Essay 4

Two Cheers for Anti-Atomism

What route would it take? Several paths lay in sight; the entrance to eachwas wide open and quite smooth; but hardly had one gone along a path thanone saw the causeway shrink, the track of the route become unclear; soonone would see no more than a narrow path half hidden by thorns, cut acrossby bogs, bounded by abysses... Where is he who would be carried through tothe end desired, who, one day, would come upon the royal way?... He whosows therefore cannot judge the value of the grain; but he must have faith inthe fertility of the seed, in order that, without fainting, he may follow thefurrow he has chosen, throwing ideas to the four winds of heaven. ---Pierre Duhem1

(i)

Throughout these essays I argue that the coarse categorizations of Theory Tthinking–viz., the logic-centered conceptions of scientific organization canonized bythe logical empiricists in the mid-twentieth century–continue to dull our sensitivitiesto the strategic subtleties of working science, in a manner that impedes probingphilosophical diagnosis, even within topics far removed from the philosophy ofscience. However, merely calling ones adversaries unpleasant names (“Oh, youlousy theory T thinker, you”) won’t improve our circumstances much. Beneath aveneer of feigned rigor, Theory T thinking is actually quite vague, taking refuge invocabularies that it rarely redeems in a concrete way.2 Hazy categorizations of thisstripe have locked contemporary philosophers into methodological dogmas of whichthey are scarcely aware. The best method I know for dispelling these diagnosticmists is to articulate clear exemplars of scientific development that do not followexpected Theory T pattern, and ask, “Have you considered these developmentalpossibilities?” I believe that the multiscalar architectures of Essay 5 and the weaksolution policies of Essay 8 serve these purposes well. Unfolding in a ratherdifferent manner, Pierre Duhem’s extensive writings on classical mechanics lay out,in admirably detailed terms, another pattern of formal development that usefulscience might pursue. Indeed, we can claim better than that, because modernprogress within the hugely successful program of twenty-first century continuummechanics has largely followed Duhem’s key structural recommendations. Of this

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Duhem

work, G. A. Maugin writes:[T]he main advance that emerged after the rejuvenation (in fact, a true“rebirth”) of [materials science] by [modern authors after 1945] was the firmgrounding of continuum mechanics in a thermomechanical framework, tothe posthumous satisfaction of Pierre Duhem. 3

Our central purpose in this essay is to explain how this alternative architectureoperates and the strategic motives behind its employment, by tracing theargumentative arc of Duhem’s The Evolution of Physics.

Because Theory T thinking is inherently vague, while deceptively persuadingits practitioners that they operate as paragons of logisticalrigor, concrete test cases provide our best tools for pryingout the unacknowledged dogmas that hide just below thebland outer surface of Theory T thinking. As we’ll latersee, one such presumption maintains that scientific theories(at least of a “fundamental physics” cast) should attempt tocapture the freely autonomous behaviors of nature withintheir mathematical netting. These formal expectations lurkin the background of the wide array of contemporaryphilosophers who freely appeal to “physical models” or“physical possible worlds,” without recognizing that thesetenets should be regarded as debatable. To my mind,Duhem’s thermal considerations supply dispositive reasons for resisting these tacitobligations. FIG: DUHEM

Unfortunately, he often cloaks his arguments in embroidered language and wemust often pause in the pages ahead and to remove these layers of extraneous outergarment. For example, Duhem characteristically claims that the task of delineating acosmology should belong to metaphysics, not science. With that oracularpronouncement, a thousand ships of misinterpretation have been launched. Understood properly, in terms of the concrete problems of developing a thermalphysics, the misty turns pellucidly clear.4 So my central task in this essay is torelocate familiar Duhemian sentiments upon the bedrock of firm scientificprocedure. With respect to methodological presumptions of this character, one of thehardest aspects of battling a dogma is that dogmatists are sometimes proved right. We hope to persuade a gambler that his rabbit foot isn’t very helpful and hereplies,“Well, I won big last week, didn’t I?” Yes, science has certainly enjoyed itssuccesses in implementing the architectural policies of canonical Theory T

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expectation and Duhem intends, in our opening motto, to acknowledge this veryfact. But, as George Elliot writes, “Among all forms of mistake, prophesy is themost gratuitous.”5 In opposing the influence of futuristic presumptions of a TheoryT cast, we should not fall into contravening dogmas of our own, for philosophyproves most useful when it enlarges our intellectual horizons rather than constrictingthem. I firmly believe that forecasting the future of science at the present time is afools’ errand, for the current fabric of mathematical physics presents us with aconfusing array of computational anomalies and oddly harmonized correctives thatblock any clear vision of its future policies. The the loose categorizations of TheoryT thinking can persuade us otherwise, by dulling our attention to the procedures weshould find puzzling. “I can see clearly ahead, for I detect no troubling clouds uponthe vistas of faraway science,” the victim of Theory T complacency proclaims. “Butthat’s only because you suffer from blurry vision,” we respond.6

Although philosophers should refrain from theoretical rhabdomancy, they cannonetheless help us better understand, through reflections upon the wider sweep ofprofitable human development, why the otherwise progressive policies of scientificimprovement sometimes deposit us in pits of conceptual confusion, on thosefrequent occasions when our computational pencils outpace our intellectualunderstanding of the underlying tactics implemented.7 Here Duhem’s discussion ofthermal physics serves us admirably, through his careful diagnosis of thecomplicated architecture that lies behind the admirable descriptive utilities of wordssuch as “temperature” and “entropy.” These are all useful and familiar words, yetfrustratingly confusing when closely probed. In contrast, contemporaryphilosophical thinking has performed rather poorly in these diagnostic regards,presuming on inadequate grounds that “temperature” is just another “natural kind”term, whose foundations have been completely clarified within contemporarystatistical mechanics.8 By following Duhem’s reasoning through, we can learnimportant lessons about the complex manners in which words sometimes meet upwith their physical correlates.

Another persuasive lesson we can learn from reading Duhem is that scienceshould cultivate methodological policies that enhance its trustworthiness, even at theprice of conceptual simplicity. The reliability-stiffening virtues of rigid constraintssupply examples to which we repeatedly return throughout these essays, for we canconfidently affirm, “Whatever else happens, this rod is unlikely to bend.” But if we

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capitalizing uponopportunity

already know something, why not exploit that fact as effectively as we can? TheEvolution of Physics documents the manner in which tricky words like“temperature” obtain their wide and remarkably effective referential ranges bybuilding from one firmly established applicational plateau to another, rather as anautilus builds up its shell one chamber at a time.9

More generally, in building up a reliable body of scientificdoctrine, Duhem recommends that successful explanatory patternwithin mathematical physics should initially fasten its hooks uponthe firmest ledges of computational opportunity and work its wayoutward from these locales, rather as mountaineers conquer widerterrains by rappelling from one rock outcropping to another. FIG:CAPITALIZING UPON OPPORTUNITY Only time can tellwhether these explorations will eventually condense uponstructures that can be agreeably repackaged within a canonicalTheory T format. But this may not happen; our improvingdescriptive methods may stabilize upon more nuanced forms ofsyntactic architecture. The purpose of this essay is to illustrate aspecific alternative of this sort, following Duhem’s own proposals. .

(ii)

As already acknowledged, Duhem is frequently a lousy writer. If anopportunity to supply an ill-conceived metaphor presents itself, he will seize it. Asterling example of an exceptionally poor choice can be found in his well-knownThe Aim and Structure of Physical Theory.10 Duhem’s underlying purpose is tointroduce his readers to a formal construction that, in modern mathematicalterminology, is called “a one-form driving force derived from a backgroundpotential field.” Some misguided muse advises Duhem that a hazy comparison withthe different “intensities of genius” encountered across the parade of mankind willaccomplish his expositional purposes. When we turn to the more technical writingsthat provide the prime sources for the reading advanced here, he often explicates hiskey conceptions in an extremely abstracted format. He makes so many demands onbackground knowledge that his account becomes impossible to follow unless thereader is already familiar, from other sources, with the materials Duhem isattempting to “introduce.” A fair amount of what I will present here is simply anattempt to translate the argumentative arc of The Evolution of Mechanics into arecognizable tongue.

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On top of this, Duhem frequently slips into writing as fierce polemicist, whileI wish to extract a genial and ecumenical tolerance from his writings, in which weattempt to liberate developing science from every form of a priorist restriction. Ourreading concentrates upon the amiable and forbearing Duhem who composed ouropening motto, not the disagreeable opponent of atomism and British science whomakes his appearance elsewhere. In an allied vein, he frequently offers sentimentsof a decidedly anti-realist cast, but, again, this is not a vein of opinion from whichwe can usefully profit.11 Insofar as I can determine, these strains within his writingslargely trace to struggles with infinitesimals common to his era, as summarized in anappendix to Essay 3. These difficulties often sparked doctrines of essentialidealization amongst Duhem’s scientific contemporaries.12 Under this label, Iarrange methodological claims to the effect that scientists must artificiallymisrepresent key qualities of physical objects, focusing instead upon “idealized”replacements that allow the tools of mathematical physics to obtain a descriptivegrip upon the target system. This is a much stronger claim than the humbleobservation that it is often fruitful to begin our investigative probes by concentratingupon structures with advantageous symmetries and purged of dissipative non-linearities. With such initial knowledge firmly gained, our investigations can laterexpand to more realistic modelings utilizing perturbations upon these preliminarystudies.13 This milder “idealization” thesis is no more surprising than therecommendation that climbers should practice on easier hills before they travel tothe Himalayas. It would be conceptually preferable if these early stages inmathematical exploration were not called “idealizations” at all, but, unfortunately,the terminology has become long entrenched in this role. As philosophers, weshould distinguish this rather innocuous observation from the fiercer essentialidealization thesis, which maintains that a modeler must permanently misdescribeher targets so that the descriptive enterprises of mathematical physics can getunderway. This is why such “idealizations” qualify as essential; they must beaccepted before the project of applying mathematics to the world makes sense at all.

Historically, such attitudes were significantly motivated by the complicationsof setting up suitable equations for flexible media in a coherent way, a task thatEssay 3 calls “the problem of the physical infinitesimal.” These difficulties largelystem from the complexities of arranging the dimensionally incongruent ingredientsof continuum physics into functional accord. With the help of extensive twentiethcentury work on measure, tensors and manifolds, Clifford Truesdell and Walter Nollestablished better foundations for continuum mechanics in the 1950s, in a mannerthat does not rely upon essential idealization interventions.14 This recasting is now

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presentation and reality

universally accepted within applied mathematics circles, although one can stillencounter reprisals of essential idealization appeal within elementary textbooks thatattempt to avoid complicated mathematics. Duhem’s lines of thought can’t be fullyfollowed without appreciating this methodological heritage, but we shall bypassthese issues here. In the appendix, however, I shall briefly discuss Duhem’s strikingcontributions to the so-called Quine-Duhem thesis, in which these issues of essentialidealization in continuum mechanics play a surprising role.

Throughout these pages, I frequently complain about loose invocations of theterm “idealization” within philosophical discussion, largely because so manydivergent threads get twisted together under this common heading. Labeling amethodological procedure as “idealized” should mark the start of a more detailedmethodological analysis; it should not qualify as a suitable conclusion.15

As a consequence of these exclusions, the anti-realist and phenomenonalistDuhem cherished by modern commentators such as Bas van Fraassen and NancyCartwright16 will scarcely make any appearance in these pages, except when I wishto remove their characteristic gloss from some otherwise helpful Duhemian passage. This is not to claim that Duhem did not endorse some of their themes; it is merelythat I have little sympathy with such threads myself. Some readers will bedisappointed with these exclusions, for Duhem’s modern acolytes often cherish hiswritings precisely for their sweeping generalities (oracular outpourings can bereadily quoted to any purpose whatsoever). Let it be stipulated, in concession tothese valid, if not especially rewarding, avenues of interpretative emphasis, that itwill be a thoroughly sober and dry-cleaned Duhemwho sallies forth in this essay, whereas our real lifeprotagonist could be an obnoxious pugilist, fond ofexaggeration and xenophobic rant.17 FIG:PRESENTATION AND REALITY

(iii)

To profit from the Duhemian insights I find valuable, we must attend to thechief scientific problematic in which he was engaged, viz., constructing a workablethermomechanics18 in which the notions of absolute temperature, heat and entropyenter on an equal footing with the familiar qualities of standard classical physics,e.g.. force, mass, potential energy, stress and strain. Duhem distinguishes severalvarieties of traditional, thermal-free mechanics and favors Lagrange’s virtual workexposition as the best exemplar of this circumscribed approach. He calls this

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coupling of thermal and mechanical effects

approach “The Old Mechanics” and proposes to erect a richer “New Mechanics”upon its foundations that embraces thermal phenomena on equal terms as well. Thecore of this essay is devoted to explaining how this enlargement proceeds and theintriguing morals Duhem extracts from the exercise. Core classical mechanicscan be approached in a variety of manners that we often regard as more or lessequivalent. But Duhem’s constructive project requires that we start from thespecific formalism favored by Lagrange within his Analytical Mechanics, for itsstatics dynamics patterns supply Duhem with the lower level platforms uponwhich he erects his thermal architecture.19 Inattention to this structural detail havelead to significant misunderstandings of Duhem’s argumentative intentions.

The advisability of bringing thermal and mechanical description into unifiedcoordination is readily apparent, for such phenomena couple to one anothercommonly in ordinary life. Consider an iron bar. If we strike one end with a mallet,we impel a pulse of compressive stress through its interior, a process governed, tofirst approximation, by the familiar wave equation. Likewise, if we heat anextremity, we send a parcel of heat across the bar, in rough accordance withFourier’s heat equation. But these two effects affect one other, greatly complicatingthe detailed flow. FIG: COUPLING OF THERMAL AND MECHANICALEFFECTS In particular, energydegradation within compressivewaves will gradually elevate thetemperature of the bar beyondFourier’s law predications andincreased temperatures will dilate thebar’s length. In many industrialsettings, coupling effects aresufficiently strong that we require aframework in which they can beconjointly addressed.20

This is not an easy task, and controversies as to execution persist to this day. In common modeling practice, frictional effects are treated entirely as dissipative

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what happens to the capacity for coherent work?

terms, with no attempt to capture the back actions that increased temperatures exertupon mechanical properties such as length. Within these treatments, the productionof heat represents a total loss in mechanical effectiveness; if a swinging pendulumloses some of motion by rubbing against the air, it will not be able to return to thesame point of highest potential energy as before. Nor will it be able to apply thesame amount of ballistic work to a target object as before. But this assessment istoo harsh; it is possible toreconvert the heatgenerated back to coherentmotion, as theaccompanying illustrationindicates. FIG: WHATHAPPENS TO THECAPACITY FORCOHERENT WORK?The silly gizmo illustratedheats water with frictional heat, which then coherently drives the turbine that createsthe friction. Without further infusion of energy, this contraption will eventually rundown, but this process takes longer than a simpler narrative of one-way energydissipation predicts. Indeed, Watt’s great improvements in steam engine efficiencystem from the fact that he realized this could be done. Typical frictional systemslose their capacities for performing coherent work as time passes, but the processreflects a two way exchange between coherent motion and heat, not the singleunidirectional process captured within simple dissipation models. In thesecouplings, a system’s abilities to perform coherent work become graduallydegraded, but not lost altogether as in our simpler dissipation models. The core ofthermodynamic thinking, developed by Rudolf Clausius, invokes the subtle notion ofentropy to codify the notion of coherent work capacity lost and regained.

Clausius and his followers addressed these issues of energetic efficiency onlyfor largish blocks of homogenous material (e.g., a gas in a flask) that were movedfrom one state of constrained equilibrium to another by external manipulations suchas increased pressure upon a cylinder filled with gas or immersion in a heat bath. Entropy and allied thermal measures were not applied to the transitory states inbetween, in which the manipulated material is internally adjusting to its enforcedalterations. But our wave transmission problem inherently reflects internal turmoilof that type--we are interested in the internal adjustments that occur within the baras a wave passes through. So the task that Duhem sets himself in his New

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Duhem’s recipe for a thermal continuum mechanics

Mechanics program is precisely that ofaddressing how these bidirectional formsof energy exchange occur within adynamical setting. To do this, he proposesthat Clausius’ thermodynamicconsiderations be applied at theinfinitesimal level, which is where thecoherent energy/ heat exchanges takeplace in simple heat transport. FIG:DUHEM’S RECIPE FOR A THERMALCONTINUUM MECHANICS

Some terminology: Duhem uses thephrase “frictional system” to marktreatments in which these two-wayentanglements are actively modeled; hedoes not intend pure mechanical formulas to which supplementary dissipative terms havebeen added, with no attempt to monitor the effects of the lost energy. I employ thephrase “pure mechanics,” Lagrangian or not, to designate any flavor of classicalphysics that eschews thermal notions as fundamental.

Duhem’s infinitesimally focused policies lies at the core of most modernforms of so-called non-equilibrium thermodynamics. This approach represents asignificant conceptual innovation for which Duhem is commonly ascribed historicalcredit. Such considerations require appeals to entropy production of some kind andintroduce great subtleties into the notion of potential energy. All of this involvesdelicate concerns with respect to the rates at which these various exchanges takeplace and disagreements about difficult cases persist to this day. Duhem himselfonly achieved partial success in these modeling efforts, a fact that he fullyrecognized (and explain the hesitant “faith in the seed he sows” remarks of ouropening epigram). Many of these difficulties trace to the fact that the back actionsof friction-generated heat against mechanical quantities such as length must bemonitored by abstract entropic considerations, due to the centrality of bidirectionalenergetic degradation within these exchanges. These descriptive challenges force aconceptual layering upon the language that Duhem ably analyzes for our benefit.

Allied cross-effects arise between pure mechanical quantities and factorsrelated to electricity and chemical composition and Duhem frequently cites theseadditional interactions as motivations for his full New Mechanics project, whichadvances proposals along all of these fronts.21 In the sequel, I shall largely

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concentrate on the more limited project of uniting traditional mechanics with thermalnotions.

(iv)

Before we enlarge upon a somewhat byzantine blur of motives, let meindicate why these seemingly narrow concerns are relevant to much larger swatchesof philosophical inquiry.22 Within contemporary philosophy of language,misapprehensions about “heat” and “temperature” have played a central role inencouraging a general complacency about linguistic meaning that has blocked manysignificant avenues of profitable advancement. This obstacle has arisen as follows. Following the suggestions of Saul Kripke and Hilary Putnam,23 thermal words arecommonly regarded as “natural kind” terms, whose semantical underpinningsappear exceptionally simple. “Temperature” is said to referentially align with themean kinetic energy per degree of molecular freedom within a target system in thesame direct way as “gold” allegedly designates the property of containingpreponderant amounts of unbonded quantities of the element with atomic number79. In its wake of this simplistic picture, various forms of ersatz apriorism haveemerged, in a manner I’ll discuss shortly.

Where do these misapprehensions come from? Standard philosophy ofscience primers claims that the slogan “temperature = mean kinetic energy perdegree of freedom” reports a straightforward property identity constructed on thepattern of “broom = brush of twigs or straw attached to a long stick.”24 But thisidentification raises many puzzles. Jiggling molecules constantly shift alter theirenergetic manifestations between kinetic and potential modes of expression, in thegeneral manner of a swinging pendulum. On very rare occasions, every moleculewithin a large ensemble may presently sit in their positions of maximal potentialenergy–does this mean that its temperature of the ensemble must have changed? Ofcourse not. Similarly, if we accelerate a block of metal rapidly along a track, itsoverall kinetic energy will increase considerably. Does this increased speed ipsofacto indicate that it has become heated? Of course not. Due to the modernfamiliarity of the word, lay people frequently develop a propensity for assuming thatnotions like “internal potential energy” denote simple kinds of stuff, akin to thecaloric of days of yore.

What has gone wrong? Natural kind stories neglect the crucial ingredientsultimately responsible for the inherent subtleties of thermal vocabulary, issues thathave surrounded thermodynamical concepts since their basic utilities were

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effectively consolidated by Clausius and Kelvin in the mid-nineteenth century. Tothis day scientific opinion remains divided with respect to the proper applicationalranges of “temperature” and “entropy.” At root, these conceptual concerns trace tothe fact, already noted, that these words serve as our chief means of numericallyquantifying the degradation in coherence that occurs when a target system storesenergy internally, e.g., the diminishments that reveal themselves within apendulum’s lessening capacities for shoving a target object in a desired direction. These work-related aptitudes supply the central criteria that distinguish a “pressure”from a “heat.” And these distinctions in turn rest upon subtle considerations ofcontrol and scale, as we shall learn in the sequel. Such concepts can be judiciouslyutilized only if their usage is carefully monitored by contextual parameters that keeptrack of the “considerations of control and scale” just mentioned. From this point ofview, the stage-by-stage assembly at the core of The Evolution of Physics correctlyidentifies much of the layered architecture that must get strung together beforethermal vocabularies can assume their full range of descriptive duties. None of thisis comprehensible on a naive, natural kinds picture.

Instead, the latter collapses “temperature”’s complex forms of word/worldentanglement into flat “Fido”/Fido simplicity, through its reliance upon themythology of “mean kinetic energy” identification. Behind these coarseningmachinations, I detect the beguiling encouragements of Theory T thinking. “Atbottom,” it coos, “all theoretical vocabularies operate in essentially the same way.” No, they don’t and the specialized duties required of our thermal vocabulariespungently illustrate why a proper semantic understanding of their underpinningsneed to be calibrated to the specific utilities they provide on our behalf.

As we’ve seen, thermal distinctions inherently rest upon abstractconsiderations involving energy degradation that are hard to capture in alternativeterms. Thermodynamics’ troubled career has been continuously plagued by naiveattempts to reduce its central notions–entropy, certainly, but temperature as well–tocontours that seem more palpably comfortable. Duhem strongly opposes suchdomesticating propensities as illusory and is absolutely correct in doing so (we’llconsider a characteristic example in the next section). I believe that the strong gripof natural kind thinking upon present day philosophical inquiry stem from thesesame tropisms and have opened many avenues of what I called erzatz apriorismearlier. If the slogan “mean kinetic energy per degree of molecular freedom” isregarded as capturing the “essence” of temperature and if we don’t inquire deeplyinto why that word “mean” is stuck in there, we can slip into a frame of mind where

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“Gee, Jimmy, look at these suckers wiggle”

we think, “Of course, once we recognize thesignificance of molecular jiggling, we canimmediately see that temperature simply has to be asreported by our kinetic energy story.” FIG: “GEE,JIMMY, LOOK AT THESE SUCKERS WIGGLE” That little “has to be” supplement has encouraged alarge body of metaphysical speculation that might beaptly characterized as conditional necessitarianism.

Once the possible world in which we live is determined, many facts about quantitieslike temperature emerge as metaphysical necessities, albeit of a tethered-to-a-possible-world character.25 But the proper morals to be learned from“temperature”’s bumpy semantic career are antipodal to such views. Useful wordsof a thermal character must work out their improving entanglements with the worldin patterns that are hard to foresee and depend centrally upon the unanticipateddescriptive opportunities that nature gradually places upon our plates.

