May 2015 Essay 3

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

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Throughout these essays we shall struggle with our computational place inthe world: in which natural circumstances and with what descriptive tools can weaddress practical problems through effective reasoning? Within the identifiably“scientific” parts of this project, many contemporary philosophers have settledfirmly, through a variety of developmental accidents, upon an optimistic appraisalof endeavor that I call the Theory T syndrome, in which central scientific endeavorcan be characterized in terms of a network of postulates that articulate the key“kind terms” (or concepts) of T along with the nest of “fundamental laws” in termsof which explanations based upon T should be constructed. The net expectation isthat a properly articulated T will delineate, through its own internal semanticengines, a range of models to one of which reality must correspond if T is tojudged wholly successful (whether this happens or not lies entirely at the hazardsof empirical fortune: man proposes, but nature disposes). As such, this portrait ispitched at a lofty level far above the grubby levels of practical informationgathering and numerical estimations, and its pleasing endorsement of descriptiveoptimism lies therein. For basic mathematical considerations with respect tocomputational complexity warn us that simple differential equation arrangementsreadily spawn solutions whose numerical values cannot be estimated withoutdevotions of inferential effort far beyond machine-assisted human capacity. Butthe theory T picture assures that we can nonetheless capture (in the sense of“semantically pick out”) the target physical circumstances ably enough; it is merely

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that our computational aids fail us in drawing out the inferential consequences ofwhat we’ve successfully captured within our theory T snares.

Developmental events in the twentieth century further suggested that T’ssyntactic structure might be compressed within the tight cocoons of first ordertheory (in logic’s sense of “theory”). In itself, this expectation would represent acomparatively innocuous requirement as long as the previous paragraph’s basicpercepts with respect to differential equation model descriptivism prove fullysatisfied. Unfortunately, this otherwise innocent logistical expectation haswrecked devastating side consequences within the realms of philosophy, for itallowed its practitioners to sort “what Science does” into broad and convenientpigeonholes without needing to know much of anything beyond a smattering ofelementary logic (this is where an mild misconception turns into a serious“syndrome”). These logic-centered categorizations have created a situation wherephilosophers now converse confidently and glibly of “laws” and “kinds” withoutentertaining any capacity to equate these convenient categories with the nitty-grittystuff of which science is actually compromised. Sometimes, I fear, that thesediagnostic tropisms enjoy their firm grip on contemporary thinking largely throughsome form of transcendental deduction that rests upon philosophy’s perceivedrequirements as an a priori academic enterprise. We would not possess our ownwell-defined subject matter unless science went about its affairs in a properlyregimented theory T way; ergo it must. Such thinking strikes me as characteristic,for example, of the analytic metaphysicians profiled in essay 6.

Well, I descend from those alternative ancestral mists in which “philosophy”does not conceive its mission in this tidy manner, but anticipates that it mustcontinually scrutinize, somewhat haphazardly, the portraits we frame of humancapacity based upon what we currently know of the world about us (so we arelikely to frame a happier philosophical assessment of our computational positionwithin nature if we think that a benevolent creator is involved in its arrangementsthan otherwise). From this vantage point, philosophers as a social entity ought tobe the last group of thinkers who should settle resolutely upon a theory Tconception of our intellectual enterprise, not the first. Perhaps, on the final day ofepistemological reckoning, all of theory T’s logical precepts will become fullyratified in developed experience, but right now the successful inferential policies ofreal life practice do not match theory T expectation very crisply. As philosophers,we should keep these objective disparities firmly in view and actively cultivatealternative possibilities of feasible intellectual arrangement.

In particular, we can learn much from earlier thinkers who have looked at

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science in less optimistic ways and ask ourselves whether some portion of theirconcerns still apply within our own circumstances. Descartes, for example,believed that the parts of nature that can be captured within clear mathematicalterms fall considerably shy of the full collection of processes encountered withinthe real world. To be sure, on favorable occasion thenatural world present us with descriptional opportunitiesto which the a priori tools of mathematical reasoning canbe ably fitted; but these circumstances are not the norm atall. For example, with mathematics’ help, we can readilyably compute the outpouring of water from a dividedpipe, based upon its single inlet flow upstream, whereasan allied calculation is generally impossible if twoindependent currents join together later. Why? Becausehe assumed that mathematics could capably describephenomena only if all of its pieces can be linked together through a common rigidrule such as a formula, whereas our two incoming flows can rendered completelyuncorrelated to one another simply by turning their respective taps independently.If so, single inlet flow offers mathematical physics a nifty descriptive opportunitythat it will never obtain within our otherwise similar two-pipe circumstances.2

We now recognize that Descartes’ appraisal of mathematics’ capabilitieswas handicapped by his ignorance of technical tools (e.g., differential equations)that were developed later. Nonetheless, his underlying concerns were well-founded in the context of his times and should be viewed as an interestingpremonition of the subtler “tapeworm” properties of analytic functions thatJacques Hadamard highlighted in the early twentieth century.3 Such inferentialqualms don’t render Descartes an “anti-realist” or even a party who seriouslyentertains the assumption that “nature works in mysterious ways” (on the contrary,Descartes’ God kindly plants the true principles of material behavior within ourheads as a priori certitudes). He has merely opined that our two inlet situationinvolves an infinity of uncorrelated factors beyond the compass of mathematics’rigid set of computational tools.

In an allied spirit (that we shall discuss more fully in the sequel), manyphysicists of the nineteenth century did not believe that mathematical physicscould offer a full “cosmology” in the modern sense: a description of the entireuniverse in full and even-handed terms. Instead, scientists must look for opportunelocations at which a “cut” can be placed that effectively effaces a detaileddescription of a “system interior” from its more crudely approximated

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environment. Typically in doing so, we turn off the mutualforce interactions demanded by Newton’s Third Law andact as if the bodies outside the “cut” (in our illustration, thesun and major planets) are not affected by the minutemovements of the “system bodies” (in our illustration, theasteroids) and concentrate solely upon how the former pushthe latter around.4 Of course, such “cuts” need to bejudiciously chosen; otherwise our problem will not simplifyin computationally helpful manner at all. To reach moreaccurate conclusions, we must push our effacing “cuts” further out in the universe,but classical physicists generally presumed that their helpful interventions couldnever be evaded completely (such presumptions lay behind the Kantian conceptionthat science only offers “regulative ideals” that proscribe behaviors within theframework of a revisable set of provisions such as our “cuts”5). Once again,science’s need to seek special “descriptive opportunities” in nature is highlighted,although in a far more benign manner than motivated Descartes.

On many occasions throughout this book I emphasize the fact that the“boundary conditions” invoked in science represent spatial locales that offerspecial opportunities for unequal data registration of a computationallyadvantageous nature (boundaries are dramatically under-described relative to theirinteriors). This simple–but often pertinent–observation has become obscuredwithin the typical theory T mangling of the proper significance of the term“boundary condition.”

In essay 9, I sketch the developmental history of the advances in algebra anddifferential equations that offered later applied mathematicians a more upbeatresolution to Descartes’ specific set of descriptive worries. Many of the oldconcerns with respect to “cuts” and “boundaries” can be addressed through similaradjustments and advances (although not, vide essays 4 and 7, always in the mannerthat the analytic metaphysicians anticipate). Nonetheless, the general character ofthe “opportunist” doubts expressed in olden days should not be forgotten as wephilosophers ponder the physical reach of mathematical description now. Asympathetic appreciation of the structural concerns of yesterday offers a helpfulprophylactic to shield us against the premature structural dogmatisms into whichwe would otherwise tumble, especially if we have also elected to not study thewinding byways of present-day reasoning with devoted attention.

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Duhem

In this vein, the celebrated physicist/philosopher Pierre Duhem’s extensivewritings on mechanics offer further penetrating reasons for exercisingcircumspection in our long range philosophical speculations, by marking, in quiteconcrete terms, potential futures for “fundamental theory” in science that do notoblige the formal whimsies of contemporary philosophers. Once again, heemphasizes the need to build upon opportunity in the general manner markedabove, but employing a notably different emphasis that focuses upon theimportance of finding crucial patterns within the messy vicissitudes of frictionaldisruption. Much of my thinking elsewhere in this book has been influenced byhis discussion and modern work in the thermodynamics of materials has largelyfollowed the developmental contours that he recommended in the final days ofclassical mechanics’ heyday.

Unfortunately, Duhem is a notably lousy writer. At virtually every juncturehis instincts betray him with respect to straightforward exposition; if anopportunity to supply a florid and an ill-conceived metaphor presents itself, he willseize it (note that this critique stems from an author–myself–who might easily befaulted for the same proclivities). My favorite example of the ill-selected metaphor(suggested to me by Sheldon Smith) can be found in The Aim and Structure ofPhysical Theory. Duhem’s underlying purpose is to introduce hisreaders to an important formal construction that, in modernmathematical terminology, is now called “a one-form driving forcederived from a background potential field.” But some unfortunateliterary muse advises Duhem that a hazy comparison with thedifferent “intensities of genius” encountered across the parade ofmankind will accomplish his expositional project. When we turn tohis more technical writings (which provide the prime sources for thereading I advance here), he articulates his key conceptions in such an abstractedformat and with so many demands upon his readers’ technical backgrounds, thathis account becomes virtually impossible to follow unless one is familiar, fromother sources, with all of the materials he is attempting to “introduce.” A fairamount of what I will present here is merely an attempt to translate theargumentative arc of The Evolution of Mechanics into a recognizable tongue.

The “anti-realist” and “phenomenonalist” Duhem cherished by moderncommentators such as Bas van Fraasen and Nancy Cartwright will scarcely makeany appearance here, except when I’m concerned to remove their characteristicinterpretative glosses from some central Duhemian passage or other (this is not toclaim that Duhem may not have intended some of those philosophical

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accouterments; it’s merely that I don’t see his special inspirational merits asaligned along those axes at all). To profit from the specific “opportunist” insights Iendorse, we must attend with a certain devotion to the chief scientific problematicin which he was engaged, viz., building up a workable “thermomechanics” inwhich the notions of absolute temperature, heat and entropy enter on an equalfooting with the familiar qualities of standard classical physics, e.g.. force, mass,potential energy, stress and strain. Duhem dubbed the latter “the old mechanics”and favored Lagrange’s approach in the Analytical Mechanics as the bestrepresentative of this circumscribed approach. Many authors writing instead selectthe formalism of Hamiltonian mechanics as their central representative of what theterm “classical physics” should embrace, although Duhem himself would haverightfully argued that such a choice is sub-optimal.6 In any event, he proposed thatall of these forms of the “old mechanics” should be amplified with the inclusion ofbasic thermal notions as additional, coequal primitives, while eschewing anyattempt to “reduce” these descriptive supplements to statistical underpinningswithin the “old mechanics.” Indeed, the underlying agenda within most ofDuhem’s extensive disquisitions on philosophy and the history of science is tocombat the hidden philosophical prejudices that he thinks made “reductionism”appear so attractive within his time, despite severe empirical indications to thecontrary. At the root of this critique lie the trenchant observations with respect to“descriptive opportunity” that I will highlight.

The need to bring thermal and “old mechanical” modes of description intounified coordination is quite palpable, for the relevant phenomena commonlycouple to one another in ordinary life. Consider an iron bar. If we strike one endwith a mallet, we will send a pulse of compressive stress through its interior, aprocess that is governed, to first approximation, by the familiar wave equation. Likewise, if we heat an extremity, we will send a parcel of heat across the bar, inrough accordance with Fourier’s celebrated heat equation. But, surely, these twoeffects will couple to each other, greatly complicating the detailed flow, becausethe compressive effort supplied to the bar willgradually elevate the temperature of the barbeyond our simple Fourier’s law expectations. Likewise, locally heightened temperatures willdilate the bar’s length, spoiling the simplepatterns of the standard wave equations. Inmany industrial settings, coupling effects aresufficiently strong that one needs to find a framework in which they can be jointly

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treated. We might parenthetically observe that, in a modern context, the coupling of

temperature to regular mechanics must be managed in a manner so that theconduction of heat proceeds at a slower pace than Fourier’s treatment describes(because of relativistic limitations on signal speed). Enforcing this modernrequirement is not easy and Duhem wrote at a time where speedier transportswould not have seemed worrisome. Because of the problem’s technicaldifficulties, such requirements are often still ignored within the most up-to-datetheories of thermodynamical behavior.

Although, for simplicity of narrative, I shall concentrate largely on theproject of uniting the “old mechanics” with thermal notions, cross-term effectsbetween electricity and chemical composition are stronger and easier to measureand Duhem often cites these kinds of phenomena in articulating his “newmechanics” project.

Duhem’s “molecular” opponents largely ignored these efforts because theyhoped that the complexities of performing a genuine thermal/mechanicalintegration might be evaded by escaping to a lower scale “molecular” realm whosesalient dynamics could be described entirely with the limited descriptive resourcesof the “old mechanics.” However, a serious problem of conceptual closure arisesas soon as one attempts this reductive project (which I’ll outline in a moment),beyond the fact that such approaches are hampered bu a host of empirical obstaclessuch as erroneous specific heats and the like. So what accounts, Duhem asks, forthe almost visceral insistence with many scientists of his era doggedly pursuedthese reductive will o’ the wisps, rather turning their attention more productively todeveloping an adequate thermomechanics (= a formalism in which thermal and“old mechanical” categories enter in a coequal and cooperating fashion)? We’llfind that, as we look into the details, Duhem was right to assume that poorlythought out philosophical prejudices were playing a substantial role in thesetheoretical preferences. Indeed, we’ll learn that surviving prejudices of an alliedilk continue to plague coherent thinking about thermal behavior to this day,especillay within philosophical circles.

