Essay 6

Is There Life in Possible Worlds?

It might be imagined that some propositions, of the form of empiricalpropositions, were hardened and functioned as channels for such empiricalpropositions as were not hardened but fluid; and that this relationaltered with time, in that fluid propositions hardened, and hard onesbecame fluid. The mythology may change back into a state of flux, theriver-bed of thoughts may shift. But I distinguish between the movement ofthe waters on the river-bed and the shift of the bed itself; though there is nota sharp division of the one from the other.

---Ludwig Wittgenstein, On Certainty1

(i)

With respect to “possibility,” we are familiar with blandishments of thefollowing sort, captured in a well-known passage from Saul Kripke’s Naming andNecessity:

An analogy from school--in fact, it is not merely an analogy--will helpclarify my view. Two ordinary dice ... are thrown, displaying two numbersface up. For each die there are six possible results. Hence there are thirty-six possible states of the pair of dice... The[es] thirty-six possible states ofthe dice are literally “possible worlds”, as long as we (fictively) ignoreeverything about the world except the two dice and what they show.... Onlyone of these mini-worlds... is the “actual world”, but the others are ofinterest when we ask how probable or improbable the actual outcome was....“[P]ossible worlds” are little more than the mini-worlds of schoolprobability blown large.2

Scott Soames enlarges upon this claim in terms of maximality:A possible world is a possibleworld-state--a way that everythingcould have been. It is, in effect, amaximal property that the universecould have had. 3

We can picture such enlargements incoverage in the manner of a familiarcinematic trope: the camera begins by

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focusing upon Beaver Cleaver’s leafy house in Anytown, USA but then graduallypulls upward until his domicile becomes a wee speck glimpsed through thestratospheric haze. This adjusted panorama then successively gives way to adiminished earth, to a tiny solar system, to a speckled cluster of distant galaxies and soforth. Just so: the narrow parochialism of our “school possibilities” of our dice tossesopens out into the amply furnished contours of a full-fledged “possible world.” I callthe latter a globalized view of “possibility.”

By the same token–this issue will prove especially salient in the sequel–, wecan also expect to elaborate our “maximal worlds” with respect to their microscopicunderpinnings as well: supplying all of the details that the Incredible Shrinking Manwill witness in his telescoping descent to oblivion.

Viewing “possibilities” and “worlds” in this expansionist mode encourages afamiliar platform upon which a flattering portrait of linguistic capacity can be erected,a prospect that many contemporary philosophers have found compelling, to the pointof unshakeable conviction. I won’t linger upon these popular constructions now,although we shall discuss a few of them at essay’s end.

But self-congratulation with respect to our logical acuity is not a good thingbecause it rests upon a tacit overestimation of human conceptual capacity. In fact, Ibelieve (but cannot prove) that some of the chief divergencies between Wittgenstein’sTractatus and his later Philosophical Investigations reflect a major recalibration in histhinking about “possibilities.” The earlier book is founded upon the percept that the“logical possibilities” comprise a well-defined collection over which any competentspeaker possesses an absolute and wholly a priori command. But the laterWittgenstein appears to think that our sundry appeals to the “possible” and“impossible” are far more localized and revisable in their proper characteristics,anchored solely in the guiding but mutable contours of our present “forms of life.” Assuch, the confines of “what seems possible” within a particular sector of our thinkingprovides a useful “streambed” along which our localized reasonings can flow, with noguarantee future speakers will feel obliged to adhere to the same restrictions. Norshould we assume that the same principles of riverbed geology pertinent to certaincurrents of our thinking will likewise undergird the other “possibilities” we invokeelsewhere. In these respects, the phrase “logical possibility” represents somewhat ofa misnomer because the underlying motives for erecting a local “possibility space”will typically vary from quarter to quarter and evidence little of the sharedcommonality that the modifier “logical” invites.4

In a celebrated metaphor, Wittgenstein further suggests: Our language can be seen as an ancient city: a maze of little streets andsquares, of old and new houses, and of houses with additions from variousperiods; and this surrounded by a multitude of new boroughs with straightregular streets and uniform houses. 5

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We can extend this analogy by installing parochial“clouds of streambed possibilities” over eachprecinct, each playing an important role in shapingthe local economy, but without any presumptionthat these circumscribed “spaces” can beproductively (or even sensibly) combined intolarger, Kripke-like “completions.” I shall argue thatphilosophical expectations otherwise rest upon asubstantive misconstrual of the pragmatic utilities that localized talk of “possibilities”commonly facilitate.

We can observe the handiwork of these misconceptions quite vividly if wesurvey some of the opinions with respect to counterfactual conditionals that prevailwithin contemporary metaphysical thinking (that is, sentences of the general form:: “IfX were altered in manner Y, condition Z would result”). The “possible worldinflationism” presently under review encourages the presumption (already surveyed inEssay 5) that such counterfactual contentions must be grounded in claims of the ilk:“In all possible worlds wi where X is Y holds in wi but wi otherwise remains closelysimilar to our own world w0, Z will also obtain in wi.” Normally the vague phrase“remains closely similar to our own world w0" embraces the proviso that these variantnearby wi’s obey the same set of “fundamental physical laws” as prevail within ouractual w0.

Generally, the contours of the “grounding relation” invoked in claims like thisare left desperately vague (imo), but the general picture suggests that any speaker whoproperly grasps the meaning of an everyday counterfactual claim must implicitlyrecognize their underlying dependence upon a battery of “fundamental laws” (even ifthey possess nary a clue as to their precise features).

Again, we might quibble about what this hazy demand concretely requires of areal life speaker but, in advancing such claims, we should bear firmly in mind thefollowing consideration to the contrary: well-selected local “possibility spaces”commonly serve the important purpose of severing macroscopic descriptive claimsfrom substantive reliance upon the complexities of microscopic fact (thisobservation encapsulates the central theme of the present essay). If we are planning abuilding and don’t want it to fall down, we should favor the collections of descriptiveparameters and computational policies that best insure against injurious collapse. Except in delicate rare circumstances requiring heavy duty computer simulation,conventional engineering wisdom dictates that we should not attempt to model ourproposed edifice upon a molecular physics basis, but should instead seek well-established cutoffs within the realm of macroscopic fact that allow us to effectivelyefface our architectural reasonings from direct engagement with the unreliablevicissitudes of lower scale speculation. That is, we require a smallish set of

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descriptive terms X1 , X2, ...that will allow us to evaluate the potential behaviors ofour building in a trustworthy manner without worrying about its lower scalepeculiarities. Unfortunately, the specific X1 , X2, ...that can best perform this“effacement from the microscopic” task often prove subtle in their conceptualcontours and need to be especially adapted to the project at hand. It is at this juncturethat our “spaces” of counterfactual possibility enter the picture. Commonly thesecollections codify some rather obvious “possibilities” that we generate through simpleinduction upon experiment. But--and this is the big surprise-- they also supply us witha kind of “topological skeleton key” that allows us to squeeze out the X1 , X2,...parameters we desire A second happy benefit of this back-handed construction is thatthe descriptive reliability of the X1 , X2,... gets certified in the process.

All of this sounds frightfully abstract, but the relevant “assemblies of insulatingcounterfactuals” are generally comprised of very humble stuff, such as the basic factsof experimental twiddling on a macroscopic scale: “if I twist this little lever to theleft, it will move but not if I attempt to twist it to the right.” Any do-it-yourselferattempting to fix a toaster knows these “spaces of possible movement” quite

intimately, for they represent the collections of manipulation data wemust first explore before we can determine what the hell thisgoddamned gizmo is supposed to do within the device we’re trying torepair (expletives are required because the relevant possibility spacescan’t be mapped out without a lot of swearing). So this is theunexpected locus where the trail to reliable descriptive parameterscommonly commences: in innocuous-looking “spaces of manipulation

possibility.” Thus we find ourselves able to redeem Wittgenstein’s “guidingstreambed” analogy in unexpectedly sharp terms.

To be sure, if one attempts to extract these methodological morals fromstandard books on applied mathematics, one will immediately encounter aconsiderable quantity of “frightfully abstract” terminology, simply because the salientfacts rest upon an extensive array of deep mathematical results. But we’ll try toexplore this basic territory in a more down-to-earth manner.

This brisk survey invites a basic philosophical concern: can the folks whomaintain that “counterfactual claims must be ‘grounded’ in engorged ‘possibleworlds’” reconcile their autocratic demands with the methodological percepts justsketched? After all, the central utilities of our localized “possibility spaces” stemprecisely from the fact that they provide pathways to reliable sets of macroscopicdescriptive parameters with lesser need to consult lower scale principles (whichusually prove extremely unreliable when applied to large systems). At first blush, itappears as if the enthusiasts of “global possibilities” have inverted the true story of“how counterfactual claims make themselves useful,” at least for the localizedcollections of “possibility” commonly invoked within applied mathematics. I have

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encountered little awareness of these significant methodological concerns within theranks of those who insist upon topsy-turvy“groundings.”

With respect to the “enlargements into worlds” of our opening paragraph, thelittle collections of “possibilities” that help us carve out effective sets of X1 , X2,...”must usually remain quite localized in their contours (they can’t perform theirreductive tasks ably if they don’t) and rarely accept free enlargement in the fashionthat Kripke and Soames anticipate.

Setting aside the vagaries of what the demands for “grounding” and“enlargement”actually require, the positive manner in which smallish spaces of“possibilities” can facilitate important linguistic objectives enjoys a patentmethodological interest in its own right and robustly supports our metaphor of the“cloud of possibilities” that benignly shelter local sectors of a city (in the next section,we shall see that Wittgenstein may have been thinking of allied utilities in rathersimilar ways).