(v)

Duhem’s rivals hope that an adequate form of thermal/mechanical integrationcan be achieved through escape to a lower-scale molecular realm whose applicabledynamics can be handled entirely within purely mechanical terms (such a doctrine isusually called the kinetic theory of heat). Duhem concedes that such accounts aretheoretically viable--indeed, he believes that they are, in principle, irrefutable–butthat they face a large host of empirical obstacles such as erroneous specific heatsand the like. Furthermore, serious issues of conceptual closure arise when weattempt to carry out this reductive project, for reasons outlined in section v. Methodologically, Duhem is troubled by the visceral insistence with which hisopponents doggedly pursue these reductive will o’ the wisps, dismissing out of handany prospect of a united thermophysics. As a contemporary specimen of theprejudices with which Duhem contends, consider Thomas Preston and J. RogersonCotter’s opinions in their contemporaneous Theory of Heat:

Motion [is] the primary basis of all phenomena. If we admit the belief whichlies at the foundations of chemical science, namely, that all materialsubstances are built up of simpler substances or elements, which maycombine in various manners, but which are unchangeable, and ever retaintheir distinctive properties, we are led to regard all changes in the universeas ultimately due to changes in the local distribution, or state of

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aggregation, of elementary matter, and therefore eventually brought aboutthrough motion. If, therefore, motion be the primary change which lies atthe basis of all other changes, the final aim of physical science must be todetermine those movements which give rise to all other phenomena, andtrace their origin and progress. The problem thus merges itself into one ofdynamics, and all the various so-called forces of nature must be estimatedby the same standard, namely, mechanical force, and this, in fact, isexpressed in the law of the conservation of energy. ... The question stillremains, what becomes of the motion when the kinetic energy of a systemdiminishes? Can motion ever be changed into anything else than motion? Ifwe assume a fundamental medium whereby to explain all the phenomena ofnature, then the properties of this medium ought to remain unchanged, andall other changes must be explained by motion of the medium. Such anassumption is quite philosophic, and the method of procedure is certainlyscientific. An evident reply to the question of what becomes of the motion ofa projectile rising upwards is that it passes into the ether... The oscillationfrom kinetic to potential, and from potential to kinetic, in the case of thependulum, is then, from this point of view, merely an interchange of energyof motion going on between the mass of the pendulum and the ether aroundit. According to this view all energy is energy of motion, and must bemeasured by the ordinary mechanical standard... [Accordingly, all energy is]probably kinetic.26

Note that Duhem’s thermomechanical suggestions have been silently set asidethrough hazy percepts of a “philosophical” kind, resting upon significant naivitieswith respect to the notion of internal potential energy. In league with otheradvanced thinkers of his time, Duhem hopes to liberate physics’ exploratoryprospects from unwanted intercessions of this character, in support of a stance oftendubbed “the free conceptual liberty of the physicist.”27

In modern commentary, however, Duhem is often cast as a conservative, anti-atomist villain to Jean Perrin’s visionary good guy in a white hat. In this stocknarrative, Perrin avails himself of Theory T liberties with respect to hypothesisformation, boldly positing unobservable entities and taking fruitful advantage of theconceptual liberties of the creative scientist. Within this narrative, Duhem is viewedas a methodological atavist who attempts to clip these fruitful flights of fancy byholding Perrin’s nose to the grindstone of phenomenological induction uponexperiment. To be sure, the hyperbolic extremities of Duhem’s prose sometimesencourage such a reading, but, for the most part, he does not argue from such a

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Mach

perch at all. Instead, he seeks a methodological absolution from the ingrainedprejudices that he detects within the obstinate core of kinetic physics preference.28 He is absolutely correct in raising these concerns, for the true utilities of thermalvocabulary cannot be correctly identified if we slip into Preston and Cottle-likerationalizations. The need for close conceptual scrutiny of the sort Duhem practicescompletely transcends all issues of whether one should postulate molecular structureor not. This observation will become clearer when we review modern multiscalarapprocahes in the final sections of this essay.

Like his comrade-in-thermal arms, Ernst Mach, he further believes thatPreston and Cottler’s preferences stem from ingrained forms of instinctivereasoning, whose guiding advice should be set aside in constructing a sounddescriptive science. He expresses his underlying agreement with the followingcharacterization of “instinctive knowledge” drawn fromMach’s The Science of Mechanics:

[I]nstinctive knowledge enjoys our exceptionalconfidence. No longer knowing how we have acquiredit, we cannot criticize the logic by which it wasinferred. We have personally contributed nothing to itsproduction. It confronts us with a force andirresistibleness foreign to the products of voluntaryreflective experience. It appears to us as something freefrom subjectivity, and extraneous to us, although wehave it constantly at hand so that it is more ours than are the individualfacts of nature. All this has often led men to attribute knowledge of this kindto an entirely different source, namely, to view it as existing a priori in us(previous to all experience).... Yet even the authority of instinctiveknowledge, however important it may be for actual processes ofdevelopment, must ultimately give place to that of a clearly and deliberatelyobserved principle.29

For Mach, “instinctive knowledge” captures the rough rules of thumb that we utilizeto plot our way through the familiar world of everyday experience. Most of thepractical lore involved was originally developed by our aboriginal ancestors whichwe learn afresh as children. FIG: MACH If we are naive, we will presume thatthese psychologically ingrained rules enjoy some privileged a priori status that Machidentifies with “metaphysical thinking”:

We are accustomed to call concepts metaphysical, if we have forgotten howwe reached them. One can never lose one’s footing, or come into collision

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with facts, if one always keeps in view the path by which one has come.30 It is ersatz prejudices of this character that persuade Preston and Cottle that theprospects of a coequal thermomechanics should be ignored altogether. Duhemglosses these Machian psychological doctrines as follows:

We are then led to give [our physical proposals] a pretended demonstration. Such a demonstration takes as axioms a certain number ... of propositionsderived from our instinctive knowledge. A prudent mind must keep itself onguard against the logical value of such demonstrations. First, it is extremelydifficult to enumerate all the instinctive knowledge which is really in play insuch a deduction; almost no author succeeds in making all of it explicitwithout any omission or repetition. Moreover, instinctive knowledge is,after all, only a confused and unanalyzed pile of experimental givensacquired at imprecise periods of intellectual development.31

Duhem rightly observes that reservations about “instinctive knowledge” allied toMach’s had become widespread in the late nineteenth century, largely driven by therecognition that they inhibit the proper conceptual liberties of the theorizing scientistor mathematician:

From the time when Mach formulated his doctrines on the nature of naturalphilosophy, thoughts more or less similar to his have been developed inEngland, Germany, and France in the writings of numerous authors. Amongthese, some were subject more or less directly to the influence of theprofessor from Vienna. Others rediscovered these already discovered ideasby their own efforts without feeling the beneficial effects of his influence.32

Indeed, Duhem is fully joined in these emancipatory sentiments by his mechanistrival Heinrich Hertz, whose ultra-purist counter-proposals are discussed in theappendix. Hertz writes:

To many physicists it appears simply inconceivable that any furtherexperience whatever should find anything to alter in the firm foundations ofmechanics... This over-favorable opinion of the fundamental laws mustobviously arise from the fact that the elements of experience are to a certainextent hidden in them and blended with the unalterable elements which arenecessary consequences of our thought. Thus the logical indefiniteness ofthe [standard] representation, which we have just censured, has oneadvantage. It give these foundations an appearance of immutability; andperhaps it was wise to introduce it in the beginnings of the science and toallow it to remain for a while... In a perfect science such groping, such anappearance of certainty, is inadmissible.

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Here Hertz complains that standard instruction within mechanics consists of amotley of intuitive tactics that appeal to both action-at-distance forces andconstraints. But these tacit ingredients do not harmonize with one another fully andcreate conceptual tensions.33

How can we counteract the unhappy shackles of traditionalist thinking, inlight of the psychological fact that we can never escape our ancestral modes ofintuitive thinking (a fact strongly emphasized by Mach)? Three distinct forms ofcorrective policy come to mind. (1) We sever all ties between scientific vocabularyand the expectation-laden classifications of everyday life by regimenting alltheoretical usage within a tightly specified axiomatic system T of the sort that DavidHilbert developed for geometry. In the jargon of the time, this axiomaticdependency allows scientific vocabulary to become “implicitly defined” by theencasing T entirely through precise formal provisos. Through achieving semanticindependence in this self-supporting manner, scientific usages can emancipate themselves crisply the cavils of traditionalist metaphysicians or philosophers whocomplain about the misuse of familiar terms. I sometimes call such a point of view a“man proposes; nature disposes” view of the theoretical enterprise, because theencompassing T must be articulated as a completed formal construction unto itself,before we can fully ratify that all bonds of loyalty to preexistent prejudice have beensevered in this internally consistent manner. Real world scientific developmentbeing as it is, such perfected pinnacle is rarely obtained, but it represents thesyntactic goal to which we should aspire.

The basic attractions of modern Theory T thinking, as we have it today, traceto these commendable aspirations for wider conceptual liberties (a doctrine to whichno credible opposition now exists). Its characteristic emphasis on logical structurebegins here as well, as any suitable articulation of rigor within an axiomatic systemrequires. But in the ensuing years, various forms of untoward supplementation creptinto Theory T thinking and framed the repository of misapprehensions that continueto misdirect philosophical inquiry. Philosophical students of science came tobelieve that they had a duty to frame a general “theory” of scientific theory--ageneral account that would suit every form of scientific inquiry, rather than trying todiagnose the strategic innovations that distinguish particular forms of scientificgambit.34 In search of this grand generality, mid-twentieth logical empiricistsborrowed vocabularies (“initial condition” and “boundary condition”) from theapplied mathematicians and reshaped them in logic-oriented terms. Theseappropriations formed part of a grand scheme for explicating notions such as “law,”“cause,” “counterfactual conditional,” etc., as structural requirements of an entirely

-17-

logical cast as well. Later philosophers came to recognize that these logicalempiricist endeavors are unlikely to succeed, but they never returned theirterminological borrowings to their original diagnostic homes. And further unhelpfulpresumptions crept into Theory T thinking in the same way–we shall investigateanother central example in section (vii).35 In this gradualist manner, a collection ofconstrictive presumptions have silently congealed that divert our attention fromimportant modes of real life applicational development--autocratic dictates hidingbehind the genial visage of a somewhat addled grandparent. These developmentsare sadly ironic, because Theory T thinking’s career began in a burst of liberationistsentiment, but gradually declined into unwanted dogmas, rather as formerrevolutionaries become stiff potentates after long years of unopposed rule.subsequent generally. The problems that beset its doctrines are similar to those thatalways spoil idealistic ambition; the ground on which descriptive language must trodis too irregular and uneven to submit to preconceived schematism. Perhaps LionelTrilling is right:

We are at heart so profoundly anarchistic that the only form of state we canimagine living in is Utopian; and so cynical that the only Utopia we canbelieve in is authoritarian.36

In any event, axiomatic embedding comprises the pathway of conceptualliberation that I have labeled as (1) above. But its demands on formalization arevery stern, and several other policies for resisting traditionalism recommendthemselves. Unwanted conceptual burdens can also be lightened through directexamination of the psychological bundles in which these ingredients have becomepacked. Such genetic disentanglement can proceed along two basic And this canbe examining how their disparate ingredients came to be united in one in the firstplace. Genetic disentanglements in this manner can assume two basic forms: (2)histories of the developmental episodes that have led to the bloated threads ofcurrent thinking and (3) narratives of psychological development that explicate whythe unwanted ingredients remain as hard-to-dislodge fossils of earlier stages ofconceptual development. Duhem and Mach pursue both lines of attack in theirattempts to dislodge kinetic prejudice, albeit in rather different manners. Withrespect to mode (3), Duhem comments:

How do we proceed, however, when we want to teach someone approachinga science such as mechanics one of these economical formulas that containthe concentrated and condensed essence of a number of facts? Will weforcefully express the relevant formula and limit ourselves to adding that thesubsequent development of the theory will always show it to be in agreement

-18-

with the facts? According to the preceding ideas, this method would belogical, but the most elementary psychology would show that it would bedeplorable. Students would see only a form devoid of all content in the lawpresented in this fashion; it would remain unknown to them. How, then, canwe prepare their minds to acquiesce to that proposition and to capture itssense? By representing a path similar to the one the inventor has followed;by examining the few facts the inventor has first studied; by reproducing theseries of analyses and extensions by which the general law was derived. Thereal introduction to the expression of a principle of physics is a historicalintroduction.37 Make no mistake; Duhem’s underlying position favors (1): “science can work

within any theoretical framework it wants, as long as that framework is clearlydelineated.” Like Mach, he ventures into historical and psychological themesbecause of the pedagogical benefits just outlined and because he is aware that thetheoretical apparatus we can articulate at a given moment within a developingscience falls short of completed axiomatic perfection. As long as incompletenessprevails, unwanted prejudices cannot be fully eradicated through type (1) appealalone and recourse to supplementary modes of defense such as (2) and (3) arerequired. Duhem hopes that these instruments of conceptual cleansing willcollectively open the hardened hearts of dogmatic mechanists to the descriptiveadvantages of thermomechanical thinking.

I believe that Duhem is deeply correct with respect to his central concerns. Thermodynamic thinking plays a more central role in our thinking about the universethan many of us acknowledge and the key to recognizing this centrality requires thatwe keep its primary objectives plainly in view, viz., the task of quantifying thedegree to which energetic coherence becomes degraded by the operations of naturalprocesses. Such concerns remain deeply entangled with some of the acknowledgedpuzzles of modern physics, such as quantum decoherence and informational loss.38 Attempts to domesticate thermal notions via simple “identifications” of a naturalkind ilk direct our attention away from very central questions with respect to thedescriptive architecture of physics in which energy degradation presently takes acentral role. Whether these identifications can be defended extensionally or not,they direct our attention away from vital issues that we don’t presently understandas well as we’d like. The Evolution of Physics deftly outlines the cautiousfundamentals that allow thermal concepts to sink their anchors initially in the solidrock of physical circumstance and then describes the delicate lifts that extend thesepreliminary entanglements out to wider swatches of mountain. We don’t properly

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understand the significance of words like “temperature” insofar as we fail toappreciate the delicacy and tentativeness of its applicational extensions.

Observe that none of this has any bearing upon the existence of moleculesand their role as carriers of incoherent motion. To be sure, Duhem often writes as if they do, but this is a simple mistake whose origins we’ll trace later.

Accordingly, I don’t look upon the great merits of Duhem’s discussion in thesame way as he does, viz., as clearing away the conceptual rubble that persuades hisfellow scientists to cling to an antiquated theory T, when Duhem has a superiorreplacement T’ to offer. I instead view his work as practicing a novel kind ofsemantic analysis that diagnoses the strategic layering responsible for the successesof thermal words as we presently recognize them, while leaving open unchartedapplicational vistas yet to come. Philosophically, we are unaccustomed to wordsthat behave as delicately as ‘temperature” or “entropy,” yet they appear everywherewithin the fabric of language. This is a central lesson that Duhem’s insights canteach us, I believe. But Duhem did not get everything right in these architecturalrespects and in the final sections of the essay we’ll consider some recent work inthermal modeling that extends his baseline diagnosis in very helpful ways.

He proceeds by assembling an applicational range for his combinedthermomechanics in stage-like steps, whose exposition will confuse the bul of pagesto follow. Since some of the details involved are a bit complex, let me outline thefour developmental platforms that he identifies as essential prerequisites for thesuccessful employment of thermal vocabulary. They are: (i) Establish theconstrained equilibrium statics of purely mechanical systems following Lagrange’svirtual work policies in his Analytical Mechanics. (ii) Following Carnot andClausius, introduce a parallel statics for purely thermal vocabulary, again only forsystems in completely constrained equilibrium. (iii) Following Lagrange once againand also Willard Gibbs, lift these two forms of statics into less constrained forms ofdynamics, which we scale down to an infinitesimal level. (iv) On this dynamicalbasis, introduce the rates of quantity adjustment that allow us frictional effects todrain coherent energy from our systems. Only at this stage of of conceptualconstruction can thermal and purely mechanical cross-effects can be meaningfullyconsidered.

Duhem then considers further platforms beyond (i-iv) to encompassirreversibility and frictional damage, but we’ll touch on these more speculativesuggestions only briefly.

These particular developmental platforms should be viewed as parochial tothe special applicational requirements of thermal words and do not generalize to

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shock wave formation

other scientific vocabularies, that may require their own varieties of strategicsubstructure. Stages (i-iv) capture the behavioral criteria that allow us to distinguishcoherent from incoherent energy storage, a distinction that occupies a uniqueposition within the overall architecture of physical thought. Beyond this, ouranalysis doesn’t provide any sweeping analysis that is applicable to other scientificwords, beyond the limp suggestion that assembling fuller applicational ranges basedupon stage-like platforms is often a good idea. Duhem himself doesn’t make theselimitations clear, a factor that has led to significant misunderstandings of his centrallines of thought.

(vi)

Before we study the details of Duhem’s constructions, there are severalfurther aspects of his thermomechanical project that possess considerablephilosophical salience in their own right. The first is the rather surprisingobservation that, within a continuum physics setting, the standard collection ofpurely mechanical notions (viz., size, mass, force, etc.) do not comprise aninternally closed set. Even if we can initially describe a physical system in purelymechanical terms, its own natural evolution will transport it to condition in whichthermal concepts are required to carry its developments forward in time. Theunderlying mathematical facts, which are fairly generic with respect to nonlinearpartial differential models, trace to Riemann, with later improvements at the handsof Rankin and Hugoniot. These “demanding further descriptive parameters”behaviors emerge with immediate centrality within gas dynamics in the form ofshock wave formation. This is a subject to which Duhem made significantcontributions.39 He is fully aware of the mathematical behaviors discussed here,although, for various reasons I’ll rehearselater, he didn’t mark their salience for hisreaders as vividly as he might. Certainly,proponents of a purely mechanical account ofnature should address these concerns clearly.

The basic problem can be illustratedwith the simplest one-dimensional model ofnon-linear gas behavior, supplied by the theinviscid Burger’s equation (u/t + uu/x =0), where u(x,t) is the local pressure at pointx at time t calibrated around its relaxed state.

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It supplies a fairly reasonable model of how a compressed gas in a long tubebehaves. FIG: SHOCK WAVE FORMATION Let’s start off the system in a sinewave pulse of compression and expansion at one end of the tube. As such, its initialstate has been specified in wholly mechanical terms, without a wisp of thermalreference. The same holds for our gas equation as well–-nothing thermal mentionedthere. But watch what happens to this starting pattern as time advances. Its curvyportions become increasingly curvy and move to the right with increasing speed. Ata critical time tc these moving pulses collapse onto the same tube position andgenerate a descriptive inconsistency.40 If we conceptualize these events from alower-scale, molecular point of view, this outcome isn’t altogether surprising. Wehave required our gas to molecules move in the manner of a traffic congestion inwhich the cars in the middle of the pack move faster than those in the front. Thepredictable upshot is a horrible pileup at time tc.

Within a continuum mechanics framework, however, we’d like to avoidshifting to a lower-scale point of view. Due to our shock wave breakdown, ourmodeling equation doesn’t tell us how the pressure will shift across the shock frontand how it fans out behind. Riemann observed, with an error later corrected byHugoniot, that a natural appeal to entropy maximization resolves both questionsuniquely, allowing us to introduce a shock wave front into our picture and continueplotting the future evolution of the gas in a successful manner. Hence the failure ofconceptual closure noted above: we started within a realm that can be capturedwithin purely mechanics language, but have been forced to invoke thermalconceptions to carry our system forward.

Unreflective mechanists, of a Preston and Cotter stripe, will undoubtedlyrespond, “Well, this simply shows that shock wave phenomena need to be treatedby purely mechanical ideas at a lower scale level.” But this thinking leads to apotentially vicious regress, reminiscent of the “labyrinth of the continuum” woes ofEssay 3. By the Victorian era, most physicists had become convinced thatmolecules themselves must be composed of flexible stuff able to flex internally inresponse to outside agitation. But virtually all plausible models for their internalbehaviors are non-linear and hence are liable to shock wave blowups identical,mutatis mutandis, to those appearing within our original gas. Indeed, if we hammerrealistic materials hard enough, shock waves generally develop in their interiors. Ifwe apply the Riemann-Hugoniot resolution to these molecular shock events, we findthat we are again invoking thermal notions within the very descriptive realm inwhich we expected to escape from upper scale thermal appeal! Plainly, a desperateregress threatens. FIG: A SHOCK WAVE REGRESS

-22-

a shock wave regress

As Duhem correctly notes, we might arrest this foundational descent ifmolecules, or some lower grade sub-mundane particle, possesses nointernal structure, in the manner ofthe point masses favored byBoscovich and the French atomists. By Duhem’s time, however, fewphysicists believed that such anapproach could explain other aspectsof matter, such as their spectra.41 Fiercely clinging to a purelymechanical alternative for no reasonother than we don’t regard thermalideas as acceptably “fundamental” represents the sort of metaphysical prejudice thatDuhem hopes to erase from science. And he is right; we should avoid thesedeceptive pits of “plausible, intuitive” conjecture, for nature works in moresurprising ways than we can ably anticipate.

Modern knowledge of shock wave behavior helps clarify the situation. Whenthe pileup occurs, close encounter interactions between molecules set in, often of achemical nature. No representatives of these processes were included within ouroriginal Burger’s equation model at all, just as the boundary conditions we normallyassign to the surface of an extended blob makes any mention of the short-rangesurface effects active within this region. If such behaviors become important to us,we must open up the lower scale degrees of freedom we had iignore din our initialmodeling efforts. So it’s not surprising that our Burger’s equation model loses itsdescriptive validity when shock wave pileups occur. In this fashion, close-rangemolecular interactions act as what Essay 1 calls “invaders from scales below”; theycollect together and overwhelm the Burger’s equation behaviors that normallydominate on a macroscopic scale. On a macroscopic scale, these molecularactivities appear in the form of an entropic driving force that fans out the gas behindthe shock front; it is not a gizmo belonging to familiar dominions of pure mechanics. This out-of-the-blue appearance of non-mechanical novelty shouldn’t be regarded asaltogether surprising. We ignored some short-range physics within our originalmodeling and they have banded together to punish us for our neglect.

A closer look at the internal physics of shock wave formation underscores thedifficulty of escaping from thermal appeal. Appropriate rules for the chemicalprocesses that become activated within shock wave compression are generally

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contact between surfaces

temperature-driven; the probabilities that new bonds form or dissociate dependdirectly upon the local thermal environment arising with our newly shock front. Once again, we haven’t eluded the long arms of thermal notions through escape to alower scale sanctuary.

Duhem hopes to convey these formal observations to his readers, but heproceeds in a very unfortunate manner. Consider this distressing passage:

But at the very threshold of [our] investigation one objection delays us: Dothere actually exist frictional systems? Are not the particularities that wehave thought to observe and which have served us to define them simpleillusions? Do they not vanish when they are subjected to a minutely detailedanalysis? According to the majority of mechanicians, a solid body whichslides or rolls on another does not rub; but a multitude of small roughnessesbristle upon the two surfaces in contact; they foul upon each other,interlock, hook, and break; and the friction is only a fiction by which oneencompasses these innumerable and complicated, imperceptible phenomenawithout analyzing them... Such an [attitude] evidently has only one aim: tosubject the whole of physics to [purist mechanical principles] ... It would havea logical value if we could recognize from another source the legitimacy ofthis goal; if we had reasons to believe that all mass systems have to yield tothe rules of this Statics and this Dynamics. But of such reasons we havenone. To define the systems that are able to yield to these rules, we have,amongst all conceivable systems, cut out a certain group; we have made thisexcision in an arbitrary way through [an hypothesis that has been posited] apriori.42

From a modern perspective, we can rightly ask, “why is this man sneering?” Theconjecture that everyday sliding friction represents the product of “a multitude ofsmall roughnesses [that] bristle upon the two surfaces in contact; they foul uponeach other, interlock, hook, and break” is readily verifiable through modernmicroscopy. The largest portion of resistence to sliding takes the form of surfaceasperities that must get out ofeach others way lest they bondtightly to one another (lubricativeeffects of surface contaminationplay a significant role as well). FIG: CONTACT BETWEENSURFACES Readings of Duhemas a methodological conservative

-24-

who balks at positing unseen entities find their grounds in appalling passages suchas this.