Unfortunately, the diagnostic depth of the central considerations headvances have been neglected within the large contemporary literature that focusesupon his ill-fated rejection of atomism and the persistent bells of anti-realism thattinkle within his writings. Perhaps Duhem even deserves his customary role asmethodological villain to Perrin’s “good guy in a white hat” due to the hyperbolicextremities of his prose. Observe,however, that Duhem’s animus towards

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“molecular hypotheses” (understood as efforts to escape into thermally freerealms) needn’t coincide with a general opposition to the postulation of minuteentities below the scale of, e.g., microscopic observation. There is nothingsubstantive in his writings or interests that precludes (insofar as I am aware) thestudy of, e.g., suspensions of very minute particles, such as contaminants in theatmosphere. It is merely that one should sometimes expect to employ thermaltools in these settings as well.

Or, at least, that is the interpretative vein I will emphasize here. We willsystematically coordinate Duhem’s main lines of methodological emphasis with thecomplex travails one encounters in developing a satisfactory extendedthermophysics. Without such moorings, I believe, the mystifying fluctuationswithin Duhem’s prose can only seem bewildering. For example, at certainjunctures, he appears to claim that “good science” should only serve as aconvenient resumé of experimental results; a page later, he will be granting thephysicist absolute license to dream up any theoretical apparatus she desires. Unless one can link these sweeping pronouncements to the odd roles that Carnotcycles (more details later) play within the development of thermal physics,Duhem’s lofty and seemingly discordant recommendations on method are hard toreconcile (rooted within thermal earth, however, they simply reflect insightfulcommonsense). To be sure, many of Duhem’s modern acolytes cherish hiswritings precisely for their expositive schizophrenia, for such an oracularoutpouring can be readily quoted to any purpose whatsoever. Indeed, the range ofdivergent readings of Duhem encountered in philosophical commentaries rivalsthose of Wittgenstein (in range, if not in quantity). As I have alreday conceded,many of these do possess substantive textual evidence in their favor (Duhem’sopposition to chemical atoms, for example, often borders on the ridiculous and willbe passed over in silence here).

So let it be stipulated, in concession to these alternative avenues ofinterpretative emphasis, that it will be a sober and dry-cleaned Duhem who sallies forth in this essay, whereas thereal personage was an obnoxious pugilist, fond ofexaggerated controversy and xenophobic rant (rather asWalt Disney’s Davy Crockett compares to the unreliablerowdy described in the Almanacs).

With these concessions im mind, let me raise my tworounds of cheer for the truly excellent things to be found in Duhem. With respectto the format that a more adequate form of “classical mechanics” should assume,

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virtually all of Duhem’s formal proposals (including those highlighted here) havebeen amply ratified by later developments. The physical realm in which classicalmechanics achieves its grand descriptive victories extends down to the nanometerlevel and our most trustworthy modern theories of matter at this scale (such asengineers employ in their planning) have generally followed Duhem’s percepts intheir avoidance of lower scale speculation (it is only through the use of computersand some advanced mathematical techniques that the traditional barriers of scalehave been somewhat transcended--see essay 4 for more on this). In other words,the formal structures that Duhem advocates have not proven to be outdatedmistakes, contrary to misleading remarks on “the failure of Duhemian energetics”that one encounters in the philosophical literature. More than that, a propermethodological appreciation of the “non-theory T” features within his ownrecommended brand of thermal endeavor is needed to make coherent sense of thetangle of doctrines that we cherish under the heading of “thermodynamics” eventoday.

Secondly, it is important to recognize that, considered in its own terms, the“old mechanics” will not form into an conceptually closed unity unless it issupplemented by additional thermal concepts introduced as primitives. The classicexemplar of this phenomenon is provided in the shock waves that often develop asnonlinear processes come to fruition. Start with a simple blast of air at one end ofa long tube. Specify its starting condition employing the usual “old mechanical”qualities of position, velocity and density. pulse of modeling. Inject a pulse ofpressure into such a gas within a tube. Assign it astandard gas dynamics, as provided in the inviscidBurger’s equation (u/t + uu/x = 0). Merelythrough adhering to the percepts laid down within this“wholly old mechanical” formula, our initial pulse willbecome transformed into a piled up singularity (a“shock front”) that leaves us with no “mechanical”criterion for deciding how the gas will distribute itselfover a fan-like region lying behind the shock front(this problem was first recognized by Riemann andDuhem contributed significantly to its development7). However, standardthermodynamics suggests the correct answer here: the density within the fanshould distribute itself in accordance with Gibbs’ celebrated rule for uncontrolledexpansion: the enclosed gas needs to maximize its entropy (= roughly, itsuselessness for organized purpose) within the allowed region.. But there is nothing

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codified within our Burger’s equation itself that suggests this specific mode ofdistribution; we must reach outside the doctrinal confines of the “old mechanics”to know how a gas (and most allied non-linear materials) will behave past the timesof shock wave development (it is sometimes said that an “entropic force” pushesthe gas into its post-shock wave configuration). Like it or not, we have becomethrust into the wider realms of thermal physics.

We can articulate this problem of conceptual closure in the form of aproblematic regress, allied to the “labyrinth of the continuum” issues that troubledLeibniz (essay 2). Suppose that, in the mode of the molecular modelers, weattempt to evade entropy’s entry into physics by dropping down a scale level to thecomponent molecules that crowd together in the shock wave region, vibrating moreviolently in a uncoordinated manner due to their cramped proximity (e.g., from amacroscopic perspective, they heat up and their entropy in Boltzmann’s senseincreases). For empirical reasons I’ll outline in a moment, such molecules need totreated as flexible blobs of some sort or another; they can’t be adequately modeledby point masses in the manner of either Boscovich or the modern classicalmechanics primers that select theirtopics according to mathematicalrequirements of quantum mechanics. If so, we must describe the behaviorsof their innards with continuumphysics equations of some kind, todetermine how our molecules willtransmit the internal stresses theyreceive from colliding into theirbumptious neighbors. Empirically,however, most matter transmitspulses in a non-linear manner,making it likely, under suitable conditions, that new shock fronts will eventuallyform within the interiors of the hypothesized molecules. Once again it looks as ifentropic considerations must be invoked to continue the molecular evolution pastthe events of shock formation. By the same reasoning as before, purist “oldmechanical” thinking is forced to retreat to some lower station of sub-molecules toavoid the dreaded intrusion of primitive thermal entropy. “And so on, adinfinitum,” in the manner of deMorgan’s famous fleas.8

The natural escape from this regress, from an “old mechanical” perspective,is to deny any internal flexibility to some range of sub-molecules, maintaining

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instead that they either act as point masses in Boscovich’s manner or as absolutelyhard rigid bodies.10 For numerous empirical reasons, none of these alternativesappealed to the physicists in Duhem’s era (such assumptions had been embracedby the earlier French atomists like Charles Navier but with mixed success). ByDuhem’s time, most scientists accepted our regress in the following fashion. Theyregarded the rigidified “particles” that appeared in their modelings as temporary“cut offs” that allow them to ignore insignificant lower scale complexities whoseexplicit treatment would only spoil the central patterns they hoped to capturewithout increasing the model’s predictive value much at all. One of the greatelasticians of the period, A.E.H. Love, articulates this approach as follows:

The necessity for a simplification arises from the fact that, in general,all parts of a body have not the same motion, and the simplificationwe make is to consider the motion of so small a portion of a body thatthe differences between the motions of its parts are unimportant. How small the portion must be in order that this may be the case wecannot say beforehand, but we avoid the difficulty thus arising byregarding it as a geometrical point. We think then in the first case ofthe motion of a point.11

In other words, by treating a smallish region of matter as if it is a simple point orminute rigid body, the larger scale behaviors of matter can be segregated fromunnecessary lower scale complexity in more or less the same way as a suitablespatial “cut off” can efface the dominant motions of the asteroids from the horrificcompilations of attending to the extremely small perturbative reactions they exertupon the sun and major planets.

Beyond these significant computational advantages, it is a brute fact that noone until much later in the twentieth century possessed the mathematicalequipment required to articulate the working principles of flexible matter in adirect way; the only available pathway to the differential equations they soughtneeded to pass through a stage of first considering little frozen-into-finite-degrees-of-freedom “elements” and only afterward obtaining the desired formulas throughsome dodgy species of “limit” argumentation (allied arguments are commonlyemployed even today in primers for beginners). And it is really impossible to seehow continuum physics could have gotten up and running if its developers had notbeen willing to engage in shaky reasoning of this “finite model to infinitesimalconversion.” As such, it illustrates the basic phenomenon of “developmentallynecessary incoherence” that represents a central topic in Essay 7.

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Except for the appendix, these issues will not prove of great moment withrespect to the key themes of this essay but it is still important to recognize that theconceptual difficulties at issue have relatively little to do with the standard“infinitesimal” worries of the calculus,as they were clarified throughWeierstrass’ celebrated δ/εconstruction. Our complications(which might be labeled “the problemof the physical infinitesimal”) insteadtrace to a variety of dimensionallydifferent ingredients) that need to bebrought into working harmony (suchclashes are entirely absent within pointmass mechanics). Notable amongstthese incommensurate gizmos are the“body” and “traction” forces describedbriefly in Essay 7 (and more fullyelsewhere) which are inherentlydimensionally incompatible and canwork in proper harness only if the latterare compressed into the peculiar objectknown as a “stress tensor” that then acts upon any of range of equally peculiarcompressed entities called “strain measures” (which, in the case of a solid, needs tobe a species of decorated “point” that retain evanescent traces of higher scale cubicsides in the manner illustrated12). And the prospects for a general thermal physicsof the sort sought by Duhem further require that the distinctive descriptiveparameters of orthodox thermodynamics (viz., pressure, volume, temperature, heatflux and chemical affinity) must coordinate with the traits of the old mechanics(force, mass, charge, momentum) within this infinitesimal arena as well (in thenext section, I’ll sketch how this works).

Let me enlarge briefly on this point, as it pertains centrally to Duhem’simportant place within the historical development of modern thermal physics. Asoriginally articulated by its founders (Clausius, Gibbs, Kelvin), “thermodynamics”deals only with largish globs of material in states of constrained equilibrium (= thelarge scale steady state into which the glob eventually relaxes after themacroscopic factors that lovk it in place are maintained long enough). As such, thesubject does not deal with “dynamics” at all, especially if that word is taken to

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connote “natural alterations in a system driven by autonomous factors internal tothe system itself.” The “changes” that orthodox thermodynamics is willing toconsider are enforced alterations of a “quasi-statical” variety, such as compressingthe volume of a gas by pressing on a piston. The fluid itself, left to its owndevices, doesn’t wish to evolve spontaneously in this manner; it is entirelyourselves, acting as meddling outside manipulators, who drive the gas’s scheduleof changes according to our own whimsies. Properly speaking, standard textbookthermodynamics doesn’t provide a “dynamics” at all and would be more justlylabeled as “thermoquasistatics” (although one can easily appreciate why that labelnever caught on).

Even today, many primers on the subject piously insist that “temperatures”should never be assigned to an object unless it is reasonably bulky and in thoroughequilibrium, even though this policy would reject everyday claims of the sort “anincreased temperature spread across the land” (more on this in the sectionfollowing). Adhering to such terminological puritanism in everyday affairs isvirtually impossible (“avoiding the big d-word is damned hard,” advises theflustered captain of the Pinafore), but substantive conceptual concerns lie behindthe unenforceable prohibitions. In this regard, Duhem himself was a great pioneerof the subject known today as “non-equilibrium thermodynamics” which seeks torelax the restrictions of Clausius-style doctrine in a manner that allows formeaningful discussion of temperature transport. Most attempts in this vein beginby presuming that thermal events on a microscopic scale will be driven by theirlocal surroundings in a manner similar to the manner in which a comparableamount of bulk material will respond to a comparable quasi-statical change inenvironment within a standard equilibrium thermodynamics setting. Obviously,these “close to local equilibrium” assumptions are quite confining and the task ofwidening the scope of non-equilibrium thermodynamics to embrace more dramaticforms of thermal exchange is not easily obtained (which is why the subject, eventoday, comes in many distinct flavors). Indeed, Duhem’s writings on methodologyadvise us (wisely) that science often works best if it lays its foundations within aset of reliable but specialized applications and strives to enlarge their outreach byworking cautiously outward, never knowing in advance when the guiding scent ofdescriptive opportunity will fade away and the trail of theoretical advance will end. At this point in scientific time, it has become evident to mosr practitioners in thefield that concepts like “temperature” must remain imprisoned forever within afading-at-the-edges landscape of this general character, in which it is not surprisingthat its useful applications eventually develop contrary flavors as they reach into

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the farthest frontiers of viable elaborations upon core “equilibriumthermodynamics” reasoning.

The “descriptive optimism” characteristic of theory T thinking does notprepare us adequately for conceptual developments like this, which don’t end in arefutation, but a whimper. And this is because it fancies that our “concepts” can besupplied at the outset with firmer rules for “truth-value” than is plausible, either ona physical or mental (= brain capacity) basis. We strive to carry on as best we can,but adventitious adaptation needs to be practiced, as we respond to causeways ofreasoning that eventually shrink and become overrun by thorns and compromisedby swampland. And this is why Duhem’s penetrating reflections on methodologywithin thermal physics should serve as a valuable corrective to the a priorioptimisms criticized on a variety of fronts throughout this book of essays. Sciencedoesn’t have to develop in the manner that Duhem outlines (and favors), but it sureas heck might.

And it is foolish to dismiss these concerns out of hand, on the flimsygrounds that “thermal physics is obviously not a ‘fundamental theory,’ but thelatter, when finally developed, will surely conform to all of the canonicalexpectations of theory T thinking.” To this, I will merely caution. Virtuallyeverything one hears of today in contemporary physics, from the broadest sweepsof cosmology through the micro-realms below, are couched in regimes oftemperature and entropy: they represent a second form of stitching that, togetherwith asymptotic linkage (see Essay 4), holds together the fabric of science today. Perhaps the sinews of thermal interlacing will eventually fade from descriptivescience, but, at present, its needlework shows little inclination of doing so. Wephilosophers, as imperfect but open-minded Cassandras of what-is-possibly-to-come, should strive to cultivate all articulate paradigms of how a useful descriptivepractice might stabilize itself with respect to reliable “truth-value” and so forth,without fully conforming to optimistic contours of theory T expectation or“classical concept” thinking. I believe that Duhem’s reflections on wisemethodology in thermal affairs (which, as I’ve already stated, have largely beenratified in later developments13) offer an extremely valuable “articulate alternative”of the sort wanted.

