Essay 7

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 possible world-state--a way that everything could havebeen. It is, in effect, a maximal property that the universe could have had. 3

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imbedding a limited possibility within a full “world”

We can picture such enlargements in the manner of a familiar cinematic trope: thecamera begins by focusing upon Beaver Cleaver’s leafy house in Anytown, USAbut then gradually pulls upward until his domicile becomes a wee speck glimpsedthrough the stratospheric haze. This adjusted panorama then successively gives wayto a diminished earth, to a tiny solar system, to a speckled cluster of distant galaxiesand so forth. Just so, the narrow parochialism of our “school possibilities” of ourdice tosses opens out into the amply furnished contours of a full-fledged possibleworld. FIG: IMBEDDING A LIMITED POSSIBILITY I call the latter aglobalized view of possibility.

By the same token–this issuewill prove especially salient in thesequel–, we can also expect toelaborate our maximal worlds withrespect to their microscopicunderpinnings as well,encompassing all of the minutedetails that the Incredible ShrinkingMan will witness in his telescoping descent into infinitesimal oblivion.

Approaching “possibilities” and “worlds” in this inflationist mode provides afamiliar platform upon which a flattering portrait of intellectual capacity can beerected, a prospect that many contemporary philosophers have found compelling, tothe point of unshakeable conviction. I won’t linger upon these popularconstructions 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 betweenWittgenstein’s Tractatus and his later Philosophical Investigations4 reflect a majorrecalibration in his own thinking about possibilities. The earlier book is foundedupon the percept that the logical possibilities comprise a well-defined collectionover which a competent speaker possesses an absolute and wholly a prioricommand. But the later Wittgenstein appears to think that our sundry appeals to the“possible” and “impossible” are far more localized and revisable in their propercharacteristics, anchored solely in the guiding but mutable contours of our present“forms of life.” As such, the confines of “what seems possible” within a particular

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Wittgenstein’s city and its localpossibilities

sector of our thinking provides a useful “streambed” along which our localizedreasonings can flow, with no guarantee future speakers will feel obliged to adhere tothe same restrictions. Nor should we assume that the same principles of riverbedgeology pertinent to certain currents of our thinking will likewise undergird the otherpossibilities we invoke elsewhere. In these respects, the phrase “logicalpossibility” represents somewhat of a misnomer because the underlying motives forerecting a local possibility space will typically vary from quarter to quarter anddisplay little of the shared commonality that the modifier “logical” invites.5

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.6

We can extend this analogy by installing parochial clouds of streambed possibilitiesover each precinct, each playing an important role in shaping the local economy, butwithout any presumption that these circumscribed spaces can be productively (oreven sensibly) combined into larger, Kripke-like completions. FIG:WITTGENSTEIN’S CITY AND ITS LOCAL POSSIBILITIES I shall argue thatphilosophical expectations otherwise rest upon a substantive misconstrual of thepragmatic utilities that localized talk of possibilitiescommonly facilitate.

We can observe the handiwork of thesemisconceptions vividly if we survey some of theopinions with respect to counterfactual conditionalsthat prevail within contemporary metaphysicalthinking, viz. sentences of the general form “If Xwere altered in manner Y, condition Z wouldresult.” The possible world inflationism presentlyunder review encourages the David Lewisian presumption that such counterfactualcontentions must be grounded in claims of the ilk: “In all possible worlds wi whereX is Y holds in wi but wi otherwise remains closely similar to our own world w0, Zwill also obtain in wi.”7 Normally the vague phrase “remains closely similar to ourown world w0" embraces the proviso that these variant nearby wi’s obey the sameset of fundamental physical laws as prevail within our actual w0.

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Generally, the contours of the grounding relation invoked in claims like thisare left desperately vague (imo), but the general picture suggests that any speakerwho properly grasps the meaning of an everyday counterfactual claim mustimplicitly recognize their underlying dependence upon a battery of fundamentallaws, even if they possess nary a clue with respect to their precise features.

Again, we might quibble about what this hazy demand concretely requires ofa real life speaker but, in advancing such claims, we should bear firmly in mind thefollowing consideration to the contrary. Well-selected, localized possibility spacescommonly serve the important purpose of insulating macroscopic descriptive claimsfrom substantive reliance upon the complexities of microscopic fact--thisobservation encapsulates the central theme of the present essay. If we are planninga building and don’t want it to fall down, we should favor the collections ofdescriptive parameters and computational policies that best insure against injuriouscollapse. Except in rare and delicate circumstances requiring heavy duty computersimulation, conventional engineering wisdom dictates that we should not attempt tomodel our proposed edifice upon a molecular physics basis. We should insteadseek well-established constraints within the realm of macroscopic fact that allow usto effectively efface our architectural reasonings from direct engagement with asmany of the unreliable vicissitudes of lower scale speculation as we can. We seeksmallish sets of descriptive terms X1 , X2, ...that allow us to augur the futurebehaviors of our building in a trustworthy manner without worrying needlessly aboutits lower scale peculiarities. Unfortunately, finding the appropriate X1 , X2, ... thatcan implement this effacement from microscopic description is rarely easy. Thetraits required are commonly abstract in their conceptual contours and need to bespecifically tailored to the system at hand. But it turns out that well-selected spacesof localized counterfactual possibilities can help us locate the recondite X1 , X2, ...that we seek. Often these guiding ensembles consist in entirely uncontroversial setsof manipulation-based counterfactuals that we can readily verify through simpleinduction upon macroscopic experiment. In doing so,--and I find this capabilityquite striking-- they can supply us with a topological skeleton key (or “norm”) thatsqueezes out the X1 , X2,... parameters we seek.

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learning counterfactuals

All of this sounds frightfully abstract, but, in applications, the relevantassemblies of insulating counterfactuals are commonly comprised of utterly humblematerials, such as the basic facts of experimental twiddling on a macroscopic scale:“if I twist this little lever to the left, it will move but not if I attempt to twist it to theright.” Any do-it-yourselfer attempting to fix a toaster knows these spaces ofpossible movement intimately, for they represent the collections of manipulationdata we first explore as we determine what the hell this goddamned gizmo issupposed to do within the device we’re trying to repair(expletives are required because the relevant possibilityspaces can’t be mapped out without a lot of swearing). FIG: LEARNING COUNTERFACTUALS So here iswhere the unexpected trail to reliable descriptiveparameters often commences: within spaces ofinnocuous-looking manipulations. Through examplesof this sort, we will find ourselves able to redeemWittgenstein’s guiding streambed analogy inunexpectedly sharp terms.

To be sure, if we attempt to extract these methodological morals fromstandard books on applied mathematics, we will encounter a considerable quantityof frightfully abstract terminology, simply because the salient facts rest upon anextensive array of deep mathematical results. But we’ll attempt to explore the basicmethodological territory in a more down-to-earth manner.

Our little survey should prompt a basic concern about philosophical practice. Can the folks who demand that counterfactual claims be grounded within inflatedpossible worlds reconcile their demands with the methodological percepts justsketched? After all, the central utilities of the possibility spaces we shall examinestem precisely from the fact that they allow us to evade reliance upon lower scaleprinciples of exactly the sort that are said to ground counterfactuals. Withinstandard engineering practice, the Lagrangian methods we shall investigate areprized precisely for their ability to eschew unreliable dependencies upon materialsof that type. From the point of view of raw descriptive accuracy, the enthusiasts ofglobal possibilities appear to gotten the the story of many counterfactual claimsmake themselves useful backwards. Yet I have never encountered any recognitionof these discrepancies amongst the philosophers who insist upon law-based

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mastering possible worlds

a topsy-turvy

groundings.With respect to the enlargements into worlds of our opening paragraph, the

collections of localized possibilities that lead us to the effective sets of descriptivevariables X1 , X2,... must usually remain compact in their contours and rarely acceptfree enlargement in the fashion that Kripke and Soames anticipate.

Setting aside the vagaries of what the demands for “grounding” and“enlargement” actually require, the positive manner in which selective spaces ofpossibilities facilitate important descriptive objectives is interesting in its own rightand the underlying considerations robustly support our metaphor of the clouds ofpossibilities that benignly shelter local sectors of a city. The chief exemplar weshall discuss is the manner in which simple macroscopic counterfactuals are centralto Lagrange’s celebrated virtual work techniques but we shall first consider asimpler example which Wittgenstein himself may have in mind.

Writing from a more general perspective, I hope that our discussion willprompt others to become more circumspect in their warm embrace of notions suchas “possible world.” We should follow the later Wittgenstein in assuming that off-handed appeals to absolutist possibilities merit critical scrutiny. After all, the chiefoccupational disease of the academic philosopher lies in our propensities forfancying that we qualify as masters of all possibilities (“just try to supply me with apossibility that I can’t already comprehend,” we challenge). Clearly, unfettered talkof possible worlds can only encourage theseregrettable tropisms, if we imagine that we canjuggle these worlds as nimbly as Charlie Chaplindoes in The Great Dictator. FIG: MASTERINGPOSSIBLE WORLDS In a related way, thedemands for “grounding in fundamental laws”displays an uncritical faith in the coarse structuraldistinctions that were canonically formulated by thelogical empiricists in the 1950's and whose obscurant foghovers over us still. I call these patterns of thoughtTheory T thinking and every essay in this book criticizesthese presumptions from one angle or another.8 Theseconceptions stem from twentieth century attempts toexplicate the central ingredients of scientific explanation

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in logic-centered terms, but the generic discriminations characteristic this thinkingdo not render justice to the strategic subtleties of real life practical inquiry. Excessive faith in Kripke-like possible worlds have only made matters worse, forthese appeals protect and preserve the basic ingredients of Theory T thinking fromcritical scrutiny.9 Approaching the counterfactual claims of science from this basicorientation quickly leads to a topsy-turvy inversion of their true purposes andobscures the scientific utilities they perform on our behalf. FIG: A TOPSY-TURVY

Let me hasten to add an ameliorating comment with respect to the usefulcharacterizations that possible world models have provided with respect to thebehaviors of many forms of modal operator, following the general policies ofTarski-style semantics (modern enthusiasms for globalized possible worlds initiatedwithin these logical investigations). Little I report here reflects negatively on theseprojects, as long as they remain confined to their original dimensions ofphilosophical salience. The range of salient possibilities contemplated within a“semantic study” of an inferential practice can perform their justificatory roles aswell (or better) if these possibilities remain kept small and focused upon theapplications close to hand. In the eyes of many contemporary philosophers,however, the phrase “Tarski style semantics” has become inflated far beyond itslegitimate ranges of utility.10

(ii)

In the discussion to come, we will adopt an investigative strategy suggestedwithin Wittgenstein’s pregnant, if not fully birthed, remarks. We shall look forlocal packets of guiding “possibilities” that appear peculiar in their contours andseek out the positive utilities that induce these oddities. By “peculiar,” I intendcounterfactual claims that ask what happens under strange antecedent circumstancesor possibilities that can’t be sensibly enlarged into a science fiction story of anykind, no matter how otherworldly. If we can successfully prosecute a variety ofinvestigations of this ilk, we shall learn how the powers of possible thinking tracedirectly to the data registration requirements of specific inferential chores, ratherthan plucked from the a priori meadows of universal logical possibility.

