1. Contact Action and Intelligibility

It is useful to contrast skepticism about bodily contact with a textbook account of collision.

Though their main concern lies with collisions between sub-microscopic particles, Halliday and Resnick begin with a pedestrian account of collision. Predicates like ‘acts for a relatively short time’ and ‘changes rather abruptly’ have a clear empirical sense. Their application presupposes nothing about the fine structure of contact – whether it entails a union, as Sextus requires, or whether one body always lies outside another, as Kline and Matheson require.

Fortunately, ‘collision’ in the pedestrian sense suffices for conceptualizing a variety of phenomena, as Halliday and Resnick proceed to show. Their treatment of collision enables us to infer final velocities from masses and initial velocities, in accordance with the laws of conservation of momentum and energy; thereby, it avoids having to measure collision forces, which ‘vary with time in a complex way that we can measure only with difficulty’.Footnote ⁵ Their approach is applicable to phenomena as diverse as collisions between bowling and beach balls and collisions between neutrons and uranium atoms during nuclear fission.Footnote ⁶

Collision is a paradigm case of bodily interaction. Making something happen, as when someone pries a nail from a board with a crow bar or pushes a wagon, is another paradigm of action.Footnote ⁷ Making something happen often involves effort, which is characteristic of simple machines to which the term ‘mechanics’ was originally applied.Footnote ⁸ Gravitational attraction is a different paradigm of action, and likewise the action of a field upon a charged body. We can observe the action of one body upon another in the cases of collision or making something happen, whereas we don't observe gravitational attraction or the interaction of field and body. There we observe motion but not bodily interaction.

The Humean will object that all we ever observe are motions; instead we infer bodily interaction, i.e., causation, after observing a series of events constantly conjoined with another. According to Hume, ‘the effect is totally different from the cause’, as I may conceive that a hundred different events follow from that cause.Footnote ⁹ We have no hint of causal interaction the first time we witness heat and flame, or weight and solidity.Footnote ¹⁰ These examples do, perhaps, illustrate Hume's claim that a hundred different events may be conceived to follow from the cause. The claim is less obvious when collision or making something happen is at issue. Thus, Louch writes,

Here a single event – a causal interaction – is observed, the crumpling of the fender; likewise, when a person uses a bat to drive a ball or a crowbar to remove a nail. As Louch puts it, ‘We may be said to see, not merely what happens in such case, but why it happens’. Granted, there is a sense in which we observe some electromagnetic interactions between bodies when we play or experiment with magnets. We might even say we observe gravitational interaction when we see lead spheres approach one another in the experiment by means of which Cavendish measured gravitational force.Footnote ¹² But these cases differ from the fender and the baseball, which are neither initially puzzling, nor suggest that an invisible medium is at work. That is, in observing magnets or Cavendish's experiment, we don't see why the movement occurs.

The explanatory force of gravitational or electromagnetic interaction resides in a general law that correlates magnitudes. Thus, effects of gravitational interaction are mathematically derived from F_grav = Gm₁m₂/d², previously determined quantities, and perhaps other laws or simplifying assumptions. The derivation provides an explanation of the effect, but a different kind of explanation from citing an observed interaction. Newton was clearly aware of this difference when, unable to specify the mechanism behind gravitation, he defended his system as follows:

Newton's laws of motion and principle of gravitation have explanatory value because they unify a great variety of phenomena. Explanation via ‘general Laws of Nature’ is, as Newton points out, preferable to explanation via ‘occult specifick Qualities’.Footnote ¹⁴ Arguably, however, explanation via description of causal interaction is still preferable to explanation via general laws, in spite of the precision, the generality, and the predictive power of the latter. Otherwise Newton would have felt no pressure to point out the virtues of his style of explanation to his mechanist critics.

