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Newton and the Billiard Ball (DRAFT)

Mark WilsonUniversity of Pittsburgh

Buttercup: Things are seldom what they seem,Skim milk masquerades as cream;Highlows pass as patent leathers;Jackdaws strut in peacock's feathers.

Captain: Incomprehensible as her utterances are, I nevertheless feel that they are dictated by a sincere regard for me.

W.S. Gilbert, H.M.S. Pinafore

(i)

In the development of science, “simple things” are seldom what they seem. Often fruitful science takes its historical origins within attempts to deal with the swampiest and most intractable forms of descriptive problem. One of the strangest aspects of the Newtonian philosophical heritage stems from the fact that the Principia is often credited with descriptive achievements that it did not, in fact, achieve. In particular, popular writings commonly characterize “Newtonian physics” as “billiard ball mechanics” or as supplying the physical principles relevant to a “clockwork universe.” Neither characterization is accurate and hasty presumptions otherwise encourage a number of unfortunate misapprehensions within philosophical circles that persist with us even today. In this note I shall concentrate upon the pool table side of the ledger.

Let us begin with Robert Boyle’s well-known extolment of “the excellency of the mechanical hypothesis:

The ...thing which recommends the corpuscular principles is their extensiveness. The genuine and necessary effect of the strong motion of one part of matter against another is either to drive it on, in its entire bulk, or to break and divide it into particles of a determinate motion, figure, size, posture, rest, order or texture.1

However, familiarity does not breed comprehension. Every modern material scientist knows that the physical issues underlying cohesion and fracture are

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extremely tricky and generally require descriptive mathematics of considerable complexity. In these regards, a compatriot such as John Locke voiced a more prudent skepticism:

The little Bodies that compose that Fluid, we call Water, are so extremely small, that I have never heard of any one, who by a Microscope, (and yet I have heard of some, that have magnified to 10000; nay, to much above 100,000 times) pretended to perceive their distinct Bulk, Figure, or Motion: And the Particles of Water are also so perfectly loose one from another, that the least force sensibly separates them. Nay, if we consider their perpetual motion, we must allow them to have no cohesion one with another; and yet but a sharp cold come, and they unite, they consolidate, these little Atoms cohere, and are not, without great force, separable. He that could find the Bonds, that tie these heaps of loose l i t t le Bodies together so firmly; he that could make known the Cement that makes them stick so fast one to another, would discover a great, and yet unknown Secret: And yet when that was done, would he be far enough from making the extension of Body (which is the cohesion of its solid parts) intelligible, till he could shew wherein consisted the union, or consolidation of the parts of those Bonds, or of that Cement, or of the least Particle of Matter that exists. Whereby it appears that this primary and supposed obvious Quality of Body, will be found, when examined, to be as incomprehensible, as anything belonging to our Minds, and a solid extended Substance, as hard to be conceived, as a thinking immaterial one, whatever difficulties some would raise against it.2

However, billiard balls only rarely shatter; indeed, they show a remarkable ability to regain their accustomed shapes swiftly after a considerable degree of buffeting. To a considerable degree of descriptive success, their two body interactions satisfy the simple rules of rebound laid down by Wren and Huygens,3which, in a Physics 101 text of today, rely upon two basic conservation principles: the conservation of linear momentum and the conservation of energy (under the assumption that its manifestation is entirely kinetic). Appealing to these principles alone, two balls of equal mass can be predicted to exchange velocities upon collision, maintaining the vectorial sum of their respective individual momentums as they do so. Such expectations are sometimes dubbed the “laws of elastic impact” in philosophical commentary and Descartes is often patronized for getting these “laws” wrong in his Principles of Philosophy.

This popular evaluation strikes me as very misleading, simply because the notion that that anything deserves the title of a “law of impact” is problematic. In

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reality, colliding balls internally distort in a very complicated way over a short temporal interval, which is ignored within the Wren-Huygens account. Newton himself was aware of many of these behavioral complexities and did not wholeheartedly subscribe to the simple picture just articulated:

Only those bodies which are absolutely hard are exactly reflected according to these rules. Now the bodies here amongst us (being an aggregate of smaller bodies) have a relenting softness and springiness, which makes their contact be for some time and in more points than one. And the touching surfaces during the time of contact do slide one upon another more or less or not at all according to their roughness. And few or none of these bodies have a springiness so strong as to force them one from another with the same vigor that they came together.4

(Later in the essay we’ll look at how Newton approached the bodies that “are not here among us,” for these are the “hard atoms” that he posits in the Opticks and elsewhere).

In truth, the standard treatment of billiard ball impact engages in a prudent form of descriptive evasion (= cheating), for incoming and outgoing events are connected together in a black box manner, ignoring the evanescent complications occurring within the actual interval of contact. In modern jargon, treatments of this character are said to sew together “far field” events using asymptotically matching, often employing simple corrective factors as stand-ins for the complications of the true physical interactions. Newton’s own innovation in this vein was to introduce a “coefficient of restitution” factor that attempts to codify the percentage of incoming kinetic energy will be maintained in the outgoing motion. Such descriptive ploys are remarkably effective in the case of billiard balls. Combined with a similar empirical factor due to Euler (that estimates how much the original energy will convert to angular spin), most run-of-the-mill pool table events can be adequately handled (a few exceptions will be discussed below).5

Asymptotic workarounds of this type are common in science, even today. Here’s an example that Descartes confronted. He believed that matter was essentially granular, with no gaps appearing between adjacent particles. If so, how can a fluid composed of such particles navigate through an abrupt constriction

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within a pipe, as illustrated? His rather surprising answer maintains that in the region of the constriction, water particles must rapidly fracture and rejoin in a manner so complicated that the finite human mind cannot follow their details. For Descartes, such “regions of indefiniteness” are fairly common in Nature. Despite these obstacles we can still connect together the upstream and downstream branches of the flow through the assumption that two vital qualities will be conserved throughout, viz., (i) total mass and (ii) Cartesian “quantity of motion,” which is roughly akin to a scalar version of our vectorial linear momentum (∑mv). This assumption allows Descartes to auger the altered velocity within the downstream flow.

However, typical Physics 101 primers that treat billiard ball collisions in a “two conservation laws” manner rarely acknowledge that they are engaging in asymptotics to a complex scattering problem. Michael Spivak, in his refreshing Physics for Mathematicians, captures the conjuring trick nicely:

Elementary physics textbooks need to provide problems that have answers, of course, so, in the manner of a host nonchalantly introducing a celebrity at a party, they will often unobtrusively insert a new definition: a collision is called “completely elastic”, if we also have conservation of kinetic energy… Consorting with this new definition we have a contrasting one: a collision is “completely inelastic” if … the two bodies stick together.…. But having a definition of the coefficient of restitution hardly tells us anything... We would like to understand why the modern definition of a completely elastic collision amounts to an idealization of the concept that lurks in the back of our minds when we think of an “elastic” body as one that pops back into shape after being squashed in a collision.6

In fact, a rational explanation of the rebound needs to be far more complicated and must consider the pressure waves that are created by the initial impact, an event that broadens the contact region from a singular point to a wider common interface (mathematically this becomes a so-called “moving boundary value problem”). These initial events send stress waves through the interiors of each ball until they are reflected off the back walls and refocused upon the impact

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region adequately enough to push the two balls away from one another. The large degree of pressure wave refocusing witnessed in a conventional billiard ball relies heavily upon its spherical shape, the fact that the waves initially spread from a very compact region and the fact that billiard ball “ivory” can swiftly transmit significant pressure waves without significant conversion to heat or suffering permanent plastic distortion. Nonetheless, a small measure of frictional resiliency is required at the interfaces to swiftly pull each ball back to their natural undistorted shapes without permanent wobbling. This resumption of natural rest state will prove a significant factor in the discussion to follow.

The mathematical notions required to formulate these interior wave expectations coherently were not available during the seventeenth century and there was no hope of simulating their complexities in numerical terms until fast enough computers became available recently. As we shall see, these obstacles didn’t prevent a Leibniz from recognizing the basic contours of the mathematical task required.

In any event, a closer inspection of slightly more complicated circumstances shows that the apparent simplicity of the canonical “two conservation law” explanation of billiard ball rebound represents a derivational illusion fueled by what we might call special assumption channeling. In particular, consider the familiar pendulum toy often called Newton’s cradle (illustrated). At first blush, the gizmo’s behaviors appear to neatly underwrite the usual “two conservation law” predictions. Thus if we begin with two balls pulled off to the left, then two balls will depart on the right, leaving the central balls motionless, just as our two equations would have predicted. But this prediction works only if there is a small gap between the balls that allows each pairwise collision to run through its compressive cycles with extreme swiftness before the struck ball collides with its neighbor to the left. When these conditions are not met, the ensemble will often wiggle and divide in unexpected manners that can only be explained by computing the locations where the pressure waves passing through several conjoined balls will refocus in a manner that can drive the balls apart.7

What do I mean by “special assumption channeling”? In a generic case of impactive interaction—two blocks of wood banging together, say--, a large number of descriptive variables are required, demanding a comparable number of governing equations to account for their interactions. By assigning our billiard

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balls wholly rigid shapes sliding upon a rigid plane, tolerating no internal potential energy storage and reducing their interface to a single point of normal contact, our two conservation laws allow us to predict the outgoing motions through Wren-Huygens asymptotics as above. Although the same two equations remain valid for our pieces of wood, we will require many further equations to resolve their behaviors adequately, due to the fact that we can’t reduce the pertinent descriptive variables so drastically as we can in the special circumstances of two billiard balls that clash head on. Allied observations explain why a Newton’s cradle will “act properly” only if small gaps secretly separate the component balls.

Later we will observe that the “explanatory illusions” engendered by tacit channeling assumptions of this character have played a significant role in generating significant forms of philosophical confusion.

(ii)

In light of this complexity, why did the scientists of the early modern era focus so intently much upon impactive events like billiard collisions?

