a loaded beam with fixed endpoints

Essay 3

From the Bending of Beams to the Problem of Free Will

Just as a finite little sum embraces the infinite seriesAnd a limit exists where there is no limit;

So the vestiges of the immense Mind cling to the modest bodyAnd there exists no limit within the narrow limit.

O say, what glory it is to recognize the small in the immense!What glory to recognize in the small the immensity of God!

---Jacob Bernoulli1

(i)

Early travelers often appreciate the charms of a landscape more vividly thanthe settlers of later years, who gaze upon the encircling splendors with a dulled andacclimated eye. G.W. Leibniz, among his many singular accomplishments, was oneof the first scientists to attempt physical modeling with equations posed at aninfinitesimal scale (i.e., differential equations) and was acutely aware of themethodological oddities involved in obtaining formulas of this intenselyconcentrated character. In particular, he was forced to confront these concerns inhis 1684 work on the elastic response of loaded beams (an important scientificsubject that Leibniz pioneered2) and many of the strangest features of his developedmetaphysics can be directly related to considerations that arise within suchendeavors. FIG: A LOADED BEAM Through inattentive familiarity we modernstend to skip over the peculiarities of thesesame procedures with scarcely a pause,because we have become accustomed tothem. Only the renewed rigors that comewith modern computing have revivedLeibniz’ old concerns. Essay 5 will revisit these same issues in exactly this light. Here we shall attend solely to Leibniz’ own context.

The mathematician J.E. Littlewood once published a wry essay entitled“From Fermat’s Last Theorem to the Abolition of Capital Punishment” whichtravels an improbable bridge between prosaic worries about mathematical functionsand weighty moral matters.3 Well, Leibniz made an analogous journey from worries

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Leibniz

about bending beams to some astonishing conclusions with respect to free will. Anyone familiar with his writings realizes that his views about the constitution ofmatter sound--let us not mince words--quite crazy, for he contends that the materialuniverse is somehow constituted of a densely packed and nested array of “monads”that don’t truly live in space and time, yet they control everything we see.

[S]hape involves something imaginary and no other sword can sever theknots we tie for ourselves by a poor understanding of the composition of thecontinuum.4

These monads behave like “little animals” in possessing desires, perceptions andactions that aim at furthering such ambitions. Furthermore, the entire materialworld--including the rocks, the iron girders and water, as well as organic stuff suchas wood, mosquitos and human beings--are controlled by these animal-like things,which congregate in great colonies ordered under obscure master/slaverelationships.

As an instructor in a public college, these are not the sorts of conclusion thatyou want your students to report about to the folks backhome (“in your philosophy class, you learned what...?”). Yet the remarkable fact about Leibniz’ thinking is thatmany of these strange opinions trace directly to perceptiveviews on sound modeling practice with respect tocontinuous materials. FIG: LEIBNIZ Perhaps a usefulway to appreciate some of Leibniz’ grander flights ofmetaphysical fancy is simply to reconnect them to thehumble–yet surprising–considerations with respect towooden and metal beams with which they were originallycommingled. If we succeed in this endeavor, ourappreciation for the remarkable depths of Leibniz’ thoughtshould increase accordingly, for modern commentators rarely link his metaphysicaloutpourings with concrete engineering technique. Instead, they often patronize hisphysics as deeply inferior to that of Newton. The supplementary consideration that,by advancing just a few steps further, a full Littlewoodian arc to the free willproblem can be completed should only increase our admiration further, through thatslightly comedic overreach that truly great philosophy often evinces.5

Of course, a proper form of Leibniz interpretation will consider all of themultifarious influences that shape his thought, whereas we will largely trace a purist“mechanical” line here, one that adheres fairly closely to the ways in whichengineers still go about modeling systems like loaded beams. Why? Because this

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aspect of Leibniz’ thought comprises a vital figure in the carpet commonlyoverlooked by scholars unfamiliar with the conceptual complexities of continuummechanics (aka the physics of continuously flexible matter).

Let me add a significant interpretative caveat before we begin. Insofar astheir physical behaviors go, the attached limb of a living tree and a steel beamdisplay elastic responses of a roughly similar character, save for the material modulithat render wood more pliant and anisotropic than steel. In particular, bothmaterials display the teleological capacity to “remember a natural state”characteristic of all solids (this is the the topic of section (ii)). We’ll later find that achief structural motivation for monads is that they supply Leibniz’ physics with thebackground support that continua require to maintain their internal coherence underdilation and compression.6 Because he also assigns his master monads variousspiritual characteristics, we reach the impious conclusion that steel beams shouldpossess lowly monadic “souls” comparable to those attached to living trees or dumbanimals. To block this unhappy conclusion, Leibniz must claim that steel beamsonly act as if they possess controlling monads in a manner we shall outline later. Because my chief purpose is to highlight the overlooked cogency of Leibniz’thinking about the structural requirements of flexible matter, I will largely ignoreLeibniz’ own discriminations between the branches of living trees and dead piecesof steel, trusting that my readers can fill in the required theological concessions forthemselves.

Once the physical and mathematical dimensions of Leibniz’ thought becomeclearly isolated in this manner, their direct affinity with many of the methodologicalconcerns raised elsewhere in this collection of essays will become manifest. Insome cases, my own route to recognizing their cogency stemmed directly fromreading Leibniz.

(ii)

The route he followed to reach a modeling equation suitable for a loadedbeam operates within the framework of explanation that Leibniz calls “the kingdomof ends or final causes,” indicating that he will employ some measure of “desiredstate” or teleology in his deliberations (later we shall consider how he intends toundergird such appeals within a parallel “kingdom of efficient causation” that

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the “memory” of a wooden beam

operates upon a lower scale size). It is in thiscontext that his odd appeals to thedesires, perceptions and actions of inanimatematerials appear. But these invocations of“personality” are not as strange as they firstappear, for there are sound structural reasons whythis vocabulary naturally suits the behaviors ofwooden and steel girders. FIG: THE “MEMORY” OF WOOD If you open anymodern primer on materials science, you will find its author describing everydaysubstances in a similarly anthropomorphized vocabulary of “memory,” “desire” and“perception” (true, no “little animals” are mentioned, but we’ll get to the monadslater). Why? Well, ignoring friction, beams behave like elastic solids-- theypossess natural equilibrium states to which they will always strive to return afterbeing bent or poked. Thus a 4x4 beam sagging under a load of rocks will struggleto regain its unloaded straight state; in lieu of achieving that, it settles for a secondbest condition called constrained equilibrium. For strategic reasons we’ll discusssoon, Leibniz’ immediate scientific objective is to characterize the staticconfiguration of this loaded equilibrium state; he does not ask how the structuremoves under its loads. If the material is afflicted with a so-called fading memory (asmany woods and plastics are), its ability to “remember” its original state of moldingdiminishes over time and it only regains a compromised and weakly curved end-state intermediate between its current loaded-with-rocks condition and its erstwhilestraight state. Even today, material scientists speak of the varying sorts of“memory” that distinguish various materials. For example, a truly elastic soliddisplays a perfect memory of its rest configuration, to which it will return to as fastas it can,7 after it is released from any binding constraints. In contrast, manymaterials display a fading memory in that, if maintained in a constrained position fortoo long, they gradually adapt to the shape that has been impressed upon them ().8 A normal fluid such as water retains no “memory” at all of its former configurations(e.g., the bottle from which it was poured), but it can detect the velocity with whichit slides by its neighbors in a manner that our perfectly elastic beam cannot. Morecomplicated materials display mixed “memories” of both types (toothpaste, forexample, retains a weak memory of the tube in which it was originally confined andgradually puffs up in a feeble attempt to resume those former proportions–“dieswell” is the official jargon for this behavior). Modern manufacture has constructedstrange “smart materials” that try to regain different forms of earlier conditiondepending upon their temperature –so they can “remember” two or more distinct

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earlier states.9 And so on. How vigorously and swiftly solids prosecute theseinherent goals reflects their defining personalities as materials. A rod of steel resistsbending far more strenuously than one of tin, although both may strive for restconfigurations that are geometrically identical. As Leibniz stresses, suchcharacterizations of “material personality” are inherently teleological in the sensethat they are determined by a system’s displacement from final ends.

Leibniz was aware of the wide range of memory behaviors witnessed in realmaterials but he fastened upon the simplest form of teleological recollection withinhis beam model: the Hookean10 capacity of a spring to exert a restoring force indirect proportion to the degree it has been stretched or compressed away from itsrest configuration. In a moment, we’ll see how he constructed a plausible beammodel upon this simple basis. To this day, the same Hookean assumptions areutilized by engineers whenever they deal with beams within the so-called linearregime where they sag only slightly.

Why did Leibniz do this? Because steel, at least, behaves in a simpleHookean way on a testing bench under moderate loads (whereas wood is stronglyanisotropic and requires more entangled equations). Mathematically, things arequickly going to become tough enough without importing significant non-linearity oranisotropy into the picture!

Leibniz viewed the behaviors of billiard balls in a similarly elastic manner,except that we are now considering objects that interact strongly in a kineticmanner–that is, through motion–rather than lying on one another statically in themanner of a quiescent beam laden with rocks (Leibniz distinguished between“living” and “dead” loads). Compressive collisions between elastic solids are verycomplex and quantitative conclusions require the assistance of a modern computer.11 Accordingly, Leibniz could not work out the dynamic behaviors of colliding billiardballs in any detail, despite the fact that their governing physics rests upon the samebasic assumptions of elasticity and memory of natural state that he utilizes in histheory of statically loaded beams. Indeed, direct observation of slow elasticcollisions--e.g., between beach balls--readily confirms that impacting balls compressand reexpand when they collide. So we moderns say that when ball A bumps intoball B, their incoming kinetic energy is temporarily converted into internal potentialenergy within the two balls, depending upon the degree to which their contours havebeen distorted. After these compressions reach a critical level, each ball will pushback against the other by transferring its unwanted strain energy into new forms ofmoving energy, including spinning. In describing these interactive events in“personality” laden terms, Leibniz correctly emphasizes the fact that notions such as

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suppressing compression in the Wren-Huygens treatment

strain energy (and potential energy more generally) are teleological through themanner in which they appeal to natural rest states for the materials.

To be sure, Leibniz’ own Aristotlean influenced vocabulary for thesearrangements is rather hard to parse:

The dynamicon or power in bodies is twofold, passive and active. Properlyspeaking, passive force constitutes matter or mass and active forceconstitutes entelechy or form... Furthermore, active force is twofold,primitive and derivative... Primitive active force, which Aristotle calls firstentelechy and one commonly calls the form of a substance, is another naturalprinciple which, together with matter or passive force, completes a corporalsubstance... [D]erivative and accidental or changeable force will be a certainmodification of its primitive power that is essential and endures in everycorporeal substance... Active force involves an effort or striving towardsaction, so that, unless something else impedes it action results. And properlyspeaking entelechy... consists in this. For such a potency involves act anddoes not persist in a mere faculty, even if it does not always obtain the actiontowards which it strives, as of course happens whenever a hindrance isimposed... Moreover, through derivative force, primitive force is altered inthe collisions of bodies, namely in accordance with whether the exercise ofprimitive force is turned inward or outward.12

I’ll explicate what some of this means in more concrete terms later on.On this score, it is important to recognize that the prevailing models of impact

favored in Leibniz’ time (i.e., as devised by Huygen, Wren and Newton himself) donot follow this compressive model at all, but practice a rather brutal form of physicsavoidance--they assume that the balls instantaneously rebound without alteration ofshape! More accurately, they employ a matched asymptotics methodology thatartificially collapses the breadth of the interval Δt* in which the impactive eventsoccur to zero duration 0. FIG: SUPPRESSING COMPRESSION WREN-HUYGENS In so doing, their scheme effectively cuts off all consideration of thedetailed events that transpire on a time scale swifter than Δt* and, at best,characterizes their effectsthrough crude rules of thumbsuch as Newton’s coefficientsof restitution.13 Suchworkaround techniques canonly be regarded as convenientcomputational tricks and

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Leibniz believes that their zero-duration-of-impact approach violates the vitalprinciple that “nature does not make jumps.” Leibniz writes to Bernoulli:

For otherwise that great, and it seems to me inviolable, axiom that governsnature,..., will not be observed [that]... I call the law of continuity. ..It says inchanges there are no leaps, and, consequently, that there is no assignablechange in an instant... [This behavior] is observed no less in changes ofdegree than in changes of place. Moreover, this avoidance of leaps in thechanges of bodies is due to an elastic force existing in them. This is how ithappens in collision that by gradual movement bodies that compress oneanother and then restore themselves yield to one another little by little andconserve their direction and force and, as you have seen demonstrated, thequantity of their motive action (which is very different from quantity ofmotion as ordinarily understood).14

Of course, fully tracking these complex interactions even within idealizedcircumstances represents a daunting computational challenge that wasn’t reallyfeasible until recent times. In the beam model we shall study, Leibniz concernshimself with an entirely static situation (the shape that a beam will eventually assumeunder a load of rocks, for example), thereby bypassing any consideration of thehorribly complex processes of energetic exchange and loss that must transpire whena beam first settles into a new position of constrained equilibrium.