But, to return to the issues immediately at hand, Duhem would have surelyacceded to the two-stage developmental pattern outlined by Love, in which acontinuum modeler starts with “points’ (or other small “elements”) conceptualizedin frozen, finite-degree-of-freedom terms and only later removes their artificiallyinstalled rigidity through shrinking the target “element” into continuously

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distributed setting. Historically, two-stage procedures of this sort have played alarge role in encouraging popular doctrines of essential idealization: that a scientistneeds to artificially misrepresent the true qualities of a target object (a“representative volume”) as a simpler gizmo (a “point”) before the apparatus ofmathematical physics can obtain a descriptive grip on the situation. This is astronger claim than the mere observation that, in reasoning about physical systems,it is often most fruitful to begin ones efforts with structures of advantageoussymmetry without dissipative terms (vide the perfect volcanos of Essay 4) andmove onto more realistic modelings through perturbations upon these preliminaryresults. It would be conceptually preferable if these initial stages of fruitfulreasoning policy were not called “idealizations” at all, but the terminology is longentrenched in that application. But a “doctrine of essential idealization” is not asimple endorsement of perturbative techniques, but a semantic thesis to the effectthat a modeler must intentionally misdescribe her target objects so that thedescriptive resources of mathematical physics can be fit to the physical realm at all(that is why such “idealizations” are essential; they must be made before theproject of applying “mathematical physics” makes any sense at all). It represents aclose cousin to the worries about mathematical applicability that troubledDescartes: the reasoning tools that mathematics offers semantically suit only therarified gizmos (e.g., Euclidean triangles and circles) belonging to its internalrarified realms: only through artificial distortion can the concept “triangle” beforced to fit the messier circumstances all around us. In my own appraisal, suchpresumptions of semantic purity are entirely mistaken, but will not argue the pointhere (they emerge, however, in a different guise within Essay 7 and 8). Few stepsare required to carry us from “science must always enforce ‘essential idealization’cutoffs within its modelings” to themes of “science can only provide regulativeideals in a neo-Kantian manner” character. More generally, the characteristicwhiffs of idealist and anti-realist fragrance that permeate nineteenth centuryruminations on scientific method appear to originate within the humble oddities ofstandard two-stage modeling technique, as Love outlines it.

From an enlightened modern perspective, the two stage “finite blob to point”reductions characteristic of Duhem’s time (and elementary textbooks to this day)stem largely from a paucity of required mathematical tools, many of which arerather sophisticated in construction.14 But Duhem did not recognize this and hiswritings are full of appeal to doctrines of “essential idealization” type. Insofar as I can ascertain, the Duhemian themes most highly prized by otherinterpreters (Nancy Cartwright, say, or Bas van Fraassen) trace, at root, to these

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methodological sources. As it happens, my original interest in Duhem (and latenineteenth century physical thought generally) was piqued by my puzzlement, as a“scientific realist,” as to “why should these guys in the late classical era wax sostrongly anti-realist in their sentiments when none of the motivating oddities ofquantum mechanics have yet peeked over the scientific transom?” (many ofEddington’s widely cited “anti-realist” sentiments of the 1930's were borrowed,virtually to the point of plagiarism, from Karl Pearson’s 1892 Grammar of Science,whose own “anti-realism” directly traces to “physical infinitesimal” struggles ofthe sort sketched here). Pace van Fraassen and Cartwright, I don’t believe that the“anti-realist” or “phenomenalist” strains within Duhem’s writings should carrysignificant probative weight today, despite their distinguished historical pedigree.

But the Duhem highlighted within our pages proves a different storyaltogether, through his perceptive comments on how one exploits the observational and computational footholds that nature offers for setting up aneffective descriptive practice. The only place where Duhem’s embrace of two-stage, “essential idealization” practices shall further concern us is within theappendix, where I outline Duhem’s chief reasons for embracing a strong“underdetermination of theory” thesis. The concrete illustrations he provides arevery instructive, but they rely crucially upon “finite blob to point” presumptions.

(iii)

Accordingly, the focal scientific task we will track is that of developing aplausible mathematical formalism capable of describing the interaction betweenthermal and “old mechanical” effects in the general manner of continuum physics,without being forced to retreat immediately to a lower scale nether world in whichthe “old mechanical” principles can autonomously reign. I stress again that quitesuccessful theories of this type have been subsequently developed and that Duhemcorrectly foresaw the non-standard formal contours that such theories commonlyassume.

Presented in naive terms, the accompanying diagram illustrates, in roughterms, the features we expect to see within a suitably integrated continuumthermomechanics. If we introduce pulses of both heat and mechanical pressureinto a dormant bar of iron, we expect that these influences will move from onelittle section of the bar to another, locally striving to lengthen the unit due to higher

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temperatures whilesimultaneously attempting tocontract it due to increasedpressures. Our central task is todevelop a more detailed pictureof how these two sorts of causalinfluence manage to work cross-effects upon one another. Butthere is a familiar bugbear thatwill complicate this integrativetask considerably. Friction--thefact that units of metal slidingacross one another typicallygenerate heat--is the obnoxiousfactor I have in mind and it willassume a rather unexpectedposition within Duhem’sdiscussion.

What procedures shouldwe follow in setting up such a theory? At first blush, one expects to adopt a simplemethodology: “Okay, lay out a set of postulates that delineate all of the possiblesystems you plan to tolerate and then test its models against nature.” In thisframing, one is allowed to introduce any new “theoretical” terminology one wants,as long as the postulates mathematically fix the behaviors of the “possiblesystems” tolerated. Of course, we have no guarantee that any real world behaviorscan be mapped onto any of these “possible world” trajectories, but such are theexpected empirical hazards of applied mathematics (“man proposes; naturedisposes”). I shall call this a “theory T picture”of scientific theorizing and itsproponents typically find strong encouragement in the “phase spaces” of orthodoxHamilton mechanics, which are mathematical constructions in which theautonomous time evolution of each tolerated “possible system” is portrayed as asingle point trajectory wandering through a high dimensional space (illustratedbelow in our pendulum portraits). These same temporal developments (= alltrajectories open to a particular system of a fixed composition) can likewisepictured as a tightly knit, high-dimensional ball of entangled strings (see diagrambelow). In a typical “phase space” context, each of these trajectories represents anautonomous evolutionary development tolerated within the framework of the

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theory: they capture how an isolated system will change over time by simplyfollowing the drummer of its own internal dynamics.

But let’s dwell a bit on that word “autonomous.” In a canonical Hamiltonsetting, these trajectories never cross or fuse, due to the fact that frictional effectsare not modeled within this branch of the “old mechanics.” In consequence, it isfairly rare that true equilibrium states exist for autonomous systems of this class,that is, we rarely find systems locked into astatic situation where its properties nolonger alter over time. In a Hamiltoniantreatment of a simple pendulum, the onlystable equilibrium point is a bob completelyat rest, around which almost all othertrajectories cycle, without ever subsidinginto rest themselves. However, if weintroduce some friction into the setting, therest position becomes an attractingequilibrium point to which almost all of the other trajectories gradually approach(depending upon the strength of the frictional braking).

Autonomous systems that gradually lose energy in this general manner arecommonly called dissipative systems. But these treatments do not permit anyreconversion of frictional energy back into large scale movement--they approachthe degradation of energy into friction as a one-way street (once Humpty Dumptyhas been broken apart by friction, you can’t put him back together again).

Fortunately, this grim portrait of our energetic capabilities isn’twholly true: we couldn’t have steam engines and electricalmotors if it was. The silly device in the diagram manages toreconvert a small amount of the heat it generates back intomechanical energy--it merely can’t execute that chore withperfect efficiency (one of James Watts’ many claims toundying fame is that he greatly increased the efficiency of

steam engines by redirecting previously lost heat towards mechanical objectives). Indeed, the whole subject of modern thermodynamics takes its historical origins inefforts to understand the principles that monitor these kinds of cross-effect and thatstory becomes rather complicated simply because the task of tolerating a limiteddegree of bidirectional energy conversion with friction is rather subtle. Inparticular, we shall need the wholly unintuitive notion of entropy to unravel what ishappening.

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Angels from the outside holding theplanets in equilibrium

I should indicate that, although we shall largely concentrate on theentanglements of heat with mechanical behaviors here, Duhem (andthermodynamics generally) is equally interested in allied couplings of the “oldmechanics’ to chemical composition and electrical condition (these issues areimportant to understanding his opposition to “atomist” thinking upon a largerscale). From an experimental point of view, these further forms of cross-termentanglement are easier to measure and were accordingly often stressed by Duhem,but, with a few exceptions to align our discussion with his text, we shall largelyignore these further dimensions of the problem here, as the structural dilemmas weneed to appreciate appear with a fully adequate ferocity within the “how can weamalgamate heat and friction into the ‘old mechanics’?” difficulties we shall probe.

Observe that is another simple way to bring a target system to equilibrium”by brute force, courtesy of some schedule of external manipulations (e.g., weimport a bevy of angels to stop the planets in their traces or we dunk a metal bar ina “heat bath” to force it to a uniform temperature). Situations of this general ilkare called controlled or manipulated systems, where the posited internal dynamicsbecome affected by outside interventions such as human-monitored magnetic fieldsor simple ad libitum taps from a hammer. Various degrees of manipulated controlare possible here. Equilibrium states proper (thatis, states whose properties do not later over time)frequently require complete outside control fortheir sustained continuance, but we shall also beinterested in semi-autonomous systems that aresubject to lesser degrees of outside intervention. Infact, Duhem’s basic mode of attack in dealing withthermomechanical systems is to first consider materials locked into tightlycontrolled equilibrium and then gradually determine how they will behave as wegradually open one or more pathways to increased autonomy (rather as a cat playswith a captured mouse). He borrows this technique from Lagrange’s methodswithin the “old mechanics,” in a manner we shall detail later.

Such manipulated systems are generally discussed as an afterthought withintypical “old mechanics” presentations (although not in Lagrange’s distinctiveapproach within his Analytical Mechanics), but the same secondary status cannotbe sustained within a thermodynamical setting, where heat baths, isothermalcontainers, diathermal partitions et al. comprise the very stuff of which a“manipulation” is made. In fact, the entire content of orthodox instruction in“thermodynamics” restricts itself to entirely static considerations involving

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Lagrange

completely controlled target blobs with little internal complexity (that is, they don’tyet divide up iron bars into little cubes as we did above). The motivesbehind these Spartan educational objectives are rarely made clear,commonly to the bafflement of the hordes of unfortunates who haveendured a typical undergraduate class in the subject.15 One of thesingular virtues of Duhem’s general approach to scientific methodologyis that it supplies cogent reasons for the structural oddities we encounterwithin thermal science. Or, to put an alternative spin on the same

situation, students find standard thermodynamics puzzling largely because theyunwittingly subscribe to “theory T thinking” expectations as to how science shouldbe presented.

Experiments, of course, inherently involve such outside interventions andloom large within Duhem’s own discussions of methodology. But the thematicblurriness within those characteristic passages have often obscured crucial featuresof his underlying intent, at least as it will be interpreted here. Accordingly, Irecommend that Duhem’s readers will do well to simply substitute the phrase“manipulated system” in most places where the Master has inscribed“experiment.”

Duhem calls an autonomous trajectory modeling for the entire universe acosmology (in agreement with its usual physical significance; I find that histerminology is sometimes misinterpreted in a more mystical manner, as signifying,e.g., the world view of the Catholic Church.). Such cosmologies contrast with themore localized modelings of physical practice (whose degrees of internal“autonomy” can only prove approximate due to the manner in which everysmallish system couples to its environment, albeit weakly). Duhem, of course, isfamously insistent that science, properly understood, should not expect to havemuch truck with such “cosmologies,” although a “metaphysician” might. Over thecourse of this paper, we will unpack the physical considerations that lead him tothis odd stance.

The expectation that any decent physical proposal should carve out simple“spaces of autonomous trajectories” is generally a hallmark of “theory T thinking”aS i employ the phrase throughout this book of essays (the label traces to thoselegions of philosophers who merrily discuss issues of methodology at a schematic,“how theory T compares to theory T’”level, while rarely examining real lifeexemplars). Certainly, writers within the so-called “semantic view of theories”contingent (e.g., Bas van Fraassen or Patrick Suppes) appear to identify “theories”outright with collections of trajectories of the sort just indicated, but even their

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dedicated “syntactic view” opponents generally expect that their formal theorieswill carve out allied “phase space” collections as models (in the sense of thelogicians). Insofar as I can determine, many of Duhem’s devoted modernchampions cling firmly to “theory T”-conceptions of formal structure, despite thefact that I believe that Duhem’s greatest single contribution to the philosophy ofscience is that of providing, in rich and amply motivated detail, an alternativetheoretical framework that might plausibly comprise the formal mathematicalstructuring of a “final physics” that also fails to supply the full bundle ofautonomous trajectories anticipated by the theory T crowd. So a robust “twocheers” for all of that!

In the same vein, the “possible worlds” beloved of analytic metaphysiciansappear to be simply “cosmologies” in Duhem’s sense: trajectories for an entireuniverse that unfold according to their own internal dynamical principles. Butmore on this later.