Like Boyle, they presumed that most physical processes operate through contact action and hence “simple” collisions between bodies that contact one another locally (e.g., at the solitary junction point between two rigid spheroids) should be investigated as emblematic of nature operating in its most elementary and foundational manner, in terms of which its more complicated entanglements could be then decomposed. The methodological percept adopted is “Find the fundamental solutions first and build up from there.”

In this spirit, these scientists naturally presumed that when billiard ball A directly affects ball B through head-on contact, some motive capacity must be transferred across the bodies that should supply the proper measure of the amount of “force” transferred. Thus the notion of a singularly acting impulse becomes central, where an “impulse” represents a primitive activity that transpires both locally and instantaneously. Even Newton adopted these priorities, for he conceptually decomposed a smoothly acting force such as gravity into a rapid sequence of impactive hammer blows.

Mathematically, however, such “fundamental solution” reductions have often proved problematic, at least from the point of view of tractable mathematical description. This is because impactive events inherently represent singular

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episodes where important rates of change become ill-defined; smoothly and continuously acting forces suit the tools of the calculus far better. A familiar historical illustration of this dilemma can be found in the Euler-d’Alembert dispute over the “fundamental solution” for a loaded string. Euler maintained that more complicated loading arrangements could be compounded from the superposition of simple triangular string distortions as illustrated. But d’Alembert reasonably objected, how can such a “solution” coherently satisfy the pertinent governing equation for its internal physics requires that the string balance the gravitation pull by curving? In the calculus, “curving” is captured by the second derivative ∂2y/∂x2 but no bending of that sort can be found anywhere within Euler’s so-called “fundamental solution.”

Leibniz’ claim that “nature doesn’t make jumps” represents a rejection of views of nature that tolerate non-smooth behaviors like Wren-Huygens impacts. In response, defenders of “fundamental solutions” often appealed to somewhat dubious constructions they dubbed “impactive integrals” (x)dt which allegedly supply the force responsible for an abrupt shift in particle velocity. Our modern assessment of such constructions: singular “fundamental solution” activities can be tolerated but only by piggybacking on smoothed out arrangements.8

It is worth observing that methodological observations later interpreted by the logical positivists and their modern descendants as objections to the unwanted “metaphysical nature” of force9 originally represented objections to bottom-up, impactive decompositions of the sort just discussed. Thus d’Alembert in the Encyclopédie:

It would be desirable if the mechanicians would finally recognize that we know nothing about movement except the movement itself. . . and that the metaphysical causes of this motion are unknown to us, that what we call causes, even of the firm kind [those of impulsion] are only improperly called causes; they are effects from which other effects result. [In a collision one body moves the other] and as a consequence the colliding body is considered as the cause of the movement of the body struck. But this way of speaking is improper. The metaphysical cause, the true cause is unknown.10

d’Alembert here refers to the fact that if we know various higher scale limitations upon a system’s overall movements—e.g., that it always moves as a completely rigid rod or ball--, we can often exploit such data in a top-down manner to calculate the local stresses within the system (knowledge of this sort is

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commonly called constraint information). In the case of certain structures such as so-called “determinate bridges,” we only have to know the loads on the assembly and the sizing of its various parts to calculate all localized stresses. If parts of the bridge are linked together with supplementary strengthening linkages, we build an over-constrained structure that requires a greater degree of interior modeling to remove the indeterminacy in stress that otherwise appears.

Although such matters cannot be adequately surveyed here, the “forces” we calculate through these distinct recipes—in “bottom up” fashion working from the smallest parts of a system and in a “top down” manner operating from prior knowledge of constraints—behave in subtly different ways that operate in conceptual tension with one another.11 These oppositions lie squarely in the background of the issues we shall discuss, because the top-down stratagems offer great reductions in computational tractability, at the cost of suppressing lower scale complexity such as our pressure waves.

As an aside, it is worth remarking that Descartes experiences problems in formulating his “laws of motion” because he attempts to register smooth, continuously acting physical interactions within the “simpler” format of singular impacts. His view of nature is essentially machine-like in conception and in such contexts one machine part A (a ball, say) will slide along a curved slot in another member B with a constant velocity relative to the track unless friction is present or the ball damages the track in some way (this natural behavioral assumption is often called “generalized inertia”). But Descartes conceptualizes this process as one that involves continuous series of zig-zag battles in which the constraining track B manages to redirects the determination of A in an impactive manner without affecting its “quantity” of motion. In my opinion, most of the implausibilities in his official “laws of motion” trace to this misguided methodological bias. Undoubtedly, his earlier successes in explaining the abrupt reflection of light upon encountering a new medium contributed to this conviction.

Newton himself proved relatively immune to these faulty expectations in the case of real life billiard balls because he correctly presumes that the springy character of such objects requires some variety of complicated supporting mechanism, involving both attractive and repulsive forces, as well as the cushioning effects of some surrounding medium to tamp down elastic jiggling and to restore the balls to their natural rest state. He freely admits that he doesn’t have

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a good model of how this all works. His friend Henry Pemberton comments on Newton’s behalf:

From numerous observations of this kind he makes no doubt that the smallest parts of matter, when near contact, act strongly on each other, sometimes being mutually attracted, at other times repelled. The attractive power is more manifest than the other, for all parts of all bodies adhere by this principle. And the name of attraction, which our author has given to it, has been very freely made use of by many writers, and as much objected to by others. He has often complained to me of having been misunderstood in this matter. What he says upon this head was not intended by him as a philosophical explanation of any appearances, but only to point out a power in nature not hitherto distinctly observed, the cause of which, and the manner of its acting, he thought was worthy of a diligent enquiry. To acquiesce in the explanation of any appearance by asserting it to be a general power of attraction, is not to improve our knowledge in philosophy, but rather to put a stop to our farther search.12

The central focus in this passage advances the positive suggestion that a significant part of the explanation of the spring witnessed in billiard ball rebound traces to action-at-a-distance type forces acting between atomic centers while they are not in contact. But these assumptions alone do not explain why the balls regain their natural rest states as swiftly as they do, which is one of the reasons why Newton seems to favor the stop-upon-true-contact hypothesis. And therein lies much of the tale we shall now unfold.

(iii)

Although Newton regards the recoil of real life billiard balls as a complex phenomenon whose operations can only be rudely approximated at the macroscopic level through appeal to asymptotics and a simplistic “coefficient of restitution,” he did believe that a different form of impact proves fundamental at the lowest corpuscular level. In particular, he maintained that atomic scale particles should prove both indivisible and shape preserving. When two of these atoms impactively collide, their summed linear momentums will remain vectorially conserved throughout the collision in exactly the manner that Huygens had diagnosed. However, Newton utterly

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rejects the additional presumption that their kinetic energy will be preserved as well. He instead augers that nature’s “fundamental particles” will stop cold in their tracks if they are ever allowed to come into true contact. In these pristine circumstances, no interval of asymptotically excised distortion arises but a wholly “inelastic” sticking together is the behavioral outcome. In an often cited speculation within the Opticks, he writes:

And thus Nature will be very conformable to herself and very simple, performing all the great Motions of the heavenly Bodies by the Attraction of Gravity which intercedes those Bodies, and almost all the small ones of their Particles by some other attractive and repelling Powers which intercede the Particles... By this Instance it appears that Motion may be got or lost. But by reason of the Tenacity of Fluids, and Attrition of their Parts, and the Weakness of Elasticity in Solids, Motion is much more apt to be lost than got, and is always upon the Decay. Impenetrability makes [hard bodies] only stop. If two equal Bodies meet directly in vacuo, they will by the Laws of Motion stop where they meet, and lose all their Motion, and remain in rest, unless they be elastic, and receive new Motion from their Spring.13

These circumstances recall the paradox at the center of the old song:

When an irresistible force such as youMeets an old immovable object like meYou can bet just as sure as you liveSomething's got to give (3).

Under Newton’s resolution, it’s the “irresistible” impactive force that must give way to the immobile object. Accordingly, his mechanics is more appropriately characterized as centered upon Nerf ball behaviors, than billiards per se.

Newton, of course, was aware that truly inelastic collisions without the assistance of adhesion represent even rarer occurrences within macroscopic nature than totally elastic reboundings, so why did he favor the former over the latter? Some of the answer traces to the fact that he believed that complex systems generally “receive new Motion from their Spring” through the assistance of the action-at-a-distance repulsive forces that generally keep his atoms from contacting one another. Insofar as I can make out, Newton further recognizes such mechanisms alone will not readily restore an ordinary, non-plastic solid to its natural rest configuration (i.e., as perfectly spherical billiard balls). To be sure, the longer range attractive interatomic forces that Newton regards as essential to the

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cohesion of matter plays an important role in underwriting this restoration of natural relaxed state, but some further mechanism is required to fully underwrite the “Attrition of Parts and Weakness of Elasticity” that allows billiard balls to stop jiggling after they have been excited. Newton’s exact opinions on the frictional mechanisms required are hard to make out, due to the fact that he also believes that interactions with a surrounding ether play some role in these proceedings as well.14 But he seems to believe that some of the requisite damping must trace to the rare occasions when his atoms actually come into direct contact with one another.

As has been amply documented,15 these speculations with respect to loss of motion generated considerable controversy in the century following, with various prizes being assigned to the “correct resolution” to our “something’s gotta give” dilemmas. Newton’s own allegiance to inelastic scattering supplied a direct rationale for his notorious presumption that divine agencies (e.g., angels giving the planets an occasional restorative boost) are required to keep the universe turning through its reliable courses (Newton himself regarded this energetic occasionalism as a theological boon).