(iii)

The central purpose of this essay is to explicate how Leibniz fits together histwo kingdoms of explanation (viz., efficient causation and teleological explanation). But such themes cohere nicely with more modern themes, such as the greediness ofscales issues discussed in Essay 5. We shall take up the two kingdoms theme insection (iv). Before doing so, let us first outline the concrete manner in whichLeibniz approaches the modeling of a loaded beam, for this exercise can greatlyilluminate many of his characteristic metaphysical categories through unpacking whatthey demand of cold steel and wood. Conceptually, Leibniz focused upon materialsthat seemingly remain continuous and flexible on every size scale, a subject that wenow call classical continuum mechanics. Unfortunately, many Leibniziancommentators on Leibniz have confused the formal requirements of flexible bodieswith a simpler form of classical physics that is based upon point masses that attractone another across distances.15 In doing so, they fail to recognize that the physics ofcontinua represents a rather subtle subject and that the proper mathematical tools for

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downward scaling to an infinitesimal level

articulating some of its requirements were not developed until far into the twentiethcentury. Leibniz is commonly patronized for supplying a “hazy metaphysics” ratherthan a “robust physics” in the manner of Newton; so it is important to recognize that,except for the final push into Monadologyland, virtually every “hazily metaphysical”notion Leibniz invokes find a match within the perfectly pellucid notions that modernapplied mathematicians have developed for the sake of beams and billiard balls. However, the detailed mathematics required to capture these ideas clearly took along time to develop. Virtually all pre-twentieth century authors–Leibniz included--were forced to cobble by employing misty appeals to “infinitesimally smallelements” and the like. Amongst other difficulties, continuous matter exhibits theunpleasant property of not becoming simpler at smaller size scales, which leads to adaunting foundational recess. To evade this difficulty, most pre-twentieth centurywriters on continua invoked some form of “science always idealizes” thesis, in amanner outlined below. From a modern vantage point, these appeals merelysubstitute dubious “philosophy” for stretches of mathematical exposition that theycouldn’t yet supply. So the chief fault with Leibniz’ physics was not that it wasinherently “hazily metaphysical,” but that, through its reliance upon infinitesimallysmall elements, it trafficked in hazy mathematical constructions that it couldn’t fullyredeem in a modern way.

Although one cannot fully appreciate the depths of his approach to flexiblematter without a firm appreciation of the traditional problems of the physicalinfinitesimal, these aspects of our discussion are relegated to an appendix, because Iwant to highlight a different aspect of Leibniz’ thought. In setting up any differentialequation modeling (of PDE type), two peculiar forms of downward scaling argument

are traditionally cited: (1) thoserequired to invoke the physicalinfinitesimals just mentioned; (2)those involving the suppression ofirregular events arising upon smallscale sizes. FIG: DOWNWARDSCALING It is the second that willlargely concern us here, but thereader should bear in mind that someof the reasons why Leibniz regards

physical description at the level of spatial points as idealized derive from (1) as well.

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three versions of beam element

For present purposes, we merely need to appreciate that, at the infinitesimallevel, the elements upon which we build our differential equation models cannot besimple points but require a fair amount of internal structure.16 In particular, theyrequire infinitesimal “sides” upon which distinct tensions can push and pull. I haveillustrated three specimen candidates for this role: (a) Leibniz’ own proposal (this isan annotated version of a diagram he supplies); (b) its later improvement at the hands

of Johann Bernoulli andLeonhard Euler and (c) ahypothetical point mass modelinvolving action at a distanceforces. FIG: THREEVERSIONS OF BEAMELEMENT (b) is widelyemployed today as the basicsimple beam model and wouldhave been happily embraced byLeibniz had he known of it (heguessed that the element’sneutral axis lies at the bottom ofthe element, but could have

been easily persuaded that this is an error). (b) assumes that the downwardgravitational force generates a torque that is balanced by the various tensions createdfrom compressing and stretching an array of Hookean “fibers” stretching across eachparallel slice of the beam. All of the subsequent argument unfolds equally well if wesubstitute the more familiar (b) for (a) in the discussion ahead. But we should notaccept any non-continuum model like (c)17 as a suitable Leibnizean element, becauseit does not contain any matter that is inherently springy (either as literal springs as in(a) or as the elastic fibers in (b)). This inherent springiness is central to Leibniz’thinking, for a Hookean spring displays the simplest form of material teleology--itimmediately strives to return to its natural state in a manner linearly dependent uponits displacement from that condition. The methodological issues discussed in theappendix already incline Leibniz to a thesis widely shared by traditional modelersworking in continuum mechanics: that the infinitesimal level of description recorded

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symbolic “elements”

in a differential equation modeling properly reflects the larger scale behaviors of thetarget system in a “symbolic” or “generative” manner.18 This is the reason, I think,why Leibniz presents his own element in the functionally apportioned manner hedoes, where the springs symbolize the elastic response; the little block, the appliedload; and the big block, the inertial “laziness.” FIG: SYMBOLIC “ELEMENTS” Allied representations remain popular in engineering today (a standard “symbolic”representation of a Kelvin-Voigtelement for viscoelastic behavior isdisplayed). In neither case should theseallocations of function be regarded asgenuine structural pictures of the targetmaterial at a lowly scale size. Much ofLeibniz’ confusing terminology withrespect to the various species of“primitive and derived force” can beclarified through these symbolicapportionments, although we shall notpursue those alignments here.

(iv)

The crucial features of beam elements (a) and (b) (as opposed to (c)) lie in thefact that they remain permanently welded together and that their potential movementsand equilibriums reflect cooperative activities stretching across the full extent of thebeam. Such cooperation reflects the statical circumstances that Leibniz investigates;we expect that the beam will settle into a condition where it can bear its burden ofrocks with the least amount of overall bending energy. In engineer’s jargon, this iscomprise’s the system’s constrained equilibrium under the assigned loading--theshape that the beam will assume once it stops wiggling. These demands are mostnaturally codified in variational terms, in which we fictitiously carve our completebeam into little elements of an equal finite length ΔL (I employ five such pieces inthe illustration).19 FIG: AN IMPROVING SEQUENCE OF APPROXIMATIONS If we consider all of the possible ways in which such a jointed assembly might sagbetween two fixed piers, we obtain a space of statical beam possibilities. As welook over these arrangements, we seek the special configurations C (there may bemore than one) which can support its allotment of weights W with the least amount

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an improving sequence of successiveapproximations

of overall bending. If we merely had to balance asingle local weight wi enclosed by two walls, wecould readily compute the required amount ofstretching from our little beam model. But our fivelittle elements are chained together, so thatminimizing the spring stretching within element Eimay easily make the tensions worse within element Ej(in modern terms, lowering the local strain energywithin Ei increases the strain energy within Ej). Suchconsiderations supply the defining condition for theacross-the-entire-beam cooperation we require. Afive-membered array of individual elementconfigurations E

should prove stable in the sense thatif any local element E

j attempts to lessen its localtensions by shifting to a more relaxed configurationE

j *, the springs within the other elements will pull jback to its original E

position. Like the ThreeMusketeers, a constrained equilibrium staterepresents a condition where the maxim “one for alland all for one” applies. Such considerations suggest

that the natural way to locate the optimized E state is through quasi-experimental

tweaking--start with an arbitrary guess E0 (illustrated in the top of the diagram) andtwiddle with some selected component E0

j to see if the chain’s overall energeticbudget can be reduced by tweaking E0

j to become E1 j. If this happens, incorporate

this alteration into a revised guess E1 and repeat the corrective testing. We do thisover and over until our successive approximations (hopefully) approach some final,optimally lower tension configuration E (this quest may require an infinite numberof steps). The old-fashioned name for this kind of approximation process is “arelaxation method” and the set of possibilities through which it searches forms arelaxation space.

Technical remark: computational circumstances of this variational characterpop up frequently throughout this collection.20 Formally, the total amount of strainenergy (= total amount of spring stretching) stored within a configuration En servesas a “norm” |En| that insures that our tweaking procedures are contractive in thesense that if a E*j guess within En is adjusted to a new value, then the norm of |En+1|will decrease. If we do this repeatedly, we should obtain our desired equilibrium E

as a fixed point, that is, a condition that cannot be improved through further

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obtaining a flexible curve in afinal limit

alterations. Reasonings of this sort are called variational because of their relianceupon step-by-step variations upon an initial guess.

Thus far we have only searched for the relaxed state of a beam divided intofive segments of a fixed length ΔL. But we immediately recognize that our beam’senergetic relaxation might be further improved if it were divided into elements ofshorter length ΔL*. So let’s now seek a better equilibriumconfiguration by searching within a relaxation space thatincludes all of these ΔL*-segmented possibilities as well, forevery length choice ΔL*. Once these supplements are allowed,we can “complete our space” by tolerating continuous beampossibilities without any segmentation or internal springs at all. FIG: OBTAINING A FLEXIBLE CURVE AS A LIMIT Inessence, we assign a continuous beam an ersatz measure ofspringiness determined by the true springiness within thesegmented possibilities to which they lie near in our norm.21 Leibniz realizes that, to equip a flexible beam with therequisite properties, we must sneak up on that condition withina relaxation space comprised of preliminary models that arenot fully flexible below an element length ΔL. Through thispeculiar bootstrapping procedure, we resolve the basic paradoxof continuous matter that has created so many headaches for the scientists who havestruggled with flexible matter in a classical context. Real world materials appear tobe smooth and throughly flexible in their qualities, yet we humans can’t obtain aworkable handle upon their governing physics without first pretending that suchmaterials decompose into artificially kinked and less pliant “elements.” In otherwords, flexible beams need to be assigned a measure of springiness even though theycontain no springs! Instead, they must borrow this measure from their artificiallykinked cousins. See the appendix and Essay 8 for more on this issue.

By shrinking progressively shorter beam models in this manner, we reach thestandard Bernoulli-Euler beam equation in the infinitesimal limit: EI d4h/dx4 = W(where “I” supplies the block’s moment of inertia and “E” is the overall modulus ofelasticity for the fibers). In plain English, this equation states that in equilibrium ourbeam will everywhere curve to the requisite degree required to balance the localweight.22

Observe that the possibilities through which we search for a constrained

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Dr. Pangloss

equilibrium are entirely static in character; our relaxation space does not contain anypossibilities that change in time.23 Appreciating how such variational considerationsoperate is important to a proper understanding of Leibniz because his critics, ineffect, frequently evaluate “best possible world” within the wrong “space.” In

Candide, Voltaire inspects the sorry trajectory that compriseshuman history and complains: “Look at all those earthquakes,plagues and wars: how could these arrangements beoptimal?” FIG: DR. PANGLOSS However, Leibnizconsiders his own optimizations in relaxation space mode. At any moment, the world is comprised of a lot of interlacedelements that strive to achieve their own localized desires inthe same manner as a beam seeks its optimal equilibriumstate. How can these sundry ambitions be mutuallyaccommodated in a maximal fashion? The “best possibleworld” reached under this scheme may scarcely prove“optimal” in a Panglossian sense; there are lots ofundeserving layabouts with rotten desires who figure equally

in the maximizing, and interlocked inertias generally make us overshoot our goals inany case. As we’ll see, this flavor of optimization is closely linked to the problem offree will. God strives to maximize everything’s desires, including those of the badpeople and the lesser “desires” revealed within the constrained equilibrium strivingsof wood, steel and rocks.24

At this stage of my argument, the suggested reading of Leibniz on optimalityprobably strikes the reader as too “cute” to be historically appropriate. After severalmore turns of the screw, we’ll be able to make closer contact with his actual remarkson necessity and continency.