We should likewise follow the later Wittgenstein in presuming that blitheappeals to absolutist “possible worlds” inherently merit critical scrutiny. After all, thechief occupational disease of the academic philosopher lies in a propensity to fancythat one qualifies as a “master of all possibilities” (“just try to supply me with apossibility that I can’t already comprehend,” they challenge). Clearly unfettered talkof “possible worlds” can only encourage these regrettable tropisms (we fancy that wecan juggle a “possible world” as nimbly as Charlie Chaplin in The Great Dictator). Likewise, airy appeals to “fundamental laws” usually betray alingering acceptance of the simplified views of “scientificexplanation” that prevailed within the 1950s (Hempel’s “D-Nmodel,” etc.), a congruence of doctrines that I call “theory Tthinking.”6 As argued elsewhere in this book (especially Essay1), such reductive conceptions can prove harmful whenever thesharp details of practical inquiry become blurred within a foggyhaze of “theory T” distinctions. We are unlikely to benefit fromthe unexpected twists and turns of our scientific past if we are prone to flatteneverything out into smug coarseness: “Oh, I understand the general principles behindall of that; it’s just blah-de-blah all over again.” Thus the under-described philosophyof science of the 1950s provides a convenient cover for the modern metaphysicianwho wishes to speculate “on what science teaches us about the world’s ontology”without needing to bother with any of the actual grimy stuff. In section (vi), we willfind that, once the thin surface of those “counterfactuals must be grounded infundamental laws” claims has been scratched, the stock repressions of “theory T”thinking will emerge in a palpably wrongheaded form, with respect to several notablestratagems for installing reliability within a modeling.

If “philosophy” is to serve a useful purpose, we should steel ourselves against

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these insidious forms of self-deception. In reviewing the grand sweep of historicalscience (and all other forms of successful human endeavor as well), we should remaincontinually astonished at the amazing range of unexpected stratagems that cleverthinkers have uncovered in the rounds of managing practical affairs effectively. Asphilosophers, we should attempt to install some clarifying methodological orderupon some of it. But not at the cost of enveloping the truly remarkable within thehaze of under-analyzed “mush.” After all, methodological issues of an unexpectedbeauty frequently hide behind the surface tedium of the grimiest engineering practice.

Our concrete examples should warn us that questions of the laws andorganizational principles that govern a material on dimensions of length below thescale upon which we presently operate are apt to prove fraught, for greatmethodological surprises commonly await us whenever we drop from one descriptiveplatform to another (this observation supplies essay 4 with its primary theme). Theasking price of an suitable set of X1 , X2, .. tools is commonly the acceptance of acertain uneven complementarity in linguistic labor that I shall expnad upon in theessay to follow. This feature usually makes it hard to anticipate what transpires withinthe physical dominions below the characteristic lengths in which we presently operate,for such “complementarity” collapses many different forms of lower scalecircumstance into the same higher scale categories. In view of the ensuing opacityacross scale lengths, any presumption that the acute philosopher can nonetheless “see”through all of these clouded layers to some hypothetical “grounding” set beyond tradesupon an inadequate appreciation of the methodological delicacies that good sciencefrequently requires. Excessive diets of logical empiricist philosophy of science canencourage “X-ray vision” fantasies to the contrary, but we should never attempt toaugur the syntactic fate of future science in such a self-confident manner (offertimorous suggestion, perhaps, but not grandiose pronouncement). As Bob Battermanremarks, very pesky devils often reside within these tricky details of “uneven dataregistration.” But this observation comprised the chief theme of the essay previous.

Let me hasten to add an ameliorating comment with respect to the positiveutilities that “possible world models” provide in supplying useful characterizations ofthe behaviors of necessity operators and allied modal claims, following the generalpolicies of Tarski-style semantics (the modern enthusiasm for globalized “possibleworlds” initiated within these logical investigations). Little I report here reflectspoorly on any of these projects, if confined to their original dimensions ofphilosophical salience (although I have cavils about standard approaches tocounterfactuals that ignore large differences within the background scientificmodeling). The range of salient “possibilities” contemplated within a “semantic”study of an inferential practice can perform their roles just as well (or better) if theyare kept small and focused upon the applications close to hand.7 Unfortunately, in theeyes of many contemporary philosophers, the phrase “Tarski style semantics”

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Reuleaux

becomes inflated beyond its legitimate measure in the manner criticized here.

(ii)

In the discussion to come, we will adopt an investigative strategy suggestedwithin Wittgenstein’s pregnant, if not fully birthed, remarks Look for local packetsof guiding “possibilities” that appear peculiar in their contours and seek out thepositive utilities that induce these oddities. By “peculiar,” I intend counterfactualclaims that ask what happens under strange antecedent circumstances or “possibilities”that can’t be sensibly enlarged into a scientific fiction story of any kind, no matter howwild or otherworldly. Wittgenstein suggests such a case from his own engineeringbackground, which can be subsequentlyextended to some considerably deeperutilities that intimately entangle withthe phenomenon of encounteringdescriptive language that is intrinsicallytethered to different characteristicscale lengths. This central concernruns through virtually every essay inthis book, especially essay 4.

If we can successfully prosecutea variety of investigations of this ilk,we shall see how the powers ofpossible thinking can directly trace tothe data registration requirements ofa specific descriptive chore, rather than plucked from the a priori meadows of “alllogical possibility.” To this end, we might start with one of Wittgenstein’s ownillustrations (if I interpret it correctly), as it is relatively easy to explain. We can then move onto a related but richer example that providespotent lessons in how an effective descriptive practice can beintelligently engineered with a carefully selected array of “possibilities”that must be kept localized. Unfortunately, the demands of “effectivelinguistic engineering” being what they are, these examples contain afair number of moving parts and I must supply enough data that diligentreaders can align my discussion with the standard materials found in agood textbook on classical mechanics.8 Indeed, one of the greatestcomplaints to be justly laid at Wittgenstein’s door is that he frequentlyarrests his own expositions at the hand-waving stage (he gave a lot of excuses fordoing so, all of which should be received with a dissatisfied snarl).

The directly Wittgenstein-related considerations I have in mind have their

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home in the manner in which various sharply defined but rather artificial-looking“spaces of possible movement” gain their centrality within the subject of machinedesign, a subject whose underlying structure was greatly clarified by Franz Reuleauxin his The Kinematics of Mechanism of 1875 (which includes a long “philosophical”preface which we know that Wittgenstein read). Here is how we typically frame thelocalized “spaces” natural to machine design. Start with a basic classification of howthe joints that hold a devices’s parts together might move across each others surface:link a may be free to rotate around b in a single plane like a hinge or slide along agroove in b or move freely in two-dimensions as in a universal joint and so forth. Let’sfix upon a particular assembly of this type that forms itself into a closed kinematicchain, such as the basic “four-bar” design illustrated.7 One of the basic tasks of a pieceof machinery is to convert a certain type of input motion into some other type, as thelinkages in a locomotive’s gear convert the back-and-forth motion of the piston intothe output circular motion of the wheels. A non-trivial problem of this general class isto devise a set of linkages that will draw a prescribed curve C when a crank is turnedin a circle. A prototype device that can often accomplish these purposes fairly well iscomprised of four bars (called “links”) that are pinned together simply with hinges,with a little appendage rigidly attached to the “follower bar,” whose tip will inscribean output curve C as the input bar A is cranked in a circle around its axis (so thedevice converts circular input motion at the crank into the varied output motion C ). The illustration shows the wide variety of curves that can be drawing by hooking up apencil to the follower link in sundry ways. An important early application of this sortis that of developing a “parallel motion”: a mechanism that converts circular inputmotion into close-to-straight-line motion over a limited span of the crank turning. One of James Watts’ most celebrated innovations in steam engine design is a pretty 4-bar contrivance (his “parallel motion”) that traces an almost perfect straight linemotion as the rank is turned.8

All of the curves that a given device can possibly draw as we turn its crankcomprises part of its so-called “mobility space,” which is obviously a rather complexgeometrical structure. In essence, this “space” supplies answers to the class ofcounterfactuals of the form “If the follower arm were equipped with a pencil on anextension arm E, the pencil will inscribe the curve C.” But Reuleaux further showedthat this “mobility space” can be further enlarged to what is now called a “control” or“design space,” in which we consider the E to C relationships of a wider family ofpossible mechanisms that differ in the comparative lengths of their component links. Within the larger gray area pictured alongside, I have sketched three different Watts-type “parallel motions” that differ only in their “sizings.” Several major features

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should be noted within this representation. (1) The wood and iron of Watts’ actualinvention (pictured as lying outside the “control space”) has been replaced with asimple stick figure Reuleaux representation common to a wide variety of mechanisms

that look quite different in appearance but operate, as Reuleaux deftly showed, in thesame fashion qua motion converters. (2) There is a natural direction of designimprovement illustrated within the space that leads to a final sizing of parts thatrepresents the optimal way of achieving circular-to-straight-line-conversion withinthis class of possible mechanisms.9 It is striking that Watt managed to implement this“perfected sizing” in his actual invention. (3) Indeed, once the “possibilities” to besurveyed become represented in Reuleaux’ stick figure manner, it becomes evidentthat an algorithm10 is available that can find these optimal improvements so long as werestrict our attention to 2-D designs (more on this in a bit). Such methods usuallyproceed by “successive approximation” trial-and-error: in Wittgenstein’s day, thesesearches for optimality were pursued through a sequence ever improving sketchesexecuted on drafting tables whereas computers nowadays perform the same chores.

We here witness a primitive form of what applied mathematicians often call the“proper setting” for a physical modeling: a well selected “space” of restrictedpossibilities pertinent to finding the proper solution to a localized modeling practice. Commonly, the practical considerations that lie behind the construction of thesefocussed “spaces” are rather subtle and more of them will be explained in the pages to

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implementations ofa commonmechanism

design in parallelplanes

come (especially essays 7 and 8). It is very common that a “norm” (or “innerproduct’) is attached to these local spaces that allows us to measure the “closeness” ofa selected possibility to others (in our Watt example, the pertinent “norm” measureshow closely an output curve approximates straight line motion). A routine for designimprovement is contractive or coersive (they are synonyms) if each stage in theprocess reduces the error measured by the norm to as small a value as possible, inwhich case the routine is said to have obtained a fixed point (= it no longerrecommends any further improvements). I repeat a figure from essay 5 forconvenience.

