XIII*—THEORY FACADES

by Mark Wilson

ABSTRACT Many common approximation methods in physics practice ‘causalprocess avoidance’ in their operative procedures and such methodologies weavedensely throughout the usual fabric of ‘classical mechanics’. It is observed thatHume was unable to find any grounding for a robust conception of ‘cause’largely because he unwittingly looked in those regions of mechanics wheregenuine causal processes had already been tacitly expunged.

I will first describe a form of mathematical schizophrenia thatcommonly emerges in the courses of practical calculation and

then revisit a familiar philosophical episode from this point ofview. More specifically, I’ll argue that several unexpected formsof mathematical strategizing lie in the background to DavidHume’s celebrated worries about causation.Let me begin with an analogy. In the days of old Hollywood,

fantastic sets were constructed that resembled Babylon in all itsancient glory on screen, but, in sober reality, consisted of nothingbut pasteboard cut-outs arranged to appear, from the camera’schosen angle, like an integral metropolis. Likewise, in scientificwork we often encounter constructions that might be analogouslylabelled theory facades: sheets of doctrine that do not truly cohereinto unified doctrine in their own rights, but can merely appear asif they do, if the qualities of their adjoining edges are notscrutinized scrupulously. That is, a ‘theory facade’ represents apatchwork of incongruent claims that might very well pass for aunified theory, at least, in the dark with a light behind it.From a computational point of view, shifting betwixt a motley

collection of ill-matched pieces frequently makes good pragmaticsense, for a direct attack on the full mathematical representationproper to a physical system is apt to leave the appliedmathematician with naught but a sense of frustrated impotence.Often the only practical recourse for making inferential headway,even in very simple circumstances, is to decompose the system’soverall behaviour into descriptive fragments where the intract-

*Meeting of the Aristotelian Society, held in Senate House, University of London, onMonday, 7th June, 2004 at 4.15 pm.

able complexities of the full problem become locally reduced tomore tractable terms by, e.g., throwing away smaller order termsor allied stratagems for reducing mathematical complexity withinshort range packets. As this happens, important mathematicalcharacteristics of the original representation may get left behindwhile alien aspects creep in. The resulting alterations inmathematical character can easily create a situation where it ishard to reconstitute the true nature of the original physicalproblem simply by examining its covering facade alone: Humpty-Dumpty can’t be easily reconstructed from his shards of shell.To consider a simple, but characteristic, illustration of this

phenomenon, suppose we have a semi-hemispherical bellwelded to a table top. The standard approach to this problemis to drop the terms that theoretically reflect the thickness ofthe bell from its proper governing equations and then, byappealing to the geometrical symmetries as well, reduce thetreatment to a very simple formula that predicts a constantstress everywhere. However, this approximation cannot becorrect near the table top where the stress in the bell actuallyrises rapidly. Accordingly, applied mathematicians enforce adifferent policy of simplification near the lip of our dome andobtain so-called ‘boundary layer’ equations that calculate thestresses near the table quite effectively. In essence, we havecovered our bell descriptively with two patches of differentmathematical types: that which handles the bulk of its interior

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and the other that treats the narrow ring of high stress near itsrim. Of course, we ensure that the numerical values supplied bythese two forms of calculation match along the circumferencewhere they overlap, but each of our localized representationsleaves out important aspects of the governing physics thatprove of vital importance in the patch next door.Imagine that we are ignorant of the original equations from

which our two local treatments descend and have learned todescribe our bell merely by quilting together our two patches on awholly empirical basis. In the history of science, situations of thislogical type commonly arise for a host of reasons, not the least ofwhich traces to the fact that the mathematics required toformulate the ‘original equations’ is often quite sophisticatedand may lie far beyond the ken of a society’s currentpractitioners, whereas the derivative covering patches oftencobble by with quite elementary mathematical resources.Human patience being what it is, the temptation soon arises topretend as if our facade patchwork provides a wholly adequatedescriptive framework solely on its own terms; that the wholestory of the bell’s behaviour has been adequately told andnothing further needs to be said. In this impatient spirit, itbecomes common to gloss over the palpable oddities in how ourpatches fit together with various forms of rhetoric, includingmuch philosophical gobbledy-gook of a pseudo-positivist ilk,such as ‘By its very nature, science traffics in intrinsicidealization’ (such themes originally became prominent inphilosophical circles as the side product of late NineteenthCentury attempts to apologize for lapses in physical argumenta-tion that were later recognized as attempts to evade moresophisticated mathematical foundations through crude bridge-work1). I once saw a children’s book illustration in which theKing’s horses and men stood triumphantly arrayed around apatently gimcrack Humpty montage, daylight streaming throughits cracks and with much leftover egg scattered on the ground.This is the image we should frame whenever we confront themethodological apologists who attempt to pass off theory facadesas unified theories. In the long run, such evasions of descriptive