(v)

Let us now turn to the scaling considerations I characterized under (2) above. The preceding methodology tacitly presumes that materials such as wood or ironrespond to pushes and pulls according to the same behavioral rules at every scalesize, no matter how minute. We might call this an assumption of complete downwardscaling. But neither Leibniz nor any modern scientist believes that real worldmaterials behave identically at all size scales. If we inspect wood or steel under amicroscope, we find that it is comprised of a maze of cells or minute grains thatindividually stretch and dilate according to far more complicated rules than reveal

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Hooke’s drawing of cork

themselves within larger hunks of the material. Small hunks of a beam no longerrespond to pushes and pulls in a simple Hookean manner. Here Leibniz’ thinking was inspired by contemporaneousdiscoveries in microscopy; he knew, from Hooke’sdrawings, that wood was composed of tiny cells, forexample. FIG: HOOKE’S DRAWING OF CORK Despite the fact that the simple stretching rules we assignto a Bernoulli-Euler beam fail after its little elements fallbelow a critical cutoff level ΔL, Leibniz correctlyrecognized that we should nonetheless push ΔL all theway to zero in extracting our finished differential equation formula, for a tractablesimplicity emerges in this asymptotic limit, just as simple probabilities also appearwithin infinite population extensions. The upshot is an odd methodologicalconundrum, closely allied to the greediness of scales concerns of Essay 5. In workingupwards from a zero-length differential equation model, its predictions will supplyquite lousy descriptions of what occurs within extremely tiny spans of wood or iron,but their successes dramatically improve once we reach larger scale lengths. Inconsequence, our model beam equation should not be viewed as a formula thatcaptures “what really happens” within a material on a very small scale, but merely asa shorthand formula that generates sound results when applied to sufficiently longscale lengths.

In other words, our beam equation represents the downwardly projectedexpression of a simplified “personality” that its target materials manifest only atconsiderably longer scale lengths. This observation can be connected to deeperconcerns involving descriptive coherence. If a beam were actually composed of littleelements of the type just examined, we might wonder whether their individualarrangements of springs and their Hookean strengths might vary haphazardly fromone element neighbor to the next. But if such variations were tolerated at theindividual element level, our composite beam would easily rip apart due to theseinhomogeneities. On the other hand, we are familiar with materials whose Hookeanelasticity varies from one point to another, e.g., a rubber sheet that stretches moreeasily within one sector than elsewhere. But this allowance reflects a continuousvariation in material properties at the macroscopic level (“nature does not makejumps”) and requires that the springs that we attribute to our little elements mustcoordinate with their neighbors in a manner that will sum to a continuous variation atthe macroscopic level.25 When Leibniz writes of materials like wood or steel as“machines,” he intends to highlight this harmonious cooperation between scale sizes.

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efficient causation account of elastic response

As we’ll later see, many of Leibniz’ strangest pronouncements about space and timetrace to insightful observations with respect to the required compatibility amongst hiselements.

This basic theme--calculus formulas represent simplified, downwardlyprojected generators of larger scale behaviors--is central to Leibniz’ elaboratemusings upon the “metaphysics of the calculus”; differential equations representshorthand, asymptotic formulas that capture a material’s characteristics only down tosome indefinite choice of scale size.26 We moderns normally view physics’ mostfundamental differential equations in a more favorable light, as directly registeringnature’s workings at an infinitesimal level. However, when we consider the standardequations for the classical continua of everyday life (beams, strings, fluids, et al.), wetacitly shift to Leibniz’ point of view, for their validity can only be understood asresulting from a downward projection of homogeneous patterns witnessed at large-to-middling scale lengths.

Leibniz’ concerns with efficient causation are directly entangled withreflections such as this. Earlier we observed that Leibniz’ model for a beam directlyembodied “final cause” factors within its construction: the fact that springs display aHooke’s law propensity to return to their natural rest state. But, like most writers ofhis era, Leibniz believed that these capacities for elastic return must be foundedwithin some assemblage of a recognizably mechanical nature. Specifically, Leibnizembraced Descartes’ “air sponge” theory of elastic rebound.27 Consider an elasticpiece of wood: from whence does it derive its“spring”? According to Descartes, its interior isriddled with penetrating pores through which asuper-mundane “air”28 continually circulates.FIG: EFFICIENT CAUSATION ACCOUNT When we squish the wood, we drive this airfrom the pores and, when we stretch it, thepassages enlarge and an excess of air rushes in. As this happens, a corrective air flow will pushthe pore walls back to their originalconfigurations, rather as a dried sponge regainsits shape when we allow water to seep into itscrevices. Here is one of the many passageswhere Leibniz endorses this account of the underpinnings of elastic rebound:

I hold all the bodies of the universe to be elastic, not though in themselves, butbecause of the fluids flowing between them, which on the other hand consist of

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air particle compression during contact

elastic parts, and this state of affairs proceeds in infinitum.29

Through this air-assisted mechanism, the wood acquires a “memory” of itsnatural equilibrium state through efficient causation mechanisms alone (i.e., non-teleological stories that rely upon geometry and conservation principles alone). Forobserve that the mechanical underpinnings of this teleological memory trace to the factthat the central bulk of our material consists in a sponge-like network of interlacedtube walls that contract or dilate according to the amount of exogenous air pumpingthrough its innards. That is, the permanent part of wood consists in a flexibleframework of attached minute pieces that can be collapsed or expanded withoutabandoning their primary attachments to one another. By retaining this connectedframework throughout the vicissitudes of exterior jostlings, our wood retains its“memory” of how its parts should be properly arranged, while relying upon thecirculating pressure in the surrounding air to provide the requisite push to restore thematerial to its teleologically desired natural equilibrium. By explicating theunderpinnings of “memory” in this lower scale manner, Leibniz has explained thematerial’s characteristic tropisms in a thoroughly bottom-up fashion based (almost)entirely on efficient causation processes, in contrast to the final causationconsiderations Leibniz invoked in constructing his differential equation model of beambehavior. Such equations superficially appear as ifthey describe events occurring at an infinitesimalscale level far below the finite scale at which our air-within-pores activities occur, but this impressionrests upon a mistaken understanding of themethodology of differential equations, according toLeibniz.

However, a good deal of further philosophylies concealed in that innocuous-appearing qualifier“(almost).” Look closely at the air/tube wallinteractions that restore our wood’s webbing to itsrest state configuration. FIG: AIR PARTICLECOMPRESSION We witness little particles of airthat bounce off walls and interchange momentum with them in the general manner of astandard billiard ball collision. But we’ve already observed that a proper account ofsuch collisions requires that both balls and tube walls must compress temporally,distributing the original kinetic energy of an incoming ball into internal energies ofdeformation. These elastic compressions will quickly push the ball away from thewall in an altered direction as the compressed bodies regain their desired shapes.

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Only under the assumption that these compressive events occur swiftly within a brieftime interval Δt* can we preserve the “inviolable axiom” of continuity that requiresthat nature never make abrupt leaps such as a radical shift in direction. We observed earlier that Huygen’s and Wren’s simple approaches to purelyelastic collisions elide over these Δt* events and compresses all of their hiddencomplexities into an instantaneous event that involves no distortion in the collidingbodies but accepts discontinuous jumps in the trajectories of the colliding balls.30 . This methodology artificially collapses the width of the short interval of time Δt* inwhich the balls actively interact down to an impulsive event lasting no timewhatsoever. In modern jargon, the approach cuts off all consideration of the detailedevents that transpire on a time scale swifter than Δt*. For Leibniz, this approach mustbe regarded as a convenient trick that works well only if one is not interested in thefiner-grained details occuring within the excluded Δt* interval. Leibniz describes thedescriptive utilities of employing such a Huygens-Wren cutoff as follows:

From this, I further showed that if we understand there to be in body onlymathematical notions, size, shape, place, and their change, or if we understandthere to be strivings for change in the body only at the very moment ofcollision, without their being any ground for metaphysical notions, namely, noground for active power in the form and laziness or resistance to motion in thematter, and thus, if it were necessary for the outcome of a collision to bedetermined by the geometrical composition of conatus alone, as we explainedearlier, then I showed that it ought to follow that the conatus of a body enteringinto a collision, however small it might be, would be impressed on the wholereceiving body, however large it might be, and thus, that the largest body atrest would be carried off by a colliding body however small it might be,without retarding it at all, since such a notion of matter contains not resistanceto motion, but indifference.31

In this last remark, Leibniz alludes to the fact that the conservation of kinetic energy(along with momentum) must be assumed in order to extract the right rules of perfectlyelastic impact.

The fact that these cutoff tactics usually supply excellent descriptive results,represents, literally for Leibniz, a Godsend. For the sake of us lowly mortalsstruggling to calibrate nature’s complex behaviors in mathematically tractable terms,the Diety has arranged nature’s behaviors so that we can usually cobble by withoutneeding to worry about the exceedingly complex events that occur within the excluded Δt* interval. Here, however, we can only say “usually,” because circumstances willsometimes amplify minute Δt* details to an observable scale.

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[B]odies can be taken as hard-elastic without denying on that account that theelasticity must always come from a more subtle and penetrating fluid whosemotion is disturbed by the tension or the change of elasticity. And as this fluidmust in its turn be composed of small solid bodies, themselves elastic, it is seenthat this interplay of solids and fluids is continued to infinity.32

But it is only by plowing over these Δt* events that we can explain the elastic behaviorof our original wooded beam in a purist efficient causation manner that speaks ofnothing but the pushing and pulling of contacting particles.

Leibniz mentions the importance of this additional, lower layer of teleology withrespect to his own work on loaded beams as follows:

All this can be clarified by the example of a hanging heavy body, or a bentbow; for although it is true that weight and elastic force must be explainedmechanically by the movement of etherial matter, it is nonetheless true that theultimate reason for the movement of the matter is the force given at creation,which is there in every body, but which is as it were constrained by the mutualinteractions of bodies.33

This is not to say that Leibniz regarded his efficient causation account of elasticity asmore fundamental than a final cause account. Quite the contrary, assumptions aboutthe future desires of springs leads, via differential equations, to a reliable modeling ofbeam behavior, whereas any lower scale efficient causation story involving air andpores is quite speculative in its details and parasitic upon the basic contours of the finalcause account:

However I find that the way of efficient causes, which is in fact deeper and insome sense more immediate and a priori, is, at the same time, quite difficultwhen it comes to details, and I believe that, for the most part, our philosophersare still far from it. But the way of final causes is easier and is notinfrequently of use in divining important and useful truths which one would bea long time in seeking by the other, more physical way; anatomy can providesignificant examples of this.34

And:From this we also understand that even if we admit this primitive force or formof substance (which, indeed, fixes shapes in matter at the same time as itproduces motion), we must, nonetheless proceed mechanically in explainingelastic force and other phenomena. That is, we must explain them throughshapes, which are modifications of matter, and through impetus, which is amodification of form. And it is empty to fly immediately, and in all cases, tothe form or the primitive force in a thing when distinct and specific reasons

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Leibniz’s two kingdoms of explanation

should be given ... for, all in all,... not only efficient causes, but also finalcauses, are to be treated in physics, just as a house would be badly explained ifwe were to describe only the arrangement of its parts, but not its use.35 When we consider billiard ball

collisions, however, preference inexplanatory tactics shifts from theteleological to the efficient causationside of the ledger, for the Wren-Huygens treatment supplies admirabledescriptive results, where any richeraccount involving elasticcompressions will require verydifficult mathematics. Accordingly,for Leibniz, neither of these storiescan be regarded as wholly correct, foreach approach must piggyback uponminor details explicable only from its rival’s point of view. FIG: LEIBNIZ’ TWOKINGDOMS

Indeed, Leibniz’ alternating levels of preferred explanation must continuallyrepeat themselves as we inspect matter upon ever lower ranges of microscopic detail. Our little balls of air will themselves require pores through which an even finer airmust circulate, which in turn will require pores of their own and so on ad infinitum, inthe manner of deMorgan’s celebrated fleas-upon-fleas analogy. Accordingly, astraightforward examination of elastic beam behavior provides us with a concreteillustration of Leibniz’ celebrated worries about “the labyrinth of the continuum”--itsmaze is comprised of a never-stabilizing hierarchy of interwoven explanatory schemes,where final cause stories forever alternate with efficient cause narratives.