Considered in their own right, without attention to thepractical objectives that frame their contours, the localized“possibilities” that Reuleaux collects together in this fashion canseem rather odd, especially if viewed from the perspective of a“possible worlds’ aficionado. Wittgenstein comments, apparentlywith Reuleaux’ techniques in mind:

And what leads us into thinking [like this]? The kind ofway we talk about machines. We say, for example, that amachine has (possesses) such-and-such possibilities of movement; we speak of theideally rigid machine that can only move in such-and-such way. --What is thispossibility of movement? It is not the movement, but it does not seem to be the merephysical conditions for moving either--as, that there is play between socket andpin, the pin not fitting too tight in the socket... The possibility of a movement is,rather, supposed to be like a shadow of the movement itself. And by a shadow I donot mean some picture of the movement--for such a picture would not have to be apicture of just this movement. But the possibility of this movement must be thepossibility of just this movement. (See how high the seas of languagerun here!) Indeed, a specific oddity of the “manipulation possibilities” we requireis that moving bars should be allowed to melt through one another asour prototype cycles through its “possible positions.” Why must weallow this? Well, at crude level, for algorithmic simplicity’s sake wecertainly want to group together devices that achieve identicalinput/output relationships even though their physical implementationslook quite different (this grouping is generally regarded as Reuleaux’great contribution to machine science). Because coersive entrapment isoften easier to achieve in two dimensions than higher, devices that canmapped onto a plane without loss offer an “easy design” opportunitythat is not readily replicated within a more tolerant class of mechanisms. In real lifeengineering, we can usually work around our “bars can’t really melt through oneanother” difficulties by implementing a two dimensional target design in offset layers:

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stacking up our linkages so that they maintain the underlying design functionalitywhile moving in parallel planes to one another. This trick allows us to not worrymuch about implementation while attempting to improve the sizing of a 2Dinvention–we can factor such worries away from the “possibilities” through which wesearch. These strategic remarks underwrite one of those astonishing apercus whichReuleaux’s work so amply provides. Why have most the mechanisms designed byman11 over the centuries operated in parallel planes? Because their devisers (Reuleaux cites James Watt as a prime exemplar) unconsciously reasoned along moreor less the same 2-D channels as our explicit improvement algorithms recommend,without realizing that they were doing so. He further stresses that some unconsciousrecognition of the formal landscape in which they should search supplies everyseasoned inventor with an “intuitive conviction” that there is both something specialabout “machines” in general12 and that certain designs are “perfect of their kind”(Wittgenstein frequently expresses an engineer’s admiration for the “perfectmechanism”). Reuleaux comments on the gradual manner in which mankind came toappreciate that special capacities for effective reasoning lay in the direction of“machinal ideas”:

Very gradually each invention came to be used for more purposes than thosefor which it was originally intended, and the standard by which itsexcellence and usefulness were judged was gradually raised. An externalnecessity thus demanded its improvement, and from this cause machinalideas slowly crystallized themselves out, and gradually assumed forms sodistinct that men could use them designedly in the solution of newproblems.13

Once the obscuring underbrush of “outer form” gets cleared away, the student ofmechanical invention will recognize the “essential nature” of a “beautiful mechanism”in all of its naked glory:

all carry[ing] on, partially before the bodily eye of the student and partlybefore the eye of his imagination, the same never tiring play. In the midst ofthe distracting noise of their material representatives they carry on thenoiseless life-work of rolling. They are as it were the soul of the machine,ruling its utterances--the bodily motions themselves--and giving themintelligible expression. They form the geometrical abstraction of themachine, and confer upon it, besides its outer meaning, an inner one, whichgives it an intellectual interest to us far greater than any it could otherwisepossess.

Note the telltale phrase “the soul of a machine.” To repeatedly arrive at close-to-optimal mechanical solutions, as Watt so frequently did, requires that the channels ofhis thinking must follow some rough hewn approximation to the pathways of searchthat Reuleaux later developed into explicit articulation. This implicit sense of “the

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Wittgenstein

right way to go about things” provides most systematic inventors with a psychologicalconviction that some underlying “machine essence” must lie in the background oftheir problems--some inarticulate notion of “optimal achievement” that makes aparticular design “feel right” or not. In my own studies, I have been frequentlyimpressed to discover that, across a wide spectrum of applications, our “commonsensethinking” has frequently anticipated the fundamental contours of many clever forms ofinvestigative strategy long before scientists have learned how the underlying searchstrategies should be articulated in formal terms (I will discuss these issues further inessay 10).

The origins of Wittgenstein’s less-than-pellucid phrase “logical grammar”appear to trace to lines of thought similar to Reuleaux’: that our intimations of“possibility” generally spring from an intuitive understanding of “the right things tothink about” in pondering a characteristic family of localized problems. In contrast,

the Tractatus was founded upon the same underlying presumptionthat animates contemporary “possible world” thinking: that we allinherently qualify as a priori “masters of absolute possibility”; thatthe very notion of the “information content” of a sentence rests uponour absolutist capacities to recognize which “possibilities” it rulesout and which “possibilities” it will accept. From this lofty point ofview, whenever we focus upon smaller “possibilities” in theconcrete and focussed manner of an inventor, we are merelygrouping together huge collections of amplified possible worlds in amanner convenient to the task at hand Such capacities rest at

bottom, upon our a priori recognition of the “variations in absolute possibility” that agiven term will accept (if the sentence “Archie loves Betty” reflects a coherentpossibility, then “Archie loves Veronica” must do so as well). In the Tractatus,Wittgenstein held that this unfettered knowledge of “possible variation” constitutes anessential aspect of the “logical grammar” of a term.14

Insofar as I can determine, the later Wittgenstein abandoned this absolutistconception of “possible variation” but continued to apply the term “logical grammar”in application to the more specialized flavors of “salient variation” that get naturallyspawned, Reuleaux-style, in the course of concrete objectives such as perfecting aroughly framed mechanical design. To become master of a trade frequently requiresthat we should, in Watt’s fashion, gain a firm control of a localized “possibility space,”even if we are unable to articulate its contours in articulate terms. Certainly, if weconcentrate upon our specific motives for “directing our thinking along locally usefulchannels,” we can better explain why it is useful to ponder “possibilities” at all, aconsideration that is not easy to address from an absolutist point of view. Regarded inthis light, the membership within our locally specialized spaces should be seen asgrounded within a variety of empirical and algorithmic considerations linked to our

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human size scale (the availability of robust solids and lubrication which allow us tobuild rough approximations to perfect pinned joints) together with the optimizingalgorithms available in 2-D for achieving practical design ambitions. From such autilitarian point of view, we shouldn’t expect that the “possibility spaces” we entertainwith respect to different aspects of the same target system will fit together nicely atall, let alone cohere into grand “possible worlds” of a Kripke-like stripe. Thus weshouldn’t expect that our parochial allowance for “parts that can melt through oneanother” will figure substantially in other discussions of “machine possibility” where“good design” no longer proves of paramount interest (and the algorithmic demands of2-D channeling no longer intrude). In this fashion, a set of practical congruenciesconnected with machine design naturally deposit a strong sense of “machine essence”and “possible variation” upon the vocabulary we employ in discussing such matters, tothe degree that we shouldn’t think that we have properly captured the “meaning” of“machine” until we have mapped out the reasons why these specific “possibilityspaces” are especially salient. Wittgenstein commented as follows:

But when we reflect that the machine could have also moved differently itmay look as if the way it moves must be contained in the machine-as-symbolfar more determinately than in the actual machine. As if it were not enoughfor the movements in question to be empirically determined in advance, butthey had to be--in a mysterious sense--already present. And it is quite true:the movement of the machine-as-symbol is predetermined in a differentsense from that in which the movement of any actual machine ispredetermined. 15

Such considerations allow Wittgenstein to retain a notion of a term’s “logicalgrammar” that still requires explication in terms of the “spaces of possibility”thatemerge in natural association with its usage, but which are now generated from someunderlying flavor of practical advantage pertinent to human proceedings , rather thanresting upon any a priori attachment to absolutist “possible worlds.”

And he can still maintain that the “grammar” of a term requires some skill ingenerating or recognizing the appropriate ranges of variation in “possibility,”even if anagent’s knowledge of these salient variations remains at the level of indistinct“instinct”as we find in Watt or Whitney.

The connection which is not supposed to be a causal, experiential one, butmuch stricter and harder, so rigid even, that the one thing somehow alreadyis the other, is always a connection in grammar. 16

By the same token, the utility of these basic channels can fade away if we somedaydiscover a pathway of ready improvement that works as effectively in threedimensions as two. Our intuitive impression of “machine essence” might shiftaccordingly and the reign of machines that work in parallel planes will come to an end. In this fashion, the channels of effective thinking may adjust over long periods of

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Lagrange

time, but they remain critical determinants of effective thinking within shorter periodsof time.

Such is my understanding of why Wittgenstein continues to employ the phrase“logical grammar”in his later writings, but I am not proposing that we should imitatethis practice. Indeed, at essay’s end I will indicate why I suspect that the phrase, inWittgenstein’s mouth, embraces themes that are quite antithetical to the point of viewI will develop here.

(iii)

An excellent exemplar of a localized set of peculiar possibilities can be foundin “virtual work” methodology briefly surveyed in the previous essay (I will not relyupon that discussion here, however). These widely employed techniques, whichrevolve around an unexpected set of “tweaked counterfactuals,” were codified byLagrange in his celebrated Mécanique Analytique, based upon the strategic insights ofmany who had come before. These methods were commonly heralded for theirintrinsic reliability by most of the great physicists of the nineteenth century and theirsuccesses made a huge impression upon philosopher/scientists such as Duhem andMaxwell.17 By “the reliability of a modeling methodology,” I simply have in mindthe sundry safeguards that insure that the buildings you design fall down less and thetheories of electricity that you develop hold up longer. As I have complainedelsewhere in these essays, reliability has not been highlighted with sufficient vivacitywithin contemporary philosophy of science, but a deep concern with trustworthinessrepresents a crucial hallmark of any enterprise that deserves the honorific science.

I will attempt to explicate the rationale underlying Lagrange’s procedures in assimple terms as I can devise but the full details may try some readers’ patience. Allthat I can offer by way of apology is the fact that a certain level of complexity may berequired in an example before the folly of inflating local possibilities into “possibleworld” monstrosities becomes patently apparent. Within science I know of manyexamples of “possibility spaces that must be kept small and controlled” (some ofthese will figure in later essays) but, of all of these, the underlying motives forLagrange’s approach are the easiest to explicate in concrete terms. Less-than-enthralled readers should be able to follow the gist of our discussionwithout mastering all of the nitty gritty details, but they shouldanticipate that the practical urgencies that render possibilityinflation an ill-conceived policy will somewhat diminish.