1. Karl Pearson’s The Grammar of Science (Bristol: Thoemmes, ND) provides anexcellent case in point.

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coherence invariably return to bite their perpetrators, forgenerally we must seek the original bell equations whenever wewish to concoct better fashions of approximation. However,undergraduate texts in physics often get away with a hugeamount of ‘patchwork masquerading as an integral theory’hokum, simply because they are not in the business of pressingcalculation to highly accurate levels.Although it would require a long discussion to make my case, I

believe many current philosophical attitudes, even with respect totopics nominally far from science, have become misguided byuncritically absorbing too much of this ‘passing off a facade for aunified theory’ double talk. The net result is a large body of faultyfolklore with respect to science that impedes our ability to thinkthrough basic problems clearly.Since I can hardly tackle such large themes in a short talk,

let me endeavour to illustrate the sorts of problems I have inmind in miniature, by reflecting upon background circum-stances critical to Hume’s celebrated worries about causation.Here, I claim, we can fairly conclude that our wily Scotsmanwas unwittingly victimized by a prominent ‘facade posing as aunified theory’ that has likely deceived most of us in othercontexts as well. The misleading patchwork I have in mind isthe collection of doctrines conventionally labelled ‘classicalmechanics’ or ‘Newtonian theory’. In truth, it scarcely takes ahigh degree of rigorous scrutiny to see that many of thedescriptive tools assigned by tradition to ‘Newtonian theory’ donot cohere easily on straightforward mathematical grounds—weshall consider several simple illustrations later. Nonetheless, therhetorical engines that stoutly pretend otherwise babble volublyon, not only in Hume’s era but even within the typicaluniversity curriculum of our day. To be sure, a palpable ‘familyresemblance’ is often discernible within scattered Newtonianapplications that do not fit together in strict terms, a potentialintimation of formalisms yet to come. Indeed, given themathematical complications that the latter typically embody,wise pedagogical policy often encourages pupils to attend onlyto the hazy commonalities and not bother unprofitably andprematurely about rigour and such. However, the line thatdivides a sensible policy of ‘Let’s let the theoreticians worryabout these mismatches’ from dogmatic misrepresentations to

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the effect that ‘all of these treatments apply basic Newtonianideas in exactly the same way’ is a delicate one and is oftentransgressed in basic physics education, at the cost of decidedlyfuzzy thinking.I happen to believe that all ‘Newtonian doctrine’, as it is

conventionally assembled, cannot find a proper ‘family resem-blance’ unification while remaining within the orbit of classicalideas alone, but the whole must instead piggyback, as a derivedfacade, upon quantum mechanics. But even if my diagnosis ofnon-internal closure is too pessimistic, it is undeniable that themathematical setting required for genuine classical physicsunification needs to be far more sophisticated than was possiblyachievable in Hume’s day or, for that matter, at least twocenturies afterward (a fair amount of functional analysis overinfinite dimensional phase spaces is clearly needed). But no onewarned poor Hume of these mathematical foibles; instead,contemporaneous poets like James Thompson sang Newton’spraise thus:

Have ye not listened while he bound the suns

And planets to their spheres! The unequal taskOf humankind till then. Oft had they rolledO’er erring man the year, and oft disgracedThe pride of schools, before their course was known

Full in its causes and effects to him,All-piercing sage.2

From such paeans to exaggerated accomplishment, an‘emperor’s new clothes’ bravado enfolds the topics native to‘classical mechanics’ in a manner that stifles doubt even whensceptical criticism is fully warranted.When I was a boy, there was a programme on American

television (‘To Tell the Truth’) wherein two impostors sought topersuade a panel of minor celebrities that they were ‘Mr X’instead of the real article, who was also on hand. Within thefull spectrum of the Newtonian facade as Hume encountered it,many genuine examples of causal processes directly treatedwithin the descriptive mathematics are provided, but these

2. ‘To the Memory of Sir Isaac Newton’ in Joan Digby and Bob Brier, eds.,Permutations (New York: Quill, 1985). For a delightful survey: Marjorie HopeNicholson, Newton Demands the Muse (Princeton: Princeton University Press, 1966).

