Essay 2

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 than thesettlers of later years, who gaze upon the encircling splendors with a dull andacclimated eye. G.W. Leibniz, among his many singular accomplishments, was one ofthe 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 intensely concentratedcharacter. In particular, he was forced to confront these concerns in his 1684 work onthe elastic response of loaded beams (an important scientific subject that Leibnizpioneered2) and many of the strangest features of his developed metaphysics can bequite directly related to considerations that arise within such endeavors. Throughinattentive familiarity we moderns tend toskip over the peculiarities of these sameprocedures with scarcely a pause, because wehave become accustomed to them. Only therenewed rigors that come with moderncomputing have brought Leibniz’ old concernsto the foreground once again and essay 4 will revisit these same issues in exactly thislight. But, here, we shall attend solely to Leibniz’ own context.

The mathematician J.E. Littlewood once published a wry essay entitled “FromFermat’s Last Theorem to the Abolition of Capital Punishment” which traveled animprobable bridge between prosaic worries about mathematical functions and weightymoral matters.3 Just so: the present essay will follow Leibniz’ analogous journey fromworries about bending beams to some of his astonishing conclusions with respect to

-2-

free will. Anyone familiar with his writings realizes that his views about theconstitution of matter sound--let us not mince words--quite crazy, for he contends thatthe material universe 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 material world--including the rocks, the iron girders and water, as well as organic stuff such as wood,mosquitos and human beings--are controlled by these animal-like things, whichcongregate in great colonies ordered under obscure master/slave relationships.

As an instructor in a public college, these are not the sorts of conclusion thatyou want your students to report about to the folks back home (“in your philosophyclass, you learned what...?”). Yet the remarkable fact about Leibniz’ thinking is thatmany of these strange opinions trace quite directly to perceptive views on soundmodeling practice that have captured the attention of engineers once again in recenttimes. Perhaps a useful way to appreciate some of Leibniz’ grander flights ofmetaphysical fancy is simply to reconnect them to the humble–yetsurprising–considerations with respect to wooden and metal beams with which theywere originally comingled. If this endeavor proves successful, our appreciation for theremarkable depths of Leibniz’ thought should considerably increase, for moderncommentators rarely link his metaphysical outpourings with concrete engineeringtechnique and often patronize his physics as deeply inferior to that of Newton. Thesupplementary consideration that, by advancing just a few steps further, a fullLittlewoodian arc to the free will problem can be completed should only increase ouradmiration further, in that slightly comedic fashion that truly great philosophy oftenevinces.5

Of course, a proper form of Leibniz interpretation will consider all of themultifarious influences that shape his thought, whereas I will largely trace a purist“mechanical” line here, one that adheres fairly closely to the ways in which engineersstill go about modeling systems like loaded beams. Why? Because this aspect ofLeibniz’ thought comprises a vital “figure in the carpet” commonly overlooked byscholars unfamiliar with the conceptual complexities of continuum mechanics6 (= thephysics of continuously flexible matter).

(ii)

-3-

Leibniz’ route to a modeling equation suitable for a loaded beam operateswithin the framework of explanation that he calls “the kingdom of ends or finalcauses,” indicating that he will employ some measure of “desired state” or teleologyin his deliberations (later we shall consider how Leibniz intends to undergird suchappeals within a parallel “kingdom of efficient causation” that operates upon a lowerscale size). It is in this context that his odd appeals to the desires, perceptions andactions of inanimate materials appear. But these invocations of “personality” are notas strange as they first appear, for there are sound structural reasons why the thisvocabulary adapts itself naturally to the behaviors of wooden and steel girders. If youopen any modern practical primer on materials science, you will find its authordescribing everyday substances in a similarlyanthropomorphized vocabulary of “memory,”“desire” and “perception” (true, no “littleanimals” are mentioned, but we’ll get to themonads later). Why? Well, ignoring friction,such members behave like elastic solids: theypossess natural equilibrium states to which theywill always strive to return after being bent or poked. Thus a 4x4 beam sagging undera load of rocks will struggle to regain its unloaded straight state; in lieu of achievingthat, it settles for a condition called constrained equilibrium (Leibniz’ immediatescientific objective is to characterize the shape of this loaded rest state, for reasons I’lloutline in a moment). However, if the material is afflicted with a so-called “fadingmemory” (as many woods and plastics are), its ability to “remember” its original stateof molding diminishes over time and it only regains a compromised and weaklycurved end-state intermediate between its current loaded-with-rocks condition and itserstwhile straight 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 in that it will always return to thatdesired state as fast as it can7 as soon it is released from any binding constraints. Incontrast, many metals display “fading memory” in that, if maintained in a constrainedposition for too long, they gradually adapt to the shape that has been impressed uponthem (secretly, they recrystallize to minimize strain energy in the new environment). A normal fluid such as water retains no “memory” of its former configurations (e.g.,the bottle from which it was poured) at all, but it can detect its present velocities withrespect to 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 and

-4-

Suppressing compression in the Wren-Huygens treatment

gradually puffs up in a feeble attempt to resume those former proportions--“die swell”is the official jargon for this behavior). Modern manufacture has constructed strange“smart materials” that try to regain different forms of earlier condition depending upontheir temperature (that is, they can “remember” two or more different earlier states).8 And so on. How vigorously and swiftly a material prosecutes these inherent goalsreflects their defining “personalities” as materials: a rod of steel resists bending farmore strenuously than one oftin, although both may strivefor rest configurations that aregeometrically identical. AsLeibniz stresses, suchcharacterizations of “materialpersonality” are inherentlyteleological in the sense thatthey 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 “memory” within hisbeam model: the Hookean9 capacity of a spring to exert a restoring force in directproportion to the degree to its is stretched or compressed away from its natural restconfiguration. In a moment, we’ll see how he constructed a plausible “beam memory”upon this simple basis (these same Hookean assumptions are generally adopted bymodern engineers when they deal with beams in the so-called “linear regime” wherethey are unlikely to sag significantly).

Why did he do this? Because steel, at least, behaves that way on a testingbench under moderate loads (whereas wood is strongly anisotropic) and,mathematically, things are quickly going to become tough enough without importingsignificant response non-linearity into his picture!

Leibniz approached the behaviors of billiard balls in a very similar way, exceptthat we are now considering objects that interact strongly in a kinetic manner–that is,through motion–rather than statically in the manner of a quiescent beam laden withrocks (Leibniz distinguished between “living” and “dead” loads). The eventstranspiring within a compressive collisions are extremely hard to calculate andinvariably require the power of modern computation (they represent so-called “movingboundary problems”) but their behaviors nonetheless rest upon the same basicassumptions of elasticity and “memory of desired state” as Leibniz utilized in histheory of beams. Indeed, direct observation of slow elastic collisions (e.g., betweenbeach balls) readily confirms that impacting balls compress and reexpand when they

-5-

collide. So we moderns say that when ball A bumps into ball B, all of the incomingkinetic energy gets converted into an work that compresses ball B into internal stress(= each ball gains a certain store of “strain energy” that it wants to discharge)determined by the degree to which the balls have become flattened. Once theseinternal compressions reach a critical level, each ball will start to push back against theother by transferring its unwanted strain energy into new forms of kinetic energy in theopposing ball (including spinning). In describing these basic events in “personality”laden terms, Leibniz correctly emphasizes the fact that notions such as “strain energy”(and “potential energy” in general) are teleological in the manner in which they appealto a “natural rest state” 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, primitiveand derivative... Primitive active force, which Aristotle calls first entelechyand one commonly calls the form of a substance, is another natural principlewhich, 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.10

I’ll parse 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 within Leibniz’ time (i.e., as devised by Huygen, Wren and Newton himself)do not follow this compressive model at all, but practice a rather brutal form of“physics avoidance”: they assume that the balls would instantaneously reboundwithout alteration of shape! More accurately, they employ a “work around”methodology that artificially collapses the breadth of the interval Δt* in which theimpactive events occur to zero duration 0. In so doing, their scheme effectively “cutsoff” all consideration of the detailed events that transpire on a time scale swifter than

-6-

Δt* and, at best, characterizes their effects through crude rules of thumb such asNewton’s “coefficients of restitution.”11 Such workaround techniques can only beregarded as convenient computational tricks and Leibniz believes that the “zeroduration” story they tell violates the scared principle that “nature does not makejumps”:

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 san 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).12

Of course, fully tracking these complex interactions even within idealizedcircumstances represents a daunting computational challenge that wasn’t really feasibleuntil recent times. In the beam model we shall study, Leibniz concerns himself withan entirely static situation (the shape that a beam will eventually assume under a loadof rocks, for example), thereby bypassing any consideration of the horribly complexprocesses of energetic exchange and loss that must transpire when a beam first settlesinto a new position of constrained equilibrium (see essay 1 for more on this).

