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Global Interaction in ClassicalMechanicsJon Pérez LaraudogoitiaPublished online: 20 Aug 2006.

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International Studies in the Philosophy of ScienceVol. 20, No. 2, July 2006, pp. 173–183

ISSN 0269–8595 (print)/ISSN 1469–9281 (online) © 2006 Inter-University FoundationDOI: 10.1080/02698590600814357

Global Interaction in Classical MechanicsJon Pérez LaraudogoitiaTaylor and Francis LtdCISP_A_181377.sgm10.1080/02698590600814357International Studies in the Philosophy of Science0269-8595 (print)/1469-9281 (online)Original Article2006Taylor & Francis202000000July 2006Jon PérezLaraudogoitiaylppelaj@vc.ehu.es

In this paper, an example is presented for a dynamic system analysable in the frameworkof the mechanics of rigid bodies. Interest in the model lies in three fundamental features.First, it leads to a paradox in classical mechanics which does not seem to be explainablewith the conceptual resources currently available. Second, it is possible to find a solution toit by extending in a natural way the idea of global interaction in the context of what iscalled interaction by impenetrability. Third, the solution presented throws light on a prob-lem posed and discussed in the recent literature in connection with the mass conservationprinciple.

1. Introduction

There are basically two ways of working on a theoretical level with a scientific theory.One way consists in focusing attention on its fundamental postulates and their conse-quences, and the other in studying concrete models of it. In classical mechanics, towhich I will restrict myself in this article, an example of the first approach is providedby the Lagrangian and Hamiltonian formulations. Examples of the second type involvethe introduction and analysis of particular sets of initial conditions whose evolutionyields interesting information about the theory, information of a kind which cannot beobtained from the general theoretical principles alone. Here, I am interested in exam-ples of this second type, among which some of great theoretical and/or philosophicalsignificance can be mentioned. Thus, Xia (1992) proved Painlevé’s conjecture aboutthe existence of non-collision singularities, and Lanford (1975) showed for the firsttime how it is possible for indeterminism to arise in the evolution of a system of parti-cles, each which is governed by strictly deterministic laws. The model I will introduceand discuss in the next sections goes along the same lines and owes its interest to three

Jon Pérez Laraudogoitia is Professor in the Department of Logic and Philosophy of Science at Universidad del PaísVasco, Spain.Correspondence to: Jon Pérez Laraudogoitia, Departamento de Lógica y Filosofia de la Ciencia, Facultad deFilología, Universidad del País Vasco, Paseo de la Universidad 5, Apartado 2111, 01006 Vitoria-Gasteiz, Spain.Email: ylppelaj@vc.ehu.es

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fundamental features. First, it leads to a paradox in classical mechanics, a radically newparadox that does not seem to be explainable with the conceptual resources currentlyavailable. Second, it is possible to find a solution to this paradox by providing a naturalextension of the idea of global interaction introduced in Pérez Laraudogoitia (2005) inthe context of what is there called interaction by impenetrability. Third, the solutionpresented throws light on a problem posed by Angel (2001) and discussed in the recentliterature in connection with the mass conservation principle.