(iv)

It will be easiest to explain the general idea behind Duhem’s approach totheory construction if we employ some convenient “pictures” from modernmathematics, particularly, in terms of “manifolds” and “submanifolds.” In themode of a standard phase “space,” we will collect together all of the possibletemporal developments that a target system of a fixed construction mightdisplay, each of which we picture as aribbon carrying the system forward in time(recall the trajectories we drew for thependulum). The entire collection (whichresembles the densely packed insides of agolf ball) constitutes our primary manifold. We observed that trajectories inside thestandard“phase spaces” of Hamilitonianmechanics only represent autonomoussystems, in the sense just specified, but in our thermal physics setting we mustinclude many additional trajectories that are weakly or strongly manipulated. Inaddition, winding as supportive skeleton throughout the whole wad of trajectoriesis a closed surface that I shall call (somewhat inaccurately16) the “Clausiussubmanifold” with respect to the target system under investigation. Inscribedwithin this submanifold are various lines that, following standard practice, we shall

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call “reversible paths,” although they actually do not represent any real form oftemporal development in themselves (we’ll expand upon this point later). However, many of the off-the-submanifold ribbons (which do symbolize genuinetemporal processes) wind fairly closely to the paths on the submanifold. Our planis to employ the mathematical landscape encountered upon the Clausius manifoldas a basis for codifying the behaviors witnessed amongst the systems that whiz pastin the greater manifold nearby (in the mode, “the space ship follows a path veryclose to the road that connects Mt. Hood to Mt. Rainer,” except that in our casetraffic flow is infinitely sluggish on the Clauius manifold itself). ”).

In Duhem’s eyes, the task of building up a satisfactory thermal theory is to(1) first specify the mathematical structure (“landscape”) living on our skeletalsubmanifold and (2) progressively construct little ropes that can reach up from theanchoring submanifold to as many of the system’s possible trajectories as wepossibly can. In doing so, we must proceed in progressive stages, beginning withvarious collections of semi-autonomous trajectories (= partially controlled by theexperimenter or known outside factors).

By proceeding in this way, can we expect to eventually obtain a tightdescriptive grip upon all of the wholly uncontrolled system developmentsencountered in the real world? Probably not. Duhem hoped that the task ofcomplete outreach might someday prove possible upon his system, but he wascertainly fully aware that the constructions he knew how to frame could notaccomplish this job fully (we will return to this point in the next section). With thegift of hindsight, we moderns can certify, with almost complete assurance, that noform of thermodynamics constructed within a wholly “classical physics” frame islikely to obtain any final condition of perfect and comprehensive closure. Rather,in its optimally developed “final physics” state, the formalism must still toleratephysical trajectories that wind through the interiors of its primary manifolds infashions that cannot be adequately tethered mathematically to the Clausiussubmanifolds within.

If, as Duhem had every right to presume, physics’ “final formulation” couldbe expected to retain the basic contours of successful “classical physics” and if ahomogenized “thermomechanics” represents a better route to that goal than shakyproposals beguiled by the deceptive charms of the “old mechanics,” then there is agood chance that physics may find itself perpetually confined to the chastenedprospects that Descartes foresaw: saddled with an optimally perfected formalismthat only captures a significant subset of the world’s anticipated behaviors inprecise mathematical terms, leaving the rest to “resemble” them in some indefinite

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way. Accordingly, part of physics’ descriptive agenda becomes one of locatingthe special opportunities within nature wherein its mathematical tools can be fittightly to the behaviors within the “opportunity.” The basic purpose of the Clausiussubmanifold is to carve out a moveable mathematical platform that can be shiftedinto convenient position as soon as an “opportunity” opens up to which itsdescriptive apparatus is well suited. After we get a few thermal specifics on board,I’ll illustrate how this moving-a-platform-into-an-opportunity operates in practice(in connection with rubber bands, see below).

On this incremental approach of “theory,” it is rash to demand that theapplicational standards for crucial terms such as “temperature” and “entropy”should be fully specified at the outset in the manner anticipated within crude“theory T” thinking (we shouldn’t fancy that we’ve managed to “implicitly define”this term). Such words need to make their way forward with the same gradualismas we progressively reach to further sectors of off-the-Clausius manifoldtrajectories.

Let us now consider why we need to set up these funny Clausius surfaces atall. The main reason is that we need to understand how a target physical systemstores and transmits potential energy. Since the appearance of friction obscuresthese patterns, we must construct a landscape whereinfriction is absent and the target system’s allocations ofenergetic storage become clear. Let’s first consider a purelymechanical system, such as the shock absorber pictured tothe right. Its natural rest position is at the top, but it can alsobe maintained in the other controlled equilibria shown if thecentral spring is stretched with an applied force of therequisite magnitude (note that it takes significantly greaterforce to maintain the spring in the bottom position than inthe middle configuration). We measure the work performedon the absorber by the directional magnitude of the force Fapplied times the distance through which the central springis elongated, which we write as Fδr. Here we regard F asan applied “intensity” (Duhem’s favored term) that drivesthe position of its application through a certain smalldistance δr (the funny “δ” symbol willl be explained insection . But these “intensity driving ‘force’/extensive reaction” pairs can assumemany different forms such as τδθ (where τ is a turning torque and θ is angle ofturning) or μδη (where μ is a chemical potential and η is the quantity of chemical

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composition that μ “drives”). When we eventually shrink our “elements” down topoints, the mechanical work Fδr performed along the bounding perimeter of alarger element through surface traction will shrink down to a single stress/straintensorial relationship σ:δε (many books in thermodynamics call the driving stress σa “generalized pressure,” but it is actually a tensor). For reasons that apparentlytrace to his Thomistic desires to credit all good ideas to Aristotle, Duhem labelsany forced alteration in an extensive quantity δY as a “motion,” even when it isonly the heat content that is “moving.”

We should note that although we are talking as if various “intensive” forcesdrive (or “cause”) an “extensive” reaction, we are actually considering completelystatic situations where nothing changes over time at all! Conceptual tensions ofthis ilk (which are inherent in any conceptualization of mechanics that makes“virtual work” a central ingredient) become quite acute within thermal physics forreasons we’ll soon rehearse.

As long as friction doesn’t raise its ugly head, we can likewise assume thatthese same XδY work pairs supply a measure of the potential energy stored in theapparatus whenever it is forced into a stretched configuration (this particular formof stored energy is called strain energy--energy stored within the strained (=distorted) configuration). As such, it stands available to perform work on itsenvironment.

As such, there are often many ways in which a device can store energy: asstrain energy (which, in a gas, is codified within its “spring,” i.e, pressure),chemical potential or simply as heat. The basic operations of a target system aredetermined by the manner in which it allocates applied work to its various forms ofpotential energy storage (this rule will eventually become codified by the system’sso-called “equation of state,” but we’re currently missing some crucial quantitiesrequired for its effective expression). Note that in thinking about a system in thisway, we only consider the manners in which we can control the energy storagethrough precise ways of holding it in equilibrium under manipulation techniques atour command: loading a spring with a measured weight, supplying a measuredblast of heat, immersing the our blob in a constant temperature “heat bath,” etc. Following our “finite blob to an infinitesimal” strategy, we will first considersystems where all of these descriptive parameters are macroscopic in character, butwe eventually plan to apply the same notions to very small units of matter when webuild up continuum models for materials by stacking a bunch of little blockstogether.

In any case, a Clausius submanifold simply represents the collection of all

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Clausius

states that a target system can displaywhen maintained in manipulatedequilibrium (in the diagram I’ve flattenedout this manifold and have drawn in afew controlling angels to remind you thatexterior manipulations must maintaineverything in equilibrium). We can nowdraw curves through these pointsindicating that the systems along theselines share some common feature, suchthe fact that no heat has flowed acrosstheir boundaries or that they are all maintained at the same temperature. Quasi-rectangular patterns made from such lines are the familiar Carnot cycles ofintroductory thermodynamics.

These curves are usually described as mysterious “processes” (=trajectories) possessing self-contradictory properties (e.g., they allegedly representa succession of unchanging states that nonetheless alter from one to another,although it takes infinitely much time to do so). Rather than talking in thisvenerable manner, let us simply state that such “lines” convey very importantinformation about how the system’s various descriptive quantities fit together. We

have already noted that these quantities align in conjugate pairs XδY,where (depending upon circumstances) the intensive quantity X drivesa change in its conjugate extensive mate. Based upon the way that theCarnot patterns must fit together within these friction-free domains,Rudolf Clausius proved (in an astonishingly abstract form ofreasoning) that an additional form of intensive potential S needed to bewell-defined upon this manifold that he called the entropy function. It

is this unfamiliar and unexpected quantity that serves to “drive” a system’salterations in overall internal energy U (including, most importantly, its stored heatenergy content). Its presence also allows use to introduce absolute temperature Tas a replacement for the cruder “thermometer temperatures” with which weactually control thermodynamic processes. S and T prove to be the missingingredients we need to complete our story of how the various quantities on theClausius surface fit together in an “equation of state”(whose purpose, you willrecall, is that of articulating the manner in which a target system will allocateapplied energy to its various modes of “potential energy” storage).

For readers familiar with modern presentations of Hamiltonian mechanics, it

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might prove useful to compare the structure of the conjugated quantities arrayedupon our Clausius surfaces with the Lie algebra geometries latent in themechanical subject. It was Clausius’ great insight to discern these systematicformal relationships between quantities (which is why I’ve named this supportivesubstructure in his honor). As noted earlier, Duhem’s own contribution lay inmapping out the key steps that need to be subsequently taken in blending Clausius’and Gibbs’ original “bulk matter only” concerns with the infinitesimal tools ofstandard continuum physics (understood within its “old mechanics” contours).

If we regard the Clausius manifold as simply a record of how basicdescriptive quantities relate to one another structurally within a selected thermalsystem, there is no reason to assign any sensible notion of “time elapsed” betweenthe points found upon its surface. Nor do we need to view the “paths” we draw insetting up the Carnot cycles as representing any kind of “physical process”requiring time to complete. Instead, our own policy will be to assign sensiblemeasures of “time elapsed” only to the trajectories within our “full manifold” thatdo not fall upon Clausius surfaces.

However, traditional ways of speaking (including Duhem’s own) do notpractice such an abstemious conceptual policy, for they instead call our paths“reversible processes” and declare that they “proceed so slowly” that they requirean “infinite time to complete” (wherein we can still submit a target system to foursuccessive processes of this extent in forming a Carnot cycle). Why they talk insuch odd ways traces to important issues to which we’ll soon return.

(v)

In any case, employing the Clausius submanifold as an anchoringframework, we can start to paint in a number of off-manifold trajectories thatgenuinely evolve in real time and also embody some measure of an autonomous,internal dynamics. That is, their movements become partially determined by theirinternal conditions, as opposed to slavishly following the static dictates of outsideintervention (all system positions upon the Clausius submanifold are of lattersort). But we must begin to draw in these off-the-manifold trajectories rathercarefully, beginning with paths that stay very close locally to the lines upon theClausius surface (more precisely, the latter should serve as limiting asymptotes to

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some of our new, less manipulated, trajectories, roughly in the same manner aslight rays appear as the natural asymptotes to collections of high frequency waves). As these partially liberated trajectories get filled in, we may gradually approach thecompletely self-sufficient dynamics trajectories of the unmanipulated naturalworld. Borrowing a grislymetaphor from the Russianopticians (of Essay 8), wegradually drape lessmanipulated “flesh” onto thebare bones of an underlyingClausius skeleton (themetaphorical variant which I’veemployed: we tie trajectoryribbons onto a coral-likeexoskeleton).

As long as we don’t attempt to depart too radically from these underlying“bones,” we can develop mathematical formalisms that can model a host of reallife situations with spectacular success (we shall examine one of these extensionsmore closely later in the next section). But as we attempt to equip our targetsystems with increasing degrees of internal dynamic autonomy, our extensionefforts become more dubious: in Duhem’s words: “the causeway shrinks, the trackof the route becomes unclear; we only see a narrow path half hidden by thorns, cutacross by bogs and bounded by abysses.” These potential barriers to completionoffer the disappointing prospect that, after the innards of our thermal manifoldhave been filled in as many threads as we possibly reach from the embeddedClausius exoskeleton, there may still be a lot of unreached paths left, windingthrough empty spaces surrounded by snarls of mathematically describabletrajectories. Unfortunately, these unreachable trajectories may represent theresidential space wherein most of the physical systems of real life experiencedwell, which may evolve in their own autonomous ways without displayingsignificant hint of the regularities we witness systems maintained under a tightersystem of thermal controls. But if a “free system” fails to retain some vestigialtraces of controlled behavior, a mathematical physics built up in Duhem’s mannermay not be able to obtain a tight descriptive grip upon its freely evolvingbehaviors. We have traveled too far from the bone and the anchoring opportunitiesthey provide have been lost.

If we replace “manipulated trajectory” with Duhem’s favored word

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“experiment,” we reach the conclusion that the proper end goal of science propershould center upon the task of cataloging “experimental results” efficiently, ratherthan feeling obliged to capture all of the licentious trajectories of unfettered natureeffectively. But employing “experiment” in this deviant manner has encouraged aprevalent reading of Duhem in which he is said to favor “phenomenological laws”in the sense of the logical empiricists and which renders his many remarks on the“liberty” of the physicist nearly contradictory. With respect to the latter, herepeatedly observes that the Clausius manifold data upon which histhermomechanics (and Lagrangian technique more generally) builds consist invirtual work pairings of the form AδB (the motives behind this notation will beexplained more fully in our optional section (vi)). In these alignments, the“virtual” side of the pairing (δB) represents a hypothesized response devised by thecreative scientist as a theory of how the driving manipulations A affect the targetsystem; the δB responses do not directly reflect real life experiment, but representa modeler’s creative riff on what those experiments might betoken (essentially, Imight add, in terms of how the hidden internal energy storage capacities of thetarget system are affected). Most readers of a “phenomenological laws”persuasion seem to have overlooked these crucial “virtual response” aspects withinhis thinking. Restored to proper argumentative position, Duhem’s admonitionswith respect to science’s obligations to “experiment” turn out to be little more thanan acknowledgment of the concrete difficulties we encounter in building up aricher thermal physics starting with the amply verified virtual work pairingenshrined upon the Clausius submanifold (which may reflect microscopicelectromagnetic pairings at scale levels far below the laboratory frame).