To this end, we can start with one of Wittgenstein’s own illustrations drawn

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Reuleaux

from his engineering background, as it is relatively easy to explain. We can thenmove onto a related but richer example that provides truly potent lessons in how acarefully pruned and tweaked array of possibilities can benefit a descriptivepractice. The demands of effective linguistic engineering being what they are, thissecond example involves a fair number of moving parts. Accordingly, I need tosupply enough background that diligent readers can align our discussion with thestandard materials found in a good textbook on these subjects.11 A just complaintto be laid at Wittgenstein’s door is that he frequently arrests his own expositions atthe hand-waving stage (he gives a lot of excuses for doing so, all of which should begreeted with a dissatisfied snarl).

The Wittgenstein-related considerations I have in mind find their home in themanner in which sharply delineated, but rather artificial-appearing, spaces ofpossible movement gain centrality within the subject of machine design, a subjectwhose underlying structure was greatly clarified by Franz Reuleaux in his influentialThe Kinematics of Mechanism of 1875, a book that Wittgenstein in known to haveread.12 FIG: REULEAUX Here is how we typically frame the localized spacesnatural to machine design. Begin with a basic classification of howthe joints holding a device’s parts together move across eachothers surface--link a may be free to rotate around b in a singleplane like a hinge, slide along a groove in b, move freely in two-dimensions as in a universal joint and so forth. We then considerthe assemblies of parts that form into closed kinematic chains, suchas the gizmo illustrated.13 Devices of this type redirect circularmotion applied to its crank into quite different output applications,just as the linkages within a locomotive’s gearing convert the back-and-forth motion of the piston into the circular motion of thewheels. A non-trivial task within machine design is that ofdevising a set of linkages can draw a prescribed curve C when a crank is turned in acircle. A prototype device that often accomplishes these purposes fairly well iscomprised of four bars (also called links) pinned together with hinges, with a little

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motion conversion in a 4 bar mechanism

appendage rigidly attached to the followerbar. Its tip will inscribe an output curve Cas the input bar A is cranked in a circlearound its axis, with the result that our 4-bar device converts circular motion at thecrank into the complex curve C. FIG:MOTION CONVERSION IN A 4- BARMECHANISM As the illustrationdemonstrates, a considerable variety ofcurves that can be drawn by hooking up apencil to the follower link in sundry ways. An important early application of thesetechniques is that of developing a so-called parallel motion, a mechanism thatconverts circular input motion into nearly straight line motion over some angularspan of crank turning. One of James Watts’ celebrated innovations in steam enginedesign is a pretty 4-bar contrivance of this type, called “Watt’s parallel motion,”that traces a nearly perfect straight line motion as the crank rocks through a limitedangle.14

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 often 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 would inscribe the curve C.” But Reuleaux furtherdemonstrated that this mobility space can be further enlarged to what we now call acontrol or design space, in which we consider the E to C relationships of a widerfamily of possible mechanisms that differ in the comparative lengths of theircomponent links. Within the larger gray area pictured alongside, I have sketchedthree different Watts-type parallel motions that differ only in their sizings. FIG:BASINS OF OPTIMAL DESIGN IMPROVEMENT Several major features shouldbe noted within this representation. (1) The wood and iron of Watts’ actualinvention (pictured as lying outside the control space proper) have been replacedwith a simple stick figure representation common to a wide variety of mechanisms

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working out thepossibilities

basins of optimal design improvement

that look quite different in appearance but operate, as Reuleaux showed, in the samefashion qua motion converters. (2) There is a natural direction of designimprovement illustrated within the control space that leads to a final sizing of partsrepresenting the optimal manner of achieving circular-to-straight-line-conversionwithin this class of possible mechanisms. It is striking that Watt managed toimplement this perfected sizing within his actual invention. (3) Indeed, once thepossibilities to be surveyed are represented in Reuleaux’ stick figure manner, itbecomes evident that an algorithm15 is available that can locate these optimalarrangements of parts as long as we restrict our attention to2-D designs (more on this in a bit). Such methods usuallyproceed by trial-and-error successive approximation. InWittgenstein’s day, these searches for optimality werepursued through a sequence of ever improving sketchesexecuted on drafting tables. Nowadays computers executethese same inferential chores. FIG: WORKING OUT THEPOSSIBILITIES The trick to finding these optimalarrangements lies in restricting our attention to thepossibilities of Reuleaux’ stick figure representations; their

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contraction to a fixed point

possible movements frame the space in which we should search for an optimalsizing. We here witness a primitive form of what applied mathematicians call theproper setting for a physical modeling--a well-selected space of restrictedpossibilities pertinent to locating the proper solution to a particularized problem. Frequently a guiding norm (or inner product) can be attached to these local spacesthat measures the “closeness” of a selected possibility to some standard. In ourWatt example, a pertinent norm should measure how ably an output curveapproximates straight line motion. A routine for design improvement is calledcontractive or coersive (these are synonyms) if each stage in the sequence ofrefinement processes reduces the measurederror to a smaller value, hopefully leading ina limit to a fixed point, where the refinementprocess no longer recommends any furtherimprovements. I repeat an illustration fromEssay 6 for convenience. FIG:CONTRACTION TO A FIXED POINT

Considered in their own right, withoutattention to the practical objectives thatframe their contours, the localizedpossibilities that Reuleaux collects together in this fashion can look rather odd,especially if viewed from the perspective of a possible worlds aficionado. Wittgenstein comments, apparently with Reuleaux’ techniques in mind:

And what leads us into thinking [like this]? The kind of way we talk aboutmachines. We say, for example, that a machine has (possesses) such-and-such possibilities of movement; we speak of the ideally rigid machine thatcan only move in such-and-such way. --What is this possibility of movement?It is not the movement, but it does not seem to be the mere physicalconditions for moving either--as, that there is play between socket and pin,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 ashadow I do not mean some picture of the movement--for such a picturewould not have to be a picture of just this movement. But the possibility ofthis movement must be the possibility of just this movement. (See how highthe seas of language run here!). 16

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design in parallel planes

implementations of a commonmechanism

Indeed, a specific oddity of Reuleaux’ manipulation possibilities is that moving barsshould be allowed to melt through one another as ourprototype cycles through its possible positions. Why shouldwe allow this? Well, for the sake of algorithmic simplicity,we want to group together devices that achieve identicalinput/output relationships despite the fact that their physicalimplementations look quite different (these groupings aregenerally regarded as one of Reuleaux’ great contribution tomachine science). FIG: IMPLEMENTATION OF ACOMMON MECHANISM Because coersive entrapment isoften easier to achieve in 2-D than in higher dimensions,devices that can mapped effectively onto a plane offer an“easy design” opportunity leading to optimal improvementthat is not readily replicated within the wider class of 3-Dmechanisms. In real life engineering, we can usually workaround any “bars can’t really melt through one another”concerns by implementing a basic 2-D design in offset layers, stacking up linkagesso that they maintain the underlying design functionality while moving in parallelplanes to one another. This trick allows us to not worry about implementation as weattempt to improve the sizing of a 2D invention–in essence, we can factorimplementation concerns out of the possibilities through which we search. Thesestrategic remarks underwrite one of those remarkable aperçus that Reuleaux’s workoften suggests. Why have most the mechanisms designed by man over the centuriesoperated in parallel planes? FIG: DESIGN IN PARALLEL PLANES Because theirdevisers--Reuleaux cites James Watt as a prime exemplar--unconsciously reasonedalong more or less the same 2-D channels as our explicitimprovement algorithms recommend, without realizing thatthey were doing so.17 In an allied vein, Reuleaux remarksthat some unconscious recognition of the formal landscape inwhich they search secretly supplies every seasoned inventorwith an intuitive conviction that there is both somethingspecial about machines in general and that certain designsare “perfect of their kind” (Wittgenstein frequently expressesan engineer’s admiration for the “perfect mechanism”).

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Reuleaux comments on the gradual manner in which mankind came to appreciatethe special opportunities for effective reasoning that lie in the direction of his“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.18

Once the obscuring underbrush of “outer form” gets cleared away, the student ofmechanical invention intuitively recognizes the essential nature of a beautifulmechanism 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 frequently did, requires that the channels ofhis thinking must follow some rough-hewn approximation to the pathways of searchthat Reuleaux later articulates explicitly. This sense of “the right way to handle amachine” provides the systematic inventor with a psychological conviction thatsome underlying machine essence must lie in the background of their problems--some inarticulate notion of “optimal achievement” that makes a particular design“feel right” or not. In my own studies, I have been frequently impressed todiscover that, across a wide spectrum of applications, our commonsense thinkingappears to have anticipated the fundamental contours of many clever forms ofinvestigative strategy long before scientists have learned how to articulate thesesame gambits in formal terms.19

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Wittgenstein

The origins of Wittgenstein’s less-than-pellucid phrase “logical grammar”appear to trace to lines of thought similar to Reuleaux’--that our intimations ofpossibility generally spring from an intuitive understanding of “theright things to think about” in pondering a characteristic family oflocalized problems. FIG: WITTGENSTEIN In contrast, theTractatus was founded upon the underlying presumption thatanimates much contemporary possible world thinking--that we allinherently qualify as a priori masters of absolute possibility,whereby the very notion of the “information content” of a sentencerests upon our absolutist capacities to recognize which possibilitiesit rules out and which possibilities it will accept. From this loftypoint of view, whenever we focus upon smaller possibilities in the concrete andfocused manner of an inventor, we are merely grouping together huge collections ofpreexistent possible worlds in a manner convenient to the task at hand In theTractatus, these hypothetical capacities are linked to our a priori understanding ofall of the variations in absolute possibility that a given term will accept (e.g., if thesentence “Archie loves Betty” reflects a coherent possibility, then “Archie lovesVeronica” must do so as well). At this stage in his development, Wittgensteinbelieves that this unfettered knowledge of possible variation constitutes an essentialaspect of the “logical grammar” of a term.20