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genuine ‘Mr X’s are arrayed without effective distinction nextto approaches that tacitly practice various policies of ‘causalhistory avoidance’: mathematical tricks that produce effectiveanswers while suppressing the very ingredients that expresscausal operations within a fuller formalism. Unfortunately,prevailing grandiloquence with respect to Newtonian perfectionstrongly encouraged Hume to select as his paradigms of‘successful treatments in physics’ two counterfeits that rely, indifferent ways, upon subtle policies of ‘causal history avoid-ance’ and to neglect, as inferior, the exemplar cases wheregenuine causal behaviours are directly described by Newtonianprocedures. Many of Hume’s characteristic observations on‘cause’ are greatly affected, I think, by this unwittingmisdirection in favour of non-causal confederates.As to the true processes, these are nicely illustrated by

conventional Newtonian treatments of planetary gravitationalsystems and falling cannon balls in a terrestrial field, processestidily described by differential equations of so-called ‘evolu-tionary type’. But such systems of equations are hard to solve andit is common to resort to various forms of ‘causal processavoidance’ to simplify the mathematics. A popular recipe fordoing this is provided by a focus on a so-called ‘equilibrium’ orfinal rest state, as when we declare to Jack and Jill before theirtreacherous descent: ‘I will make no attempt to trace theconvoluted causal process in which you two will now engagebut I can confidently predict the final state in which you’ll landand the condition in which you’re apt to find yourselves. So I’llmeet you both at the bottom with a medical kit.’ That is, Jackand Jill’s bruised outcome can be augured well enough withoutneeding to trace the causal history that leads them there. In itsmathematical guise, such a treatment corresponds to dropping alltemporal terms from a set of equations and then posting alocalized patch over the bottom of the hill. In essence, this trick isformally comparable to the boundary equations we framed forour bell near its rim. But it is precisely the terms dropped thatembody all of the dynamics pertinent to Jack and Jill’s situation.Such ‘causal process avoidance’ is routine fare in many forms ofmechanical calculation.As an aside, it is worth remarking that anytime we see an

appeal to Poisson’s equation, some allied ‘avoidance’ lurks in

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the background, a fact that philosophers frequently overlook.3

It would take us too far afield to discuss the situationadequately, but related problems afflict Bertrand Russell’scelebrated critique of causation in physics:4 he wasn’t able tolocate any because he focused upon replacement equationsfrom which the notion had already been leached away byvarious means. These issues are further complicated by the factthat words like ‘cause’ and ‘force’ display an uncannyinclination to reinsert themselves into descriptive contextsfrom which they have been strategically purged (in a Poisson’sequation situation, we commonly consider the ‘causal processes’operating upon small particles by the large bodies ‘frozen’ intheir vicinity). In a book just completed,5 I have argued thatthis syntactic phenomenon of illicit return is quite commonamongst large classes of descriptive words, but I lack the spaceto explain why this is so here. Generally, such re-entrant wordsmean something different when they reappear in this peculiarfashion, but such shifts are often overlooked at the cost of theconfusions that we witness in, e.g., Russell’s discussion ofcausation.In any case, descriptive strategies that tacitly appeal to

equilibrium can only describe a limited portion of a system’sbehaviour and, if we incorrectly regard the ploy as forming partof a seamless account of the system’s behaviour, we will havemistaken a portion of patchwork facade for a genuine unifiedaccounting.As I stated, there are two specific pretenders that divert

Hume’s attention from the genuine causal processes displayed inthe Newtonian treatment of a falling cannon ball. Each achievesits misdirection in a somewhat different manner. Our first fakercites the intuitive notions natural to how we think about machinedesign—how we plan effective inventions built of gears, rods andcams—to discredit the adequacy of the ‘causation’ on display in a

3. A good example of this can be found in Hartry Field, Science Without Numbers,(Oxford: Blackwell, 1980).

4. ‘On the Notion of Cause’ in Mysticism and Logic (Garden City: Doubleday, 1957).Cf. Sheldon R. Smith, ‘Resolving Russell’s Anti-Realism about Causation’, TheMonist, April 2000 Volume 83, Number 2.