2. Essential Dynamics

In order to introduce my model, it will be convenient first to have a look at a simpleexample of binary collision typical of textbooks for engineers. Let D be a rigid andcircular homogeneous disc, of unit mass, at rest, whose centre coincides with the originof a Cartesian system of coordinates and which lies on plane Y–Z. We will assume thedisc to have unit radius. Let A be a rigid homogeneous ring, of unit mass and radius 1/2, whose geometrical centre moves at constant velocity along axis X towards disc D andwhich is always on a plane perpendicular to this axis. For the sake of simplicity, we willassume that both A and D have nil thickness, but if you have misgivings about whattheir mutual contact at the time of collision could imply, you can imagine they have anarbitrarily small finite thickness (in which case A would be a torus). Finally, we will alsoassume that A and D are smooth—not rough—solids, and so, as the surfaces that comeinto contact during collision are smooth, there is no friction between A and D, forwhich reason the linear momentum exchanged between them does not have a non-nullcomponent parallel to plane Y–Z but is directed in the direction of axis X. In otherwords, the linear momentum exchanged is normal to the contact plane (defined as thetangent plane shared by both solids at their points of contact and which, in our case, isobviously plane Y–Z). As regards the dynamics of the situation, I will assume that theimpact between A and D is elastic. The elasticity condition of the impact is generallyexpressed in terms of the condition of conservation of the kinetic energy of translation(kinetic energy of translation before = kinetic energy of translation afterwards), butthere is an implicit, and most of the times elementary, derivation in this. Strictly speak-ing, the elastic impact is defined as that at which the restitution coefficient ε is unity. Inthe case of central (binary) collisions with ε = 1 between smooth rigid bodies—whichwill obviously not be point particles, as in the situation which we will soon deal with—not only is the kinetic energy of the system conserved but, more specifically, the kineticenergy of translation is conserved. It follows from this that the degrees of freedom ofrotation present in the problem will not be excited and, as a most important conse-quence, that the collision will be deterministic. All this is true when it is two points (onein each of the bodies that collide) that come into contact, in which case the impact iscentral if the centres of mass of the bodies lie on the impact line (defined as the straightline normal to the contact plane that passes through both of the points coming intocontact). In our case of impact between A and D, there are not just two points thatcome into contact but a non-numerable infinity of point pairs (again, one in each ofthe bodies that collide). As a consequence, there is not a single impact line, for which

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reason the notion of central impact is not well defined. We will next see, however, thatin spite of this, the elasticity condition ε = 1 guarantees that the kinetic energy of trans-lation will be conserved and that the impact will therefore be deterministic.1 Readersnot interested in the details of this proof may move on directly to Section 3 of the paper,‘The Model and the Paradox’, assuming that, after the collision, D and A simplyexchange speeds.

Let us suppose that the linear momentum transferred to disc D (and which, as wealready know, is normal to the contact plane Y–Z) has a non-null torque with respectto the centre of mass of the disc (which coincides with its geometrical centre). Thistorque, which is nothing but the angular momentum of D after collision, has radialdirection, which means that D turns around one of its diameters d* (which is a prin-cipal axis of inertia) with constant angular velocity. Since the total angular momen-tum (null before the impact) must be conserved, ring A must turn the other wayaround one of its diameters a* parallel to d*. Let P and Q be the points of A locatedon a*, and R and S the corresponding points of D with which they respectively comeinto contact. Likewise, let T and U be the points located half-way between P and Qon the ring, and V and W again the corresponding points of D with which theyrespectively come into contact. We will use PRa (and analogously for the other pointpairs: Q and S, T and V, U and W) to denote the module of the component normalto the contact plane of the relative velocity of points P and R immediately after theimpact (and obviously the component of such relative velocity parallel to the contactplane is null immediately after the impact). Of course, PRa = QSa. But if, after theimpact, A and D turn around their corresponding diameters a* and d* in oppositedirections (as we are assuming happens), then necessarily TVa ≠ UWa, so that not allof the equalities in the sequence PRa = QSa = TVa = UWa can be true. Notice,though, that, given the initial conditions before the impact and if we use PRb (andanalogously for the other point pairs: Q and S, T and V, U and W) to denote themodule of the component normal to the contact plane of the relative velocity ofpoints P and R immediately before the impact, PRb = QSb = TVb = UWb. But we areassuming that the restitution coefficient ε (which is defined as the quotient, takenwith positive sign, between the components normal to the contact plane of the rela-tive velocities of the corresponding points coming into contact, immediately afterand immediately before the impact) is unity, so that (PRa / PRb) = (QSa / QSb) =(TVa / TVb) = (UWa / UWb), and given that all the denominators are equal, so mustthe numerators. In particular, then, TVa = UWa, which contradicts the inequalityTVa ≠ UWa previously obtained. This is a reductio ad absurdum of the startingassumption that after collision, D and, consequently, A have a non-null intrinsicangular momentum. Therefore, during the collision between D and A, with theinitial conditions stipulated, the degrees of freedom of rotation are not excited,2 andfor this reason the collision can, dynamically speaking, be considered as the elasticcollision (ε = 1) of two rigid spheres of equal mass (just as the masses of D and A areequal) whose centres of mass move along the same common line (just as in the caseof D and A). The result of this instantaneous interaction is well known: the twobodies simply exchange velocities, so that finally, it is the ring that ends up at rest,

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while the disc moves with the initial velocity of the ring. Let us keep this result inmind for the analysis of the model that comes next.