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 are naturallyspawned, Reuleaux-style, in the course of concrete objectives. To become masterof a trade frequently requires that we should, in Watt’s tacit fashion, acquire firmcontrol of a localized possibility space, even if we are unable to delineate itscontours in articulate terms. Certainly, if we concentrate upon the specific utilitiesconnected to “directing our thinking along locally useful channels,” we can morereadily explain why pondering possibilities is useful at all, questions not easilyaddressed from an absolutist point of view. From this pragmatic perspective, theutilities of a specific specialized space are often closely linked the empirical andalgorithmic opportunities that recommend various forms of confined search to us asuseful patterns of reasoning. From this utilitarian point of view, there is no a prioriexpectation that the distinct possibility spaces we frame within different

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Watt

applicational circumstances will fit together nicely at all, let alone cohere into grandpossible worlds of a Kripke-like stripe. In fact, we should expect that incongruentconceptions of possibility will arise even within the limited dominions of machinedesign, for we can’t expect that our parochial allowances for “parts that can meltthrough one another” can be tolerated in contexts where optimal design is no longerof paramount interest. Nonetheless, the sharp utilities attaching to Reuleaux’special spaces of possibilities seem to be chiefly responsible for depositing a strongsense of “machine essence” and “possible variation” upon our thinking aboutmechanisms, to such an extent that we don’t feel that we have demonstrated aproper understanding of the “meaning” of “machine” until we can map outReuleaux’ spaces in the general manner required. Wittgenstein comments on theseimpressions 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.21

As I understand him, such considerations allow the later Wittgenstein to retain anotion of a term’s “logical grammar” that he can still explicate in terms of the“spaces of possibility” associated with a usage. But now these spaces of variation

are regarded as arising from some flavor of practical advantagepertinent to human proceedings, rather than stemming from an apriori attachment to absolutist possible worlds. He can stillmaintain that the knowing the “grammar” of a term requires someskill in generating or recognizing the appropriate ranges of variationin possibility, even if the agent’s knowledge of these salientvariations remains at the level of indistinct sensibility as we find itin Watt or Whitney. FIG: WATT

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.22

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Bernoulli and Lagrange

By the same token, the utility of these basic channels can fade away if we somedaydiscover an alternative pathway of machine improvement that works as effectivelyin three dimensions as two. Our intuitive impression of “machine essence” mightshift accordingly and the long reign of machines that operate in parallel planes willfinally end. In this manner, the channels of effective thinking about possibility maydramatically adjust over long periods of time, but their localized control may remaincritical determinants of effective thinking within shorter spans of time.

Such is my tentative understanding of why Wittgenstein continues to employthe phrase “logical grammar” in his later writings, but I am not proposing that weimitate this practice. Indeed, at essay’s end I will indicate why I suspect that thephrase, in Wittgenstein’s mouth, embraces further themes that are antithetical to thepoint of view developed here.

(iii)

An excellent exemplar of a localized set of peculiar possibilities can be foundin virtual work methodology briefly surveyed in Essay 6 (I will not rely upon thatdiscussion here, however). These widely employed techniques, which revolvearound an unexpected set of tweakedcounterfactuals, were codified by Lagrange in hiscelebrated Mécanique Analytique,23 based uponstrategic insights developed previously by theBernoullis and others. FIG: BERNOULLI ANDLAGRANGE These methods were commonlyheralded for their intrinsic reliability by most of thegreat physicists of the nineteenth century and theirtop-down successes made a huge impression upon philosopher/scientists such asDuhem and Maxwell. By “the reliability of a modeling methodology,” I have inmind the sundry safeguards that insure that the buildings you design fall down lessand the theories of electricity you develop hold up longer. As I have complainedelsewhere in this book, 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

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as simple terms as I can devise, but the full details may try some readers’ patience. All that I can offer by way of apology is the fact that a certain level of narrativecomplexity is required before the follies of inflating Lagrangian possibilities intopossible world monstrosities becomes evident (the complications largely come inexplaining why “virtual manipulations’ are required). Every modern engineeringtext is full of function spaces that represent special domains of possibilities that needto be kept small and controlled.24 but the underlying motives for Lagrange’sapproach remain the easiest to explain in concrete terms. I hope that less-than-enthralled readers will be able to follow the gist of our discussion without masteringevery gritty details, over which they may lightly skip.

Other essays have observed that a wise policy for heightening modelingreliability is to inject as much higher-scale physics avoidance into our modelingtechniques as we can, following the recommendation, “if you already know thatcertain things are certain to happen within a complex system, utilize that partialknowledge as directly as you can.” Unwisely attempting to recalculate what wealready know runs the significant risk of introducing unnecessary errors that canspoil our endeavors entirely. To be sure, there are circumstances where thispreference for trustworthy results is not paramount, but most engineering practice tothis day adheres to this percept (you can be sued if you don’t). The multiscalemethods of Essay 5 demonstrate the great computational advantages of mixing dataobtained drawn from different choices of scale size, whose successful blendingrequires the employment of fancy homogenization techniques. In essence,Lagrange’s virtual work methods execute a similar task in data amalgamation, butwithin a somewhat simpler setting where search within a restricted space of tweakedpossibilities effectively blends data types that otherwise do not fit together well. Inthis approach,25 we begin within configuration spaces allied to the mobility spacescanvassed in the previous section, but in which we introduce applied forces able topush the parts of a system around. This combination demands a deeper symbiosisbetween a specialized space of counterfactual claims and the practical utilities theyunderwrite.

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constrained bead and wire systems

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 beads-and-wire circumstances pictured supplystandard illustrations. FIG:CONSTRAINED BEAD AND WIRESYSTEMS The two connecting barsin figure (b) constitute constraints thatkeep their beads at a constant distancefrom one another without restrictingthe beads’ capacities to follow thewire.26 The inflexible shape of theunderlying wire in both figuressupplies an additional constraint thatdemands that all beads, wherever theymay travel, must remain upon the wire(allowing us to label their positions with arc length q-numbers as illustrated27). Intruth, no real wire, no matter how stout, can avoid wiggling in small measure when areal bead thing slides by. However, we will ignore these “back action” effectsentirely. Upon a macroscopic scale of observation, the wire never moves at all.

For variety’s sake, I have accorded bead 4 with an additional freedom fortwisting about its spring (θ), denied to the others because of the constraints. Weshall also assume that our beads can slide frictionlessly along their respective wires. Again, such behavior is impossible for real materials, but well-lubricated ceramicbeads can approximate such conditions ably over appreciable intervals. Later in thepaper we shall scrutinize the formal role of these “idealizations” more closely.

Thus far, our invocation of constraint knowledge is rather similar to thestatical considerations of our previous example, but we will eventually ask how ourknowledge of constraints can be exploited within dynamical circumstances, in whichmoving beads scuttle along their constraining wires under the influence of gravityand other outside forces. But to do this we first have to construct an appropriatenotion of available potential energy for our system

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tthe active part of an applied force

In the absence of friction, the total energy within a system should remainconstant and if bead situation (a) arises within a constant gravitational field, then itssingle bead should continually shuttle energy between kinetic energy (= movingenergy) and potential energy expressions. More specifically, any time the beadclimbs up the wire through a vertical distance Δh, the whole system gains anincreased total potential energy of amount mgΔh, where m is the mass of the beadand g is the terrestrial gravitational constant. If not obstructed, this stored energy

will attempt to accelerate the bead by an amount -V(q)/q, leading to an increased kinetic energy. But theavailable portion V(q) of this total energy depends uponthe wire’s orientation with respect to the vertical y atposition q, for only the “active part” of the gravitationalforce can move the bead forward along the wire. FIG:THE ACTIVE PART OF AN APPLIED FORCE Insofaras the wire runs perpendicular to the earth, gravity willeffectively pull upon the bead to a lesser degree. Why?Because its perpendicular constraint force component only

pushes the bead into the wire and is hence wasted (in standard jargon, it “performsno work”28). If the wire’s shape is supplied by a sufficiently simple formula, thencalculating V(q) from the unfettered gravitational potential increase mgΔh is a fairlysimple affair.

Note that the two locales at which our bead-and-wire can remain stably at restare easy to locate--they occur at the two locally lowest points where the wire turnsaround. If our bead contains no surplus kinetic energy, it will remain at one of theseequilibriums forever, despite the fact a bead captured within the energy well on theleft stores a more gravitational energy than a bead captured on the right. This isbecause this excess trapped energy remains unavailable to the bead until it istransferred into the right hand energetic basin, or otherwise released from the wire’sconfines. Theoretically, a bead situated in the maximal potential position at the topof the central hump should also remain stationary, but this is regarded as an unstableequilibrium because the bead is readily jostled out of these circumstances. So muchfor the unforced equilibrium configurations of our solitary bead and wire. Byapplying an additional applied force to our bead, we can hold our system in a state ofconstrained equilibrium at any location q, simply by supplying an opposing force of

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virtual work of force F3

lower scale factors we avoid through virtual work suppression

the precise magnitude required to oppose the pull of gravity at q. This criterion helpsus experimentally map out our system’s available energy reserves by directlymeasuring the strength of the force required to keep the bead stationary at everyposition. This rather innocuous observation shall 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 uponthe more complex situation illustrated in figure (b), by seeking an available energyfunction V(q1, ...q4, θ) that can govern the system in a manner comparable to thesimple V(q) just examined.29 Our new system’s connecting bars and springs

complicate the situation because the beads are coupledtogether and lowering any one of them (say, bead 3)usually alters the overall available energy by pushingother beads upward or downwards and by stretching orcompressing the two connecting springs. Accordingly,the amount of effort F3 required to push bead 3 forwardthrough a distance Δq depends upon faraway features ofour combined system. FIG: VIRTUAL WORK OFFORCE F3 Accordingly, we must usually include all of

our q coordinates within our V-function, for the available energy now represents aholistic characteristic of the entire system, depending upon how its sundry parts linktogether.30

Suppose that we attempt to compute the appropriate V(q1, ...q4, θ) in a naive,bottom-up manner by directly summing the effective “active force” that applies toeach bead at every possible position within the mechanism, an approach sometimeslabeled as”Newtonian.” Ouracross-the-systementanglementsimmediatelygenerate horrendousdescriptiveobstacles because,e.g., bead 3'spositioning

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static balance under a schedule of applied forces

indirectly supplies restraining forces upon all the other beads, through excitingintermediary forces within the wire, springs and rods that connect the beads together. To calculate this force transference directly, we would need to decompose ourerstwhile constraints into little force centers bound together by extremely strongbonds (symbolized in the picture with a large set of additional springs). And we alsoneed additional gunk to prevent the beads from adhering to the wire by the sameattractive factors that hold them together internally.31 FIG: LOWER SCALEFACTORS WE AVOID THROUGH VIRTUAL WORK SUPPRESSION It is quiteunlikely that a lower-scale modeling of this type can be constructed at all but, if itcan, it will contain many speculative details likely to introduce great errors within ahorrific set of difficult calculations. This is a prime example of our earlier adage, “ifyou already know that certain things are certain to happen within a complex system,utilize that partial knowledge as directly as you can.” We already know that stiffwires and ceramic beads don’t flex appreciably when excited to a moderate degreewithin systems of this type. So let us frame a sound policy of physics avoidance thatallows us to bypass these lower scale complexities through a deft exploitation of thealready known behaviors of our wire and connecting rods.