Of course, Newton's predecessors proposed and used general laws, some mistaken (Descartes') and others not (Huygens'). But they were put forward as descriptions rather than explanations. Mechanist philosophers sought explanation, but for them explanation resided in models based upon observable cases of causal interaction. Boyle's defense of the mechanical philosophy rests upon the (not implausible) assumption that, ‘the curiosity of an inquisitive person … seeks not so much to know what is the general agent that produces a phenomenon, as by what means, and after what manner it is produced’.Footnote ¹⁵ A demand for the manner and means by which a phenomenon is produced is not a demand for a general law. The explanation of a magic trick consists in revealing the unseen; it is the particulars of the setup, rather than a general law, that explains the trick. Likewise, Boyle illustrates mechanical explanation with the production of an eclipse by interposing a body between two others; it is a case for which we can observe (in a model) the causal interaction.Footnote ¹⁶

To appreciate further how seventeenth century philosophers could put stock in observed interaction, when it is so plainly nonsense for the Humean, we must look more closely at causation. Holland finds an important difference between paradigm cases of causation and the abstract idea of causation.

The fundamental cases of causation, then, involve relations between material objects – the sun melted the snow, etc. Curiously, though, the verb ‘cause’ does not indicate a relation between material objects; it requires an event as one of the relata – the sun caused the melting of the snow. As Holland puts it, the snow has ‘been melted away by a grammatical sleight of hand’.Footnote ¹⁸ Subsequently our taste for symmetry leads us to substitute another event, the shining of the sun, for the cause.Footnote ¹⁹ Once a causal relation is expressed in event-language, there is no longer a term which denotes an observable interaction. Observable interactions require verbs signifying causal relations between material objects – a bat strikes a baseball or A crumpled B's fender. When we abandon observable interactions we abandon the paradigms by means of which causality is intelligible in the first place. ‘Cause’ and ‘effect’ are, as Holland points out, primarily useful when we are looking for or ignorant of an explanation rather than giving one.

The ideal of intelligible physical explanation arises from paradigm cases of the sort with which Halliday and Resnick begin their treatment of collision. The skeptical argument purports to establish that these paradigm cases are an illusion, and the Humean analysis of causation conspires toward the same end. Lange argues that field theory can render contact action intelligible once again. In the next section we examine his case for this thesis and argue that fields are not up to the task.

2. Fields and Contact Action

According to Lange, a moment's reflection shows that causal influence remains mysterious unless cause and effect are spatially and temporally local to one another.

Plainly the ball could have nothing to do with the shattering if it were never anywhere near the window. To shatter the window, the ball must strike the window, and the ball can't strike the window if it is nowhere near it. In light of this example, the thesis that causes must be local to their effects seems uncontroversial. It becomes controversial in the light of apparent violations of locality, as when bodies separated by a finite distance affect each other gravitationally, electrically, magnetically, and – in light of the skeptical argument – even by collision.

Lange accepts an updated version of the skeptical argument (7ff.).Footnote ²¹ Consider two billiard balls. Presuming the balls have radius r with distance 2r between their centers upon collision, three cases are possible, each with undesirable consequences. (i) If each sphere consists of points at distance ≤r from its center, then upon collision the balls penetrate one another, contrary to our conception of solid objects. (ii) If each ball consists of points at distance <r, then at the moment of collision there is a gap – of measure 0 – between the balls. This entails that colliding billiard balls have influence at distant locations without having effects at all of the locations between them. (iii) The preceding consequences are avoidable if one ball consists of points at distance ≤r from its center and the other of points at distance <r. However, this case entails that some but not all billiard balls are capable of colliding with one another. Lange focuses his attention upon case (ii), presumably because physics offers a neat solution to the apparent violation of spatiotemporal locality but not the difficulties entailed by (i)Footnote ²² and (iii).

By immersing bodies in fields of force, Lange claims, it is possible to avoid violations of spatiotemporal locality. While fields enjoy distinct advantages over central forces, they can't dispel the mystery surrounding apparent violations of locality. Lange's ‘is located at the site of’ and ‘is local to’ occupy a different logical category from ‘strike’, ‘touch’, and ‘be in contact with’. Whereas ‘strike’, ‘touch’, and ‘is in contact with’ denote relations between material objects, ‘is local to’ and ‘is located at the site of’ denote relations between events, which are in a different logical category from material objects.