3. The Model and the Paradox

The model that I will introduce now is reminiscent of Pérez Laraudogoitia (1997).There, we had a denumerable infinite set of point particles Pi (i ∈ {1, 2, 3, …}) of equalmass distributed along axis X and moving leftwards (that is, towards the region of nega-tive X coordinates) in such a way that at t = 0, P1 was at X1 = 2 with velocity V1 = 1, P2at X2 = 4 with velocity V2 = 2, … and in general Pn at Xn = 2·Xn−1 + n − 2 with velocityVn = 2n−1. We will now have a denumerable infinite set of rigid rings of null (or, in anycase, small enough) thickness Ai (i ∈ {1, 2, 3, …}) of equal mass and radii Ri = i / (i +1) distributed along axis X (the centre of ring Ai lies on axis X at the point we shall callXi, and the plane of the ring is always perpendicular to that axis; for the sake of brevity,we shall say that Ai is at Xi) and moving leftwards in such a way that at t = 0, A1 was atX1 = 2 with velocity V1 = 1, A2 at X2 = 4 with velocity V2 = 2, …, and in general An atXn = 2·Xn−1 − n + 2 with velocity Vn = 2n−1. If this were all, nothing interesting couldbe expected, for each ring would perpetuate its own inertial motion forever as it cannotcollide with any other taking it over or being taken over by it (as opposed to whathappened with particles Pi). Therefore, let us assume that at t = 0, there is also a disc Dwhich is homogeneous, rigid, circular, of unit radius, and of mass equal to that of ringAi, at rest with its geometrical centre at X0 = 0 (coinciding with the origin of a three-dimensional Cartesian coordinate system) and located on plane Y–Z. It is easy to imag-ine the resulting configuration: for each i, the pair formed by D and Ai is a physicalsystem identical to that considered in the previous section, which was formed by D andA. It is easy to check that at t = 0, An is at Xn = 2n−1 + n and that, therefore, before t =1 disc D’s state of rest will not be affected. What happens after t = 1? Let there be acertain instant t* > 1. If D is anywhere, it must be at some point X* of axis X (that is,with its geometrical centre at X*), because it can only collide with the Ai, and becauseof what we saw in the previous section, those collisions preserve the unidimensionalcharacter of motion. Let D be at X* at t* > 1 (at a distance |X*| from the origin of coor-dinates), and let p be the smallest positive integer such that (t* − 1)2p−1 > |X*| + p, thatis, such that t*·2p−1 > |X*| + 2p−1 + p. This implies that at least Ap must have collidedwith D before t* (since the velocity of Ap is 2p−1) and so transmitted its velocity to it—recall the previous section again—and that, consequently, the distance between D andthe origin of coordinates at t* must be of at least t*·2p−1 − (2p−1 + p) > |X*|. But thismeans that at t* > 1, D cannot be at X*. As this is valid for any negative number X*,3 itfollows that the starting assumption, according to which D is somewhere at t* > 1, leadsto a contradiction. Consequently, disc D cannot be anywhere after t = 1.

The previous argument assumes (very naturally) that the world tubes of connectedmaterial bodies, as subsets of the topological space R3 × R with the usual topology, arepath-connected (Buskes and Van Rooij 1997). This avoids (whether physically ormetaphysically) undesirable possibilities, like disc D disappearing from space for anarbitrary interval of time so that a numerable infinity of the Ai does not interact with

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it, and reappearing later and colliding with some Ak. Indeed, the path-connectednessof a world tube guarantees a very intuitive aspect of the continuity which lies at thebasis of the metaphysical requirement of identity throughout time for a connectedmaterial body: if such a body disappears from space, whatever body may reappear later(if any actually does) will already be a different body.