As I understand him–or, at least, how I understand the better angels of hisargumentation–, Duhem is merely attempting to illustrate the follies of prejudice-founded conjecture, of the sort we witnessed in Preston and Cottle. But he hasselected a lousy case for illustrating these concerns; we scarcely want to excludeimportant details of surface chemistry from the annals of science.

Our shock wave considerations suggest a better recipe for extracting a valid,if milder, Duhemian lesson from this passage, one better suited to real-lifethermodynamic practice.43 Specifically, scientists freely utilize thermodynamicnotions to characterize how the microscopic effects responsible for sliding frictionactually work. For example, the asperities that inhibit sliding alter their shapes asextremely high local temperatures force their tips to melt. Likewise, entropic forcesinduce flows within the thin lubricating layers that coat these surfaces and facilitatesliding. And so forth. We do not escape thermal consideration even when we probethe smaller scale details of the frictional processes that degrade upper-scale coherentmotion into upper-scale incoherent heat. Note that these issues of size scale leadphysicist to apply the term “temperature” in a markedly scale-sensitive manner: “Ata normal temperature of 50oC, a sliding force of 2 nts proves sufficient for heatingthe surface asperities of a steel plate to their melting temperatures of 10,000oC and,accordingly, begin to impede lateral movement less.” So we require anunderstanding of “temperature’s” applicational significance that explains how thesame tiny sector of metal can be assigned two simultaneous temperatures, 50oC and10,000oC.

For various reasons to be discussed later, Duhem doesn’t provide a scale-sensitive treatment of “temperature” and this decision represents a significantoversight on his part. But such accounts can be constructed on architecturalprinciples similar to his. We’ll return to these intriguing issues towards the end ofthe essay.

Here is Duhem’s central point. To establish a descriptive repertory adequateto the requirements of scientific practice, thermal notions should be considered onall fours with those of pure mechanics. He should have kept his aversion tomolecular surface chemistry out of the mix.

It would have certainly proved helpful to his readers had Duhem presentedthe shock wave blowup in standard Riemann-Hugoniot form, as presented here.Why didn’t he? Here is my hypothesis. As soon as we acknowledge that long-termshock wave belongs within the wider conceptual realm of thermomechanics, we

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naturally seek supply plausible models for the two-way frictional exchanges thatmust occur, if energy is to be properly conserved. Duhem did this for shock wavesappearing within controlled environments and his proposals represent a significantcontribution to the subject. But the coupling effects he introduces retard the peakingof the wave front enough that the Riemann-Hugoniot singularities never form. Asresult, our original claim, “pure mechanics is incomplete because it can’t handle theshock waves that develop according to its own principles” must be weaned to “puremechanics is incomplete because it can’t handle the development of shock frontsadequately.” The salient phrase “according to its own principles” has beendropped. We arrive at a clash that sounds like a simple matter of “thus far, you’venot modeled shock fronts as well as me,” to which an opponent naturally replies,“Oh, just give me a little more time.” The internal failure in descriptive adequacycharacteristic of purely mechanical vocabulary becomes lost in Duhem’s recasting.44

One also wonders why Duhem didn’t insist more strongly on the lack ofempirical evidence for temperature regimes. Much of this stems from the fact thathis approach to the infinitesimal elements required in continuum mechanics force atheme of “there’s always room at scale levels below” upon him. We’ll considerthese issues in the appendix.

The observation that temperature attributions are sensitive to size scale neverappears in Duhem’s writings, insofar as I am aware. Such claims become centralwithin the multiscalar modeling techniques we shall discuss later and considerablyclarify the valid core of his own argumentation

(vii)

We noted that Duhem believes that rigorously delimited syntacticalframeworks represent the optimal format in which physics’ anointed missionsshould be pursued, although he acknowledges the fact that developing sciencesusually falls short of this ideal. In this stress upon rigor and implicit definability,Duhem’s views are very much in agreement with standard Theory T thinking.45 Butwe observed earlier that supplemental illiberalities have gradually crept into thelatter collection under the shade of logic-addled vagaries. Many of these accretionsDuhem would stoutly–and, rightly--reject. Among these is the presumption that it isthe duty of so-called “fundamental science” to delineate models of an evolutionaryand autonomous stripe, generally of a cosmological cast as well. Here“evolutionary” simply means “temporally progressing” and “autonomous” indicatesthat the manner in which a target system alters over time can be specified in an

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establishing a descriptive arena

isolated and unforced manner, entirely following the guidance of an internallydetermined dynamics.46

Plainly, working science cultivates many forms of physical description that donot pretend to be fully autonomous in the manner described, but treat localizedsubsystems as partially controlled by outside agencies or non-autonomously linkedto driving factors that operate upon higher or lower size scales. Here’s a typicalexample. A standard policy for dealing with the complexities of the solar system isto factor the celestial bodies into two classesaccording to their massiveness. FIG:ESTABLISHING A DESCRIPTIVE ARENA Insuch treatments, the larger bodies are treated ascontrolling the smaller ones, while the backactions of the latter upon the former are ignored. This ploy allows us to introduce the potential fieldcreated by the salient larger bodies as an interiorarena in which the smaller planets and theasteroids can dynamically interact with each other,in a manner non-autonomously conditioned by thelarge body gravitational field in which they frolic. Mathematically, this factoring strategy allows us to attribute a quasi-staticalgravitational field to the larger planets and sun, calculated from Poisson’s equation,which involves no consideration of time whatsoever.47 In doing so, we haveintroduced two convenient “cuts” that turn off the mutual force interactions betweenlarge and small demanded by Newton’s Third Law and we proceed as if the largebodies outside the cut are not affected by the movements of the little bodies. I haveseen the term “effacement” employed in this context–the cuts efface a dynamicinterior from an environment that can be characterized in quasi-statical terms. Ofcourse, the placement of these cuts need to be judiciously chosen; otherwise ourproblem will not simplify in computationally helpful manner at all.

Confronted with circumstances like this, we have a natural disposition toproclaim, “Oh, that sort of descriptive architecture is merely the product of a usefulapproximation technique; treating the same problem from the vantage point of morefundamental theory, viz, Newton’s unreduced law of gravitation, these cuts willutterly disappear as artifacts derived from an approximation scheme.” When weadvance assertions of this character, we must bear in mind the degree to which thesehypothetical fuller embeddings represent our aspirational hopes, rather thanaccomplished tasks. Sometimes these futuristic expectations are reasonably based,

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factoring S + SE into controlled trajectories

but they are often more conjectural than their proponents recognize.48 As we’ll latersee, basic confusions about the proper scope of the term “statistical mechanics”trade upon aspirational confusions of this type.

In the circumstances of celestial mechanics, Newton’ original law ofgravitation ably provides an autonomous modeling of all of our celestial bodies,large and small, to which our Poisson’s equation factoring merely represents aconvenient approximation technique.49

These musing suggest a general investigative recipe: if a description of systemS represents its target as behaving non-autonomously, then we should anticipate thatS can be imbedded within a larger grouping S + SE that evolves together in a fullyself-contained manner and where the controls exerted by SE on S merely representan effaced approximation to the fuller S + SE evolution. FIG: FACTORING INTOCONTROLLED TRAJECTORIES The result is a picture as shown, where we startwith a so-called phase space in which the autonomous time evolution of eachtolerated system is portrayed as a single point trajectory wandering through a highdimensional space. From this full manifold,we extract the special trajectories that toleratean approximate factoring into S and SE piecesin which S travels close to an SE base,regarded as a controlling manifold unto itself. In the thermal circumstances we shall soonconsider, reversible S’s will allegedly traveldirectly on SE at an infinitely slow pace,whereas more realistic trajectories will onlyhover near it.

We can turn these aspirationalexpectations into a formal demand of thischaracter: fundamental theories in physicsshould aim towards delineating autonomoussystems of this character, consigning all non-autonomous specializations to the dustbin ofconvenient approximations. Like Duhem, Iregard this organizational demand as both incautious and unproven. When we turnto thermal behaviors, we’ll find that such cautions are well-motivated, for significantforms of non-autonomous concern weave through classical thermodynamics’ basicforms of descriptive policy.

A related supplement can be added: “fundamental physics” will strive to

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supply us with a set of equations capable of governing the entire universe withoutany intermediary stitching. Duhem characterizes models of such arrangements as “cosmologies.”

I see this demand for autonomous description as mistily, yet deeply, ingrained within the contours of basic Theory T thinking, revealing itself in the lackof proviso for explanatory architectures that rely upon non-autonomouscircumstances in a more robust manner. Confronted with this question, manywriters whom I would regard as paradigmatic Theory T thinkers may denyvehemently that they accept the factoring assumptions outlined. My evidence to thecontrary is provided by a wide variety of suspicious clues, such as their penchant fordescribing physical theories as universally “delineating state spaces,” which isphraseology commonly reserved for autonomous systems within mathematicalphysics. Likewise, their unshaded appeals to “models” of “laws,” with no mentionof boundary conditions or other significant contributing ingredients, strikes me as acontinuing reflection of the naive architectural expectations encouraged by “lookingat differential equations through ODE eyes.”50 Many Theory T thinkers furtherpresume that “fundamental physics” must seek unblemished cosmologies inDuhem’s sense. These are represent autonomous trajectory articulated on thegrandest scale possible.51 Writers who unreflectively presume that a perfectedphysical theory will delineate complete “possible worlds” appear to have silentlyadded “csosmological coverage” to their tacit set of formal expectations.

But these are merely clues. Because Theory T thinkers employ terms like“law” and “boundary condition” in such hazy ways, convicting any party of theprecise crime of harboring a demand for autonomous description becomes difficult. So without accusing any particular author of autonomist presumption, let usacknowledge that such issues should be chased into brighter daylight of activemethodological consideration. On this score, Duhem’s writings prove particularlyhelpful.

In Essay 6, I complain of authors who freely appeal to the alleged contours offuture “fundamental physics” to justify the metaphysical categorizations they areeager to lay down in the here and now. I point out that such expectationsunwittingly sweep from the table substantive forms of alternative descriptivearchitecture that deserve more careful cultivation. I think that Duhem’s vision of aproper framework for non-equilibrium thermomechanics supplies a suggestivearchitecture of exactly this character, albeit arranged along different axes ofemphasis from the examples of other essays. None of this is to suggest that themetaphysicians should cease their endeavors, but they should recognize that their

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the proper arena for futuristicprognosis

tasks require deeper consideration of appliedmathematical technique than they presently presume. Daydreaming about unclouded Theory T eventualitiesmay strike us as pleasant and entrancing, but confidentclaims of futuristic prognosis should be left tophilosophers of the cracker barrel school. FIG: THEPROPER ARENA FOR PROGNOSIS

(viii)

With respect to thermal vocabulary, Duhem makes a mistake inwholeheartedly signing onto “conceptual liberty of the physicist” tenets, because hisactual procedures point to deeper aspects of the specific tasks that thermal wordsmust accomplish on our behalf. The little failures of kinetic theory that he presentsalong the way–e.g., that a proper theory of sound requires a temperaturecorrection–scarcely faze his kinetic theory opponents, because they typicallypresume that notions like that will disappear in some richer, “fundamental theory”yet to come. Duhem believes that he cannot rule out these pie-in-the-skyexpectations on straightforward empirical grounds, so he shifts to offeringmethodological pronouncements with respect to sound inductive policy that are fartoo restrictive. We witnessed an example in his dismissal of reasonablespeculations with respect to the underlying mechanisms of surface friction. Theseare the passages upon which readings of Duhem as a methodological conservativecommitted to phenomenological generalizations upon laboratory experiment arefounded. And it is undeniable that Duhem often drifts in those unfortunatedirections.

But I do not want to follow him there, for we miss most of what is valuable inhis concrete discussion. In real life, the fruitful employment of thermal termspresents us with striking applicational puzzles to which Duhem’s discussioncontributes greatly, not by supplying a grand thermomechanical housing in which allsuch usages become securely housed, but by stressing the developmental stagesrequired to build in that direction, with an expectation that our natural lines ofapplicational extention may break down before we reach the autonomoustrajectories of Theory T anticipation. To understand why this is so, we need toappreciate the tasks and considerations at the heart of thermodynamic thinking, as itwas first consolidated within the remarkable, but puzzling, work of Rudolf Clausius.

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projecting a timeless base manifold from special trajectories

His lines of scrutiny warn that we should become far more circumspect with respectto the applicational range of “temperature” than we would have otherwise dreamed. Inspection of the strictures articulated in every standard primer of equilibriumthermodynamics indirectly52 insist that we should never credit any solid object suchas a heated bar with a “temperature” or a “heat content.” Nor should we engage inloose, out-of-equilibrium chatter such as “the temperature is currently falling.” Ofcourse, these claims are impossibly restrictive and completely at variance witheveryday practical discourse. Nonetheless they rest upon substantive issues ofoperational concern and we require a non-trivial approach to the semantics ofthermal vocabulary that properly acknowledges these concerns, yet can reach out tofar wider swatches of the universe. As we trace these matters through followingDuhem’s insightful instruction, we find that our thermal vocabularies play a moredistinctive role within our overall descriptive practices than we shall anticipate if wedwell excessively upon soothing nostrums such as “‘mean kinetic energy per degreeof molecular freedom’represents the underlying referential essence of‘temperature.’” As the old song says, that ain’t the half of it, dearie. Not to bemysterious about all of this, the distinctive role we shall isolate reflects the fact thatthermal distinctions serve as our central descriptive vehicles for assigningquantitative measures to the processes of energy degradation that transpire allaround us, across a large spread of characteristic scale sizes. And virtually all ofthis is firmly contained within the pages of The Evolution of Mechanics, albeitdisguised in an unwanted tissue of methodological rant.

The best route to these conceptual conclusions is simply to follow along withDuhem as he builds up broader applications for physical vocabulary through asequence of stagelike constructions, briskly outlined a few pages back. He employs(without using this terminology) a widely employed form of mathematicalconstruction called a lift that allows us to erect a dynamical modeling over atimeless behavior base.

To explain what I have in mind, letus consider an example that is brieflydiscussed in Essay 2: the steady stateflow of a fluid around an obstacle. Steady state flows represent an extremelyspecialized class of autonomoustrajectories–they are not controlled in anyway, but are special because successiveparticles of fluid must follow the same

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lifting a base manifold into a dynamic manifold

streamlines as their predecessors. Particles that move in these orderly patternsdeposit crisp patterns upon a timeless representation called a base manifold, ratherin the manner of a tidy time exposure on film. FIG: PROJECTING A TIMELESSBASE MANIFOLD But these projected adornments occur only if the particle flowis very regular--if particle A follows after particle B, then A’s image will not messup the projection come from B, but merely intensify its strength. If the flow isn’tsteady, we find nothing but blur on our photographic plate.53

In these circumstances, we view the steady state flows as specializedbehaviors extracted from a much larger group of trajectories, most of which will notdeposit clear patterns upon a 3D base manifold. But it turns out that there are asurprisingly large number of circumstances in physics where a contrary procedure isadvisable. In these we begin with a timeless base manifold and build up dynamicflows by lifting base manifold patterns into a larger temporal manifold, whose pieceswe then connect together with temporal arrows by some procedure or other.54 Wethen explore whether this restricted dynamical space can be further enlarged withtrajectories that are not steady, possibly through perturbative techniques of somekind. Lifted constructions of this general character are surprisingly common inmathematical physics and can create considerable conceptual confusion if theirstructural contours are not properly recognized. FIG: LIFTING A BASEMANIFOLD INTO A DYNAMIC MANIFOLD

We’ll now see that Lagrangebuilds up his celebrated analyticalmechanics in this distinctive fashion,a structural feature that is essential toDuhem’s thermal enterprise. We’llexplain why when we turn toClausius’ work. As notedpreviously, for Duhem the phrases“Lagrangianmechanics,”“Lagrange’s methods,”“analytical mechanics” and “The OldMechanics” all refer to a two-stageconstructive technique articulated in Lagrange’s Analytical Mechanics. It involves atwo-stage process that employs the principle of virtual work to capture themotionless statics of a target system, upon which d’Alembert’s principle issubsequently employed to introduce its dynamics. I remind my readers that thephrase “Lagrangian mechanics” often codifies a different foundational technique

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constrained equilibriums

Lagrange

that will not work for Duhem’s purposes. In the sequel, I shall always employterminology in Duhem’s manner.

Let us now explicate these techniques in thelingo of lifted manifolds. When we consider thestatics of a target system, we are concerned with itsconstrained equilibriums: the arrays of externalforces that can hold the unit in a stable position. Consider the shock absorber pictured. FIG:CONSTRAINED EQUILIBRIUMS Its natural,unforced equilibrium is shown at the top left; ifplaced in that position, it will remain steady withoutany assistance from the outside. But a force of adeterminate magnitude F is required to maintain thegizmo in the second position pictured; any applied force smaller than F will allowthe device to relax more. By the same token, a greater degree of internal springtension is needed to balance a larger applied force, as shown at the bottom of theillustration. We can prevent our shock absorber from moving with a single appliedforce, but, in more general circumstances, a variety of applied forces and torqueswill be required for this purpose. So we can associate each state of constrainedequilibrium with an array of applied forces and torques.55

Why do these static states offer exceptionally useful descriptiveopportunities? Stated in post-Lagrangian terms, knowledge of its constrainedequilibriums allow us to adjudicate, in an experimentally effective way, how adevice stores potential energy internally, e.g., as the collective strain energycaptured with its springs. Through these means, we only need to attend to thesystem’s effective degrees of freedom–the modes in which its parts collectively

remain free to wiggle. Here’s the general idea. Ignoring forthe moment the degradations of energy produced by frictionand other factors, a mechanical device will store anycoherent energetic effort (usually known as “work”) appliedto it in some internal fashion, which it can later reapply to itsenvironment as circumstances permit. FIG: LAGRANGEConsider any location upon our device that we are able towiggle. Measure the work accomplished by the formulaFδr, where F represents the amount of applied force weapply to our position and δr reflects the displacemen throughwhich it travels in response to F (I’ll explain the funny “δ”

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symbol in a moment). Accordingly, if a stretching force of strength F extends ourshock absorber through a small vertical distance δr, Fδr represents the workaccomplished. These work relationships always occur in pairings, that can assumemany distinct forms depending upon the manner of generalized “force” utilized ininducing these changes For example, if some knob on our device can be turnedthrough an angle δθ, we apply a torque τ to do, resulting in applied work measuredby τδθ. In non-mechanical contexts, analogous work pairings such as Tδr (where Tis temperature and s is entropy) or μδη (where μ is a chemical potential and η is thequantity of compositional change that μ “drives”) appear. These additional workpairings are central within Duhem’s conception of an enlarged “New Mechanics.” 56

The principle of virtual work maintains that a device’s special states ofconstrained equilibrium represent circumstances in which a gizmo’s internaldispositions for acting against its environment are exactly checked by the scheduleof applied forces we employ to restrain these capacities.57 Lagrange’s virtual workprinciple captures this balancing through a stability criterion--any external attempt towiggle the device away from a constrained equilibrium is exactly countered by aninternally generated restorative response. The companion insight codified withind’Alembert’s principle (to be considered shortly) maintains that, when we release atarget system from the constraints we have imposed, it will begin moving in amanner in which the internal energy it has stored while constrained becomesreleased in the form of kinetic movement. Once we let go our shock absorber, itbegins to move, with a whim determined by how severely it had been compressed orstretched beforehand. This is why the constrained equilibriums of Lagrange’s offerdirect insight into a device’s capacities for storing potential energy.

There are several formal aspects of this procedure of which we should takenote. First, the potential energy evaluations offered are holistic, in the sense thatthey reflect across-the-entire-device coordinations without delving into the details ofhow these coordinations are internally implemented (in the jargon we don’t worryabout the forces of constraint that bind the system together). This abstraction frominternal detail is one of the celebrated advantages of virtual work technique and itpops up quite frequently in other essays.58 As we’ll soon see, the holistic nature ofthe evaluations supplied by the technique will prove crucial in Duhem’s laterextensions, due to the centrality of system homogeneity within thermal reasoning. Secondly, input energy retains all of its original coherence when it becomesinternally stored in this manner. Duhem will later employ Lagrange’s measure ofperfectly warehoused coherent energy as a benchmark against which he cancalibrate the degradations characteristic of real life storage.

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principle of virtual work

Let us briefly pause to note a few technical peculiarities required to make thispoint of view precise, although these further details will not prove especiallyimportant for our purposes. There are a number of reasons where the fanciedresponse δr must be conceptualized as “virtual,” rather than actual. As justremarked, some amount of any real effort exerted upon a target system dissipatesinto heat, but we don’t want to bother with these processes at present. We areprimarily interested in the manner in which the various configurations of our devicereflect its internal capacities for storing restorative capacity; we’re not currentlyinterested in the extraneous processes that prevent our external injections ofcoherent effort from fully reaching these internal energy coffers. Put another way,virtual work thinking tries to estimate the size of the warehouses in which we storeour wheat; we’ll worry later about how much grain the mice eat before it gets there. As we’ll see, this policy of “estimate storage capacity first and worry about themice/friction later” is central to Duhem’s policies for introducing thermal notionsinto his New Mechanical thinking.