Lange follows common philosophical practiceFootnote ²⁴ in holding that events rather than material objects stand in causal relations. But his concept of locality also requires this analysis: For a body to feel a force at the same place and time as its accelerating in response cause and effect must be events. The locality of cause and effect is not just being ‘somewhere near’ or touching; rather locality demands that cause and effect occupy the very same place at the very same time, and events, but not material objects, can satisfy that criterion.

Occupying the same place at the same time is quite different from Aristotle's idea that the mover must be together with that which it movesFootnote ²⁵ or even Hume's idea that cause and effect must be contiguous in space and time.Footnote ²⁶ Hume does not suppose contiguity to consist in occupying the same time and the same place, for the second rule for judging causes and effects requires that ‘the cause must be prior to the effect’.Footnote ²⁷ Moreover, Hume is comfortable identifying causes and effects by means of material objects. After affirming that causality consists in particular objects being conjoined with one another, he observes that Adam ‘could not have inferred from the fluidity and transparency of water that it would suffocate him, or from the light and warmth of fire that it would consume him’.Footnote ²⁸ ‘Suffocate’ and ‘consume’ exemplify Holland's doing-to verbs; they are paradigms of causation, consisting of a relation between material objects that are spatially contiguous without occupying the same place. Thus, even if Hume ultimately identified events as causes and effects, he still wouldn't have required that causes and effects occupy the same places at the same times.

Of course, Lange has a reason to abandon traditional notions of locality, the skeptical argument: If the spatiotemporal location of causes and effects were determined by the material objects involved, then there would be spatial and temporal gaps between cause and effect. He avoids violations of locality by simply surrounding material objects with repulsive fields.

Since fields are physical objects that can occupy the same location as the bodies they affect, they can insure the spatiotemporal locality of events that stand in causal relations. To paraphrase Lange: The window's feeling the baseball's repulsive force is an event occurring at the same place and time as the window's shattering in response. Presumably the window's feeling the repulsive force results from the condition of the baseball's field at that time and location. With the introduction of fields, then, contact action – indeed, bodily interaction – at last becomes intelligible.

But what of the intelligibility of field theory itself? Are its concepts so transparent that they remove the mystery from bodily interaction? On the one hand, field theory's claim to intelligibility arises from the paradigms of action that underlie ordinary causal discourse; this is apparent from presentations of field theory. On the other hand, the existence of disanalogies between these paradigms and field-body interaction summon new mysteries. This is not to disparage the use of analogies in presenting physical theory. But the existence of disanalogies ought to give one pause about appealing to field theory in order to demystify collision phenomena when the theory itself relies upon analogies with collision phenomena. Let us make good on these claims.

Lange's presentation of field theory uses familiar language from physics texts. Since he wants to say that a field can exert a force on a body, he begins by saying what a force is: ‘A force is a vector: it has a magnitude (that is, an amount) and it pushes or pulls in a certain direction’.Footnote ³⁰ ‘Push’ and ‘pull’ are doing-to verbs. Paradigm cases include a person pushing a block of wood across a floor, a horse pulling a cart, and a person pulling a spring connected to a body, all of which involve bodily contact. It is useful, naturally, to apply ‘push’ and ‘pull’ by analogy to cases in which a force is apparently exerted but without bodily contact; it renders the case less puzzling by grouping it with familiar cases of causation. Thus, when we read that ‘a field surrounding a given particle then causes an immense repulsive force on any particle that comes very near’, we have an analogy by means of which we can picture the action. But we should not suppose that analogies can render such a theory intelligible rather than merely serviceable, since there are crucial disanalogies too. For example, paradigm cases of pushing or pulling involve observable contact between the object exerting the force and the object feeling the force. But the cases that lead us to invoke these analogies in the first place – gravitational and electromagnetic interaction – involve body forces, which act throughout the volume of a body rather than at its surface. Neither pushing nor pulling throughout the volume of a body – let alone at a distance – are among the experiences which enable us to catch on to ‘push’ and ‘pull’ in the first place. Thus, body forces lack the intelligibility of contact forces.