(iii)

The central purpose of this paper is to trace how Leibniz proposed to fit togetherhis “two kingdoms of explanation” (viz., efficient causation and teleologicalexplanation), for such themes clearly anticipate the “greediness of scales” issues thatwill concern us in a modern guise elsewhere in the book. We shall return to the “twokingdoms” topic in section (v). Before doing so, I will first outline the concretemanner in which Leibniz approached the modeling of a loaded beam, for such anexercise can greatly illuminate many of Leibniz’ characteristic metaphysical categoriesby unpacking what they demand in terms of cold steel and wood. Depending uponscientific background, many readers may find some of these evanescent details toocompactly elucidated. Unfortunately, classical continuum mechanics (= the science ofmaterials continuous and flexible on every size scale) represents an intrinsically

-7-

complex subject and the proper mathematical tools for articulating some of itsrequirements were not developed into far into the twentieth century. I have expandedupon these issues more fulsomely elsewhere.

In fact, a comparatively hazy acknowledgment of the concerns raised within thenext two sections is all that will be required to grasp the basic themes pertinent to therest of this book (where they will be introduced independently, in the manner promisedin the preface). However, Leibniz is so commonly patronized for supplying a “hazymetaphysics” rather than a “robust physics” in the manner of Newton, I think it is quiteimportant to recognize that, except for the final push into Monadologyland, virtuallyevery “hazily metaphysical” notion he invokes along the way corresponds quite directlyto the basic, entirely unhazy, notions that a modern engineer invokes in setting up abasic continuum mechanics modeling of the target systems such as beams and billiardballs.

At the same time, familiarity with modern work in mechanics impresses upon usthe extreme complexity of the mathematical apparatus required to capture everydayintuitions about the behavior of ordinary flexible objects in a coherent fashion. We allknow that billiard balls compress when they collide; we are only surprised to hear thatthe mighty Newton lacked tools for treating such events directly. So the chief faultwith Leibniz physics was not that it was “hazily metaphysical,” but that, through itsreliance upon intuitive notions of compression and stress, it was forced to appeal hazilyto mathematical constructions it wasn’t able to yet redeem, let alone compute with.13

Brief philosophical digression: Such intellectual circumstances raise a generalconceptual puzzle that I’ve attempted to illuminate within other studies: how can wesuccessfully juggle descriptive terminology that we cannot inferentially manage in anycomplete or practical fashion? How do we stitch together the patches of computationwith which we can successfully operate into fuller facades that we view asdescriptively adequate? The answer, I think, often trades upon background contextualsafeguards whose vital connective operations we often overlook. I believe that a fullnarrative of the tangled tale of “understanding billiard ball behavior” during thecenturies after Leibniz and Newton would only reinforce these methodological morals.

As essays 5 and 7 will recount, a recent movement within self-styled “analyticmetaphysics” has, improbably, attempted to explicate the “conceptual requirements” ofclassic flexible matter employing naught but the intellectual equipment of everyday“intuition” plus a smattering of point-set topology. In my assessment, suchpresumptions betray an utter failure to recognize the subtle ministrations of backgroundcontext that allows “everyday intuition” to skirt the shoals of difficult mathematicsotherwise required in our subject. End of philosophical digression.

-8-

The key problem that Leibniz must address concerns the rather subtle “physicalinfinitesimals” required in modeling truly continuous matter effectively, for we do notwant to treat matter as a swarm of disconnected point masses interacting at a distance(an approach that is often called “Newtonian” even though Newton himself did notembrace it). The problem traces to the fact that, as soon as we consider trulycontinuous matter, two kinds of affective forces come into view, the second of which isentirely absent from the “Newtonian” framework just considered. Consider a completewooden beam, pinned between two endpoints. Obviously, strong “traction forces”must somehow attach the beam firmly to these posts, while gravity will pull its insides downward according to how heavy the beamhappens to be at the position of interest. Obviously, the final sagging shape of thebeam will be determined by how these two factors--endpoint traction and gravitation--work together in view of the beam’s internal capacities for resisting displacement fromits favored equilibrium condition (= the natural rest configuration that it always“remembers” and to which it continually strives to return). Obviously, the answer willprove rather complicated, especially if the local mass varies as we move along thebeam or its local elastic strengths differ as well. So the natural suggestion thatprompts the employment of infinitesimally focused equations in the first place is thatphysics proves simpler in the small14.: if we first consider a narrow span of beam, theacross-the-whole-beam variations just mentioned should vanish, leaving us with just asingle local mass and elastic strength to deal with. Following usual terminologicalpractice, let’s call the smallish bit of beam we nowwant to consider an “element,” However, despitethese simplifying advantages, we realize that we arestill considering a physical problematic with respectto our “element “of essentially the same character aswe had posed with respect to the complete beam:how will a flexible stretch of beam when sag when itis pulled downward by gravity yet held fixed bytractions along its sides? In fact, our problem hasgrown slightly worse because previously we knewthat our pinned endpoints would remain upright butour element’s local neighbors will sag along with therest of the beam, so we really don’t know in whatdirections the endpoint traction attachments will act.

If we look at these matters more closely, we will observe that a deepmathematical incongruity distinguishes the two types of “force” just mentioned. The

-9-

Bernoulli-Euler element

Leibniz element

gravitational force supplied by the load W directly pullson points inside each element body (hence theterminology: “body” or “volume” force) whereas thetensions that arise from the beam’s endpoint pinnings gettransmitted through the beam by acting upon theboundary surfaces where our element joins its neighbors(hence the usual modern classification: “contact” or“traction” force).15 This distinction sounds prima facieinsignificant, yet these two “forces” act upondimensionally incongruent locales: gravitation pulling on points while tensions tugupon surfaces. Ipso facto, our two “forces” must be of different grades of infinitesimalsmallness, which we now enshrine in the different measure theoretic “densities” weassign to points and surfaces. But to obtain a workable physics for continuous body,

we must persuade these dimensionally incompatible critters tooperate in harness (in modern terminology, we must convertthe boundary region tractions into an internal stress capable ofinteracting with the body forces and the element’s inertialresponse). This, in a nub, is the “physical infinitesimalproblem” mentioned earlier; it rests upon worries about thecoordination of “forces” that reach considerably beyondsimple Cauchy/Weierstrass concerns with mathematicalinfinitesimals.

. As just noted, physics should become “simpler in the small” and oneintuitively hopes that, if the breadth of our little elements could only be reduced toever smaller dimensions, then the traction and body forces will wind up applying to thesame “infinitesimal point” and we will find ourselves able to compute the coordinationbetween force types that alludes us at every finite scale size. In short, we seek a rulethat (after a limit operation) will tell us how a single gravitational tug acting upon apoint-sized beam element point will force the walls of the element to fan out, rather inthe manner of the baby’s jumping jack toy pictured. In particular, we want to find theconstrained equilibrium in which the little springs inside the element have stretchedand compressed to the exact degree required to balance the downward gravitational tug. A short portion of beam that behaves like this (rotating around a bending but non-stretching line called “the neutral axis”) is called a Bernoulli-Euler element after thelater mathematicians who devised it.

In point of fact, Leibniz’ own resolution of our force coordination problemproceeds in a slightly different manner than suggested, for he tacitly incorporates an

-10-

additional modeling assumption that simplifies the subsequent mathematics butengenders incorrect behaviors: he assumes that his beam element will fan out aroundits lowest point as a fulcrum, instead at some higher point.. This lowest hinging pointcan be easily discerned within the “element” diagram that Leibniz supplied, which Ihave reproduced with a few explanatory label supplements. But why should this be? Isn’t it more plausible that the twisting should instead transpire around some medianpoint within the material (traditionally called its “neutral axis”)? Yes, and if we makethese corrections, we obtain the Bernoulli-Euler element suggested above (where ourregular notion of “mass” must be replaced by “moment of inertia” to capture theelement’s “passive resistence” to the twisting pull properly).

Incidently, we should observe that it is no longer common to segregate theinterior ingredients into blocks and springs in quite this manner and a more commonrepresentation of the Bernoulli-Euler element is one where the posited elasticityappears in the guise of Hookean fibers that stretch or contract along parallel curves. Through affecting this change, we remove the asymmetries within Leibniz’ diagram,where it appears as if the neighboring element to the left of our target element shouldbe a vertical wall, rather than another element that fans out bythe same rules as govern our focal slice of beam. But thesedistinctions wash out once we try to make a full beam out ourlittle element pieces (in the denouement the only portionsrequired to be upright walls will be the two the far endpointposition where our beam is rendered vertical through itspinning).

Leibniz clearly did not view his element diagram asproviding a correct spatial portrait of what happens in a beamat a small scale size, but regarded its rotating block and littlesprings as a merely symbolic representation of the chief factorsoperating within (rather as mechanicians sometimes deploy“spring and dashpot” schemes even today). Thus, the driving engines of our element’steleological impulses have been completely consigned to thelittle springs that run along the right hand side of our blockwhile the block itself embodies the magnitude of its resistiveinertia (= Leibniz’s “force of passive resistence”). Byconsidering these ingredients carefully, one can make concretesense of the otherwise mysterious distinctions that Leibnizinvokes in the long quotation of the previous section. His“primitive active force” corresponds to the permanent capacities

-11-

of the little springs, as registered within the Hooke’s law strengths of the varioussprings. The downward tug of gravitation becomes internally “perceived” as amodification of the restorative impetus that these springs would otherwise supply if itsparts were fanned out into the configuration supplied, but where the action on the blockcould proceed at full strength. This scaling back of the element’s full restorativestrength supplies us with the residual “derivative primitive force” that will activelywork to return the block to its desired rest state. If this residual strength exceeds thegravitational pull, the element will be able to straighten itself out but if its puissance isfeeble, the element will continue to fan out until its springs become sufficientlycompressed and stretched16 to arrest the gravitational intrusion (this is what Leibnizmeans when he writes above of “the exercise of primitive force [being] turned inwardor outward”).