(vi)

Descartes had apparently presumed that his ambient air could straightforwardlyreinflate a collapsed elastic structure, but, taken at face value, this supposition isridiculous; bagpipes don’t blow themselves back up after their wind has beenevacuated. Why? Because air pressure normally acts equally in all directions and itrequires a piper of the Black Watch or similar directive agency to puff them up again(sponges can restore themselves only because capillary forces draw the water inward). The only way that air movements could regularly supply a Cartesian springiness is if

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friendly demon supporting an elastic response

some Maxwell’s Demon providentially directs themotions of the ambient air in exactly the right waythat they will collectively knock the elastic materialback towards its original configuration. FIG:FRIENDLY DEMON SUPPORTING ELASTIC

As I understand him, this is precisely whatLeibniz believes occurs, for he posits anaccommodating Deity that plays precisely the role ofour benevolent Demon in supporting the satisfactionof our human desires. Earlier in the essay, weexplicated a beam’s behavior through teleologicalappeal to final causes, specifically, the unstressedequilibrium conditions that wood and iron strive toregain, with different degrees of vim depending upontheir elastic “personalities.” God attends to the final causation desires of beams alongwith the parallel desires of everything else in the macroscopic world. He thenoptimizes their joint satisfaction in the same democratic manner that He employs ininsuring that the linked segments within a beam reach an internal “all for one; one forall” harmony. Once this grand optimization of worldly “desire” has been determined,God fills in the full physical universe at the microscopic level by directing the ambientair molecules in exactly the right way that the final state desires of the middle-levelobjects become maximally accommodated.

It is therefore infinitely more reasonable and more worthy of God to supposethat, from the beginning, he created the machinery of the world in such a waythat... it happens that the springs in bodies are ready to act of themselves asthey should at precisely the moment the soul has a suitable volition or thought;the soul, in turn, has this volition or thought only in conformity with theprevious states of the body.36

Accordingly, if we probe our wood at a scale level below the critical length ΔLC whereit stops behaving in a homogeneous fashion, we will observe air bumping into porewalls within the wood along the exact efficient causation trajectories required to knockthe distorted beam back to its relaxed shape. Viewed from this efficient causationperspective, we don’t witness any springy teleology in play, for God’s providentplanning has supplied the beam with a micro-mechanism that allows it to regain its restshape through air contact action alone.

[E]lastic force is essential to every body, not in the way that the force is someinexplicable quality but because every body, however small, is a machine from

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whose structure a recoil must arise when this is required to conserve force. Moreover, this should not seem surprising to anyone who considers the actualdivision of the parts of this matter into parts exceeding every number.37

This, I believe, represents the proper physical reading of Leibniz’s famous“preestablished harmony”: behaviors that can be explained in a top-down manneraccording to final causation narratives can also be addressed from below withimpeccable efficient causation underpinnings.

Nonetheless, these efficient causation mechanisms shouldn’t persuade us tobecome rank materialists, for none of this providential pushing and pulling could havehappened if Someone Swell hadn’t designed the lower scale world for the benefit ofthe wood. Recall:

[A]ll in all,... not only efficient causes, but also final causes, are to be treatedin physics, just as a house would be badly explained if we were to describeonly the arrangement of its parts, but not its use.38

And thus we appreciate how Leibniz’ two wondrous kingdoms of explanation meshtogether:

I have shown that everything in bodies takes place through shape and motion,everything in souls through perception and appetite; that in the latter there is akingdom of final causes, in the former a kingdom of efficient causes, which twokingdoms are virtually independent of one another, but nevertheless areharmonious.39

In other words, pore-centered physics at a lower, efficient causation level will beincredibly complicated and its structure will be partially determined, through God’spreplanning, by the final states that macroscopic level “souls” strive to reach. In otherwords, God has planned the world largely for the sake of the monads in the middleranks and then arranges the rest of the stuff around them.40

A convenient side effect of this provident structuring is that it allows thedescriptive cutoffs and scaling extensions we have discussed to prove trustworthy ablymost of the time. We can usually reason ably about billiard balls if we suppress theirelastic behaviors through Wren-Huygens rules and we can usually reason ably aboutwooden beams through smoothly extending their higher level proclivities down to theinfinitesimal level of differential equations. Both approaches can be registered inmathematical vocabulary, albeit not at the same time and not within a commonformalism. Neither mathematized portrait should be regarded as supplying a fullpicture of reality, because on rare occasions lower scale complexities will spoil thecutoffs and extensions on which we normally rely. But if we ignore the manners inwhich we patch over these occasional breakdowns in setting up our two self-enclosed

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kingdoms of explanation, we will trust our descriptive mathematics excessively andpresume that material objects genuinely possess wholly objective spatial shapes:

It is the imperfection of our senses that makes us conceive of physical things asMathematical Beings, in which there is indeterminancy. It can bedemonstrated that there is no line or shape in nature that gives exactly andkeeps uniformly for the least space and time the properties of a straight orcircular line, or of any other line whose definition a finite mind cancomprehend.41

In truth, every attribution of a geometrical characteristic to a material object merelyrepresents a false-but-useful projection of idealized behavior based upon its smoothed-over upper scale activities. Thus:

[Matter] has not even the exact and fixed qualities which could make it pass fora determined being... because in nature even the figure which is the essence ofan extended and bounded mass is never exact or rigorously fixed on account ofthe actual division of the parts of matter to the infinite. There is never a globewithout irregularities or a a straight line without intermingled curves or acurve of a finite nature without being mixed with some other, and this in itssmall parts as in its large; so that far from being constitutive of a body, figureis not even an entirely real quality outside of thought. One can never assign adefinite and precise surface to any body as could be done if there were atoms. I can say the same thing about magnitude and motion.42

In my view, such unexceptionable considerations concerning descriptive practicewithin applied mathematics lie at the base of Leibniz’ strange insistence that space andtime are “merely ideal”--he is observing, quite correctly, that every practicabledescription of everyday matter utilizing geometrical vocabulary secretly incorporates afair degree of fictitious projection to unwarranted size scales. With this feignedhomogeneity comes the presumption that a thoroughly continuous material can bepotentially divided into every possible scale length ΔL. As such, the doctrineoverreaches, yet these same fictitious “possibilities of division” can be exploited toprovide a relaxation space pathway to differential equation models like our beamformula. Such methodological tactics still characterize large sectors of successfuldescriptive policy within modern applied mathematics and deserve greater attentionfrom philosophers than they have typically received.43 We shall find ourselvesfrequently returning to Leibniz’ underlying concerns within our other essays.

But few of us are likely to wholeheartedly embrace Leibniz’ own remedies forresolving these tensions, for this is where his peculiar monads enter the scene. Hebelieves that material behaviors require firm underpinnings within exterior reality (no

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a well-founded phenomenon

phenomenalist he). But we can never reach a stable descriptive accounting as long aswe insist upon describing matter in spatial and temporal terminology, despite the factthat mathematical physics requires these very modalities for its operations. In contrast,a substance’s powers in resisting alterations and seeking goals represent morefundamental characteristics than those of geometrical depiction.44 When we describematter within mathematical physics, we treat spatial displacement as if it supplies thecause of internal change; a weight’s propensity to fall within a gravitational field iswhat makes a beam element upon which it acts fan apart. From the perspective of thetrue monadic order, however, these dependencies operate obversely; the attachedweight correlates with an element’s internal inability to return to its desired naturalstate as swiftly as it otherwise might. But the cause of this internal resistance comesdirectly from God and stems from his policies for maximizing the teleological tropismsof all matter, including the rocks that sit on top of our beam and the neighboringelements that lie along its breadth (recall the “all for one; one for all” cooperation thatgoverns constrained equilibrium). In such circumstances, a local element A must oftenendure large amounts of internal tension due to the optimized desires of its neighboringelements. If elements B and C lying far away in the beam can resist their rock loadsbetter than A–that is, if their interior springs are stiffer than those found in A–, thenGod’s optimization forces A to deviate more from its desired rest state.

These required harmonies are closely connected with our earlier observationthat, within a continuous hunk of flexible matter, the “personalities” of the componentelements must coordinate with one another continuously, lest the material splinter intodisconnected shards.45 But any spatial partitioning of our beam into segments involvessome degree of artificial projection on our parts, for the behaviors captured withinthese mathematical ascriptions inevitably blur over finer scale behavior through scalingover-extension. The descriptive arsenal of mathematical physics therefore lacks acapacity for consigning the full powers of an extended piece of wood to precise spatiallocales. The restricted set of mutual coordinations that we capture within a solution toour beam equation are accurate insofar as they go, but their localized elements don’talign with the true power relationships within thematerial in a simple way. Leibniz often compares ourperception of a body’s spatial characteristics to themanner in which we see rainbows. Individual drops ofrain are responsible for the redirection of light thatoccurs but our eyes cannot discriminate these droplets onthe basis of the combined light that reaches us, becausethe originating sources lie too near one another and alter

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preformationwithin a sperm

cell

the light in very similar manners. FIG: A WELL-FOUNDED PHENOMENON Likewise in mathematical physics, when a beam is described as curved in somedefinite mathematical pattern such as a catenary, its lower scale complexities havebeen artificially suppressed for the sake of a convenient representation of how itsupper scale behavioral propensities relate to one another. When section A of a beamis described as “more strongly curved due to the weight pressing upon section B,” weare actually claiming that A’s powers of influence have partially come under the swayof B’s. Leibniz writes:

However, it must be confessed that the continuous diffusion of color, weight,malleability and similar things that are homogeneous only in appearance ismerely apparent and cannot be found in the smallest parts [of bodies]... It isonly the extension of resistance, diffused through body, that retains thisdesignation (extension) on a strict examination. You ask now what is thatnature whose diffusion constitutes body?... [W]e say that it can consist innothing but the dynamicon or the innate principle of change of persistence.46

And: I had to look more deeply into the notion of corporeal substance, which I holdto consist more in the force of action and resistence than in extension, which isonly the repetition or diffusion of something prior, namely that force.47

Indeed, an object’s visible “size” merely reflects the fact that it will “look large” if itscomponent parts possess a capacity for making large quantities of light alter their owninternal dispositions in corresponding ways, e.g., they acquire a greater disposition todiverge from one another. Portions of objects that strike us as different in colorindicate that their local powers exert different controlling powers over the ambientlight, which, in turn, then affects our retinas in corresponding ways.