Returning to our “innocuous observation” with respect to constrainedequilibriums, observe that we can still freeze our complex system in any position welike by applying an array of additional forces to counter the system’s internalinclinations for forcingthe beads to move(simile: these appliedforces hold theapparatus in check inthe same manner thatthe little boy holds backthe ocean by insertinghis finger in the dike). FIG: STATIC BALANCE UNDER A SCHEDULE OF APPLIED FORCES Let uscall a suitable array of applied forces <F1*, ...F4*, τ*> a “sweet spot” configurationfor our system.32 But should we estimate the magnitudes of the forces required? Here Lagrange offers a strategic recommendation widely employed in science--characterize a sweet spot configuration through consideration of what happens under

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small perturbations of its circumstances. What distinctive behaviors should a sweetspot force array <F1*, ...F4*, τ*> reveal under possible manipulations? Well,suppose we displace bead 3 a short distance Δq3 from its equilibrium point. To doso, we must apply a small amount of work F3*Δq3 to our system. To be atequilibrium, the system’s internal responses need to generate a restorative effort ofexactly the same magnitude, in order to pull bead 3 back to its equilibrium locationq3. This reasoning applies to all of our beads, and supplies us with an importantcollection of counterfactual information about our system:

Fi applied at location qi represents the correct balancing force for holding thedevice in a sweet spot configuration <F1, F2 , ..., Fn> iff the work required tomove bead i through a distance Δqi will be exactly matched by opposingamount of work arising from the available energy within the system giventhat the other applied forces Fj remain as specified.

This collection of hypotheticals is usually called the principle of virtual work,although I haven’t yet explained the “virtual” aspect of it. Notice that such claimsreport on behaviors that would arise if the system’s sweet spot parameters weremanipulated in simple ways.

These counterfactuals don’t present us with a sweet spot force balancedirectly, but they supply us with a suitable search space and norm in which we canfind them. We execute this quest through a series of successive approximations of ageneral type encountered in our other essays, viz., beginning with an arbitrary guess<F1

0, ...F40, τ0> with respect to a hypothetical sweet spot array, we apply a small

perturbation Δq3 to some coordinate, say, q3. Unopposed, this adjustment will pullthe other beads over to new positions <Δq1,..., Δq5>, which we can calculate through acombination of rigid rod geometry and knowledge of the Hooke’s law behaviors ofthe springs. If the total reaction created by these linked adjustments does not exactlybalance the work we supplied (i.e., F3

0 Δq3), then our initial guess <F10, ...F4

0, θ0>could not have represented a true sweet spot configuration. So we should reset ourF3

0 value to a new value F31 and try again, working with a replacement guess <F1

0, ..,F3

1, F40, τ0>, until we reach a viable F3 value. But we’re not yet done, for equilibrium

stability requires that we enforce a policy of equal justice for every bead. Accordingly, we must apply the same test to bead 4, say, and make alliedcorrections. In the course of doing so, we will likely need to revisit bead 3's situationonce again. The end result is a lengthy set of improving guesses Gi (usually non-

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following a norm in the right search space

terminating) that should bring us ever closely to our desired sweet spotarrangements.

Although this refinement process sounds laborious, it is child’s play comparedto solving all of the simultaneous equations that we confront in a Newtonianapproach to the same circumstances. This is why engineers highly prize Lagrangiantechnique for its significant physics avoidance efficiencies.

For our purposes, we want to focus upon the intriguing manner in which thesecounterfactual possibilities guide us to the V(q1, ...q4, θ) characteristic we seek. Inthe jargon of the mathematicians, our virtual work tests supply us with a norm |Gi|that measures how far a given guess Gi departs from storing potential energy in aminimal manner. By repeating this norm-guided search over and over, we construct alengthy sequence of improving guesses thatgradually squeezes in on a sweet spot forceallocation as an eventual fixed point. As ananalogy,33 consider the manner in which thewhisperings of conscience lead us along theStraight and Narrow. It constantly asks us toevaluate a lot of counterfactuals--“If you wereto do that, what will the harvest be?” “Ick,” wesay and alter course. FIG: FOLLOWING A NORM IN THE RIGHT SEARCHSPACE Repeated sagacious steering of this type should convey us along a zig-zagpath to Goodness. A comparable beads-and-wire norm is captured within the virtualwork requirements outlined above, for they evaluate possible configurations of oursystem in terms of manipulationist counterfactuals of the form, “if bead bi werewiggled by an amount Δqi, the adjustment would make the overall energetic situationbetter/worse for the rest of the system.” Such is the core data that we collect into apossibility space suitable for Lagrangian searches. . For strategies of this character to operate as intended, we must ensure that theonly possibilities we consider are of an appropriate kind (if the Devil is allowed tothrow in some of his own “possibilities,” we may find ourselves on the road toPerdition). If we tolerate adjustments that do not conform to the mobilityrestrictions imposed by our constraints–we allow the wire to bend, for example--, wewill lose the wherewithal to say “Ick!” at the right times.

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converting static information to dynamic purposes

Plainly, the guidance inherent within these searches resembles the tactics thatlead old-fashioned engineers like James Watt to optimal machine improvements byexamining trial sizings repeatedly upon a drafting table. So the computationalguidance supplied by our virtual work norms provides another concrete illustration ofhow macroscopic counterfactuals of great certitude can supply us with “channels ofreasoning” that guide us in establishing other facts about our target system that are“not hardened but remain fluid.” Through these quasi-a priori channels we locate thephysics avoidance pathways that allow us to escape the computational extremities ofmodeling our system’s complexities in full microscopic detail.

Lagrange showed how our sweet spot data can be further exploited to computethe behaviors of our beads-and-wire gizmo when it is left to its own autonomouinclinations and outfitted with a surfeit of conserved energy.34 FIG: CONVERTINGSTATIC INFORMATION TO DYNAMIC PURPOSES How? Start with thesystem at t0 in initial configuration <q1,..., qn>. Calculate its corresponding “sweetspot” force allocation <F1, ,..., Fn> and replace each of these by an accelerativefactor Fi/mi where mi is the mass (or moment of inertia) of freedom i. Then employ astandard marching method algorithm to estimate where the beads must wind up aftera time interval Δt.35 If we keepour step size Δt small enoughand reiterate this marchingmethod computation over andover, we can plot the evolvingbehaviors of our beads-and-wiregizmo over long spans of time,without needing to bother withthe largely conjectural molecularforces that bind our systemtogether in tight constraints.

These technicalpreliminaries established, let us now consider the particular aspect of Lagrangiantechnique that clearly indicates the follies of inflating useful local possibilities intogargantuan possible worlds. In truth, the collection of counterfactuals required forLagrangian purposes should not be comprised of exactly the ones I have invoked, butmust be slightly tweaked to include the supplementary qualifier “virtual.” That is,

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difference between real and virtual work

they should assume the form, “if bead bi is virtually lowered through a distance δq,the energetic situation will become virtually better/worse for the other beads.”36 Assuch, “virtual movements” and allied notions represent the conceptual bane of everynovice in elementary physics, for it is a rare beginner who hasn’t wondered, “Whatthe hell is going on?” when they first learn of these corrections. Such cavils usuallyget suppressed under the heavy imperatives of “get your homework in on time,” butbetter answers can be provided. Why must these odd qualifications enter our guidingcounterfactuals?

The answer stems from the fact that, via Lagrangian guidance, we are secretlyattempting to evaluate a system’s internal ability to perform useful work37 when itssundry beads are located at specified locations. Let’s again consider a solitary beadsitting near the top of a curved wire. If we try to extract work from our bead byallowing it to move forward along the wire, we will only obtain a feeble response. Although the bead’s absolute gravitational energy is high at this point, most of

gravity’s attractive force points into the wire andcan only contribute a small portion of availablework in pulling the bead forward along the wire. But if the bead moves through the full arc δq, theactive gravitational component quickly becomeslarge, eventually becoming nearly equal to itsunfettered gravitational value. So if we naivelyadd the actual work supplied in moving a beadfrom q to q + δq (viz., q q + dq Fdq), we willobtain a totally inappropriate estimate for the

available work back at q. FIG: DIFFERENCE BETEEN REAL AND VIRTUALWORK But we must wiggle our bead somehow–our sweet spot stability standards restupon manipulations of that character. So we should instead compute a virtual workvalue by adding up constant copies of the feeble force contribution acting at q alongthe full δq arc, for the corrected integral q q + δq Fδq now provides a more accuratemeasure of the work potential available back at q.38

To function correctly, our various possibility counterfactuals must therefore betweaked into a virtual work format that effectively compromises between twofunctional demands.39 (1) They must exploit the reliable manipulation data that we candirectly glean from experiment or from obvious facts about the geometrical behavior of

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rigid rods. (2) They must be able to guide researchers to correct evaluations of thesystem’s hidden work capacities in a zig-zag fashion. This remarkably meldedreasoning combination allows us to sidestep the insecure speculations that would havearisen had we tried to model all of the subterranean parts and forces that actually holdour gizmo together.