In essence, this portrayal conceptualizes the internal operations of an “element”in what we would now regard as “energetic terms”: fanned away from its rest position,our element acquires a potential energy which it tries to convert into moving (=kinetic) energy as rapidly as its inertia and the resisting gravitational pull permit. Unfortunately for its teleological ambitions, the same “passive force” of inertia willforce our element to overshoot its desired rest position once it reaches it, just as apendulum bob can’t halt at its desired point of lowest potential energy, because it ismoving too fast when it gets there (lacking some additional intervention from friction,our pendulum will never come to rest, but will merely swap its energetic allotmentforever between potential and kinetic energy registrations).

Accordingly, Leibniz’ understanding of the dynamics of a moving elastic beamis very close to the patterns that Lagrange later canonized under the anachronisticheading of “d’Alembert’s principle.” It should be remembered, however, the calculushad not yet developed to a state where these energetic exchanges could be clearlyexpressed in mathematical terms (we must codify how spatial variation relates totemporal variation and this involves a partial differential relationship in a manner thatwasn’t adequately understood until the 1750's). This is why the beam equation weshall develop only applies to the static circumstances where our beam has managed tolocate a constrained equilibrium in which it has become flexed to the requisite exactdegree to balance the downward pulls of gravitation and will not begin to move of itsown accord.

In any case, Leibniz’ “symbolic ingredient” approach to his modeling elementsrepresents but one of a wide range of traditional responses to a basic dilemma thatotherwise threatens the coherence of true continuum mechanics. It might be called theproblem of unprofitable descent, for, as we’ve seen, the mere fact that the basic

-12-

objects of continuum mechanics are bodies that remain prove continuously flexible onany choice of scale size indicates that, no matter how finely we inspect theirmicroscopic behaviors, we will continue to see different force types operating withindimensionally distinct locales. Until adequate theories of measure and integrationwere developed in the twentieth century, most methods of halting this regress appealedto some substantive recasting of “small elements” that deprives them of their fullposited flexibility (thus no portion of an Euler-Bernoulli beam, no matter how small,will ever prove as restricted in its movements as the modeling element we haveassigned to the beam). To justify these weird methodological procedures, the everpopular philosophy of “essential idealization” was born: to make its tools applicable tothe real world, applied mathematics must always misrepresent worldly behaviors insome quasi-rigidified manner such as we have just considered.17

We might also note that Leibniz’ rather strange talk of the element developingan “internal perception” of its exterior circumstances (the fact that its boundaries arepushed on by both gravity18 and tractions from its neighboring elements) appears todisplay a rather remarkable understanding of the basic conceptual dilemma ofcontinuum mechanics: how to persuade surface region traction forces to coordinatesensibly with body forces and the internal inertial reaction.

Leibniz’ willingness to tolerate “purely symbolic” representations of hiselements is also deeply affected by worries about how teleological explanations (= theappeals to little springs that attempt to regain their rest states) can be made to accordwith an “efficient causation” account of the same events. This is the conundrum mostcentral to this essay and we’ll consider Leibniz’ astonishingresolution in section (vi).

(iv)

At this stage, elementary physics texts frequentlysqueeze their representational elements down toinfinitesimal dimensions via some ill-defined "limiting procedure." But better approaches approach this shrinkingprocess in a so-called "variational" framework that betteraccords with Leibniz' own thinking on these matters.19 Let'sbegin by fictitiously carving our complete beam into littleelements of an equal finite length ΔL (I employ five suchpieces in the illustration). If we consider all of the possibleways in which such a jointed assembly might sag between

-13-

two fixed piers, we obtain a “space” of “statical beam possibilities.” As we look overthese “possibilities,” we seek the special configurations C (there may be more thanone) which can support its allotment of weights W with the least amount of overallbending measured by the degree to which their components springs have been set intocompression or tension. If we merely had to balance a single local weight wi enclosedby two walls, we could readily compute the required amount of stretching from ourlittle beam model. But all of our little elements are chained together, so thatminimizing the spring stretching within element Ei may easily make the tensions worsewithin element Ej (in modern terms, lowering the local strain energy within Ei

increases the strain energy within Ej). Thisconsiderations supplies the defining condition for theequilibrium states we seek: an array of individualelement configurations E

i that are stable in the sensethat if any local element in condition E

i lessens itslocal tensions by shifting to the configuration to E i *,the springs within the remaining elements will pull itback to E

i (like the Three Musketeers, equilibriumrepresents a mutually regarding “one for all and all for one” condition). The easiestway to locate this optimized state is by experimental tweaking: start with an arbitraryguess E0 (illustrated in the top of the diagram) and twiddle with a selected componentE0

j to see if we can improve the overall energetic budget of our chain by tweaking E0 j

to E1 j. If so, build this change into a revised guess E1 and repeat the twiddling.

Continue until the approximations (hopefully) approach some final, optimally lowtension configuration E (this may require an infinite number of steps). The old-fashioned name for this kind of searching is “relaxation method” and the set ofpossibilities through which we search forms a “relaxation space.”

We shall often visit computational circumstances of this general characterthrough this book. We employ the total amount of strain energy (= total amount ofspring stretching) stored within a configuration Ei as a “norm” |Ei| and insure that ourtweaking procedures are “contractive” in the sense that if we tweak a E*j guess to E**j, then the norm of |E** j| will decrease. If we do this repeatedly, we should obtain ourdesired equilibrium E

i as a “fixed point”: that is, a condition that cannot be furtherimproved through tweaking. Reasoning of this sort is called “variational” because ofits reliance on the quasi-experimental tweaking of an initial guess. We shall revisit thepolicy in more sophisticated garb within other essays.

-14-

Thus far we have only searched for the relaxed state of a beam divided into fivesegments of a fixed length ΔL. We immediately recognize that our beam’s“relaxation” might be further improved if it were divided into elements of shorterlength ΔL*. So let’s now seek a better equilibrium configuration by searching within arelaxation space that includes all of these ΔL*-segmented possibilities as well, forevery length choice ΔL*. Once these supplements have been allowed, we can“complete our space” by tolerating continuous beam possibilities without anysegmentation or internal springs at all (their ersatz “springiness” becomes measured bythe “springiness” in the segmented possibilities to which they lie near in our norm20). Although I’ve employed some modern ideas here (which will frame the principaltopics of essay 7), such thinking captures the underlying manner in which Leibnizaddresses a significant conceptual oddity of matter that remains continuously flexibleon all size scales: to articulate the desired behaviors in a coherent manner, we mustfirst walk through a “relaxation space” comprised of preliminary models that are notfully flexible below the element length ΔL. Through this peculiar bootstrappingprocedure, we confront the mathematical signature of the paradox of continuousmatter that have created so many headaches for the scientists who have worried aboutmatter in a classical context: real world materials appear to be smooth and throughlyflexible in their qualities yet we humans can’t obtain a workable handle upon theirgoverning physics without first pretending that such materials decompose intoartificially kinked and less pliant “elements.”

If we piece these ingredients together, we obtain the standard Bernoulli-Eulerbeam equation in its “weak” or “variational” form: d2h/dx2EId2δh/dx2 = Wδh.21 Superficially, this formula appears tosupply infinitesimally localized description of beam behaviorbut, in fact, the formula invokes many tweaked neighborhoodsof a target region through the variational term δh (such issueswill of prime concern in essay 7). In plain English, ourequation expresses the condition that in equilibrium our beamwill everywhere curve (at small angles the curvature of theneutral axis is closely approximated by d2h/dx2 ) to therequisite degree required to balance the local weight modulothe additional proviso that any tweaking of this curvature willspoil this balance.

Observe that the “possibilities” through we search areentirely static in character; our relaxation space does notcontain any “possibilities” that are changing in time (these

-15-

Hooke’s drawing of cork

behaviors will eventually piggyback upon static concerns in the “d’Alembert’sprinciple” manner sketched above). Recognizing this distinction is important to aproper understanding of Leibniz because his critics, in effect, frequently evaluate “bestpossible world” within the wrong “space.” In Candide, Voltaire inspects the sorrytrajectory that comprises human history and complains: “Look at all those earthquakes,plagues and wars: how could this path be optimal?” However, Leibniz considers his“optimization” in relaxation space mode: at any moment, the world is comprised of alot of interlaced elements that strive to achieve their own localized desires in the samemanner as a beam seeks its optimal equilibrium state. How can these sundry ambitionsbe mutually accommodated in a maximal fashion? The “best possible world” reachedunder this scheme may scarcely prove “optimal” in a Panglossian sense: there are lotsof undeserving layabouts with rotten desires who figure equally in the optimization andinterlocked inertias usually make us all overshoot our goals in any case. As we’ll see,this kind of optimization is closely linked to the problem of free will: God strives tomaximize everyone’s freedom to choose, including the bad people and the lesser“desires” displayed within the constrained equilibrium strivings of wood, steel androcks. 22

At this stage of my argument, the suggested reading ofLeibniz on optimality probably strikes the reader as too “cute” tobe historically appropriate. After several more turns of the screw,we’ll be able to make closer contact with his actual remarks onnecessity and continency.