Leibniz conceives the hierarchical relationships of continuityrequired amongst these background power centers in intriguing terms thathe adopted from the developmental biology of his day.48 On microscopicevidence it was commonly believed that complex organisms originatefrom “seeds with all of their organs folded up,” albeit greatly shrunken insize. FIG: PREFORMATION WITHIN A SPERM CELL Normalbiological development consists largely in these primordial organs takingon enough food to eventually assume adult proportions (for Leibniz, upondeath these same parts relinquish their hold upon fleshy matter and shrinkback to their original, minuscule proportions). To make sense of theserelationships, the monadic units corresponding to each bodily part must

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freedom from enslavement

belong within a master and slave hierarchy where the teleological requirements of anentire animal body determine the desires of its heart which in turn fix the ambitions ofits left ventricle. And so on, in a classic “the nest on the branch, the branch on thelimb, the limb on the tree” manner:

[A] natural machine has the great advantage over an artificial machine, that,displaying the mark of an infinite creator, it is made up of an infinity ofentangled organs. And thus, a natural machine can never be absolutelydestroyed just as it can never absolutely begin, but it only decreases orincreases, enfolds or unfolds, always preserving, to a certain extent, the verysubstance itself and, however transformed, preserving in itself some degree oflife or, if you prefer, some degree of primitive activity. For whatever one saysabout living things must also be said, analogously, about things which are notanimals, properly speaking.49

Indeed, an allied hierarchy is required to keep the nested teleological ambitions of asimple continuous body like a plank of wood coherent, where each component hunk ata critical ΔLC scale must cooperate in a slavish fashion with the natural state desires ofthe beam as a whole. Indeed, it is striking (although I’ve not found anywhere whereLeibniz argues accordingly) that a small arc a of a wooden ring A molded undertension must agreeably cooperate with the natural state desires of the whole ring A aslong as a remains enslaved to A. FIG: FREEDOM FROMENSLAVEMENT But once a is “liberated” from its chains(e.g., we cut a free from A), it will freely display its own setof natural state desires.

As remarked previously, conclusions such as this teeteron the edge of impiety (i.e., attributing organizing “souls” topieces of dead wood or iron) and the historical Leibniz onlyclaims that the top-down organization displayed withininorganic materials and wooden beams provides only a simulacrum of the monadicmaster-slave relationships operative within living biological systems (although everymaterial system contains some measure of animal-like monads). Our wooden beammerely evidences an apparent “dominant monad” behavior in the manner of a flock ofsheep

where the sheep are so tied together that they can only walk with the same stepand cannot be touched without the others bleating.50

The herd looks like a single animal, but this is simply the product of enforcedcooperation (which, as we’ve seen, God is happy to arrange). Presumably, Leibniz

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evokes these imitations of monadic life to avoid the theologically unwelcomeconclusion that sawed-off boards and rocks possess immortal souls. Whatever itsultimate origins, a behavioral coherence amongst the top-down scaling behaviors withinany form of continua, biological or not, is crucial to their successful treatment withinmathematical physics.51 I again stress that my purpose in this essay is to articulatesome important (but generally overlooked) lines of physical reasoning within histhought as cleanly as I can, rather than attempting to follow the many further strandsthat pull Leibniz’ doctrines in more complicated directions.

Although Leibniz’ preformative analogy strikes us fanciful, the underlyingrecognition that the physics of continua must display a tight pattern of continuity acrossscale sizes is not; it remains fully enshrined within the compatibility equations found inany modern rigorous foundations for these topics. We have already highlighted theprofound physical insight that lies concealed within the simple requirement that, in theabsence of rupture or fusing, the pieces of a continuous body remain firmly attachedthroughout all of their distortions. To enforce this condition, an organized integrationof behaviors across all scale sizes in the top-down manner characteristic of ourrelaxation space constructions (and modern measure theory in general) is required. Yetwe have also observed that this complete scale invariance is only apparent, as otherprocesses become secretly active below the cutoff level ΔLC. The multiscalarmodelings of Essay 5 utilize clever techniques for addressing these discordantmethodological concerns.

It is fortunate that master and slave monads cooperate so agreeably with respectto their larger scale teleological ambitions, for the resulting upper-scale behavioralinvariance supplies limited intellects such as ours with convenient footholds on thematerials of daily life. We couldn’t forge a mathematical pathway to reliable beammodelings if smoothed-over teleological constructions such like our Bernoulli-Eulerelements don’t provide us with reasoning patterns that work pretty well most of thetime. As inherently limited calculators, it is best that we rarely detect themicrostructures within the world around us, for that awareness might encouragequixotic searches for bottom-up modeling ambitions that can never be completelyfulfilled due to labyrinth of the continuum considerations.52 Indeed, for most purposeswe should conceptualize the world about us with the unalloyed mixture of efficient andfinal cause stories that Leibniz constructs to understand how a flexible beam behaves. The behavioral harmonies on which these techniques rely trace entirely to the kindlyactivities of the Deity who has arranged matters thus.

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preordained harmony

In fact, God does a good deal more than this, for he plans his universe from oursize scale outward, in the sense that he first maps out how macroscopic objects(including human souls) should interact on their own characteristic scale level–how abeam or billiard ball should bend, for example, or what happenswhen I order a martini. FIG: PREORDAINED HARMONYGod then directs his lower scale air through all the right pores toback up this macroscopic world:

[I]t is just as if [someone] who knows all that I shallorder a servant to do the whole day long on the morrowmade an automaton entirely resembling this servant, tocarry out tomorrow at the right moment all that I shouldorder; and yet that would not prevent me from orderingall that I should please, although the action of theautomaton that would serve me would not be in the least free.... The knowledgeof my future intentions would have actuated this great craftsman, who wouldaccordingly have fashioned the automaton: my influence would be objective andhis physical. For in so far as the soul has perfection and distinct thoughts, Godhas accommodated the body to the soul, and has arranged beforehand that thebody is impelled to execute its orders. And in so far as the soul is imperfect andas its perceptions are confused, God has accommodated the soul to the body, insuch sort that the soul is swayed by passions arising out of corporealrepresentations.53

The provident manner in which God constructs a fuller universe around the desires ofthe composite objects dominating our own scale level emerges as central in the storythat Leibniz subsequently supplies with respect to human free will.

When we inquire into a material’s “possibilities of division,” what do we seek? Two Leibnizian answers suggest themselves: (i) “possibilities” as they pertain togranular monads of the real universe and (ii) “possibilities” as they apply to thesmoothed-over continua that we attribute to the world for the sake of descriptive utility. Insofar as (i) is concerned, a proper answer must be secretly determined by the full“personality” rules that determine when one range of monadic influence cooperateswith another and when not. If I read Leibniz correctly, he fancies that if, perimpossible, we could actually learn these rules in all of their infinitely complex glory,we would discover that only one overall outcome was “possible.” On the other hand,according to (ii), the notion of “being divisible into segments of length ΔL” central toour beam modeling techniques merely represents a fictive aspect of our restricted upper

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length scale knowledge, albeit a form of projection vital to descriptive success withinmathematical physics. It is this second notion of “possibility” that allows us to declare,“Of all of its possible configurations, a loaded beam chooses the shape that supports itsloads with the least overall of internal strain energy.” Leibniz explains suchdistinctions as follows:

[I]n actual things, there is only discrete quantity, namely a multitude of monadsor simple substances, indeed, a multitude greater than any number you mightchoose in every sensible aggregate. That is, in every aggregate correspondingto phenomena. But continuous quantity is something ideal, something thatpertains to possibles and to actual things considered as possible. Thecontinuum, of course, contains indeterminate parts. But in actual thingsnothing is indefinite, indeed, every division that can be made has been made inthem.... As long as we seek actual parts in the order of possibles andindeterminate parts in aggregates of actual things, we confuse ideal things withreal substances and entangle ourselves in the labyrinth of the continuum andinexplicable contradictions. However, the science of continua, that is, thescience of possible things, contains eternal truths, truths which are neverviolated by actual phenomena, since the difference [between real and ideal] isalways less than any given amount that can be specified. And we don’t have,nor should we hope for, any mark of reality in phenomena, but in the fact thatthey agree with one another and with eternal truths. 54

By these lights, a trait qualifies as “necessary” only if it holds within all “possibilities”of our second class, a standard that liberates most human actions from the burden ofappearing “necessitated.”

(vii)

With these props in place, it is but a short step to Leibniz’ remarkable defense offree will. We humans possess a range of personality-driven desires which we can actupon as freely as the middle-level constraints of the world permit, including theopposing desires of other agents. God optimizes the world with these middle-rangeconstraints in view and then fills in the rest with enough directed air that the “springs ofbodies” act in the fashion that our operative final cause demands require. So we aregenuinely free to make all sorts of dumb decisions in accordance with our ownpersonalities; God has merely crafted the lilliputian air that puts the “spring” into ourfreely chosen steps.

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our hero

For this very reason the choice is free and independent of necessity, because itis made between several possibles, and the will is determined only by thepreponderating goodness of the subject... There is therefore a freedom ofcontingency ... but there is never any indifference of equipoise, that is, where allis completely even on both sides, without any inclination towards either. Innumerable great and small movements, internal and external, co-operate withus, for the most part unperceived by us. And I have already said that when oneleaves a room there are such and such reasons determining us to put the onefoot first, without pausing to reflect. For there is not everywhere a slave, as inTrimalchio’s house in Petronius, to cry to us: the right foot first.55 Because we normally “see” our surroundings only in macroscopically smoothed-

over terms, our operative notions of “contingency” and “necessity” ipso facto reflectour middle-scale placement within the cosmos. From this point of view, the onlyavailable explanation for our normal activities X is that we choose them--we desire Y,X seems a suitable method to reach Y and nothing prevents us from executing X. True,if we could inspect the microscopic workings of our neurons carefully, we wouldobserve God’s preplanned air particles shunting along tubular walls in an impeccableefficient causation manner. But this fact in no way impugns our actions as not, at core,entirely free, for God has cleverly plotted the harmonies of the world to optimally fulfilldesires working from our size scale outward.

Of course, no one would swallow this exuberant fantasy in its entirety today, butI want to strongly underscore how many of Leibniz’ motivating concerns stem fromsound observations with respect to the complex practices involved in treating mattercoherently from a continuum physics point of view. Admittedly, Leibnizian monadsneedn’t be invoked in addressing these concerns, but the essential ingredients of“natural state” striving and behavioral coherence across size scales must. Our presentday policies for describing everyday materials in flexible matterterms tacitly rest upon subtle contextual controls that fully meritmuch closer inspection. Many contemporary philosophersappeal glibly to “the possible worlds of classical physics,” whileignoring the modern analogs of Leibniz’ methodologicalconcerns.56 In these respects, it is startling to realize that it isonly through quantum physics that we moderns can halt Leibniz’labyrinth-of-the-continuum regress, for classical point-massmechanics lacks adequate resources for rendering matter stableat an atomic level. In other words if we do not invoke

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intrinsically quantum behaviors involving a hazily delineated “effective size” to halt thenon-terminating descent, the upper-scale coherence of classical materials will pull usinto the same foundational maze that troubled Leibniz (quantum physics raises manyparadoxes of its own, but at least it rescues us from these worries).57 But this liberationis achieved only at the cost of denying well-defined shapes and trajectories to the“particles” that inhabit the lower tiers of the space-time arena. Or so it may seem. FIG: OUR HERO

Accordingly, we needn’t go so far as to agree fully with Leibniz:[Matter] has rigidity as well as fluidity everywhere and no body is hard or fluidto the ultimate degree, that is, that no atom has insuperable hardness, nor isany mass entirely indifferent to division.58

But we must surely acknowledge that his accompanying insights into the conceptualcomplexities of flexible bodies have proved deep and prophetic.

Appendix: The Problem of the Physical Infinitesimal

As soon as we consider matter that is truly continuous, we confront thefoundational difficulty that two distinct kinds of affective forces appear, one of thementirely absent within the point mass framework characteristic of Newtonian celestialmechanics.59 Naive attempts to bring these two types of force into cooperativeengagement generates a serious regress that we might call the paradox of unprofitabledescent. Consider a complete wooden beam, pinned between two endpoints. Obviously, strong traction forces must attach these ends firmly to their respective posts,while gravity pulls their interiors downward according to the beam’s resistence tocurvature and the loads it bears. The sagging configuration of the full rod is determinedby how these two factors--endpoint tractions and interior gravitation--work together inview of its internal capacities for resisting displacement from its favored equilibriumcondition, that is, from the natural rest configuration to which the wood continuallystrives to obtain. But this final figure can be quite complex, especially if the beam’smass60 varies from point to point as we move along its breadth or if the local elasticstrength varies in this manner as well. But this is why we model a beam withdifferential equations in the first place; we hope that, by first describing its behaviors atan infinitesimal level, we will be able to work our way to its more complex larger scale

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continua don’t simplify at smaller scales

Bernoulli-Euler element

behaviors through integrations, following the adage that “physics is simpler in thesmall.”61 FIG: CONTINUA DON’T SIMPLIFY AT SMALLER SCALES Unfortunately, as we consider ever smaller sections of our rod, we find ourselvescontinually confronted with the same division betweenendpoint tractions and interior forces. Consider a smallsection of beam at the level of an “element.” What dowe find? (A) Gravitational forces pushing directly uponits interiors (hence the modern terminology: “body” or“volume” force), as well as the inertial responses tothese loads (which Leibniz dubbed “changeable activeforce”). (B) Tension forces transmitted by contactaction to the two sides of our element that twist it ininfinitesimally distinct ways (hence the modernclassification: “contact” or “traction” force).62 Primafacie, this distinction seems insignificant, but,mathematically, these two species of “force” act upondimensionally incongruent locales: gravitation pulling on points while tensions tug uponsurfaces.63 To obtain a workable physics for continuous body, we must persuade thesedimensionally incompatible creatures to work together in harness.64 Section (x) ofEssay 8 supplies details and references in a higher dimensional context.