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On a number of occasions we have observed that a wise stratagem forheightening modeling reliability is to inject as much higher scale “physics avoidance”into our modeling techniques as we can, following the percept “if you already knowthat certain things are certain to happen within a complex system, utilize that partialknowledge as directly as you can, rather than insecurely attempting to recalculatewhat you already know with a significant risk that unwitting errors may be introducedthat spoil your proceedings entirely.” To be sure, there are circumstances where thismight not prove a wise policy, but most engineering practice to this day adheres to thispercept (you can be sued if you don’t). The “multi-scale” methods surveyed in Essay4 demonstrate the great computational advantages of mixing data drawn fromindependent sources (once “homogenization” tools have been found that permit thesea consistent sharing of information between these data pools). Well, Lagrange’s“virtual work” methods perform a similar task, but within a simpler setting wherea special set of “tweaked possibilities” allows us to effectively blend together datadrawn from two different scale sizes. In these treatments,18 we begin within aconfiguration spaces allied to the “mobility spaces” canvassed in the previous section,but in which we introduce applied forces to push the parts of a system around. whichembody a deeper symbiosis between a specialized “space” of counterfactual claimsand the practical utilities they facilitate.

The “higher scale knowledge” relevant here is registered in the form of so-called constraints: geometric restrictions on the possible mobility of the sundry partsof the target system. The two connecting bars in figure (b) constitute constraints thatkeep their pins at a constant distance from one another without restricting their abilityto follow the wire (the spring between beads 2 and 3 does not comprise a constraintbecause its length alters under applied forces and because it exerts active forces of itsown). The inflexible shape of the underlying wire in both figures supplies anadditional constraint that demands that all beads, wherever they may travel, mustremain upon the wire (allowing us to alwayslocate their positions with arc length q-numbers illustrated19). In truth, no real wire,no matter how stout, can avoid wiggling insmall measure when a bead-like thing slidesby but we intend to suppress such “backaction” effects entirely (certainly, upon amacroscopic scale of observation, the wiredoesn’t move at all).

For variety’s sake, I’ve furtheraccorded bead 4 an additional freedom totwist about its spring (θ), denied to the othersbecause of the constraints. We shall also

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assume that our beads can slide frictionlessly along their respective wires. Again,such behavior is impossible for real materials, but ceramic beads approximate suchconditions fairly ably over appreciable intervals. Later in the paper we shall scrutinizethe formal role of these “idealizations” more closely.

Thus far, our invocation of constraint knowledge is rather similar to that employed in the previous section, but we now want to see how this knowledge can beexploited within dynamic circumstances where we want to know how our beads-and-wire apparatus will move under the influence of gravity and other outside forces.

In the absence of friction, the total energy in the system should remain constantand if situation (a) sits within a constant gravitational field, then its single bead shouldcontinually shuttle energy between kinetic energy (= moving energy) and potentialenergy expressions. More specifically, any time this climbs up the wire through avertical distance Δh, the whole system will gain an increased potential energy V ofamount of mgΔh (where m is the mass of the bead and g is the terrestrial gravitationalconstant). If not obstructed, this stored energy will attempt to accelerate the bead by anamount -V(q)/q, leading to an increased kinetic energy. But the magnitude of thisconvertible V(q) energy will depend upon the wire’s orientation with respect to thevertical y at position q, for only the “active part” of the force illustrated can move thebead forward along the wire (its perpendicular “constraint force” component onlypushes the bead into the wire and hence becomes wasted and “performs no work”20). If the wire’s shape is supplied by a sufficiently simple formula, then calculating V(q)from the unfettered gravitational potential increase mgΔh is a fairly simple matter.

Note that the two locales in which our bead-and-wirecan remain stably at rest are easy to locate: they are found atthe two lowest points where the wire turns itself around. Ifour bead contains no surplus kinetic energy, it will stay ineither of those equilibrium positions forever, even though abead within the well on the left stores a bit moregravitational energy than a bead on the right, it will not beable to harness this energy unless it is released from thewire’s confines. Theoretically, a bead situated in themaximal energy storage position at the top of the centralhump might also remain stationary, but this is regarded asan unstable equilibrium because the bead is readily jostled out these circumstances. We should further note that by subjecting our target system to an additional appliedforce, any bead location whatsoever can be converted to an equilibrium position,simply by applying a resistance force to our bead of exactly the same magnitude as the“active force” generated by the potential energy, but pointing in the opposing direction. Indeed, we can experimentally map out the strength of our bead’s energy storagecapacities through directly measuring the strength of the force required to keep the

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bead stationary at every position.These rather innocuous observations will eventually serve as the basis of

the Lagrangian methodology under review. So far, so good. But let us attempt to bring this same perspective to bear upon

the more complex situation illustratedin figure (b) by seeking a suitablepotential energy function V(q1, ...q4, θ)that can govern the system in acomparable manner to our simple V(q)(the local acceleration applying to qishould be extracted V(q1, ...q4, θ)through partial differentiation in the qidirection). But the various connectingbars and springs complicate the

situation because our beads are now coupled together and lowering any one of them(say, bead 3) may sometimes increase the overall potential energy by pushing otherbeads upward and stretching or compressing the two connecting springs. This is whywe must usually include all of our q coordinates within our V-function, for storedpotential energy usually represents a holistic characteristic of the entire systemdepending upon how its sundry parts lock together.21

[Aside: readers who have read essay 4 may recall that the standing wave modesof a vibrating string likewise represent “locked together” behaviors that stretch acrossthe entire spring. The chief difference between those circumstances and the presentexample is a string possesses a large number of independent repositories into whichinput energy can be independently stored, whereas our beads-and-wire system (usually)has only one. In both cases, the “characteristic length scale” of the energetic storage isas wide as the entire target system.]

Let us first attempt to compute the appropriate V(q1, ...q4, θ) in a naive, bottomup manner through directly adding up the effective “active force” that would apply toeach bead at every possible position within the mechanism. Our across-the-systementanglements immediately generate a horrendous descriptive problem because, e.g.,bead 1's position will indirectly supply restraining forces upon all the other beads, byexciting intermediate forces within the wire, springs and rods that eventually reachtheir cousin beads. To calculate this force transference directly, we would need todecompose our erstwhile constraints into little atomic centers that are held together byextremely strong forces (which I’ve symbolized in the picture by a bunch of additionallittle springs), as well as adding in some additional microscopic gunk that can preventthe beads from adhering to the wire through the same strong forces that hold themtogether internally.22 A thoroughly force-based modeling of this type (if it could beproduced at all) involves many extremely speculative elements and will prove highly

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susceptible tonumerical error, for welack plausible classicalmechanical models forinterconnective forcesof this stripe. All thatwe confidently know istheir summed“dominant behavior”effects: that stiff wiresand rods don’t bend visibly if excited moderately within a target system of the typeconsidered. We’ll return to the problems of modeling the system in this entirelybottom up manner later in the essay.

Let us instead convert our “innocuous observations” concerning macroscopicrigidity into a sound policy of “physics avoidance” that allows us to bypass these lowerscale complexities through a deft exploitation of the already known behaviors of ourwire and connecting rods (strictly speaking, we only know their “dominant on amacroscopic scale” behaviors, because real rods will ripple minutely in a“subdominant” manner whenever a sound wave passes through). Observe that we canstill freeze our system into a state of static equilibrium (consistent with the constraints)

by applying enough additional forces23 to counter thepotential energy accelerations that would otherwise forceour beads to move (simile: these locally applied forceshold the whole apparatus in check in the same manner asthe little boy held back the ocean by inserting his fingerin the dike in the requisite way). Let us call this array ofextra forces <F1*, ...F4*, τ*> a “sweet spot”configuration for our system. How might we find themagnitudes of the forces required? Let us adopt a

strategic recommendation that often proves useful in science: characterize a localbehavior through consideration of what will happen under small perturbations of itscircumstances. Since a “potential” represents “a capacity for performing work ifunhindered” and because mechanical work is measured by the applied force multipliedby the displacement Δq through which it acts, let’s allow each Fi* to operate in over aperturbed distance Δqi. Focus first on bead 3. If the new applied force F3 (marked ingray in the diagram) possesses its requisite “sweet spot” value, consider what musthappen if bead 3 is wiggled by a small displacement Δq3 along the wire in eitherdirection. Answer: the old forces responsible for the potential energy storagecapacities of our system must conspire to push our target bead exactly back to q3. Articulated another way, if left unopposed, our new force F3 would perform a small

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amount of work F3 Δq3 in moving the bead to q3 + Δq3, but our device’s potentialenergy is precisely of a strength that can exactly counter this movement. As I stressedearlier, do-it-yourselfers commonly search for these “sweet spot” force balances whenthey twiddle with an unfamiliar mechanism. Accordingly, the “possibilities” withwhich are concerned are closely accessible to experimental manipulation (that factcomprised a chief theme within Essay 5).

The long and the short of all of this is that we possess a very robust (= reliablytrue) set of counterfactual claims about our device:

Fi applied at location qi represents the correct balancing force for holdingthe device in a “sweet spot” configuration <F1, F2 , ..., Fn> iff the workrequired to move bead i through a distance Δqi will be exactly matched byopposing amount of work arising from the potential energy of the system andthe other applied forces Fj already possess acceptable “sweet spot” values.

(strictly speaking, this “no net work” requirement needs to be tweaked slightly byadding the qualifier “no net virtual work,” for reasons I’ll explain in a moment)..