In these atomist speculations, it prima facie appears as if Newton is merely engaging in loosey-goosey, armchair speculation about the unseen levels of nature in a manner that he rigorously eschews within the methodologically sterner Principia. Clifford Truesdell voices this evaluation in his usual pungent manner:

Whatever may have been the speculations Newton chose not to publish, the works he did release do not justify imputation to him of so narrow a view of nature [as his hard atom speculations suggest]. In the first place, it would have to rest largely on one sentence in the preface to the Principia and on the comments at the end of the Queries in the Opticks. The exaggerated weight often laid upon these few pages may reflect the fact that while a mathematician may need a day to get through a page of the Principia, then he has it beyond quibble, but anybody can read the Queries in ten minutes and go on arguing about them forever. The Queries are not science; they show Newton from a side he withheld usually and would have been better advised to have withheld without exception; at best, they are prophecies of science… Second, even if we must tarry over these fleshless speculations, we cannot justly read into them atomism in the narrow sense later given to the word. Newton’s fancied atoms were “solid, massy, hard, impenetrable, movable”. Of these five adjectives, three are not punctual. The repulsion experienced by two solid, hard, impenetrable grains, however small, when

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they strike each other, is not an action at a distance but a contact force. Contact forces, if crudely handled by Newton, are nevertheless nowhere excluded by him.16

Newton’s atomist speculations appear to be “armchair a priori” in exactly the worst manner that one encounters in Descartes, whom Newton often criticizes on this very methodological score.

In fact, Truesdell overlooks an arresting passage within the Principia itself that suggests that Newton’s hard atom speculations play a more central structural role within that work. Specifically, in the midst of Newton’s “third rule for reasoning in (natural) philosophy,” the hard atom speculations appear, unexpectedly mixed together with his justifications for approaching gravitation in his distinctive, “I do not frame hypotheses” manner:

Those qualities of bodies that cannot be intended and remitted [i.e., qualities that cannot be increased and diminished] and that belong to all bodies on which experiments can be made should be taken as qualities of all bodies universally.

For the qualities of bodies can be known only through experiments; and therefore qualities that square with experiments universally are to be regarded as universal qualities; and qualities that cannot be diminished cannot be taken away from bodies. Certainly idle fancies ought not to be fabricated recklessly against the evidence of experiments, nor should we depart from the analogy of nature, since nature is always simple and ever consonant with itself. The extension of bodies is known to us only through our senses, and yet there are bodies beyond the range of these senses; but because extension is found in all sensible bodies, it is ascribed to all bodies universally. We know by experience that some bodies are hard. Moreover, because the hardness of the whole arises from the hardness of its parts, we justly infer from this not only the hardness of the undivided particles of bodies that are accessible to our senses, but also of all other bodies. That all bodies are impenetrable we gather not by reason but by our senses. We find those bodies that we handle to be impenetrable, and hence we conclude that impenetrability is a property of all bodies universally.17

This passage certainly suggests that Newton wishes to assign the same methodological approbation to his speculations about hard atoms that he attributes to his prodigiously well-confirmed—and, philosophically, greatly admired--rules

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for gravitation. But why on earth should Newton claim that inelastic non-repelling at a microscopic level represents as securely grounded an aspect of “universal experience” as gravitation, given that most of us, if asked to make rash inductions from macroscopic collision experiences, would have instead favored the Wren/Huygens behaviors?

Here’s a conjecture as to why Newton proceeds otherwise. The conservation of total linear momentum for a complex but isolated system represents a very important conservation law for Newton, for it justifies his strong belief in what we now call the “Galilean relativity” of classical physics, a behavioral assumption already highlighted by Descartes and Huygens (the accompanying illustration that experiments unfold identically whether one stands on dry land or in a steadily moving boat comes from Huygens’ Du Motu Corporum ex Percussione). Allied strands of reasoning lie in the background of many of Newton’s central methodological tactics within the Principia, e.g, with respect to the importance of a body’s center of mass (most of his modeling efforts in celestial mechanics rely upon the fact that most astronomical entities will move very much like its center of mass if we only consider gross gravitational effects).

(iv)

From a modern point of view, the conservation of total linear momentum is of a fundamentally different nature than the conservation laws that Early Modern scientists generally sought. Thanks to Emmy Noether, we now recognize that conservation of total linear momentum follows from the fact that the behaviors of complete physical systems remain invariant under rigid translations within a Euclidean space. Conserved quantities of this class derive from the global behaviors of the target systems, rather than necessarily stemming from features that will seem evident on a localized level. But Early Modern thinkers typically presumed that such globally seated conservation principles must directly stem from simpler, lower scale behaviors in some amalgamative and bottom up fashion. Here are two illustrations:

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Example 1: Why do large systems preserve their masses? The traditional answer maintains that their component atoms preserve their individual masses through all interactions.

Example 2: Descartes’ non-vectorial conception of a system’s “quantity of motion” essentially represents a stored capacity for performing work, in the manner in which a rotating flywheel can store energy for later use. He believes that this ability is locally conserved: the “quantity of motion” can be drained from one such device only if the same work capacity is transferred to adjacent physical systems. His complex entangled vortices will preserve their individual allotments of “quantity of motion” except on the occasions when they rub against one another and exchange work capacity. So Descartes’ conservation law for his “quantity of motion” also proves additive from small systems upwards, in the general manner of traditional mass conservation.Following these bottom up patterns, if we can confidently assume that the

only forces acting between bodies are two body action-at-a-distance forces that act along the line connecting their centers in a manner that is not velocity sensitive, then we can indeed derive total momentum conservation in an additive, bottom up manner. As already indicated, Physics 101 primers supply similar arguments for momentum conservation to this day, following lines of thought similar to what Newton actually provides. Doing so requires the restrictions just mentioned--that forces that act along the line connecting their centers in a manner that is not velocity sensitive--, assumptions that are sometimes called “the strong reading of Newton’s Third Law.” These are the ingredients that Helmholtz later exploited to argue for the conservation of energy (the original notion of “potential energy” relies centrally upon these assumptions).

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However, as Truesdell correctly observes, Newton himself did not fully ascribe to these stringent assumptions. The doctrine that classical physics only needs to worry about action-at-a-distance forces is that of Boscovich, some of the later French atomists and modern physics writers unconcerned with the behaviors of extended bodies. Newton did not embrace such a pristine landscape and allowed his extended atoms to occasionally come into direct surface contact with one another, for the reasons sketched above. Accordingly, to reach total momentum conservation, Newton must introduce a second thesis into his Third Law restrictions to dictate how the forces arising within direct contact should behave, in the general form: at every point at which body A exerts a traction force fAB on body B across the tangent plane to the point of contact, body B will exert a reciprocal forces –fBA on body A across this same plane. Newton appeals to collision experiments to justify this contact postulate and simultaneously extends its application to action-at-a-distance forces through a somewhat hokey argument that we shall discuss later.

So if Newton can reasonably contend that all forces acting in Nature ultimately prove pairwise balanced at base in one of these two basic manners, he can establish total momentum conservation for all systems, no matter how complex, relying upon a cancelation argument that exploits resting the equal but opposite forces appearing between his paired off bodies (indeed, this is exactly how he proceeds). For these deductive purposes, it doesn’t really matter whether the contact actions between atoms operate in the Wren-Huygens “elastic” manner or in Newton’s alternative, stop-dead-in-their-tracks fashion. As we’ve just seen, he opts for the second alternative.18

But he clearly requires some form of pairing off doctrine for contact actions to reach these objectives, which explains why his peculiar “hard atoms” assumptions play an important structural role within the overall plan of the Principia, despite their comparative non-centrality elsewhere.

Many contemporary commentators on this subject either ignore or diminish the contact force aspects of Newton’s discussion and write as if the only forces with which he needed to contend are entirely of an action-at-a distance character, operating between bodies that are forever forbidden from coming into true contact with one another. Under this reading, elastic rebound on a fundamental level becomes entirely a matter of what physicists now call “elastic scattering” (i.e., interactions that conserve kinetic energy). Not only has our intuitive understanding of “elastic bodies” as systems that pop back into shape after being squashed in a collision (quoting Spivak again) vanished from view, but any

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allowance that such bodies can make contact with each other at all. So if central forces between point mass centers comprise the only interactive phenomena contemplated within “Newtonian mechanics,” the popular stereotype that the subject concerns itself with everyday “billiard ball behavior” becomes completely unjustified. Indeed, from this point of view, it is hard to see how Newton’s physics remotely ratifies any of Robert Boyle’s original “mechanist” expectations.

But this action-at-a-distance-forces-between-point-masses recasting of Newton is entirely anachronistic, largely influenced by the fact that elastic scattering remains the central behavior of interest to modern quantum theorists. Newton’s own justifications of his Third Law emphasize contact action behaviors more centrally than his action-at-a-distance applications. In point of fact, through the examples he suggests, Newton de facto extends the alleged reach of his Third Law beyond either of the two categories (contact action and action-at-a-distance) just surveyed. These extensions have induced great confusions in the subject, a point to which we will quickly return.

Before we do so, some general observations on the notion of “contact forces” are warranted. Insofar as I can determine, Newton’s original contact action principle was viewed as only applying to forces that appear along the outer surfaces of extended bodies, not to any tractions that might arise within their interiors. But Euler and Cauchy later extended this postulate to the traction forces that arise within the interior of an extended flexible body when it is stressed. More exactly, if we draw an arbitrary plane P through any interior point p of a larger flexible body, the material on each side of P will exert a local traction force (density) on the opposite side and that these two traction forces must balance one another in a fAB and –fBA manner (this posit is usually called “Euler’s cut principle”19). Euler originally presumed that these forces would always operate normally to the dividing plane P, indicating that the point p is subjected only to a compressive or dilative interior pressure. But Cauchy later allowed the fAB and –fBA to operate obliquely across P, shearing p locally as they do so. In doing so, Cauchy conceptualized our modern notion of stress, a tensor object that captures how all of these local traction forces relate to one another as we adjust Euler’s dividing plane P to different orientations around point p. When a flexible body is distorted (= strained, in contemporary vocabulary) from its natural rest state (if, like a billiard ball, it has one), the local stress will increase proportionately by some kind of constitutive

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relationship such as the generalized Hooke’s law: the stress components are linearly linked to the strain components by two material constants E and . In these terms, the “pressure waves” we discussed earlier with respect to billiard balls will consist in trackable pulses of heightened stress that progressively move across the interior of a flexible body.20 As already emphasized, a fully rational explanation of impactive rebound requires tracking how these waves pass through the insides of the affected balls.