5. Wandering Significance (Oxford University Press, forthcoming). Chapter Eightenjoys some overlap with the material presented here.

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robustly causal gravitational treatment. The Swiss engineerFranz Reuleaux called such notions ‘machinal ideas’ in hisgreat Kinematics of Machinery6 of 1875 and I shall often followthis terminology here. As is well known, Newton’s action at adistance account of gravitation was frequently criticized in histime (and afterwards) as insufficiently embodying the intelligible‘causal processes’ we think we witness in a watch or steam enginefitting (this is the vein of appeal that lies at the heart of thepopular complaint that Newtonian gravitation represents areturn to the ‘occult qualities’ of Mediaeval physics). In truth,Reuleaux’s penetrating analysis of the situation (which seemsinsufficiently known in philosophical circles) shows that it isprecisely from the realm of standard ‘machinal ideas’ thatcausation proper has been effectively purged. In short, ourmachinal impostor makes a true causal process look poorlybecause the former promises wondrous but bogus features thathumble real causation cannot adequately match.However, before I enlarge upon these observations further, let

us inspect the otherNewtonian deceiver on our panel: the standardmethodology brought to bear in Hume’s time in the treatment ofimpact—e.g., the collision of two billiard balls. Once again, thisapproach pretends to be cut from the same causal cloth as a fallingcannon ball, but, in hard fact, secretly practices a policy of‘avoidance’ that silently skips over the critical causative events.Hume focuses upon billiard ball impact in his discussion of ‘cause’and finds no robust notion of causal interaction revealed there. Butthis isn’t surprising; the standard Newtonian approach descrip-tively skirts the crucial events at issue.Since the methodological rationale for evading genuine

causation is easier to explain in the collision case than in‘machinal ideas’ circumstances, let me enlarge upon its para-meters first. But before we consider the ‘cause avoiding’ tactics oftraditional Newtonians, let us first survey the daunting array ofmathematical elements that will be required in any direct andrational approach to real life impact, so that we can moresympathetically appreciate the reasons physicists of Hume’s eraneeded to treat billiards as they did. (1) Describing the mutuallyinduced distortions of the colliding balls requires, at a minimum,

6. Trans., Alexander Kennedy (New York: Dover, 1963).

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some means for treating the induced internal stresses, eventhough the relevant mathematics (partial differential equations)had not been invented in Hume’s era. (2) Further technicalcomplications require that some more complicated functionalanalysis-type framework be employed to extend the partialdifferential equation picture to encompass the formation ofshock waves and complicated reflections (this is TwentiethCentury stuff). (3) As a part of (2), some consideration of heatloss needs to be brought into our picture as well. (4) Boundaryconditions need to be articulated that can be reasonably expectedto hold during the interval of contact. Given the intrinsicallyabrupt nature of the collisions, these are extremely hard toformulate. Although enormous progress on all of these fronts hasbeen achieved, it is presently dubious7 —it is certainly unclear—whether any mathematical formalism wholly adequate to ourintuitive expectations about what happens in a billiard collision(even in midair!) has yet been articulated.In the face of these rather frightful mathematical challenges,

how did the treatment of impact standard in Hume’s era unfold?Devised by Newton himself, the basic trick is to almost—but notcompletely—cover the history of our colliding balls with twodescriptive patches, one devoted to the balls as they approach thecollision and the other as they scatter away from it. But the actualevents of compression and re-expansion that occur when our twoballs contact one another are set in a little window that ourmethod does not attempt to describe. Instead, it patches over thishiatus by matching our incoming and outgoing sheets accordingto a rule of thumb involving gross energetic qualities and acrudely empirical ‘coefficient of restitution’. The rough reason-ableness of such approximation can be justified by Jack and Jill-like considerations, but it is plain that we’ve simply skirted thechief causal events as we normally conceive them. Indeed, byciting some important ideas of Euler’s involving rigid bodybehaviour, this Newtonian ‘blanking out the collision events’picture can improved to handle oblique impacts with tolerablesuccess and supply predictions more or less adequate to most—but not all!—standard billiard table events. But, in the long run,

7. Cf. Werner Goldsmith, Impact (New York: Dover, 2001) and Michel Fremond,Non-Smooth Thermo-Mechanics (Berlin: Springer, 2002).