A second aspect of the “virtual” modifier worth mentioning. Consider theconstrained equilibrium positions of an old-fashionedseesaw. The traditional law of the lever says that, inorder to hold hefty Jack in equilibrium, Jill must choosethe location L* such that W.L = W*.L*. But a bit oftrigonometry expresses this same fact in terms of thearcs, θ and θ* through which our children will turn ifdisplaced, giving rise to the “virtual work” standard ofequilibrium: the seesaw is in equilibrium if and only ifany attempt to displace Jack through an angle δθ wouldinduce a corresponding Jill displacement of amount δθ*such that the work performed in moving Jack (W.δθ) isexactly canceled by the contrary work required to moveJill (W*.δθ*). But a subtlety complicates this simplepicture.59 FIG: PRINCIPLE OF VIRTUAL WORK IfJill turns through a real angle θ, the effective forceinducing her turning motion will no longer be the full gravitational force W*,because, as she turns, some of that becomes wasted in pushing her parallel to theplank. The actual work expended in moving Jill through the angle θ is thereforesupplied by a quantity of force We less than the full original W* (and is rather hardto compute). But to mark out the equilibrium position of our seesaw correctly, we

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turning on a dynamics with d’Alembert’sprinciple

want to retain W and W* at their original, pointing entirely downward, values,which are signalized by the “virtual expressions” W.δθ and W*.δθ*, indicating thatwe don’t want to consider how Jill turns through a real angle, with theaccompanying diminishment of force. Elementary textbooks often codify theserequirements through loose appeals to “we shallonly consider modifications δθ that are so short thatthey qualify as infinitesimal.” Such glosses do notrender justice to their proper strategic rationales.60

(ix)

Let’s return to our central thread. As alreadyintimated, Lagrange injects dynamic change into hispicture by releasing constraints and followingd’Alembert’s celebrated principle, viz. anyunopposed degree of freedom r within a connectedsystem must display a corrective accelerationd2r/dt2 such that the quantity mr

. d2r/dt2 fills in forthe missing F that would otherwise hold the systemin a motionless constrained equilibrium. Here mrrepresents the measure of “inertial laziness”associated with the generalized coordinate r, suchas a mass or a moment of inertia. This provisosupplies our shock absorber with a natural dynamicbehavior as follows. FIG: TURNING ON ADYNAMICS WITH D’ALEMBERT At time t0 andstarting from rest, let us inject a certain amount ofkinetic energy into the device by impelling itsmoveable center briskly downward, as illustrated atthe top of the diagram. This supplies the apparatuswith a total amount of injected energy symbolizedby the gray bar. No internal reactive is generatedwithin this position, allowing the center to travel thefull distance δr in time Δt required by d’Alembert’sprinciple. Once it reaches this new position,however, the springs inside our gizmo begin to act

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lifting a statical base manifold into a dynamic realm

against the kinetic motion (i.e., mc. d2r/dt2) and counter its accelerative demands.

When the moving center reaches its maximal extension at its turnaround point, all ofits original kinetic energy has become converted into internal strain energy and thedevice will stop moving momentarily. Since the center is unopposed by anyconstraint, d’Alembert’s principle demands an accelerative reaction which nowpoints upward, forcing the center to move in that direction. The acquired kineticenergy causes our moving center to overshoot its natural equilibrium point at thebottom of the diagram, proceeding onward until it reaches a compressed positionwhere sufficient internal energy operates to halt the movement. No frictionalresistance has been included in this model, so our shock absorber will cycle throughthese same positions forever.

In this instance, our specimen displays a single degree of released movement,but d’Alembert’s principle allows us to ascertain how a system with many degreesof freedom will behave when any selection of its binding constraints are released.61 If we remove them all, we obtain an account of how a wholly autonomousmechanism will evolve, once we know the initial positions and momenta of itsvarious degrees of freedom.

Let’s now recast these Lagrangian arrangements in lifting-from-a-base-manifold terms. FIG: LIFTING A STATIC BASE MANIFOLD INTO ADYNAMIC REALM We begin with a base manifold comprised of the collection ofconstrained equilibriumcircumstancesappropriate to our shockabsorber, calculatedaccording to Lagrange’sprinciple of virtual work. No temporalconsiderationswhatsoever have beenregistered upon this initialmanifold. Nor do anyautonomous systemsappear there–every position on the base manifold represents a state of whollyconstrained equilibrium. If we want, we can connect together these points in waysthat are mechanically informative, by studying how the constrained equilibriums willalter if some of its constitutive parameters are altered.62 But our base manifold tellsus nothing about kinetic movement whatsoever. However, d’Alembert’s principle

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dissipative effects upon autonomous behavior

allows us to lift these base manifold states into a genuine dynamic arena, by freeingthem from their constraints and connecting together the lifted states with temporalarrows, whose lengths are determined by the inertial sluggishness of the system. Aswe do so, we liberate our target specimen from the constraining forces that hold it instatic configuration and allow it–if the harness of all applied forces is loosened--toevolve entirely autonomously according to its internal dynamics.

We can now model frictional effects by adding supplementary terms to ourmodeling equations that gradually sap the device of its energy stores. This incursiontypically makes the system’s phase trajectories collapse into one another, creatingthe motionless equilibriums we expect to see when Jack and Jill reach the bottom oftheir hill. The diagram shows how the familiar phase portrait of a pendulum altersafter some degree of friction is turned on. FIG: DISSIPATIVE EFFECTS UPONAUTONOMOUS BEHAVIOR Frictional effects are intrinsically linked tovelocities and we can’t talk about its activities sensibly until we reach our dynamicmanifold, because velocity is a notion that depends upon the temporal arrows thatwe paint in as we lift our constrainedequilibria into the freer setting of a dynamicmanifold. At this point, Duhem introducesa further stage-like construction upon whichwe won’t greatly elaborate, although itaffects the details of his discussion insignificant ways. Once dissipative frictionenters the scene, we find ourselves able tohold our systems in a much wider array ofstatic configurations.63 With respect to ourshock absorber, if friction inhibits the recoilof its springs, we will be able to hold the gizmo in an extended position with asmaller force than we would otherwise require, because the frictional resistencehelps us maintain our mechanism in position.64 Based upon these considerations,Duhem projects a wider region of so-called false equilibria onto our original basemanifold, a maneuver that allows him to introduce extra descriptive parameterssuitable for irreversible processes. We won’t worry about further details here; infact, I find his discussion hard to follow.

Let us observe, however, that this purely mechanical frictional plateau isunsatisfactory in the sense that the conservation of energy has not been enforced.Dissipation is presently addressed as a single-directional process that pays noattention to where the lost energy goes once it is drained from the purely mechanical

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behaviors of the system by friction. But this energy needs to be retained somehow,in the form of heat, chemical change, radiation or other storage capacity. And wehave also observed that these additional energy stores can exert back-effect actionsupon the system’s mechanical behaviors, as occurs when we turn a turbine by aheated furnace. Accordingly, further layers of conceptual structure are required toimplement these bidirectional energetic demands. Accordingly, we need to return toour static base manifold and consider systems that can be maintained in constrainedequilibrium through thermal factors as well. We will study how this is done soon.

This characteristic progression--constrained equilibria behaviors ofincreased autonomy dissipative behaviors--captures the central architectural planthat Duhem expects to follow within the wider arena of thermal behaviors. However, the bugbear of frictional damage creates greater headaches when weattempt this project, for reasons we’ll examine later.

The crucial posit behind this three stage construction is this. Until friction orother irregular effects enter the picture, a system will display tidy transfers ofconserved energy between one storage facility and another, resulting in the whollyreversible behavior of our shock absorber (it is completely indifferent to thedirection in which it is initially pushed). But friction spoils these tidy exchangepatterns by robbing our shock absorber of some of its kinetic energy, so that it nolonger possesses the energetic oomph required to return the device to its originalconditions. Nonetheless–and this is our central posit–,we normally presume that ourdevice’s internal capacities for storing potential energy will not have becomedamaged by the friction; it is merely that frictional processes leach away some of thedevice’s energy as it attempts to shift from one mode of energetic expression toanother. Analogy: a saving banks still retains all of its original vaults, despite thefact that outlaws have robbed the Brinks truck that was supposed to fill them up. Intruth, various physical processes such as hysteresis and plastic flow genuinelydamage internal energy capacity and Duhem expects to add further layers to hisarchitecture that can accommodate this destructive frictional processes. I’ll returnto this point briefly later.

I submit (but won’t develop at any length) that Duhem’s rather mystifyinginvocations of “natural classifications” very much reflect these lower layers ofdescriptive architecture–the characteristic intensive/extensive Fδr and Tδs pairingsof our base manifolds reflect nature’s central linkages between its various qualities,whose workings are obscured through the meddlesome handiwork of friction. Suchsentiments were common in Duhem’s era. The “father of modern mechanism”(Franz Reuleaux) writes of machines in these terms:

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[Once friction’s obscuring veil is cleared away, the student of mechanicalinvention will be able to see the essential nature of these devices in theirnaked glory]: all carry[ing] on, partially before the bodily eye of the studentand partly before the eye of his imagination, the same never tiring play. Inthe midst of the distracting noise of their material representatives they carryon the noiseless life-work of rolling. They are as it were the soul of themachine, ruling its utterances--the bodily motions themselves--and givingthem intelligible 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.65

And of the intellectual processes required to clear away this underbrush: Goethe,--who had so great an interest in the inner nature of everything thatcould enlarge the circle of our ideas,--expresses himself in the followingnoteworthy sentence: “Everything we call Invention, discovery in the highersense, is the ultimate outcome of the original perception of some truth,which, long perfected in quiet, leads at length suddenly and unexpectedly toproductive recognition.”66

In our context, we never witness a device’s capacities for internal energetic storagein perfect operation–some mice always eat a bit of the grain along the way–but itnonetheless represents one of the device’s most important characteristics. Adiscerning eye needs to pierce peer through the frictional fog.

It is through these themes that many of Duhem’s seemingly contradictoryassertions can be reconciled, e.g., how the free conceptual liberty of the physicist iscompatible with the claim that “theory” should somehow represent a resumé ofexperiment. As I understand him, an “experiment” represents the study of a targetmaterial under changes that are entirely in the control of the experimenter; theemphasis rests upon “controlled manipulation,” rather than “studied on a laboratoryscale,” as Duhem is often interpreted. A scientist must insert creative abstractionsthat can sort these behaviors into informative layers, in the same manner asLagrange builds up a richer architecture starting from a totally controlled basis.

Duhem believes that “the best science” works upward from totally controlledcircumstances in this fashion as best it can. If the resulting architecture fails toproduce the fully autonomous trajectories of Theory T expectation, so much theworse for those expectations. Such demands do not stem from the scientificenterprise per se, but from a priori convictions as what a preferred account shouldlook like. To be sure, within the limited domains of pure mechanics, Duhem grantsthat Lagrange’s methods come close to fulfilling these dreams.67 But the project of

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reaching autonomy encounters greater obstacles when we consider the widercanvass of thermal behaviors. We’ll return to this point later.

(x)

When Duhem turns to the task of assembling a wider thermomechanics inwhich the interrelationships between pure mechanics and thermal events can bediscussed, he follows the same stagewise recipe: statics reversible dynamics irreversible behavior. But substantive obstacles stand in his way. These stem fromthe fact that, according to thermodynamic orthodoxy, notions like temperature andentropy only apply to a hunk of block after it has been allowed enough time to reacha homogeneous condition after being initially manipulated in some manner or other(the interval required to reach this equilibrium is usually a relaxation time). Müllerand Weiss write:

In general working and heating on the boundary of a body will create havocinside the body: a turbulent flow field and strongly inhomogeneous fields ofmass density, pressure and temperature. We may say that the internalequilibrium of the body is disturbed. But if the working and heating areapplied carefully and slowly, there will no appreciable flow field in theinterior and pressure and temperature will be essentially homogeneouswhile slowly changing in time. The internal equilibrium will thus prevailduring the slow application of working and heating.68

I’ll return to the caveat about “slow and careful application” in a minute. But thereasons for these classificatory precautions are evident: entropy is intended tocapture the proportion of degraded energy that is currently unavailable for coherentwork. But if identifiable flows remain active within the interior, the relevant energyhasn’t become fully degraded–we might be able to recover a greater amount ofuseful work by delicately aligning little paddles with the directions of flow. So wemust wait until these forms of intermediate intervention are no longer possible. Byignoring the very processes of flow that carry a material from one homogeneousstate to another, we will have removed all of the “dynamics” from“thermodynamics,” leaving behind a subject that some authors call “thermostatics”and which is only concerned with the equilibrium states that appear after decentrelaxation times. And orthodoxy is happy with this restriction. Herbert B. Callen inhis classic textbook explains:

If a closed composite system is in equilibrium with respect to internalconstraints, and if some of these constraints are then removed, certainpreviously disallowed processes become permissible. These processes bring

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the system to a new equilibrium state. Prediction of the new equilibriumstate is the central problem of thermodynamics.69

Although these doctrines represent conventional opinion, such strictures areferociously draconian in their terminological consequences. By such loftystandards, ordinary solids such as rocks and rubber bands should never be creditedwith temperatures or heat contents, because none of these materials satisfy thetextbook requirements for equilibrium. Properly speaking, we should be unwillingto talk about a diamond’s temperature until the billions of years pass required forour jewel to stagger its way to its room temperature equilibrium as graphite. Likewise we should have refused to talk of heat transferring along a rod--if itsinnards still displays a flow, it is not ready for thermodynamic classification. HansU. Fuchs complains of these absurdly circumscribed terminological policies:

[T]he classical field of thermodynamics and the subject of heat transfer forma natural unity. Despite all the claims in books on thermodynamics thatheat transfer cannot be included, and of books on heat transfer thatthermodynamics is an altogether different thing, we do not artificially haveto separate the two... Continuum processes are clearly irreversible. Whilethe quest for a description of reversible processes is understandable, andwhile theories of the dynamics of such processes can be built and appliedsuccessfully, the belief that reversibility requires equilibrium in the sense ofstatic conditions has led classical thermodynamics into a tight corner out ofwhich it can only escape if the lessons of continuum mechanics are accepted. 70 Of course, the very textbooks that lay down these prim terminological

restrictions violate their own prohibitions at the first opportunity (sometimes on thevery page where the stern terminological admonitions have been laid down). Everybody approaching the subject in practical terms wants to say that heat isgradually driven along our bar by an “entropic force”-- that is, a “force” that drivesour disturbed, out of equilibrium bar closer to a genuine equilibrium where the barhas the same temperature everywhere. Allied “forces” increase the entropy of adiamond when it is immersed in a hot liquid and drive an expansion wave to fan outbehind a shock front. Officially, however, all of these “forces” qualify as fake,fictitious or metaphorical.71 The anonymous author of the Wikipedia entry onnon-equilibrium thermodynamics writes:

[E]ntropy, though it may be defined for a non-equilibrium system, is, whenstrictly considered, only a macroscopic quantity that refers to the wholesystem, and is not a dynamical variable and in general does not act as alocal potential that describes local physical forces. Under special

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prolongation of usage

circumstances, however, one can metaphorically think as if the thermalvariables behaved like local physical forces. The approximation thatconstitutes classical irreversible thermodynamics is built on this metaphoricthinking.72

Adopting the same point of view, textbooks solemnly inform us that neither entropynor entropic forces can meaningfully “drive any kind of change” since theyrepresent concepts pertinent to conditions of strict equilibrium only. Yet, fiveminutes later, these same terminological scolds merrily discourse of “moleculardriving forces.”

The characteristic double-talk associated with the notorious “Second Law ofThermodynamics” follows a similar pattern. It is initially articulated as a claim thatthe total entropy of an isolated system always increases over time, or remainsconstant within an idealized reversible process. But we are then informed thatentropy talk only makes sense for systems that have been allowed to relax afterbecoming subject to fresh set of external controls. So both the “isolation” and“autonomous increasing” of the original statement have been banished from ourofficial vocabularies?73 What’s going on?

In the roughest terms, conventional thermal equilibrium talk only makes senseagainst a wider backdrop of non-equilibrium processes.74 But discovering a stableand universally accepted codification of this wider field has proved very elusive,leading to a plethora of incomplete and non-equivalent accounts within the largeliterature devoted to the subject. Duhem’s constructive labors offer deep insightinto the origins of these terminological delicacies, which trace to the fact that thebaseline notion of internally stored energy is a holistic conception, demanding alarge degree of settled cooperation amongst a target system’s component parts, inthe basic manner that Lagrange highlights within the Analytical Mechanics. Duhemshows that we can effectively extend “temperature”’s applications to wider arenasthan orthodox thermostatical doctrine allows, but this outreach needs to beaccomplished delicately and in stage-like sections,always with an eye to the inhomogeneities thatClausius worries about.

On many occasions throughout this collectionand WS, I have commented on the fact that manywords gain fuller meanings by reaching from oneapplicational patch to another through naturalprocesses of prolongation. FIG: PROLONGATIONOF USAGE Frequently, these advances transpire without anyone noticing andsubstantial misunderstanding can later arise when these constructive layers are not

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wave movement via d’Alembert’s principle

recognized and we seek elusive “core meanings” for the terminology involved.75 Duhem’s analysis of the stage-by-stage architecture required for sound thermalusage strikes me as an admirable exemplar of how such developmental processesunfold. Thermodynamical discourse is filled with mysterious talk of “infinitelyslow processes,” “inexact differentials” and untenable instructions for never talkingof ordinary solids in thermal terms. Many a beginner in the subject has torn out hisor her hair trying to make coherent sense of it all.76 I believe that Duhem’s stage-by-stage reconstruction successfully codifies the underlying strategies that explicatethe rather weird locutions that populate the surface grammar of thermal discourse.

(xi)

In a passage quoted earlier, Hans Fuchs claims (paraphrasing slightly):classical thermodynamics can escape the tight corner into which it has beenpressed through accepting the lessons of continuum mechanics.

By this he means that a thermodynamics suitable for heat transport should beassembled at the infinitesimally local level of continuum mechanics. To understandwhat he has in mind, let’s linger a bit longer within the dominions of pure mechanicsand consider how a wave dynamics for a flexible bar might be assembled from theingredients of Lagrangian mechanics. Supposethat we hit one end of our bar with ahammer–how do these processes unfold from avirtual work point of view? For simplicity, let’sconcentrate upon a simple Hookean rod so thatwe needn’t to worry about complex shears andtorques. Divide the entire beam into a sequenceof little blocks so small that they qualify asinfinitesimal and consider an early moment inwhich the remainder of the rod is completelyrelaxed and only a small portion of its right handflank has become compressed by our blow,largely from the right. FIG: WAVEMOVEMENT VIA D’ALEMBERT’SPRINCIPLE Along each of the boundaries of ourlittle blocks, calculate the amount ofsupplementary traction force required to holdeach little block in its current state ofcompression forever. In our wave’s simple

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Clausius base manifold

circumstances, the only little block requiring supplementary retaining forces is theblock that contains our compressed wave; everything else is completely relaxed. I’ve symbolized this state of affairs by two little men who compress the affectedblock by different degrees on each side (stronger on the left). Through thesedissective policies we divide our rod into a collection of constrained equilibriumsapplied locally to small elements. Applying d’Alembert’s principle77 within thispartitioned format, we remove the constraining tensions and presume that the twoliberated sides of our compressed little block will accelerate over a time step Δt insuch a way that our block expands with a kinetic energy equal to the compressivework just released. By the end of Δt, the two neighboring blocks will have becomecompressed by the movements of their own boundaries and the transmission processwill continue, proceeding at a rate determined by the inertia of each block, initialconditions and the Hookean strengths with which they resist compression.78 Bythese means, we can persuade a simple compressive wave to travel along our rod.

Reasonings such as this are said to rest upon presumptions of localequilibrium. All forms of non-equilibrium thermodynamic thinking with which I amfamiliar adopt this general policy, while differing in the details of how it isprosecuted.

With this policy in mind, let’s return to the original arena of thermodynamicreasoning as it was codified by Rudolf Clausius. We start by laying out an enrichedbase manifold consisting of states completely constrained by thermal processes aswell as mechanical ones. I’ll call this a Clausius base manifold. FIG: CLAUSIUSBASE MANIFOLD A few experimenting scientists are included to remind readersthat these states are maintained in equilibrium through exterior manipulation. Wecan now draw curves through these points indicating that the systems along aspecific lines share some commonfeature, such the fact that no heatflows across their boundaries but thatpressure steadily increases. Quasi-rectangular patterns constructed fromsuch lines are the familiar Carnotcycles of introductorythermodynamics.

These connecting curves arecommonly described as mysterious“processes” (= trajectories)possessing self-contradictoryproperties, e.g., that they represent a

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Clausius and Gibbs

succession of unchanging states that nonetheless alter from one to another, althoughit takes infinitely much time to do so. Avoiding this language, let us simply state thatwe can arrange constrained equilibriums in these patterns, in which every systeminvolved shares the same internal construction. Based upon the way that the Carnotpatterns must fit together within these friction-free domains, Clausius thenestablished, through an astonishingly stretch of abstract reasoning, that an additionalform of potential function S becomes well-defined upon this manifold called theentropy function. It is this unfamiliar quantity that determines how much of thesystem’s confined energy is currently available as a source of coherent work. Itspresence allows use to introduce an absolute temperature T as a replacement for thecruder “thermometer temperatures” with which we actually control thermodynamicprocesses.79 These innovations supply the missing ingredients required to fill out theequation of state that articulates the manner in which a target system allocatesapplied effort to its various warehouses of stored energy. Conceived in this manner, entropy is a concept that only makes sense if theinternal processes within its target applications have settled into a constrainedequilibrium of just the type that the standard tracts demand. However, if a largeobject can be divided in a hypothetical array of local equilibriums of this character,we obtain a base manifold from which adynamics can be lifted in an approximatelyd’Alembertian manner. To perform thisfunction, Duhem adopts a thermal principledue to Willard Gibbs. FIG: CLAUSIUSAND GIBBS The resulting lift will notconvey us into an arena of fullyautonomous processes or of frictionalprocesses either, but into a wider array ofreversible behavior that unfold under lessconstrained circumstances, among which are rods that are simultaneously hammeredand heated.

Gibbs’ rule holds that if two or more systems (A, B and C, say) maintained intheir own individual equilibriums become joined together and are allowed to reach amutual equilibrium as the combined system A + B + C, they will select, amongst allof the possible equilbria on the manifold to which they could evolve, the finalcondition that displays the highest value of entropy consistent with whateverenvironmental controls remain in place. That is, we examine all the variations ofentropy δS and their affect upon the system’s equation of state (viz., (1/T)δU +εi,/T:δ σi - (υij/T) δNij) and compute the arrangement of block states (U, σi ,Nij)

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lifting a Clausius base manifold into a dynamic realm

whose combined entropy is greatest.80 As such, this determination tells us nothing asto how long it will take before the newly joined system A + B + C reaches this newequilibrium (in fact, to avoid turning on frictional effects, it would take forever). FIG: LIFTING A CLAUSIUS BASE MANIFOLD INTO DYNAMIC REALM But

we can imitate our earliertechnique of drawing timearrows between statesthrough appeal to thesystem’s inertial lazinessof response by invokingthe empirically measuredrelaxation times (orcapacities) associatedwith the non-mechanicalintensive/extensivequantity pairings of oursystem, e.g., the rate at

which blocks at differing temperatures correct their thermal differences or the rates atwhich different concentrations of chemicals reach combinational accord. As before,we predict, that over a unit of time Δt, our A + B + C chain of blocks will move asclose to Gibbs’ maximized entropy condition as it can achieve in view of thesluggishness of the system’s kinetic response when it becomes unconstrained. Uponreaching time ti + Δt, we repeat these calculations to carry our bar to the time ti +2Δt. And so on. Procedures of this sort are called local equilibrium assumptionsbecause they presume that each small block of our material never gets pushed too farfrom equilibrium at a local level. In this manner, the Gibbs rule paints dynamicarrows between states within our lifted manifold in manner very similar to theLagrangian policies we employed in pure mechanical circumstances. Indeed, Duhem often stresses that d’Alembert’s principle can be subsumed under the widerGibbs rule. He writes:

The spirit of the methods of Lagrange’s Statics has therefore passed in itsentirety into the general Statics, the conception of which will forever redoundto the glory of J. Willard Gibbs; but in passing from one to the other theyhave evolved; the germs sown by the author of Analytical Mechanics owetheir ample and full development to the physicist who treated the Equilibriumof Heterogeneous Substances.81

We obtain the standard equations of reversible thermomechanics by smoothing outthese trajectories by pressing our partitioned blocks to infinitesimal levels and

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allowing Δt to go to 0. As the total entropy within our blocks is maximized, each lefthand term in the

expression (1/Ti)δUi + εi,/Ti:δ σi - (υij/Ti) δNij will drive an alteration in theterm with which it is paired (which will operate differently in materials of differentcompositions). As observed before, the current stress εi,/Ti in each block (divided bythe current temperature) serves as a “force” that the strain to alter and similarly forthe chemical potentials codified in the third term. But now consider the first term inthe Gibbs rule. Formally, the inverse temperature 1/T acts as a parallel “force” thatdrives an increase in the “heat energy”component Q within the internal energy U(1/T appears because the higher T becomes, U settles upon a higher value for Q,other factors being equal). Factors of this origin are commonly called “entropicforces” and will prove of interest to us later.