With these modern extensions, we have moved far beyond Newton’s vaguer conception of “pressure,” which he seems to conceive of as some averaged representative of the surface contact forces arising within a granular medium such as a gravel or fine sand. In a modern presentation of continuum physics, “Newton’s Third Law” is interpreted as enforcing a fAB and –fBA discipline upon the traction vectors appearing within Euler’s cut principle. As it happens, Newton’s allied restrictions on direct action-at-distance forces are often ignored within these same contexts because (1) they usually prove unimportant in applications and (2) because they generate confusion with respect to a body’s inertial reactions.

(v)

But Emmy Noether has shown us that, amongst all conservation laws, “some of these principles are not like the others” (to paraphrase the old Sesame Street song), in the sense that some appear to be top-down in their general import. Indeed, in modern formulations of continuum mechanics, relying upon the generalized notion of internal traction force just reviewed, conservation of total momentum is a top-down manner entirely. Truesdell comments:

In continuum mechanics central forces, and indeed mutual forces of all kinds, play no special part, and the general approach of analytical dynamics is untypical and next to useless.21

From this point of view, Newton’s attempts to reach global conservation of momentum via bottom-up molecular activities represents a misapplication of his Third Law as properly understood. What has gone wrong?

As Truesdell remarks, modern formulations of continuum mechanics do not attempt to decompose their fundamental stress/strain relationships into separate relationships between two bodies directly acting upon one another (i.e., the subject does not demand mutual forces22). From the continuum point of view, Newton’s bottom up argument for conservations is an artifact of special assumption

18

channeling of the sort that we discussed in relationship to the behaviors of Newton’s cradle. In particular, Walter Noll proved a pretty theorem that shows that if we pull the connected blobs of regular continuum physics apart and condense them into singular points, then basic top-down considerations leave the forces between them no choice but to act upon one other in a strong Third Law fashion.23 From the point of view of continuum mechanics, this apparent reduction merely represents a degenerate case of the general framework, which explains why Truesdell remarks that “the general approach of analytical dynamics is untypical and next to useless.”

Is this top-down mode the proper way to adjudicate the status of conservation of momentum in classical mechanics? J.G. Papastavridis offers a helpful assessment:

From such a continuum viewpoint, a particle is viewed not as the building block of matter, but as a rigid and rotationless body! As Hamel (1909, p. 351) summarizes: “What one understands, in practice, by particle mechanics (Punktmechanik) is none other than the theorem of the center of mass (Schwerpunktsatz).” Both models of a body--that is, the one based on the atomistic hypothesis (body as a finite, discrete set of material points, or particles; namely, small hard balls with no rotational characteristics) and the other based on the continuity hypothesis (body as a family of measurable sets, with associated additive set functions representing the mass of that set)—-have advantages and disadvantages; and both are useful for various purposes. The sometimes (in some engineering circles) fierce debate for/ against one or the other viewpoint, we consider counterproductive and petty hairsplitting; and so we will use both models as needed. Such dualisms are no strangers to physics (e.g., particles/corpuscles vs. waves/fields in atomic phenomena) and constitute a creative, dialectical, stress in it.24

We here witness a sterling exemplar of the fact that doctrines such as the Third Law and the Conservation of Momentum do not possess firm semantic content in their own right, but remain hostage to the collection of middle scale interactions that we tolerate elsewhere within our mechanics. If we believe that all of these can be successfully reduced to mutual force relationships between paired bodies in Newton’s manner, we can regard his bottom-up “derivation” of momentum

19

conservation as successful; otherwise, we should declare that Newton was misled by covert special assumption channeling. Unfortunately, Nature itself does not take clear sides in this dispute; it goes fuzzily quantum mechanical on us in just the places where it might resolve our question.

Certainly, the special assumption channeling characteristics of pendulum-based collision experiments (short interval of head on contact largely at a single locus in a line) persuaded Newton himself that he directly witnessed the operations of some general and far-reaching physical principle within these experimental behaviors, despite their complex nature. On this, Spivak comments:

Newton took a basic experimental fact, the conservation of momentum in collisions, and recast it as a corollary of an apparently equivalent formulation, Newton’s Third Law. Nowadays, physics texts may allude to the essential role played by the third law—“forces always appear in pairs”—but they give scant attention to the fact that Newton’s decision to cast those experimental results in terms of the third law was an incredibly audacious generalization! Based on results involving the completely unknown repulsive forces between colliding bodies, Newton hypothesized a more general law concerning all forces… One might well wonder why critical readers readily accept so general a law buttressed by so little experimental evidence, as if it somehow expresses a morally compelling symmetry. Perhaps it’s just because it’s so easy to confuse laws of nature expressing the symmetry of space with other laws that merely seem to.25

The “audacity” of this induction is manifest when one realizes that the “general principle” tenets that he extracted from his experiments actually represent radical hypotheses about unseen things, viz., atoms that stop cold upon contact and action-at-a-distance forces operating upon this same microscopic level that obey strong third law balancing requirements.

Neither of these requirements directly apply to the complex interactions that we witness in the macroscopic world about us; specifically, they do not pertain to real billiard balls in a straightforward manner, for their impactive interactions appear cloaked in a good deal of structural complexity. And it is here that the conjectural connection that Newton drew between these microscopic Third Law demands and the global requirements of Galiliean relativity misled him, for it led him to “see” the operations of Third Law balancing as directly implemented within the middle-range operations of familiar experience, despite the fact that these interactions fail to fit either his contact action or his action-at-a distance paradigms.

20

And here commences the long traditions of textbook mumble-jumble and doctrinal blurrings that Spivak correctly labels as the Third Law’s “numerous misuses and invalid analogues.”26 These are the dodgy Third Law extensions that render elementary mechanics so confusing to the logically minded. Here’s a typical example, drawn from an otherwise excellent modern primer in classical physics. The author (David T. Greenwood) argues for a version of the virtual work principle as follows:

Assume that two particles are connected by a rigid massless rod... Becauseof Newton’s third law, the forces exerted by the rod on the particles m1 andm2 are equal, opposite and collinear. Hence R2 = - R1 ... as shown.Furthermore, since the rod is rigid, the displacement components in thedirection of the rod must be equal or e.δr1 = e.δr2 [where e is a unit vectorpointing in the direction of the rod]. Therefore the virtual work of theconstraint forces is zero: δW = R1.δr1 + R2.δr2 = 0.27

But there are nominally three bodies involved, albeit one that is “massless.” Why should the collinearity of the rod have anything to do with the Third Law? If the rod were bent by an angle in the middle, shouldn’t the “response force” R2 show a corresponding shift in direction?28 Suppose that mass m2 rests lightly upon its end of the stick, whereas m1 is firmly glued. Why should there be any long distance relationship between the local force pairs involved (i.e., the differing adhesions that hold m1 and m2 to their respective locales of rod attachment)?

In fact, a completely different set of physical considerations has tacitly entered Greenwood’s scenario that has no direct connection with the Third Law as it has been specified above. Solid objects generally possess a natural equilibrium rest state to which they closely adhere—that is, a steel rod possesses a natural relaxed state of fixed length and shape.29 This is the key presumption that lies behind Greenwood’s casual invocation of a rigid rod between m1 and m2. But the mere fact that a bar is capable of equilibrium doesn’t insure that it is in equilibrium. And here begin the off-handed appeals to the “internal forces that perform no work” that so confused me when I was a student in Physics 101 long ago. Suppose that we push on the rod at the m1 end with an applied force F and divide our rod into little “atoms” a1, a2, etc. We are supposed to presume that

21

because the internal forces fi,i+1 and fi+1,i attaching atom ai to its neighbor ai+1 and vice versa are always equal, then the separated interatomic forces fi,i+1 and fj,j+1 must be equal along the whole rod as well. If not, it is often argued, the rod itself would spontaneously move to correct the imbalance, which is deemed as ridiculous.30 But, in point of fact, that is exactly what happens: when we push a1 closer to a2, the activity generates a stronger internal force upon a2 which then passes along the rod as a pulse that eventually moves the far end at m2. In that manner, the increased internal force f1,2 will certainly “perform work” on a2 by moving it through a small distance whereas the original force F that we applied to a1 will not directly reach to a2 at all. Nonetheless, Physics 101 students are invariably encouraged to view F as directly acting upon the far endpoint m2. The seemingly innocuous invocation of our rod’s “rigidity” suppresses all of the local activity occurring in between.

Newton himself was directly responsible for many of these dubious forms of Third Law appeal, which continually enlarged over time as later writers applied his descriptive vocabulary to wider applicational arenas.31 It’s not possible to trace all of these variegated windings within this essay, but the graveyard of classical physics is littered with these phony “Third Law” look-alikes, along with a legion of double-talk defenses of these extensions accompanied by contrary attempts to supply Newton’s “action equals reaction” with some semblance of precise content.

In point of pure pragmatics, classical physics would have never reached its astonishing levels of descriptive and predictive success had it not engaged in many of these dodgy “Third Law” appeals with respect to middle-sized objects. The art of successful description within classical practice relies mightily upon our abilities to “cut off” lower scale complexity through appeal to various fast time exigencies such as “rods resume their natural rest states very quickly.” The asymptotic matching involved within the Wren-Huygens collision rules offers a sterling exemplar of how a swiftly regained natural equilibrium state can be fruitfully employed to elide over a good deal of transitory complexity.

From this point of view, the “practical meaning” of Newton’s “Third Law” may largely lie within the confusing jumble of applications that Physics 101

22

students learn to master in spite of their murky logical interconnections. Indeed, it could be fairly argued that our “precise rendering” of the Third Law in terms of local contact actions doesn’t render proper justice to this wider conception of its “applicational meaning” at all. We shall return briefly to these issues within our concluding section.