Lange uses analogies with familiar cases of contact action once he turns explicitly to fields. In order to clarify the manner in which a charge affects its surrounding field he writes,

There are at least three analogies here to help us to understand the operation of a field: a trace left by distant charges, a pebble causing concentric ripples, and the news (i.e., an event) traveling. Here too, the analogies enable us to conceptualize the action involved only to the extent that one forgets the disanalogies.

A footprint in the snow is a trace left by a passing animal, and cave paintings are traces of ancient humans. The news of a charge's magnitude, location, velocity, and acceleration at an earlier moment is, let's say, a trace of a charge. We can observe a foot making a footprint and a painter making a painting. We can't observe a charge leaving a trace in the surrounding field, but more importantly we have no idea of the manner in which the charge leaves its trace. It can't be an instance of a charge exerting a force on the surrounding field since, as Lange will point out in a subsequent chapter, only bodies can feel forces.Footnote ³²

In that chapter Lange also invokes ripples: ‘[A] field's strength and direction here now is given by events (the ripples) that earlier began spreading out from bodies as a result of their charges then’.Footnote ³³ One can observe a dropped pebble producing ripples on the surface of a pond; it is a paradigm case of causation, and it involves contact between material objects. It begins with the pebble displacing water as it comes in contact with the surface of the pond. Since a field is an immaterial medium, the pond analogy is misleading; there is no material that, by its displacement, causes a ripple. The spreading out of ripples in a pond is a complicated issue because it involves action within a continuous medium. But prima facie the movement of the ripple results from matter exerting force on matter that is contiguous with it. Here too, there is a significant disanalogy between the news moving through the field and the ripple moving across the pond; for there is no mechanism for transmission. Even worse, while a field may be local to a body in Lange's sense, this will not be true of adjoining regions of a field through which the news is traveling. There will be no pair of overlapping events as Lange requires for cause and effect; the transmission of a causal signal, then, will not be a causal process. This is not to deny that the condition changes in a manner that can be represented by a wave, but only to say that the change itself remains mysterious.

Finally there is the news analogy, which appears earlier in Lange's discussion of Coulomb fields.Footnote ³⁴ For Lange, the traveling of the news is a useful stand-in for an event's spreading out from a charge; it is useful because it refers to a familiar phenomenon, unlike spreading out of an event.Footnote ³⁵ To describe fields in terms of the news traveling and their action in terms of the news arriving is to employ a second order analogy; for the news travels only by analogy with some material object moving from one place to another. By likening the field's action to the arrival of an object, e.g., the arrival of a wave at the shore, Lange again invokes a causal paradigm that implies bodily contact. But plainly the news (i.e., the field's condition at a specific location and time) is not a material object, and unlike a material object it changes systematically as it moves away from its source. Hence, it is misleading to construe the news – or the event – as the same individual occupying different locations, similar to a wave whose arrival might knock someone down. Moreover, unlike a wave at the shore, a field's condition cannot affect a body until it has interpenetrated the body; for the force it exerts is a body force.

The analogy with one object exerting a force on another is sufficiently pervasive in presenting the field concept that it's worth looking at another, more subtle illustration. The following is a ‘typical excerpt’ that Lange quotes from one of his college physics texts.

Lange's interest in this passage is different from ours. He criticizes Tipler's attempt to establish the reality of fields for presuming – without argument - that distance action is a problem.Footnote ³⁷ And perhaps Lange does not think of the assertion that a field exerts a force on a charge as employing an analogy. After all, it is a standard and helpful way to think about fields. But there is an analogy here. Compare a field exerting a force on a charge with Halliday's and Resnick's ‘the bat exerts a large force on the ball’, a paradigm case of one thing exerting a force on another. Cases like a bat exerting a force on a ball are observable interactions; we can see why the ball moves away from the bat after the collision. They are the cases that give sense to ‘exert a force’ and similar expressions, like ‘produced by’, when they are applied to fields.