The disappearance of material bodies from space–time is nothing strange or alien toclassical mechanics (see note 3), at least since we have known the escape solutions tothe equations of motion (of which Xia 1992 is just one of many examples). Even theprinciple of conservation of mass can be expressed in such terms as to be compatiblewith such disappearances. In the formulation of Earman (1986), it reads thus: ‘particleworld lines do not have beginning or end-points and mass is constant along a worldline’ (one can apply this principle to extended bodies inasmuch as they can be consid-ered to be made of particles, that is, arbitrarily small elements of matter). If we acceptthis form of the principle of conservation of mass (I will henceforth simply call it ‘theprinciple of conservation of mass’), an interesting consequence follows for my model.Since D is at X0 = 0 before t = 1 and is nowhere at t > 1, it follows that it cannot exist att = 1 either, as its world tube (and, therefore, the world lines it is made of) would other-wise have end-points. And now, here is the paradox: we are faced with a very strikingform indeed of sudden disappearance from space. Disc D exists only at instants of timet < 1, at which it occupies position X0 = 0 permanently. Its acceleration is null at eachand every instant of its history, which means that it never undergoes interactions of anykind. What is then the cause of its sudden disappearance by t ≥ 1? The cause seems tobe the peculiar set of movements of rings Ai, but if that is true, this is at first sight a caseof causation without interaction. As causation without interaction, the effect cannot bea dynamic effect, and it certainly is not: neither D nor the Ai undergoes a change in itskinematic state at any moment in their existence. On the other hand, the effect is nodoubt physical: one of the material bodies involved disappears suddenly from space.The paradox then consists in having apparently found a form of physical causation inclassical mechanics, which is not a case of dynamic causation.

4. Solving the Paradox: Global Interaction in Classical Mechanics

By definition, a world-line-particle will be a world line (that is, a line in space–time) ateach one of whose points there is one particle (it is not presupposed that it is the sameparticle that is at all of them). I will assume that the continuity (in the topological sense)of a world-line-particle L guarantees the identity throughout time of a particle that, atany instant for which L is defined, is on L, in such a way that one and the same particleis at all points of a continuous world-line-particle.4 Obviously, a continuous world-line-particle is a conventional particle world line. For the case of extended materialbodies, we have body world tubes in space–time, where a body world tube is consti-tuted by a certain set of particle world lines, all of which are defined for at least one andthe same instant of time.5 I shall call such an instant in time maximal instant for thebody world tube in question (and, by extension, for the material body in question).Naturally, and along the same lines as the previous assumption, it is the continuity (in

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the same topological sense as before) of each of the particle world lines belonging to acertain set S (a set which is a body world tube) that guarantees the identity throughouttime of the extended material body (‘extended’ in the sense that it is not simply a pointparticle) that, at any instant for which each element of S is defined, is on the world linesof S and only on the world lines of S. In this way, the extended material body persists(that is, preserves its identity) even at instants of time (if there are any) at which not all(but at least one) of the world lines of S are defined.

A material body, in the sense in which I am using the term here, is simply a physicalsystem, and one must not confuse the identity throughout time of a physical systemwith the identity throughout time of the object (or objects) formed with it. This latteris a subtle question that has traditionally concerned philosophers (‘The Ship ofTheseus’; see, for instance, Clark 2002), but that need not worry us here. If an explosionpartially destroys Houdon’s bust of Voltaire, it is arguable whether what is left of it isor is not a bust of Voltaire, but there will be no doubt whatsoever about the persistencein time of the physical system used in the sculpture. Nevertheless, the criterion of iden-tity throughout time for extended rigid bodies has applications less trivial than this, andit is precisely those non-trivial applications that will enable us to find a solution to theparadox introduced in the previous section. In particular, one must also not confusethe identity throughout time of a physical system with the identity throughout time ofthe defined set of material particles that constitute it at a particular instant. Let ussuppose, for example, that at instant t (we shall assume maximal for F) physical systemF consists of the set of particles P = {P1, P2, P3} and that all that happens between t1 andt2 (with t2 > t1) is that P2 disappears from space suddenly at some t, t1 < t < t2 (it is notrelevant for the present discussion to know whether such a thing is physically possibleor not). It is obvious that particle set P does not persist throughout time at interval [t1,t2], because P2 is not there at t2, but physical system F does persist at this interval, forat least one of the particle world lines (in actual fact two) on which F is at t1 reaches t2.Analogously, particle P1 also persists at [t1, t2] because the particle world line on whichP1 is at t1 reaches t2 (at least). As can clearly be seen from this example, a material body(physical system) is uniquely specified at each instant of time by giving a set of particleworld lines (each of which, as we already know, is continuous) which are defined for atleast one and the same instant of time. In our example, it is clear that F is uniquelyspecified at each instant of time by giving the set of world lines of P1, P2, and P3.