These lifted trajectories, however, are still reversible and we can’t subject oursystems to dissipation until entropy production terms are added to our dynamicequations analogous the frictional forces we applied to our Lagrangian trajectoriesafter they first constructed. These production terms can operate in different ways butreflect the manner in which the rate of a particular process, such as temperaturechange, increases the entropy of the system above its reversible process values. Asbefore, temporal arrows must be assigned to succeeding states before any talk of the“rates” of thermal driving forces makes mathematical sense. In this fashion, thecoherent modeling of dissipative thermal processes is parasitic upon an underlyingplatform of reversible thermodynamic trajectories. As noted before, this constructiveapproach makes sense of Duhem’s repeated claims that the vital connectionsbetween the “qualities of nature” are articulated at the reversible level, before theobscuring veil of friction enters the scene.

In this setting, the Second Law of Thermodynamics obtains its expectedsignificance: entropy production terms are always positive.82 The construction alsoexplains why the Carnot cycle “paths” inscribed upon the base manifold should notbe viewed as peculiar “physical trajectories” that unfold at infinitely glacial speeds. Properly speaking, the positions on this underlying manifold merely reflectstraightforward reports on the wholly constrained equilibriums that we can readilyinduce within a laboratory setting (usually), supplying us with very reliableinformation about the system’s internal responses to external driving forces. Talk of“processes” only makes sense as we gradually relax the binding constraints andintroduce plausible assumptions (d’Alembert’s principle and Gibb’s rule) about howthe system will release its internal behaviors into behavioral changes such asmovements or increased temperatures. Due to friction’s meddling interventions, wewitness these processes most directly when the rates of active energetic conversion

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are not high, because these rates excite the entropy productions that bedims theunderlying patterns at work. Characterizing reversible processes as the “idealizedlimits of real processes” obscures these basic conceptual dependencies, in a ratherunhelpful way.

As suggested before, I believe that this Duhemian analysis correctly identifiesthe central strategic shifts that support the applicational utility of classificatory wordssuch as ”heat,” “temperature” and “entropy.” We find ourselves in peculiarconceptual muddles if we attempt to parse such usages in single layer terms, e.g., astalking about “processes” in some unitary but baffling manner.

(xii)

To summarize, these step-by-step stages of lifting carry us from systems heldin totally immobile constraint by outside, often experimental, controls to conditionsof increased autonomy. Generally, these enlargements operate by releasing onebinding constraint one at a time, holding others in place. These restrictions arecharacteristic of successful thermodynamic thinking; we plot out a plan for installinguseful forms of frozen order with steel by devising manufacturing processes thatapproximate as closely to single-variable manipulations as we can.83 Within the fullyautonomous processes of nature, behaviors are not so tidily delineated as this,leading us to wonder if we can reach any account of fully autonomous behaviorthrough lifting procedures of a Duhemian stripe. Certainly, many significantobstacles stand in its path. For example, practical uses of the Gibbs rule appeal toone of thermodynamics’ four basic forms of thermodynamic potential to obtain clearpredictions for how internal changes in a material affect each other.84 But theoptimization procedures associated with each of these four potentials depend uponthe controlling factors we maintain as fixed, allowing other features to vary. Wereach a position where thermomechanical cross-effects can be modeled ably inDuhem’s manner so long as our target system is maintained at, e.g., a constanttemperature. But what should happen when those sustaining controls are notmaintained remains uncertain.85

This does not entail that more liberal answers to these concerns can’t befound, but we lose the firm guiding hand of established thermodynamic practice atthis point. And we have reasons to worry about our capacities for furtherimprovement in these respects. The optimizations associated with our four speciesof thermodynamic potential trade upon a natural complementarity amongstexperimental manipulations. If we maintain a system’s temperature under tightcontrol through constant immersion in a heat bath, the amount of energetic exchangewith its surrounding becomes uncontrolled in the process. After a sufficient relation

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The Slough of Despond

time, we may generally presume that maximal degradation in coherence hastranspired along this uncontrolled channel of exchange, leading to an increase inheat. But if too many channels of this character are left open, our confidence inenergetic allocation becomes less certain.86

In addition, our enlargement recipe has also presumed that the processesinvolved leave the underlying system essentially undamaged, in the sense that thetarget material can generally be restored to its original starting condition. ButDuhem recognized that such restoration is often impossible and identified severalimportant centers of concern: permanent structural damage, as when a paper clipplastically assumes a new rest configuration if bent with sufficient strength;hysteresis, when the clip breaks after being flexed repeatedly; and frozen disorder,the fact that the internal arrangements of its component steel are not arranged in thetidy crystalline arrays required for the material’s conventional chemicalequilibrium.87 We possess no guarantee that we will find ourselves able to reachthese phenomena by elaborating further upon Duhem’s constrained equilibrium basis.

For all of these reasons, Duhem believed our opening motto applied to his ownambitions as well:

the causeway shrinks, the track of the route becomes unclear; we only see anarrow path half hidden by thorns, cut across by bogs and bounded byabysses. FIG: THE SLOUGH OF DESPOND

These barriers of permanent damage suggest the unhappy eventuality that, even afterwe have reached the highest descriptive plateaus attainable through lifted manifoldconstructions over a Clausius base, there may be plenty of natural behaviors leftunreached. Ranging too far from our fully constrained, statical basis, the anchoringopportunities provided may supply insufficient purchase upon less constrained formsof physical behavior. Disappointed in our ambitions,we may derive some solace from the fact that ourautonomous trajectory rivals don’t appear to makeadequate contact with all of nature’s varied behaviorseither, a point that Duhem frequently emphasizes.

To be sure, he personally hopes that acompleted thermomechanical framework adequate toall natural behaviors will be eventually built throughextrapolation from the platforms he has managed toreach, in a manner that continues to emphasize “driving force” combinations of aTδS character.88 He offers speculative suggestions as to how this goal might bereached through appeal to additional intervening variables, in a manner that we’ll

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discuss later. In an fully autonomously, liberated physics, these fleshed out TδS-style pairings can be plausibly regarded as reflecting nature’s own operativeprocesses, active in circumstances outside of the controlled relationships for whichwe can concretely test within our experimental manipulations. For Duhem, such anoutcome would provide both a cosmology and a pleasant scientific ratification of themetaphysical world view of St. Thomas Aquinas. But science itself falls under noobligation whatsoever to fulfill these longings. This is why Duhem constantlyreminds his readers that supplying full cosmologies–that is, fully autonomoustrajectories for the entire universe unfettered by monitoring controls or significantenvironmental constraints–does not represent an inherently scientific objective, butstems from convictions of “metaphysical” origin.

In my recasting, articulated in terms of lifts from one descriptive platform toanother, these Duhemian remarks reflect a vivid appreciation of the varied andlayered forms of explanatory architecture that a useful descriptive practice mightemploy, without fulfilling the autonomous trajectory demands of Theory T thinking. A proper appreciation of the substantive empirical obstacles that lie in the path of anautonomous thermomechanical completion should strengthen these cautions. Aswe’ve seen, orthodox instruction in equilibrium thermodynamics worries about whatterms like “temperature,” “heat” and “entropy” might possibly signify with respect tosystems presently roiling in large degrees of internal turmoil. These concerns traceback to Clausius’ original ruminations, as he looked for descriptive tools that couldmonitor the amount of coherent work that can be expected within a system at fixedenergy. Excessive degrees of out-of-equilibrium agitation may indicate that thesystem has not yet settled into a condition where the required distinctions betweencoherent pressures and incoherent heats can be meaningfully drawn. The work ofDuhem and his modern successors have demonstrated that the referential reach of“temperature,” etc. can be extended to a much wider array of applicationalcircumstances than characterize the target systems of traditional equilibriumthermodynamics concern, but the original limiting worries remain and need to beaddressed cautiously, in exactly the one-platform-at-a-time manner than Duhemadopts.

From the very beginning89, of course, ordinary employments of “temperature”have reached, quite properly, far beyond the strictures laid down withinthermodynamic orthodoxy. The word could have never established its foothold inpopular usage had we fastidiously refrained from assigning solid objects“temperatures.” But this coarse, yet real, applicational utility does not entail that thestrategic underpinnings of thermal usage don’t require careful scrutiny, attending tothe conceptual warnings supplied within the better primers on orthodox equilibrium

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thermodynamics. Although Duhem and his modern confederates have constructeddescriptive architectures that more closely approximate the freewheeling thermalappeals of untutored common sense, these advances remain mindful of the legitimateapplicational worries that stand behind the subject’s original equilibrium-onlystrictures.

Although Duhem harbors expectations with respect to rigorous scientificendeavor that are closely allied to the “free creativity of the scientist” themes thatalso stand at the headwaters of Theory T thinking, his robust motivations forexcluding autonomous trajectory demands derive from a finely observed appreciationof the hard-to-anticipate limitations on applicational outreach characteristic of theever-evolving careers of “temperature” and “heat.” For reasons discussed elsewherein this book (and WS as well), modern portraits of word reference based upon coarse“natural kind or “possible world” conceptions display little awareness of the cautionsthat lie behind these deepest veins within Duhem’s writings. I hope that our attemptsto reduce Duhem’s often oracular pronouncements to their thermomechanical rootsmay help Theory T thinkers better appreciate the unwitting dogmatisms into whichthey have fallen. They should ask themselves, “How did autonomous trajectoryexpectations become a tacit but canonical requirement upon ‘fundamental theory’within my thinking?” In one of those nice ironies of developmental fate, a creed thatbegan as a utopian quest for conceptual liberty has mutated into a collection ofautocratic demands upon scientific architecture in which nature itself may be refusedaccommodation.

(xiii)

Other parts of this book, particularly Essay 5, focus upon the problems ofcapturing the behavior of materials that contain large amounts of significant structureat intermediate size scales, e.g., the structural features that distinguish one igneousrock from another; the out-of-equilibrium formations that blacksmiths hammer, heatand fold into their steels. Almost all solid materials display some degree of “frozenorder” that prevent them from relaxing into their theoretically computed equilibriumsquickly. A classic exemplar of long preserved frozen order is the diamond, whichrepresents a crystalline form of carbon whose predicted energetic equilibrium atroom temperatures should be that of graphite (= ordinary pencil lead). Strongenergetic barriers within the gemstone prevent the stone from randomly wiggling itsway back to low pressure graphite, resulting in an extremely long relaxation time.90 In the same, albeit less dramatic, ways, most solid materials display little inclinationfor maximizing their entropies, if we adjudicate “entropy” according to the

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ensembles that we can construct upon a molecular basis, imitating the calculationsthat work ably for simple gases. It is only comparatively recently that modelershave found ways to elude the barriers that frozen order creates for the conventionaltechniques of equilibrium thermodynamics. We shall return to these circumventionsshortly.

Duhem’s response to these difficulties, if I understand him correctly,91 is tohope that such phenomena could be handled by crediting a target material withadditional variables that capably lock ordinary solids into genuine states ofconstrained equilibrium. He frequently asserted that the “conceptual liberty” of thephysicist permits the free invocation of extra variables of this sort, which he oftenidentified loosely as “virtual variations”:

Now is it absurd to accept the existence of a change during which all thepoints of the system remain at the same position? Clearly not; in physics weare sometimes led to conceive of such changes... [Worries to the contrary]only apply to real changes. In the course of a virtual change, the values of[the state variables] cannot be regarded as functions of time.... [So] it is clearthat any sequence of equilibrium states of a system, provided that it iscontinuous, can always be envisaged as forming a virtual change of thesystem.92

In this fashion, a block of carbon might be assigned one or more structural variablesγ associated with a hypothetical generalized “force” that captures the internalresistance that a diamond displays with respect to any γ-parameterized set ofmanipulations leading to graphite.93 He recognized that his statistical mechanicalopponents would instead claim that diamonds are not in true equilibrium at all, butmerely heading to that destination at an incredibly sluggish rate. Duhem devotesmany pages of polemics to arguing that such an equilibrium-eschewing approach isless scientific than his, based upon the fact that his approach remains truer to thelong established Lagrangian successes of a mechanics founded upon a basis of fullyconstrained behaviors. He further complained (rightly) that the not-really-in-equilibrium approach was unable to put reliable numbers on the empirical strength ofresistance to structural change.

G. A. Maugin has recently offered a valuable evaluation of Duhem’s work: Although Duhem was not equipped to solve the problem of what he called the

“nonsensical” (abérrantes) branches of mechanics—those branches wheredissipation is the most important mechanism at play, one may find in some ofhis penetrating writings, ... the germ of the notion of internal variable ofstate. Without digressing too much on this we simply note that these areadditional variables of state whose introduction reflects our lack of complete

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higher and lower level structure in rubber

control of microscopic mechanisms (e.g.dislocation movement) which are responsible forsome macroscopically irreversible manifestations(e.g. plasticity and hardening in metals, magnetichysteresis in ferromagnets). Althoughmeasurable by a “gifted” experimentalist oncethey have been identified (this is the crux of thematter), these variables are not controllable sothat they clearly distinguish themselves from themore classical observable variables of state thatare controlled by body or surface actions.94

But at this juncture, modern work in thethermal behavior of solids departs from Duhemianexpectations in a significant way. In this quotation,Maugin refers, in part, to structural changes thatinvolve forms of frozen order that arise on scalelevels intermediate between the macroscopic and the molecular. Here’s acharacteristic example, which carries us into the dominions of multiscalar behaviordiscussed in Essays 1 and 5. Consider an ordinary rubber band. This materialconsists of highly flexible polymer chains pinned together at various points but whichotherwise alter their shapes constantly with little energetic cost. FIG: HIGHERAND LOWER LEVEL STRUCTURE IN RUBBER We speak of the“temperatures” of rubber bands without apologetics and report that they becomeslightly hotter when stretched. For the reasons traced above, such discourse shouldbe rejected as illicit if conventional thermodynamic stipulations with respect toequilibrium are followed. That is because the equilibrium state for rubber at roomtemperature is a disjointed polymer soup with severed cross-linkages, a conditionwhich we are likely to see after a relaxation time of several hundred years. Tadmorand Miller write:

Recall that a standard statistical mechanics phase average integrates over allconfigurations accessible to the system regardless of the amount of time thatone would have to wait to reach such a configuration. The resulting phaseaverage will therefore provide a smeared view of the liquid-like behavior thata solid will exhibit over infinite time.95

However, let us look at our rubber band upon its next lowest stage of significantRVE behavior.96 There we can assemble a restricted form of statistical ensemble thatconsists in all the higher scale shapes that rubber’s cross-linked chains mightpossibly assume, reflecting the multiplicity cage in which these chains can rattle

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homogenization of a rubber band

around freely. This ensemble supplies us with mesoscale-centered surrogates forboth entropy and temperature using standard statistical mechanics techniques. Onthis basis, we can then apply the Gibbs rule to successfully calculate the mean restlength LR the chains will prefer at a given temperature, as well as the amount ofoutside force F needed to stretch the band to LR + ΔL. Tadmor and Miller continue:

[T]he basic idea [of the restricted ensemble approach] is to restrict thestatistical mechanics by only including configurations that are consistentwith the metastable state being considered ... These properties suggest that it

should be possible to use statisticalmechanics to compute the properties of asystem in metastable equilibrium, suchas its temperature, free energy, stressand so on. Strictly speaking, these arenot thermodynamic state variables, sincethe system is not in thermodynamicequilibrium, but they play the same role. Penrose and Lebowitz refer to them asanalogs for the thermodynamicvariables.97

Indeed, it is only at this relatively elevated mesoscopic scale that the truesources of rubber’s elastic behaviors are found, not at the atomic levels of the atomsthat comprise the polymer chains. These considerations suggest that the problems ofsolid thermal behavior might be profitably addressed following the multiscalarmodeling policies outlined in Essay 5. On this approach, we should assign ourrubber band an initial array of descriptive variables based upon standard thermalcontinuum mechanics technique operating on a macroscopic level, but we shouldthen check these determinations for self-consistency against suitable lower scalesubmodels of local polymer structure. In this manner, we should verify that thetemperatures we assign to our rubber band on a macroscopic level agrees with theensemble distributions we encounter at the characteristic scale level of the polymerchains. In the same way we also verify that the restoring forces witnessed on amacrolevel agree with the maximized “entropy” increase appearing at the polymerlevel. If these assessments don’t agree, we correct our estimates until they come intocross-scalar harmony. Lower levels of rubber structure may require consultation inan allied manner. Communications between these levels of submodeling transpirethrough homogenizations, in the general manner that the ensemble wigglings of thecross-linked polymers become averaged into suitable scale-linked analogs oftemperature and entropy. FIG: HOMOGENIZATION OF A RUBBER BAND

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However,–and this is an important departure from previous practices–the exactchoice of homogenization technique utilized should reflect the physical manners inwhich lower scale behaviors harmonize with, or overwhelm, the dominant behaviorsnormally witnessed on the scale length above. In rubber’s circumstances, usefulhomogenization employs averaging tactics very similar to those employed for simplegases, albeit with an adjustment in the ensemble consulted. But suitable“temperature” analogs in other materials may require more nuanced forms ofhomogenization averaging, selected to imitate the manner in which coherent energyon a higher scale degrades to less organized forms of energetic expression in thedomain below.

This vantage point suggests that we look upon the descriptive utilities of“temperature,” “heat” and “entropy” in a different light.98 Recall that the essentialdifferences between a pressure and heating reflect the degree of coherence retainedin the material’s response to some exterior energetic effort. When we apply adirectional force to a spring, it retains most of the work applied in the form ofmolecular strain energy that allows the spring to eventually regain its normal relaxedcondition. This is a coherent form of energetic storage, but, insofar as our effortraises the temperature of the steel without elongating it, that portion of our originaleffort has become degraded into random wiggling at the molecular level. With arubber band something surprising occurs when we stretch it; our endeavor decreasesthe mesoscale “entropy” corresponding to the fact the polymer chains have fewerways of wiggling than before. When we let go of the band, the chains wigglethemselves back to an arrangement where the linked chains can wiggle as freely asthey are able, thereby maximizing the mesoscale “entropy” that captures thisfreedom. Even so, the cross-linking of the polymers retains a fair capacity forperforming coherent work along the central axis of the rubber band; we can stilldirect their motions towards our third grade classmates. But the more closely ourrubber approaches its official room temperature equilibrium as a disorganized soup,less of this mesoscale capacity for retaining energetic coherence is retained. Suchconsiderations show that, in most solids, coherence to incoherence degradationstranspire along trickle-down hierarchies linked to natural cause and effectrelationships between RVE scales. Coherent work at the macroscale becomesincreasingly degraded as it drops through a hierarchy of size scales, but this processcan be greatly retarded if some robust level of frozen order intervenes along the way. Loss of macroscopic coherence needn’t manifest itself as increased wiggling uponthe molecular level, as happens within an ideal gas due to its lack of retardingmesoscale structure.99 These are the factors that Maugin has in mind in the quotationabove, when he talks of lower scale factors that can be measured by a skilled

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experimentalist but are not easily controlled by macroscopic manipulations upon thetarget system. In modeling such a hierarchy of trickle-down causes and effects,homogenization relationships model the manner in which higher level coherencebehaviors become gradually lost within a material, without immediately appearing asfully randomizing wigglings at the molecular level. Allied considerations of thermal environment affect the permanent forms ofdamage that can arise upon variety of intermediate size scales, e.g., at the level of thechemical bonds that sustain the cross-linkages between the polymer chains. As localtemperatures become higher, these bonds are more likely to break, leading tohysteresis effects analogous to those discussed in Essay 5 (overly-flexed rubberbands eventually snap through accumulated chemical damage). But these lowerscale events are generally controlled by local temperature and entropy “analogs” thatdiffer significantly from the notions applicable at other scales. Earlier I invoked ascale-sensitive thermal analog when I wrote of surface asperities that deformplastically at elevated local “temperatures” (children don’t increase the macroscopictemperature of a playground slide much as they slide along, but they heat theirregular surface protrusions within the metal to tremendous local “temperatures”). To capture these hierarchial relationships properly, we must develop useful“analogs” for temperature and entropy that can capture the coherence/incoherencerelationships that act across RVE scales. Generically, these these ΔLhigher scale“thermal analogs” are obtained by “averaging” (= homogenizing) over the freedomsof movement witnessed upon an adjacent scale size ΔLlower, but the exact choice ofhomogenization technique employed is rarely the same from case to case. Eachaveraging policy should mimic the precise character of the cascade of increasedincoherence that connects ΔLhigher to ΔLlower within nature.

At this point let us pause to correct a descriptive mischaracterization that Ihave carried over from Tadmor and Miller (and, indirectly, Penrose and Lebowitz). We should not have spoken of scale-sensitive analogs to “temperature” and“entropy”; we should have couched our discussion in terms of the context sensitivityof these two words. Many scientists tacitly presume that “temperature”’s localadaptation to ideal gases somehow captures its proper core reference, but privilegingthat particular adaptation doesn’t render the glorious descriptive reach of the termsufficient justice. It merely reflects the historical order in which the randomizationprocesses active within various materials have been successfully investigated. Ofcourse, the simpler materials lacking significant mid-range structure were conqueredfirst, but the word “temperature” was in active use long before that. More than that,the coherence/incoherence focus of thermal words makes the project of unravelingthe strategic choices that underwrite its proper word/world attachments extremely

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difficult. As I said before, even within the philosophy of langauge we’re not gottenused to words that behave like this. And so we shouldn’t expect that greatphysicists will prove equally attuned to the complexities of real life semanticattachment.

The net result may be that, looking across the broad sweep of their physicalapplications, there is no simple answer as to what “temperature” and the otherthermal notions should concretely pick out; these notions obtain their robust physicalcorrelates family-resemblance style within the context of elaborate forms ofhierarchical modeling. When we talk of a “temperature,” we implicitly invoke somediscourse dependent size scale to outfit our word with referential underpinnings of anadequately robust character. Ignore the background explanatory architecture inwhich the word appears and you abandon all hope of understanding its semanticbehaviors.

It might be objected, “But don’t the successes of statistical mechanics firmlyestablish that ‘heat’ and ‘temperature’ are firmly correlated with molecularmotions?” Here we must be careful about what we mean by “statistical mechanics.” Certainly the successes of our higher scale referential “analogs” for these notionsinvolve inherently statistical ingredients (e.g., averaging over suitable ensembles andall that), but we apply these statistical tools to behaviors that arise on a variety ofhigher characteristic size scales. Moreover, no single flavor of “statistics” isemployed–different crystal laminates require different forms of homogenizationrelationships to capture their interscalar relationships properly. All of this deservesthe label “statistical mechanics,” but it is not “molecular level statistical mechanics.” Hence considerable ambiguity attaches to offhanded appeals to “the successes ofstatistical mechanics.”