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the dodge is too crude to handle the blows encountered in, e.g.,sophisticated aircraft design, where the whole army of causallydescriptive mathematics (partial differential equations et al.)needs to emerge from its prior centuries of facade-stylesuppression.This example affords an excellent specimen of a phenomenon

of which we philosophers should be aware. When we hope toascertain ‘the world view’ of some branch of physics, we typicallylook at the mathematics provided in a generic undergraduate orbeginning graduate level text. But commonly the mathematics ondisplay is merely some patchwork approximate to some frame-work not provided (and, possibly, as yet unformulated). Indeed,if we trace the trail of footnotes that ingenuously declare ‘For amore detailed treatment, see . . . ’, we often encounter, not merelymore ‘details’, but radically dramatic shifts in mathematicalsetting altogether. The upshot is that assessments of ‘world view’based upon freshman texts are apt to prove highly misleading.In point of historical fairness, it is important to understand

that most Newtonians did not clearly realize they were evadingthe crucial interval of causal interaction because many of them(including Newton himself) also believed that collisions betweenballs that retain their shapes throughout a collision needed to betolerated as theoretically possible (leading to the old conundrumof what happens when a perfectly immobile object is struck by anirresistible force, the subject of both Johnny Mercer’s song and aprize essay set by the French Academy in 17248). In particular, itwas expected that discontinuous, hard sphere impact mightprovide a paradigm for the causative infinitesimals they believedwere manifested in a smooth gravitational interaction (that is, itwas common to picture a falling cannon ball as continuallystruck by little hammers of gravitational impulse). But all of oursubsequent experience with measures and mechanics has sug-gested quite otherwise: the non-smooth interaction can besuccessfully approached only by piggy-backing, functionalanalysis style, upon the smooth interaction, rather than viceversa. However, we can scarcely expect this to have been

8. Cf. Wilson L. Scott, The Conflict between Atomism and Conservation Theory 1644–1869 (London: MacDonald, 1970), p. 22, and Piero Villaggio, ‘A Historical Survey ofImpact Theories’ in Becchi, Corradi et al. (eds.), Essays on the History of Mechanics(Basel: Birkhauser, 2003), pp. 223–234.

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appreciated by an Eighteenth Century Newtonian (although Ibelieve Leibniz’s insistence that ‘nature does not make jumps’ isquite prescient on these topics). Certainly, these faulty expecta-tions with respect to infinitesimals supplied impactive phenomenawith a centrality in Seventeenth and Eighteenth Century physicsthat, by all rights, they shouldn’t have possessed. ‘All the betterto deceive you with, my dear’, coos our billiard ball wolf to hisvictim Hume.As we’ll see, the faulty dominance given to ‘machinal ideas’

(which mainly exploit the ‘rigid body’ idea) help make thisassumption of perfectly hard impactive infinitesimals seemirresistible in mechanical tradition.Thus encouraged by the prevailing, albeit misleading, char-

acterization of Newtonian physics as, at heart, ‘billiard ballmechanics’, Hume understandably fastened upon pool tablecollision as providing the ‘typical physical interaction’. Accord-ingly, he found no robust notion of ‘causal process’ revealedthere for such processes had been methodologically excludedfrom their standard treatment:

Here is a billiard ball lying on the table, and another ball is movingtowards it with great rapidity. They strike; and the ball which was

formerly at rest acquires a motion. This is as perfect an instance of therelation of cause and effect as any which we know either by sensationor reflection. Let us therefore examine it. It is evident that the two balls

touched one another before the motion was communicated, and thatthere was no interval between the shock and the motion.9

Note that Hume characterizes impact as ‘as perfect an instanceof the relation of cause and effect as any which we know’. Hecontinues:

Were a man such as Adam created in the full vigour of understanding,without experience, he would never be able to infer motion in thesecond 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 . . . But noinference from cause to effect amounts to a demonstration. Of whichthere is this evident proof. The mind can always conceive any effect tofollow from any cause, and indeed any event to follow upon another:

whatever we conceive is possible, at least in a metaphysical sense; but

9. ‘An Abstract of a Treatise of Human Nature’, in An Enquiry concerning HumanUnderstanding (Indianapolis: Bobbs-Merrill, 1955), pp. 186–7.