(v)

The preceding methodology appears to presume that materials such as wood oriron respond to pushes and pulls by exactly the same rules down to an extremelyminute level (we might call this a doctrine of complete downward scaling). Butneither Leibniz nor a modern scientist believes that such assemblies behave identicallyat all size scales. If we inspect wood or steel under a microscope, we find that it iscomprised of a maze of cells or minute grains that individually stretch and dilateaccording to far more complicated rules that reveal itself within larger hunks of thematerial (the simpler behaviors at large scales arise partially through “law of largenumbers” randomization–see essay 4). Leibniz’ own thinking was frequently inspiredby contemporaneous discoveries in microscopy: he knew, from Hooke’s drawings, thatwood was composed of tiny cells, for example. Despite the fact that the simplestretching rules we assign to a Bernoulli-Euler beam fail once its “little elements” fall

-16-

below some critical “cut off” level ΔL, Leibniz correctly recognized that we shouldnonetheless push ΔL all the way to zero in extracting our finished differential equationformula, for a tractable simplicity emerges in this asymptotic limit (rather as simpleprobabilities emerge in infinite population limits). The upshot is an oddmethodological conundrum, closely allied to the “greediness of scales” concerns surveyin essay 4: in working upwards from a zero length differential equation model, wediscover that its predictions supply quite lousy descriptions of what occurs within veryshort spans of wood, but their successes improve dramatically after we reach largerscale lengths. In consequence, our model beam equation should not be viewed as aformula that captures “what really happens” in the material at a “length zero”scale, butrather as a shorthand formula that generates sound results when applied tosufficiently long scale lengths.

In other words, our beam differential equation represents a downwardlyprojected expression of a simplified “personality” that the target material manifestsonly at reasonably long scale lengths. This observation can be connected to deeperconcerns involving descriptive coherence. If a beam were actually composed of little“elements” of the type just examined, we might wonder whether their individualarrangements of springs and their Hookean strengths might vary haphazardly from oneelement neighbor to the next. But if such variations were tolerated at the individualelement level, our composite beam would easily rip apart due to theseinhomogeneities. On the other hand, we are quite familiar with materials whoseHookean elasticity varies from one point to another (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 still requires that the springs that we attribute to our little elementscoordinate with their neighbors in a manner that will sum to a continuous variation atthe macroscopic level.23 When Leibniz writes of materials like wood or steel as“machines,” he means to highlight this “harmonious cooperation” between lower scaleparts. As we’ll later see, many of Leibniz’ strangest pronouncements about space andtime trace to insightful observations such as this.

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 merelyrepresent shorthand, asymptotic formulas that capture a material’s properly only downto some indefinite choice of scale size. We moderns normally view physics’ mostfundamental differential equations in a more favorable light as directly capturingnature’s workings at an infinitesimal level. However, when we consider the standard

-17-

equations for the classical continua of everyday life (beams,strings, fluids, et al.), we tacitly shift to Leibniz’ point ofview, for their validity can only be understood as resultingfrom a downward projection of homogeneous patternswitnessed at large-to-middling scale lengths.

Leibniz’ concerns with “efficient causation” aredirectly entangled with reflections such as this. Earlier Iemphasized that Leibniz’ model for a beam directlyembodied “final cause” factors within its construction: thefact that springs display a Hooke’s law propensity to return to their natural rest state. But, like most writers of his era, Leibniz believed that these capacities for elasticreturn must be founded within some assemblage of a recognizably mechanical nature. Specifically, Leibniz embraced Descartes’ “air sponge” theory of elastic rebound.24 Consider an elastic piece of wood: from whence does it derive its “spring”? Accordingto Descartes, its interior is riddled with penetrating pores through which a super-mundane “air” continually circulates. When we squish the wood, we drive this “air”from the pores and, when we stretch it, the passages enlarge and an excess of “air”rushes in. As this happens, a corrective “air” flow will push the pore walls back totheir original configurations, rather as a dried sponge regains its shape when we allowwater to seep into its crevices. Here is one of the many passages where Leibnizendorses this account of the underpinnings of elastic rebound:

I hold all the bodies of the universe to be elastic, not though in themselves,but because of the fluids flowing between them, which on the other handconsist of elastic parts, and this state of affairs proceeds in infinitum. 25

Through this “air” assisted mechanism, the wood acquires a “memory” of itsnatural equilibrium state through what Leibniz calls “efficient causation” mechanismsalone (i.e., non-teleological stories that rely upon geometry and conservation principlesalone). For observe that the mechanical underpinnings of this “teleological memory”trace to the fact that the central bulk of our material consists in a sponge-like networkof interlaced tube walls that contract or dilate according to the amount of exogenous“fluid” pumping through its innards. That is, the permanent part of wood consists in aflexible framework of attached minute pieces that can be collapsed or expandedwithout abandoning their primary attachments to one another. By retaining thisconnected framework throughout the vicissitudes of external fortune, our wood retainsits “memory” of how its parts should be properly arranged, while relying upon thecirculating pressure in the surrounding “fluid” to provide the requisite “push” to restorethe material to its teleologically desired natural equilibrium. By addressing the

-18-

“Fluid particle”compression duringcontact

underpinnings of “memory” in this lower scale manner,Leibniz has explained the material’s characteristictropisms in a thoroughly “bottom up” fashion based(almost) entirely on “efficient causation” processes, incontrast to the “final cause” explication Leibniz invokedin his differential equation model of beam behavior. Superficially, such equations appear as if they describeevents that transpire within the wood at an infinitesimalscale level far below the finite scale at which our “ballswithin pores” activities occur, but this impression restsupon a mistaken understanding of the methodology ofdifferential equations, as Leibniz saw it.

However, a good deal of further philosophy lies concealed in that innocuous-appearing qualifier “(almost).” Look closely at the “air”/tube wall interactions thatrestore our wood’s webbing to its rest state configuration. We witness little particles of“air” that bounce off walls and interchange momentum with them in the generalmanner of a standard billiard ball collision. But we’ve already seen that a properaccount of such events requires that both balls and tube walls must compresstemporally, distributing the original kinetic energy of an incoming ball into internalenergies of deformation. However, these represent elastic compressions which willpush the ball away from the wall in an altered direction as the compressed bodiesregain their desired shapes. Only under the assumption that these compressive eventsoccur swiftly within a brief time interval Δt* can we preserve the “inviolable axiom”of continuity that requires that nature never make abrupt “leaps” such as a radical shiftin direction.

We observed earlier that Huygen’s and Wren’s simple approach to “purelyelastic collision” elides over these Δt* events and compresses all of their hiddencomplexities into an instantaneous event that involves no distortion in the collidingbodies but tolerates a discontinuous “jump” in the path of the incoming particle (onthis treatment, one prolongs the trajectories of the clashing particles through a“collision singularity” by relying upon conservation laws in the stock mannerexplicated within every freshman physics text). This methodology artificiallycollapses the width of the short interval of time Δt* in which the balls actively interactdown to an “impulsive event” lasting no time whatsoever (in modern jargon, theapproach “cuts off” all consideration of the detailed events that transpire on a timescale swifter than Δt*). For Leibniz, this approach must be regarded as a convenienttrick that works well only if one is not interested in the finer grained details that occur

-19-

within the excluded Δt* interval. Leibniz describes the descriptive utilities ofemploying such a “Huygens-Wren cutoff” in this manner:

Accordingly, if we think of bodies only under mathematical concepts like size,shape, place and their modification and introduce the modification ofvelocity only at the instant of collision, without resorting to metaphysicalconcepts, i.e., therefore, without going into what form has to do with activeforce and matter with passive force--in other words, if we must determine thedata of collision only through geometrical configurations of the velocities,the result will follow, as I have shown, that the velocity of the smaller bodywill be imparted to a much bigger body that it meets.26

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“perfectly elastic” impact.

The fact that these “cut off” tactics usually supply excellent descriptive results,represents, literally for Leibniz, a Godsend because, as a lowly mortals struggling togauge nature’s complex behaviors in mathematically tractable terms, these shortcutusually save us from needing to worry about the exceedingly complex events that occurwithin the excluded Δt* interval (again, we can only say “usually,” becausearrangements can be made–e.g., looking closely at the collisions through amicroscope–that will amplify the minute Δt* details to an observable scale).