Let us see now how the discussion above helps us to resolve the paradox we areconcerned with. We shall give the name DA1 to the physical system which at instant t= 0 (we shall assume maximal for DA1) consists of disc D and ring A1 (any other ringAi would do, and even any finite set of rings, as we will see) and whose linear momen-tum (in the system of reference R which I have used to discuss my model in section 3)is of m units towards the left.6 At instant t = 2, say, DA1 consists of ring A1 exclusively,but its linear momentum (in R) is still the same. Now, let R* be a frame of referencethat moves with unit velocity towards the right with respect to R. In R*, the linearmomentum of DA1 at t = 0 is 3 m, but at t = 2 it is 2 m. This change in momentumreveals (‘by definition’, it may be said) the presence of an external force (disturbance)on DA1, and since the change in momentum is formally due to D’s disappearance, one

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can reasonably say that this external force is the cause of the disappearance. It is ratheran enigmatic force because, to begin with, it does not have the same value in all inertialframes (precisely in R it is null, and this is vaguely reminiscent of gravitational force ona particle, which is also null in the system of reference accompanying it). But if what wewish to do is to explain (and so to resolve) the paradox with its help, what really mattersis understanding its genesis: where can such a force (interaction) upon DA1 comefrom? The key to the answer lies in an at first sight counterintuitive observation justi-fied in Pérez Laraudogoitia (2005), namely, that the assumption that if object A inter-acts with object B, then for all divisions in parts of B, A interacts with at least one ofthem, is false. When A interacts with the set of parts PBi into which B has been dividedwithout interacting individually with any of the PBi, Pérez Laraudogoitia (2005) saysthat A interacts globally (collectively) with the set of PBi. In contexts in which there areno particles appearing or disappearing from space–time, physical systems (materialbodies) persist throughout time in an identical manner to the sets of parts into whichthey can be divided. In this way, they can be identified with one another at no risk (atleast for the type of questions dealt with in this paper). But, as was seen above, such anidentification is not reasonable in more general contexts, in which not all particles areeternal entities. Therefore, the idea of global interaction must now be generalized toinclude also these new contexts (in which the parts comprising a physical system, for agiven decomposition of that system in parts, may not be the same at different instantsof time) in such a way that the central idea is preserved that what interact are alwaysphysical systems. We will say, then, that physical system A interacts globally with phys-ical system B if and only if A interacts with B without interacting individually with anyof the parts PBi (which are also physical systems) into which B has been divided (forsome division into parts of B). Although almost nothing changes in the form fromPérez Laraudogoitia (2005), there are, crucially, changes in the content, because now,by virtue of the criterion of individuation referred to in note 5, a physical system retainsits identity throughout time, even when many of its parts do not persist throughouttime. This criterion of individuation has been relevant before, in that it enabled us to‘individuate’ the physical system DA1 and to detect an interaction on it. Now, itbecomes relevant once again, by enabling us to ‘individuate’ the physical system DA1and confirm the absence of interaction on its parts (and this for any division of DA1 inparts). In the case of our paradox, therefore, the force upon DA1 comes from thephysical system A− formed by rings A2, A3, A4, …, and the fact that A− does not interactindividually with any of the parts into which DA1 can be divided (remember thatneither the linear momentum of D nor that of A1 is altered) does not lead to a contra-diction at all but to the verification that A− interacts with DA1 globally.7 The (global)interaction here is obviously also an interaction by impenetrability (see, especially,Pérez Laraudogoitia 2005, note 7): if it did not take place, all the rings Ai wouldinterpenetrate with D at some moment or other. However, as we have seen, it is a qual-itatively different form of global interaction from the types of global interaction admit-ted in Pérez Laraudogoitia (2005). Moreover, there it was accepted that if we get rid ofgravitation, then all the interactions in Newtonian dynamics are interactions underconditions of contact (even if they are not all interactions by impenetrability).