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

Of course, “obvious facts” display an annoying propensity to break down if wewait long enough. After an excessive amount of bead shuffling, our erstwhile rigidwire will sag and our beads will distort, at which point we must open up the “frozendegrees of freedom” ignored within our rigid wire and rigid beads thinking. Themultiscale methods of Essay 5 represent the preferred modern methodology fortackling these lower-scale damage problems without plunging directly into theunmanageable insecurities of full Newtonian modeling. But our focus in this essay hasbeen on the strong descriptive grip that engineers can obtain upon complicated systemsthrough following the guidance of carefully circumscribed spaces of manipulationistpossibility.40

(iv)

Observe that our virtual possibilities stoutly resist ready enlargement into richerpossible worlds of the type discussed at the beginning of the essay. The sentence“Jean-Louis applied virtual work to the pump handle” is unlikely to appear withineven the wildest extravaganzas of science fiction fantasy (unless the author simplydoesn’t understand what the term “virtual work” means41). This is because the originsof these categories of possibility trace directly to the modeling utilities they serve,rather than appearing as redactions from wider collections of a priori possibility. Accordingly, they illustrate the investigative recommendation we extracted from thelater Wittgenstein: “Look for the local packets of guiding ‘possibilities’ that appearpeculiar in their contours and trace these oddities to the linguistic utilities that theyfacilitate.” The Tractatus had maintained that there is a great a priori universe oflogical possibility from which any localized specimen has been plucked, and furtherclaimed that we aren’t ready to employ language at all until a complete mastery of thisgrand space of permissible variation has been acquired. The latter Wittgensteindecides that this narrative gets the true origins of possibility talk backwards. Non-

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degenerate possibilities are bred and nurtured within the local river beds of directiveguidance that endow them with their idiosyncratic utilities. Their occasionally oddmarkings--such as needing to be “virtual”–reflect the strategic contours of the streambeds in which they have been spawned. They look peculiar only as they areencountered within a vast ocean of amalgamated possibilities fed by many inletstreams.

This Wittgensteinian sentiment, registered in ichthyological metaphor,epitomizes my own reluctance to rely heavily upon possible world tools in attemptingto understand real life linguistic usage. For example, Essay 6 complains that theLewis-Stalnaker approach to counterfactual conditionals and causation blurs togethermodeling tactics that applied mathematicians carefully distinguish. If we ignore theirsignificant structural differences within an ambitious quest for a “general theory ofcounterfactual conditionals,” we are likely to have washed away the very factors thatexplain why we care about such claims at all. Our little study of Lagrange’s methodillustrates the moral--it is only through an appreciation of the detailed manner in whichthe technique deftly exploits higher-scale constraint knowledge that we can understandhow we assign salient truth-values to its otherwise peculiar “virtual work”counterfactual conditionals at all.

All modern books on mathematical modeling that pretend to any degree of rigorset their problems within carefully delineated “function spaces” (which is the officialtitle of the possibility spaces I have described). Unfortunately, they are often lessforthcoming about why they do this, anticipating that mathematically sophisticatedreaders will see through their morass of technicalities to the motivating reasonsbeneath. Often unwanted possibilities--which might prove desirable in othercontexts–must be kept out of a space lest its guiding norm get led astray. Essay 8illustrates the specialized cooperative family relationships that sometimes must bindlocal species of possibility together into a coherent descriptive unit42; the functionspaces” of the textbooks reflect these localized considerations as well. Thesecooperative family concerns reinforce the importance of studying possibilities in termsof the strategic functionalities that they provide as localized collections, most of whichbecomes obscured when they are unwisely deposited within the amorphous vastness ofextended possible worlds.43

Accordingly, the skillful exploitation of well-crafted spaces of localizedpossibilities directly founded upon our upper-scale knowledge of material behavior

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the splendid but obsoleteseagoing clock

allows us to identify important physical structures, such as available work potential, more readily than if we had attempted to uncover these quantities through puristbottom-up endeavor. This lesson in hand, let us return to the broader philosophicalthemes with which this essay opened. Consider Wittgenstein’s remarks on the long-term mutability of “channels of thought” that strike us as obligatory at the presentmoment. An excellent policy for rendering a strategic pattern obsolete is to uncover analternative methodology that operates more effectively over a broader range ofpractical purposes. Sometimes the utilitarian salience of formerly vital computationscan vanish altogether. Thus, the modern availability of GPS technology has largelyobliterated the “position at sea” concerns that once dictated how wars were waged andgreat fortunes accumulated. FIG: THE SPLENDID BUTOBSOLETE SEAGOING CLOCK Perhaps in years to come, ourfussings with virtual displacements and delta function impulseswill appear as puzzling to future archeologists as the centralpatterns within Babylonian time reckoning now seem to us. Theactive shelf life of a vibrant “necessity” is not forever, despitewhatever possible world enthusiasts might fancy to the contrary. And so “the river-beds of thoughts may shift,” Wittgenstein tellsus. Allied considerations lie at the root of Quine’s skepticismswith 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 first consider themore urgent question of why so many contemporary philosophers have returned to aTractatus-like faith in possible worlds standing fully amplified in all of their details. We are, after all, intellects of a restricted computational compass. Any conceit that wecan juggle huge data sets pertinent to possibilities on such a grand scale comprises asan odd fairy tale we have concocted to make ourselves feel good through exaggeratingour mental capacities. A simple count of available brain cells suggests that we insteadcultivate a host of localized and mildly disconnected skills, in which we juggle smallersets of possibilities in a more tractable manner. The fact that the smallish possibilitiesthat assist our projects most effectively–e.g., counterfactuals about virtual variations--do not happily enlarge into grander possibilities provides a worthy symptom thatsomething is deeply amiss in grander portraits of our linguistic capacities.

So where does the illusion that we actively traffic in grander possibilities come

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acquiring the early apriori

from? Some of it traces to the simple fact that the standard possible world analyses ofmodal logic have assisted us considerably in unraveling common patterns of linguisticbehavior, with respect to how we talk of necessity and possibility. Surely, it willasked, possible world considerations must have latched onto important aspects of ourusage?

I agree, but my more constrained diagnosis unfolds as follows. We certainlyrequire juvenile training in the rudiments of how useful appeals to possibility andnecessity can be invoked and monitored within our commonsensical thinking. Essay 6

characterizes the set of psychological skills acquired in thismanner as the early apriori. FIG: ACQUIRING THEEARLY APRIORI Typically we operate on several tracksin reasoning about these matters. (1) We learn directinferential rules for reasoning about local concerns, alongwith management capacities for modifying these structuresas needed.44 (2) Alongside these reasoning rules, we learn tocultivate relevant counterexample spaces as correctivechecks upon these deductive procedures learned in manner(1), in which we test our reasoning patterns against a set of

potentially “relevant possibilities.” In this basic two-pronged manner, computerscientists take their algorithms for trial runs over a representative sample of well-understood situations to see if any unanticipated glitches threaten their proposedreasoning procedures. Well, we are taught to do the same in our early training, for weacquire none of our initial skills in perfected form, and rely on each to correct theother. By leap-frogging forward between (1) and (2), we can gradually adjust ourreasonings to suit nature’s complexities.

Essay 1 argues that significant evolutionary advantage attaches to our capacitiesfor assembling novel banks of inferential principle swiftly, even if these techniques arenot completely accurate at first. Acquiring a fairly elaborate reasoning routine in aquick and dirty way is often wiser than striving for unblemished perfection right off thebat, for we can usually straighten out the ensuing kinks later and delicately nuancedroutines are not readily learned. The same allowances for swift learning attach to ourcapacities for framing helpful counterexample spaces, for these can also berecalibrated for better results later on. Accordingly, we initiate our children into thearts of good reasoning in rough and generic ways, anticipating that skills more finely

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what is this thing?

adapted to specialized circumstance may be required later on.Among biological creatures, homo sapiens are notable for their astonishing

conceptual plasticity and our early apriori training in words like “possibility” plays asignificant role in facilitating this versatility. This juvenile instruction supplies us witha basic inferential repertory that can be swiftly adapted to novel purposes, somtimesaltering the referential focus of the affected terminologies considerably in the process. Often a strong psychology of classificatory confidence accompanies these adaptiveextensions, in the sense that we are predisposed to assume that we already possess asuitable label for everything we encounter, no matter how unfamiliar. Confronted with

a random blot, we confidently label it as “two spider-like things.”45

FIG: WHAT IS THIS THING? We fancy ourselves greatconceptual “deciders,” in the unhappy mode of those scions ofprivilege who are certain that their “intuitions” will allow them torun a country capably even if they don’t understand internationalrelations nor how an economy works. Possibly these forms of

classificatory hubris enjoy their own evolutionary advantages--‘tis often better toboldly classify than linger in indecision. These psychological vanities encourage thephilosophical conviction that our early apriori “necessities” count for more over thelonger run than they properly should. As we scrutinize our improved inferentialtactics closely, we detect the telltale symptoms of adaptive modification, shaped by therequirements of improved practical advantage. In this manner, we find that weinstinctively consult different collections of “relevant counterexamples” withindifferent species of applicational endeavor. We likewise attend to altered forms of“causal relationship” in the context of varied explanatory architectures. And so forth. In formalized science, these differences are often reflected within the specializedfunction spaces that applied mathematicians assign to specific classes of problem. Indeed, this essay began life in musings about why such complications are required. So I believe that we retool our commonsensical classificatory categories in theessentially same ways as the applied mathematicians, although our innate classificatoryhubris often makes us overlook the referential significance of the alterations.