Rather than repeating the technical discussion of that essay, we can gain anintuitive appreciation of our coordination problem by dissecting a Bernoulli-Eulerelement into Leibnizian ingredients. We find ourselves with an element that needs tooperate rather like some jumping jack toy that balances an interior tug through springsthat are attached to the toy’s arms and legs. FIG: BERNOULLI-EULER ELEMENT Plainly these springs must be internally connected in astructural manner that supplies a upward effective force thatcan oppose gravity along exactly the same vertical line. The array of springs within our specimen beam element areattached to adjacent sections of the wood along their mutualboundaries and are joined together in the element’s interiorby a rigid block. This internal block supplies a simplemechanism that sums and reorients the lateral tractions ofthe springs to supply a net upward force opposing gravity’sbody force pull. When beam locales are characterized bydifferential equations at an infinitesimal level within physics, each locale must be

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credited with a non-trivial interior mechanism that can sum and redirect traction forcesin a manner such as this. This is the basic reason why I earlier remarked that physicsconfronts problems with its parochial “infinitesimals” that are significantly different inunderlying character from the δ/ε issues with which Weierstrass was concerned.

Until we can somehow resolve this force incompatibility problem, we willconfront a paradox of unprofitable descent. Why? Every small section of a flexiblebody remains affected by body forces within its interior and by traction forces along itsbounding surfaces.65 From a formal point of view, this is exactly where we started;the relevant physics hasn’t simplified at all. No matter how finely we inspect ourmaterial, we will continue to find different species of force operating upondimensionally incompatible locales.

Note, however, that this resolution of our summation problem attributes a fairnumber of artificial ingredients (including some rigid blocks) to the interior of ourelement, despite the fact we had originally assumed that our beam is fully continuousand flexible at all size scales! What justifies these artificial insertions, other than theyprovide differential equations with a foothold upon the behavior of a flexible body? If Iunderstand Leibniz’ thinking on this score, he presumes that we should always expectto approach nature’s complex behaviors through artificially finitary means, just as wemust locate a beam’s fully optimized bending by sneaking up on that configurationthrough a sequence of artificial relaxation space segmentations.

Until adequate theories of measure, integration and tensorial objects weredeveloped in the twentieth century, traditional methods of blocking this regressinvariably appeal to some substantive recasting of their elements that deprives them ofof their originally posited flexibility. Accordingly, their infinitesimal innards arealleged to act like (i) sets of attracting point masses, (ii) rigid pieces or (iii) rigid piecesconnected by simple springs and the like (Leibniz’s own policy). To justify these weirdmethodological procedures, the ever popular philosophy of “essential idealization” wasborn--to render its tools applicable to the real world, applied mathematics must alwaysmisrepresent worldly behaviors in some quasi-rigidified manner such as (i) -(iii). In1892 the celebrated statistician Karl Pearson expressed the thesis in the following form:

I feel quite sure that to assert the real existence in the world of phenomena ofall the concepts by aid of which we describe phenomena--molecule, atom,prime-atom--even if [they be admitted] ad infinitum, will not save us fromhaving to consider the moving thing [we utilize within our mathematicaltreatments] to be a geometrical ideal, from having to postulate [a fictitiousentity which] is contrary to our perceptual experience [of a continuous world].66

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Or Max Planck in 1932:Nature does not allow herself to be exhaustively expressed in human thought. Itis the most important and at the same time the most difficult task of thetheoretical physicist, when finding a mathematical formulation of the problemwhich he is attacking, to introduce just those simplifying assumptions which areof characteristic importance for the properties which interest him in thephysical processes which he is investigating, and to neglect all influences of alower order of magnitude which can produce no essential change in the mainresult and which enter into the argument as mathematical ballast. An importantand inevitable postulate is that the different hypotheses introduced for dealingwith different problems should be compatible with each other.67

As it happens, Pearson maintains that one must always expect to begin with masspoints in a mathematical treatment of nature, although, as more precise experimentalinformation is gathered, the size scale at which these ideal elements will be introducedwill require readjustment to ever lower levels (from “atom” to “prime atom” andbeyond). This suggests that we must build up our modeling elements in manner (i)above, through imposing rather implausible constraints upon a collection of pointmasses. To this day, many philosophers presume that “classical mechanics” invariablyconstructs its models in Pearson’s point mass manner, but this is simply amisunderstanding. Writers within the broader continuum physics traditions pioneeredby Leibniz have traditionally favored approaching these problems in a top-downmanner where some milder policy of small element rigidification is invoked to resolvetheir body force/traction force resolution problems.68

The twentieth century studies of Clifford Truesdell, Walter Noll and their schoolhave clearly established that the essential idealization thesis merely serves as aconvenient excuse for bypassing some tricky mathematical arrangements involvingtensors and so forth that need to be directly confronted in any case. Suchdevelopmental episodes are common in the history of science: some significantmathematical barrier is initially evaded through rough methodological appeal to some“Oh, physics always idealizes” alibi. Only much later, pressured by the necessities ofgreater applicational rigor, do clever applied mathematicians figure out the truestructural underpinnings behind these strange appeals. Such developmental exigencies(and the fact that freshman level physics texts often appeal to antiquated forms of hazyhandwaving) have left permanent scars upon modern philosophy of science; the allegedmethodology of essential idealization is frequently invoked to justify many dubiousforms of philosophical analysis.

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1.“Proportiones arithmeticae de seriebus infinitis earumque summa finata” in D.Struik, A Sourcebook in Mathematics (Cambridge: Harvard University Press, 1969),p. 320.

Such concerns make it very difficult to assign a firm “ontology” (in Quine’ssense) to most historical writers who write about classical matter, simply because theyrarely identify the rigidified ingredients employed utilized in their differential equationconstructions with the actual objects found in nature itself. J.H. Poynting captures thisdisparity admirably in his 1899 address to the British Association for the Advancementof Science:

While the building of nature is growing spontaneously from within, the model ofit we seek to construct in our descriptive science, can only be constructed bymeans of scaffolding from without, a scaffolding of hypotheses. While in thereal building all is continuous, in our model there are detached parts, whichmust be connected with the rest by temporary ladders and passages, or whichmust be supported till we can see how to fill in the understructure. To give thehypotheses equal validity with the facts is to confuse the temporary scaffoldingwith the building itself.69

Let me close with several philosophically pertinent remarks with respect to ourproblem of the physical infinitesimal. (1) It has little to do with standardCauchy/Weierstrass concerns with respect to limits and “infinitesimals” quamathematical object. The problem of what sorts of localized structures (viz., tensors ofwhat class?) should be induced upon the points within a flexible, continuous bodycodify intrinsically physical considerations.70 (2) Problems of the physical infinitesimalcomplicate most historical discussions of matter across the entire era of classicalmechanics’ reign. (3) The force-reconciliation difficulties at the base of these tensionscan be now be addressed with more sophisticated mathematical tools. (4) Theseproblems should be sharply distinguished from the Leibnizian worries about the over-extension of scaling assumptions that are central within the main body of this essay. The latter continue to affect methodological practice within present day science, asother essays discuss under the heading of “the greediness of scales.” That Leibnizpresciently anticipated these important concerns within his odd musings about monadsis further proof of his incisiveness of his thought with respect to continuous matter. Endnotes:

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2. “New Proofs Concerning the Resistance of Solids,” Acta Eruditorum., July,1684. I am indebted to Clifford Truesdell’s excellent discussion of this essay andthe allied literature in The Rational Mechanics of Flexible or Elastic Bodies 1638 -1788 (Basel: Birkhäuser, 1980), pp. 59-64. See also Edoardo Benvenuto, AnIntroduction to the History of Structural Mechanics, Pt I (New York: Springer-Verlag, 1991), pp. 268-271.

3. To this end, Littlewood relied upon Russell’s incorrect parsing of “determinism”as articulated within his “On the Notion of Cause” in Mysticism and Logic(NewYork: Doubleday, 1963). Littlewood’s essay can be found in AMathematician’s Miscellany (London: Methuen: 1963).

4. “A Specimen of Discoveries of the Admirable Secrets of Nature in General” inRichard T. Arthur, ed. and trans., The Labyrinth of the Continuum (New Haven:Yale University Press, 2001), p. 315.

5. Bertrand Russell; “The point of philosophy is to start with something so simple asto not seem worth stating and to end up with something so paradoxical that no onewill believe it.” The Philosophy of Logical Atomism (LaSalle: Open Court, 1985),p. 53.

6. In modern presentations of continuum physics that follow Walter Noll, a so-calledreference manifold outside of space and time is employed to enforce these structuralrequirements. For a good presentation, cf. Clifford Truesdell, A First Course inContinuum Mechanics (New York: Academic, 1977).

7. This is usually determined by the speed of sound within the material; variousrubbers are quite poky in regaining their desired end state.

8. Secretly, these materials slowly recrystallize to minimize strain energy in the newenvironment

9. Among these “smart materials” are the nickel-titanium alloys utilized in antennasintended for outer space use that can “remember” two or more natural rest states. Such apparatus will rest docilely in a folded up condition while riding to itsdestination in a rocket, but as soon as the gizmo is released into the interplanetaryvoid, it senses the heightened cold which then jogs its “memory” of the prioroccasion when it had been molded into a stretched out configuration under chilly

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conditions. Thus our smart antenna “perceives” the cold which awakens a“striving” to return to a different rest state than it had “desired” while it felt warmer. Watching one of these devices unpack itself is rather unsettling, for it looks likesome creepy insect slowly bestirring itself.

In truth (as Leibniz probably recognized), there are no pure “solids” or“fluids” existing in nature; every real material displays some slight measure of beingable to detect its neighbors’ internal conditions.

10. Leibniz acquired this principle from Edne Mariotte, who had discovered itindependently and first applied such thinking to beams. He also informed Leibnizthat a simple relationship was only valid experimentally within the range of smalldeflections. Cf. J.F. Bell, The Mechanics of Solids: Volume I: The ExperimentalFoundations of Solid Mechanics (Berlin: Springer, 1984).

11. These comprise so-called moving boundary problems, which invariably requiremodern numerical methods..

12. “Of Body and Force: Against the Cartesians” in Leibniz: Philosophical Essays,R. Ariew and D, Garber, trans. (Indianapolis: Hackett, 1989), pp. 252-4.

13. Wren and Huygens’ rules for impact presume perfect elasticity (rather than thefrictional losses crudely captured within Newton’s coefficient of restitutionsupplements). Leibniz invariably alludes to the former treatment in contrasting suchefficient causation accounts with his own teleology-based account, so I shall omitNewton’s name in this connection henceforth. Unfortunately, later philosophicalthinking about matter and causation often carelessly presumes, as Hume, forexample, did) that Newton captured the “correct physics” for billiard ball interactionwithin the Principia (for fuller comments on this situation, see WS, Chapter 9). What should be properly said is that Newton and his followers practiced anadmirable restraint in their descriptive ambitions, by substituting a crude but reliablewalkaround method for a very difficult moving boundary computation. Even today,modern models of impact follow a Newtonian pattern whenever they can get awaywith it, but delicate cases sometimes force practitioners to reopen the smoothed overΔt* impact events and investigate the sundry compressions and expansions thattranspire therein in elaborate detail. The first serious attempt to estimate thecompression of contacting billiard balls (in static circumstances only) traces toHeinrich Hertz’s work in 1886.