Duhem plainly hoped that a throughly complete thermomechanicalframework would be eventually constructed on his principles, but his appreciationof its reliance upon extended opportunity played a significant role in his repeatedinsistence that the requirements of science cannot demand such completeness (incontrast to the a priori expectations of the theory T crowd). To be sure, Duhemallowed that a scientifically well-informed “metaphysics” might permissiblyadvance such completeness claims, glibly embracing sweeping characterizations ofhow unmanipulated nature freely unfolds everywhere (such views he called“cosmologies” and his reservations on their scientific viability are relate dto the“cut off” concerns voiced above). He believed that his own Catholicism providedhim with a metaphysical basis (courtesy of St. Thomas’ strong endorsement ofAristotle) for anticipating that the virtual work pairings he favored wouldeventually be found to govern nature everywhere, but he stressed that these

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convictions stem entirely from faith, rather than the scientific enterprise proper. And he likewise insisted that opponents who rejected primitive thermal notions infavor of a purist “old mechanics” were following a “religion”/”metaphysics” oftheir own, stemming from convictions equally outside the sphere of scientificdeliberation proper. As the title of Essay 5 indicates, I detect a parallel analogiesto religion within the theory T dogmatisms of today’s “analytic metaphysicians.”

In more concrete terms, Duhem’s reservations are anchored in deepconcerns about the role of scales and controllable energy in science and appear inmany guises throughout these essays. As a matter of contemporary assessment, itis unlikely that classical thermal physics can be fully “completed” with respect toevery autonomously evolving trajectory in the manner we (or Duhem) mightideally prefer. The chief villain enforcing these limitations is humble friction,which represents an uncontrolled release of system energy to the adjacentenvironment and to scale levels below. Insofar as we have successfully managedto incorporate these rude intrusions within our mathematical models, we haverelied upon coarse descriptive measures (tools such as simple linear or statisticaldependencies) that are likely to fail us in delicate situations. Duhem was fullyaware of these challenges and the dialectic he lays out in The Evolution ofMechanics is shaped by the ongoing struggle with friction and hysteresis. The besthope for surmounting difficulties is to escape into nether realms in whichtemperature and disorganized energy no longer represent significant factors and, aswe have noted, the stitching of effective physics may not permit such a thermaldisengagement. Perhaps the best that mathematical physics can expect to achieve,while working within the general vicinity of significant frictional effect, is to locatevarious special anchors of descriptive opportunity upon which a viable formalismcan be usefully erected to parochial purpose (we observed that many usefulmodern forms of “extended thermodynamics that have been proposed followingthe general contours of Duhemian methodology, but none of them claim to reachevery form of frictional behavior with untarnished success17). If such proves ourdescriptive fate, we confront a methodological endpoint similar in character to thatDescartes envisaged: although we possess a generally adequate inventory of thebasic processes active within nature, many of the free trajectories of autonomoussystems will forever elude our ability to track them in precise mathematical terms. As stressed previously, these deflated expectations do not constitute an “anti-realism” in any evident sense, but simply a sad acknowledgment of a shortfall withrespect to our available tools of reasoning.

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It strikes me as patently evident that, in circumstances like these, it isultimately an wholly empirical matter how far the reach of a useful descriptivepractice might extend and it is not useful to pretend, as philosophers commonly do,that “theories delineate sets of mathematical models and either the entire universefits onto one of these structures or the theory utterly fails.” These folks generallyassume that, as soon as we grasp the concepts of “temperature” or “entropy,” wemust have firmly fixed the subset of “possible worlds” in which those notions willprove viable and the complementary class where they do not. But Duhemianconsiderations indicate that, at present, we remain unsure how far into the realm ofuncontrolled processes the classificatory utilities of these same useful words can bestretched, even within the narrow confines of everyday terrestrial occurrence. Indeed, for reasons we’ll soon survey, crediting a humdrum, everyday solid (suchas a rock or table) with a “temperature” is often a tricky business. Philosophersdon’t like concepts whose “extensions” peter out into inapplicability in a whimper,rather than a bang, but effective science is full of them.

Insofar as I can see, these struggles with words that don’t behave as wemight like account for the pedagogical “weirdness” that every sentient studentexperiences within their first course in “thermodynamics.” Due to the unclaritiesas to how far away from the Clausius manifold the terms “entropy” and“temperature” can be usefully prolonged in a “non-equilibrium thermodynamics”vein, orthodox primers in the subject often adopt the radical “cure” of denying thatsuch terms possess any meaning except in cases of macroscopically maintainedtotal equilibrium. That is, the proper subject of “thermodynamics” should be theClausius manifolds pure and simple, without any notion of how they should beextended to richer applications. If a rock fails to satisfy its complete requirementsfor equilibrium, so much the worse for crediting the stone with a “temperature,”everyday assertion be damned. This draconian corrective leads these same booksto claim that the “paths” inscribed upon the Clausius surface describe the“reversible processes” found in nature, while simultaneously insisting that no realsystem will ever follow any of these paths, no matter how much manipulativeeffort we devote to the task (surely, this is to take away with one hand what hasbeen given by the other). Hans U. Fuchs rightly complains of these pedagogicalpolicies:

[T]he classical field of thermodynamics and the subject of heat transferform a natural unity. Despite all the claims in books on thermodynamicsthat heat transfer cannot be included, and of books on heat transfer that

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thermodynamics 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. 18

In this last remark, Fuchs alludes to the consideration already noted: to get thermaland mechanical effects to couple with one another plausibly (in the manner of ourheated bar struck on one end with a hammer), one needs to retreat to theinfinitesimal level in exactly the manner that Duhem pioneered.

Of course, these elementary primers violate their own conceptualprohibitions at the first opportunity (sometimes on the very page where the sternterminological admonitions have been laid down). Everybody in the subject wantsto say that heat is gradually driven along our bar by an “entropic force”-- that is, a“force” that drives our disturbed, out of equilibrium bar closer to a genuineequilibrium where the bar has the same temperature everywhere. Allied “entropicforces” cause the gas behind a shock front to spread out in the predicted patterns. Officially, however, all of these “forces” qualify as fake, fictitious ormetaphorical.19 We are then solemnly informed that neither entropy nor entropicforces can meaningfully “drive any kind of change” since they represent conceptspertinent to conditions of strict equilibrium only. And five minutes later, thesesame terminological prudes are speaking of “molecular driving forces.”

These schizophrenic prohibitions are motivated in part by the fact thatstudents eventually must appreciate that the realms of “non-equilibriumthermodynamic behavior” cannot be settled readily and that serious ambiguities ofconcept extension can arise when one strays too far from the motherland of “bulkmaterial in thorough equilibrium.” But to insist that “temperature” and “entropy”make “clear sense” only within the narrow confines of our Clausius manifolds is totell a semantic story that makes the physical utility of these same words utterlymysterious: how can such words only make sense for “processes” that aren’t reallypossible at all? Following Duhem, the proper response to these confusions is toobserve that it is precisely the “nearby trajectories with dynamic arrows paintedon” that supply “thermodynamics” with its descriptive targets and they cannot beexpelled from the semantic landscape pertinent to these words, despite the fact thatthe exact boundaries where these non-equilibrium behaviors end remain unsettled.

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Partially releasing a blob from the controls of two different outside agencies

Analogy: the fact that plants and animals expire within the thinner air of thestratosphere doesn’t alter the fact that these same critters require the thicker layersof the lower troposphere for their viability.

In short, some unsustainable “philosophy” of “define your terms clearly”seems responsible for an educational policy that renders the wise methodologicalcautions ingrained within thermodynamic thinking incomprehensible to a dutifulstudent. Sometimes nature doesn’t present us with “descriptive opportunities” thatsubmit to “define your terms clearly” precepts. I firmly believe that a just appreciation of the developmental considerationsoutlined by Duhem in The Evolution of Mechanics (and recounted here, in my ownvariant terminology and metaphor) should help to greatly diminish the semanticdogmatisms that lock large amounts of contemporary thinking into unimprovingstasis.

(vi)

. Note to the reader: The material in this section is intended to flesh outimportant details in Duhem’s procedures, so that a closer concordance between hisphilosophical pronouncements and the nitty-gritty specifics of his chief scientificproject can be reached. Insofar as the chief objectives of this essay go, thediscussion in this section is entirely optional.20

Accordingly, we regard the “paths” upon a Clausius manifold, not as asuccession of states that closely model any kind of real world process at all, but assimply providing a mathematical platform that captures how the basic descriptivequantities pertinent to a target thermal system interrelate to one another. Thevarious “reversible paths” invoked within conventional thermodynamic instructionrepresent nothing beyondthe equilibrium states S*that are like S except thatthe controls on one or morevalues (temperature, say, orpressure) have been set todifferent values. Our taskin creating a viablethermomechanics is to allowour target systems someincreased degree of internal

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autonomy that allows them to escape from the Clausius surface and behave as agenuine, evolving under relaxed controls, trajectory. Let’s consider two simpleexamples of how this partially easing of constraints should look. First, if we relaxonly the tight pressure and volume controls that maintainkeep a blob of gas inequilibrium, while keeping its interior at a constant temperature throughout, weshould expect to see waves of compression moving across the interior of our blobin the manner of the “old mechanics.” Let’s picture these developments inevolving stages Bm

1, Bm2, Bm

3, .... If our mechanically-liberated blob is essentiallyelastic in behavior, it may oscillate like a spring between two bulges that appear atcorresponding extremities (the bulge is largest on the left at stage Bm

1; largest onthe right at Bm

4). However, we might alternatively relax the temperature controls,keeping pressures and volumes constant, and generate a sequence Bt

1, Bt2, Bt

3, .... We then expect to witness the pure movements of heat described by Fourier’sequation--viz., our heat bath-liberated blob will move towards a condition where itsinternal temperatures equalize In both of these simplified situations, we possessgood intuitions as to how our partially emancipated blobs should behave. Ourcentral task is to figure out how these two processes should couple to one another.

Each of the little blobs Bmi, or Bt

i we pull off the Clausius manifold willpossess a preimage back on the manifold that keeps track of how the physicalquantities pertinent to our blob relate to one another. What this manifold does nottell us is how we should tie together our evolving blob states into trajectories. Metaphorically, we must find a trustworthy criterion that can paint little “dynamicarrows” between our various off-the-manifold blobs. Since there is no notion oftime properly registered within the Clausius manifold, we should anticipate thatfresh considerations from some another quarter will be required before we acquirethe tools needed to treat any form of temporal evolution whatsoever.

But how can we paint plausible “dynamic arrows” upon our liberated blobsso that they can serve as models of partially controlled, but genuinely evolving,physical systems? The basic prototype for this emancipation was pioneered byLagrange within the context of the “old mechanics.” Paralleling Duhem’s owndiscussion in The Evolution of Mechanics, let’s first survey how this process ofLagrangian liberation works and then observe how Duhem generalizes theprocedure to handle thermally emancipated events as well. As indicated before,embedding traditional “thermodynamics” into a wider “installing dynamic arrows”context goes a long way in rendering the purposes of the subject rationallycomprehensible.

To do this, we must, as Hans Fuchs suggested, bring continuum mechanics,

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with its attendant infinitesimals, into our picture. As noted before, we shouldpartition a target system into a tiled set of tiny blocks that fit together tightly. Eachof these little blocks will comprise a “representative volume element” that we willlater shrink to infinitesimal size accordingly to the venerable “finite blob to point”policies discussed earlier. In doing so, we must describe the mechanical featuresof our blocks in standard continuum mechanical terms rather than relying upon thecrude surrogates usually emphasized in elementary courses on thermodynamics,where the “mechanical variables’ acting upon a blob are usually restricted topressure and volume, without any particular indication of where that “pressure” isapplied. But we observed that these descriptive simplifications are plausible onlyfor a gas or fluid, for in a solid we expect that we will be able to push or tug uponits various sides in different ways (many materials--e.g., wood--respond to thesevaried tractions in a non-isotropic manner). So we need to start with a“representative volume element” that is surrounded by a large pack of differentsurface force vectors, pointing in different directions and often of different“generalized” natures (e.g., torques). When we add in thermal phenomena later,we must replace the solitary “heat bath” of traditional thermodynamics with asurface array of temperature differences that indicate how strongly heat is likely toflow across one boundary location or other.

Even within these “old mechanical” circumstances, we require a rule,specialized to suit the particulars of the “element” under consideration, thatdetermines the relevant strengths of the potential energies associated with theavailable modes of exerting generalized forces upon our representative blob. Thisis what Lagrange provides in his celebrated “principle of virtual work”: ourelement will remain in controlled equilibrium if and only if the “virtual works” ofall of the applied “forces” exactly balance out (the exact “rule” that suits asupposed element is determined by its internal composition, e.g., the stiffness ofthe various springs inside our shock absorber element). As explained before, afterour element gets reduced to an infinitesimal “point,” these “finite blob” rules willsimplify into a so-called stress-strain constitutive modeling rule for the element. As such, the stress σ winds up serving as a driving intensity that strives to alter theconjuagte “virtual” strain δε. In typical circumstances, this stress is “derivablefrom a potential” in the sense that there is a scalar function V of ε and alliedcoordinates such that the strength of the stress in the direction r is supplied byV/r. It is this “potential function” that justifies the sense in which a strainedblob stores a certain degree of “stress energy.”

In short, Lagrange’s “virtual work” techniques supply a finite-scale

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blueprint for how each of the little blocks without our off-the-manifold targetsystems will respond to applied mechanical forces. But we can extend theapplication of this rule to determine how a connected array of such blocks willreach an equilibrium if they are connected to one another and press on each otheracross their connected surfaces. That is, we claim that our entire collection ofconnected elements should locally compress or expand according to the overallpattern of altered blocks that best minimizes the summed strain energy of theentire, connected array.21 These composite equilibrium systems can be viewed asalso sitting on the Clausius surface, because they remain, overall, within tightmanipulated control through their bounding surfaces (= boundary conditions).