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wherever a demonstration takes place the contrary is impossible andimplies a contradiction.

In particular, the Newtonian has purged all the delicate traitsof elastic strain, stress and heat dissipation required to makerational sense of the concussion and retains only the incomingand outgoing traits of asymptotic shape, momentum and energy.These quantities are plainly inadequate to rationalizing the causalprocesses occurring within a collision:

But as to the causes of these general causes, we should in vain attempttheir discovery, nor shall we ever be able to satisfy ourselves by any

particular explication of them. These ultimate springs and principlesare totally shut up from human curiosity and inquiry. Elasticity,gravity, cohesion of parts, communication of motion by parts—these

are probably the ultimate causes and principles which we shall everdiscover in nature; and we may esteem ourselves sufficiently happy if,by accurate inquiry and reasoning, we can trace up the particularphenomena to, or near to, these general principles . . . Thus the

observation of human blindness and weakness is the result of allphilosophy, and meets us, at every turn in spite of our endeavours toelude or avoid it.10

Plainly Hume is arguing to a stronger conclusion than simplythat the laws of physics need to be empirically confirmed; heclaims that we lack the requisite concepts to make what occurs inthe breach intelligible (allied, but milder, themes appear in Locke,for closely related reasons11). Hume writes:

[T]he necessary conclusion seems to be that we have no idea ofconnection or power at all, and that these words are absolutely withoutany meaning when employed either in philosophical reasonings orcommon life.12

Indeed, if science could weave no better tale of impact thanNewton’s coefficient of restitution story, Hume would be right.Let us now return to our ‘machinal ideas’ impostor, for, as I’ve

indicated, it represents the key background factor that serves bothto discredit conventional gravitational interaction as a satisfactorycausal process and to encourage the assumption that abruptimpact can serve as a paradigm for what occurs during a smooth,

10. Enquiry, p. 45.

11. Locke approaches Hume’s assessment fairly closely: An Essay Concerning HumanUnderstanding, Book IV, Chapter III (New York: Dover, 1959).

12. Enquiry, p. 85.

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infinitesimal interaction. Of course, ‘machinal ideas’ (where themain objects under consideration are perfectly rigid gears, rodsand hinges) have been prominent within physical thinking sincethe Greeks and, prima facie, we possess an admirably intuitiveunderstanding of such devices superior to any we enjoy for anyother sort of physical system. Consider, in this light, a prototypical‘four bar linkage’ (the diagram is extracted from Reuleaux). If weturn the crank rod (marked ‘a–d’) around its base (‘a–b’), we feelthat we can perfectly anticipate how every other portion of thedevice will move as well. Superficially, this circumstance seems likea perfect case of ‘determinism’, but it isn’t, at least in theconventional ‘past fixing the future’ sense. What we insteadwitness is what might be dubbed ‘here versus there determinism’: alocalized movement at the crank is prescribed along an extendedtemporal interval, which then supplies a rationale for computingthe movements of the linkage away from this locale over exactlythe same time span (formally, we have a singular ‘boundarycondition’ rather than ‘initial values’). A little thinking shows thatthe relationship of this flavour of ‘determinism’ to the conven-tional sort (a so-called ‘Cauchy problem’) must be quite subtle andtacitly involve a good deal of suppression of complex detailsthrough approximation.When we engage in the design of machinery, a pronounced

emphasis on rigid parts sliding or rolling across each other’ssurface in a frictionless way emerges into the foreground foressentially computational reasons, although, on any realisticmechanical appraisal, such circumstances are hard to implementphysically (parts must be carefully machined, linked andlubricated). Furthermore, its components can be joined togetheronly in very special ways (specified by the so-called ‘Grublercriterion’) to allow our contraption to display the internalmobility exemplified in the ‘here versus there determinism’ of ourfour-bar linkage (in other words, our pieces can neither be over-