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Significantly, the paradox analysed in this paper shows that the opposite is alsopossible in Newtonian dynamics, as it shows global interactions (by impenetrability)that do not happen in conditions of contact to be possible: A− and DA1 interact globallyin spite of the fact that their mutual distance is of one unit at the moment (t = 1) of theinteraction.

In general, given what we have seen above, it is clear that the physical system that, atinstant t = 0 (we shall assume maximal for it) is formed by any (non-zero) finitenumber of rings together with disc D, interacts globally by impenetrability at t = 1 (atunit distance) with the physical system formed by the rest of the rings (or by anyinfinite set of rings belonging to the rest). The former finiteness is obviously requiredso that in the inertial referential R* (or any other different from R), a change in linearmomentum can be defined which reflects the presence of a (global) interaction thatcorresponds to (and, in that sense, explains) the sudden disappearance of D. The latterinfinity is necessitated by the fact that, as we know, there is no alteration in the kine-matic state of any of the rings at any moment (for this, see note 7). This unavoidablearbitrariness in the specification of exactly which physical system interacts with which(at t = 1) is a necessary (and plausible) consequence of the global (and to that sameextent not clearly located, as we are dealing with an interaction at a distance) characterof the interaction.

5. On the Principle of Conservation of Mass

In the paradox in Section 3, we specified an initial state at t = 0 and indicated that t = 1is the first instant of time at which D does not exist. At first sight, there is somethingpuzzling about this. We know that at t = 0, An is at Xn = 2n−1 + n moving leftwards withvelocity Vn = 2n−1, while D is at rest at X0 = 0. From here, it all seems to be a matter oftrivial mechanics, all movements being inertial, to predict the state of system at t = 1:An will be at Xn = n moving leftwards with velocity Vn = 2n−1, while D must continueat point X0 = 0. However, this contradicts the previous conclusion that D does not existat t = 1. As the non-existence of D at t = 1, arising making use of the principle of conser-vation of mass, goes against the predictions of elementary dynamics, should we notdrop such a principle? (Let us make it clear that here we are referring to Earman’sformulation of it, which leaves the possibility of alternative formulations open.)Although my aim in this paper has not been to defend it, I would like to finish by point-ing out that the clash between it and the predictive content of the most elementarydynamics is only apparent. Indeed, the laws of dynamics, in virtue of their predictiveand retrodictive powers, enable us to determine the different states of a physical systemF from the state specified at a given instant, whereas the principle of conservation ofmass sets constraints on what type of physical system is to be allowed in classicalmechanics, by ruling (among other things) that the particle world lines that univocallyspecify it ‘do not have beginning or end points’, that is, that they are topologically open.Thus, there is really no clash at all between the principle of conservation of mass andthe predictive content of classical mechanics because they operate at different levels. Theformer does globally, by characteristically laying down constraints of a topological

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nature on the type of physical systems that can be allowed in classical mechanics. Thelatter does at a local level, at which the (point-by-point) constraints on the class ofadmissible physical systems are laid down by the differential equations of motion.Referring now to our particular case, the absence of conflict in our paradox is evident.What dynamic laws really lead to is that D will be at x0 = 0 in t = 1, supposing that Dexists at t = 1! But dynamic laws cannot assure or predict the existence of D at t = 1; thatis not in fact their mission. That is why there is no contradiction between them and theprinciple of conservation of mass, when, thanks to the latter, we can reduce to the absur-dum the assumption that D exists at t = 1. Indeed, in our paradox, D exists exactly forinstants t < 1 (not at t = 1) and, throughout its entire existence, is a free body in inertialmotion: this remains perfectly compatible with the requirements of dynamic laws.