[B]odies can be taken as hard-elastic without denying on that account thatthe elasticity must always come from a more subtle and penetrating fluidwhose motion is disturbed by the tension or the change of elasticity. And asthis fluid must in its turn be composed of small solid bodies, themselveselastic, it is seen that this interplay of solids and fluids is continued toinfinity.27

But it is only by plowing over these Δt* events that we can explain the elasticbehavior of our original wooded beam in a true “efficient causation” manner (=invoking only the pushing and pulling of contacting particles).

Leibniz mentions the importance of this additional, lower layer of “teleology”with respect 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 thatthe ultimate reason for the movement of the matter is the force given atcreation, which is there in every body, but which is as it were constrained bythe mutual interactions of bodies.28

-20-

This is not to say that Leibniz regarded his “efficient causation” account of elasticity asmore “basic” or “fundamental” than a “final cause” account. Quite the contrary,assumptions about the“final causes” of springs leads, in the manner traced above, to aquite effective modeling of beam behavior, whereas any lower scale “efficientcausation” story involving “‘air’ and pores” type will prove quite speculative andparasitic upon the basic contours of the “final cause” 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, ourphilosophers are still far from it. But the way of final causes is easier and isnot infrequently of use in divining important and useful truths which onewould be a long time in seeking by the other, more physical way; anatomycan provide significant examples of this.29

And:From this we also understand that even if we admit this primitive force orform of 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 reasonsshould 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 explainedif we were to describe only the arrangement of its parts, but not its use.30

Indeed, he might well have contrasted his own treatment of beams to that a Newtonianmight have produced.

When we consider billiard ball collisions, however, this preference inexplanatory tractability reverses, for the Wren-Huygens treatment supplies admirableeffect descriptive results, where any richer account involving elastic compressionswould require very difficult mathematics obtainable only with the use of moderncomputing equipment. However, in truth, none of these stories can be regarded aswholly correct, for each approach needs to piggyback upon minor details explicableonly from its rival’s point of view.

Accordingly, Leibniz’ alternating levels of preferred explanation will continuallyreoccur as we inspect matter upon ever improving degrees of detail. Our little balls of“air” themselves require pores through which an even finer “air” must circulate, whichin turn will require pores of their own and so on ad infinitum, in the manner of Swift

-21-

and deMorgan’s celebrated fleas-upon-fleas analogy. Accordingly, a straightforwardexamination of elastic beam behavior provides us with a concrete illustration ofLeibniz’ celebrated worries about “the labyrinth of the continuum”: its maze iscomprised of an never stabilizing descending hierarchy of interwoven explanatoryschemes, where “final cause” stories forever alternate with “efficient cause” narratives.

(vi)

Descartes had apparently presumed that hisambient “air”could straightforwardly reinflate acollapsed elastic structure, but, taken at face value, thissupposition is ridiculous: bagpipes don’t blowthemselves back up after their wind has been evacuated. Why? Because air pressure normally acts equally in alldirections and it requires a piper of the Black Watch orsimilar directive agency to puff them up again (spongescan restore themselves only because capillary forcesdraw the water inward). The only way that “air”movements could produce a Cartesian springiness is ifsome Maxwell’s Demon could providentially direct the motions of the ambient “air” inexactly the right way that they can collectively knock the elastic material back towardsits original configuration.

As I understand him, this is precisely what Leibniz believes occurs, for he positsan accommodating Deity that plays precisely the role of such a benevolent Demon insupporting the satisfaction of our human desires. Throughout the first part of thisessay, we explicated a beam’s behavior through teleological appeal to what Leibnizconsiders as “final causes”: the equilibrium states wood and iron strive to regain, withdifferent degrees of vim according to their specific elastic “personalities.” God takesthe “final cause desires” of these beams into consideration along with the “desires” ofeverything else in the macroscopic world and optimizes their satisfaction in thedemocratic manner we modeled for a specific beam within our relaxation space. Oncethis grand optimization has been settled, God fills in the rest of the complete physicaluniverse at the microscopic level by directing the ambient “air” molecules in exactlythe right way that the “final state” desires of the middle level objects will be maximallyaccommodated.

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 as

-22-

they should at precisely the moment the soul has a suitable volition orthought; the soul, in turn, has this volition or thought only in conformity withthe previous states of the body. 31

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 toknock the distorted beam back to its relaxed shape. Viewed from this efficientcausation perspective, we don’t witness any springy teleology in play, for God’sprovident planning has supplied the beam with a micro-mechanism that allows it toregain its rest shape through “air” contact action alone.32

[E]lastic force is essential to every body, not in the way that the force is someinexplicable quality but because everybody, however small, is a machine fromwhose structure a recoil must arise whenthis is required to conserve force. Moreover, this should not seem surprisingto anyone who considers the actualdivision of the parts of this matter intoparts exceeding every number. This, I believe, represents the properphysical reading of Leibniz’s famous“preestablished harmony”: behaviors thatcan be explained in a top down manner according to “final causation” narratives canalso be addressed “from below” with impeccable “efficient causation” accounts.

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.33

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

I have shown that everything in bodies takes place through shape andmotion, everything in souls through perception and appetite; that in thelatter there is a kingdom of final causes, in the former a kingdom of efficientcauses, which two kingdoms are virtually independent of one another, but

-23-

nevertheless are harmonious.34

In other words, pore-centered physics at a lower efficient cause level is incrediblycomplicated and its structure is partially determined, through God’s preplanning, by the“final states” that macroscopic level “souls” strive to reach. In other words, God hasplanned the world largely for the sake of the monads in the middle ranks and thenarranges the rest of the stuff around them.

A convenient side effect of this provident structuring is that it allows the various descriptive “cutoffs” and “extensions” that we have previouslyconsidered to work ably most of the time. We usually reason fairly well about billiardballs if we cut off their elastic behaviors with Wren-Huygens rules and can reasonfairly well about wooden beams if we smoothly project their high level behaviors downto an infinitesimal level via Leibnizean techniques. As such, both approaches can becaptured in mathematical terms ably, although not at the same time in a commonformalism. Yet neither portrait can be regarded as supplying a full picture of reality,because on rare occasions lower scale complexities will pierce through the cutoffs andextensions on which we otherwise rely. But if we ignore the manners in which wehave patched over these occasional breakdowns in erecting artificially self-enclosed“kingdoms of explanation,” we are likely to trust our descriptive mathematicsexcessively and presume that material objects genuinely possess wholly objectivespatial shapes:

It is the imperfection of our senses that makes us conceive of physical thingsas Mathematical 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.35

In truth, every attribution of a geometrical characteristic to a material object merelyrepresents a false-but-useful downwardly projection based upon its homogeneouslarger size scale behaviors. Thus:

[Matter] has not even the exact and fixed qualities which could make it passfor a determined being... because in nature even the figure which is theessence of an extended and bounded mass is never exact or rigorously fixedon account of the actual division of the parts of matter to the infinite. Thereis never a globe without irregularities or a a straight line withoutintermingled curves or a curve of a finite nature without being mixed withsome other, and this in its small parts as in its large; so that far from beingconstitutive of a body, figure is not even an entirely real quality outside of

-24-

thought. One can never assign a definite and precise surface to any body ascould be done if there were atoms. I can say the same thing about magnitudeand motion.36

In my view, such unexceptionable considerations concerning descriptive practicewithin applied mathematics lie at the base of Leibniz’ strange insistence that “space”and “time” are “merely ideal”: he is merely observing, quite correctly, that everypracticable description of everyday matter utilizing geometrical vocabulary secretlyincorporates a fair degree of fictitious projection to unwarranted size scales. With thisfeigned homogeneity comes the presumption that a thoroughly continuous material canbe potentially divided at every scale length ΔL. As such, the doctrine overreaches, yetthese same fictitious “possibilities of division” get exploited whenever we follow adownward seeking “relaxation space” pathway to valuable equational models such asour Bernoulli-Euler prototype. The methodological strategies employed in thesetechniques have undeniably led to great descriptive victories yet, in many respects,they remain puzzling from a philosophical point of view. As such, we shall findourselves continually returning to many of Leibniz’ underlying concerns in the essaysahead.

But few of us are likely to wholeheartedly embrace Leibniz’ own policies forresolving these tensions, for this is where his peculiar monads enter the scene. Hecontends that material behaviors require firm underpinnings within exterior reality (nophenomenalist he), but we’ll never provide a wholly stable answer as long as we insistupon describing matter in spatially dominated terms (although this supplies thedescriptive mode in which a successful physics must operate). Instead, he maintainsthat a substance’s powers to resist alteration and to seek goals (e.g, the manner inwhich wood exhibits inertia and strives to recapture its natural rest state) representmore fundamental descriptive characteristics than geometrical extension (he usuallyarranges such “power”-related strivings under the heading of “entelechy”–we mightemploy “material personality” as shorthand for these behavioral tropisms). In astandard physics treatment, where the notions of “position” and “velocity” rule asprimary, a material’s “personality” rules will usually be expressed in a functionalformat that privileges spatial displacement. A free weight will fall through a certaindistance in a certain time in a gravitational field and we regard this potentialdisplacement as a measure of the same weight’s capacity to fan out a Bernoulli-Eulerelement to which it is attached in equilibrium circumstances (in modern terms, thisdownward force performs work in fanning out the element). Within the real, monadicorder of things, such dependencies operate obversely, for an attached weight inhibits

-25-

the element’s natural capacities to seek its natural state and the accompanying internaltensions in turn affect the restorative capacities of the neighboring elements to whichour target element is attached. Recall the “all for one; one for all” nature ofconstrained equilibrium within a loaded beam. In such circumstances, local element Amay be forced to store a large amount of strain energy because other elements B and Cfar away in the beam display a greater capacity to resist gravity’s intrusions (= theircomponent springs are stiffer than those found in A). These represent judgements ofcomparative power, rather than simple curvature per se.