None of this implies that workers in statistical mechanics haven’t constructedsuccessful up-from-the-molecules models for simple substances such gases andperfect crystals, but these materials don’t present the same difficulties of supplyingsuitable mathematical surrogates for the complex environmental baths in which smallhunks of typical solids typically sit. At the present time, our most successful modelsfor these materials function in the multiscalar fashion just outlined. Anticipatingworkable up-from-the-molecules models for these layered materials must beregarded as an aspirational hope at the moment, rather than an accomplished reality.

In an allied vein, modern physicists following Einstein now propose broadscale cosmologies for the entire universe in a manner that Duhem and many of hiscontemporaries would not have anticipated. At the present moment, however, theyapproach the task more in the spirit of a multiscalar modeling than in the simple, fullyautonomous trajectory mode of Theory T expectation. Specifically, our current

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cosmologies articulate broad scale models that supply local temperature and entropyenvironments in which smaller scale events unfold in the self-consistent manner of atypical multiscalar scheme within materials science. As such, thermal relationshipscontinue to comprise an important part of the asymptotic stitching that holds theirvarious patches of descriptive fabric together in cooperative harmony. Will suchpatchwork organizational patterns persist in future “fundamental science” or willthey be eventually replaced by something else, including the simpler syntacticcontours of a Theory T scheme? I don’t know and, in agreement with Duhem, don’tbelieve that we should lay down a priori strictures in these syntactic respects.100

I stress that these speculative cavils largely revolve around our abilities tocapture natural processes fully within our computational webs. At present werecognize that notions of energy degradation and information loss somehow play animportant role in the overall story of our universe, but we remain unsure of exactlyhow these processes should be acknowledged within our mathematical models. Right now we employ statistical constructions (coupled with a lot of asymptoticappeal!) to introduce workable surrogates for these processes into our moresuccessful models but I doubt that anyone today fully understands why we need todo this. My own suspicion is that some kind of cooperative family harmonizationbetween descriptive elements is involved, of the sort surveyed in Essay 8.101 But thisis merely a half-baked hunch, not a well-worked out proposal.

In any event, the object of this essay is not one of preaching hazy dogmas ofmy own, but to lessen the limiting strictures that Theory T dogmas have unwittinglyplaced upon developing science, in the same way that Duhem and Mach sough tofree physics from the manacles of thinking about heat in the cocksure manner ofPreston and Cotter, quoted above. As Essay 9 emphasizes, our exact computationalplace in nature remains unsettled and we don’t fully know how ably natural physicalprocesses can be captured within the webbing of our mathematics. We certainly doknow that, starting from opportunistic descriptive perches anchored in characteristicscales, boundaries and temperature regimes, we can construct precise mathematicalmodels that do astonishing well by nature’s lights. But we don’t yet know that thesepatches of successful description can be welded into a unified mathematical settingthat can provide fully autonomous, cosmological trajectories of a possible world ilk. This isn’t a fault of the world, but it may represent an inadequacy in ourcomputational resources.

In any case, I do not strive to act as a prophet in these regards; I am merelyreminding other philosophers that one of our chief tasks is to keep intellectualoptions open, rather than closing them prematurely through unacknowledgeddogmas.

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(xiv)

My Wandering Significance devotes many bloated pages to arguing that trustin accepted standards of rigor should be seasonal--there are times when suchstandards should be sternly obeyed and other times when they should be blithelydisregarded. This is because their reliability largely depends upon whether thesemantic circumstances of the vocabularies to which we apply our reasonings havebeen rightly conceived. Sometimes we frame incorrect pictures of no fault of ourown, because effective usage presents us with many semantic mimics that look asour words have entangled themselves with the world in manner X, when, in fact, theyactually operate in manner Y. Commonly, X is strategically simpler than Y, so wenaturally opt for X, having little reason to suppose otherwise. Indeed, this is how webecome fooled by conjurer’s tricks. Through the atmospherics of setting and patter,we are coaxed into presuming that process X occurs when it’s really process Y.

In other essays, I argue that words often shift their manners of worldlyattachment as they become applied within enlarged applicational settings and theirdetailed contours adapt to local opportunity. These silent developments offer ripeopportunities for semantic confusion and misdiagnosis. Despite his deep insightsinto thermal reasoning, I believe that Duhem eventually loses his way thoughinsisting upon a certain flavor of rigor beyond its rightful expiration date. Hesatirizes proponents of the kinetic theory of heat for their irregular invocations ofprobability:

These simple pointers announce sufficiently the extreme difficulty that weregoing to be met by physicists when they wanted to use the kinetic hypothesesas the point of departure for rigorous deductions; these difficulties aresummarized in these two words: approximation, probability.102

If our recent observations are well-founded, adaptivity in selection ofhomogenization technique represents precisely the descriptive fine-tuning we shouldexpect to apply to the subtly different patterns in which coherent energy degradationoccurs within different materials, according to their individual hierarchies of higher-level structure. Once a suitable appraisal of how the cross-scalar energetic cascadesoperate within a target material has been obtained, a corresponding homogenizationstudy should be pursued as rigorously as possible according the standards of rigorinternal to functional analysis. But we must get the operative physics rightbeforehand; the merits of rigorous inquiry remain hostage to the accuracy of thephysical diagnosis.103

On the other hand, Duhem’s criticism of up-from-the molecules approaches

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may be entirely correct. Suitable probabilistic surrogates for our macroscopic level“temperature” classifications may not be available at this lowly scale level, for thereasons surveyed in Essay 5; the mechanisms of upper scale coherence maintenancecan’t be recognized from that lowly perch. A significant methodological controversyraged in the nineteenth century over whether continuum materials were betterapproached in Navier’s up-from-the-molecules manner or to follow Cauchy, Greenand Stokes and operate from the macroscopic scale downward.104 Eventually thelatter approach won out and the methodological lessons learned lie behind Duhem’sscorn for bottom-up modeling approaches like those of conventional kinetic theory. He is excessively extreme, of course, in his dismissive edicts, but he is right topresume that, in application to complex solids, that such approaches must frameexcessively speculative guesses as to what their higher-level structures look like froma molecular point of view. The modern multiscalar techniques I have cited suggestan interesting mixed-level compromise. Work outward from an array of middle levelsubmodels and make sure they exchange energy with one another in appropriateways. From this vantage point Duhem has gotten the overall architecture of thehouse of “temperature” right, but not the decoration within the individual rooms.

Such considerations illustrate why we should be wary of absolutist appeals to“rigorous reasoning,” of the sort captured in this characteristic Duhemian rant:

In the course of its development, a physical theory is free to choose the pathit pleases, providing that it eschews all logical contradictions; in particular,it has to take no account of experimental facts. [But] this is no longer sowhen the theory has attained its complete development. When the edifice hasreached its peak, it becomes necessary to compare with the set ofexperimental facts the set of propositions obtained as conclusions from theselong deductions.105

He employs such sermons as a mace of rigorism with which he bludgeon hisopponents, especially the hapless English physicists whom he maligns as haplesspurveyors of inconsistency and unmonitored intuition. But we’ve just noted thatformal rigor can prove a tricky weapon to wield judiciously, in circumstances whereless constrained forms of exploration must ascertain how the going of it goes,paraphrasing Oliver Heaviside. Science’s ultimate responsibilities, after all, lie withnature’s behavioral patterns, not with the structures that mathematics can articulatemost readily. Other essays in this collection have suggested that “stupidity ofmathematics” considerations may demand rough-hewn correctives to maintainreasoning with applied mathematicians along nature-suited trajectories. And ourproblems in capturing energy degradation in any form other than homogenizationaveraging may represent one of those corrective interventions.

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Truesdell

This basic dialectic shows up in an interesting wrinkle within the moderndevelopment of continuum thermomechanics, at exactly the stage where progressbegin following a somewhat different path than Duhem’s otherwise propheticsuggestions indicated. As noted before, the mid-twentieth century researches ofClifford Truesdell, Walter Noll and their school developed Duhem’s suggestions invery effective ways. And Truesdell especially entertained conceptions of rigor verymuch like Duhem’s, to such an extent that, on a blind testing, one can’t distinguishone author from the other. In coauthorship with R.A. Toupin, Truesdell writes in1960: FIG: TRUESDELL

Experiment, indeed, is a necessary adjunct to a physical theory; but it is anadjunct, not the master.... [We follow] the postulational standpoint,according to which physics, as an abstract discipline, may employ anyvariables and any consistent initial assumptions or “laws” which areconvenient. In construction of this mathematical system it is not necessary tomaintain contact with experiment at every stage. The system is an abstractmodel, designed to represent some of the observed phenomena of thephysical universe, but directly concerned only with ideal bodies. Some few ofthe properties of these ideal bodies are postulated; the numerous remainderis to be derived mathematically. Whether these derived propertiescorrespond with physical observation is a separate question to be decided bysubsequent comparison with experiment.106

But multiscalar technique departs from this demand for single-level purity. Thesuperior performance of multiscalar modelings that don’t adhereto Truesdell’s stern admonitions prompts G.A. Maugin tosardonically comment:

The bias, shortcomings and audacity [of Truesdell’sapproach] illustrates the elegance and temptations of allformal logico-deductive approaches: it is veryattractive!107

As a result, many useful modern forms of extendedthermodynamics that have been proposed following the generalcontours of Duhemian (or Truesdellian) methodology, but none of them claim toreach every form of frictional behavior with untarnished success. And there is nopressing reason why they should.

I want to underscore the fact that I do not write as a blowsy opponent of rigor,in the manner of Paul Feyerabend, say. Sharply specified axiomatic rigor stillprovides the same utilities as when such methods were originally devised: to keepunwanted ingredients from surreptitiously compromising our models through the

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vagaries of intuitive thought. The byways of conceptual development being as theyare, we don’t want the prejudices sneaking in, but we should welcome theinnovations. Unfortunately, distinguishing the two isn’t always easy.

In many ways, the Theory T strands within Duhem’s thinking clash with thediagnostics procedures I have emphasized within this essay, viz., the wisdom oflocating reliable referential attachments for “temperature” by assembling its usagethrough carefully moderated prolongations across modulated layers, rather thanboldly attempting to lay out a “complete postulated theory” in full Theory T regalia. True, the latter policies can offer a valuable logical perspicuousness, whereas thegradualist products of reliable assembly often result in semantic arrangements ofconsiderable complexity. We are often confused by the latter arrangements, just asbeginners usually find thermodynamics baffling. Nonetheless, these awkwardnessesshould be regarded as neither surprising nor distressing. In the final analysis, ‘tisbetter to trust a somewhat muddled political leader who governs by warm instinct forhuman vicissitude rather following someone who invariably obeys the strict logic ofrigorist tenet.

As noted before, Duhem can be quite snippy about the adventitious manner inwhich anti-thermal mechanists invoke shifting notions of “probability” and“averaging” to advance their programs. The modern descriptive successes of the“thermodynamic analogs” utilized within modeling suggest that shiftingspecializations may supply the proper recipe for coping with the semanticperplexities of “temperature” and “entropy.”108

Although Duhem rejects autonomous trajectory demands, he does insist uponwhat might be called single-level descriptive demands.

[T]heoretical physics is [not] beyond the laws of logic. It deserves the nameof science on the condition of being rational. He is free to choose itshypotheses as he pleases, provided that these hypotheses are not redundantor contradictory; and the chain of deductions that connects to the hypothesesthe truths of the experimental order must contain no link of dubiousstrength... If various parts of physics are represented by theoriesunconnected with each other, or even by theories that contradict each otherwhen they meet in a common domain, the physicist must regard this disparityand contradiction as transitory evils; he must endeavor to substitute unity forthe disparate, logical agreement for contradiction; he should never have totake sides.109

Surely, this is not a reasonable demand to make of theories that deal with complexsolids, for it demands that lower-scale events need to find adequate expression inparameters at the macroscopic level. The successes of multiscale modeling trade on

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the fact that they eschew this demand, in favor of self-consistent and homogenizedcommunications between submodels. He quotes Henri Poincaré on his behalf,writing of Maxwell:

The English scholar does not seek to build a building unique, definitive, andwell ordered edifice. It seems rather that he raises a large number ofprovisionary and independent constructs, between which communications aredifficult and sometimes impossible.... Two conflicting theories can,indeed—provided they do not mix and that are not seeking the bottom ofthings—be both useful instruments of research, and perhaps reading Maxwellwould be less suggestive if it had not opened both new and divergentpathways.

Characteristically, Poincare is gentler and nuanced than Duhem, for he restricts hissingle-level expectations to fundamental theories that “seek the bottom of things.” Even here, we might note that homogenization techniques provide an alternativemanner in which syntactically unamalgamated subtheories might not mix, yet still canstill exchange communications with one another in homogenized manners thatsupplies us with a fairly satisfactory portrait of nature. This degree of satisfactionwill be enhanced if it turns out that the barriers between our packets of descriptiveeffort should be aligned along the natural hierarchies of scale encountered in nature,which seem to be carved out by considerations of energetic coherence andinformational degradation. If so, the barriers to amalgamated theoretical descriptionof a Poincare stripe may reflect the fact that mathematics only offers us statisticaltools for capturing these relationships.110

At this point, I have ventured further into the speculative mists than I prefer togo. Let us therefore stop and thank Duhem for his profound insights into thesupportive architectures that underlie our eminently useful, yet deeply confusing,thermal vocabularies. Ersatz rigorist propensities blocked him from recognizing thewider architectural opportunities that multiscalar elaborations offer, so I will limit myglasses of cheer to two, rather than three. His diagnostic efforts did not fully heraldthe shape of thermal physics yet to come. But they came remarkably close.

Appendix. Hertz’s Challenge and the Underdetermination of Theory

Contemporary readings generally trace Duhem’s anti-realist strains to hisarticulation of what is now known as the Quine-Duhem Thesis: the claim that anycherished hypothesis H can be protected against empirical disconfirmation throughblaming some other hypothesis H' utilized in its problematic applications. Indeed,Duhem sometimes writes as if a simple modus tollens is sufficient to establish this

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fact: (H & H') E E H v H' Yet, surely, this innocuous logical observation cannot adequately support anysweeping claim that the choice of a physical ontology represents, in the finalanalysis, a matter of metaphysics rather than physics proper. Yet it is precisely thesestrong philosophical contentions that inspire the modern anti-realists who evokeDuhem for intellectual support.

However, if we inspect, not The Aim and Structure of Physical Theory, butDuhem's prior and more technical work, The Evolution of Mechanics, we find thatDuhem further believed he had established the stronger thesis known today as theobservational underdetermination of theory--the supposition that two or morecompletely distinct theories can organize all possible observational evidence withequal adequacy. He further believed that he could demonstrate this claim with aconcrete example. He wrote:

Whatever may be the form of the mathematical laws to whichexperimental inference subjects physical phenomena, it is alwayspermissible to pretend that these phenomena are the effects of motions,perceptible or hidden, subject to the dynamics of Lagrange.111

As was his wont, Duhem did not explain what he had in mind as pellucidly as hemight. The purpose of this appendix is to supply a crisper explication of his thinking.

It would be a matter of some moment for contemporary philosophy of scienceif Duhem’s example could be fully redeemed. That distinct theories exist which areobservationally equivalent in some strong sense represents a doctrine to which manythinkers cling passionately but, arguably, represents a thesis for which noindisputably sound examples have yet been proposed. Most proposed cases ofwhich I am aware either turn upon some misunderstanding of the relevant physics orinvolve factors that should seem innocuous to the average scientific realist.112 Manyphilosophers of the logical empiricist era--W.V. Quine representing a primeexample--endorsed the undetermination thesis in the absence of concrete illustrationslargely because they believed that any collection of "observational consequences"can be supplemented in many ways by inequivalent extensions. But a “theory” isjust such a logical extension, hence underdetermination follows. Today most of usbelieve that this logic-based defense of undetermination rests upon an untenablepicture of how real life theories produce their "observational consequences." Itwould be preferable if defenders of the claim could produce some convincingexamples from real life science. Without attempting to evaluate the merits of other

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Hertz

contenders here, I will simply state that Duhem's suggested pair of allegedlyobservationally theories represents as good an illustration of the expected behavioras I have encountered.

However, when we inspect his chief example closely, we find that a key stepin Duhem’s reasoning relies upon the outmoded assumption that a policy of essentialidealization must be invoked in setting up the equations for a continuous bodies. These concerns trace to a fundamental difficulty that arises within in continuummechanics and is central to the topics of Essay 8. Continuous matter ismathematically self-similar on every size scale, in the sense that any smaller piece ofa homogeneous material behaves exactly like any larger piece. Differentialequations supply us with convenient reasoning tools because they normally describesystems on size scales at which the interactions that characterize their governingphysics directly operate. But when we attempt to gain descriptive purchase upon aflexible string through these tactics, we find ourselves once again confronted withsmaller versions of our original problematic, with no helpful simplification gained at

all. To halt this unprofitable regress and to locate a platform uponwhich trustworthy reasoning can be founded, scientists of Duhem’sera presumed that flexible matter should be intentionallymisdescribed at some small size scale to allow Newton’s laws andallied mechanical principles a foothold.113 Often one encountersdubious remarks to the effect that “these little parts are so small,they will scarcely differ in their behavior from a flexible object atthat size scale.” We reasons about these little “elements” for awhile at a finitary level and then shrinks down to a differentialequation level conclusion through taking a loosely defined “limit.”

But of what stuff should these finitary replacements be made? A popular veinof response favors a finite array of interacting point masses, in the mode of S.J. Boscovich and the French physical atomists who wrote the beginnings of thenineteenth century. Even today one often finds this mass point-centered policyemployed within textbooks that are written for students of quantum mechanics, forpedagogical reasons that are entirely irrelevant here. Such techniques often produceimproper results and introduce lead to further puzzlements of their own.114

A more rational tactic begins with “elements” that correspond to extended,finite-degrees-of-freedom systems, that either contain no flexible parts whatsoever ora limited selection of simple continua such as springs and dashpots. The behaviorsof these little assemblies can be monitored by any set of finitary principles applicableto connected mechanisms of this kind, e.g., rigid body mechanics, Hooke’s law forsprings and so forth. If we take the further precaution of limiting our “our elements

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cyclic coordinate

stored energy due to hiddenmotions

are so small, they will scarcely differ in their behavior from a flexible object at thatsize scale” appeals to circumstances of static constrained equilibrium, then thesecurity of our procedures will become enhanced as well.115

Duhem finds his descriptively equivalent theories within different approachesto these elements-before-they-are-shrunken-to-limits. On one side of the ledger heplaces any approach that monitors active force strength with its elements throughappeal to stored potential energy (his own thermomechnicalapproach fits this bill, but also standard Lagrangian puremechanics technique). On the other side of the ledger, heconsider the ultra-mechanical proposals advanced by HeinrichHertz in his Principles of Mechanics116 of 1894. FIG: HERTZ This approach is ultra-mechanical in the sense that Hertz bars all potentials within hiselements altogether, precisely implementing the popularpresumption that we encountered earlier within the quotationfrom Preston and Cottle: potential energy, at root, is merelykinetic energy in disguise. But Hertz advances beyond theirvague speculations, through appeal to some striking results obtained earlier byRouth, Helmholtz and James Thomson. Suppose that some subset of thedescriptive parameters pertaining to the large-scale behavior ofa mechanical system are cyclic, in the sense that their spatialpositions do not effect the total energy of the complete system. FIG: CYCLIC COORDINATES A classic exemplar issupplied by the spinning inner ring of a gyroscope. As soon asone particle within the ring rotates out of its present position,its place becomes immediately occupied by some similarneighbor. For all intents and purposes, the overall behavior ofthe gyroscope is indifferent to the positions of the particleswithin the ring, although their collective angular velocitiesremain important, for they determine how difficult shifting the‘scope to another orientation will prove. The faster its ringspins, its resistance to displacement will increase–the spinningcreates a simulacrum to inertial sluggishness.

This ability to imitate the effects of mass intrigued figures like Kelvin, whoemployed the idea centrally within his speculations on the vortex atom, then popular. This approach does not eliminate the need for mass within mechanics altogether, butcreates a situation where what we normally measure as “mass” isn’t a true mass atall, but rather a gyroscopic mimic. But Hertz adopted these considerations to

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eliminate potential energy-derived forces from his elements altogether. Let’s put agyroscope in a box and hook up the arms of a visible puppet to its spinning axis. FIG: STORED ENERGY DUE TO HIDDEN MOTIONS If we are unable toobserve the insides of our gyroscope directly--its innards are encased within anopaque box, say--, the internal whirling will appear to supply a hidden source ofpotential energy that makes moving the arms harder than it would have otherwisebeen. But this resistence stems entirely from the fact that trying to wiggle thegyroscope affects its kinetic energy of spinning. In more formal terms, this amountsto the observation that any potential energy term V(x,y) encountered within a normalLagrangian can be fleshed out with a further cyclic variable z such the magnitude ofV(x,y) becomes exactly replicated by a new factor that calculates the kinetic energyof z based upon z’s placement relative to x and y. This is the tactic captured in thephrase “Hertz’ technique for eliminating forces through appeal to hidden masses.” Since this mimicking capacity is quite general and can be captured within an easytheorem, it looks like we have found our desired set of observationally equivalentpairs, essentially echoing the braggadocio of Annie Get Your Gun: "Anything youcan do, I can do smaller." And Hertz is also prepared to imitate Duhem’scherished forms of thermodynamic potential in the same manner.

A scientific realist would prpose that we want to ascertain whether there’s apendulum inside the box, send in a probe to find out. But such tests are precluded bythe special idealized character of our elements–they don’t correspond to tangible,worldly structures, yet they make our theory pairs look different in their theoreticalcontent. This is why Duhem suggests that any preference of one over the other musttrace to extra-physical considerations.

However, all of this rests upon a methodological mistake–the physicists ofDuhem’s era appealed to a stretch of “philosophy” (viz., essential idealization) tofill a gap in their mathematical procedures that they were otherwise unable to repair. The continuing popularity of underdetermination themes within philosophy, largelyunsupported by specific exemplars in Duhem’s manner yet still invoking hisconclusions as an argument from authority, indicates that, in a very real sense,modern philosophy remains haunted by the ghosts of departed physicalinfinitesimals. But the proper moral we should extract from Duhem’s example is notanti-realism, but patience. Methodological puzzles often take a long time beforetheir proper underpinnings can be fully rationalized.

ENDNOTES:

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1. P.M. Duhem, The Evolution of Mechanics, J.M. Cole, trans. (Berlin: Springer:1980), p. xl.

2. For example, little effort is devoted to aligning its loose talk of “laws” and “initialconditions” with the ingredients found in a standard physics textbook.

3. Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics(London: Chapman and Hall, 2010), p. 2. See Maugin’s longer appraisal offeredin The Thermomechanics of Nonlinear Irreversible Behaviours (Singapore: WorldScientific Series, 1998).