All of these factors conceded, one might still wish that stock Physics 101 instruction weren’t so darned dogmatic. In these regards, Spivak rightly comments:

[The] physics textbooks …that do discuss [this problem] engage in the usual maddeningly nonchalant assumption that any fool would know how to analyze it.32

Indeed, the great charm of Spivak’s survey lies in its willingness to declare that typical Physics 101 bombast often trades on arguments that depend upon special assumption channeling tricks that fall apart when our attention shifts to slightly varied physical circumstances (his willingness to “speak truth to authority” is the chief reason why I have quoted Spivak liberally here).

For present purposes, readers should recognize that when we link the Third Law to macroscopic features such as rigidity in these characteristic ways, we are apt to suppress crucial details that will be needed to make the phenomena involved truly comprehensible. As we saw, a proper appreciation of billiard ball recoil demands that we pay some attention to the internal pressure waves that move back and forth within the balls, as complex as these processes are. If we nonchalantly equate all of the fi,i+1 and fj,j+1 pairs within our ball in Greenwood’s manner, we will have ipso facto suppressed the internal pressure differences required to track these complex causal processes. If we descriptively freeze our balls into rigid solids, the pertinent notions of “traction force,” “pressure” and the “speed of sound” are likely to lose coherent meaning.33 Suppressing their fast time activities with the casual presumption that “the balls retain their equilibrium shapes throughout the collision” renders basic aspects of their interaction easy to calculate, but leaves their details disconcertingly mysterious.

For these reasons, crediting Newton with a “billiard ball” conception of the universe is a thesis that is hard to defend. We shall return to this issue as well in our concluding section. (vi)

23

The themes just examined can be greatly illuminated if we pause to survey Leibniz’ astonishing reflections upon the “springs in nature,” for his alternative modes of thinking directly emphasize the natural equilibrium states that occasion so many difficulties for Newton’s Third Law.34 He often starts with the ”teleologies” that springs and beams natively exemplify, viz., they can remember a desired natural rest state and display capacities for returning to the state depending upon how each system views it current displacement from that favored state. On this basis, he develops the first differential equation modeling of a wooden beam, based directly upon the inherent springiness of the fibers that stretch and contract along its length when distorted. In the simplest circumstances, the stress/strain relationships within these fibers will behave in a simple Hooke’s law fashion acting upon a differential size scale.

The characteristically Leibnizean employment of psychological terminology italicized in the previous paragraph is apt to sound crazy at first acquaintance but it neatly encapsulates the constitutive ingredients one employs to model spring-like behavior in an equilibrium-centered partial differential equation. In modeling a springy material, we first locate a natural rest state for the target system and describe its behavior in terms of how the system responds to local deviations from that condition. Indeed, modern books in material science still speak in this psychologically inflected manner (a true elastic solid fully remembers its natural rest state whereas more complex materials such as toothpaste or paint only recall desired conditions slowly and with fading memory). Readers should recognize that observe that early writers in a Leibnizean tradition often employ terms like “passive force” or “inertia” in non-Newtonian ways to further embrace the body’s internal reaction to factors that disturb it from its natural state, so that a wooden beam is said to develop an increased amount of “passive force” when bent by an outside agency, which the beam can later convert to active movement when it is liberated from the constraints. As such, the notion encapsulates an early recognition of the condition that we now call potential strain energy.

As it happens, Leibniz’ specific modeling proposal didn’t get the continuum physics of a flexible beam entirely right (he mistakenly located the neutral axis along the bottom of the beam) but he set the basic prototype for the improved approaches developed soon after by the Bernoullis, Euler and others.

24

Another intriguing aspect of Leibniz’ thinking traces to the fact that it is conceptualized in a top-down manner, in the sense that a local element’s ability to respond to its local weight will be materially constrained by its structural linkages to the other springy elements within the beam. When a wooden rod is placed in constrained equilibrium due to a bunch of applied weights, the whole beam should minimize the overall strain energy encountered across its full span, requiring element A to bend more than it otherwise would so that some companion element B far away might bend less. Typically we find the proper equilibrium balance by trial and error improvements upon an initial proposed solution (one locates the natural equilibrium of a set of linked seesaws loaded with an assortment of children through a similar sequence of successive approximations). Indeed, this global cooperation shows the curved configuration of a loaded beam represents “the best of all possible beams” relative to its external burden of weights. This unexpected linkage between the global optimization of beam behavior and the more Panglossian aspects of Leibniz’ metaphysics is no joke! (although it is amusing that such a parallel exists).

Under this recipe, a scientist first locates the constrained equilibriums of a target system in static circumstances (= where no movement appears) and subsequently “turns on a dynamics” by presuming that each element will begin to convert its stored passive force into active movement seeking (but often overshooting) its unconstrained natural rest state. The canonical exemplar of such a two-tiered approach can be found in Lagrange’s Analytical Mechanics, but its historical antecedents far predate both Leibniz and Lagrange.35 For our purposes, it is important to observe the globalized and top-down character of the approach, a point to which we will return in section (vi).

If Leibniz were content to leave matters here, he would have traveled far along the road to a modern understanding of how the “constituitive equations” within a continuum mechanics approach to elasticity should operate. However, like Newton, Leibniz was uncomfortable with appeals to an inherent springiness

25

that lack lower scale mechanical underpinnings altogether. But he entirely rejected the action-at-a-distance repulsive forces to which Newton vaguely appeals, due to his allegiance to contact forces only. This predilection drives Leibniz to locate the source of natural “spring” within the operations of a “pressure” engendered by a body’s immersion within an ocean of surrounding super-mundane particles, that I’ll call an “air” here. In particular, Leibniz borrows an “efficient causation” theory of elasticity from Descartes in which an ambient “air” continuously flows through the pores within an elastic body. The Cartesian Jacques Rohault explains this theory as follows:

But because the subtle Matter which passes through the Pores which are so very small cannot endeavor to wear the Particles of the hard Body through which it passes, but it must at the same time endeavor to restore the same Particles to the State they were in before the Body was bent, it follows, that this ought to make the Body grow straight again. And thus we experience the Property which is called Stiffness, and which Workmen Call the Power of Springing…

The Force with which a Body unbends itself, depends partly upon the Swiftness of the Motion of the subtle Matter, and partly upon the great Number of Pores through which it passes at a Time. But it depends chiefly upon the Disposition of these Pores as they become insensibly straighter and straighter. For by this means, that which gets into them ought to have the same Force, and to produce the same Effect, as a Body which passes between two others, whose Superficies are almost parallel. Now according to the Laws of Mechanics, though the Body which thus passes between two others be very small, and move but slowly, it will notwithstanding, have an incredible force to separate those two from each other.36

Unlike the Cartesians I have consulted, Leibniz recognizes that this “air” flow must be non-randomly directed if the larger body is to regain its natural rest configuration. Bagpipes, after all, don’t automatically reinflate when collapsed because atmospheric air pressure is the same in all directions and a piper of the Black Watch is required to direct the air stream in a univalent direction. Rohault has overlooked the fact that the rushing “air” on the other side of each pore should push its walls in

26

opposing ways. With respect to our earlier discussion, the Cartesians have confused a notion of “pressure” that is inherently hydrostatic in conception (= the fully equalized and entirely isotropic condition one finds in a flask of gas at equilibrium) with the localized dynamic pressure differences that convey causal effects across the gas and allow it to reach equilibrium. In these terms, Leibniz is pointing out that locally coordinated pressure waves are required if an elastic body is to regain its natural equilibrium condition.

It is at this exact juncture that Leibniz’ physics begins to take on its many of its most unusual aspects, for he claims that God’s provident direction supplies the directionality required in the “air” flow. Specifically, the Good Lord recognizes the teleological “desires” of a wooden beam with respect to its “spring” and providently orients the super-mundane particles in directions that generally—but not inevitably—carry the beam back to its natural rest state. In this fashion, Leibniz’ world is largely configured from our own macroscopic size scale outward; we collectively attempt to fulfil our large scale teleological ambitions as best we can, relying upon the predictable springiness of the wooden beams and springs all around us, and God generally backs up these expectations in favorable ways. I believe that, in physical terms, this is what Leibniz’ notorious “preordained harmony” represents: higher scale teleology-based explanatory patterns should normally comport closely with lower scale efficient causation patterns. The preplanned orientations of the “air” within elastic rebound illustrate the coordination in behaviors required.

However, Leibniz’ thinking adds a further unexpected feature to this scheme, in a manner that returns us to the asymptotic matching inherent in the Wren-Huygen’s approach to billiard ball impact. In particular, Leibniz realizes that the lower scale efficient causation patterns witnessed in his ambient “air” secretly require small amounts of inherent elastic spring to work intelligibly. For how does each “air” particle manage to push away from the pore wall that it pushes ahead? Answer: it must momentarily compress in the manner of a macroscopic billiard ball and regain its normal equilibrium shape after a short interval. To be sure, for the purposes of everyday prediction, we can profitably ignore these

27

springy events by asymptotically papering over their details in Wren-Huygens manner. But full explanatory completeness requires that we investigate the lower scale underpinnings of this microscopic spring, leading to an unending hierarchy of alternating levels of teleological and efficient causation explanation that runs on forever, in patterns that Leibniz dubs “the labyrinth of the continuum.”

In Leibniz’ final assessment, practical science requires that we should invoke scale-based “cutoffs” whenever we can, whether these prove of a Wren-Huygens asymptotic character (in efficient causation circumstances) or arise in the form of primitive appeals to natural equilibrium state (in circumstances where teleological appeals assist us most). Either way, we can’t claim to understand nature properly unless we recognize that Leibniz’ two basic modes of explanation forever complement one other by filling in the explanatory gaps that we would otherwise plow over by trusting excessively to convenient asymptotics.