Angel (2001) put forward an initial condition similar to that which I have derived att = 1 using elementary dynamics, that is, he proposed something like the following asan initial condition: An is at Xn = n moving leftwards with velocity Vn = 2n−1, while Dis at X0 = 0. He then inferred from this that the principle of conservation of mass isfalse. The argument I develop in this section clearly shows that in situations like this,one does not really infer the falsity of such a principle but is merely presupposing it.Had he started ‘a bit earlier’, as I do in this section (at t = 0), he would have recognizedthat it is the principle of the conservation of mass itself that requires the non-existenceof D at t = 1. And also, if he had started a bit earlier, he might perhaps have recognizedthe global character of the interaction at a distance involved, as is patent from the anal-ysis in Section 4. It is doubtful, though, that he could have done this latter. His modelhas the serious drawback of involving (potentially) triple collisions between rigidspheres, and as triple collisions are dynamically indeterminate, we cannot thereforepredict the dynamic state of the spheres still remaining in the final state (we cannoteven predict which ones will remain in the final state). In these conditions, the possi-bility of revealing the existence of a global interaction vanishes. To be able to do thisclearly, one must eliminate from the model used any possible source of indeterminism,and to do this latter, one must be sure that the dynamic processes (potentially) involvedare strictly deterministic. This also was the aim of the detailed discussion in Section 2.

To end, I would like to mention a point highlighted interestingly by an anonymousreferee. Recently, Angel (2005) called into question the plausibility of Newtonian colli-sion mechanics on the basis of characteristics such as the lack of an upper bound onspeed. Can the model presented in this paper therefore be discarded as implausible? Itis not necessary to go into this here, as the value of my model is independent of theanswer to this question. If the answer is affirmative, the paper will have helped toconsolidate the implausibility that Angel (2005) mentions, by lighting upon a ‘para-doxical’ example of physical causation in classical mechanics that is not a case ofdynamic causation. If it is negative, the paper’s contribution would consist in an exten-sion of the idea of global interaction that allows us to clarify not only the metaphysicalrelevance (as has occurred until now) but also the physical relevance of consideringphysical systems (e.g. particles) as tetra-dimensional objects. I think the reasons infavour of the second alternative are much more interesting and convincing than thoseAngel (2005) gives, but that, as they say, is another story.

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Notes

[1] If we simply laid down the conservation of kinetic energy as a condition we would not be ableto avoid the indeterminacy that arises from the possibility that the kinetic energy of transla-tion varies at the expense of the kinetic energy of rotation. If we imposed the conservation ofkinetic energy of translation as a condition directly, we would be excluding a priori arbitrarilyprecisely that possibility. What I am saying is that the condition of elastic impact (ε = 1)eliminates both the indeterminacy and the arbitrariness.

[2] Since elasticity (ε = 1) implies that the degrees of freedom of rotation are not excited, theexcitation of the degrees of freedom of rotation implies that there is no elasticity. This is natu-ral. In general, non-elasticity implies a violation not of the conservation of the energy but ofthe kinetic energy of translation, precisely because a part of this latter is invested in energy ofthe internal degrees of freedom of the system. In our case, the excitation of the degrees of free-dom of rotation would be an excitation of the internal degrees of freedom, which would makethe collision inelastic not because the total kinetic energy would not be conserved but becausethe kinetic energy of translation would not be conserved.