(v)

Returning to the question of why possible world structures prove so helpful inmodern linguistics, I can suggest the following hypothesis. As we saw, our initial

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an ersatz possibility

training in handling the basic vocabularies of “possibility” and “necessary” should berather generic in its contours, just as we first learn the word “force” in a looselyfocused manner. But we imbibe these early apriori reasoning skills at the same time aswe master our language’s chief grammatical structures, so it is not surprising thatcertain basic inferential patterns will find reflection within the grammatical distinctionsmarked in our language. Since these reasoning capacities also include our juvenileskills in searching through relevant spaces of salient possibilities, the contents of thesespaces should integrate with basic grammatical reasoning in roughly the syntax-linked-to-possible-world-semantics patterns of the linguists.46

Yet none of these considerations should be regarded as especially probativewith respect to longer swatches of human experience, for essentially the reasons thatQuine and Wittgenstein have suggested. The computational demands of nature herself--the modes of manipulating syntax that operate effectively as instrumentalities fordealing with the external world--often run counter to our originalist expectations. Andso early a priori presumptions frequently need to be altered or overwritten, in classicNeurath’s boat mode. My favorite example47: we swiftly acquire an earlyunderstanding of the word “rainbow” by assimilating its usage to the geometricalreasoning patterns we associate with words like “arch.” But these inferential expectations must be corrected by thetime a speaker becomes an adult, for the underlying physicalphenomena demand a different set of reasoning policies(unless we like to waste time searching for pots of gold). Making these corrective adjustments isn’t difficult, but werequire phraseology through which we can mark theadjustments. With respect to “rainbow,” we need to suppress the many ersatzpossibilities that loom large within our child’s current conceptual frame. FIG: ANERSATZ POSSIBILITY If we ask her, “Can anyone climb on a rainbow or find goldat its foot?,” she will reply, “Sure, Polychrome and Flax have done it,” referring to thestorybooks from which she has received her early instruction in these issues. Werequire a linguistic vehicle for weaning her rainbow possibilities away from thestorybooks and onto to a collection based in sounder meteorology and optics. So wesay, “Reaching a rainbow’s foot isn’t a real possibility that you should be thinkingabout, although totally red bows remain conceivable.”48 In other essays, I call phraseslike “real possibility” tools of linguistic management; they provide the verbal

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instruments that can explicitly redirect patterns of usage as circumstances require.Through these corrective agencies, our conceptions of “what is possible” significantlyenlarge and contract over time.

We shouldn’t presume that all of these newly congregated possibilities havelain latent within our linguistic skills from their inception. Just because we havebought enough insurance to cover a future car accident doesn’t indicate that we havefully anticipated the wreckage. I believe that conceptual back-dating of this sortcomprises one of the central misconceptions that the later Wittgenstein lays at theTractatus’ door. And this is also why static views of classificatory semanticsencounter significant problems whenever they consider word behavior over evolvingspans of applicational development. To be sure, we rarely notice these inferentialretoolings in our everyday talk of household furniture and the like, because theyappear disguised within our usual manners of conceptualizing the world in scenario-centered terms.49 In contrast, closely analyzed scientific exemplars–-like the carefullymonitored variations of Lagrangian technique–-display the importance of parochialrealms of local possibility in more sharply etched terms. These valuable examplesshow that the truth-values of the restricted class of counterfactual possibilitiesexploited in Lagrange’s manner commonly stem from direct induction uponmanipulative experimentation, such as the tinkerings of our do-it-yourselfer. Claiming--as the metaphysicians of Essay 6 do--that the truth-values of these counterfactualsmust be dependent upon the very laws that they are designed to circumvent strikes meas strange. If we turn Lagrangian techniques on their head in that grounding-in-lawsmanner, we won’t be able to understand the reliability-enhancing utilities of thispopular form of scientific reasoning at all.

It is hard to be sure, but it strikes me that academic philosophy’s currentattraction to inflated possibilities traces to a thesis about language that is central withinthe Tractatus as well. This is the notion that the truth-conditions of descriptivesentences ultimately reflect the manners in which they segregate the absolute a prioripossibilities into two groups, the worlds they accept and the worlds they reject. Fromthis informational perspective, we then anticipate that the underlying semantics forcounterfactual claims must be directly anchored within allied discriminations amongstpossible worlds. If so, any English-speaker competent in counterfactual constructionmust recognize how this informational basis operates. Throw in an additional dash ofTheory T philosophy of science and we’re well on our way to regarding a Lewisian

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account of the “meaning” of counterfactual claims as conceptually necessary, to whichwe briefly referred in section (i) and discuss more fully in Essay 6. Our examinationof Lagrangian technique suggests that this popular view of information and truthconditions often inverts the true dependencies. The information we exploit within thetechnique reports on macroscopic experiment, not upon sub-atomic carrying-ons. Andtheir corresponding truth-values should be viewed in the same vein. When wecharacterize wires and rods as rigid, we apply applicational standards appropriate tothe macroscopic scales of length and time scale on which we speak. These everydaycategorizations should be regarded as completely true, despite the fact that, if we shiftour attention to other characteristic scales, we might describe the same physicalcircumstances in alternative ways. As Essay 1 argues, contextually based evaluationsof this character bring great linguistic efficiencies in their train, although this fact isoften overlooked within philosophical discussion. For allied reasons, the “truthconditions” of our virtual work conditionals should be regarded as informationallyrooted within the manners in which we decide upon their truth: viz., direct inductionsfrom experiment, subsequently altered to correct for virtual work effects. It is only byworking downward, in alpine mountaineer fashion, from these experimental anchorswithin upper scale rock that science earns its justified reliability. Tractatus-likeconceptions of “informational content” obscure these epistemological dependencieswithin fairy tale depictions of our mental abilities, as if our brains could possibly storeenough data to “segregate the absolute a priori possibilities into two groups.”

In other words, when we learn to frame our peculiar virtual work possibilities asuseful guideposts to hidden physical structure, we hereby enlarge our priorconceptions of “what is possible” in novel ways, just as children eventually learn tocorrect the class of guiding possibilities associated with “rainbow.” Philosopherssometimes argue that we must have been tacitly aware of these unexpectedpossibilities all along, simply because we can appreciate their utilities when someoneexplains them to us. But conceptual pre-dating of this ilk is the characteristichallmark of the bumptious chump who believes that he has already dreamed ofeverything within his philosophy. Such vanities underestimate the strategicinnovations that our clever forebears have already uncovered and the comparableimprovements yet to be mined from other opportunities yet latent in the world beforeus.

To be sure, I’m not entirely sure what the “grounding” claims of our

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metaphysicians amount to. Sometimes all that is involved appears to be some fuzzyexpression of scientism with which I might well concur. If that is so, we findourselves in circumstances reminiscent of the following. When I was a kid, our smalltown paper published any letter to the editor it received, with the net effect that lonelyatheists from Prescott, Arizona regularly wrote to proclaim that “Darwin hasdemonstrated that man is but a lowly evolved worm.” These missives were promptlyaddressed by a local contingent from Eagle Point that retorted, “No, man is not a lowlyworm, for John 43:2 saith....” And so it went on for years. Although I would quibblewith their exact biological allegations (proper lines of descent probably do not linkhominoids with Oligochaeta), I am not religiously inclined and would most likelysympathize with the Arizona contingent in some bottom line fashion. But I am puzzledby their motives in maintaining such a prolonged correspondence.

In the same way, I might agree with some rock-bottom naturalist sentimentunderpinning the claim that “counterfactual behaviors are grounded in ‘fundamentallaw,’” while rejecting most of the accompanying commentary on scientific procedureas naive and inappropriately focused. But I don’t understand how such weak tea canmerit the voluminous philosophical literature it has generated.

However, as soon as we rise above this level of banal scientism, we activelyignore crucial policies of localized linguistic engineering that merit careful scrutiny. We shouldn’t dilute the focusing capacities of Lagrange’s special counterfactuals bydrowning them within a common ocean of possibilities drawn from every walk of life. Nor should we inflate their localized utilities into possible world enormities byimitating the policies that science fiction writers employ in expanding small conceitsinto developed novels. As we do so, we deprive ourselves of the diagnostic resourcesneeded to unravel standard forms of philosophical puzzle. For these typically demanda critical probing of the underlying strategic factors that supply our local invocations ofpossibility with their applicational utilities.

Examples from other essays indicate that the organizational principles thatgovern a material on dimensions below a current scale length ΔL can be hard toanticipate, for substantive methodological surprises often await whenever we dropfrom one descriptive platform to another. In light of this opacity across scale lengths,any presumption that an acute philosopher can nonetheless “see” through all of theseclouded layers to some level of hypothetical “grounding” strikes me as a blunt denialof plain historical fact. Indeed, Lagrangian technique illustrates how carefully

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nurtured ΔL level possibilities can be cultivated precisely as a defense againstunreliable speculations of this ilk.

(vi)

Although these last conclusions reflect, in modern dress, the apparentresonances between my own proposals on possibilities and the later Wittgenstein’s, Imust acknowledge that significant temperamental differences distinguish our two veinsof thinking. Indeed, it may well be that I have been quoting scripture entirely for myown purposes, for I scarcely pretend to understand Wittgenstein in large bulk. 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 strategiesof integrated linguistic employment that allow us to reach practical conclusionseffectively and swiftly within a suitable environment. Lagrange’s techniques forexploiting the partial information of macroscopic rigidity provides an excellentillustration of how nature’s fortuities can be mined for fun and profit. Likewise, thecrisp reasoning pathways that Reuleaux extracts from the intuitive wilderness ofeffective machine design trace to similar opportunistic underpinnings. At the sametime, the prevalence of semantic mimics within the history of science warns that greatsurprises potentially await whenever we attempt to improve a usage beyond the triedand true. So the history of developing words represents a continuing struggle betweenthe angels of applicational advancement and the devils of misleading picture. Theappendix illustrates what I have in mind.

Thus I would equate Wittgenstein’s “channels of thought” and possibly 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 donot pretend 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, my restricted themes would have struck the laterWittgenstein as mortifying exemplars of the wretched “scientific” impulse to “explainhuman behavior,” whereas a philosopher proper should only seek “elucidations” of ourpractices. Insofar as I can determine, Wittgenstein is here captivated by the semi-

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ur-plant

mystical vision of the descriptive enterprise articulated byGoethe in The Metamorphosis of Plants50--the proper method ofappreciating nature is to isolate a hypothetical archetype (the“ur-plant”) and to view all varieties of botanical expressions asvariations upon that core theme. FIG: UR-PLANT A propergrasp of the permitted range of variation allows us to reach tothe generative heart of Nature herself, but “explains” nothing. From this point of view, the philosopher should only catalog thevariations native to the sundry “forms of life” which seeminglyrepresent the human analogs of Goethe’s natural categories for Wittgenstein,eschewing any misbegotten attempt to ground these policies within environmentalsetting and computational capacity, as I have done.

Well, this divergence in attitude isn’t surprising, given that I prefer Darwin formy biology to Goethe. But sixty years of inconclusive Wittgenstein scholarshipsuggest that the articulation of scientifically well-understood examples of “hardenedpropositions” of the sort investigated here might advance our understanding in moreprofitable ways, even if we never reach the master’s lofty, and possibly misty, heightsvia such a pathway. Surely no legitimate Wittgensteinian imprimatur can be attachedto the opinions advanced here; their merits must stand or fall entirely on whether Ihave successfully explicated how certain classes of hardened counterfactuals earn theirdescriptive spurs.