To be sure, all of this collectively constitutes a wild metaphysical picture, but Leibniz’ underlying considerations are generally sound and help us vividly appreciate the great structural tensions that seemingly innocuous assumptions about billiard ball behavior introduce into our “mechanical” patterns of thinking.

W.L. Stronge characterizes our contemporary situation with respect to impactive collision as follows:

When bodies collide, they come together with some relative velocity at an initial point of contact…. This reaction force deforms the bodies into a compatible configuration in a common contact surface that envelopes the initial point of contact. Ordinarily it is quite difficult and laborious to calculate deformations that are geometrically compatible, that satisfy equations of motion and that give equal but opposite reaction forces on the colliding bodies. To avoid this detail, several different approximations have been developed for analyzing impact: rigid body impact theory, Hertz contact theory, elastic wave theory, etc.37

Each of the listed schemes practice various cut-off strategies that reduce the complexity of the calculations required, at the cost of suppressing further layers of subtler effect.

28

In the next two sections, I’ll offer a few additional reflections on why the subsequent history of billiard ball science has proved so perplexing. The unconcerned reader may wish to skip directly to section viii, in which I document how these old issues remain of considerable concern to philosophical thinking even today.

(vi)

Flickering recognitions that global, top down principles are wanted within mechanics often emerge prior to their explicit canonization as integral principles within the late nineteenth century. In Letters to a German Princess, Euler offers these remarkable observations:

[A]s this impenetrability of bodies exerts always and universally a force which prevents all penetration, it is not at all surprising, then, that we should observe perpetual changes in the state of bodies, though every one has a tendency to preserve itself in the same state… While they can continue in the same state, without any injury to impenetrability, they then exert no force, and bodies remain in their state; it is only to prevent penetration that impenetrability becomes active, and supplies a force sufficient to oppose it. When, therefore, a small force suffices to prevent penetration, impenetrability exerts that and no more; but when a great force is necessary for this purpose, impenetrability is ever in a condition to supply it.

Thus, though impenetrability supplies these powers, it is impossible to say that it is endowed with a determinate force; it is rather in a condition to supply all kinds of force, great or small, according to circumstances; it is even an inexhaustible source of them… [W]hen two billiard balls clash, … they act upon [each other only] insofar as their state becomes contrary to impenetrability; from whence results a force capable of changing it, precisely so much as is; necessary to prevent any penetration; so that a small force would not have been sufficient to produce this effect. It is very true, that a greater force would likewise prevent the penetration; but when the change produced in the state of bodies is sufficient to prevent mutual penetration, the impenetrability acts no farther, and there results from it the least force is capable of preventing the penetration. Since, then, the force is the smallest, the effect which it produces, that is, the change of state which it operates, in order to prevent penetration, will be proportional; and, consequently, when two or more bodies come into contact, so that no one

29

could continue in its state without penetrating the others, a mutual action must take place, which is always the smallest that was capable of preventing penetration.

You will find here, therefore, beyond all expectation, the foundation of the system of the late Mr de Maupertuis, so much cried up by some, and so violently attacked by others. His principle is, that of the least possible action; by which he means, that in all the changes which happen in nature, the cause which produces them is the least that can be.38

Euler believes that all forces are of contact origin39 and in this passage makes an observation about the proper assignment of forces to a complex structure that closely resembles Leibniz’ recognition that the stresses across a loaded beam in equilibrium must be assigned in manner that globally minimizes the total strain energy. Euler appears to be thinking of the top-down manner in which local stresses are computed with respect to a bridge or similar structure, in which we begin with the overall geometry of the bridge (how its parts are fastened together) and its assigned loads (e.g., a number of heavy trucks) and works backward to the local stresses, which are assumed to be minimized at equilibrium. Danilo Capecchi explicates this methodology as follows:

[In] study[ing] the equilibrium of a constrained system, …. it is assumed that there are known external forces, named active forces, and forces due to the constraints, named reactive forces or constraint reactions, the presence of which should be inferred indirectly from the empirical evidence that motions of the material points of a constrained systems are different from those registered without constraints. The value of constraint reactions is not [directly] given, [but] depend[s] on the geometry of constraints and the active forces.40

At this juncture, one of the great conceptual tensions that troubles the entire history of classical mechanics makes a silent entrance, for those top-down induced “constraint forces” cannot obey the strong Third Law assumptions to which we appealed earlier in reaching the conservation of momentum (as noted previously, constraint forces must usually prove velocity sensitive).

Commentators often attribute eighteenth century interest in Maupertuis’ principle to its theological suggestiveness alone. Although these motivations are genuine, its virtues as a vehicle for an important form of top-down reasoning within mechanics should not be overlooked.

(vii)

30

A seemingly paradoxical aspect of late nineteenth century thought traces to the fact that most scientists of that era firmly rejected point masses as physically realistic, yet continued to build up the foundations of their subject in a point-based manner. A. E. H. Love summarized the former opinions as follows:

The hypothesis of material points and central forces does not now hold the field. This change in the tendency of physical speculation is due to many causes, among which the disagreement of the rari-constant theory of elasticity with the results of experiment holds a rather subordinate position. Of much greater importance have been the development of the atomic theory in Chemistry and of statistical molecular theories in Physics, the growth of the doctrine of energy, the discovery of electric radiation. It is now recognized that a theory of atoms must be part of a comprehensive theory, which must include also sub-atomic constituents of matter (electrons), and that the confidence which was once felt in the hypothesis of central forces between material points was premature.41

Thus Maxwell compared real life atoms to bells: continuum physics structures whose vibrational spectra directly reflect their detailed shapes:

We may compare the vibrating molecule to a bell. When struck, the bell is set in motion. This motion is compounded of harmonic vibrations of many different periods, each of which acts on the air, producing notes of as many different pitches.42

Yet Maxwell, in company with Lord Kelvin and P.G. Tait, was a strong proponent of the opinion that Newton successfully encapsulated the entirety of classical mechanics in his original laws of motion, a contention that encourages the dubious readings of the Third Law that we have already surveyed.43 What accounts for this conceptual discrepancy?

Insofar as I can see, it traces largely to the Victorian’s inability to see how adequate foundations for mechanics might be laid down except in an “elaborate simple constructs first” fashion. Often this point of view assumes a neo-Kantian cast, as in Karl Pearson’s influential Grammar of Science:

Whenever we … assert reality for the mechanisms by aid of which we describe our physical experience, then we are more likely than not to conclude with an antinomy, or a conflict of rules. For such mechanisms are constructs largely based on conceptual limits, which are unattainable in the field of perception. When we consider space as objective and matter as that which occupies it, we are forming a construct largely based on the

31

geometrical symbols by aid of which we analyze motion conceptually. We are projecting the form and volume of conception into perception, and so accustomed have we got to this conceptual element in the construct that we confuse it with a reality of perception itself. When we go a stage further in the phenomenalizing of conceptions, and postulate the reality of atoms, the antinomy becomes clear. If bodies are made up of swarms of atoms, how can they have a real volume or form?

He regards point-masses as the most elementary form of the “mental constructions” that mathematicians employ to bring the real world under the control of their reasoning processes. More complex forms of infinitesimal description should be reached by squishing together point mass models, in the mode of Charles Navier’s approach to elasticity. None of these constructions should be viewed as “realistic”; they merely represent artifacts of the manner in which we fit our mathematical tools to the data of experience. If we presume a greater descriptive realism, we are apt to fall into conceptual confusion.

As an example of such an “antinomy,” Pearson supplies a Leibniz-like regress argument:

It might seem at first sight easier to explain why two adjacent ether elements “move each other” than why two distant particles of matter do. The common-sense philosopher is ready at once with an explanation: They pull or push each other. But what do we mean by these words? A tendency when a body is strained to resume its original form; a tendency in a certain relative position of its parts to a certain relative motion of its parts. But why does this motion follow on a particular position? It is the old problem over again, with the difference that relative position now involves small instead of large distances. It will not do to attribute it to the elasticity of the medium; this is merely giving the fact a name. We do indeed try to describe the phenomenon of elasticity conceptually, but this is solely by constructing elastic bodies out of non-adjacent particles, the changes of position of which we associate with certain relative motions. In other words, to appeal to the conception of elasticity is only to “explain” one “action at a distance” by a second “action at a distance.” If the ether-elements owe their elasticity to such an arrangement, we shall want another ether to “explain” the motion of the first, and the process will have to be continued ad infinitum. … We cannot proceed forever “explaining” mechanism by mechanism. Those who insist on phenomenalizing mechanism must ultimately say: “Here we are ignorant,” or, what is the same thing, must take refuge in matter and force.

32

In this peculiar fashion, a Boscovichean core (point masses and action-at-a-distance forces only) becomes selected as the proper “foundations” for “classical mechanics” with little concern for the descriptive accuracy of such tools. Allied points of view linger with us to the present day.

In the twentieth century, the cavalier attitudes of the Victorians with respect to the construction of suitable infinitesimals for continuum mechanics needed to be drastically rethought in the face of complex materials such as rubbers and non-Newtonian fluids. This process of renegotiation required a more careful consideration of the top-down reasoning processes mentioned previously. But we cannot continue the history of these developments here.44

(viii)

Let us finally turn to some far-reaching effects upon general philosophy that stem from this unfortunate tangle of descriptive confusions.

Suppose that, as a devoted Anglophile such as J.T. Desaguliers, you presume that Newton has successfully puzzled out the “system of the world”:

Newton the unparallel’d whose NameNo Time will wear out of the book of fame…Nature compelled, his piercing mind obeys,And gladly shows him all her secret ways;‘Gainst Mathematics she has no Defense,And yields t’ experimental consequence.45

Suppose further that, like David Hume, you believe that Newton’s achievements represent a physics soundly based upon an underlying “billiard ball physics” of an impactive character. What will you make of the “cause and effect” patterns so exemplified?