It is also worth comparing a field exerting a force with a similar claim in a distance action setting. In explaining a Coulomb field, Lange writes

Here too a familiar expression enables us to conceptualize an otherwise puzzling phenomenon. The Coulomb field is nonetheless mysterious because the forces are exerted from a distance, unlike the case of the bat and the ball. However, the action of a field upon a charge is doubly mysterious: The force is exerted not only by an immaterial object, but also by an object capable of being in the same place at the same time as the charge. Not only is there no observable interaction between the parties, but also one of the parties acts only after penetrating the other. There is simply no familiar precedent upon which we can ground action of this sort.

Lange's text begins with a joke from which he draws a metaphysical moral.

Lange comments: ‘Of course, the joke is that Ivan doesn't succeed in making the radio less mysterious’, and he draws the moral that the original mystery remains because ‘without the giant dog … there is no connection between the sender and receiver’.Footnote ⁴⁰ Lange is right about the joke, but wrong about the moral. When Ivan removes the giant dog, he removes not only an object that occupies the gap between sender and receiver, but also an object to stand in a paradigm causal relation, viz., being stepped upon. Unless sender and receiver connect by a medium capable of standing in some intelligible causal relation, Piotyr's puzzlement will remain. If Ivan had continued, ‘… and in the dog's place insert an immaterial physical object capable of occupying the same location as a material object’, Piotyr would still want to know how (i.e., the means by which) the transmitter affects this medium and how different portions of the medium affect one another.

More importantly, if one takes seriously the skeptical argument, even with the giant dog there is a gap between shoe and tail, and so a gap between sender and receiver. By those lights, there is still a mystery; Piotyr, a peasant, simply lacked the sophistication to understand that Ivan's original analogy entails the same difficulty as his explanation. In the final section I shall defend contact action from the skeptical argument and show that it deserves its role as a paradigm causal interaction.

3. The Intelligibility of Contact Action

The impossibility of bodily contact is supposed to follow from the impossibility of either of the ways in which it might occur, viz., by the overlap of two bodies or by the adjacency of two bodies. On the contrary, I believe we can make sense of adjacency. Lange's formulation of the skeptical argument (as well as Kline's and Matheson's) supposes that bodies occupy positions in continuous space. This supposition is supposed to rule out adjacency: Points in a continuum are dense; between any two points, there is a third. Therefore, between points of distinct bodies there is always an unoccupied point, and that entails that the bodies are not adjacent to one another.

The skeptical argument is just another instance of deriving a conundrum from a mathematical continuum. It's similar to Zeno's extension paradox, that a finite extension composed of dimensionless points has length 0. Philosophers have attempted to refute this paradox using modern mathematical tools, but these simply avoid the paradox by ad hoc stipulations.Footnote ⁴¹ The skeptical argument is also similar to the paradox of stopping and starting, that there is no consistent and non-arbitrary way to determine whether, say, a train that leaves the station at noon is at rest or moving at the instant of noon.Footnote ⁴² Conundrums like these arise from representing continuous magnitude by means of a mathematically dense structure. This representational technique is invaluable to classical mathematics and physics, which focus upon continuous magnitude. However, given that the technique generates paradoxes, it seems reasonable to look for an alternative conception of continuous phenomena.

Körner offers a clear, empirical, but non-mathematical analysis of continuity (1962).Footnote ⁴³ His analysis requires a series of inexact predicates, P₁, …, P_n – e.g., a series of colors, a series of lengths, or a series of temperatures. Predicates are inexact insofar as they admit borderline cases or neutral candidates. A neutral candidate for a predicate is an object for which it would be linguistically correct to assert the predicate and linguistically correct to deny the predicate, though not for the same person to both assert and deny the predicate at the same time. ‘Green’ is an inexact predicate, because ‘green’ can be correctly applied or correctly withheld from turquoise objects. Inexactness is characteristic of empirical predicates.