Indeed, the appearance of “hardened propositions” in a book entitled OnCertainty suggests that Wittgenstein’s conception of logical grammar may bear a moreintimate kinship to the notion of a revisable a priori championed by the early HansReichenbach and, in a different form, by Michael Friedman.51 Their discussions focusexclusively upon linguistic determinations at the lofty level of “what space-time metricshould we employ?,” rather than upon the narrower task-focused concerns I havediscussed. Like Wittgenstein himself in the Tractatus, they view the permissiblepossibilities tolerated by a choice of geometry as providing an encompassing “net” towhich the linguistic moves we regard as “empirical” must be referred, even thoughfuture generations may someday favor an alternative choice of webbing. This granderperspective might legitimate some of the connotations of Wittgenstein’s use of“grammar” in a more substantive manner than I can, for I regard that term asinappropriate to our humbler examples. For myself, I am not convinced of the merits

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inserting Lagrange multipliers to compute internal stresses

of “net” thinking and so few vestiges of neo-Kantian doctrine or talk of grammarattach to the localized assemblies of controlled possibilities that I emphasize. I fearthat if we attempt to generalize our diagnosis of the strategic utilities of Lagrange’sspecial set of counterfactuals to suit the higher levels of methodological maximpursued by Reichenbach and Friedman, we may wind up with rather thin gruel, of thesort that, if we’re setting out on a carpentry project, we’re better off if maintaining aselective set of instruments in our toolbox and not mix in a lot of kitchen utensils. Ican’t detect the shaping hand of effective computational strategy within advice asgeneric and broad-based as that. I rather seek the concrete opportunities thatgenuinely call for an auger and a specific set of bits, for that is the nitty-gritty levelupon which nature directly rewards and punishes us for our linguistic forays.

To be sure, every once in a while a dinner fork turns out to be just the ticket forprying unwanted gunk out of a crevice and with those unanticipated adaptations newvistas for invention and improved craftsmanship open.

Appendix: The Origins of Conceptual Clashes II

There is an additional aspect of Lagrangian technique that is not especiallygermane to the foregoing essay, but supplies an important illustration of other themespursued in this collection (and may even possess some salience with respect to wideraspects of Wittgenstein interpretation). As we’ve seen, the basic genius of Lagrangianmethods lies in the fact that they exploit our upper-scale knowledge of rigid wires andbeads to effective physics avoidance purposes. In particular, they allow us to cutoffany need to model the complex processes arising inside our wire as it is buffeted byscuttling beads. Although we want to avoid directly modeling these interior behaviors,we may still wish to estimate theirfeatures. Wires, after all,sometimes break under excessiveloads and it is helpful to knowhow great their internal stressesmay become. But we’d still preferto not calculate these stressesusing more elaborate--andcomputationally less

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tightening constraint forces to satisfy upper-scale demand

reliable–modeling techniques. Lagrange proposed a clever method for addressingthese concerns without needing to approach our system in a fully bottom-up manner. Here’s the general strategy. FIG: INSERTING LAGRANGE MULTIPLIERS Insertvarious little springs into the wire at the positions where we’d like to estimate thestresses. Mark the strengths of these hypothetic springs with a set of variables λ1, λ2,... called Lagrange multipliers. Work out the motion of the whole system followingstandard virtual work routine except that the λ1, λ2,.. get treated as undeterminedconstants within our algebraic computations. With respect to the illustration, we areeffectively modeling our previously rigid wire as three separate pieces joined togetherby springs of an unknown strength. If these little λ springs are feeble, then our revisedbeads-and-wire-broken-into-pieces system will behave very differently than when thewire stays thoroughly rigid. But we have no intention allowing that to happen, for wewill now assign our multipliers the requisite force strengths needed to pull our threepieces of wire into compliance with our original requirement of complete rigidity. That is, the strengths of the λ1 and λ2 multipliers supply the exact forces required topull the three pieces of wire back together, no more and no less. For this reason, theserigidity-induced “forces” are traditionally called constraint forces. As such, the λi’ssupply our desired estimates of the stresses in the wire at the points of interest. If anyof these stresses becomes too large, we should worry that the wire may distort orbreak under the imposed load, rather than continuing to act as a rigid constraint. Inone of those unexpected inversions characteristic of clever physics avoidancetechnique, Lagrange has added λi supplements to his virtual work models as internaltests of whether his basic techniques retain their validity in the circumstancesconsidered. In other words, the magnitudes of the λi provide canary-in-a-coal-minewarnings that something may have gonewrong with the presumption that our wireshall remain rigid.

Formally, these multipliers serve asa cross-scalar “handshake” trick thatallows our upper-scale constraints tocommunicate on a common mathematicallevel with the other forces involved in ourmodeling effort (viz., gravity and theoriginal two springs). The λi’s serve as a

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vital communication link between asymptotically separated scale sizes52 in the generalmanner of the homogenization techniques of Essay 5. When we assign concretevalues to our λ’s at the end of our computations, we algebraically tighten up our lower-scale description to suit the demands of our upper-scale knowledge, which I’vesymbolized in the illustration as macroscale tightening winches. FIG: TIGHTENINGCONSTRAINT FORCES TO SATISFY UPPER-SCALE DEMAND

At first glance, it appears as if the constraint forces calculated by these methodsrepresent unproblematic cousins to the other forces considered in our modeling, viz.,gravitation and the Hooke’s law spring forces. But there is a crucial difference. Thestrength of the latter must be specified before we calculate the configuration of oursystem whereas we operate oppositely with our constraint forces. We know theirstrength of these only after we determine the system’s configuration. In other words,the strength of the regular forces are generated by straightforward force laws, whereasthe constraint forces are not. Worse yet, when we investigate these matters in adynamic context, we find that the constraint forces must adjust their attractivestrengths according the velocities whereby the scuttling beads whiz by. Thisobservation historically engendered a great instability in mechanics because Helmholtzhad originally derived the conservation of energy from the presumption that no forcescould be velocity sensitive. Foundationally, this discrepancy led his student, HeinrichHertz, to write his celebrated Principles of Mechanics in 1891, a book to whichWittgenstein often refers.

Thus nineteenth century classical physics confronted a significant foundationaldilemma--either exclude all constraints from physics as convenient approximations orwork only with constraint forces within our physics and obtain the conservation ofenergy via another route. This second task becomes easy if all Newtonian style forcescan be removed from our picture.53 In his Principles, Hertz adopted the second policy,probably because earlier mechanists such as Maxwell had relied heavily uponconstraints in assembling their models of the electromagnetic aether. He commentedupon 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.54

If I understand him rightly, Hertz rejects the notion that anything useful can be gainedby internally scrutinizing the concept of force beyond categorizing its descriptive

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anholonomy

utilities and installing whatever repairs are needed to insure that clashes of the sortconsidered can be avoided.

Here we witness a common developmental phenomenon discussed at(excessive) length in my Wandering Significance. Initial employments of a word havebeen naturally extended in a manner that turns out, upon closer examination, to clashin subtle ways with its parent applications. Mathematicians call such “failures to returnto home unaltered” anholonomies, of which the Riemann surface for a typical complexfunction supplies a splendid paradigm. FIG:ANHOLONOMY This is what has occurred to thenotion of “constraint force”–it represents anabsolutely natural extension of the word “force”from its normal, Newton-style applications.55 PaceHertz, a better resolution to our regularforce/constraint force dilemma is not to ban eitherusage from mechanical discourse altogether, but tosimply keep better track of their underlyingconceptual dependencies.

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 been misapplied through some violation of its absolutist, possible-world-dividing significance, but simply because the shifting contours of extended applicationhave invested the word with altered inferential requirements.

In any case, if Wittgenstein didn’t view natural developmental patterns in thislight, I certainly do! Conceptual paradoxes frequently emerge as the collateral sideeffects of otherwise heathy patterns of linguistic improvement and a hallmark of truephilosophical insight lies in our capacities to explicate the developmentalunderpinnings of longstanding muddles.

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1. Ludwig Wittgenstein, On Certainty, trans. by Denis Paul and G.E.M. Anscombe(Oxford: Blackwell, 1969), §§96-7. With respect to my title, long ago I assignedthis label to a popular lecture I had prepared on modal logic. As it happened, thistalk had to be cancelled. Afterwards, I received a note from the University’s eventoffice asking when it would be rescheduled, for they had received many requestsfrom the Extraterrestrial Club on that score. At the time, I was too timid to employthe title again, but now, in my older years, I have decided to be bolder. I hope theClub likes it!

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

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

4. Tractatus Logico-Philosophicus (Abindon: Routledge, 2001) and PhilosophicalInvestigations (New York: Wiley-Blackwell, 2009).

5. 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 governsthese parcels 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 formalconsiderations we shall canvass.

6. Investigations, op cit. §18.

7. David Lewis, Counterfactuals (New York: Wiley-Blackwell, 2001). Essay 6scrutinizes these proposals from an applied mathematics point of view.

8. In his celebrated report on turbulence, Osborne Reynolds writes: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.

(“On Vortex Motion” in Papers on Mechanical and Physical Subjects Vol. I(Cambridge: Cambridge University Press, 1900), p. 185). An allied “pride in logic”appears responsible for the Theory T shortsightedness of much contemporaryphilosophy.

9. Although, for historical reasons, I align Saul Kripke’s name with these worlds,the real architect of “possible worlds in the philosophy of science” is the late DavidLewis. He is also the prime founder of the school of analytic metaphysiciansdiscussed in Essay 6.

Endnotes:

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10. For more on this, see Essay 9 and my "Can We Trust Logical Form?," Journalof Philosophy XCI, 1994.

11. E.g., Donald T. Greenwood, Classical Dynamics (New York: Dover, 1997).

12. Franz Reuleaux, The Kinematics of Machinery, A. Kennedy, trans..(New York:Dover, 1963). It includes a long philosophical preface that is worthy of carefulattention.

13. The surprising subtleties of the term “mechanism” are not adequatelyappreciated by many philosophers, who would benefit from Reuleaux’s deeperinsights into the subject. I amplify on these issues in Essay 2, appendix 2 and in“What is ‘Classical Mechanics’ Anyway?” in Robert Batterman, ed., OxfordHandbook for the Philosophy of Physics (New York: Oxford University Press,2013).