This is the case when both the cause and effect are present to the senses. Let us now see upon what our inference is founded, when we conclude from the one that the other has existed or will exist. Suppose I see a ball moving in a straight line towards another, I immediately conclude that they will shock and that the second will be in motion. This is the inference from cause to effect, and of this nature are all our reasonings in the conduct of life: on this is founded all our belief in history; and from hence is derived all philosophy, excepting only geometry and arithmetic. If we can explain the inference from the shock of two balls, we shall be able to account for this operation of the mind in all instances.46

33

And

Were a man, such as Adam, created in the full vigor of understanding, without experience, he would never be able to infer motion in the second ball from the motion and impulse of the first. It is not anything that reason sees in the cause which makes us infer the effect... The mind can always conceive any effect to follow from any cause, and indeed any event to follow upon another: whatever we conceive is possible, at least in a metaphysical sense; but wherever a demonstration takes place, the contrary is impossible, and implies a contradiction. There is no demonstration, therefore, for any conjunction of cause and effect. And this is a principle which is generally allowed by philosophers.47

But Hume’s (and Adam’s!) inability to track the causal details characteristic of billiard ball collision arises largely because his hero Newton has asymptotically erased the very causal processes that render billiard ball rebound comprehensible. After all, the causal processes we find most intuitively satisfying often rely upon wave motion convection in some form, despite the fact that physics textbooks often adopt descriptive stratagems that suppress these movements in favor of rigid body or allied assumptions.

Of course, Hume’s many modern defenders will still argue that, despite his frequent reliance upon billiard ball collision as a central example, his basic “empiricist” percepts remain valid. I will only remark that common stereotypes of where the empiricist/rationalist divide falls do not strike me as just or accurate with respect to the contrasts that distinguish Newton from Leibniz with respect to billiard ball impact.

In passing we might also observed that Kant’s attempts to recast classical physics as regulative principles demanded by the understanding apparently strive to justify a notion of “internal pressure” along the contours sketched above.48 If so, he has provided a retort to Hume much along the lines canvassed here: a rational understanding of billiard ball rebound demands attention to the causal processes that distribute pressure waves across the interiors of the balls involved.

In light of the history just surveyed, I am astonished to find how quickly commentators who should have known better reverted to thumbnail descriptions of “classical mechanics” as “billiard ball physics.” For example, we find Arthur Eddington writing in 1928:

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The Victorian physicist felt that he knew just what he was talking about when he used such terms as matter and atoms. Atoms were tiny billiard balls, a crisp statement that was supposed to tell you all about their nature in a way that could never be achieved for transcendental things like consciousness, or beauty or humor.49

Eddington was evidently familiar with The Grammar of Science, for he liberally borrows metaphors from it. How could any reader of that book seriously maintain that “the Victorians knew all about the nature of what they were talking about”? Quantum mechanics had been in full bloom for only about three years at this point. How quickly do we attribute simple-mindedness to the thought processes of our elders!

Some of these misconstruals also trace to the fact that the early quantum theorists preferred to construct their own models by “quantizing” simple point mass models of an ODE character. Whatever the merits of that rather peculiar gambit, it should not serve as a motive for rewriting the developmental history of classical mechanics in its reflected image. However, the standard primers of a modern Physics 101 course do exactly that, for the same easiest-path-to-quantum-mechanics sort of reasons. One only learns to correct these stereotypes when one takes a serious upper division course in material modeling, probably in an engineering department.

Oddly enough, loose stereotypes of “classical mechanics” as “billiard ball physics” continue to pull wide swatches of philosophical thinking in unfortunate directions even today, in patterns reminiscent of the traps into which Hume fell long ago. This is because many contemporary writers regularly appeal to the purported formal characteristics of a somewhat mythical entity they call “our best fundamental theory of the world T.” They recognize that current science is far from settling upon such an empirically adequate T, but they maintain that it continually strives to do so and that “classical mechanics” exemplifies a failed attempt to provide such a codification. In these respects, their thinking follows the following pattern: (i) the tenets of any fundamental physical doctrine should come formatted in the strict form of Theory T axiomatics; (2) classical mechanics provides a sterling example of a properly codified theory T (albeit one that is not empirically confirmed); (3) because this T merely incorporates familiar doctrines of a billiard ball character, the exact formalities of any purported classical mechanics codification T don’t require detailed study. Along this garden path,

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loose familiarity with pool table behavior convinces many contemporary philosophers that they fully understand, in principle at least, “how scientific theories manage to do their jobs.” In doing so, they have unwittingly fallen prey to illusions of clear comprehension closely allied to those that mislead Boyle long ago in his praise of the “excellency of the mechanical hypothesis.” In my opinion, false confidences of this general character are rampant throughout contemporary philosophy and seriously inhibit its future prospects.

In underestimating the humble billiard ball, philosophy fails to avail itself of an excellent exemplar of its proper diagnostic duties. What Hilbert writes about science is true of human thought generally:

The edifice of science is not raised like a dwelling, in which the foundationsare first firmly laid and only then one proceeds to construct and to enlarge the rooms. Science prefers to secure as soon as possible comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, it sets about supporting and fortifying them. This is not a weakness, but rather the right and healthy path of development.50

Descriptive technique in most walks of life must begin in what initially appear as “comfortable places,” even when those same locales later prove dauntingly complex. Recapitulating this etiology of opportunism, the traditional training exercises of Physics 101 textbooks are apt to seem perversely obscure to the logically inclined. But we should pardon these developmentally necessitated confusions with the indulgent tolerance of the Captain of the Pinafore: “As incomprehensible as these utterances are, I nevertheless feel that they are dictated by a sincere regard for my scientific development.”

1 Robert Boyle, “The Excellency and Grounds of the Corpuscular or Mechanical Philosophy” in M.R. Matthews, ed, The Scientific Background to Modern Philosophy (Indianapolis: Hackett: 1989), p. 113.

2 John Locke, An Essay Concerning Human Understanding (Oxford: Oxford University Press, 1979), p. 310.3 John Wallis should be assigned much credited as well, for his views in many ways anticipated Newton’s. See J.F. Scott. The Mathematical Work of John Wallis (New York: Chelsea, 1981).

4 Isaac Newton, Laws of Motion Paper, quoted in W.L. Stronge, Impact Mechanics (Cambridge: Cambridge University Press, 2000), p. 116.

5 Gaspard-Gustave Coriolis (Mathematical Theory of Spin, Friction and Collision in the Game of Billiards, David Nadler, trans., (San Francisco: Nadler Publications, 2005), p. 56) ably recognized the special assumption cutoffs that render the physics of the billiard table especially tractable to descriptive analysis:

Suppose that there were a second momentum, due to elasticity, to be added to the principal momentum, due to collision during the compression of the tip. However much this second momentum might differ in direction from the main momentum, it would to that extent be impossible to play with any confidence... [B]ut we recognize from experience that [this does not happen]. Although it is impossible for current science to establish this proposition by theory, we may nonetheless take it as sufficiently proved by experience, and assume it as a basis for calculations.

Newton and his followers (e.g., Colin MacLaurin in MacLaurin’s Physical Dissertations (London: Springer, 2007), pp. 66-8) make some attempt to deal with oblique collisions, largely through assuming that the impact only cancels the normal component of momentum. To do a better job, Euler needed to develop the notion of angular momentum and to explicate how a suitable coefficient of surface roughness could quantify how much of the incoming energy will become reallocated as spin.

6 Michael Spivak, Physics for Mathematicians, Mechanics I (United States: Publish or Perish, 2110).

7 Although I here gesture at a full ab initio continuum mechanics explanation for the rebounding patterns witnessed in this apparatus, every concrete discussion I have located within the scientific literature employs one of the radical “cut off” techniques enumerated by W.L. Stronge in the quotation below. For the basic anomalies in behavior, see Seville Chapman, “Misconceptions Concerning the Dynamics of The Impact Ball Apparatus,” American Journal of Physics 28 (1960) and, for the basic operations of wave reflection, David Auerbach, “Colliding Rods: Dynamics and

Relevance to Colliding Balls” American Journal of Physics 62 (1994), who comments with respect to billiard balls: “I have neither seen any solution to [this problem] nor an explanation as to how kinetic energy comes to be conserved: a singular situation for this experiment used so often to justify conservation principles” (p. 525). Wayland C. Marlow in his The Physics of Pocket Billiards (Palm Beach Gardens: Marlow Advanced System Technologies, 1995) likewise encounters explanatory difficulties in dealing with the break shots in pool, although most other phenomena covered succumb to Euler’s rules and simple frictional hypotheses.

8 For an overview, see my “Semantic Mimicry” in Physics Avoidance and Other Essays (Oxford: Oxford University Press, 2017). For a detailed exposition, see Bernard Brogliato, Nonsmooth Mechanics (Cham: Springer, 2016).

9 I have in mind glosses such as the following (from Max Jammer, Concepts of Force (New York: Dover, 1999), p. 242):

With the rise of Newtonian dynamics and its interpretation along the lines of Boscovich, Kant and Spencer, the concept of force rose almost to the status of an almighty potentate of totalitarian rule over the phenomena. And yet, since the very beginning of its early rise to power, revolutionary forces were at work (Keill, Berkeley, Maupertuis, Hume, d'Alembert) which in due time led to its dethronement (Mach, Kirchhoff, Hertz). This movement in mathematical physics, from the time of Newton onward, was essentially an attempt to explain physical phenomena in terms of mass points and their spatial relations. For it became increasingly clear that the concept of force, if divested of all its extrascientific connotations, reveals itself as an empty scheme, a pure relation.

I submit that this popular evaluation is wrong or misleading in almost every particular. Unfortunately, such assumptions still motivate contemporary projects such as Bas van Fraassen’s “constructive empiricism” (The Scientific Image (Oxford: Oxford University Press, 1980)), which attempt to consign notions like “force” to lesser semantic categories.