4. I shall trace the essential ingredients in a later footnote.

5. Middlemarch (Oxford: Oxford University Press, 1997), p. 82.

6. In the manner of the redoubtable Mr. Magoo, we might add.

7. Euler is reputed to have observed that his pencil appeared to surpass himself inintelligence.

8. Since the 1950s, traditional questions of scientific realism have becomeunfortunately entangled with simplistic theories of linguistic reference, mostnotoriously in the Richard Boyd/Hilary Putnam form that “terms in a mature sciencetypical refer” (cf., Putnam, Meaning and the Moral Sciences (London: Routledgeand Kegan Paul, 1978), p. 20). Here their conception of “reference” demands thesimple contours of “”Fido”/Fido alignment. The layered architecture of Duhem’sanalysis shows that simplistic encoding of this nature do not suit the utilities ofthermal words ably at all. If “semantic” doctrines of this stripe come into tensionwith a realistic view of science, then the semantic doctrines should take the fall, notthe realism. WS discusses these issues at considerable length, and they alsoreemerge within the discussion of Paul Benacceraf’s views in Essay 9.

In an allied vein, apriorist claims that “applied mathematics works” throughexploiting “isomorphisms” between abstract structures and physical relationshipsstrike me as similarly naive and motivated by a desire to cloak complexrelationships within the ill-fitting garments of a simplistic “resolution.” In all ofthese cases, coarse Theory T thinking encourages a jolly disregard for theoperational details it regards as picayune. “Don’t bother with all of those pettycomplexities, for such pockmarks on practice will surely vanish in our scientificfuture.” In that same manner, the Great Oz advised Dorothy to pay no attention to

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the little man behind the curtain.

9. Later in the essay we will show how Duhem’s account can be further improved byexploiting the multiscalar techniques outlined in Essay 5.

10. Note that this critique stems from an author–myself–who can be readily faultedfor the same proclivities. Sheldon Smith directed my attention to this particularexemplar of inept analogy. It comes from The Aim and Structure of Physical Theory(Princeton: Princeton University Press, 1954). p. 116. In footnotes yet to come, Ideal with a large number of common misconceptions of Duhem’s central concerns,largely generated by the deficiencies of his exposition. I recognize that these smallprint elaborations quickly become annoying, but incorporating them in our centralnarrative would complicate our main lines of thought unnecessarily (they areelaborate enough as they stand). On the other hand, these misunderstandings havebecome so firmly entrenched in philosophical lore that serious confusion will ensueif these interpretative issues are left unaddressed.

11. Psychologically, I sympathize with Duhem’s occasional irritabilities. Theopponents who favored a purely mechanical approach to thermal behavior oftendismissed his cogent observations out of hand without much consideration. In aparallel manner, I am often frustrated by philosophers, especially of the analyticmetaphysics school, who dismiss substantive features of scientific practice onsimilarly flimsy grounds. In response, my prose grows strident even when I’mattempting to promote a counsel of ecumenical tolerance.

12. My interests in classical continuum physics trace to a historical puzzle. Thescientists that present day anti-realists most often cite as motivational authorities areMach, Duhem and Hertz. Yet they all lived in the Victorian period, at the allegedpinnacle of classical mechanics success. This observation made me wonder, “Whyshould these folks, of all people, exude anti-realist sentiment?” As this essay makesclear, their motives were not as portrayed by their modern apologists.

13. This is the exact moral of the perfect volcanos analogy of Essay 5

14. A good source: Truesdell’s A First Course in Rational Continuum Mechanics(New York: Academic, 1977).

15. In fairness, my own concerns with the awkward manners in which mathematicscaptures dominant behaviors and energetic degradation can be recast in a limited“essential idealism” vein, although I do not favor that mode of expression myself.

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16. Bas van Fraassen, The Scientific Image (Oxford: Oxford University Press, 1980)and Nancy Cartwright, How the Laws of Physics Lie (Oxford: Oxford UniversityPress, 1983).

17. I am reminded of the contrast between Walt Disney’s Davy Crockett and theunreliable rowdy of the Almanacs.

18. I employ this term to designate what might be more fulsomely designated asnon-equilibrium thermal continuum mechanics (“non-equilibrium thermodynamics”alone won’t do, as that label sometimes embraces approaches that fail to reckonwith purely mechanical crossover effects). Terminologies in this branch of physicsare generally misbegotten, due to the fact that orthodox “thermodynamics” doesn’tsupply a dynamics at all, but merely comprises a specialized department withinstatics.

19. J.L. Lagrange, Analytical Mechanics (Dordrecht: Springer, 1997). Readersshould beware of confusing the book’s two-stage, virtual work policies with themore restricted formalism commonly called “Lagrangian mechanics” today, which isfounded upon Hamilton’s principle as a single-level variational standard. Lagrangediscusses this second approach in his great tome, but ultimately favors alternativefoundations based upon virtual work d’Alembert’s principle lifts. When Duhemwrites about “the Old Mechanics,” he generally intends the latter. The formermethods does not handle non-holonomic constraints ably, but these allowances arecentral within the macrolevel equilibriums that thermodynamists following Clausiusinvoke in justifying their thermal supplementations.

20. Mathematically, the problem of integrating thermal and mechanical effects isrendered difficult by the fact that Fourier’s heat equation is parabolic in signature,whereas the waves of pure mechanical experience rest upon evolutionary equationsof a hyperbolic character. As Duhem himself notes, simple accounts ofmechanical/temperature exchange such Laplace’s theory of sound do not retard theinteractions strongly enough to support identifiable wave fronts within theirmaterials. Modern requirements on modeling descending from relativity’slimitations on signal speed demand the latter, so modern research in this area isoften subtle (see Brian Straughan, Heat Waves (New York: Springer, 2011)). Forreasons we’ll examine later, cross-effect issues become increasing difficult as alarger number of channels of energetic exchange open, as occurs even within thelimited confines of an anisotropic, three-dimensional elastic bar.

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21. Although we concentrate exclusively upon the entanglements of heat with purelymechanical behavior in this essay, Duhem’s own interests (and those of practicalthermodynamics, generally) focus more intently upon the further couplings of theseparameters to chemical composition and electrical condition–at what temperaturesand at what pressures will two substances combine, dissociate or otherwise changephase? From an experimental point of view, these strong cross-over effects areeasier to measure and, oftentimes, easier to model convincingly (for dimensionalreasons, inter alia). Accordingly, they frequently appear as central motivationalexamples within Duhem’s methodological narrative. For simplicity of discussion,we shall ignore these further dimensions of Duhem’s unificationist project, exceptfor the bits of occasional mumbling required to align our myopic discussion withDuhem’s actual text.

22. As I stress in my preface, readers new to thermodynamics’ shores should notexpect to absorb fully every shaping consideration we will outline, but I hope thegeneral moral of our discussion remains clear. The supportive semantics of thermalvocabulary rests upon elaborate and hard-to-foresee strategic underpinnings, duelargely to the fact that the central phenomena of energetic (and informational)degradation resist easy mathematical modeling in significant ways. Thesecomplications, I think, are characteristic, in spirit if not detail, of the generalentanglements of descriptive opportunity that link most successful forms ofdescriptive usage to the world in effective ways.

23. Saul Kripke, Naming and Necessity (Cambridge: Harvard University Press,1980) and Hilary Putnam, “The Meaning of ‘Meaning’” in Philosophical PapersVol. 2 (Cambridge: Cambridge University Press, 1975). The doctrines in turnencourage David Lewis’ “natural properties” doctrines, which have shaped recentmetaphysical endeavor in unfortunate ways (“New Work for the Theory ofUniversals” in Papers in Metaphysics and Epistemology, Vol. 2 (Cambridge:Cambridge University Press, 1999). Putnam himself (whose student I am) regardsthese claims of simple word/world alignment as largely empirical contentions andsubject to correction under more exacting investigation.

24. A canonical source: Ernest Nagel, The Structure of Science (New York,Harcourt, Brace and World, 1961). A lengthy, if rather naive, literature has sprungup with respect to whether these “identities” are “reductive” or not. In fairness toNagel, scientists frequently employ the mean kinetic energy jingle as a mostlyharmless way of correcting coarse misapprehensions with respect to temperature.

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But the formula does not enjoy the scientific centrality with which is improperlycredited.

25. As I understand him, this point of view is central to David Chalmers’Constructing the World (Oxford: Oxford University Press, 2012). In my “DavidChalmers Versus the Boll Weevil,” (Philosophy and Phenomenological Research,89 (1) (2014)), I raise some of the issues presented here in a severely compressedformat. In his reply, Chalmers rejects their salience out of hand. Insofar as I candetermine, he doesn’t appreciate the gravity of the example.

26. Thomas Preston and J. Rogerson Cotter, The Theory of Heat (New York:Macmillan, 1904), pp. 82, 91-2. Note the complacency with which they address thenotion of internal potential energy, which, like entropy, represents a subtlethermodynamic conception. These reductive impulses strike me as reminiscent ofthe philosophical impulses for rendering thermal notions “palpably comfortable.” Disagreements on these tricky issues persist to this day and the so-called Swinburneentropy debate represented a famous controversy at the beginning of the twentiethcentury.

27. Allied claims are advanced, mutatis mutandis, for the liberties of themathematician and these two veins of toleration intimately commingle in latenineteenth century thought. For a survey of the mathematical side of the ledger, seemy "Frege's Mathematical Setting" in Michael Potter and Tom Ricketts, eds.Cambridge Companion to Frege (Cambridge: Cambridge University Press, 2011).

28. These reductive tropisms date back to Robert Boyle’s “Considerations Aboutthe Excellency and Grounds of the Mechanical Hypothesis” (in M.A. Stewart, ed.Selected Philosophical Papers of Robert Boyle (Indianapolis: Hackett, 1991)) andallied sources.

29. “Analysis of Mach’s The Science of Mechanics” in Essays in the History andPhilosophy of Science, R. Ariew and P. Barker, eds. (Indianapolis: Hackett, 1996),p. 116. The original appears on pp. 93-4 of The Science of Mechanics (LaSalle:Open Court, 1989).

30. History and Root of the Principle of the Conservation of Energy (Cambridge:Cambridge University Press, 2011), p. 17. Mach’s Principles of the Theory of Heat:Historically and Critically Elucidated (Dordrecht: Reidel, 1986) is extremelyinteresting with respect to the topics surveyed here.

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31. Evolution, op cit., p. 116. In the main, Duhem envelopes his psychologicalthemes with in his silly–and obnoxious–characterizations of various forms of“national mind.” We will pass over these passages in embarrassed silence. Mach’s suggestions along these diagnostic dimensions are greatly superior and myown Essay 6 remarks on the early apriori very much follow in his footsteps.

32. Duhem, “Analysis”, op cit., p. 113. He further remarks:Please allow us to excuse in this way the absence of the name of Mach frompublications in which we have sometimes put forth thoughts that had morethan mere similarity with his.

33. He is right about this–the appendix to Essay 7 discusses the formal problems ofharmonizing constraints with regular forces. Commentators frequentlymisunderstand this celebrated passage, presuming that Hertz attempts to purgemechanics of the notion of “force” due to its occult, or otherwise fuzzy, conceptualstatus. Quite the contrary, Hertz makes no claims whatsoever about force’sconceptual underpinnings, except to indicate that the current amalgamation ofinstructions for its employment point in different directions that induceinconsistencies. The passage comes from Heinrich Hertz, The Principles ofMechanics, Jones and Walley, trans. (New York: Dover, 1956), p. 9.

34. In turn, these individualized investigations allow us to understand why we oftenfall into conceptual confusions through no fault of our own, when we first stumbleacross novel computational routines that suit nature well but operate in morecomplex way than we anticipate. “Temperature”s complex underpinning supply avivid illustration of the phenomenon, I contend. This emphasis on the value ofindividualized corrective diagnosis, rather than sweeping (and rather uninformative)theories of “theory,” reappears as a constant motif throughout all of these essaysand aligns my attitudes with respect to philosophical mission with the“unsystematic” twentieth century diagnosticians praised in the preface.

35. Viz., the requirement of fully autonomous modeling.

36. Partisan Review 50th Anniversary Edition, William Phillips, ed. (London: Steinand Day, 1985).

37. Duhem, “Analysis,” op cit., p. 117. Unfortunately, his policies for employinghistorical developments as weapons against anti-thermal reductionists lead tomodern portraits of Duhem as an anti-innovative conservative. In particular, his

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commentaries assume the form: “If we retrospectively reconsider the historicalsteps of widening applicational success within physics, we see that they largelyinvolve statics to dynamics lifts. Hence reasonable canons of scientific inductionshould favor my so-constructed thermomechanics over the shoddily assembledproposals of my anti-thermal rivals.” If we ignore the particularist details of thisargumentation and convert it into general percepts, we obtain a methodologicalpicture much like that proposed by Rudolf Carnap in “Testability and Meaning,”(Philosophy of Science 3 (4), 1936)), in which a “good science” builds itself upinductively in increasing hierarchies of “experimental generalization.” This leads tothe “phenomenological laws” readings of Nancy Cartwright and others. To besure, Duhem often gravitates to strictures like these, especially when he is mostfiercely engaged in pro-thermomechanical battle. But then, recalling his“conceptual liberty of the physicist” sympathies, he usually backpedals, “Of course,those English physicists, with their despicably shoddy inductions, have sometimesreached very worthy conclusions.” The proper remedy, in my opinion, is to followthe recommendations of this essay and recast Duhem’s thermal observations as anaccurate diagnosis of the patch-to-patch extensions that have allowed the notion of“temperature” to extend to wide physical applicability. On this approach, Duhem isviewed as practicing a novel manner of “semantic” analysis, rather than preachingsome general methodological lesson.

38. One sometimes hears philosophers confidently pronounce, “Science hasestablished that notions such as temperature have no place amongst the fundamentalconstituents of the universe.” What this murky assertion actually means is unclearto me, but, insofar as it implies that we have reached collective agreement on howenergetic degradation should be properly conceptualized, it should be rejected asplainly false. Such misguided assertions strike me as paradigmatic of the irrelevantshifts of attention that Duhem regards as characteristic of unfounded forms ofkinetic appeal. He is excessively harsh in these assessments, but not by as much ascontemporary philosophy often presumes.

39. Peter O. K. Krehl, History of Shock Waves, Explosions and Impact (Berlin:Springer, 2008) details Duhem’s important contributions to shock wave research.

40. In the language of wave front characteristics, these curves intersect with oneanother at tc. Each such curve tells a different story as what the pressure will like atthe critical juncture. Mathematicians call such behaviors blowups.

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41. Victorian speculations on material substance can be found in B. Stewart andP.G. Tait, The Unseen Universe; or Physical Speculations on a Future State (Kyiv:Leopold Classic Library, ND) and J.C. Maxwell, “Atom” in The Scientific Papersof James Clerk Maxwell (New York: Dover, ND).

42. Evolution, op cit., pp. 161-2. In context, Duhem is concerned with the utilitiesof a regarding a friction purged version of thermomechanics as the fundamentalscience, rather than a mechanics purged of both friction and temperature as weconsider here. In both instances, Duhem’s response is the same. The two purgedforms of mechanics capture important connections between physical qualities thatmust to be cleanly isolated conceptually–a coherent understanding of internalpotential energy requires this--, but we should subsequently build up a richer scienceembracing temperature and friction on top of these preliminary platforms. Most ofthe present essay is devoted to explaining, roughly, how this stage by stage processis supposed to work. From this point of view, neither temperature nor friction isfictitious, but both operate in a manner that obscures important basic relationshipsbetween important physical qualities.

43. Insofar as I can see, Duhem is willing to talk of structures that only exist uponextremely small scale sizes. One of the earliest employments of Gibbs’ phase rulewas in the manufacture of steel, in which small portions of material are forced intophase changes within temperature or pressure regimes very different from roomtemperature. Afterwards, these extremely small structures are locked in placethrough rapid freezing or decompression. Duhem appears to be so impressed by thecogency of Cauchy-Green-Stokes methods for setting up continuum mechanics in atop-down way that he has also become convinced of the superiority of formalismsthat register every conceivable physical effect within the parameters of someextremely accommodating, yet single-leveled, descriptive format. Such expectationsbroaden the range of permissible constitutive equations to uselessly wideproportions, which is a major reason why recent authors have turned to multiscalarmodeling techniques instead. It should be acknowledged that single-level modes ofdescriptive offer many convenient utilities, but determining their required parametersfrom lower scale considerations usually represents a wiser policy than attempting toaccommodate everything required a priori, in classic Duhem/Truesdell fashion.

Some of the issues brutally telescoped in this footnote should become clearerbyessay’s end. The fact that Duhem consistently jumbles distinct considerationstogether in his attacks on speculative lower-scale modeling makes excavating thegold from the dross within his writing much harder. But that is the task I’m

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attempting here.

44. Stanley L. Jaki (Uneasy Genius: The Life and Work of Pierre Duhem(Dordrecht: Martinus Nijhoff, 1987), p. 273) cites a passage that clarifies Duhem’sthinking on this score:

In the kind of propagation which applies to the motion of sound in air, ... theexistence of waves and the existence of a velocity of propagation are merelyappearances and approximations.

Here Duhem employs “wave” in the Hugoniot sense of a propagating singular wavefront and he claims that, in a proper formulation, temperature effects inhibit theformation of crisply identified fronts of this type. He does not deny that studyingsingular fronts is not mathematically useful; indeed, his friend Hadamard contributedmuch to the modern theory. In 1816 Laplace modified Newton’s equations forsound with thermal correctives, so Duhem frequently cites Laplace’s approach as anargument in favor of an expanded thermomechanics. In doing so, the surprising lackof conceptual closure within pure mechanical description becomes obscured,although Duhem clearly recognizes that it occurs.

45. These “rigorist” presumptions will be criticized later in the essay on the basis oflater work in continuum mechanics.

46. These presumptions strike me as similar to the presumptions that LauraRuetsche labels as “pristine interpretation” in Interpreting Quantum Theories(Oxford: Oxford University Press, 2011), p. 3.:

Underlying this methodological ideal of pristine interpretation is adistinction between what holds at each world of which a theory T is true andwhat varies.... The class of what applies in all settings where T appliesincludes T’s laws, as well as metaphysical, methodological andmathematical truths; the class of what changes from setting to settingincludes initial/boundary conditions, as well as practical considerationsparochial to the settings which give rise to them.

She merely intends to capture prevailing philosophical opinion; Ruetsche herselfexpresses merited skepticism about these goals as her discussion unfolds.

Due to the infelicitous manner in which Duhem articulates his key insights,many of his fervent admirers have embraced themes that he would have stronglyabjured. I have particularly in mind the “semantic view of theories” advocated byBas van Fraassen, which strikes me as founded upon the very autonomous trajectorydemands that Duhem rejects.

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47. By quasi-statical, I mean that the environmental setting can be allowed to adjustover time, but in generally in a slower manner than the “fast time” activities withinthe interior core. Mathematicians usually articulate such dependencies in controlspace terms. In this regard, the “gravitational field” we invoke is not a proper fieldin the physical sense (i.e., an entity that can sustain waves and such), but an artifactdependent upon where our effacing cuts are placed. In these regards, I find thatphilosophers commonly misunderstand the status of Poisson’s equation. It does notcapture the contents of Newtonian gravitation fully; it merely outlines a static arenawithin which a dynamics for the interior planets still needs to be specified. Observethat the formula is of elliptic–hence, timeless--signature, whereas the fullergravitational dynamics from which it descends via base manifold projection arestandardly expressed as evolutionary ODEs. I should further remark that we haveexcluded Mercury and Venus from our representation altogether, due to their feeblegravitational influence.

48. A common philosophical mistake of this sort is to claim loosely that regularsolids can be autonomously modeled within statistical mechanics because webelieve that we can theoretically write down Hamiltonians for their behaviors. Butwe must look rather carefully at the residual potential energy terms before we makesuch claims. Will they include terms of the absolutist format V(r) or can they all berepresented as interactive terms V(q1, q2)? Only if the latter holds will the evolutionqualify as autonomous. Due to the various forms of higher scale frozen disorderfound in most solid materials, calculating a workable effective potential V(r) directlyfrom molecular considerations has proved an elusive goal, whereas the multiscalemethods of Essay 5 have produced better results. At present, handling typical solidsin a fully autonomous manner remains at the purely aspirational stage.

Essay 2 discusses a number of allied cases where we presume that supportiverationales for common forms of non-autonomous modeling assumption related tostrut equilibrium should be forthcoming but where we are presently at a loss fordetails. Substantial misunderstandings can arise when the merely aspirational isconfused with the actually accomplished.

49. Although autonomy expectations appear reasonable within this specific context,most physicists working within the nineteenth century would have nonethelessrejected the demand even here, insofar as I can determine. This is because theyviewed the cuts as encapsulating the averaged environmental factors that frame thedistortions and simplifications inherent within the descriptive tools they apply withinthe inner, active realms. On these grounds, they were not troubled by the various

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forms of descriptive breakdown that attend point mass gravitational modeling,maintaining that, upon close encounter, unmodeled physical factors emerge to arrestthe singularities. Likewise, Olbers-type paradoxes involving infinite universes aretamed by the consideration that, whatever exactly goes on in those nether realms,we know that their accumulated effects remain finite when we reach regions ofsuitable cuts. And so on. In this manner, many authors–there were certainlyprominent exceptions--didn’t regard physics as obliged to supply full cosmologies inDuhem’s sense.

50. This is a common manner of characterizing faulty expectations about sciencereached by over-generalizing upon the structural features of Newton-style celestialmechanics. See Essay 9 for more on this topic.

51. Duhem rejects the presumption that science falls under any obligation to supply“cosmologies” in scathing terms. In fact, a preponderance of physicists in his erawould have agreed, contending that it is impossible to cleanse physics of non-autonomous cuts of the sort indicated. Since Einstein, we have become accustomedto cosmologies, but, at present, these are not free of the prohibited forms of internalstitching. Instead, we witness a number of submodels tied together throughasymptotics calibrated to regimes of temperature and scale (whereas temperatureshould presumably have no place within a truly autonomous, unstitched cosmology). The more speculative aspects of this essay can be viewed as a suggestion that theasymptotic ensemble averagings that we employ to connect these regimes togethermay represent our best mathematical surrogates for the otherwise elusive notions ofenergy and information degradation. If so, presumptions that physics will produce asingle set of equations to govern the entire universe autonomously should beregarded as predictively premature.

52. Texts differ in how willingly they explicitly bite these astonishing bullets, butthey all follow from their claims that thermal notions pertain to equilibriumcircumstances only.

53. Incidently, other types of flow can engrave interesting patterns upon a basemanifold of a less repetitious character–the phenomenon is called hyperbolicscarring. If the complete state of a target system (viz., its various generalizedpositions and momenta) is represented by a single point within a space of highdimension, its timeless base manifold projection always displays an informativepattern. Indeed, this base projection is exactly what a phase space in themathematician’s sense connotes, while the dynamic arena from which these patterns

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are projected is called an extended phase space.

54. In our fluid’s circumstances, the streamlines of the base manifold support anatural velocity field v, which we can utilize to construct an appropriate temporalarrow connecting a fluid particle presently located at p to a future position p* over atime step Δt. Namely, p* is the position reached if the particle presently at p travelswith the velocity v(p) over the interval Δt. A simple example is supplied in the firstappendix to Essay 2. According to the standard mathematical distinctionsadvocated in that same essay, the evolving behaviors in the lifted manifold aredescribed by evolutionary equations, generally of hyperbolic signature, whereas thepatterns within the base manifold are captured by elliptic formulas containing nomention of time whatsoever. Generally, the latter are easier to deal withmathematically and shuttling between the two representations represents a usefulform of computational tactic. Lifted manifold constructions are also central to theunderpinnings of modern optics in very interesting ways, a topic to be surveyed in afuture essay entitled “How ‘Wave Front’ Found its Truth-Values.”