Let me pause to parenthetically explain why the “work” in Lagrange’sprinciple needs to be expressed in “virtual” terms. Consider the equilibriumposition of an old-fashioned seesaw. The traditional law of the lever says that, inorder to hold hefty Jack in equilibrium, Jill must choose the location L* such thatW.L = W*.L*. But a bit of trigonometry expresses this same fact in terms of thearcs, θ and θ* they would move through if displaced, giving rise to the “virtualwork” standard of equilibrium: the seesaw is in equilibrium if and only if anyattempt to displace Jack through an angle δθ would induce a corresponding Jilldisplacement of amount δθ* such that the work performed in moving Jack (W.δθ)would be exactly canceled by the contrary workrequired to move Jill (W*.δθ*). But a slight subtletyintrudes upon this simple picture (first noticed byDescartes, apparently). If Jill turns through a realangle θ, the effective force inducing her turningmotion will no longer be the full gravitational forceW*, because, as she turns, some of that becomeswasted in pushing her parallel to the plank. Theactual work expended in moving Jill through the angleθ is therefore supplied by a quantity of force We lessthan the full original W* (and is rather hard tocompute). But to mark out the equilibrium position ofour seesaw correctly, we want to retain W and W* attheir 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.22

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But the more vital need for approaching equilibrium in “virtual” termsemerges when we hook up lots of simpler mechanical systems into chains orlattices. To calculate the rest position of the entire array under assigned boundaryconditions, we must search through a large space of “virtual variations” to locatethe overall condition that minimizes the collective energy of the entire ensemble.23 How do we introduce some dynamic effects of an “old mechanical” stripeinto this picture? Answer: at each time ti, let’s now assume that each block sitscurrently in local equilibrium with itself according to the Clausius submanifoldrules, but not with its neighbors. To remain within the purified realms of “oldmechanical” behavior, we must keep the thermal properties of all of our blocksunchanging. As a result, our “local equilibrium” assumption largely24 reduces tothe fact that the summed traction vectors appearing on the boundary join betweenneighboring elements A and B will prove unbalanced at time ti. Clearly, the sidesof A and B are likely to move in such an unbalanced state. To important dynamicchange into his picture, Lagrange invoked d’Alembert’s celebrated principle: anyunbalanced side of a block must display an acceleration d2r/dt2 such that thequantity mr

. d2r/dt2 corrects for the missing deficit in the “virtual work” rule forethe system’s behavior at equilibrium. Here mr represents the measure of “inertiallaziness” associated with the generalized coordinate r, such as a mass or a momentof inertia. As such, the mr’s set the rates at which an individual block can adjustthe positions of its various boundaries over a time increase Δt. If we apply theseconsiderations to our connected partition of blocks, we reach the conclusion that,beginning at the time ti, our entire collection will have adjusted their boundarieswithin the short time period Δt in accordance with the pattern that best corrects forthe deficit in the minimized “virtual work” of the whole ensemble. Due to the

inertial sluggishness of their responses,each component block is unlikely toreach complete equilibrium by the time ti

+ Δt, but it will have managed to shiftsome share of its out-of-equilibriumproblems onto its neighbors. And werepeat this calculation again to advance to

the time ti + 2Δt, etc It isthrough such “spoil yourneighbor’s contentment”processes that pulses ofmechanical compression or

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expansion travel along a hammered rod (because we have not yet provided anyprovision for friction in these proceedings, the rod as a whole will never reach anyfinal equilibrium but will simply shift around little pockets of strain energy in theform of moving pulses). When our blobs are finally shrunk to “points,” this talk of“altered boundaries” turns into the strain (= distortion in both volume andskewing) exhibited within the infinitesimal unit. The rate at which the resultingwaves of compression travel along the rod traces to the “inertial laziness” of thematerial and supply the material with a characteristic “speed of sound” (which candiffer along different orientations).

Such is the procedure whereby standard Lagrangian techniques “paintdynamic arrows” onto our mechanically liberated system: we break our rod intolittle blocks and units of passing time Δt so that it will alter its condition as timepasses, providing us with a genuine “trajectory” (which become smooth as weshrink Δt to 0 and our blocks to infinitesimals). Note that the proper temporalmeasure of how rapidly these processes unfold derives from the “inertial laziness”whereby our blocks display accelerations; it is not information that lies encodedwithin our starting “virtual work” considerations. This is the formal considerationI had in mind earlier when I declared, pace the strange policies otherwise found inthe thermodynamics primers, that no real measure of “passing time” is definedupon the Clausius surface at all; that notion gets added later when we fill in the“dynamic arrows” for an evolving system. Instead, the states depicted on thesubmanifold merely serve as preimages to the stitching together of dynamic statesperformed off the manifold. (which is one of many reasons that we shouldn’tregard Carnot cycles as describing any kind of “process,” super-idealized or not).

In such a manner, we can spin off the continuous systems of the “oldmechanics” from the Clausius manifold. Since all of their thermal qualities haveremained under tight control, these trajectories all qualify as reversible: if system Scan mechanically evolve from condition A to condition F by passing through theintermediate states B, C, D, E, then another system S* can evolve in an exactlyopposite manner. However, all real life systems display some coupling betweenmechanical and thermal processes, in a manner that destroys this “idealized”behavior. Duhem, of course, believed that many scientists were attracted to“atomism” largely because of the aesthetic appeal of this simple “mechanics only”picture (“if we can only escape to the nether realms of frictionless behavior, lifewould be so sweet”). His most salient response to this inclination is to point outthat there is a plausible generalization of d’Alembert’s principle that can reach to amuch wider range of target systems without embarking upon a fairytale journey

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Gibbs

into lower scale mechanical purity.Specifically, to construct a more general thermomechanics (indeed, the

technique readily carries an enlarged chemical mechanics in its wake), Duhemproposes that we employ a thermal rule proposed by Willard Gibbs in the place ofd’Alembert’s principle. But obeying its dictates requires that we are able to linkour descriptive quantities together with the assistance of entropy in thestandard mode of orthodox thermostatics. But there’s the rub,because our old nemesis, friction, quickly threatens to intrude. Indeed, the real reason why we need to set up the Clausiussubmanifold at all is because we require a landscape in which entropycan link our descriptive variables together in the patterns we need freeof frictional meddling. So the extension techniques Duhem proposeswill not work plausibly for all conceivable physical trajectories, butwill function admirably for a significant number of evolving systems that stayfairly close to the Clausius manifold in their successive states.

How does he propose doing this? In analogy to Laplace’s employment ofthe “virtual work” principle, we should rely upon the energy allocations capturedwithin the system’s “equation of state” (which require appeal to the new notions ofentropy and temperature25 for their correct formulation). So far we have notintroduced any notion of “elapsed time” into our considerations. Gibbs’ rule, asoriginally articulated, claims that if two or more systems (A, B and C, say)maintained in their own individual equilibriums are joined together and are then allowed to reach a mutual equilibrium as the combined system A + B + C, theywill select, amongst all of the possible equilbria on the manifold to which theycould evolve, the final condition that displays the highest value of entropy thatmaintains whatever environmental controls still remain in place. That is, weexamine all the variations of entropy δS (= (1/T)δU + εi,/T:δ σi - (υij/T) δNij)and compute the arrangement of block states (U, σi ,Nij) whose combined entropyis greatest (note that this determination reflects a global variational principle,allied to the manner in which we employed Laplace’s principle of least work).

As such, this determination tells us nothing as to how long it will take beforethe newly joined system A + B + C subsides into equilibrium (in fact, to avoidturning on frictional effects, it would take forever). But we can imitate our earliertechnique of introducing a time measure by appealing to the empirical laziness ofinertial response by invoking the empirically measured relaxation times (orcapacities) associated with our non-mechanical intensive/extensive quantitypairings, e.g., the rate at which blocks at differing temperatures correct their

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differences or differentconcentrations of chemicals reachaccord. Just as before, we claim,that over a unit of time Δt, our chainof blocks will move as close toGibbs’ maximized entropy conditionas it can achieve given the material’ssluggishness of response whenconstraints on its behaviors arelifted (once again these involve aminimization over the whole bar).After reaching time ti + Δt, weperform these same calculationsover again to carry our bar to thetime ti + 2Δt and so on. Procedures of this sort are called “local equilibrium”assumptions because they presume that each small block of our material will neverbe pushed too far from equilibrium locally (small bits of matter straighten out theirthermal irregularities very quickly, whereas significant inhomogeneities can persistfor very long periods within large blocks of material.

By appealing to empirically determined relaxation times in this manner,Duhem can employ the Gibbs rule as a vehicle for painting genuine “dynamicarrows” on off-the-manifold-states in a manner similar to that employed in the “oldmechanics” (indeed, suitably interpreted, the old use of d’Alembert’s principle canbe subsumed under the wider Gibbs rule, as Duhem often stresses). Once again,our new dynamic trajectories become smooth as our partition blocks becomeinfinitesimal and Δt goes to 0. As the total entropy within our blocks ismaximized, each lefthand term in the expression (1/Ti)δUi + εi,/Ti:δ σi -(υij/Ti) δNij will drive an alteration in the term with which it is paired (which willoperate differently in materials of different compositions). As observed before, thecurrent stress εi,/Ti in each block (divided by the current temperature) serves as a“force” that the strain to alter and similarly for the chemical potentials codified inthe third term. But now consider the first term in the Gibbs rule. Formally, theinverse temperature 1/T acts as a parallel “force” that drives an increase in the“heat energy”component Q within the internal energy U (1/T appears because thehigher T becomes, U settles upon a higher value for Q, other factors being equal). Factors of this origin are commonly called “entropic forces” and will prove ofinterest to us later.

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In his own words, here is how Duhem himself articulates the blueprint of the“New Mechanics” program just outlined:

The notion of “virtual modification” lay at the root of the mechanics ofLagrange just as it remains at the base of the New Mechanics but how muchmore general it becomes in the latter context than the former! The onlyvirtual alterations that were previously considered were the alterations ofshape and position within the various parts of the system but alterations ofmany other sorts become considered within the New Mechanics... The latterdoes not restrict its attention to the study of what the Peripatetics called“local motions” [ = changes in bodily location at different instants] but italso studies many other kinds of motion [= changes in a quantity] whose richvariety enlarges the basic idea of “motion” to the huge extension thatAristotle recognized... This extension in the idea of “motion” requires anallied extension of its opposite, the idea of equilibrium... for one will nowconsider not only of the spatial equilibrium of a configuration, but also itsthermal, electrical and chemical equilibria as well. Thus generalized, thisextended notion of equilibrium will be the object of the New Statics.26

Unfortunately, we are not yet done, for we haven’t yet included thedistressing effects of friction within our picture. Now that we have managed toinstall a measure of passing time within our rod, we can start measuring the fluxesof quantities that flow between the blocks A and B, such as the rate dQ/dt at whichheat Q moves from block A to B or the so-called “rate of working” that registershow rapidly block A is making block B skew. Empirically, as these rates becomeswifter, additional entropy gets pumped into the outcome state (that is, each inputflux dQ/dt will produce extra entropy within block A according to some correctiverule of the form f(dQ/dt). This increased entropy adjusts the position of theoutcome state selected by Gibbs’ rule, an adjustment that tends to direct a greaterproportion of the outside work exerted upon our blocks into higher internaltemperatures (if controlling precautions otherwise are not taken). Such are theeffects we normally associate with viscosity and sliding friction and all realmaterials display some resistence to skewing (molasses does this more vigorouslythan water) and become hotter when rubbed. So our “entropy production rules”supply vital, irreversible corrections to our original Gibbs rule computations. Inmany circumstances, this entropy-derived resistence to skewing will look verymuch as if a regular mechanical force is acting (which is how friction was naïvelytreated in the simple “one-way dissipation” models considered earlier). But note acrucial formal difference: the strengths of these new friction-related “forces”

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usually depend upon a velocity (= flux) whereas the more basic forces treated uponthe Clausius surface do not (they are all “derivable from potentials” in the senseexplained previously).27 And it is exactly these new entropy-production “forces”that render the dynamics of real life systems irreversible.

By constructing a “dynamics” for our off-the-manifold trajectories in thismanner, the notions of “entropy” and “temperature” continue to make sense evenfor systems that are not in equilibrium; indeed, it is natural to say that entropyproduction represents the prime factor that drives real life systems towards theirequilibrium states (an assertion that corrects the simpler idea that frictional forcesin themselves do this). It also supplies a sound sense to the many “loose”formulations of the Second Law of the general form “the entropy of an isolatedsystem never decreases.” Furthermore, we now possess the conceptual tools tocoherently ask: “how closely can real life processes mimic the paths from A to Bupon the Clausius manifold?” Answer: as closely as we’d like in a limitingprocedure where all entropy production is reduced by slowing the rates of thefluxes. In my opinion, this asymptotic construal is the proper way to make senseof standard, but close to contradictory, claims that the paths upon the Clausiusmanifold represent “quasi-statical processes of an infinite duration.”

Observe that the platform upon which all of our constructions rely--theClausius submanifold--does not attempt to describe real life systems (even as“idealizations”), but instead articulates a network of descriptive entanglements thatprovide a skeletal mathematical framework upon which a large number ofprocesses can be draped. By such means, the thermomechanics of irreversibleprocesses deftly marks out a profound computational opportunity within thenatural realm that can bring a huge number of natural systems within the reach ofprecise mathematical tools. Duhem expresses this sentiment as follows:

Experiment proved that [our postulation of Carnot cycles and the like] wasuseful, that it was not a vain whim without a real object; that the line ofdemarcation it traced--and which could only enclose the smallest parcel ofland--delimits a vast and fertile domain.28

As I understand him, no real life trajectories live on the Clausius manifold itself,but, considered as an exoskeleton that winds through the full space of real lifesystem trajectories, it provides a supportive frame upon which “a vast and fertiledomain” of dissimilar processes can be successfully arranged.