To capture Leibniz’ thinking on this score more exactly, recall our earlierinsistence that the “personalities” of adjacent elements must coordinate with oneanother continuously, less the material splinter into disconnected shards. But anypartitioning of our beam into spatially delineated “elements” involves some degree ofartificial projection, for the behaviors we capture thereby pertain only to the largerscale activities of our beam (far above the size scale at which its pores and “air”become prominent), so we lack any precise means of assigning the distinctive powerswithin an extended piece of wood to precise locales (the power relationships wewitness are absolute in character but they can’t be aligned with definite spatial homes). He often compares our perception of a body’s length to themanner in which we see rainbows: individual drops of rain areresponsible for the redirection of light that occurs but our eyescannot discriminate these droplets on the basis of the combinedlight that reaches us because the originating sources lie too nearone another and alter the light in very similar manners. Inmathematical physics, when we describe a beam as shaped into some mathematicalconfiguration such as a catenary, we have artificially suppressed its lower scalecomplexities for the sake of a convenient representation of its upper scale propensitiesto affect or be affected by other systems within its vicinity. 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.37

And: I had to look more deeply into the notion of corporeal substance, which I

-26-

preformedsperm

hold to consist more in the force of action and resistence than in extension,which is only the repetition or diffusion of something prior,namely that force.38

Indeed, an object’s visible “size” merely reflects the fact that itwill “look large” if its component parts possesses the capacity tomake large quantities of light alter their inertial dispositions insimilar ways, whereas the components parts of a nearby objects ofa different color will alter similar light in patterns that affect our eyes quite differently.

Leibniz conceived the hierarchical relationships of continuity required amongstthese background “power centers” in intriguing terms that he adopted from thedevelopmental biology of his day.39 Specifically, it was commonly believed, onmicroscopic evidence, that complex organisms originate from “seeds with allof their organs intact,” albeit greatly shrunken in size. Normal biologicaldevelopment consists largely in these primordial organs taking on enoughfood to eventually assume adult proportions (for Leibniz, upon death thesesame parts relinquish their hold upon fleshy matter and shrink back to theiroriginal, minuscule proportions). To make sense of this, the monadic unitscorresponding to each bodily part must belong within a “master and slave”hierarchy where the teleological requirements of an entire animal bodydetermine the needs of its heart which in turn fix the ambitions of its leftventricle. And so on, in a classic “the nest on the branch, the branch on the limb, thelimb 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 arenot animals, properly speaking.40

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 hunkabove the critical ΔLC scale must cooperate in a slavish fashion with the “natural state”desires of the beam as a whole. Indeed, it is striking (although I’ve not foundanywhere where Leibniz argues accordingly) that a small arc a of a wooden ring A

-27-

molded under tension will agreeably cooperate with the “natural state” desires of thewhole ring A as long as a remains enslaved to A. But once a is “liberated” from itschains (e.g., we cut a free from A), it will freely display its own set of “natural state”desires.

Unfortunately, this last conclusion teeters on the edge of impiety and Leibnizonly claims that the “top down” organization displayed within inorganic materials andwooden beams provides only a simulacrum of the monadic “master-slave”relationships operative within living biological systems (although every materialsystem contains some measure of animal-like monads). Our wooden beam merelyevidences an apparent “dominant monad” behavior in the manner of a flock of sheep

where the sheep are so tied together that they can only walk with the samestep and cannot be touched without the others bleating.41

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, Leibnizevokes these imitations of monadic life to avoid the theologically unwelcomeconclusion that sawed-off boards and rocks possess immortal souls. Whatever itsultimate origins, the “top down” scaling behaviors common to everyday forms ofcontinua, biological or not, remain crucial to their successful treatment withinphysics.42

Although Leibniz’ preformative analogy strikes us fanciful, the underlyingrecognition that the physics of continua must be controlled through a tight pattern ofdownwardly directed integration between scale sizes is not: it remains fully enshrinedwithin the modern rigorous foundations of the subject. We have already highlightedthe profound physical insight that lies concealed within the simple requirement that, inthe absence of rupture or fusing, the pieces of a continuous body must normally remainfirmly attached throughout all of their local distortions. To enforce this condition, wemust demand an organized integration of behaviors through all scale sizes in the “topdown” manner characteristic of our relaxation space (and modern measure theory moregenerally). As we observed, this complete scale invariance is only apparent, as otherprocesses become secretly active below the cutoff level ΔLC.

It is truly fortunate that master and slave monads cooperate so agreeably withrespect to their larger scale teleological ambitions, for the resulting upper scalebehavioral invariance supplies limited intellects such as ours with convenient footholdson the materials of daily life (such as wooden beams), for we couldn’t forge a path toreliable physical rules if smoothed over models of a Bernoulli-Euler character didn’twork pretty well most of the time. Likewise, as finite calculators, it is better that we

-28-

rarely detect the microstructure within materials, for such awareness might encouragequixotic searches for “bottom up” modelings that can never be completely fulfilledbecause of “labyrinth of the continuum” considerations. Indeed, for most purposes it isbest to conceptualize the world about us with the mixture of “efficient and final cause”thinking that Leibniz exploited when he constructed a workable model of a loadedbeam. But, of course, the behavioral harmonies on which these techniques relyentirely result from the kindly activities of the Deity who has arranged matters thus.

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) need to interact on their own level–when a beam or billiardball should bend, for example, or I order a glass of iced tea. God then directs his lowerscale “air” through all the right pores to back up this macroscopic world:

[I]t is just as if [someone] who knows all that I shall order a servant to do thewhole day long on the morrow made an automaton entirely resembling thisservant, to carry out tomorrow at the right moment all that I should order;and yet that would not prevent me from ordering all that I should please,although the action of the automaton that would serve me would not be in theleast free.... The knowledge of my future intentions would have actuated thisgreat craftsman, who would accordingly have fashioned the automaton: myinfluence would be objective and his physical. For in so far as the soul hasperfection and distinct thoughts, God has accommodated the body to thesoul, and has arranged beforehand that the body is impelled to execute itsorders. And in so far as the soul is imperfect and as its perceptions areconfused, God has accommodated the soul to the body, in such sort that thesoul is swayed by passions arising out of corporealrepresentations.43

This picture of how God has constructed a universe around thecomposite objects that dominant the scale level we inhabit is centralto the story he will tell about 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 to grainy monads of the real universe and (ii)“possibilities” as they apply to the smoothed continua that we attribute to the world forthe sake of descriptive utility. Insofar as (i) is concerned, a proper answer must besecretly determined by the full “personality” rules that determine when one range ofmonadic influence “cooperates” with another and when not. If I read Leibniz correctly,

-29-

he fancies that if, per impossible, we could actually learn these rules in their infinitelycomplex glory, we would discover that only one overall outcome was “possible.” Onthe other hand, according to (ii), the notion of “being divisible into segments of lengthΔL” which we utilized in building up our beam model represents a fictive extension ofour restricted upper length scale knowledge, albeit a form of “projection” vital toeffective descriptive procedure within physics. It is this second notion of “possibility”that allows us to declare, “Of all its possible configurations, a loaded beam chooses theshape that supports its load W with the least expenditure of internal energy.” Leibnizexplains such distinctions as follows:

[I]n actual things, there is only discrete quantity, namely a multitude ofmonads or simple substances, indeed, a multitude greater than any numberyou might choose in every sensible aggregate. That is, in every aggregatecorresponding to phenomena. But continuous quantity is something ideal,something that pertains to possibles and to actual things considered aspossible. The continuum, of course, contains indeterminate parts. But inactual things nothing is indefinite, indeed, every division that can be madehas been made in them.... As long as we seek actual parts in the order ofpossibles and indeterminate parts in aggregates of actual things, we confuseideal things with real substances and entangle ourselves in the labyrinth ofthe continuum and inexplicable contradictions. However, the science ofcontinua, that is, the science of possible things, contains eternal truths, truthswhich are never violated by actual phenomena, since the difference [betweenreal and ideal] is always less than any given amount that can be specified. And we don’t have, nor should we hope for, any mark of reality inphenomena, but in the fact that they agree with one another and with eternaltruths. 44

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 a remarkable defense of freewill. We humans possess a range of personality-driven desires which we can act uponas freely as the middle level constraints of the world permit (which includes theopposing desires of other agents). God optimizes the world with these middle range

-30-

constraints in view and then fills in the rest with enough directed “air” that the “springsof bodies” act in the fashion that our operative “final cause” demands require. So weare genuinely free to make all sorts of dumb decisions in accordance with our own“personalities”; God has merely crafted the lilliputian “air” that puts the “spring” intoour freely chosen steps.