[3] X* can just be negative at most because, if it moves, D can do so only leftwards, and the laws ofdynamics make it impossible for it to remain at X0 = 0 after t = 1. On the other hand, it is anelementary algebraic exercise to check that in region X < 0, D could never simultaneouslycollide with more than one ring, and so we are justified in appealing, as we have done in thetext, to the result of the transmission of velocity obtained in Section 2. The circumstances inwhich physical objects (usually particles) disappear from space are not unknown in the litera-ture. Examples can be found in Xia (1992), Pérez Laraudogoitia (1997), Angel (2001), andothers. Obviously, disappearances are never obtained as solutions to the differential equationsof motion (they could not be obtained this way, as the equations always provide positions inspace as a function of time). It would be interesting to have a broader concept of ‘solution’that would enable us, in particular, to specify the circumstances in which disappearances (andappearances, if we assume temporal reversibility) take place. Unfortunately, none of this isavailable to us at present, and we have to make do with justifying specific ‘solutions’ of disap-pearance for specific cases. In all cases (at least in all those I know of), these justifications takethe form of reductio ad absurdum arguments. For example, for Xia (1992), the reductiomakes use of the continuity of the particle world lines involved. As we have seen, the reductioin the case analysed in the present paper makes use of the previously established character ofthe (potential) collisions between the rings Ai and the disk D. In Pérez Laraudogoitia (1997),the reductio is of the same type as in Xia (1992). This underlines the fact that the modelpresented in this paper is not a minor variation with respect to Pérez Laraudogoitia (1997),which the following observation makes explicitly clear: neither the paradox presented herenor the consequences of the solution that I propose can be manifested in the model intro-duced in the former paper. The nature of this essential difference can be seen to lie in thedifferent form of the reductios involved. With respect to Angel (2001), see Section 5. The factthat a solution of disappearance is justified by a reductio ad absurdum enables us to under-stand why, when accepting solutions of disappearance, we are not opening the door to allkinds of monsters. For example, with regard to the model in the present paper, disk D will notdisappear before t = 1, and none of the Ai rings will ever disappear. Any reductio posited tojustify such disappearances would have to be based on specially introduced ad hoc hypothesesand therefore would have no better justification than those hypotheses.

[4] It should be clear that I am assuming that continuity guarantees identity throughout time, notthat continuity may be used to determine identity throughout time. Consider the case of thecollision of two identical point particles. The continuity of all the world-line particles involvedhere guarantees that the particles are the same after the collision as before the collision butobviously does not allow us to determine which is which (in intuitive terms, do the particlesbounce off each other or do they pass through each other?).

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[5] Of course it is not required that all the particle world lines be defined at the same instants oftime, but if there were no instant of time at which all the particle world lines are defined, thenit would in principle be possible that the body world tube of a connected body C would not bepath-connected (for instance, one half of the particle world lines of C could be defined at timeinterval I1, and the other half at I2, without I1 and I2 having any adherent points in common).In any case, the requirement that all the particle world lines of a body world tube should bedefined for at least one same instant in time implies the introduction of a criterion of individ-uation for body world tubes (and therefore for material bodies), which will be crucial to thesolution of the paradox.

[6] DA1 is uniquely specified at each instant of time by the set union of the set of particle worldlines that uniquely specify disc D with the set of particle world lines that uniquely specify ringA1.

[7] It goes without saying that DA1 also interacts globally with A− and does not interact individu-ally with any of the Ai (i ≥ 2). This explains why the finite momentum transmitted from DA1to A− does not affect the kinematic state of any of the Ai (i ≥ 2), given the infinite mass of A−.

References

Angel, L. 2001. A physical model of Zeno’s dichotomy. British Journal for the Philosophy of Science 52:347–358.

Angel, L. 2005. Evens and odds in Newtonian collision mechanics. British Journal for the Philosophyof Science 56: 179–188.

Buskes, G., and A. van Rooij. 1997. Topological spaces: From distance to neighborhood. New York:Springer.

Clark, M. 2002. Paradoxes from A to Z. London: Routledge.Earman, J. 1986. A primer on determinism. Dordrecht: Reidel.Lanford, O. E. 1975. Time evolution of large classical systems. In Dynamical systems, theory and

applications, edited by J. Moser. New York: Springer.Pérez Laraudogoitia, J. 1997. Classical particle dynamics, indeterminism and a supertask. British

Journal for the Philosophy of Science 48: 49–54.Pérez Laraudogoitia, J. 2005. An interesting fallacy concerning dynamical supertasks. British Journal

for the Philosophy of Science 56: 321–34.Xia, Z. 1992. The existence of non-collision singularities in Newtonian systems. Annals of Mathemat-

ics 135: 411–68.

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