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Contrary to popular assessment, Leibniz was not a “purely philosophicaldreamer” who mused airily about physical circumstances that he could not covertinto concrete mathematics (in the manner of Boscovich, say). His own beam theory(and most of the improvements that follow thereafter) rely upon a host ofmacroscopically-based constraints (i.e., that beam movements transpire mainlyalong fibers and in plane sections) that cleverly open viable pathways to practicalcomputation. Physicists before Mariotte and Leibniz (like Galileo) approached thebreaking load of a gradually weighted wooden beam as an instantaneouscatastrophic event very in much the “compressed interval” fashion of Newton’sapproach to billiard balls. “No,” Leibniz replies, “you’ll not obtain trustworthyresults until you insert some form of internal elastic response linked to flexure intoyour models, because you must first estimate the gradual increase in internal stressthat leads to material failure.” This recognition represented an important landmarkin improved practical modeling.

14. “Letter to Bernoulli, September 30, 1698" in The Leibniz-De VolderCorrespondence, Paul Lodge, trans. (New Haven: Yale University Press, 2013), p.11.

15. Often anachronistically identified as “Newtonian physics.” See my “What isClassical Physics Anyway?” in R. Batterman, ed., The Oxford Handbook inPhilosophy of Physics (Oxford: Oxford University Press, 2012).

16. The fact that Leibniz models his beam in a reduced dimensional manner, as aslender one-dimensional curve rather than a fatter three-dimensional prism, rendersthe internal nature of his “elements” more complex than is required within 3Delasticity. More modern approaches arrive at the Bernoulli-Euler element by takingthin slices of a three-dimensional prism and arguing their way to the reduced one-dimensional formula d2(EI d2h/dx2)/dx2 = l through an appropriate shrinkingargument (relying upon additional assumption such as “plane sections remainplane”). The unavailability of PDE representations within Leibniz’ time (not tomention unforgiving natures of the proper equations for a 3D elastic solid) forcedmathematicians of Leibniz’ era to approach flexible 3-D bodies through variousmore or less plausible policies of 1D ODE decomposition.

17. In truth, (c) is far too underspecified to qualify as a proper model (what forcesbind our element together?). I have included this dubious alternative because many

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philosophical commentators off-handedly presume that “this is the manner in whichNewtonian classical physics models matter.” In point of fact, articulating requisiteforce laws to carry off this modeling project is fraught with difficulties and, quitepossibly, cannot be secured within the realms of non-quantum modeling at all. Forpresent purposes, I merely stress that Leibniz’ thinking becomes unintelligible if oneinsists upon approaching “classical mechanics” from a point mass perspective like(c).

18. More formally, our physical descriptions become “realistic” only after weintegrate over finite spans of space and time.

19. In searching through a “possibility space” in this manner, we are tacitlyappealing to variational considerations in the general mode of the “virtualvelocities” (later, “virtual work”) criteria developed by Leibniz’ friends, theBernoullis, and later canonized within Lagrange’s Mécanique Analytique. Twoexcellent histories are P. Duhem, The Origins of Statics (Berlin: Springer, 1993) andAgamenon Oliveira, A History of the Work Concept (Berlin: Springer, 2013). Inlight of Leibniz’ frequent insistence that calculus ideas should be understood interms of what we would now call its finite element truncations (see Essay 8), ourappeals to an increasingly refined element grid suits his thinking ably as well. Inthis regard, Leibniz derives Fermat’s allied “Principle of the Least Time” in opticsthrough an allied element-like decomposition into stages–see Jeffrey McDonough,"Leibniz on Natural Teleology and the Laws of Optics," Philosophy andPhenomenological Research 78 (3) (2009)). Hamilton himself derived his“principle” within mechanics by imitating the piecewise deflections of the opticalpath one witnesses inside a telescope with a lot of mirrors and lens, according toDarryl D. Holm, Geometrical Mechanics, Pt 1 (London: Imperial College Press,2008), chapter 1. Essay 7 is devoted to “possibility spaces” of this type.

20. E.g., the shooting method of Essay 2.

21. Leibniz’ views on how differential equations relate to their discretizations aredeeply entangled with his reflections on “progressions of the variable,” which Iwon’t attempt to explain here. See H.J.M. Bos, “Differentials, Higher-orderDifferentials and the Derivative in the Leibnizian Calculus, Arch Hist Ex. Sci 14,1974. Chapter 3 of D. Bertoloni Meli’s Equivalence and Priority (Oxford:Clarendon Press, 1993) also contains an useful discussion of these topics.

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Formulating transparent equations for continua requires partial differential operators,but most early continuum models work in stages with ordinary differential equationsby evoking symmetries to reduce dimensions and then employing some form ofspatial integration to convert an array of distributed forces into a bending moment orallied sum (this is the task Leibniz attempted in his original article and falteredslightly in several of his intervening stages). The resulting infinitesimal elements arethen assembled into a final ODE that describes the one-dimensional displacement ofthe target object from the x-axis. Only when we “turn on motion” via d’Alembert’sprinciple that partial derivatives enter our picture. S.B. Engelsman in his Families ofCurves and the Origins of Partial Differentiation (Amsterdam: Elsevier Science,1984) claims that Leibniz and Johann Bernoulli had PDE equivalents available by1697, but these represent technical historical issues that I cannot evaluatecompetently. Without a doubt, Leibniz had a rough conception of the multivariantcalculus even if his implementations understandably failed to exhibit theirrequirements clearly. Justifications for the dimensional reductions common withinthe study of elastic bodies comprise another aspect of the essential idealizationtradition, but we won’t pursue these issues here.

The “completion of a space” tactics utilized here frame the principal topics ofEssay 8. Although I have employed modern terminology to explicate the underlyingformal problem, Leibniz clearly appreciates the conceptual issues involved in theseextension techniques.

22. Under small deflections, the curvature of the neutral axis is closelyapproximated by d2h/dx2. From a modern point of view, our shrinking argumentsupports what mathematicians now call the weak or variational form of the beamequation: d2h/dx2EI d2δh/dx2 = Wδh. Here the role of our varied tweakings areexplicitly registered through the inclusion of the variational term δh. Theseemendations strengthen the scope of the principle in a significant way (the revisedequation is said to be “weak” because it places weaker demands upon the objects towhich it applies, allowing it to model a wider class of target beams). Assumingsuitable smoothness and homogeneity in our beam, our new equation reverts to itsclassical form: EI d4h/dx4 = W.

23. Dynamic behaviors can be piggybacked upon static concerns in Lagrange’smanner as discussed in Essays 4 and 8. Leibniz seems to have viewed dynamicalprocesses in roughly this manner, but lacked the PDE mathematics required toimplement the approach precisely.

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24. Essay 6 criticizes the standard Lewis-Stalnaker approach to counterfactuals fornot attending to the significant structural differences between equilibrium-focusedappeals, such as we consider here, and evolutionary settings where time plays amore central role. I submit that we can’t understand Leibniz’ own appeals topossible worlds properly without attending to this difference, so vital to the greatmechanical traditions, culminating in Lagrange, that build up mechanics on aninitially statical basis.

25. Within regular three dimensional elasticity, so-called compatibility equationsare required precisely to ensure that infinitesimal strains sum to macroscopicdisplacements. That infinitesimal and extended modes of description need toharmonize in this non-trivial manner is often overlooked by Leibniz’ commentators.

26. The strange methodological policies traditionally invoked to validate thepeculiar infinitesimals of continuum mechanics practice can elicit similar morals andLeibniz would not have segregated these concerns. In the body of this essay Iconcentrate upon the methodological oddities involved when higher scalingbehaviors are artificially extended to infinitesimal levels at which they cannot apply,for these are concerns that remain with us yet–vide Essay 5. In contrast, Leibniz’original concerns with respect to physical infinitesimals can be rectified usingmodern techniques and are discussed in the appendix.

27. “Letter to Jacob Bernoulli, 1702,” quoted in D.B. Meli, Equivalence andPriority (Oxford: Oxford University Press, 1993), p. 55. See Rene Descartes,Principles of Philosophy (Dordrecht: Kluwer, 19910, pp. 42-2. Leibniz employs thesame sponge metaphor in “Of Body and Force”:

For, even if some bodies appear denser than others, this is only because thepores of the former are filled to a greater extent with matter that belongs tothe body, while, on the other hand, the other rarer bodies have the makeupof a sponge. (Ariew and Garber, op. cit., p. 252)

I am surprised to find Euler subscribing to similar views in Letters of Euler onDifferent Subjects Addressed to a German Princess (New York: Harper andBrothers, 1840), p. 236.

28. Leibniz often calls this stuff an “etherial fluid,” but I want to emphasize theirparticle-like constitution.

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29. [S]ince the Cartesians recognized no active, substantival and modifiableprinciple in body, they were forced to remove all activity from it and transferit to God alone, summoned ex machina, which is hardly good philosophy.(“On Body and Force,” Ariew and Garber, op. cit., p. 254).

30. Mathematicians say that we “prolong the trajectories of the clashing particlesthrough a collision singularity” by relying upon conservation laws in the stockmanner explained within every freshman classical physics text.

31. “Specimen Dynamicum” in Ariew and Garber, op. cit., p. 123. Descartes’proposed rules of collision included an additional “big ball/little ball” anomaly thatLeibniz also criticizes here, but this feature was not present in the Wren-Huygensapproach and is unimportant for our concerns.

32. Passage translated by Freda Jacquot in Rene Dugas, Mechanics in theSeventeenth Century (Neuchatel: Editions du Griffon, 1958), p. 478). In the jargonof the times, “hard” indicates “resists change of shape” (for a good survey of theseissues, see Wilson F. Scott, The Conflict Between Atomism and ConservationTheory, 1644-1860 (London: MacDonald, 1970)). The imposition of the rigidityconstraint converts a word (“elastic”) that originally signifies “ability to regainoriginal shape” into a term signifying “energy conservation within a collision.”

33.“Reflections on the Advancement of True Metaphysics and Particularly on theNature of Substance Explained by Force” in Woolhouse and Francks, op cit., p. 31.

34. Discourse on Metaphysics, Ariew and Garber, op. cit., pp. 54-5. Here “force”indicates Leibniz’ “primitive force”: the spring-like restorative capacities inherent inthe internal construction of the air particles and pore walls.

35. Ariew and Garber, op. cit., p. 254. Note that “the use of a house” corresponds,within our beam, to the end state condition it tries to fulfill.

36. “Letter to Arnauld” 4/30/1687 in Ariew and Garber, op. cit., p. 84. I shouldindicate that what I describe as an optimization for the sake of middle-level objects”actually represents “an optimization for the sake of souls” for Leibniz (these soulspotentially exist at every conceivable scale size). I have set these theologicallymotivated considerations aside, because I’m mainly concerned to link Leibniz’approach to optimization with the tactics of energy minimization across a connected

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system characteristic of variational methods within physics. Many commentatorsappear to have overlooked this methodological affinity, which immediately placesLeibniz’ physical thought on sounder ground.