For this very reason the choice is free and independent of necessity, becauseit is 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, whereall is completely even on both sides, without any inclination towards either. Innumerable great and small movements, internal and external, co-operatewith us, for the most part unperceived by us. And I have already said thatwhen one leaves a room there are such and such reasons determining us toput the one foot first, without pausing to reflect. For there is not everywherea slave, as in Trimalchio’s house in Petronius, to cry to us: the right footfirst. 45 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 choosethem: we desire Y, X seems a suitable method to reach Y and nothingprevents us from executing X. True, if we could inspect themicroscopic workings of our neurons carefully, we would observeGod’s preplanned “air” particles shunting along tubular walls inimpeccable efficient causation manner. But this fact in no wayimpugns our actions as not, at core, entirely free, for God has cleverlyplotted the “harmonies” of the world to optimally fulfill desires working from our sizescales 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 upon the puzzling and complex practices involved in treatingmatter coherently from a continuum physics point of view. True, Leibnizian monadsneedn’t be invoked to this purpose, but the essential ingredients of “natural state”striving and careful policies of behavior coherence and scaling must. Furthermore, oursuccessful everyday policies of describing everyday materials in continuum physicsterms tacitly rest upon subtle contextual controls involving tricky homogenizations and

-31-

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

2. “New Proofs Concerning the Resistance of Solids,” Acta Eruditorum., July, 1684. Iam indebted to Clifford Truesdell’s excellent discussion of this essay and the alliedliterature in The Rational Mechanics of Flexible or Elastic Bodies 1638 - 1788 (Basel:Birkhäuser, 1980), pp. 59-64. See also Edoardo Benvenuto, An Introduction to theHistory 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” asarticulated within his “On the Notion of Cause” in Mysticism and Logic, Littlewood’sessay can be found in A Mathematician’s Miscellany (London: Methuen: 1963).

other forms of interscalar relationship that most philosophers of science ignore butmerit much closer inspection (such issues will emerge in many of the essays yet tocome). Even Leibniz’ “labyrinth of the continuum” concerns ought to trouble themodern metaphysicians who blithely prattle about “the world view of classical physics”requires, without seriously attempting to catalog these necessities in a coherentmanner. In this regard, it is startling to realize that it is only through quantum physicsthat we moderns are able to halt Leibniz’ regress, for classical mechanics appears tooffer inadequate tools for rendering matter stable. In other words if we do not invokeintrinsically quantum behaviors involving a hazily delineated “effective size” to haltthe unterminating descent, the scale invariance of classical continuous materials is aptto pull us into the same maze that Leibniz feared (quantum physics raises manyparadoxes of its own, but at least it rescues us from these worries). 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.

Accordingly, we needn’t go so far as to agree fully with Leibniz:[Matter] has rigidity as well as fluidity everywhere and that no body is hardor fluid to the ultimate degree, that is, that no atom has insuperablehardness, nor is any mass entirely indifferent to division.46

But we must surely acknowledge that his insights into the conceptual complexities offlexible bodies were deep and prophetic. Endnotes:

-32-

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: YaleUniversity 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 one willbelieve it.” The Philosophy of Logical Atomism. (LaSalle: Open Court, 1985), p. 53.

6. I have strongly protested on many occasions the unfortunate inclination to confusethe subtle conceptual contours of continua with the much simpler patterns of pointmass thinking in a quasi-Newtonian (more properly, Boscovichian) vein.

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

8. 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 prior occasionwhen it had been molded into a stretched out configuration under chilly conditions. Thus our smart antenna “perceives” the cold which awakens a “striving” to return to adifferent rest state than it had “desired” while it felt warmer. Watching one of thesedevices unpack itself is rather unsettling, for it looks like some creepy insect slowlybestirring 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 being able todetect its neighbors’ internal conditions.

9. Leibniz acquired this principle from Edne Mariotte, who had discovered itindependently and first applied such thinking to beams. He also informed Leibniz thata 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).

10.“Of Body and Force: Against the Cartesians” in Philosophical Essays, p.252-4.

-33-

11. Wren and Huygens’ rules for impact presume perfect elasticity (rather than thefrictional losses crudely captured within Newton’s “coefficient of restitution’supplements). Leibniz invariably alludes to the former treatment in contrasting such“effective causation” accounts with his own “material memory” based account, so Ishall omit Newton’s name in this connection henceforth. But it is important torecognize that one of the worst conceptual calamities to afflict later philosophicalthinking about matter and causation is to carelessly presume (as, for example, Humedid) that Newton had managed to capture the “correct physics” for billiard ballinteraction within the Principia (for fuller comments on this situation, see “What is”and WS). What should be properly said is that Newton and his followers displayed anadmirable restraint in their descriptive ambitions, through substituting a crude butreliable walkaround method for a very difficult and shaky “moving boundary”computation. Even today, models of impact follow the Newtonian pattern wheneverthey can get away with it, but delicate cases force us to reopen the Δt* impact intervaland investigate the sundry compressions and expansions that transpire therein inelaborated detail. The first serious attempt to estimate the compression of contactingbilliard balls (in static circumstances only) traces to Heinrich Hertz’s work in 1886.

I do not want to reinforce the common impression that Leibniz was a “purelyphilosophical” dreamer who mused airily about physical circumstances that he couldnot covert into concrete mathematics (in the manner of Boscovich, say). In point offact, his own beam theory (and all of the improvements later to follow) rely upon alarge host of macroscopically derived constraints (i.e., that beam movements mainlytranspire along fibers and in plane sections) introduced precisely as a means ofopening a path to a tractable estimation (in contrast, to fashioning a simple 3Delasticity model in Navier’s mode, which will prove essentially intractable withoutmodern computing techniques). Physicists before Mariotte and Leibniz attempted toapproach the breaking load of a gradually weighted wooden beam as an instantaneouscatastrophic event in much the “compressed interval” fashion of Newton’s treatmentof billiard balls. “No,” our two friends replied, “you’ll not obtain trustworthy resultsuntil you consider models that exploit known constraints on fiber movements in apractical manner that allows us to estimate the gradual build up of internal stress in themoments just before the beam fails.” And they were absolutely right in this: what’ssauce for the goose isn’t always suitable for the gander.

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

-34-

13. Is it Leibniz’ fault that he didn’t invent the modern, integrated circuit computer?He certainly tried!

14. This is a methodological motto that I have often heard, but when I try to track itdown on Google, I only come across references to my own writings!

15. In describing matters thus, I employ distinctions that only became canonical later. Within his vortex theory of gravitation, Leibniz regarded gravitation’s influence asoperating entirely as another traction force tugging on the boundary of an element,rather than a “body force” as treated here. But he tacitly assumed that the interior of asufficiently small element would somehow “perceive” these tensions as a singlevectorial “force” operating to “modify its primitive force.” In modern terms, thisamounts to the assumption that we don’t need to integrate the gravitational tractionsaround a “sufficiently small” element but such integrations will still be required whenwe deal with the tractions tracing to its endpoint pinnings. In this manner, Leibnizresolves our force reconciliation problem through a distinction between forces that“require summing” or not.

Similar remarks apply to Leibniz’ view of how “primitive force” (= elasticstrain energy) converts itself into the kinetic energy of movement, where he is plainlyconceives of these conversions according to the basic contours of “d’Alembert’sprinciple.” Once again, he presumes that one can regard our element as a point centerin computing the resulting accelerations (which is why modern writers such asClifford Truesdell classify the element’s inertial reaction (= Newton’s m.a) as a “bodyforce”).

Technical remark: One should recall that it is difficult to follow Leibniz’thought very far along kinetic lines simply because partial differential equations in anadequate form had not yet been invented and Leibniz could only effectively modelbehaviors one dimension at a time, which precluded any close study of how a flexiblebeam might simultaneously arrange itself spatially while moving forward in time. This is why equilibrium questions of the sort we are presently considering loom largein the continuum mechanics that he was able to develop in concrete mathematicalterms.

16. That is, within the Bernoulli-Euler element. The springs within Leibniz’ ownmodel (implausibly) only elongate. To state the distinction marked here in moreprecise terms: our body and traction “forces” represent “force densities” defined overvolumes and surfaces respectively. Accordingly, these measures must be integrated

-35-

by different procedures to produce genuine (finite) forces.

17. The twentieth century studies of Clifford Truesdell and his school clearlyestablished that the “essential idealization” thesis merely serves as a convenientexcuse for bypassing some tricky mathematical arrangements that need to be directlyconfronted at some point in any case. Historically, techniques for resolving ourcontact/body force coordination problem through quasi-rigidification assume abewildering number of forms, some of which are nicely surveyed in James Casey,“The Principle of Rigidification” Arch Hist Exact Sci 43, 4 (1992). See also my“What is ‘Classical Physics’ Anyway?” Proper reasons for maintaining this “requiredmisrepresentation” thesis may arise within other arenas of physical technique (somepossibilities will be considered in other essays) but not here.