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or under-constrained, although these situations would be thenatural norm). To the modern engineer, the phrase ‘the theory ofmechanism’ applies solely to the very special physical systemsthat roughly approximate such requirements; from this point ofview, there are very few ‘mechanisms’ proper to be foundanywhere in the known universe, except on the surface of earth(which has become considerably cluttered with such gizmosthrough man’s handiwork and the activities of natural selection).The special mathematical feature that makes this limited speciesof assembly so salient to our interests lies in the designopportunities offered in their reduced patch description: byappealing heavily to their rigidity, we are able to move into avery simple mathematical domain (basically, matrix algebra) thatallows us to readily perfect the performance of any invention wemight crudely plot. In fact, it is finding this improved design thatrepresents the important task for the inventor, not the accurateprediction of its detailed causal behaviour. It is Reuleaux’sspecial merit to have discerned this important fact clearly: that,through a collusion of special purposes (design improvement)and simple mathematical opportunity (matrix algebra), the‘theory of mechanism’ effaces itself from orthodox physics andbecomes a topic of a completely different character altogether. Ashe observed:

[T]he sense of the reality of this separation [of the theory of mechanismfrom general Newton-style mechanics] has been felt not only by

engineers or others actually engaged in machine design, but also bythose theoretical writers who have had any practical knowledge ofmachinery, in spite of the increasing tendency in the treatment of

mechanical science to thin away machine-problems into those of puremechanics.13

The relationship of ‘machinal’ calculations to the processesactually responsible for the causal behaviour of a mechanism areeven more oblique here than in the billiard ball case.Reuleaux’s explications of why ‘machinal ideas’ loom so large

in our physical thinking, although plainly sound once itscontours are brought plainly in view, were not, of course,appreciated in Hume’s era nor, unfortunately, even by mosthistorians of ideas working today. Instead, the presumption

13. Kinematics, p. 30

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remains abroad that, in some ill-defined way, appeal to machine-like structure must lie at the core of classical physical thinking,rather than merely representing an opportunistic mathematicalpatch spanning a very specialized subdomain. Or, as Humeframed the erroneous expectations in his Dialogues concerningNatural Religion:

Look round the world: Contemplate the whole and every part of it:You will find it to be nothing but one great machine, subdivided into

an infinite number of lesser machines, which again admit ofsubdivisions, to a degree beyond what human senses and facultiescan trace and explain. All of these various machines, and even their

most intimate parts, are adjusted to each other with an accuracy,which ravishes into admiration all men, who have ever contemplatedthem.14

But the proper, Reuleaux-inspired retort to this ‘argumentfrom design’ is ‘Well, there really aren’t many such arrange-ments to be found in nature apart from those encountered onearth.’More generally, the notion of rigid body—say, in the form of

the ‘constraints’ so central in Lagrangian approaches—repre-sents an exceptionally productive, yet ultimately venomous,viper that mechanical tradition has long clasped fondly to itsbosom. This form of reductive trick is quite productive, in thatotherwise intractable problems are commonly rendered viable,but it proves poisonous in that its alien notions cannot bewholly harmonized with the notions of internal stress and strainotherwise required in a reasonable approach to macroscopicbodies. ‘Rigid body behaviour’ needs to be regarded as a pieceof a patchwork covering facade, housed within a discordantmathematical framework, that hovers over the wider range ofgeneric mechanical behaviours merely as a localized approx-imation. Worse yet, for the reasons already sketched, once therigid body is welcomed as a respectable citizen within thekingdom of mechanics, it is practically inevitable that the non-distorting collisions of perfectly hard billiard-ball spheroidsseem as if they must embody the essence of mechanicalinteraction, rather than merely representing pallid and gappysubstitutes for the genuine article. Once we are tricked, as

14. (Indianapolis: Bobbs-Merrill, 1956), p. 143.

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Hume surely was, into assuming that collisions and ‘machinalideas’ comprise the core of mechanical doctrine, we will noteasily locate the causal processes robustly embodied withinclassical physics, because we will have examined specializedtechniques that have been sanitized of true causal processesfrom their very inception.15

Department of PhilosophyUniversity of Pittsburgh1001 Cathedral of LearningPittsburgh, PA 15260–6299mawilson+@pitt.edu

15. I began musing about these topics after discussing causation with Sheldon Smithand by reading Graciela De Pierris, ‘Hume’s Pyrrhonian Skepticism and the Belief inCausal Laws’, Journal of the History of Philosophy, Vol. XXXIX, No. 3, July 2001. Iam also indebted to the NSF for research support with respect to the foundations ofclassical mechanics.

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