55. The linked seesaws of Essay 6 provide an exaple with a larger number ofdegrees of freedom. Note as well that the “forces” involved must be generalized toinclude torques and so forth. In Duhem’s usage the term also embraces drivingfactors of a non-mechanical nature, such as an entropic forces or a chemical affinity.

56. Duhem calls these intensive driving force/extensive reaction pairings. His ownmetaphysical hopes for developing a portrait of nature’s workings rest upon thehope that an autonomous physics of quality and quantity can be constructed aroundthese pairings. We’ll discuss these extra-scientific ambitions later.

57. The virtual work principle captures this balancing by the stability requirementssupplied in Essay 7--any external attempt to wiggle the device away from aconstrained equilibrium will be exactly countered by an internally generatedreversive response.

58. The great reliability that Lagrangian methods offer is stressed in other essays(e.g., 6 and 7) and constitutes an important theme within Duhem’s writings as well. Indeed, my own appreciation of the centrality of such concerns within any “science”worthy of the name largely came from reading Duhem. In this essay, I insteadhighlight the fact that his standards offer a means of capturing coherent energystorage in a manner that is calibrated to scale size. The importance of these remarks

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should become clear later.

59. According to Duhem’s great Origins of Statics (Dordrecht: Kluwer, 1991), thisdiscrepancy was first noticed by Descartes in a letter to Huygens: ThePhilosophical Writings of Descartes, Vol. 3 (Cambridge: Cambridge UniversityPress, 1991).

60. To fully capture this stability criterion, we must look at the second variations ofthe displacements, but we’ll ignore this complication here. I include these detailsbecause Duhem views the employment of virtual terms as emblematic of the non-descriptive devices that a physicist characteristically employs in her theorizing. ThusEvolution, pp. 114-5:

The use of these virtual modifications is an artifice of reasoning, acalculational process; it is therefore useless for a virtual modification tohave a physical meaning.In an allied vein, Duhem frequently praises the substitution of “virtual work”

for “actual work” as a glorious example of the “free creativity” of the physicist infull blossom. To the uninitiated eye, this transduction looks like minor tweaking,rather than exemplifying “free creativity” in some grandly Romantic manner. Within thermal contexts, Duhem is correct in the sense that these little conversionsconnote a vastly original conception of how internal potential energies comport withentropies, considerations utterly lost on readers who don’t appreciate the deepsubtleties of these Clausian conceptions. But we’ve already noted that Duhem isfrequently injudicious in his choice of illustrative examples.

61. To do so requires further information about how the system apportions its storedenergy responses into different kinetic channels. If we press on a material from acertain direction, will it remember the directional coherence of our impulse longenough that it will push back in that direction when released in the manner of anelastic ball? Or will it respond in a less directed way, as when we compress a blobof rubbery fluid? These matters have to be settled within the so-called constitutiveequations for the material at hand.

62. When a manifold is investigated in this manner, it is usually called a controlspace and the altered parameters are called control variables. They may have somekind of clock associated with their changes, but it is not the clock that drives thedevice’s autonomous movements. The celebrated “reversible trajectories” ofclassical thermodynamics are really control variable curves of this nature.

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63. He calls these “false equilibriums”; more often they are called metastable states.

64. Duhem provides a good exposition in Thermodynamics and Chemistry (NewYork: John Wily and Sons, 1903), p. 398-402.

65. The Kinematics of Mechanism, Alexander Kennedy, trans. (New York: Dover,1963), p. 85. Reuleaux’ own intellectual context is outlined in Essay 7; it tiestogether mechanical issues, Wittgenstein and Goethe in an intriguing way.

66. The Kinematics of Mechanism, op cit. p. 21.

67. On pp. 29-30 of Evolution, op. cit., Duhem correctly observes that commonpractice of exploiting constraints within Lagrangian applications introduces“fictitious” (his term) constraint forces into the picture that, arguably, compromisethe full autonomy of the system studied. He appears to be thinking about theseconcerns very much in the manner of the Appendix to Essay 6.

68. Ingo Müller and Wolf Weiss, Entropy and Energy (Berlin: Springer, 2005), p. 3.

69. Herbert B. Callen, Thermodynamics and an Introduction to Thermostatistics(New York: Wiley, 1985), p. 26.

70. Hans U. Fuchs, The Dynamics of Heat (Berlin: Springer, 1996) pp.

71. It should be remarked that the non-thermal trajectories of the “old mechanics” allqualify as reversible paths, but, on Duhem’s way of thinking, these paths are addedin later, appearing as the constant temperature asymptotes to curves that carry aGibbs principle installed dynamic arrows (all of this terminological gobbelty-gookwill be explicated later). The “constant temperature” proviso means that only thed’Alembertian part of the Gibbs rule remains active.

72. http://en.wikipedia.org/wiki/Non-equilibrium_thermodynamics

73. Clifford Truesdell captures the situation aptly: “In thermodynamics today,Clausius’ greatest discovery is incomprehensible because the words needed toexpress it have become dirty words.” (Rational Thermodynamics (New York:Springer, 1984), p. 9).

74. There is a large philosophical literature that attempts to rationalize the SecondLaw solely in terms of its anemic equilibrium thermodynamic contours, whiledisdaining non-equilibrium practice on the grounds of its confusing “lack of rigor.”

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Such efforts strike me as failures of diagnostic insight–here is a situation wherephilosophers can assist the physicists by schooling them in the uncanny ways ofwords.

75. Mathematically, the distortions are similar to what happens when tries to flattena Riemann surface into a single sheet–see WS, pp. 313-9.

76. With respect to his early training in thermodynamics and statistical mechanics,Marc Kac comments : “[Such talk] rendered the subject unpalatable and evenrepulsive to a young mind already conditioned to look for clarity and rigor.” (InStephen Körner, The Pleasures of Counting (Cambridge: Cambridge UniversityPress, 1996), p. 176).

77. We also need the constitutive principle that the metal completely remembers thedirection from which it was originally compressed. When the block converts itspotential energy back into kinetic energy, it directs its kinetic response along thesame direction in which it had been pushed. Accordingly, the big and little men inthe diagram each push their own wave in opposite directions along the bar. Indumber forms of material, the little man robs the big man of some of his originalimpetus and a certain amount of directional coherence becomes lost in the exchange.

78. The degree to which the block to the right of our wave will compress depends,other conditions being equal, on the rod’s initial conditions, which I symbolized by asmaller little man on the right. For reasons of narrative simplicity, I’ve suppressedthe fact that d’Alembert’s principle is a demand on accelerations, not velocities, andthat we should consider our rod over two time steps to augur how it will behave. Likewise, to compute how preexistent compressions within multiple blocks willcompete with each another, we must return to the “one for all; all for all” aspects ofvirtual work computation, as discussed in Essay 7. I feel that addressing thesecomplicating concerns fully might only confuse readers unfamiliar with suchconsiderations, without supplying significant additional insight. In truth, virtualwork principles don’t strike me as the optimal method for articulating the basicphysics of continua. But it was Duhem’s way and so we follow it here. But thesetechniques invoke the sundry puzzles of “physical infinitesimals” surveyed in theappendix to Essay 3. As this essay’s own appendix indicates, Duhem’s deepestworries about the underdetermination of theory trace to this source.

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79. Many presentations employ an empirical thermometer temperature as an initialdescriptive variable and replace it later by absolute temperature after the latter hasbeen established as an integrating factor for the entropy through standard Clausius-style considerations. This complication is not important for our purposes but it isneeded to appreciate Duhem’s lengthy discussion of the gap between “temperature”and thermometer reading in The Aim and Structure of Physical Theory (Princeton:Princeton University Press, 1962). These passages are often supplied with anexcessive Quinean gloss within modern commentaries.

80. This determination reflects a global variational principle, allied to the manner inwhich Laplace’s principle of least work in purely mechnaical circumstances.

81. The Evolution of Mechanics, op cit, p.136

82. In contrast, its contents become oddly rarified (e.g., Carathéodory’sformulation) when we attempt to push its strictures back to their ramifications withrespect to the underlying Clausius base manifold.

83. At many points in this book, close contacts with Jim Woodward’s studies ofmanipulated causal processes naturally emerge. Present ruminations support mybelief that Woodward’s diagnostic tools offer significant insight into the historicalturning points that frame the evolving architectures of effective science. I havecertainly relied upon Woodward in my own thinking in that manner.

84. Namely, internal energy, Gibbs energy, enthalpy and Helmholtz free energy.

85. Formally, these problems are similar to those that we briskly considered inconnection with d’Alembert’s principle. If we free up too many channels of kineticresponse at a time, d’Alembert’s principle alone can’t tell us how the system willallocate its stored energy to accelerate the opened degrees of freedom. Suchmatters have to be settled by the constitutive properties we assign to the mediumitself. In a purely mechanical context, these allocation policies can usually beresolved fairly readily, but in thermal circumstances they seem more elusive. As wenoted, thermodynamics’ greatest predictive successes often require that we onlyrelease a limited number of constraints, so that the channels of thermodynamicresponse remain clearly focused.

86. Another significant worry traces to the fact that in this methodology we brutallycompress our little blocks of material to infinitesimal length and tacitly assume thatthey reach their local equilibriums extremely quickly. Operationally, we are

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presuming that the relaxation times for these local processes are much swifter thanthe clock that monitors that the transport of our non-equilibrium blast of thermallyaffected wave along the bar. Sometimes those assumption are untenable but wehave covered them up via shrinking assumptions akin to those discussed in Essay 4. In this manner, Clausius’ original worries about whether thermal ideas make goodsense for systems in internal agitation still create significant problems for extending“temperature”’s applicational reach as widely as we might expect. These are alsothe basic reasons why non-equilibrium mechanics remains such a controversialsubject even today. Nonetheless, it’s not a subject that can be ignored simplybecause it’s hard to articulate crisply. I earlier complained of the philosophers whoonly study equilibrium thermodynamics on the grounds that it alone has beenclarified enough to qualify as a “rigorous theory.” In such tropisms they havebecome so bedazzled by the attractions of Theory T thinking that they have lostcontact with the mysteries of their target subject as it proves useful within real lifeapplication. In my harshest moments, I feel that these ersatz rigorists don’tunderstand what “philosophy” is properly about at all.

87. Duhem uses the term “false equilibrium” for these behaviors. He was keenlyinterested in–and frustrated by–hysteresis phenomena. Essay 5 sketches howmultiscalar approaches approach these issues.

88. He calls intensive/extensive pairings and their hypothetical fulfillment willprovide a “science of quantity and quality.”

89. Modern usage begins around 1600. See W. E. Knowles Middleton, A Historyof the Thermometer and Its Uses in Meteorology (Baltimore : Johns HopkinsUniversity Press, 2002).

90. To the relief of jewelers everywhere.

91. He sketches his proposals in such an abstract manner that even experts like G.A.Maugin find him hard to follow. I am certainly not prepared to evaluate what heconcretely achieved at the advanced end of his New Mechanics enlargements.

92. Commentary, pp. 93-4. If we replace “constrained equilibrium” and “controlledtrajectory” with Duhem’s favored word “experiment,” we reach the conclusion thatthe proper end goal of science proper should center upon the task of cataloging“experimental results” efficiently, rather than feeling obliged to capture all of thelicentious trajectories of unfettered nature effectively. But employing “experiment”

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in this deviant manner has encouraged a prevalent reading of Duhem in which he issaid to favor “phenomenological laws” in the sense of the logical empiricists andwhich renders his many remarks on the “liberty” of the physicist nearlycontradictory. Readers of this persuasion have not looked deeply what Duhemincludes under the heading of “virtual modifications” (which is understandable;Duhem expresses himself in an extraordinarily opaque manner). If we look into thedetails of the γ-variable “forces” he is willing to tolerate under the heading of a“virtual modification,” we soon realize that most of them lie at great remove fromany conceivable experiment. To be sure, they “control” the substance in a veryabstract way–e.g., by preventing a diamond from collapsing into graphite–, but these“controls” remain as far from experimental check as the widest postulations of aTheory T free thinker.

What we do find in Duhem that we also encounter in the Clifford Truesdellschool discussed later. nace in these virtual response aspects within Duhem’sthinking. Restored to proper argumentative position, Duhem’s admonitions withrespect to science’s fealty to “experiment” turn out to be little more than a tribute tothe reliability inherited in schemes that attempt to build up a richer infinitesimalthermal physics starting with the amply verified data enshrined with our initialClausius base manifold. With respect to the latter, he repeatedly observes that theClausius manifold data upon which his thermomechanics (and Lagrangian techniquemore generally) builds consist in virtual work pairings of the form AδB (the motivesbehind this notation will be explained more fully in our optional section (vi)). Inthese alignments, the “virtual” side of the pairing (δB) represents a hypothesizedresponse devised by the creative scientist as a theory of how the drivingmanipulations A affect the target system; the δB responses do not directly reflectreal life experiment, but represent a modeler’s creative riff on what thoseexperiments might betoken (essentially, I might add, in terms of how the hiddeninternal energy storage capacities of the target system are affected).

93. That is, in conjunction with normal temperatures and pressures, the γ-“force”manages to hold the gem in constrained equilibrium as a diamond.

This may be a good point to discuss a common misunderstanding of Duhem’sintent. If we replace “constrained equilibrium” and “controlled trajectory” withDuhem’s favored word “experiment,” we reach the conclusion that the proper endgoal of science proper should center upon the task of cataloging “experimentalresults” efficiently, rather than feeling obliged to capture all of the licentioustrajectories of unfettered nature effectively. But employing “experiment” in thisdeviant manner has encouraged a prevalent reading of Duhem in which he is said to

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favor “phenomenological laws” in the sense of the logical empiricists and whichrenders his many remarks on the “liberty” of the physicist nearly contradictory. Readers of this persuasion have not looked deeply what Duhem includes under theheading of “virtual modifications” (which is understandable; Duhem expresseshimself in an extraordinarily opaque manner). But if we look into the details of theγ-variable “forces” he is willing to tolerate under the heading of a “virtualmodification,” we soon realize that most of them lie at great remove from anyconceivable experiment. To be sure, they “control” the substance in a very abstractway–e.g., by preventing a diamond from collapsing into graphite–, but these“controls” remain as far from experimental check as the wildest postulations of thefreest Theory T thinker.

What we do find in Duhem is a methodological restriction that we alsoencounter in the Clifford Truesdell school discussed later, viz. any modelingconsideration employed within continuum mechanics must display itself in somekind of macroscale parameter such as our diamond-preserving γ. I call this aninsistence upon single-level, top-down modeling. Historically, this preference tracesto the clear superiority of top-down methodologies within classical elasticity overmolecular approaches, as mentioned elsewhere. But it leads to a program whereone attempts to codify continuum mechanics in a “rigorous” manner that anticipatesevery conceivable γ-parameter that a modeler might ever wish to use. In the nameof this same “rigor,” continuum mechanics was forced to consider tolerances ofpreposterous allowance. Eventually, practical modelers grew sick of thesedemands, thinking, “Gee, if I want a parameter that can reflect lower scaledislocation movements is an effective higher scale way, why don’t I attempt to findsome averaging that supplies plausible results and tinker with continuum mechanicsprinciple to make it fit?” Such lines of thought are closely allied with the mixedlevel approaches characteristic of multiscale modeling.

Apropos of virtually nothing beyond the fact that it constitutes a pricelessrelic of our cultural heritage, I will remark that, in the Superman serials of myyouth, its eponymous protagonist was able to crush a piece of coal into a diamondthrough tremendous hand pressure (he needed to surreptitiously replace a gemstoneeye stolen from an idol cherished by an indigenous population).

94. Gérard A. Maugin, Continuum Mechanics Through the Eighteen and NineteenthCenturies (Cham: Springr, 2014), pp. 103-4.

95. Ellad B. Tadmor and Ronald E. Miller, Modeling Materials (Cambridge:Cambridge University Press, 2012), p. 553-4

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96. RVE= Representative Volume Element–see Essay 5 for a fuller explanation.

97. Modeling Materials, op cit., p. 554.

98. Essay 7 supplies several circumstances in which the word “force” attache stosurprising referents upon lower scale examination. The restorative force we detectwithin a springy metal bracelet traces (mainly) to the strong forces that hold itsmolecules together but the comparable “force” of the rubber band stems almostentirely from polymer chain entropy maximization.

99. These blockages explain why the old “mean kinetic energy per molecular degreeof freedom” jingle doesn’t supply extensionally correct underpinnings for ournormal “temperature” attributions at the macroscale. The question of how ablycoherence work capacity is retained as structures at higher scale lengths affect theirlower scale companions strikes me as a characteristic illustration of the centrality ofthe factors that Jim Woodward has highlighted within his studies of controlledmanipulation along cause and effect chains. Similar concerns arise frequentlythroughout these essays.

100. Certainly unqualified claims that “scientists seek a wave equation that cangovern the entire universe” should be viewed with suspicion.

101. I have written largely of increased energetic incoherence here, rather thaninformational loss, but the two notions seem intimately entangled. In both cases, weface the dilemma that, although we believe that these degradations reflect objectivephysical process, as soon as we assemble detailed mathematical models for therelevant events, the degradation itself seems to disappear from our efforts. The chiefmanner we have discovered for persuading degradation to make a mathematicallyrecognizable appearance is to wrap our modeling efforts within the fog of statisticalconsideration. From this point of view, the invocation of statistics should not beviewed as a descriptive incapacity on our parts, but as providing a needed correctiveto dominant behavior models that were excessively detailed in the first place. Suchattitudes largely reflect a chastened faith in applied mathematics’ capacities formodeling physical processes accurately, without injections of compensatingcorrectives of an unequal data registration character (viz., boundary conditions, cuts,upper scale constraints, etc.). I find echoes of allied speculations in many authors,but rarely in a fully articulated manner.

102. Evolution, op cit., p. 50

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103. How far we will get must remain an open question at present. In the words ofthe redoubtable Oliver Heaviside, we must “first find out what there is to find out” (Electromagnetic Theory Vol. 2 (New York: Chelsea, 1971), p.3). Heaviside’sinnovations with respect to differential equations supply WS’s central illustration ofthe seasonality of rigor within scientific development.

104. Specifically, many authorities were much influenced by the fact that Cauchy’stop-down approach to continuum physics supplies more accurate estimates of theparameters required to capture elastic behavior than Charles Navier’s bottom-uppolicies. In the literature of the time, these conflicts show up as debates between“rari- and multi-constant” approaches to the subject. Good sources for backgroundare A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity (New York:Dover, 1944) and Robert Batterman, “The Tyranny of Scales” in R. Batterman, ed.,The Oxford Handbook in Philosophy of Physics (Oxford: Oxford University Press,2012).

105. Evolution, op cit., p. 112.

106. C. Truesdell and R. Toupin, “The Classical Field Theories” in S. Flügge,Handbuch der Physik (Berlin: Springer-Verlag, 1960), p. 229.

107. G.A. Maugin, Thermodynamics of Nonlinear Irreversible Behaviors (Singapore:World Scientific, 1998), p. 12. See also R.S. Rivlin, “Red Herrings and SundryUnidentified Fish in Nonlinear Continuum Mechanics” in Collected Papers Vol.2(New York: Springer, 1997).

108. Note that this retort to Duhem does not straightforwardly endorse the popularTheory T contention that “modern science has shown that ‘temperature’ and‘entropy’ represent eliminable features of the universe.” That I regard as a claimwith respect to the syntax of future science, but mathematicized science may verywell continue to require direct appeal to thermal vocabulary in the manner sketchedhere. The situation strikes me as reminiscent of the manner in which physicists ofthe nineteenth century approached the issue of eliminability of the notion of“gravitational potential.”

109. The Electric Theories of J. Clerk Maxwell, Alan Aversa, trans. (Dordrecht:Springer, 2015), p. 8. The Poincaré quotation comes from Électricité et Optique I.

110. Walter T. Grandy, Entropy and the Time Evolution of Macroscopic Systems(Oxford: Oxford University Press, 2012) supplies a great amount of worthy advice

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on the applicational range of thermal ideas, in a manner much influenced by thewritings of E.T. Jaynes. Characteristically, this school insists that entropy and alliednotions reflect conditions of human knowledge, which, in turn, leads manyphilosophers to reject their suggestions out of hand as excessively “subjective.” Ioffer the tentative suggestion that many of these insights might be repackaged interms of the difficulties of capturing information degradation in mathematical terms,rather than reflecting issues of human knowledge in any significant way.

111. Evolution, op. cit., p. 78.

112. For my own thinking on these topics, see “The Observational Uniqueness ofSome Theories” Journal of Philosophy 77 (1980) and “The Double Standard inOntology” Phil. Studies 39 (1981).

113. Articulated another way, strings represent infinite degrees of freedom systems,whereas standard physical principles only make sense directly for finite-degrees-of-freedom systems, either containing no flexible parts whatsoever or a limited numberof simple continua such as springs or dashpots.

114. Here’s one that bothered me when I was taught elementary physics in thismanner. “The gaps between these points are much larger in measure than the pointsthemselves. How does it happen that, in this limit, the gaps entirely disappear whilethe points themselves fus einto a smooth continuum.” The reason quantummechanics oriented textbooks favor these models, is because standard “firstquantization” techniques form their Schrodinger equation PDE differential operatorson a starting basis of classical point-based ODE operators. Such utilities do notshown that such an approach is a good way to set up classical mechanics,considered in terms of its won descriptive virtues.

115. Procedures of this sort rest upon the so-called “principle of rigidification”(despite the fact that springs and dashpots commonly included within its ambit aswell). References are supplied in Essay 8. Duhem himself follows this statics-centered approach, as do Thomson and Tait in their Treastise on NaturalPhilosophy. The following passage from The Evolution of Mechanics (op cit., p 30)makes Duhem’s presumptions clear:

To bring to light the foundations of Lagrange’s Statics, we have considereda system whose state is fixed entirely by a larger or smaller, but limited,number of independent variables. All systems cannot be so defined; this isso of continuous media, which have to be decomposed into an unlimited

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number of infinitesimal elements contiguous with one another, each of theseelements depending upon a limited number of variables. Amongst thesecontinuous systems some, such as filaments or elastic rods, extend in onedimension; others, such as membranes or sheets, spread over twodimensions; yet others, like fluids or elastic solids, have finite extension inall dimensions. The principles whose main features we have just describedwill be applied to such systems with no need to modify them greatly.

In Duhem’s case, this tactic is critical because he wishes to apply the finitarythermodynamic reasoning patterns developed by Clausius and Gibbs to his rigidifiedelements as well. Hertz is a stranger case, although he doesn’t discuss thesecontinuum mechanics limits much at all. Essentially, he wants his elements toconsist of little mechanisms held together entirely by constraining ties, with noactive forces such as springs or action-at-a-distance ingredients entering at all. Why? Because he wants to eliminate the conception tensions that arise when activeforces and constraint forces are mixed together–see the appendix in Essay 7. But hethen adopts the further expedient of replacing the conventional degrees of freedomof rigid body mechanics with peculiar unit mass points, which again renders theconsequent limiting processes quite opaque.

116. Principles of Mechanics (New York: Dover, 1956).