14. In Watt’s steam engine, the conversion operates in the converse direction:straight line motion delivered by the piston gets converted into a circular arc at thepump arm.

15. For details, see S. Molian, Mechanism Design, (Oxford: Elsevier Science,1997), pp. 15-8. Close study reveals that the output tracing is not perfectly straight,but only supplies the best fit to be achieved within this class of possible inventions(“best fit” can be plausibly measure in several different “norms” but they all agreeon the optimality of the Watts’ sizings). In the 18?, the French Engineer Peaucilliershowed that theoretically perfect conversion can be achieved employing the sixcomponent mechanism pictured alongside Watt’s in a different “basin of mechanicaldesign.” One would greatly like a “perfected design” search algorithm that couldleapfrog out of the Watt design space into that of Peaucillier, but, to the best of myknowledge no such method has ever been found. Even back in the 1870s JosephSylvester commented (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.

Eugene S. Ferguson, Kinematics of Mechanisms from the Time of Watt(Washington, DC: Smithsonian, 1962), p. 30.

16. Wittgenstein, Investigations op cit, §194.

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17. The links that connect isolated thoughts seem indeed to be almost entirelydestroyed,--we have to reconstruct them. We see the whole picture before usonly like a faintly outlined or half-washed-out picture, and the painterhimself can hardly furnish us with any better explanation of it than we candiscover for ourselves. Indeed, the comparison holds good in more than onepoint. In each new region of intellectual creation the inventor works as doesthe artist. His genius steps lightly over the airy masonry of reasoning whichit has thrown across to the new standpoint. It is useless to demand fromeither artist or inventor an account of his steps.

Reuleaux, op cit, p. 6. Such opinions are very much in accord with the overarchingthemes of my own work.

18. Reuleaux, op cit, p. 231. The subsequent quotation is from p. 85.

19. Essay 1 explores these methodological parallels more fully.

20. Notoriously, in the Tractatus, the importance of this “grammar” can only“show”itself, it can’t be coherently talked about. In his later thinking, an allied“importance” reveals itself more locally within the salience that particular “forms oflife” offer to us. I’d recommend not employing this terminology in our ownendeavors.

21. Wittgenstein, Investigations, op cit. §§193-4.

22. Ludwig Wittgenstein, Remarks on the Foundations of Mathematics (Oxford:Blackwell’s, 1964), §128.

23. J.L. Lagrange, Analytical Mechanics (Berlin: Springer Verlag, 2001). Portionsof our discussion will be patterned after Duhem–see Essay 3 for background. Thegreat advantages, with respect to modeling security, of framing physics “from thetop down”–that is, forming its constitutive principles based upon direct laboratorytest rather than errant molecular speculation, made a great impression on the greatmethodologists of the nineteenth century, such as Duhem, Green and Stokes (thepalpable superiority of Cauchy’s top down methods with respect to elasticity incomparison with Navier’s bottom up efforts played a significant role in thisthinking). However, though blending information drawn from different size scales isprudent policy from the point of view of modeling security, small disharmonieswithin this data can generate significant conceptual confusion. In a nutshell, that isa major theme of this book rendered concise.

24. In particular, any rigorous text on continuum mechanics is filled with specializedspaces of bewildering subtlety, whose purpose is arrange various modelingingredients in harmonious accord. Essay 8 addresses concerns of this type.

25. Classical mechanics of a Lagrangian cast works with a combination ofNewtonian forces (operating on a point mass scale) and applied constraints (such as

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rigid rod connections or sliding along a wire), which bind these centers togetheralong higher dimensional lines or surfaces. As indicated in Essay 2, thesedimensional disparities create data amalgamtion difficulties akin to those attendingconventional multiscalar modeling. To stress these affinities, I sometimes describeour points-mixed-with-curves problems as involving “asymptotically separatedscales”; the reader may freely reject this characterization if it doesn’t seem helpful. Throughout this essay, “Lagrangian technique” is restricted to the virtual work +d’Alembert’s principle tactics of the Analytical Mechanics, and does not apply tothe less general “Lagrangians” associated with Hamilton’s principle and modernquantum physics. The latter rarely involve constraints in the manner we require andhence do not illustrate the cross-scalar applications that supply classical Lagrangianmethodology with its peculiar strengths.

26. In contrast, the spring between beads 2 and 3 does not constitute a constraintbecause its length alters under applied forces and because it exerts active forces ofits own.

27. In wider applications, the terms “coordinates,” “mass” and “force” need to begeneralized for Lagrangian techniques to work. I’ve tacitly included a torque, arotation and a moment of inertia on bead 4 to illustrate this enlargement.

28. I’ve always regarded this phrase as misleading: it indicates that reaction forcesinternal to the device exactly cancel the outside force’s capacities to move partsagainst the constraint.

29. We’ll later see that the local acceleration attaching to qi should be extractedfrom V(q1, ...q4, θ) through partial differentiation in the qi direction.

30. Duhem emphasizes the degree of “theory” involved in assigning a complexmechanism a capacity for potential energy storage in this manner, because in reallife, we lose a tolerable of inputted work to friction whether we move our beadsaround. As Essay 4 explains, Duhem regards it as a chief hallmark of a goodtheorist that he or she should be able to see through the degradations of friction tothe more fundamental patterns of energy exchange operating with the target system. We might further observe, in connection with other Essay 2 themes, that attendingto the collective character of this energetic locking together means that we suppressthe compression waves that rapidly pass through the wires whenever we jiggle ourbeads in any manner. Formally, the holistic character of standard potential energyallocation produces a descriptive situation in which such energies can rarely becorrelated with specific spatial locations and can be extracted as workinstantaneously from any position anywhere within the span of the mechanism, inthe same manner, that pushing on one end of a rigid rod must, mathematically,induce an immediate response at its other extremity.

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31. In fact, without intervening lubrication, metal beads will bond tightly to metalwires. As remarked elsewhere, orthodox classical mechanics of a mass pointcharacter probably lacks the modeling capacity to set up a plausible microscopicmodeling for a bead sliding along a wire. A central advantage of Lagrangiantechnique is that it effaces our descriptive endeavors from foundational concernssuch as this.

32. We’re actually only interested in the stable forms of constrained equilibriumbalance, as will emerge in the details to follow.

33. Pilgrim’s Progress, after all, serves as a leit motif throughout these essays.

34. These policies rely upon d’Alembert’s principle and are discussed at greaterlength in Essay 4. As explained there, the construction can be profitably viewed asa lift from one possibility space to another.

35. The finite difference method described in Essay 2 is the simplest scheme of thistype.

36. This is why Lagrange’s form of analytical mechanics is usually called a virtualwork or a virtual velocities technique.

37. We moderns have become accustomed–perhaps, overly accustomed–to thenotion of potential energy but, historically, this was not so; Lagrange wrote beforethe conservation of energy was established upon a potential energy basis. Indeed,the notion of potential we employ here–a system’s internal storage capacity forperforming external work–is a subtle, holistic notion. Readers of Essay 5 willrecognize that the standing wave modes of a vibrating string likewise representholistic, locked together behaviors that stretch across its entire length. The chiefdifference between our string and the present circumstances is that the formerpossesses a large number of independent repositories into which input energy can beindependently stored, half of the time as strain potential energy, whereas our beads-and-wire system (usually) has only one. But in both cases, the characteristic lengthscale of the energetic storage is as wide as the entire target system.

38. 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.

39. This duality of purpose explicates the Essay 4 mystery of why Duhem claimsthat sound physical practice should simultaneously serve as an epitome ofexperiment yet illustrate the free creativity of the scientist involved.

40. I have not explicitly mentioned Jim Woodward’s work on manipulationistcounterfactuals in this essay, but they frame central topics within Essay 6. It is easyto see that present considerations strongly bolster the argument against theadvocates of “counterfactual grounding” supplied in that essay.

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41. I write this knowing full well that some devotee of the genre will take up thechallenge, if they ever stumble across this passage. Science fiction writing beingwhat it often is--imaginative expansions of ur-philosophical conceits, no matter howdefective--, someone might construct a narrative in that vein, but I think it willsignify little. One can also write long novels about the fairies who live on rainbows,but these efforts tell us little about the useful meteorological information we conveywhen we talk of “rainbows.”

42. For example, persuading the possible boundary conditions to harmonizeproperly with the possible interior behaviors often requires very delicatearrangements.

43. As Essay 4 remarks, many philosophers of science loosely presume thatphysical theories produce “models” that represent complete “cosmologies” inDuhem’s sense: full universes replete in detail on every scale of size and time. AsWS, pp. 193-203,argues, these goals are probably unattainable within the closedorbit of “classical” modeling ingredients, for all known methods of constructingstable universes from stable molecules rely upon deeply quantum mechanicalphenomena such as tunneling. Presumably, inflationist thinking with respect to thenotion of “model” parallel to that witnessed within Kripke and Soames has played atacit role in misconstruing the mathematical structures delineated within real lifephysics in this manner.

44. E.g., provisos that prevent problematic data migrations from one strategiccontext to another: “You can’t appeal to a situation like that to show that p must bepossible in present circumstances; you’re confusing apples with oranges.”

45. In the Mel Brooks/Ernie Pintoff cartoon from which I borrowed my blot, theeponymous “Critic” identifies the figure as “two things having sex.”

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

47. WS, pp. 22-4.

48. In the spirit of old soldiers never die, but merely fade away, storybookpossibilities rarely relax their grip on “rainbow” completely, but their capacity toarrest sound reasoning as “counterexamples” lessens. See WS for more on thistheme.

49. See Essay 1 for these “scenarios” and considerable efficiencies achieved byapproaching semantics in a contextually sensitive manner.

50. J.W. von Goethe, The Metamorphosis of Plants (Cambridge: MIT Press, 2009).

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51. Reichenbach, The Theory of Relativity and A Priori Knowledge (Berkeley:University of California Press, 1965) and Friedman, Dynamics of Reason (PaloAlto: Center for the Study of Language and Information, 2001).

52. I employ the phrase “asymptotically separated scales” to mark the differencebetween the mathematical entities that need to work together, viz. the 0-dimensional“beads” (modeled as points) and the 1-dimensional wire (modeled as a curve).

53. Hertz employs Gauss’ least work principle to this purpose.

54. Heinrich Hertz, The Principles of Mechanics, translated by D.E. Jones and J.T.Walley (New York: Dover, 1952), p. 8.

55. See Essay 6 for allied behaviors of this sort.