10 Thomas L. Hankins, Jean d’Alembert: Science and the Enlightenment (Oxford: Oxford University Press, 1970), p. 158.

11 Among other matters, the forces derived from constraints are generally velocity-sensitive in a manner inconsistent with the strong Third Law restrictions discussed above (for a discussion, see my Wandering Significance (Oxford: Oxford University Press, 2006), pp. 369-70). One of Heinrich Hertz’ prime objectives in The Principles of Mechanics (New York: Dover, 1956) is to resolve this tension in favor of constraint forces. Josh Eisenthal is currently working on a detailed examination of Hertz’ motives in this rather opaque work.

12 Henry Pemberton , A View of Sir Isaac Newton’s Philosophy (London: S. Palmer, 1728), pp. 406-7. 13 Isaac Newton, Opticks (New York, Dover, 1952), pp. 397-8.

14 In Book II, Newton distinguishes three classes of surrounding dissipative medium, which George E. Smith explicates as follows:

Section 7, in all three editions, distinguishes among three theoretically defined types of fluid. A rarified fluid consists of particles spread out in space, with inertial resistance arising from impacts between these particles and macroscopic bodies, as if the body were moving through debris in empty space. An elastic fluid is a rarified fluid with repulsive forces among the particles and thus between the particles and the body as well. A continuous fluid consists of particles so packed together that each particle is in contact with its neighbors.

(“The Newtonian Style in Book II of the Principia” in J.Z. Buchwald and I. B. Cohen, eds., Isaac Newton’s Natural Philosophy (Cambridge: MIT Press, 2001), p. 267). However, Newton’s third alternative is not a fluid in the sense of modern continuum mechanics capable of sustaining a true internal pressure. What the Third Law is supposed to demand in such circumstances strikes me as unclear.15 Wilson L. Scott, The Conflict Between Atomism and Conservation Theory (London: MacDonald, 1970).

16 Clifford Truesdell, Essays in the History of Mechanics (Berlin: Springer-Verlag, 1968), p. 176.

17 Isaac Newton, The Principia, I.B. Cohen and A, Whitman, trans. (Berkeley: University of California, 1999), pp. 795-6.

18 Pierre Simon Laplace (in The Mechanics of Laplace, J. Toplis, trans. (London: Longmans Brown and Co, 1814), p. 71) is explicit on these topics. He agrees with Newton that direct pairwise contact between simple bodies causes them to lose motion, but in more complicated interchanges only a return to “equilibrium” may be presumed:

What we have said, is on the supposition that bodies are composed of similar material points, and that they differ only by the respective positions of these points. But as the nature of bodies is unknown, this hypothesis is at least precarious; and it is possible that there may be essential differences between their ultimate particles. Happily the truth of this hypothesis is of no consequence to the science of mechanics, and we may make use of it without fearing any error, provided that by similar material points, we understand points which by striking each other with equal and opposite velocities, mutually produce equilibrium whatever may be their nature.

19 More accurately, Reint de Boer (Theory of Porous Media: Highlights in Historical Development and Current State (Berlin: Springer, 2000), p. 7) comments:

In his [Mechanica, Euler] also addressed the cut principle. However, his reflections have little in common with the formulations used today, which state that all balance equations are valid not only for the total body but also for every part which is imaginarily separated from the bulk body, where the freed interactions on the cut surfaces are to be attached as external interactions. In the course of his long research career, Euler never gave a clear statement of the cut principle. However, in his considerable works he employed the cut principle with mastery.

20 In billiard ball circumstances, a further measure of strain energy can be associated to these moving pulses of localized distortion, considered as a localized form of potential energy storage.

21 Clifford Truesdell, A First Course in Rational Continuum Mechanics, Vol. 1 (new York: Academic, 1991), p. 54.

22 Note that the Euler’s cut principle does not make such a demand.

23 Of course, Noll’s downward proceeding argumentation requires some intermediating doctrinal tissue to connect together the local with the global, largely utilizing a second physical principle that is sometimes mistakenly viewed as a Third Law variant. I have in mind the general principle of balance of linear momentum, reformulated to suit extended bodies (along with its cousin, the balance of angular momentum). It maintains that the net action of the forces stemming from the exterior B* of a smooth but otherwise arbitrary interior blob B will act upon B itself in a manner that will accelerate the center of mass of B in a F = ma manner. As such, it plausibly represents a generalization of something like Newton’s original Second Law, but in a manner that must explain how the traction forces operating around the boundaries of B can coordinate with whatever body forces might reach directly into B’s interior (the two kinds of force are dimensionally mismatched and apply at different sites, so some care is needed).

24 J.G. Papastavridis, Analytical Mechanics (Oxford: Oxford University Press, 2002), pp. 100-1. Spivak’s comments on Noether’s principle (pp. 469-70) complement Papastavridis’ remarks.

25 Spivak, pp 22-3.26 Spivak, pp 22-3.

27 Donald T. Greenwood, Classical Dynamics (New York: Dover, 1997), pp. 16-8.

28 Indeed, most “Third Law” discussions of how cords behave upon pulleys blithely disregard the directions of forces treated as “mutual.”

29 In the next section, we’ll observe the centrality of natural equilibrium states within Leibniz’ quite different approach to these issues.

30 E.g., Principia, pp. 427-8, a passage that Spivak characterizes as “may well be the silliest thing that Newton ever said, at least among scientific statements” (p. 276).

31 What we shouldn’t want to do—although opposing assumptions are frequently made—is to regard blob B’s interior response (in the guise of its center of mass movements, often called its “inertial reaction”) as a “Third Law response” to the combined set of applied forces that we have aligned with the exterior region B*. B’s “reaction,” after all, isn’t a true force at all, but an acceleration, often labeled in many modern books as a “pseudo-force.” To be sure, there an another striking physical symmetry now called “objectivity” or “material frame indifference” that marks the fact that most materials will respond to a sequence of inertial changes by the same constitutive rules as govern their responses to a sequence of applied body forces of the same strength. These behavioral affinities prompt many authors into counting our “inertial reactions” amongst the “forces” generally, a shift in usage that generates many common forms of Third Law confusion.

32 Spivak, p. 246.

33 More exactly, the relevant tensors become indeterminate until further degrees of lower scale physical description are supplied. In an allied manner, crediting a fluid with “incompressibility” simplifies the mathematical description of a target system significantly at the cost of suppressing the fast time pressure waves required to explain how distant parts of a pipe flow affect one another.

34 For a fuller treatment of the themes in this section, see my “From the Bending of Beams to the Problem of Free Will” in Physics Avoidance.

35 Two good sources: Pierre Duhem, The Origins of Statics (Berlin: Springer, 1991) and Danilo Capecchi, History of Virtual Work Laws (Heidelberg: Birkhuser, 2012),

36 Jacques Rohault, Rohault’s System of Natural Philosophy, Vol 1, John Clarke, trans. ((London: James Knapton, 1723), pp 133-4

37 Stronge, p. xvii.

38 Leonhard Euler, Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess I, David Brewster, trans. (New York: Harper and Brothers, 1854), pp. 263-5.

39 See Curtis Wilson, “Euler on Action-at-a-distance and Fundamental Equations in Continuum Mechanics” in P.M. Harman and A.E. Shapiro, eds.,The Investigation of Difficult Things (Cambridge: Cambridge University Press, 1992).

40 Capecchi, op cit, p. 18.

41 A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (New York: Dover, 1944), p. 14. The mention of “rari-constant elasticity” refers to a celebrated methodological dispute between the followers of Charles Navier and those of George Stokes over whether the subject should be elaborated via bottom up or top-down modeling techniques. The philosophical opinions of writers such as Pierre Duhem were greatly impacted by the superiority of Stokes’ approach. 42 J.C. Maxwell, “Atom” in W.D. Niven, ed., The Scientific Papers of James Clerk Maxwell (New York: Dover, 1965), pp. 483-4. Maxwell concludes with a sly cosmological flourish:

Atoms have been compared by Sir J. Herschel to manufactured articles, on account of their uniformity… [I]t seems … probable that [Herschel] meant to assert that a number of exactly similar things cannot be each of them eternal and self-existent, and must therefore have been made, and that he used the phrase “manufactured article” to suggest the idea of their being made in great numbers.

43 See J.C. Maxwell, Matter and Motion (New York: Dover, 1991) and William Thomson (Lord Kelvin) and P.G. Tait, A Treatise on Natural Philosophy Vol. 1 (New York: Dover, 1962). Tait, in his later Newton’s Laws of Motion (London: Adam and Charles Black, 1899), pp. 30-1, encapsulates Newton’s purported intent as follows:

[Newton’s] form of words … represents two different Laws:—which may be stated thus:—

Momentum is transferred (without change) from one body to another.Energy is transferred (without change of amount) from one body to another.

The first is called the Conservation of Momentum, the second the Conservation of Energy.

In his well-known Newton for the Common Reader (Oxford: Oxford, 1999), S. Chandrasekhar praises these exaggerated readings as “particularly insightful” (an assessment I find surprising).

44 G.A. Maugin, Continuum Mechanics Through the Ages - From the Renaissance to the Twentieth Century (Cham: Springer, 2016).

45 Margorie Hope Nicolson, Newton Demands the Muse (Princeton: Princeton University Press, 1966), p.136. See also B. J. T. Dobbs and M. C. Jacobs, Newton and the Culture of Newtonianism (New Jersey: Humanities Press, 1995).

46 David Hume, An Enquiry concerning Human Understanding (Oxford: Oxford University Press, 1999), p. 111. 47 David Hume, “An Abstract of a Treatise of Human Nature” in David Fate Norton, ed., A Treatise of Human Nature, vol. 1 (Oxford: Oxford University Press, 2007), p. 405. For a fuller discussion of these issues, see Wandering Significance, pp. 589-598.

48 For details, see Michael Friedman, Kant's Construction of Nature (Cambridge: Cambridge University Press, 2013).

49 A.S. Eddington, The Nature of the Physical World (London: MacMillan, 1928), p. 259.

50 Quoted in Leo Corry, David Hilbert and the Axiomatization of Physics (1898–1918) (Berlin: Springer, 2004), p. 127.