37. “Letter to Fontanelle, 1704" in A. Foucher de Careil, Leibniz (Paris;BiblioBazaar, 2008) p. 234. Leibniz’ full views on the exact nature of the efficientcausation rules we will observe at this level of analysis are rather sophisticated. Ata truly microscopic level, we will not discover the fictitious little “springs” of ourBernoulli-Euler element, but instead witness God’s providently directed air particlesrebounding from the cell walls of the wood. But how do we account for theirrebound? In two ways, Leibniz asserts. First, in these special circumstances, wecan assume that the elastic rebound of the air is close to perfect and we canaccordingly evoke conservation rules to handle the bouncing in the standard mannerthat perfectly elastic scattering are still treated within freshman physics texts to thisday (Leibniz correctly observes that conservation of vis viva is critical to this story--cf. Daniel Garber, “Leibniz: Physics and Philosophy” in Nicholas Jolley, ed., TheCambridge Companion to Leibniz (Cambridge: Cambridge University Press, 1995),pp, 316-7). As such, the collisions are handled through efficient causationprinciples alone, without appeal to teleology. On the other hand, it violates therational principle that “nature does not make jumps” for bodies to recoil withoutdistorting in the process, so if we scrutinize our air/wall collisions more closely, wewill find that both parties alter their shapes very rapidly, first compressing and thenreturning to their original geometries. But how do these dumb objects remembertheir original configurations? We are naturally returned to the explanatory domainof “final causes” (i.e., desire for original shape), now operative at scale sizes farbelow the conventionally “microscopic.” However, these fresh forms of teleologicalappeal can be once again supported by efficient causation mechanisms operating ateven lower scales, courtesy, as before, of God’s benevolent engineering. And everdownward this explanatory duality alternates, into the bottomless depths of Leibniz’celebrated “labyrinth of the continuum.”

38. “On Body and Force: Against the Cartesians” in Ariew and Garber, op. cit., p.252.

39. “Against Barbaric Physics” in Ariew and Garber, op. cit., p. 319.

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40. As noted in an earlier footnote, Leibniz actually maintains that God constructsHis efficient causation fill-ins for the sake of the “souls” he believes are scatteredthroughout the universe at all scale levels. Sticking with my resolution to outline thecentral physical considerations tangled up within Leibniz’ lines of thought, Ihighlight the fact that, unless catastrophic lower-scale events such as fractureintervene, flexible bodies must retain their material coherence upon all higher sizescales. This is a non-trivial structural demand that becomes registered in concretemathematical terms within modern texts on continuum mechanics. That Leibnizclearly anticipates these conceptual requirements is impressive proof of his deepmathematical insight.

41. “Letter to Princess Sophie,” 1705 cited in Samuel Levey, “Leibniz on PreciseShapes and the Corporeal World” in Donald Rutherford and J.A. Cover, eds.,Leibniz: Nature and Freedom (Oxford: Oxford University Press, 2005), p. 82. Thesuggestions in the present essay are quite congruent with Levey’s readings, inopposition to the many commentators who instead view Leibniz as embracing someflavor of phenomenalism.

42. “Letter to Arnauld” 10/09/1687 in Philosophical Papers and Letters, LeroyLoemker, trans. (Dordrecht: D.Reidel, 1969), p. 343.

43. In Essay 6, I complain that the analytic metaphysicians who considerthemselves Leibniz’ modern heirs ignore these motivating concerns.

44. He usually discusses such “power”-related strivings under the heading of“entelechy”; I employ “material personality” to capture the intended behavioraltropisms.

45. Las previously noted, eibniz’ harmonization requirements find mathematicalexpression in the fact that additional compatibility equations must be imposed toinsure that the infinitesimal relationships captured in our equations for stress andstrain can be integrated coherently into a continuous displacement field. Withoutthese supplementary demands on compatibility, our material could develop gapsbetween its elements like Swiss cheese.

46.“Of Body and Force,” op cit., p. 251.

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47.“Letter to Basnage” 1/03/96 in Woolhouse and Francks p. 64. See also:[I]n unraveling the notion of extension, I noticed that it is relative tosomething that must be spread out and that it signifies a diffusion orrepetition of a certain nature...[which is] the diffusion of resistance.Theodicy, trans. by E.M. Huggard (Eugene: Wipf and Stock, 2001), p. 251.

48. Cf., Richard T.W. Arthur, “Animal Generation and Substance in Sennert andLeibniz” in Justin Smith, ed., The Problem of Animal Generation in Early ModernPhilosophy (Cambridge: Cambridge University Press, 2006). For present purposes,Leibniz employs the topdown teleology of an animal body as a structural means forenforcing the topological prerequisites required to keep basic continuum quantitiessuch as mass and force distribution coherent between size scales (Chapter 1 ofTruesdell, op cit., provides a good overview of the sorts of measure theoreticcontrol required).

49. “Of Body and Force,” op. cit., p. 253.

50. “Letter to Arnaud,” Ariew and Garber, op. cit., p. 465.

51. I reiterate that the main object of this essay is to trace the hard-headedcontinuum physics considerations that run through Leibniz’ thinking, not to followits otherwise motivated strayings in detail.

49. Characterized in the Appendix as “the paradox of unfounded regress.”

53. Theodicy, op cit., pp.157 and 159. Leibniz’ final remark alludes to theCartesian doctrine that when a material system contains harmful factors (e.g.,sources of food poisoning) lodged at size scales beneath our normal levels ofsmoothed over discernment, God often attaches emotional warning signals to ourperceptions (“Ick; that hamburger smells disgusting!”).

54. “Letter to de Volder” 1/19/1706, in Ariew and Garber, op. cit., pp. 185-6. Theinfluence of Aristotle’s views on potential division is evident.

55. Theodicy, op cit., pp. 148-9.

56. In other words, the original “father of possible worlds” did not look upon their“possibilities” in the modern vein at all, and was deeply concerned with the subtleepistemological concerns that attach to mathematical modeling within a continuum

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physics environment.

57. Although it is common to label point-mass modelings as “Newtonian,” thehistorical Newton did not endorse such an attitude, which was pioneered by laterwriters such as Christian Wolff, R.J. Boscovich and the French atomists. Newtonhimself opined that matter was probably composed of extended rigid bodiessurrounded by oceans of intervening fluid. To be sure, he (mostly) modeled thesolar system as a collection of point masses in ODE fashion, but this policy does notjustify assigning a point mass “ontology” to Newton’s thinking. Moderncommentators frequently overlook the widespread resistance to point masses (whichprevailed throughout most of the era of classical physics dominance) due to the“physics always idealizes” appeals outlined in the appendix and becauseconventional instruction in “classical physics” took a decidedly ODE-favoring turnin the 1920's due to the formal requirements of quantum mechanics.

59. New Essays on Human Understanding (Preface) in Ariew and Garber, op. cit., p. 299.

60. More exactly, its moment of inertia.

61. This is a methodological motto I have often heard, but when I try to track downa source on Google, I only find references to my own writings! Employing ageographical analogy from Essay 9, a description of a beam at the differentialequation level can be viewed as an “out of country” extension element that opens upprofitable inferential pathways, which are unavailable to us if we restrict ourattention to beam behaviors witnessed at finite scale lengths. Leibniz clearlyregards his calculus equation models as functioning in this extension elementfashion.

62. In describing matters thus, I employ distinctions that become canonical onlylater. Within his vortex theory of gravitation, Leibniz regards gravitation’s influenceas operating as another form of traction pressure tugging on the boundary of anelement, rather than a body force as treated here. But he tacitly assums that theinterior of a sufficiently small element will unproblematically perceive gravity’scombined tractions as a single vectorial influence acting to “modify its primitiveforce.” In contrast, the contact forces operating inside the beam cannot be

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addressed in the same naive manner. Leibniz furthermore expects that its inertialreaction can be codified in a body force manner, as standard applications ofd’Alembert’s principle require. In mathematical terms, Leibniz assumes that wedon’t need to integrate the gravitational tractions around an element that issufficiently small but we must still perform surface-based summations when weconsider the contact tractions that attach our unit to its neighbors. Through thesedistinctions, Leibniz confronts the problem of how to reconcile element forces thatrequire summing and those that do not. I have employed modern renderings of thedifficulty to render its formal demands pellucid.

Similar remarks apply to Leibniz’ view of how “primitive force” (= elasticstrain energy) gets converted into the kinetic energy of movement, which heconceptualizes according to the basic statics-to-dynamics contours of d’Alembert’sprinciple (a tenet that considerably predates d’Alembert himself).

Following Leibniz’ thought along kinetic lines is difficult simply becauseadequate representations of partial differential equations had not yet been devisedand Leibniz could only effectively model behaviors one dimension at a time. As aresult, the equilibrium considerations belonging to one-dimensional statics representthe modeling regimes in which his physical thought can be most readily suppliedwith concrete mathematical readings.

63. Ipso facto, our two “forces” need to be of different grades of infinitesimalsmallness, which we now enshrine in the different measure-theoretic densities weassign to points and surfaces.

64. In modern terminology, the boundary region tractions must be contracted into aninterior stress capable of interacting with the body forces and the element’s inertialresponse

65. Physical tradition presumes that elements are so small that a single gravitationalforce and a single inertial reaction act inside, but the traction forces upon theelement’s outer surface cannot be treated in this same way, for it is the differencesamongst these bounding tractions that generate the interior stresses needed tobalance the applicable body forces.

66. The Grammar of Science (London: Thoemmes Continuum, 1992), p. 298. Onpp. 335-6, he claims that essential idealization is “the logical right of the physicist”(a theme that Pierre Duhem frequently echos):

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If we take a piece of any substance, say a bit of chalk, and divide it intosmall fragments, these still possess the properties of chalk. Divide anyfragment again and again, and so long as a divided fragment is perceptibleby aid of the microscope it still appears chalk. Now the physicist is in thehabit of defining the smallest portion of a substance which, he conceives,could possess the physical properties of the original substance as a particle. The particle is thus a purely conceptual notion, for we cannot say when weshould reach the exact limit of subdivision at which the physical propertiesof the substance would cease to be. But the particle is of great value in ourconceptual model of the universe, for we represent its motion by the motionof a purely geometrical point. In other words, we suppose it to have solely amotion of translation...;we neglect its motions of rotation and strain. ...Whatright has the physicist to invent this ideal particle? He has never perceivedthe limited quantity, the minimum esse of a substance, and therefore cannotassert that it would not produce in him sense-impressions that could only bedescribed by the concepts spin and strain. The logical right of the physicistis, however, exactly that on which all scientific conceptions are based. Wehave to ask whether postulating an ideal of this sort enables us to constructout of the motion of groups of particles those more complex motions by aidof which we describe the physical universe. Is the particle a symbol by aidof which we can describe our past and predict our future sequences of sense-impressions with a great and uniform degree of accuracy? If it be, then itsuse is justified as a scientific method of simplifying our ideas andeconomizing thought.

67. The Mechanics of Deformable Bodies (London: MacMillan and Company,1932), p. FIND PAGE

68. Historically, techniques for resolving our contact/body force coordinationproblem through quasi-rigidification assume a bewildering number of forms, someof which are nicely surveyed in James Casey, “The Principle of Rigidification” ArchHist Exact Sci 43, 4 (1992). Also see my “What is ‘Classical Physics’ Anyway?” inR. Batterman, ed., The Oxford Handbook in Philosophy of Physics (Oxford: OxfordUniversity Press, 2012). The two basic approaches to building elements I highlight(viz. point mass based and induced through downward scaling) are intimatelyentangled with the great nineteenth century controversy between “rari-constant” and“multi-constant” approaches to elasticity, associated with the derivational methods

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pursued by Navier and Cauchy respectively (Pearson, judging by his comments inHistory of Elasticity, never accepted the empirical disconfirmation of the formerapproach to the subject). Similar oppositions arise earlier, as witnessed by thescathing manner in which Euler dismisses the point-mass approach (which heidentifies with the “monadic” views of Christian Wolff):

Finally, let those philosophers turn themselves which way so ever they willin support of their monads, or those ultimate and minute particles divestedof all magnitude, of which, according to them, all bodies are composed, theystill plunge into difficulties, out of which they cannot extricate themselves.They are right in saying that it is proof of dullness to be incapable ofrelishing their sublime doctrines; it may however be remarked that here thegreatest stupidity is the most successful.

Letters to a German Princess, vol 2, ed. by David Brewster (New York: Harper andBrothers, 1835), p. 42).

69. Science X (247) 1899, pp. 392-3.

70. Cosserat media (commonly employed to model liquid crystal behavior) illustratethe manner in which the descriptive degrees of freedom attributed to theinfinitesimal elements of continuum mechanics directly reflect the intrinsicallyphysical considerations embodied within the basic balance relationships assigned toa material.