1122 (2021). https://doi.org/10.1007/s11229-019-02086-z
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DOI: https://doi.org/10.1007/s11229-019-02086-z