A Non-reductive Model of Component Forces and Resultant Force

Abstract

While there are reasons to believe that both component forces and a resultant force operate on a body in combined circumstances, the threat of overdetermination largely prevents adoption of this view. Accordingly, a lively debate has arisen over which force actually exists and which force is eliminated in combined circumstances, the components or the resultant. In this article I present a non-reductive model of resultant force which ensures the existence of both the resultant force and the component forces without overdetermination. According to the non-reductive model, the resultant force is the summation of component forces, where summation is interpreted as meaning that the component forces, when temporally and spatially overlapping, form a superposed mixture which is the resultant force, but which is not identical to the conjunction of component forces taken with temporal and spatial independence.

Acknowledgements

I should like to thank Jessica Wilson for valuable discussions and suggestions that significantly improved this article. I should also like to thank two anonymous referees of this journal for a number of valuable comments. This research was supported by the Social Sciences and Humanities Research Council of Canada.

Notes

J. Wilson (2009, 547) concedes that it is ‘natural enough’ to use the arm-wrestler case as proof that we experience both the resultant force and the component forces. However, she argues that since the arm-wrestler case does not imply the elimination of the resultant force, we are left with both a resultant force and a number of component forces. This appears to render the effect overdetermined, which calls into question the viability of countenancing both the component forces and a resultant force. I will consider this overdetermination problem in section 7.

Bigelow and Pargetter (1990, 284–285) agree with this model's preference for the aggregation of component forces over a singular resultant force, though they eliminatively reduce the resultant force to the aggregation of the component forces rather than conservatively reduce the resultant force in combined circumstances.

The phenomenon of multiple realizability, as related to forces, is similar to the so-called law of transmissibility, according to which the motion of a (rigid) body is unchanged if the force acting on a body is replaced by another force of the same magnitude and direction acting on the body.

Russell makes a related point, contending that the resultant force must be a simple, new entity (say, D), which is not identical to its components B and C, or the conjunction thereof: ‘Let there be three particles A, B, C. We say that B and C both cause accelerations in A, and we compound these two accelerations by the parallelogram law. But this composition is not truly addition, for the components are not parts of the resultant. The resultant is a new term, as simple as its components, and not by any means their sums’ (Russell 1903, 477; cf. J. Wilson 2009, 541). The resultant force, in so far as it is simple (i.e. not the multiplicity which the component forces are), and, in so far as it has an ontological status of its own (i.e. it persists through change to the component forces), is a new entity unto its own, distinct from the conjunction of the component forces.

In a number of places, Cartwright interprets Mill along these lines (Cartwright 1994, 179; 1980, 78–79). This interpretation, however, is not without controversy. It is possible, for example, to interpret Mill's statement that ‘both causes have their full effect’ as simply endorsing (b _t ), the view that the boat will end up in the same location, whether the forces are applied with overlap or independence (Mumford and Anjum 2011, 41). Yet while Mill clearly endorses (b _t ), he also argues that (a _t ) the boat will travel the same distance whether the forces are applied with overlap or independence; a possibility that the case of balancing forces appears to rule out. It is also possible to interpret Mill's statement that the first, northern individual cause is only ‘tending to drive [the boat] to the north’ as indicating that while the northern cause tends to drive boats north, or disposes boats to move north, in combined circumstances it does not necessarily do so (McLaughlin 1992, 60–61). Mill seems clear, however, that in combined circumstances both causes continue to have the full effect they would have had independently. For example, Mill argues that ‘even when the concurrent causes annihilate each other's effects, each exerts its full efficacy according to its own law, its unique law as a separate agent’ and ‘even if the two causes which are in joint action exactly annul one another, still the laws of both are fulfilled … each agent produced the same amount of effect as if it had acted separately’ (Mill 1843, III.6). Besides these textual difficulties with this interpretation, there are three philosophical problems as well. First, as discussed in detail by Bigelow and Pargetter (1990, 282–283), any account appealing to forces as dispositional entities is problematic. Secondly, even among those who read Mill as endorsing causal laws as tendencies, there are those who cannot help but read him as insisting that these tendencies must be activated in every case. Bhaskar, for example, complains that ‘Mill's mistake here is to suppose that whenever a tendency is set in motion the effect must be in some sense … occurring’ (Bhaskar 1978, 99). Finally, the wider context of Mill's argument in section III.6 tells against this interpretation. In this section Mill is deeply interested in introducing the influential distinction between homeopathic causes and heteropathic causes. Homeopathic causes exert their causal efficacy the same in isolation as in combined circumstances, and as such exert their causal efficacy independently of other causes present. Heteropathic causes, on the contrary, act differently in combined circumstances than they do in isolation. With this taxonomy in mind, Mill argues that those entities obeying the rule of the composition of causes are homeopathic causes, so these entities must operate as independent causes when in combined circumstances. Thus, those objects that obey the composition of causes have laws ‘which are rigidly fulfilled in every combination into which the objects enter’ (Mill 1843, III.6).

I should like to thank an anonymous referee for suggesting this point.

For Lombard, events have two constitutive properties, which are the property the object instantiated before the change and the property the object instantiated after the change. For example, the water's being hot is a constitutive property of the event of the water's becoming cold, and the water's being cold is a constitutive property of the event of the water's becoming cold. This model is endorsed because events are typically construed as changes, and changes involve an object's losing and/or gaining a property. Thus, the event is the water's entire change from hot to cold, a process that happens over a time. For Kim, events have only one constitutive property and one constitutive time. However, Kim construes the category of time broadly enough so as to allow lengthy events that occur over a period of time to be counted as one event. For example, he speaks of the event of the leisurely stroll (Kim 1976, 42–43), which, presumably, happens over a lengthy period of time.

At this point in the argument it does not matter whether the individual is the binary product or the binary sum.

If the mixture cannot be partitioned into its component forces then we would have a case of fusion. Should we perhaps endorse the eliminative fusion of the component forces into the resultant force in combined circumstances (cf. Humphreys 1997)? There are several problems with this model. First, the resultant force is the summation of component forces, not simply a summation of nothing, so the component forces must continue to exist. Second, the elimination of the component force in combined circumstances is counter-intuitive, given the results of section 3. Third, endorsing fusion implies that the individual components stop existing at the moment of fusion only to begin existing when the fusion ends. Finally, if the component forces exist evenly across more space than the fusion occupies (i.e. the wind is blowing on more than the boat), then the component forces will exist in this space, but not in the space where the current is simultaneously active. For these reasons, it is better not to endorse a fusion of component forces into a resultant force.

This is akin to saying that Theseus's ship, made of planks ABC, is the same as Theseus's ship, made of planks XYZ, which, though subject to controversy, is a plausible position.

Traditionally, an objection to the non-reductive suggestion that underlying entities, whether they be component forces or something similar, are individually insufficient is that this move violates physical causal completeness. Physical causal completeness suggests that every event has a sufficient physical cause, and is supported by an accumulation of scientific evidence. This worry is moot in this setting. To say that the component forces, as referred to in Newton's law of universal gravitation and Coulomb's law of electrostatic force, are insufficient due to the fact that the resultant force, as referred to in Newton's second law, is required as well is not to advert to non-physical causes.

Besides, if this is a reductive position (the semantics of the situation is not of great concern), this does not imply the model is incorrect. Further argumentation would be required in order to prove the model false.