One of the major reasons that Duhem’s murky prose is often interpreted incoarse ways (as “instrumentalism” or “structural realism”) is that he needs to walka fairly narrow methodological path in recommending the virtues of his new

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thermal physics: On one hand, he must defend the theoretician’s right to employstructures as odd and “unphysical” as Carnot cycles within her reasonings; But hemust also capture the sense in which the cautious manner in which Clausius andGibbs argue for the employment of entropy is far superior as trustworthy science tothe errant “theorizing” we find in Boltzmann or Maxwell.

(vii)

Duhem observes: The mechanics of systems of reversible modifications is shown to be apt forrepresenting, to a sufficient approximation, a great number of physicalphenomena. Does that authorize us to think that all the phenomenaproduced in inanimate Nature have to be set in order by this Mechanics?29

We have already noted that he answers “no” to his own question, although hestrives to reach as many physical phenomena as he can, starting from the basicClausius submanifold (including some further stages that I’ve not outlined here). Any dogmatism on this score can only derive from some “metaphysical” source, ofa possibly religious nature. And we will not have failed in our scientific missionfor all of that, as the engineering successes of modern extended thermomechanicsabundantly establish. Nor should we anticipate that descriptive constructions of athermal character will ever be totally abandoned in our scientific future: they maybe fortified with stronger ties to lower scale modelings (in the fashion of Essay 4,say), but not replaced.

In sum, Clausius and his successors have located a set of firm but mobilemathematical platforms from which descriptive tendrils can be extended to reachdeeply into the world’s many behaviors without pretending, necessarily, to capturethem all. On such a developmental policy, we might well wind up in a “finalphysics” situation like the one that Descartes anticipated: we enjoy an excellent setof descriptive tools that can ably capture the partially manipulated trajectories oflaboratory experiment and allied circumstances within a net of precisemathematics, yet which remain only hazily suggestive of the freely evolvinggeneric processes encountered in untamed nature. Almost certainly, this was thedescriptive endgame to which classical mechanics was irrevocably headed and wemoderns were spared that chastened fate only through the surprising interventionof quantum mechanics. But it would be a great mistake to presume dogmaticallythat the latter science (or its successors) will fully fulfill the formal expectations ofthe “theory T” crowd either (I noted earlier that there are counter-indications to the

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contrary in contemporary practice). Duhem’s narrative of out-of-equilibrium development also warns us to be

wary of contemporary assumptions with respect to the semantic fixedness ofthermal terms. A moment ago I described the Clausius manifold as providing a“firm but mobile platform” from which mathematically precise tools can bedescriptively extended. Why did I include the word “mobile”? Answer: becausewe must often scour nature to locate the special opportunities that can support theextensions of core thermomechanics and these can be found at virtually every scalesize. In this regard, we must correct, with an example, a serious philosophicalmythology widely accepted within philosophy today: the conceit that scientistshave discovered that the venerable notion of temperature can be equated with the“natural kind” having k ergs of mean kinetic energy per degree of freedom. Butthese same scholars rarely ask themselves “over what set are we taking a mean?”or“what qualifies as a degree of freedom?”30 And a formal answer to our “what set?”question requires that we must select, in accordance with the backgroundconditions of manipulation to which the target system is subject, one of the several“ensembles” found in the standard literature of statistical mechanics, along withadditional constraints expressed as Lagrange multipliers.31 As such, these“constrained ensembles” often consist of system variations that arise at a muchhigher scale level than occur within the simple interactions of molecules within anideal gas (which is the chief instance in which the crude “kinetic energy”identification works properly).

Why? The Gibbs’ rule of section (v) predicts, other things being equal, that atarget system will evolve to a condition of maximal entropy with respect to theapplied constraints--in those situations where the notion of entropy has found asecure perch. To understand why we must search for “secure perches,” let us askourselves, “Why is it likely that our Duhemian program for thermodynamics, assketched above, is likely to fail for many real life systems, unless its procedures arealtered in some substantive manner? And the answer lies in the “local equilibrium”assumptions we made in order to paint the “dynamic arrows” on our little boxesusing the Gibbs rule. Such presumptions are often not realistic:

In general, working and heating on the boundary of a body will createhavoc inside the body: a turbulent flow field and strongly inhomogeneousfields of mass density, pressure and temperature. We may say that theinternal equilibrium of the body is disturbed. But if the working and heatingare applied carefully and slowly, there will no appreciable flow field in theinterior and pressure and temperature will be essentially homogeneous

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while slowly changing in time.32

In fact, whether the tools developed here suit a real life process or not will dependcrucially on how whether the microscopic “relaxation times” of the variousadjustments occurring inside our blocks are fast enough in comparison to the largelaboratory scale times ΔtL whereby we clock the material’s measurable evolution. Such limitations don’t indicate that some distinct variety of “extendedthermodynamics” won’t be able to get a descriptive grip on some of these moreturbulent circumstances, but it’ll need to rely upon different bridging assumptionsthan “local equilibrium.” I’ll return to these issues in a moment.

Furthermore, it is a brute fact that many (indeed, most!) materials show verylittle disposition to “maximize entropy” if we judge “entropy” according to theensembles that we might construct at a molecular level (i.e., in the fashion thatworks ably for simple gases). This is because most solid materials display somedegree of “frozen order” that prevent them from “relaxing” into their molecularlycomputed “equilibriums” quickly. The classic exemplar is the diamond: a form ofcrystalline carbon whose predicted “equilibrium configuration” is that of graphite(= pencil lead). Strong energetic barriers internal to the gemstone make it veryhard for the material to randomly wiggle its way to graphite, resulting in anextremely long “relaxation time” (to the relief of jewelers everywhere). If onehopes to employ the tools of thermodynamics to materials affected by substantialportions of “frozen order,” one must construct appropriate “ensembles” basedupon configurational variations that occur at higher scale levels than the lowlymolecular.

A substance that illustrates the availability of such a “higher scale perch” ina remarkably simple way is an ordinary rubber band. This material consists ofhighly flexible polymer chains pinned together at various points but whichotherwise alter their shapes constantly with little energetic cost. So the statistical“ensemble” one constructs for rubber consists in all the higher scale shapes thatthese cross-linked chains might assume (we can think of the appropriate“ensemble” as a kind of “multiplicity cage” in which our chains can rattle aroundfreely). We can then apply the Gibbs rule to successfully calculate the “mean” restlength LR the chains will prefer at a given temperature. And the same rule will tellus the amount of outside force F needed to stretch the band to length LR + ΔL. Inconsequence, the band will pull on our fingers with that same force F. At firstblush, this phenomenon appears exactly the same on a macrolevel as the muchgreater elastic restorative force one experiences in dealing with a stretched steel

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band (don’t pull too hard or the metal will ceaseacting elastically). But there’s a surprise hiddenhere. The laboratory scale restorative force in ametal bracelet traces (mainly) to the strong forcesthat hold its molecules together but thecomparable factor within the rubber band stemsalmost entirely from its desire to maximize itsentropy. If we trace these effects to the Gibbsrule, we find that our two “forces” coordinate withtwo completely different terms, which is why therubber band force is often labeled as “entropic.”

Challenge in solids possessing other formsof “frozen disorder.”

Essay 5 documents a broad school ofcontemporary philosophy that has fallen victim to my “theory T syndrome”without recognizing in any way that they are ill. For such folks, a betterappreciation of Duhem’s core methodological cautions, purged of the crude “anti-realisms” and “instrumentalisms” in which they are commonly draped, may offer abracing tonic to diminish these dogmatisms and open a window to larger vistas ofpotential conceptual development. Of course, no one at present (certainly, notme!) can reliably augur whether a contrite “descriptive opportunism” of the sortsketched here will be required upon science’s final day of epistemic tally or not. But, surely, we shouldn’t wager resolutely on a more optimistic nag simplybecause we hope to found a present day “metaphysics” upon our future winnings.

Appendix. Hertz’s Challenge and the Underdetermination of Theory

Contemporary readings generally trace Duhem’s anti-realism strains to hisarticulation of what is now known as the "Quine-Duhem Thesis": the claim thatany cherished hypothesis H can be protected against empirical disconfirmationthrough blaming some other hypothesis H' utilized in its problematic applications. Indeed, Duhem sometimes writes as if simple modus tollens was sufficient toestablish this fact: (H & H') E E H v H'

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Yet, surely, this innocuous logical observation cannot adequately support Duhem'ssweeping claims that the choice of a physical ontology must represent, in the finalanalysis, a matter of metaphysics rather than physics proper. Yet it is preciselythese strong philosophical contentions that inspire the modern anti-realists whoevoke Duhem 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 conceit 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 ofmotions, perceptible or hidden, subject to the dynamics of Lagrange.33

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

It would be a matter of great moment for contemporary philosophy ofscience if Duhem’s example could be fully redeemed. That distinct theories existwhich are observationally equivalent in some strong sense represents a doctrine towhich many thinkers cling passionately but, arguably, represents a thesis for whichno indisputably sound examples have yet been proposed. Most proposed cases ofwhich I am aware either turn upon some misunderstanding of the relevant physicsor involve factors that should seem innocuous to the average scientific realist.34

Many philosophers of the logical empiricist era--W.V. Quine representing a primeexample--endorsed the undetermination thesis in the absence of concreteillustrations largely because they believed that any collection of "observationalconsequences" can be supplemented in many ways by inequivalent extensions. Buta “theory” is just such a logical extension, hence underdetermination follows.Today we generally feel that feel that this logic-based defense of undeterminationrelies upon an unsuitable picture of how real life theories produce their"observational consequences." It would be preferable if defenders of the claimcould produce some convincing examples from real life science. Withoutattempting to evaluate the merits of other contenders here, I will simply state thatDuhem's suggested pair of allegedly observationally theories represents as good anillustration of the expected behavior as I have encountered.

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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 ofessential idealization needs to be invoked in setting up the equations for acontinuous physical system property. These presumptions were surveyed in thebody of the essay in connection with the two-stage “finite blob to continuousmatter” techniques favored by A.E.H. Love and virtually every other methodologistof Duhem’s era (including Duhem himself). Continuous matter is mathematicallyself-similar on every size scale and if if one attempts to find guidance in setting upthe equations for, e.g., a flexible string, by inspecting smallish bits of it, one findsoneself merely inspecting a smaller version of the original problematic, withoutseeing any of the simplifications one usually hopes to find according to the popularmotto “physics is simpler in the small.” To halt this unprofitable regress and tolocate a platform upon which trustworthy reasoning of a finite-degrees-of-freedomcharacter can be practiced, scientists of the era assumed that we must intentionallymisdescribe matter’s qualities at a small size scale to permit the application ofNewton’s laws and allied mechanical principles. In particular, small “elements”within our string need to be falsely described as if they more rigid than they trulyare. Essay 7 discusses the provenance of these characteristic “rigidification”themes in greater detail.

As we also noted in the body of the paper, Duhem strived to articulate abroadly based thermal doctrine wherein thermodynamic notions such astemperature, entropy and internal energy could enter physics as primitive notionscoequal with mechanical concepts like momentum and kinetic energy. This“thermomechanical” framework should be sharply contrasted with the approachfavored in elementary modern texts designed for physicist,35 in which classicalmatter as treated as swarms of isolated point masses bonded together throughaction at a distance forces only. Within this rarified setting, temperature and itscousins are statistically (and somewhat shakily) founded upon more basicmechanical concepts. The point mass approach was overtly first formulated by S.J.Boscovich in the 1740's and was pursued, with varying degrees of loyalty, by theFrench "physical atomists" working near the beginnings of the nineteenth century. But Duhem and contempoaries usually dismissed the Boscovichean approach asempirically unsound, for reasons well appreciated in the late nineteenth century(e.g., one obtains the wrong number of constants in Navier's equations for anelastic solid). When Duhem writes of "the effects of motions, perceptible orhidden, subject to the dynamics of Lagrange", he alludes to a different variety of“old mechanical” approach, distinct from Boscovich's, that Duhem believes can be

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cyclic coordinate

rendered observationally indistinguishable from his own approach. Theorganization of conventional mechanics that Duhem has centrally in mind is due toHeinrich Hertz in his Principles of Mechanics of 1894 (although Duhem'sdiscussion can be extended to a wider collection of allied approaches). So let usreview a few characteristics of Duhem’s own work and contrast those with Hertz’approach.

Duhem's characteristic pathway to thermal physics relied upon extending thevenerable “principle of virtual work” to accept primitive conjugate pairs ofquantities that represent inherently thermodynamic forms of "work", such as T δS(where T is temperature and S is entropy) or μ δn (where μ is a chemical potentialfor some compound and n is its concentration).36 An “old mechanist” such asHertz will attempt to cobble by with just the pairings familiar from mechanicaltradition--F δy (force through a virtual distance) as witnessed above in the stringexample) and T δθ (torque through a virtual rotation). In other words, an element"free body diagram" for Duhem is allowed to be more"abstractly specified" (his term) in its behaviors thanstrict mechanists permit. As just noted, Hertz andDuhem first consider these variable pairings in arigidified finite-degrees-of-freedom setting beforemoving onto true continuously matter in an ill-definedlimit. I again stress that these methodologicalinvocations of “essential idealization” werecommonplaces within the era.