None of these counterfactuals allow us to find an overall “sweet spot” forcebalance directly, but they provide us with a suitable “search space” and a “norm” inwhich to find them. We do this through successive approximations of a type oftenencountered within these essays: start with an arbitrary initial quess <F1

0, ...F40, τ0> and

apply a small perturbed displacement Δq3 to coordinate q3, say. That shift will pull theother beads to new positions <Δq1,..., Δq5> (whichwe can calculate through a combination of rigidrod geometry and our knowledge of the Hooke’slaw behaviors of the springs). If the total workcontributed by these linked adjustments does notbalance to 0, then the other parts of the systemcannot hold our system in “sweet spot”equilibrium under the <F1

0, ...F40, θ0>

assignment and we should adjust our startingguess to <F1

0, .., F31, F4

0, τ0>, where F31 is a new

guess for the #3 bead that lessens the amount ofthe virtual work calculation. But we’re not yet done, because we must practice a policyof equal justice for every bead and apply the same test to the other components of our“sweet spot” force guess (returning once again to bead 3 after these adjustments havebeen installed). The result is a lengthy set of improving guesses Gi (usually non-terminating) that brings us ever closely to our desired “sweet spot” values.

In the jargon of the mathematicians, our virtual work test supplies us with anorm |Gi| that measures how far a given guess Gi departs from storing potential energyin a minimal manner

If we repeat this norm-guided computational routine over and over, we produce

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an improving sequence of guesses that gradually squeezes in on a desired “sweet spot”force allocation as a final “fixed point”limit (a “fixed point” is simply a position thatour norm is happy with). As an analogy, consider the manner in which the whisperingsof conscience can guide us along the Straight and Narrow: “Before you do that, haveyou contemplated this possibility?” “Ick,” we say and alter course. Repeatedsagacious steering of this type should convey us along a zig-zag path to Goodness. Thebeads-and-wire evaluative “norm” analogous to this “voice of conscience” is capturedwithin the biconditionals outlined above, for they compare possible configurations ofour system through the standard “if bead bi is wiggled by an amount Δqi, theadjustment will make the overall energetic situation better/worse for the rest of thesystem.” Such is the core data that we should collect into an appropriate “localpossibility space.” . Obviously, for strategies of this character to operate as intended, we must ensurethat the only “possibilities” we look at are of the right kind. If the Devil is allowed tothrow in some of his “possibilities,” we may find ourselves on the road to Perdition. The same reservations apply to the whisperings of our Lagrangian counterfactuals: ifwe tolerate any possibilities that do not conform to the mobility restrictions imposed byour constraints, we will lose the wherewithal to say “Ick!” at the right times.24 Andthere are some other restrictions upon the “possibility spaces” in which we conduct our“sweet spot” searches which I’ll mention later.

Plainly, the “guidance” inherent within these successive approximation searchesvery much resembles the backgroundconsiderations that allow an old-fashionedengineer to uncover an optimalimprovement of a machine design throughrepeatedly examining revised sizings upon adrafting table, in the manner sketched insection (ii). Accordingly, our “virtual work”norms provide a second concrete illustrationof how, e.g., our macroscopic certitude thatconnecting rods remain rigid (on amacroscopic, “dominant behavior” scale) supplies us with “channels” in which weallow us to establish other facts about our target system that are “not hardened butremain fluid.” Through these quasi ”a priori” channels we find the “physicsavoidance” tools that allow us to escape the computational extremities of modeling allof our system’s microscopic details from the bottom up.

Indeed, Lagrange showed us how our “sweet spot” data can be further exploitedto compute the behaviors of our beads-and-wire gizmo when it is autonomously left toits own devices and supplied with a surfeit of conserved energy (these policies relyupon the so-called “d’Alembert’s principle).25 How? Start with the system at t0 in

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initial configuration <q1,..., qn>. Find its corresponding“sweet spot” force allocation <F1,,..., Fn> and replace each of these by an accelerative factor Fi/mi where mi is the mass(or moment of inertia) of bead i. Then employ a standard “marching method”

algorithm26 to estimate where the beads must wind upafter a time interval Δt. If we keep our “step size”Δtsmall enough and reiterate this “marching method”computation over and over, we can plot the evolvingbehaviors of our beads-and-wire gizmo over long spansof time, without needing to bother with the largelyconjectural molecular forces that bind our systemtogether in tight constraints.

These preliminaries established, let us turn to theparticular aspect of Lagrangian technique that most clearly indicates the follies ofinflating useful local “possibilities” into “possible world” monstrosities. In truth, theset of counterfactuals we need for purposes is not exactly comprised of those that Iprovided, but need to be tweaked replacements with the puzzling qualifier “virtual”stuck in: “if we virtually lower bead bi through a distance δq, you’ll make the energeticsituation virtually better/worse for the other beads.” Now “virtual movements” are theconceptual bane of every novice in elementary physics and it is a rare beginner whohasn’t wondered, “What the hell is going on?” when the topic is initially broached (tobe sure, such cavils get quickly crushed under the heavy imperatives of “get yourhomework in on time”). Why must these odd qualifications enter our guidingcounterfactuals?

The answer lies in the fact that we are attempting to evaluate the potential energycontained in the system when our target bead is located at q, where the activegravitational contribution may be very small (most of its attractive force points into thewire and hence can only contribute a small portion of work in pushing the bead forwardalong the wire. In real life, however, if the bead is moved through the full arc δq, theactive gravitational force grows quite large (becoming nearly equal to its unfetteredgravitational value in our illustration) and so if we add up gravity’s actual work inmoving the bead from q to q + δq (viz., q q + δq Fdq), then we get an inappropriate valuefor estimating the value of the potential energy back at q. What we should properly dois add up constant copies of the original feeble force acting at q as we move through theδq arc, as this “virtual work” contribution (q q + δq Fδq27) provides a more accurateimpression of gravity’s potential energy contribution at q. And we must make similaradjustments for all of the other factors that contribute to the local potential energystrength (in our example, for the “virtual work” supplied by the two springs).

In other words, to operate correctly our various ”possibility counterfactuals”must be altered into a “virtual work” format that efficiently compromises between twofunctional demands: (1) to exploit reliable manipulation data that can be gleaned

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directly from experiment or from obvious facts such as the geometry of rigid rods and (2)to guide researchers to correct evaluations of the system’s hidden energy storagecapacities in a zig-zag fashion. This remarkable tweaked combination allows us tosidestep the insecure speculations that would have arisen if we had tried to model oursystem in purist “Newtonian” mode (that is, model all of the subterranean parts andforces that actually hold our gizmo together).28

Reliable answers--that’s not a virtue to be lightly sneezed at if you’re designing abuilding or bridge!

Of course, we are all aware that “obvious facts” display an annoying propensity tobreak down if we wait long enough. Certainly demon friction will slow down our beadsin fairly short order. However, it is easy enough to incorporate simple models forfriction into our Lagrangian scheme (I didn’t for simplicity’s sake) but other forms of“obvious fact” degradation cannot be so readily handled. After an excessive amount ofbead shuffling, our erstwhile rigid wire will begin to sag and our beads will begin todistort, in which case we must “open up the frozen degrees of freedom” we ignored inour “rigid wire and beads” calculations and consider more elaborate lower scale modelsof why the erstwhile “rigid part” assumptions have lost their behavioral dominance. Themulti-scale methods of Essay 4 represent the preferred modern methodology for tacklingthese lower scale damage problems without plunging directly into the unmanageableinsecurities of full “Newtonian” modeling.

(iv)

But notice: our tweaked “virtual possibilities” stoutly resist ready enlargementinto richer “possible worlds” of the type we discussed at the start of the essay. Thesentence “Jean-Louis applied virtual work to the pump handle” is unlikely to appearwithin even the wildest extravaganzas of science fiction fantasy (unless the authorsimply doesn’t understand what the term “virtual work” means29). This is because theorigins of these categories of “possibility” trace directly to the modeling utilities theyserve, rather than emerging as straightforward redactions from wider collections of apriori possibility. As such, they effectively illustrate the investigative recommendationwe extracted from the later Wittgenstein: “Look for the local packets of guiding‘possibilities’ that appear peculiar in their contours and trace these oddities to thepositive linguistic utilities that they facilitate.” As observed previously, the Tractatushad maintained that “There is a great a priori universe of ‘logical possibility’ from whichany localized specimen can be plucked” (and further asserted that we aren’t ready toemploy language at all until a complete mastery of this grand space of permissiblevariation has been acquired). But the latter Wittgenstein decided that this narrative getsthe true origins of “possibility talk” backwards, for non-degenerate possibilities must bebred and nurtured within the local channels of directive guidance that endow them with

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their practical utilities. Their occasionally odd markings (such as needing to be“virtual”) simply reflect the strategic contours of the stream beds in which they havebeen spawned and appear peculiar only if they are first encountered within a vaster oceanof amalgamated “possibilities” that has been fed by many inlet streams.30

This Wittgensteinian sentiment, registered in ichthyological metaphor, epitomizesmy own reluctance to rely heavily upon “possible world” tools in attempting tounderstand real life linguistic usage. I have already complained (in Essay 5) that theLewis-Stalnaker approach to counterfactual conditionals ignores important divergenciesin effacement policy to which applied mathematicians carefully attend (is the modelingof evolutionary character or does it concentrate upon equilibrium or steady stateattainment? How are the boundary and interfacial conditions set? Does the problemadmit of significant factoring? etc.) But if we wash away these significant structuraldifferences within an overly ambitious quest for a “general theory of counterfactualconditionals,” we are likely to erase the very factors that cogently explain why we careabout such claims at all. Such a circumstance should be evident from our study ofLagrangian method: it is only if we appreciate the detailed rationale whereby thetechnique exploits higher scale partial knowledge about energetic capacity that we canunderstand the utilities of framing these otherwise peculiar “virtual work” conditionals atall.

In point of fact, all modern books on mathematical modeling that pretend to anydegree of rigor at all set their problems within carefully delineated “function spaces”(which is the official title of the “possibility spaces” I have been discussing). Unfortunately, they are generally less forthcoming about why they do this, anticipatingthat mathematically sophisticated readers will see through the morass of technicalities tothe motivating reasons beneath (such as the desire to render the problem “well-set” in itsboundary conditions, which becomes a non-trivial concern when sophisticated notion of“trace deposit” are required–see essay 7 for more on this). As remarked before,unwanted “possibilities” (which might prove quite desirable in other contexts) need tobe kept out of the space lest our guiding “norm” get led astray. And the chief topic inessay 7 to follow is a peculiar “cooperative family” relationship between descriptiveelements that relies centrally upon a well-behaved norm and the topology it inducesupon a carefully selected “possibility space.”