18. Recall that Leibniz regards gravity or “load” as a surface traction rather than a“body force” in the modern sense; for him only the element’s inertial response fallsinto the latter category.

19. In searching through a “possibility space” in this manner, we are tacitly appealingto variational considerations in the general mode of the “virtual velocities” (later“virtual work”) criteria developed by Leibniz’ friends, the Bernoullis (their integralreformalization in the guise of the familiar “Hamilton’s principle” applies to aconsiderably smaller set of mechanical circumstances, which is why Lagrange reliedupon “virtual work” in his Mécanique Analytique). Two excellent Histories are P.Duhem, The Origins of Statics and NEW BOOK. In light of Leibniz’ frequentinsistence that calculus ideas should be understood in terms of what we would nowcall its finite element truncations, our appeals to an increasingly refined element gridsuits his thinking ably as well. In this regard, Leibniz derived Fermat’s allied“Principle of the Least Time” in optics through an allied element-like decompositioninto stages--Jeffrey McDonough has a nice discussion of the importance of thisalternate form of variational principle for Leibniz’ thinking in his "Leibniz on NaturalTeleology and the Laws of Optics," Philosophy and Phenomenological Research,forthcoming. In fact, Hamilton himself derived his “principle” within mechanics byimitating the piecewise deflections of the optical path one witnesses inside a telescopewith a lot of mirrors and lens, according to Darryl D. Holm, Geometrical Mechanics,Pt 1 (London: Imperial College Press, 2008), chapter 1.

20. Leibniz’ views on how differential equations relate to their discretizations aredeeply entangled with his reflections on “progressions of the variable,” which I won’t

-36-

attempt to explain here. See H.J.M. Bos, “Differentials, Higher-order Differentialsand the Derivative in the Leibnizian Calculus, Arch Hist Ex. Sci 14, 1974. Chapter 3of D. Bertoloni Meli’s Equivalence and Priority (Oxford: Clarendon Press, 1993) alsocontains an useful discussion of these topics. Formulating transparent equations forcontinua requires partial differential operators, but most early continuum models workin stages with ordinary differential equations by evoking symmetries to reducedimensions and then employing some form of spatial integration to convert an array ofdistributed forces into a “bending moment” or allied sum (this is largely the taskLeibniz attempted in his original article and failed to do it entirely correctly). These“elements” are then assembled into a final o.d.e. that describes a one-dimensionaldisplacement of the target object from the x-axis. Only “turning on motion” viad’Alembert’s principle demands partial derivatives. S.B. Engelsman in his Familiesof Curves and the Origins of Partial Differentiation (Amsterdam: Elsevier Science,1984) claims that Leibniz and Johann Bernoulli had a p.d.e. equivalent available by1697, but these represent very technical historical issues that I cannot evaluatecompetently. Without a doubt, Leibniz had a rough conception of the multivariantcalculus even if his execution (understandably) faltered. Justifications for thedimensional reductions common within elasticity comprise another aspect of the“essential idealization” tradition, but we won’t pursue these issues here. In essay 7,we shall investigate the essential role that the our bending “norm” plays in assigning ameasure of “springiness” to continuous beams whose internal springs have alldisappeared within an infinitesimal limit. In many ways, Leibniz’ thinking on thisscore looks quite modern.

21. “I” supplies the block’s moment of inertia whereas “E” is the overall modulus ofelasticity for the springs. Assuming suitable smoothness, this converts to its classicalform: EI d4h/dx4 = W.

22. Essay 5 criticizes the standard Lewis-Stalnaker approach to counterfactuals for notattending to the significant structural differences between equilibrium-focusedappeals, such as we consider here, and evolutionary settings where time plays a morecentral role. I submit that we can’t understand Leibniz’ own appeals to “possibleworlds” properly without attending to this difference, so vital to the great mechanicaltraditions, culminating in Lagrange, that build up mechanics on an entirely staticalbasis first.

-37-

23. Within regular three dimensional elasticity, so-called “compatibility equations”are required precisely to ensure that infinitesimal strains sum to macroscopicdisplacements.

24. “Letter to Jacob Bernoulli”, 1702 quoted in D.B. Meli, Equivalence and Priority(Oxford: Oxford University Press, 1993), p. 55. Find Cartesian reference. I derive thesponge metaphor from Leibniz himself (Of Body and Force,” p. 252):

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.“

25. [S]ince the Cartesians recognized no active, substantival and modifiableprinciple in body, they were forced to remove all activity from it and transfer it toGod alone, summoned ex machina, which is hardly good philosophy: “On Bodyand Force,” p. 254.

26. Specimen Dynamicum in Philip P. Wiener, ed. Leibniz Selection (New York,1951), p. 129. Descartes’ proposed rules of collision included an additional “bigball/little ball” anomaly that Leibniz also criticizes here, but this feature was notpresent in the Wren-Huygens approach and is unimportant for our concerns.

27. Passage translated by Freda Jacquot in Rene Dugas, Mechanics in theSeventeenth Century (Neuchatel: Editions du Griffon, 1958), p. 478). In the jargon ofthe 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 regain originalshape” into a term signifying “energy conservation within a collision.”

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

29. Discourse on Metaphysics, AG, pp. 54-5. Here “force” indicates Leibniz’“primitive force”: the spring-like restorative capacities inherent in the internalconstruction of the “air” particles and pore walls.

-38-

30. Garber and Ariew p. 254. Note that “the use of a house” corresponds, within ourbeam, to the end state condition it tries to fulfill.

31. “Letter to Arnauld” 4/30/1687, AG, p. 84.

32. Leibniz’ full views on the exact nature of the efficient causation rules we willobserve at this level of analysis are rather sophisticated. At a truly microscopic level,we will not discover the fictitious little “springs” of our Bernoulli-Euler element, butinstead witness God’s providently directed “air” particles rebounding from the cellwalls of the wood. But how do we account for their rebound? In two ways, Leibnizasserts. First, in these special circumstances, we can assume that the elastic reboundof the “air” is close to perfect and we can accordingly evoke conservation rules tohandle the bouncing in the standard manner that perfectly elastic scattering are stilltreated within freshman physics texts to this day (Leibniz correctly observes thatconservation of vis viva is critical to this story--cf. Daniel Garber, “Leibniz: Physicsand Philosophy” in Nicholas Jolley, ed., The Cambridge Companion to Leibniz(Cambridge: Cambridge University Press, 1995), pp, 316-7). As such, the collisionsare handled through efficient causation principles alone, without appeal to teleology. On the other hand, it violates the rational principle that “nature does not make jumps”for bodies to recoil without distorting in the process, so if we scrutinize our “air”/wallcollisions more closely, we will find that both parties alter their shapes very rapidly,first compressing and then returning to their original geometries. But how do thesedumb objects remember their original configurations? We are naturally returned tothe explanatory domain of “final causes” (i.e., desire for original shape), nowoperative at scale sizes far below the conventionally “microscopic.” However, thesefresh forms of teleological appeal can be once again supported by efficient causationmechanisms operating at even lower scales, courtesy, as before, of God’s benevolentengineering. And ever downward this explanatory duality alternates, into thebottomless depths of Leibniz’ celebrated “labyrinth of the continuum.”

33. “On Body and Force: Against the Cartesians,”AG, p. 254.

34. “Against Barbaric Physics,”AG, p. 319.

35. “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. The suggestionsin the present essay are quite congruent with Levey’s readings, in opposition to the

-39-

many commentators who instead view Leibniz as embracing some flavor ofphenomenalism.

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

37.“Of Body and Force,” p. 251.

38.“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.

39. 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 ofClifford Truesdell, A First Course in Continuum Mechanics (New York: AcademicPress, 1977) provides a good overview of the sorts of measure theoretic controlrequired). The fact that these requirements are not trivial is shown by the fact thatadditional “compatability equations” must be imposed to prevent a material thatappears well-behaved at a stress-strain level from displaying holes like Swiss cheesein its macroscopic force to distortion relationships.

40. “Of Body and Force: Against the Cartesians,” AG, p. 253.

41. “Letter to Arnaud,” AG, p. 465.

42. I reiterate that the main object of this essay is to trace the hard-headed mechanicalconsiderations that run through Leibniz’ thinking, not to follow its otherwisemotivated strayings in detail.

43. Theodicy P.157 and 159. Leibniz’ final remark alludes to the Cartesian doctrinethat when a material system contains harmful factors (e.g., sources of food poisoning)lodged at size scales beneath our normal levels of smoothed over discernment, Godoften attaches emotional warning signals to our perceptions (“Ick; that hamburger

-40-

smells disgusting!”).

44. “Letter to de Volder” 1/19/1706, AG, pp. 185-6. The influence of Aristotle’sviews on potential division is evident.

45. Leibniz, Theodicy pp. 148-9

46. New Essays on Human Understanding (Preface) Translated by Roger Ariew andDaniel Garber in Philosophical Essays, (Indianapolis: Hackett Publishing, 1989), p.299.