Hertz’ characteristic approach to mechanicsfocusses upon these virtual work pairings (Hertz himselfconsiders only “old mechanics” pairings such as F δy butDuhem sees that Hertz’ reasoning will extend to his ownthermal pairings as well). Developing earlier themes found in Routh, Helmholtzand J. Thompson, Hertz considered the large scale behavior of a mechanicalsystem if some subset of its descriptive parameters are cyclic--that is, their spatialpositions do not effect the total energy of the complete system. Example: considerthe spinning ring within a gyroscope. As soon as one "particle" within the ringrotates out of its present position, its place becomes immediately occupied by somesimilar neighbor. For all intents and purposes, the overall behavior of thegyroscope is utterly indifferent to the positions of the particles within the ring,although their collective angular velocities remain important, for they determinehow difficult shifting the ‘scope to another orientation will prove (the faster its ring

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spins, its resistance to displacement will increase–the spinning creates asimulacrum to inertial sluggishness). Hence the formal definition of a "cyclicvariable": a quantity whose velocities but not positions appear in a Lagrangiansuitable for the composite system. If we are unable to observe the insides of ourgyroscope directly--its innards are encased within an opaque box, say--, theinternal whirling will appear to supply a hidden source of potential energy thatmakes the box harder to move than it would otherwise prove. Without opening thebox, we can’t tell whether its resistence to movement is caused by some novelexternal force field or simply due to its hidden gyroscopic whirling. In otherwords, the unseen "hidden motions" of the gyroscope’s cyclic quantities display aremarkable capacity to imitate the behavior of an externalized force potential. Wemight parenthetically observe that cyclic behaviors of this kind were much studiedat the time–smoke rings were a staple of classroom demonstration–because popularspeculations with respect to molecular constitution (such as Kelvin’s “vortexatom” model) relied upon these inertia-like imitations.

We can generalize these remarks further. Let us place a puppet on theexterior of our box whose arms are somehow linked to the insides of our box.Suppose we find it is moving the puppet’s two hands closer together is unusuallydifficult. Does this resistance trace to an invisible action-at-a-distance force operating across the gap between the twoarms, to some connecting spring inside, or to the fact thatthe gyroscope acts as a flywheel that stores the energyrequired to oppose our attempts to press the puppet’s handstogether? Without opening the box, we can’t tell--thesethree arrangements serve as perfect mimics of one another atthe level of the puppet’s overall movements.

Hertz exploits the last alternative in his approach tomechanics. Take any standard Newtonian "action at adistance" force F derivable from a potential energy functionV (i.e., Fx = V/x). Its large scale behaviors canalways–note the generality of the claim!--be adequately imitated by positingsuitable hidden cyclic elements of flywheel type at a size scale below the one wepresently consider. In other words, Hertz hides the innards of his “puppet arm”action-at-a-distance imitators at a very small scale, rather than in some overt blackbox. He can then maintain that any observed “active force” is merely themacroscopic manifestation of whirling “hidden masses’ operating upon a lowersize scale. In consequence, the only “forces” that basic mechanics requires

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consider are the entirely "reactive" contact forces associated with geometricalconstraints (see Essay xxx for more on these). This reduction is conceptuallyquite important, because Hertz was troubled by the fact that contemporaneousderivations of the conservation of energy relied centrally upon “derivable from apotential” assumptions (closely aligned with Newton’s rather nebulous Third Law). But the “forces of constraint” commonly invoked in discussions of contactingmachinery (such as the mechanical underpinnings that Maxwell provided for theaether) generally need to be velocity-sensitive and cannot be derived fromconventional potentials at all. To remove this conceptual contradiction inmechanics’ foundations, Hertz exploited standard results on cyclic coordinates toshow that any apparent derivable-from-a-potential force could be adequatelyimitated by an additional set of hidden flywheel arrangements providing asimulacrum of “potential energy storage” such as the kinetic whirling of the hiddengyroscope supplies for our puppet’s arms. Often Hertz’ program is looselydescribed as "doing without forces," but his strategy is better described as one of“making do with constraint forces only.”

Such techniques pose a considerable challenge to Duhem's own relianceupon a broader class of potential energy functions. Can’t any of Duhem'sintrinsically thermodynamic forms of potential (say, the chemical potential μ) belikewise mimicked at a smaller size scale by invoking hidden “old mechanics”flywheels at a characteristic scale smaller than where Duhem has elected to"freeze" his finite-degrees-of-freedom “elements”? Can’t Hertz invoke thebraggadocio of Annie Get Your Gun: "Anything you can do, I can do smaller"? Or, to quote Duhem’s own words once again, isn’t it “always permissible topretend that these phenomena are the effects of motions, perceptible or hidden,subject to the dynamics of Lagrange?” On a macroscopic level, each setup imitatesthe other perfectly. And thus we have Duhem’s central exemplar of theunderdetermination of theory under all possible evidence.

Of course, Duhem regards implementing Hertz’ hidden arrangements as acolossal waste of time, but the “observational equivalence” of the two approachesshows that he cannot supply irrevocably empirical grounds for discouraging theapproach. Such nitty gritty reflections best explain why Duhem maintains that“taste” in the fundamentals of a physical theory ultimately represents a matter of“metaphysics” rather than empirics.

The fly in the ointment to all of this lies in the two-stage approach tocontinuum modeling that Duhem and Hertz both embrace. If the finite-degrees-of-freedom stages within their modeling procedures are regarded as reflecting true

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

2. Descartes: “Geometry should not include lines (or curves) that are like strings,in that they are sometimes straight and sometimes curved, since the ratios betweenstraight and curved lines are not known, and I believe cannot be discovered byhuman minds, and therefore no conclusion based upon such ratios can be acceptedas rigorous and exact.” Mathematics in Civilization p. 263.Find originaldiscussion. He considers curves generated by two or more independent parameters(such as spirals) to be ”purely mechanical” and beyond the reach of mathematics(Henk J.M Bos, Redefining Geometrical Exactness: Descartes' Transformation ofthe Early Modern Concept of Construction (Berlin: Springer, 2013)). His troubles

physical reality, the we simply require lower scale probes to determine empiricallywhich mechanisms are operative on those levels (or inside our puppet’s box). Onthe other hand, if our competing “force potential” rivals merely comprise “essentialidealization” way stations on route to genuine continua lacking true internal grainof the same nature, then both derivational pathways lead to exactly the same finaltheory and don’t represent competitors in any evident sense. In consequence, nohard bitten scientific realist should be troubled by Duhem’s central examples,because they trade essentially upon such methodological considerations connectedto continuous matter that we no longer accept, thanks to a good deal of twentiethcentury clarification at the hands of sophisticated applied mathematicians.

Nevertheless, Duhem’s concrete proposals for underdetermination illustratethe nitty-gritty manner in which useful discussion of methodological issues indescriptive mathematics should be prosecuted. As it happens, traditional “essentialidealization” theses stem from historically natural misapprehensions with respectto the mathematical treatment of continua, but broader unclarities with respect tothe descriptive capacities of applied mathematics remain afield (vide Essay 9).

That allowance being offered, the continuing popularity ofunderdetermination themes within philosophy, largely unsupported by specificexemplars in Duhem’s manner yet still invoking his name as an argument fromauthority,37 indicates that, in a very real sense, modern philosophy remains hauntedby the ghosts of departed physical infinitesimals beyond the reach of simple δ/εclarifications.

Endnotes.

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with “indefiniteness” in material flow are closely related (see Essay 9).

3. Hadamard’s views are discussed in essay 1, as well as XXX of WS.. In point offact, a standard treatment based upon steady state flow does impose an “analyticfunction” connection between the two pipes that seems excessive. Such concernsemerge in a quite troubling way within the optical circumstance sof essay 8.

4. In terms of the technical distinctions whose utilities I praise in essay 1, thesedifferences are formally marked by interactive potential energy terms V(q1, q2)centered upon material particle location and the more absolutist form V(r)articulated in terms of spatial location.

5. Michael Friedman, Kant's Construction of Nature (Cambridge: CambridgeUniversity Press, 2015) is excellent on all of these matters.

6. Modern approaches to mechanics based upon Hamilton’s principle supply thewrong variational principles required to handle non-holonomic constraints suitably. For Duhem, these technical differences become non-trivial because his owntechniques for introducing the arrows of dynamic change into previously staticcontexts build crucially upon the wider set of variational considerations favored byLagrange himself (which are virtual work based, not modern “Lagrangianmechanics”).

7. Peter O. K. Krehl, History of Shock Waves, Explosions and Impact (Berlin:springer, 2008) details Duhem’s important contributions to shock wave research. Insofar as I am aware, Duhem nowhere articulates the never-ending foundationalworries in terms of entropy and shocks in the manner presented here, but is contentto observe that one can’t estimate the speed of sound in air correctly withoutappeal to temperature. That observation is correct, but lacks the foundational biteof our unavoidable shock wave considerations.

8. Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum.From Augustus deMorgan, A Budget of Paradoxes; it is a recasting of a coupletfrom Jonathan Swift.

10. Observe that the “points” within a continuous material need to be denselyarranged with respect to one another, rather scattered about in the isolated andnon-contacting manner of Boscovich’s point masses. Articulating the behaviors of

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the former in a mathematically coherent manner is a far more exacting chore thanthe latter require and one automatically misjudges many central tensions within thehistory of classical mechanics if one blithely blurs these conceptual affairs togetherunder some vague rubric of “it’s all classical physics, isn’t it?”

11. Theoretical Mechanics (Cambridge: Cambridge University Press, 1906), p. 2. Love is a good example of an author who appears to espouse a Boscovichean pointof view, yet allows his "particles" to exert contact "reactive stresses" upon eachother, a posit that is not consistent with a strict mass point of view (cf. pp. 347-352). Love's later A Treatise on the Mathematical Theory of Elasticity (New York:Dover, 1944) is clearer on fundamentals and he admits, "The hypothesis ofmaterial points and central points does not now hold the field" (p. 14). His "NoteB" (pp. 616-27), however, suggests a lingering personal nostalgia for mass points. His "Historical Introduction,” incidently, provides an admirable précis ofnineteenth century controversies on thesew issues.

12. In the diagram, I’ve included some typical boundary movement constraintswhich occasion further conceptual difficulties that I’ll not trace here.

13. Gerard A. Maugin, The Thermomechanics of Nonlinear IrreversibleBehaviours (Singapore: World Scientific Series, 1998).

14. Complicating the conceptual problematic even further are the “greediness ofscales” concerns raised in Essay 4, which, in part, need to be addressed through thethemes of Essays 7 and 8.

15. Mark Kac complains: “ In kinetic theory volumes Δv ‘small enough to be takenas elements of integration yet large enough to contain many particles’ rendered[thermodynamics] unpalatable and even repulsive to a young mind alreadyconditioned to look for clarity and rigor. Quoted in T.W. Korner, Fourier Analysis(Cambridge: Cambridge University Press, 1989), p. 176.

16. Inaccurately, because the Clausius structure is not truly part of the broaderlandscape through which it wanders, but stands as a limiting locus to variousdriven and autonomous trajectories encountered within that target arena. For thisreason, the “reversible processes” that comprise the Clausius structure itself arenot true “processes” at all in any reasonable sense of the term, but merely theasymptotic traces of a large cluster of them.

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17. References on non-equilibrium mechanics

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

19. It should be remarked that the non-thermal trajectories of the “old mechanics”all qualify as reversible paths, but, on Duhem’s way of thinking, these paths areadded in later, appearing as the constant temperature asymptotes to curves thatcarry a Gibbs principle installed dynamic arrows (all of this terminologicalgobbelty-gook will be explicated later). The “constant temperature” provisomeans that only the d’Alembertian part of the Gibbs rule remains active.

20. I apologize for the repeated warning, which reminds me of the illustratedpicture book/sound recording combinations that I owned as a kid (e.g., “Bozo atthe Circus”). At the outset, the narrator would instruct us, “whenever Bozo blowshis horn, turn the page.” But every time that happened, he would immediately add,“That means ‘turn the page,’ boys and girls,” never trusting his audience toperform as instructed. Certain readers--I won’t tell you who you are, boys andgirls--always ignore my “you can skip this section” permissions on the dubiousgrounds that they “can understand anything.” But thermodynamics is aninherently tricky subject and may require fuller instruction for adequatecomprehension than I can supply here. And similar remarks apply to othertechnical passages that I must include for accuracy’s sake, yet may easily arrestnormal readers in their tracks if attended to wuth excessive diligence. Omission isnot always a mortal sin.

21. Since this equilibrium reflects an integrated pattern stretching across the wholeof S, it ia best specified by a variational principle such as the “virtual work”principle that Lagrange deploys.

22. To fully capture this stability criterion, we must look at the second variations ofthe displacements, but we’ll ignore this complication here.

23. I include these details because Duhem views the employment of “virtualterms” as emblematic of the non-descriptive devices that a physicistcharacteristically employs in her “theorizing.” Thus Evolution, 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.

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24. If significant “body forces’ such as electricity enter the picture, furthercomplications ensue, which I’ll ignore.

25. 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 PhysicalTheory (Princeton: Princeton University Press, 1962). These passages are oftensupplied with an excessive Quinean gloss within modern commentaries.

26. Evolution, pp. 116-7 I have silently modified Cole’s translation for sake ofreadability.

27. These technical issues become salient in the appendix to this essay, as well asthe additional discussion of constraint forces in the attachment to Essay 6.

28. Evolution, p. 165.

29. This continues the passage previously cited.

30. Nor have they asked, “Why should kinetic energy serve as the appropriatemeasure?” In fact, that simple identification depends upon “equipartition”assumptions that often fail in real life materials, in ways that suggest otherplausible platforms upon which the word “temperature” can effectively rest..

31.The continuum physics analog to these ensembles are the “spaces of variation”over which the Gibbs rule is applied.

32. Ingo Müller and Wolf Weiss, Entropy and Energy: A Universal Competition(Berlin” Springer, 2005) p. 3.

33. Evolution, p. 78.

34. 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). Also WS.

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47 Continued interest in classical continuum mechanics has become (except for afew special cases) the dominion of engineers and applied mathematicians, ratherthan physics departments per se. Hence the textbooks of the latter often fail toreflect the conceptual requirements of the former.

36. Pierre Duhem, Mixture and Chemical Combination (Dordrecht: Kluwer, 2002)and Thermodynamics and Chemistry (New York: John Wiley, 1913).

37. Bas van Fraassen, I am looking at you.