In any event, insofar as the limited objectives of the current essay go, thesesupplementary “cooperative family” concerns again illustrate the importance of studying“possibilities” in terms of the descriptive functionalities that they provide as localizedcollections, most of which becomes substantially obscured if they are unwisely depositedinto the amorphous vastness of a universe of “possible worlds.”

As a further consideration along these lines, inflationist metaphysicians such asKripke or Soames appear to presume tacitly that classical physics models of the limitedscope we have considered can be amplified into classical universes replete on every

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the splendid but obsolete seagoingclock

scale of size and time. As I have argued elsewhere,31 this goal is probably unattainable ifwe solely rely upon “classical” modeling elements now familiar, for all known methodsof building up stable universes from stable molecules rely upon deeply quantummechanical phenomena such as tunneling to hold regular matter together.

In a nutshell, then, this is the chief methodological moral to be extracted from ourmorass of technical details: the skillful exploitation of well-crafted “spaces of localpossibility” directly founded upon our upper scale knowledgeoften allows us to identify important coherent structureswithin a target system far more easily than if we hadattempted to uncover them through purist bottom-upprocedures alone. That lesson in hand, let us return to thebroader philosophical themes with which this essay opened. Consider Wittgenstein’s remarks on the long term mutabilityof “channels”of directed thought that appear obligatory at thepresent moment. Plainly, there’s no better way to render acomputational pattern obsolete than uncovering newmethodologies that operate better over the same set ofpractical purposes. Sometimes the utilitarian salience offormerly vital computations can vanish altogether (the modernavailability of GPS technology has largely obliterated the“position at sea” concerns that once dictated how wars were waged and great fortunesaccumulated). Perhaps in years to come, our fussings with virtual displacements anddelta function impulses will seem as puzzling to future archeologists as the centralpatterns within Babylonian time reckoning now appear to us. The active shelf life of avibrant “necessity” is not forever, despite whatever “possible world” enthusiasts mightfancy to the contrary. And so “the river-beds of thoughts may shift.” Alliedconsiderations plainly lie at the root of Quine’s skepticisms with respect to both“permanent necessity” and “absolute possibility.”

Why didn’t Wittgenstein himself cultivate these Quinean consequences morevigorously himself? I’ll suggest a tentative answer in a bit, but let’s consider the moreurgent question of why so many contemporary philosophers have once again returned toa Tractatus-like faith in “possible worlds” that stand fully amplified in all of their details. We are, after all, creatures of a relatively small brain and the conceit that we cansomehow manufacture or otherwise control huge data sets of an infinite and a prioricomplexity should strike us as an odd fairy tale we have concocted, perhaps, to makeourselves feel good through exaggerating our mental capacities.32 A simple count ofavailable brain cells renders it more likely that we cultivate a host of localized andmildly disconnected skills in juggling various species of limited variation in a mannerthat advances our practical objectives usefully. The fact that some of the “localpossibilities” that aid us most--counterfactuals about virtual variations--do not happily

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enlarge into grander possibilities provides a significant symptom that something isdeeply amiss in grander accounts in which we somehow grasp dauntingly cosmic“possible worlds.”

So where does the illusion that we can do better comes from? Some of it traces tothe simple fact that sundry modifications of the standard “possible world” analyses ofmodal logic help us considerably in understanding common forms of linguistic behavior,with respect to both the grammatically of everyday sentences (which do, after all, talk of“necessity” and “possibility” a lot) and the inferential connections that obtain betweenthem. Surely, “possible world” talk must have latched onto to an important aspect oflinguistic usage?

I agree, but my more crabbed diagnosis would unfold as follows. There is agenuine set of psychological factors that I dubbed “the early a priori” in essay 5: theycapture the various forms of apparent “conceptual necessity” that can be associated withthe standard grammar of our language. As I stress on several occasions throughout theseessays, significant evolutionary advantages attach to an abilities to acquire large banksof inferential principle swiftly, usually in league with a paired set of “search forpotential counterexample” skills designed to evaluate whether such reasoning toolsmight lead us astray if applied in lengthy combinations (computer scientists often try outtheir algorithms across a range of well-understood situations to see ifany unexpected hazards in reasoning might appear).33 Acquiring afairly elaborate inferential technique swiftly in a quick and dirty way isoften superior to aiming for unblemished perfection right off the bat,for we can usually straighten out the ensuing kinks later. By the sametoken, it can prove advantageous to learn the majority of thesereasoning rules in a generic manner, allowing specializations to topic to develop later. Inthe light of the benefits of early uptake, it is not surprising that our acquisition of thesepreliminary reasoning frameworks should be inculcated in tandem with our mastery ofconventional grammatical parsing. Such an amalgamated inferential package lays downcomputational foundations that can be effectively adapted to novel purposes quiteswiftly. Amongst biological creatures, homo sapiens are notable for their astonishingconceptual plasticity and the inferential underpinnings that customary linguisticinstruction provides must play a significant role in facilitating this versatility. Whenthese grammar-enshrined interconnections get expressed in the form of sentences, theyare apt to strike us as indubitable, tracing to their underpinnings within our early a prioriarchitecture. Typically a strong psychology of classificatory confidence accompanies this sortof training, in the sense that we are predisposed to assume that we possess a suitablelabel for everything, no matter how unfamiliar. Supplied with a random blot, weconfidently label it “two entangled spider-like things.”34 We fancy ourselves greatconceptual “deciders,” in the unhappy mode of those scions of privilege who are certain

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that their “intuitions” will allow them to run a country capably even they don’t reallyunderstand how a lot of things work. Most likely this form of classificatory hubrisenjoys its own set of evolutionary advantages, but it also encourages the conviction thatour early apriori “necessities” count for more over the longer run than they properlyshould.

With respect to our everyday talk of “possibility” and “necessity,” the genius ofTarski-style “semantics” is that it adjudicates whether a desirable harmony between theinferential principles one accepts and the range of potential counterexamples one regardsas pertinent to the reasoning obtains or not. If your inferential permissions advise you toinfer S from S, but that transition appears problematic within your set of potentialcounterexamples, then an undesirable tension needs to be resolved, if possible. But the“necessities” with which we are largely concerned in this essay are more stronglylocalized in import (viz, our constraints: “this wire must stay rigid throughout ourproceeding”) and the potential counterexamples to be weighed in the short term aregenerally the limited “possibilities” cultivated within our humble–yet strategically well-motivated–localized “search spaces.” Viewed from this vantage point, the motives forstudying inferential behavior from a “semantic” point of view are, at root, no differentfrom the reasons why sophisticated modern engineers now set their modeling problemswithin carefully delineated “function spaces,” so that we apply the strongest “a priori”safeguards (in the humble scientists’ sense of “what we presently think we know”)available to protect against the potentially wild “conclusions” we may otherwise reachwhen we turn over large blocks of reasoning to our computers.

Such are the psychological facts on which conventional linguistics duly reportsand it is not surprising that the modeling apparatus of Tarski-style semantics prove aptfor the purpose.35 Yet none of these considerations alone should be regarded asespecially probative with respect to longer swatches of human experience, for essentiallythe reasons that Quine and Wittgenstein suggest. After all, the computational demandsof nature herself (= the modes of manipulating syntax that will operate most effectivelyas a means for dealing with the external world) often run counter to ouroriginalist expectations and so our “early a priori” presumptions oftenneed to be overwritten, in classic Neurath’s boat mode. My favoriteexample36: we acquire an early “understanding” of the word “rainbow”very swiftly through assimilating our employments to the generalgeometrical patterns we associate with words like “arch.” But these sameinferential expectations must get corrected by the time a speaker becomesan adult, for the underlying physical phenomena demand a different set of reasoningprinciples (making the corrections isn’t very hard). Most of the time, we rarely noticethese “inferential retoolings” in our everyday talk of household furniture and the like, butthey arise in a very robust way as soon as we survey the tergiversating behaviors ofclassificatory phrases such as “temperature” and “force” over larger stretches of scientific

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time. And this is why conventionally static views of “classificatory semantics”encounter significant problems whenever they deal with word behavior over evolvingspans of conceptual development.

Such corrective necessities also requires language to contain phraseology tofacilitate these requirements of linguistic adjustment (what I called “tools of linguisticmanagement” in Essay 5). With respect to “rainbow,” say, we must eventually shift achild’s conception of its salient “possibilities” from the standard storybook variations toa collection based in sounder meteorology (“‘reaching a rainbow’s foot’ isn’t apossibility applicable to real life bows, but appearing totally red is.”37). In this way, ourconception of “what is possible” significantly enlarges over time.” But it is a mistake topresume that all of these newly congregated “possibilities” must have lain latent in ourlanguage from its inception (just because you have enough insurance to cover a future caraccident doesn’t indicate that you have fully anticipated the wreckage). I believe thatthis is one of the key misconceptions that the later Wittgenstein laid at the Tractatus’door.

It is hard to be sure, but it strikes me that academic philosophy’s current attractionto inflated “possibilities” traces to a thesis about language and “truth conditions” that iscentral within the Tractatus: the “truth-conditions” of descriptive sentences ultimatelyreside in the manners in which they segregate the a priori possibilities into two groups:the “worlds” they accept and the “worlds” they reject. From this vantage point, itbecomes natural to expect that the underlying “semantics” of counterfactual claims mustbe anchored in similar discriminations amongst “possible worlds” and that any English-speaker competent in this mode of sentence construction must recognize how this“anchoring” proceeds. Throw in a dash of 1950's philosophy of science and you’re wellalong your way to a Lewisian account of the “meaning” of counterfactual claims.38 Butour focus on the canny injection of reliability into science suggests that this popular viewof “truth conditions” is improperly inverted. Adjudicated according to the tacit standardsof length and time scale that monitor their utterances, our everyday classifications ofwires and rods as rigid should be regarded as completely true, despite the fact that, if wedecide to shift our attention to other characteristic scales, we might describe the sameobjects in alternative ways (these shifting descriptive policies comprise the centraltheme of Essay 4). In an allied manner, the proper “truth conditions” of our peculiar“virtual work” conditionals should be seen as rooted in the manner in which we decideupon their truth: viz., inductions from experiment and simple upper scale calculationswhich are subsequently tweaked for “actual versus virtual work” effects. After all, it isonly by working downward, in alpine mountaineer style, from these firm anchors withinupper scale rock that science earns its justified trust.