To constitute a continuous series, predicates must be connected in a particular way: All and only adjoining predicates share neutral candidates, but no positive (i.e., uncontroversial) candidates. ‘Green’ and ‘blue’ are connected in this sense, and their shared neutral candidates constitute what Körner calls their connector class, C(green, blue).Footnote ⁴⁴ In order that P₁, …, P_n, constitutes an empirically continuous series, neither the individual P_i's nor their connector classes can be, in light of one's discriminatory apparatus, further divisible into connected subclasses. Thus, 1mm, 2mm, …, 1000mm would constitute an empirically continuous series presuming that neither my eyes nor my instruments were able to make finer discriminations.Footnote ⁴⁵

A set of inexact predicates that are connected and not divisible into further inexact predicates satisfies all of Dedekind's postulates for a mathematically continuous series except the postulate of mathematical density.Footnote ⁴⁶ In particular, such a set satisfies the completeness postulate:

Mathematical density, of course, is not possible for a finite series. Instead of being mathematically dense, an empirically continuous series is empirically dense: there is no empirically distinguishable class, other than a connector class, between P_i and P_i+1.Footnote ⁴⁷ The series is dense in the sense that there is no room for further empirical distinction. The neutral candidates which adjoining predicates share express the continuity of the series.

The preceding analysis avoids both Zeno's extension paradox and the start/stop paradox. It avoids the former because an empirically continuous magnitude is not infinitely divisible, and so its parts are finite in number and magnitude; it avoids the latter because the transition from rest to motion involves a connector class consisting of intervals for which it will be equally correct to say that the train is moving and the train is not moving. Körner's analysis also enables us to understand empirical contact. Contact in the mathematical sense occurs when objects share a point, a line or a surface; this is not an option for material objects. Rather, material objects are in contact when, as Aristotle put it, their extremities are together, i.e., when there is nothing between them.Footnote ⁴⁸ Körner's connector class gives sense to extremities being together without overlapping. Consider empirically the distances from some reference point at which two billiard balls collide. We assign a distance to the right hand extremity of the left ball and a distance to the left hand extremity of the right ball. As material objects, the balls cannot share their extremities, but the distances of those extremities from a reference point can share the same connector class. Both measurements will be neutral candidates for a pair of adjoining distance predicates. Measuring as accurately as we can, the extremities of the balls will, so to speak, fall on the border between two possible assignments of distance. There is nothing between the balls in the sense that we are empirically incapable of dividing further the connector class. Adjacency of material objects consists, then, in their distances from some reference point belonging to the same connector class. This is what it means – empirically – to say there is no gap between the balls.

If we choose to represent the collision by a continuous mathematical space, then, among other things, we use the same real number in assigning a distance from a reference point to the extremities of the balls. But doing so does not entail that the balls occupy the same location; to draw that inference is to conflate a mathematical representation and an empirical phenomenon. We commit the same mistake if we insist that we cannot understand collision without invoking fields, for the skeptical argument fails unless we conflate an empirical phenomenon with its mathematical representation. That is, it fails unless we suppose that a billiard ball is really a mathematically continuous collection of point particles, at a given distance from some central point, capable of moving in a mathematically continuous space.

To add weight to our criticism, it's useful to consider another discussion of the topic. Smith offers a searching criticism of the skeptical argument.Footnote ⁴⁹ He demonstrates that the concepts of impenetrability, contact, and body that underlie the skeptical argument are simply ‘not relevant to the physics of continuous media’.Footnote ⁵⁰ Moreover, the actual contact of distinct bodies is consistent with continuum mechanics, the relevant physics.

Smith does not deny the viability of the field theory to which Lange appeals; he also grants that the body forces in field theory are ‘in many ways … closer to the sorts of forces that are appealed to in, say, a quantum mechanical modeling of particles’.Footnote ⁵² But, he claims, physicists would not use quantum mechanics to model billiard ball collisions.

I won't attempt to duplicate Smith's careful explication of the continuum mechanical concepts of contact, impenetrability and body. However, it's important to have some feel for the representation of contact action within continuum mechanics. Smith describes one response to the skeptical argument as a bite-the-bullet approach, viz., allowing the balls to overlap at a point without interpenetrating.Footnote ⁵³ Continuum mechanics opts, in a sense, for biting the bullet, but since the move is independently motivated – rather than introduced ad hoc to avoid the skeptical argument – ‘no bullets have to be bitten’.Footnote ⁵⁴ The independent motivation rests on the observation that ‘contact actions are quite routine and a great deal is known about them and the mathematics they involve’.Footnote ⁵⁵ The continuum mechanical notion of contact arises from this knowledge of physics and applied mathematics: ‘[T]wo bodies are in contact if they have a non-empty contact set’, where the contact set is ‘given by the intersection of the reduced boundaries of the two bodies’.Footnote ⁵⁶ According to Smith, this concept of contact is presupposed by the Gauss-Green theorem, which is crucial:

Smith notes further that the area measure of the reduced boundary is not generally the same as the area measure of the topological boundary, and that only the reduced boundary can appear in the surface integral that represents contact forces in the Gauss-Green theorem.Footnote ⁵⁸ In short, the actual physics of contact action depends upon an applied mathematical technique in which the boundaries of bodies – defined in that theory as boundaries of open sets – intersect.Footnote ⁵⁹

The skeptical argument treats billiard balls as continuous sets of points and their impenetrability as the impossibility of points from different bodies occupying the same location. Owing to the density of mathematical continua, this treatment entails a gap between distinct bodies, and so a violation of locality. Delving into continuum mechanics, Smith finds alternative mathematical representations of contact, bodies, and impenetrability that do not entail a gap. He reminds us that the choice of a particular mathematical representation is a pragmatic one that depends, among other things, upon satisfying the conditions of theorems that enable us to organize and predict phenomena. In other words, a particular mathematical representation reflects a solution to a set of problems faced by applied mathematicians, a solution constructed from the tools of their craft.

Smith complains that Lange's representations of bodies and impenetrability are not grounded in a contemporary scientific effort to understand contact action:

Thus, Lange's appeal to fields solves a pseudo-problem, a problem that arises not from the phenomena, but from a particular, unnecessary choice of a representation for them.

According to Smith, continuum mechanics not only performs better than field theory but it even outperforms quantum mechanics if we are concerned to ‘take into account and predict the largest range of the features of actual billiard balls’.Footnote ⁶¹ While he acknowledges that one would have to turn to quantum mechanics to know the actual relations holding between colliding billiard balls, he argues that the salient features of a collision are not likely to be revealed by quantum mechanics, owing to the ‘enormous n-body problem’ involved.Footnote ⁶² Thus, continuum mechanics provides the best available means for modeling collision phenomena. But Smith goes beyond this result in singing the praises of continuum mechanics.

These remarks go beyond the stated goal of his essay – resolving an ‘alleged tension between contact and impenetrability’Footnote ⁶⁴ , and they put Smith in an awkward position. Since he grants that billiard balls are not continuous in the way continuum mechanics supposes, he has to formulate his continuum mechanical realism counterfactually:

This is a rather weak claim. Even if quantum mechanics gave better predictions than continuum mechanics for collision phenomena, the counterfactual claim could still be true. But in that case the continuum mechanical model would be of little interest to us. Our interest in continuum mechanics stems from its superior empirical adequacy, not from its capacity to illuminate a non-actual world.

More importantly: It is one thing to say that billiard ball collisions are best modeled within continuum mechanics, but another thing entirely to say that billiard balls, if they were continuous, would actually be open sets or that their being in contact would consist in the non-empty intersection of their reduced boundaries. If the set-theoretic apparatus underlying continuum mechanics is taken to describe billiard balls rather than merely to model them, then we have to attribute to billiard balls some rather unusual properties. For instance, given the Banach-Tarski theorem, we would be committed to saying that one of them could be separated into a finite number of pieces and reassembled in such a way as to create two solid balls, each identical in shape and volume to the original.Footnote ⁶⁶ Not surprisingly, the theorem is known also as the Banach-Tarski paradox; it is paradoxical if we imagine that it applies to bodies we experience, rather than their idealizations. The Banach-Tarski paradox – not the Banach-Tarski theorem - is another paradox that arises from conflating the empirically continuous world with its representation by means of mathematical continua. It involves the same conflation that tempts us to take seriously the skeptical argument. By concluding that continuum mechanics tells us what contact action amounts to, Smith is guilty of the same conflation. Of course, physicists no more worry about the Banach-Tarski paradox than they worry about Zeno's paradox of extension; they employ the mathematics of continua only insofar as it serves their ends.