In other words, when we learn to frame our peculiar “virtual work” possibilities asuseful guideposts to hidden coherent structure, we have thereby enlarged our priorconceptions of “what is possible” in novel ways, just as children eventually learn to

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correct the class of guiding “possibilities’ that they associate with “rainbow.” Pretendingthat we were aware of all of these possibilities beforehand simply because we are ableto appreciate their rationale once it is explained to us comprises the characteristichallmark of the blowhard who believes that he has already dreamed of everything in hisphilosophy. Such self-congratulation underestimates both the astonishing strategicinnovations that our clever forebears have uncovered and the comparable innovations yetto be mined from the further computational opportunities available in the world all aboutus.

However, as I’ve already confessed, I’m not entirely sure what the “grounding”assertions of our metaphysicians amount to. Underneath them all, I suppose, may onlylie some fuzzy expression of scientism in which I might well concur. If so, we confrontcircumstances reminiscent of the following. When I was a kid, our small town paperpublished any letter to the editor it received, with the net effect that lonely atheists fromPrescott, Arizona regularly wrote to proclaim that “Darwin has demonstrated that man isbut a lowly evolved worm.” These missives were promptly addressed by a localcontingent from Eagle Point that retorted, “No, man is not a lowly worm, for John 43:2saith....” And so it went on for years. Although I would quibble with their exactbiological allegations (proper lines of descent probably do not link hominoids withOligochaeta), I am not religiously inclined and would most likely sympathize with theArizona contingent in some bottom line fashion. But I am puzzled by their motives inmaintaining such a prolonged correspondence.

In much the same way, I might agree with some rock bottom naturalist sentimentthat underpins the claim that “counterfactual behavior is grounded in ‘fundamental law,’”although their accompanying commentary on scientific procedure seems invariably naiveand inappropriately focused. But I don’t understand how such weak tea can merit thevoluminous philosophical literature it has generated.

But as soon as we rise above this base level of banal scientism, we run the risk ofignoring crucial policies of effective linguistic engineering that require careful scrutiny. If we merrily dilute the focusing capacities of Lagrange’s special counterfactuals bydrowning them in a common ocean of “possibilities” drawn from every walk of life andinflate everything into “possible world” monstrosities through imitation of the policiesthat science fiction writers employ in expanding a wee conceit into a developed novel,then we have deprived ourselves of the resources needed to unravel philosophicalpuzzles that demand a more intense probing of the concrete methodologies thatunderwrite the invocations of “possibility.”

(v)

Although these conclusions reflect, in modern dress, the apparent resonancesbetween my own proposals on “possibilities” and the later Wittgenstein’s, I should

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ur-plant

immediately acknowledge that significant temperamental differences distinguish our twoveins of thinking. Indeed, it may well be that I have been quoting scripture entirely formy own degraded purposes, for I scarcely pretend to understand Wittgenstein in largebulk. Perhaps a concluding comment on how I would optimistically minimize the mostglaring discrepancies might prove useful. Elsewhere in these essays I talk much of thedescriptive opportunities that nature makes available to us: the patterns and strategies ofintegrated linguistic employment that allow us to reach practical conclusions effectivelyand swiftly within a suitable environment. Lagrange’s skillful techniques for exploitingthe strong partial information available to us in the simple fact that rods, strong wires andbeads don’t distort very much under the duress of everyday terrestrial traffic provides anexcellent illustration of how to mine nature’s proffered opportunities for fun and profit. Likewise the crisp reasoning pathways that Reuleaux extracted from the intuitivewilderness of effective invention trace to similar opportunistic underpinnings. At thesame time, the common prevalence of “semantic mimics” within the history of sciencewarns us that great surprises potentially await whether we attempt to improve a usagebeyond the tried and true (the appendix illustrates what I have in mind).

Thus I would typically equate Wittgenstein’s “channels of thought” and even his“language games” with the reasoning opportunities that nature makes available to us inthe manner of a Lagrange or Reuleaux, although such nuts-and-bolts identificationsundoubtedly fall far short of Wittgenstein’s overall vision of language (for which I do notpretend to offer any analog–I’m solely concerned with how effective descriptivepractices tend to form, evolve and sometimes fade away within the realms of materialclassification). No doubt, accounts of this “naturalistic” stripe would have struck thelater Wittgenstein as mortifying exemplars of the wretched “scientific” impulse to“explain human behavior,” whereas a philosopher proper should only seek“elucidations” of our practices. Insofar as I can determine, Wittgenstein is herecaptivated by the mystical vision of descriptive enterprise developed by Goethe in TheLife of Plants: the proper method of appreciating nature is to isolate a hypotheticalarchetype (the “ur-plant”) and to view all varieties of botanicalexpressions as variations upon that core theme. A proper grasp of thepermitted range allows us to pierce to the generative heart of Natureherself, but “explains” nothing. From this point of view, thephilosopher should only catalog the “variations” native to the sundry“forms of life” which seemingly represent the human analogs ofGoethe’s natural categories for Wittgenstein, eschewing anymisbegotten attempt to ground these policies within environmentalsetting and computational capacity, as I have done.

Well, this divergence in attitude isn’t surprising, given that Ivastly prefer Darwin for my biology to Goethe. And sixty years of inconclusiveWittgenstein scholarship suggest that the cultivation of scientifically well-understood

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examples of “hardened propositions” of the sort embodied within a Lagrangianconstraint might help focus our attention better, even if we never reach the master’s lofty(and probably rather mystical) heights. But the net conclusion is that no legitimateWittgensteinian imprimatur can attach to the opinions advanced here; their merits muststand or fall entirely on whether I have successfully explicated how certain classes of“hardened” counterfactuals earn their descriptive spurs.

Indeed, the appearance of “hardened propositions” in a book entitled On Certaintysuggests that Wittgenstein’s conception of “logical grammar” bears some strongerkinship to the notion of a “revisable a priori” as championed by Hans Reichenbach and,in a different form, by Michael Friedman (whereas I find the invocation of “grammar”for humbler purposes such as our own misplaced). But their emphases focus uponlinguistic determinations at the level of “what space-time metric should we employ?,”rather than upon the lowly task-focused concerns I have explored. Specifically, likeWittgenstein himself in the Tractatus, they view the permissible possibilities toleratedby a choice of geometry as providing an encompassing “net” to which the linguisticmoves we regard as “empirical” must be referred, even though future considerations maysomeday lead us to favor another choice of webbing. Such a point of view mightlegitimate some of the connotations of Wittgenstein’s use of “grammar” in a moresubstantive manner than I can. For myself, I am not convinced of the merits of “net”thinking and so few vestiges of neo-Kantian doctrine attach to the locally controlledranges of “possibility” I have stressed here. As a result, my own diagnosis of theprobative merits of Lagrange’s special set of partial knowledge counterfactuals, if raisedto methodological maxim at the elevated levels pursued by Reichenbach and Friedman,amounts to little more than the advice that, if you’re setting out on a carpentry project,you’ll be better off if you maintain a selective set of instruments in your toolbox anddon’t mix in a lot of kitchen utensils.

But every once in a while a three-pronged fork turns out to be just the ticket forprying unwanted gunk out of a crevice and with that discovery open new vistas forinvention and improved craftsmanship.

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Appendix: Lagrange multipliers and the origins of conceptual clashes.

There is an additional aspect of standard Lagrangian technique that is notespecially pertinent to the themes of the foregoing essay but which supplies an importantillustration of other themes pursued in this collection (and may even possess somesalience with respect to Wittgenstein interpretation). As we’ve seen, the basic genius ofLagrangian methods is that we canexploit our upper scale knowledgeof rigid wires, beads and so forth asan astute “physics avoidance” devicethat allows us to not worry aboutmodeling the complex internalprocesses that occur within our wireas it gets buffeted about by scuttlingbeads. But although we want toavoid modeling these interior features, we may still wish to estimate their features.Wires, after all, sometimes break under excessive loads and it would be greatly helpful ifwe had some means of reckoning the local stresses inside the wire under the burden ofour scuttling beads. Lagrange proposed a clever way of addressing these concernswithout falling into the “tyranny of scales” morass of modeling our system in acompletely bottom up manner. Here’s the general idea. Insert a little spring or two inthe spots in the wire where we want to estimate the local stresses and mark the strengthsof these hypothetic springs with a set of variables λ1, λ2, ... called “Lagrange multipliers.” Work out the motion of the system following the routine described above except that wetreat λ1, λ2,.. as undetermined constants in our computations (in the illustration ourpreviously rigid wire is now allowed to flex in three separate pieces). Now if these littleλ springs were very feeble, then our revised beads-and-wire-broken-into-pieces systemmight behave very differently than we had calculated before. However, we have nointention allowing that to happen, for we will now assign our multipliers the exact forcestrengths they require to pull our apparatus back into compliance with the rigid wireconstraints we had imposed originally. In other words, the strength of the λ1 springforce should be exactly the force required to pull the two ends back together, no moreand no less. The magnitude of these forces supplies our desired estimate of how stronglystressed the wire will be at the point of investigation. If λ1's magnitude grows too large,we should worry that the wire may distort or break under the imposed load, rather thancontinuing to act as a rigid constraint.

Formally, our multipliers serve as a cross-scalar “handshake” trick that allows ourupper scale constraints to communicate on a common mathematical level with the other

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forces (e.g., gravity and the original twosprings) invoked within our original modeling. In other words, our multipliers serve as a vitalmeans of communication between two scalesizes in somewhat the manner of the“homogenizations” discussed in essay 4. When we assign concrete values to our λ’s atthe end of our computations, we aremetaphorically tightening up our lower scale

description to suit the demands of our upper scale knowledge (which I’ve symbolized inthe illustration as higher scale tightening winches).

At first glance, it appears as if the “constraint forces” we can calculate throughthese methods should represent unproblematic cousins to the straightforward other forceswe have considered in our modeling, viz., gravitation and the Hooke’s law spring forces. But there is a crucial difference. The strength of the latter must be calculated before wecalculate the configuration of our system whereas we do exactly the opposite with ourconstraint “forces”: we know their strength only after we settle upon the system’sconfiguration. In that sense, “constraint forces” appear as after-the-fact hangers on. Or,to put the same issue in variant terms, the strength of the regular forces are generated bystraightforward force laws whereas the constraint “forces” are not. Worse yet, when weinvestigate these matters in a dynamic context, we find that our “constraint forces” mustadjust their attractive strengths according the velocities whereby the scuttling beads whizby. Foundationally, this requirement creates great headaches because the orthodoxassumption that Helmholtz (= Heinrich Hertz’ teacher) originally employed to derive theconservation of energy presumed that all forces could not be velocity sensitive.

What has occurred is a common developmental phenomenon that I discuss at(excessive) length in my Wandering Significance: initial employments of rule-governed“force” have become naturally extended to accept a projected-downward-from-an-upper-scale-constraint usage that turns out, upon close examination, to clash in subtle wayswith its parent applications (mathematicians callsuch “failures to return to home unaltered”anholonomies, of which the typical Riemann surfacefor a complex function supplies a splendidparadigm).

Nineteenth century physics then confronted adilemma: either exclude all constraints from physicsas merely “convenient approximations” or work onlywith constraint forces in ones physics and reach theconservation of energy via another route (which iseasy if the Newtonian style forces can be removed from our classical picture). In hisPrinciples of Mechanics, Hertz adopted the second policy, probably because earlier

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mechanists such as Maxwell had relied heavily upon constraints in assembling theirmodels of the electromagnetic aether. He commented upon his ministrations as follows:

When these painful contradictions are removed, the question of the nature [offorce] will not have been answered, but our minds, no longer vexed, will ceaseto ask illegitimate questions.39

If I understand him rightly, Hertz rejects the notion that anything useful can be gained byinternally scrutinizing the “concept of force” beyond considering its descriptive utilitiesand installing whatever repairs will insure that clashes of the sort mentioned can beavoided.

It is revealing, I think, that, for most of its career, the intended motto forWittgenstein’s Philosophical Investigations was the Hertz quotation just cited. Theunderlying sentiment seems to be that philosophical clashes arise, not because someword has gotten misapplied according to some violation of its absolutist, “possible worlddividing” significance, but simply because the shifting local contours of extendedapplication invest the word with altering inferential requirements.

In any case, if Wittgenstein didn’t view natural developmental patterns in thislight, I certainly do!

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1. On Certainty, trans. by Denis Paul and G.E.M. Anscombe (Oxford: Blackwell,1969), §§96-7.

2. Naming and Necessity (Cambridge: Harvard University Press, 1972), pp. 16-8.

3. Philosophical Analysis in the Twentieth Century, vol. II (Princeton: PrincetonUniversity Press, 2003), p. 355.

4. Insofar as I understand him, Wittgenstein employs the term “logic” in a fashionthat consigns the interconnections between a particular “space of possibilities” and itsconjugate descriptive parameters to the “logical grammar” that allegedly governs theseparcels of language. I don’t find such phraseology useful and fail to see how “logic”(in any reasonable sense) plays a substantive role in the formal considerations we shallcanvass.

5. Philosophical Investigations

6. In his celebrated report on turbulence (“On Vortex Motion” p. 185), OsborneReynolds wrote:

It would seem that a certain pride in mathematics has prevented thoseengaged in these investigations from availing themselves of methods whichmight reflect on the infallibility of reason.

An allied “pride in logic” appears responsible for the “theory T” shortsightedness ofmuch contemporary philosophy.

7. Correctness proofs

8. Greenwood

7. The surprising subtleties of the term “mechanism” are not adequately appreciatedby many philosophers, who would benefit from Reuleaux’s deep insights into thesubject. I’ve amplified on some of these issues in my “What is Classical MechanicsAnyway?”

8. In Watt’s steam engine, the conversion operates in the converse direction: straightline motion delivered by the piston gets converted into a circular arc at the pump arm.

9. Close study reveals that the output tracing is not perfectly straight, but only suppliesthe best fit to be achieved within this class of possible inventions (“best fit” can beplausibly measure in several different “norms” but they all agree on the optimality ofthe Watts’ sizings). In the 18?, the French Engineer Peaucillier showed thattheoretically perfect conversion can be achieved employing the six componentmechanism pictured alongside Watt’s in a different “basin of mechanical design.” One would greatly like a “perfected design” search algorithm that could leapfrog out

Endnotes:

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of the Watt design space into that of Peaucillier, but, to the best of my knowledge nosuch method has ever been found. Even back in the 1870s Joseph Sylvestercommented (as condensed by Eugene Ferguson)

The perfect parallel motion of Peaucellier looks so simple and moves soeasily that people who see it at work almost universally expressastonishment that it waited so long to be discovered... [But reflecting furtheron the problem, one] wonders the more that it was ever found out, and cansee no reason why it should have been discovered for a hundred years tocome. Viewed a priori there was nothing to lead up to it. It bears not theremotest analogy (except in the fact of a double centring) to Watt's parallelmotion or any of its progeny."

Sylvester quoted in Eugene Ferguson Pamphlet on mechanism

10. S. Molian, Mechanism Design, (Oxford: Elsevier Science, 1997), pp. 15-8.

11. Natural selection. Manmade exceptions to our parallel plane policy were oftenborrowed directly from the 3-D prototypes that nature managed to find through bruteforce, “hook or crook” exploration.

12. Reuleaux on Newton; pertinence to other issue sin this book.

13. p. 231

14. Notoriously, the importance of this “grammar” can only “show”itself, it can’t becoherently talked about.

15. Philosophical Investigations §§ 193-4.

16. Remarks on the Foundations of Mathematics §128.

17.many features of our discussion will be patterned along lines that Duhemarticulates in his classic The Evolution of Mechanics standard continuum mechanicstries to incorporate as much data extracted from direct laboratory testing into itsmodeling efforts as it can, due to the palpable superiority of Cauchy’s “top down”methods displayed over Navier’s cognate “bottom up” efforts.

18. I.e., a classical mechanics modeling involving both Newtonian forces and appliedconstraints (being connected by a rigid rod, sliding along a wire). The “Lagrangians”of quantum physics rarely involve “constraints” in this sense and hence do notillustrate the cross-scalar applications that supply classical Lagrangian methodologywith its peculiar strengths.

19. Later we will also allow our bead to rotate and this supplies it with an additionaldegree of freedom θ. Center of mass and infinitely small. Discuss.

20. Always considered this a misleading phrase.

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21. note on instantaneous energy drainage As such, this “holistic” behavior simplyreflects the fact that, in classical physics, potential energies can rarely be correlatedwith specific spatial locations and can be extracted as work from any positionanywhere within the span of the mechanism.

22. In real fact, without intervening lubrication, metal beads actually bond tightly tometal wires. It is common to refer to such a force-based modeling as “Newtonian,”although Newton himself frequently invoked constraints in a time tested “physicsavoidance” ways. Internal forces are “Balanced and do no work”

23.“Forces” is set in quotation marks because a central advantage of Lagrangianmethods is that they can treat applied pushes that are not true forces as if they were (inwhich case, they are called “generalized forces”). Thus the “τ” coordinate in our 5-tuple < F1, F2, F3, F4, τ> reflecting bead 4’s ability to rotate must be a torque, not aforce, and its corresponding “mass” will be a moment of inertia.

24. Continuing this theological analogy, in the next essay we’ll find that often themembership of the sequences that lead most directly to the desired fixed point cannotenter this blessed kingdom themselves, but, like Moses (or Ethan Edwards in TheSearchers), can only guide other pilgrims there. A proper “norm” is central to thiscommon “cooperative family” behavior.

25. In standard jargon, Lagrangian methods employ “d’Alembert’s principle” toconvert knowledge of a system’s possible static behaviors into knowledge of its freemotions.

26. Such as the simple Euler’s rule provisions of Essay 1.

27. In the jargon of the mathematicians, “Fdx” signals an “exact differential” in whichF’s value remains a function of q, whereas “Fδx” turns off this coupling.

28. Reference Duhem here

29. Of course, science fiction being what it often is--imaginative expansions of sundryur-philosophical conceits, no matter how defective--, someone will be sure to doexactly that, if they ever stumble across this particular passage of mine

30. All of this suggest that “Which naturally occurring forms of localized “possibility”readily accept enlargement into fuller stretches of science fiction?” represents anintriguing psychological question (which I can’t answer). But one must first recognizethat it requires a nontrivial diagnosis..

31. Me on stability Leib book

32. add some comment about second pilgrim

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33. In more concrete terms, our understanding of “how to reason in this language”represents a rather unruly mixture of rules of permitted syntactic manipulation,counterexample safeguards against trusting rash conclusions reached in this manner(such safeguards form the proper wellspring of “semantics” in my opinion) and roughheuristics for harnessing the foregoing tools to useful purpose.

34. In the Mel Brooks/Ernie Pintoff cartoon from which I borrowed the blot, the“Critic” in question identified it as “two things having sex.”

35. As noted above, I feel less sanguine about the Lewis-style “semantics” oftenproposed for counterfactuals, at least for the classes under consideration here.

36. Wandering Significance (Oxford: Oxford University Press, 2006), pp. 22-4.

37. In the spirit of old soldiers never die, but merely fade away, storybookpossibilities rarely relax their grip on “rainbow” completely, but their capacity to arrestsound reasoning as “counterexamples” lessens. WS

38. Perhaps a brisk recollection of one of the incidents that prompted this essay mightprove helpful. My colleague Jim Woodward told me that his fine work on causalreasoning is often disparaged by “metaphysicians” because his basic manipulationcounterfactuals are not properly “grounded in fundamental law” (Jim’s target claimsare closely allied to the “mobility spaces” we have studied). And I thought, “Butthat’s crazy; the strengths of those statements reside precisely in the fact that they canbe induced fairly directly from experiment and thus serve as a vital instrument forinjecting firmer reliability into our scientific contentions.”

39. Hertz Look UP